sổ tay cdt Chuong 14- He CDien TT&Quay

8/6/2019 s tay cdt Chuong 14- He CDien TT&Quay

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14Cc h c-in tnh tin v quay: tng hp,

vi ch to, phn tchv ti u MEMS

Sergey Edward LyshevskiPurdue University Indianapolis

14.1Gii thiu ...........................................................................1

14.2Phn loi v tng hp cu trc vi thit b chuyn ng MEMS ................. ......2

14.3Ch to MEMS ..................................................... ......... .....4

14.4C s in t v m hnh MEMS ............................... ........6

14.5M hnh ton hc ca MEMS .............................................814.6iu khin MEMS ...................................................... ......17

14.7Kt lun .............................................................................27

14.1 Gii thiu

MEMS da trn in t c s dng rng ri trong cc ng dng cm bin v chp hnh. Vi cc MEMS ny, cc vithit b chuyn ng tnh tin v quay c cn c ly tng, thit k, v iu khin. Chng ta a ra mu phn loi thc hin s tng hp cu trc ca MEMS da trn cc c tnh in t. Nh cc vi thit b chuyn ng c pht minh,vn sau y c nhn mnh: m hnh ha, phn tch, m phng, iu khin, ti u, v ph chun. Cc kt qu mi c

nghin cu p dng cho cc khi nim phn loi, tng hp cu trc, thit k, phn tch, v ti u c pht trin. S cn thitvi cc phng php tch hp mi nhm thc hin vic phn tch ton din, m hnh c chnh xc cao, v thit k caMEMS c nhng tin b l thuyt trong hnh nh tng th ca khoa hc v k thut. Chng ny a ra mt hng vi cccng c nhm thc hin s tng hp cu trc, m hnh ha, phn tch, ti u v iu khin MEMS.

Cc h vi c in t tch hp cc vi cu trc chuyn ng v thit b nh cc IC trn mt chip n hay trn mt chip thp. ch to MEMS, ngi ta dng cc cng ngh, k thut, qui trnh, v vt liu vi c in t cao cp. Do vic dngcc cng ngh in da trn cht bn dn oxit kim loi b sung (CMOS) trong cc vic ch to cc vi cu trc, vi thit b, v IC,MEMS thc y ngnh vi in t.

nh ngha sau v MEMS c thy trong [1]:

Cc thit b c micro c ch to theo khi (cc IC v cc vi cu trc chuyn ng) bin i cc thng s vt l thnhcc tn hiu in v ngc li, v thm vo , cc c trng c micro ca cc thnh phn, kin trc, cu trc, v tham s cv in l cc thnh phn quan trng ca hot ng v thit k ca chng.

Phm vi ca MEMS ngy nay m rng ti vic ngh ra mu mi, m hnh chnh xc cao tch hp cp h thng,phn tch d liu tp trung, iu khin, ti u, ch to v thc thi. Nh vy, chng ta nh ngha MEMS nh sau:

Cc h c micro c ch to khi (cc vi thit b/vi cu trc nng lng pht tn v chuyn ng cc mch dnng/nhn bit cc IC iu khin/x l) m

1. bin i cc kch thch, s kin, v tham s thnh cc tn hiu in v c, v ngc li,

2. thc hin vic cm bin v chp hnh

3. gm c cc c tnh iu khin (tr thng minh, to quyt nh, kin thc tin ha, thch nghi, t t chc, v.v..),chn on, x l tn hiu, v thu thp d liu

v cc c tnh c micro ca cc thnh phn c in t, in t, quang hc, v sinh hc (cc cu trc, thit b, v h thngcon), kin trc, v nguyn l iu khin l c s hot ng, thit k, phn tch, v ch to ca chng.

Vic thit k tch hp, phn tch, ti u ha, v to mu o ca MEMS thng minh v hiu sut cao, tr thng minh h

thng, kin thc, thch nghi, to quyt nh, v t t chc c th c nh r, nghin cu, v gi quyt thng qua vic dngcc l thuyt c-in tin tin, phn cng tin tin, cc cng ngh mi, v phn mm leading-edge. Nhiu vn trongMEMS c th c lm thnh cng thc, tn cng v gii quyt dng vi c-in t. C th, vi c-in t gii quyt cc vn chun v cp bch trong k thut, khoa hc v cng ngh in-c-my tnh tch hp. Vi c-in t l s thit k tch hp,

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S tay C in t

phn tch, ti u ha, v to mu o ca MEMS thng minh v hiu sut cao, tr thng minh h thng, kin thc, thch nghi,to quyt nh, v iu khin thng qua vic dng phn cng tin tin, phn mm leading-edge, v cc cng ngh v qui trnhch to mi. Cc c tnh a ngnh tch hp tip cn mt cch nhanh chng, v vi c-in t din ra.

Cc cng c thit k c tr gip ca my tnh c yu cu h tr phn tch, m phng, thit k, ti u ha, v ch toMEMS. Nhiu n lc c trin khai nhm t c c trng ng lc v trng thi n nh ring ca MEMS png cc yu cu v tiu chun c p t. Hin nay, MEMS c thit k, ti u, v phn tch bng cch dng cc gi phnmm sn c da trn phn tch trng thi n nh v tuyn tnh. Tuy nhin, nghin cu ny c tp trung trn m hnh ton

hc c chnh xc cao, phn tch su sc d liu, v cc m phng phi tuyn, cng nh iu khin (thit k ca cc thutton iu khin nhm t c hiu sut nh mong i). S tng hp, m hnh ha, phn tch, m phng, ti u ha, v cckhi nim, cng c, v mu iu khin c cng b m bo mt gii php hiu qu v c th c dng m bo tomu nhanh cho MEMS tin tin hiu sut cao. iu thng rt kh khn, v i khi l khng th c, gii quyt mtmng ln cc vn phn tch v thit k phi tuyn cho cc vi thit b chuyn ng dng cc phng php truyn thng. Cnphi c cc khi nim, phng php, v cng c mi h tr mt cch y s phn tch, m hnh ha, m phng, iukhin, thit k, v ti u ha. Cc cng ngh ch to c dng trong MEMS c pht trin [2,3], v cc cng ngh vigia cng c miu t trong chng ny. Chng ny gii quyt mt s vn tn ti cho MEMS da trn in t.

14.2 Phn loi v tng hp cu trc vi thit b chuyn ng MEMS

C th nhn mnh rng nh thit k phi thit k MEMS bng cch ngh ra cc vi thit b chuyn ng hiu sut cao, phntn cc vi thit b nng lng, mch dn ng/nhn bit c micro, v iu khin/x l cc IC. Mt th tc tng bc trong

thit k cc vi thit b chuyn ng l: nh ngha ng dng v cc yu cu m trng

nh r cc thng s k thut hot ng,

ngh ra cc vi cu trc v vi thit b chuyn ng, phn tn cc vi thit b nng lng, mch dn ng/nhn bit cmicro, v iu khin/x l cc IC,

pht trin qui trnh ch to dng cc cng ngh vi gia cng v CMOS,

thc thi bin i in t, nng lng, c hc v cc c lng c/kch thc,

thc hin thit k in t, c hc, dao ng, v nhit ng lc hc vi phn tch kh nng v d on kt qu,

xc minh, chnh sa, v ci tin thit k vi cc mc tiu v i tng c bn nhm ti u hot ng

Trong phn ny, s thit k v ti u cc vi thit b chuyn ng c trnh by.

Nhm minh ha th tc ny, hy xem xt cc vi my dng khng khe ng b vnh cu 2 pha nh thy trn hnh 14.1.Hin nhin rng h thng in t ny l endless, v cc hnh dng khc nhau c th c dng nh thy trn hnh 14.1.

Ngc li, cc vi my ng b tnh tin (tuyn tnh) l h thng in t open-ended. Cc n lc nhm phn loi cc thit bchuyn ng vi c-in c thy trong [1,4,5]; tuy nhin, phn tch ton din v s lng v cht lng phi c nghincu.

Cc h thng in t v hnh dng vi cu trc chuyn ng phi c tch hp trong s tng hp, phn tch, thit k, vti u ha. Cc vi cu trc c th c hnh tm, cu, torrioidal, nn, tr, v khng i xng. Dng cc h thng in t v hnhdng khc nhau ny, chng ta c th phn loi c MEMS. tng ny rt hu ch trong vic nghin cu s tn ti MEMScng nh trong vic tng hp mt s lng v hn vi thit b chuyn ng mi. C th, dng cc h thng in t v hnhdng c th (endless, open-ended, v tch hp), ngi ta c th tng hp c MEMS mi hiu sut cao.

Cc vi my (vi thit b) in t c bn c xem xt di kha cnh l dng in mt chiu v xoay chiu, cm ng vng b, quay v tnh tin (tuyn tnh). Phn loi cc vi thit b dng mt b phn loi c bn

{ : }Y y y Y =

Cc vi thit b chuyn ng c phn loi theo hnh hc (tmP, hnh cu S, torroidalT, hnh nnN, hnh tr C, hay hnhkhng i xng A) v phn loi h thng in t (endless E, open-ended O, hay tch hp I). Phn loi vi thit b, thy trnbng 14.1, c chia thnh 3 hng v 6 ct , v bao gm 18 phn, mi phn c nhn dng bi cc cp ch ci c th t,nh (E, P) hay (O, C).

BNG 14.1 Phn loi cc vi thit b in t s dng b phn loi h in t - hnh dng

G

M

Hnh dng

Tm,P Hnh cu, S Torroidal, T Nn,N Tr, CKhng ixng,A

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C cu chp hnh

Hint

Endless(Closed),E

Open-ended,

O

Tchhp,

I

HNH 14.1 Vi my ng b nam chm vnh cu vi cc hnh dng khc nhau

Trong mi cp th t, mc t th nht l ch ci c chn t tp h thng in t gii hn

{ , , }M E O I=

Mc t th hai l ch ci c chn t tp hnh hc

{ , , , , , }G P S T N C A=

Tc l, vi cc vi thit b in t, tp h thng in t-hnh hc l

{( , ),( , ),( , ),....,( , ),( , ),( , )}M G E F E S E T I N I C I A =

Ni chung, chng ta c

{( , ) : }M G m g m M andg G =

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S tay C in t

C th p dng cch phn loi khc. V d, ngi ta phn loi cc thit b n pha, hai pha, ba pha, v nhiu pha theo mtb phn loi pha

{ : }H h h H=

Nh vy, {( , , , ) : , , }Y M G H y m g h y Y m M g G andh H =

Ngi ta cng c th phn loi d dng hnh hc topo (xuyn tm hoc trc) hnh dng cc cc nam chm vnh cu (di,

cung, a, hnh ch nht, hnh tam gic, hay cc hnh dng khc), cc c tnh nam chm vnh cu (ng cong kh t BH,tch s nng lng, vng gng tr), s o mch, phn b emf, lm mt, cng sut, m men, kch c, c tnh mmen-tc, cng nh cc c tnh khc ca vi thit b.

Tc l, ngi ta c th phn loi cc vi thit b in t mi bng mt N-tuple nh

{loi vi thit b, h thng in t, hnh hc, tp, pha, vnh, u ni, lm mt}.

Dng b phn loi nh cho trn bng 14.1, v mt h thng in t-hnh hc, nh thit k c th phn loi cc vi thit bchuyn ng hin nay cng nh tng hp cc vi thit b hiu sut cao, mi. V d, hnh 14.2 minh ha cc dng hnh cu,hnh nn, v hnh tr ca mt vi thit b ng b nam chm vnh cu 2 pha.

HNH 14.2 Hnh dng vi thit b (vi my) ng b nam chm vnh cu hai pha

Phn ny cho thy cc kt qu mi trong tng hp cu trc c th c dng ti u ha hiu sut ca vi thit b. Hnh14.2 minh ha cc hnh dng vi thit b dng nn (hin nay) v cu-nn (sp ti). Dng hnh cu-nn mi khc rt nhiu sovi hnh dng nn hin nay, n lm tng chiu di hot ng L, v ng knh trung bnhDr. Vi cc vi thit b dng xuyntm, m men in t Tc t l vi ng knh rotor bnh phng v chiu di trc. C th l

2c T r r T k D L= , y Tk l hng

s. T mi lin h trn, hin nhin cc vi mt cu-cn c mmen in t cao hn so vi thit k truyn thng. Thm vo ,

vic ci tin h thng lm mt, gim cc thnh phn mmen khng mong mun, cng nh vic tng kh nng chu ng vbn vng gp phn tng kh nng hin thc ca gii php a ra. Nh vy, dng m hnh phn loi trn, ngi ta c th tora cc vi thit b mi vi kh nng tt hn.

14.3 Ch to MEMS

Vi c in t tch hp vi c hc v vi in t i hi cc cng ngh ch to nng sut cao, chi ph thp, c iu kincho php gia cng cc thit b v cu trc 3D c micro. Vi gia cng l mt cng ngh ch to ch yu cho cc cu trc, thitb v MEMS c micro. Cc cng ngh ch to cc h thng vi c in t c th chia thnh 3 loi: gia cng khi, gia cng bmt v k thut LIGA (gn LIGA) [1-3].

Gia cng khi

Vi gia cng khi v b mt u da trn CMOS ci tin v c bit l qui trnh vi gia cng c thit k. Vi gia cng khisilicon dng k thut n mn t v kh chung vi cc mt n n mn v cc lp-dng-n mn nhm pht trin cc vi cutrc t nn silicon. Cc vi cu trc c x l bng cch n mn cc vng ca nn silicon nhm to ra cc vi cu trc 3Dmong i. Trong vi gia cng khi, ngi ta thng s dng rng ri cc qu trnh n mn t ng hngv d hngcngnh cc k thut n mn ph thuc tp trung. Cc vi cu trc c hnh thnh bng cch n mn khi mng silicon nhm tora cu trc 3D mong i. Gia cng khi vi cu trc tinh th v cc qu trnh n mn dopant-dependent, khi c t hp vilin kt mng vi mng, to ra cc vi cu trc 3D phc tp vi hnh dng mong mun. Thng qua vi gia cng khi, n ch tocc vi cu trc bng cch n mn su vo trong mng silicon. C mt s cch n mn mng silicon. n mn d hngdng cht n mn nhm n mn cc hng cu trc tinh th vi tc khc nhau. Thng qua n mn d hng, ngi ta chto cc cu trc 3D (cons, chp, lp phng, v cc rnh trong b mt mng silicon). Ngc li, n mn ng hngn mntt c cc hng trong mng silicon vi tc ging nhau (hoc gn ging nhau), v nh vy, c th to ra cc cu trc bncu v tr. n mn ion phn ng su dng plasma n mn cc cu trc vch trc tip (lp phng, ch nht, tam gic,v.v..).

Vi gia cng b mt

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C cu chp hnh

Vi gia cng b mt tr thnh cng ngh ch to chnh trong nhng nm gn y do c th ch to c cc cu trc vthit b 3D c micro phc tp. Vi gia cng b mt vi silic n tinh th, a silic, nitrit silic, oxit silic, v ioxit silic (nh ccvt liu lp cu trc v hy sinh c kt ta hoc n mn) c s dng rng ri ch to cc cu trc v thit b c microtrn b mt mng silicon. Cng ngh nng sut cao, chi ph thp ny c tch hp vi cc qu trnh ch to IC m bo tnhtng thch ch to vi cu trc IC cn thit. Cc cng ngh dng cho vic kt ta v to cu trc mng mng c dng sn xut cc vi cu trc v vi thit b phc tp trn b mt mng silicon (vi gia cng silicon b mt) hoc trn b mt ca ccnn khc. Cng ngh vi gia cng b mt cho php to cc cu trc nh l cc lp mng mng. Cng ngh ny m bo s chto vi thit b 3D vi chnh xc cao, v vi gia cng b mt c th c gi l qui trnh mng mng. Mi mng mngthng c gii hn dy ti 5 mm dn n vic gia cng cc vi cu trc v thit b dng phng hiu sut cao. u imca vi gia cng b mt l vic dng qui trinh v tin nghi trong gia cng CMOS chun cng nh s tng thch vi IC. Bivy, cng ngh ny c dng rng ri sn xut cc vi c cu chp hnh v cm bin (cc vi thit b).

HNH 14.3 Vi gia cng b mt

HNH 14.4 S mt ct cho vi ng c nam chm vnh cu khng chi qut, khng khe vi cc IC

Vi gia cng b mt da trn ng dng lp hy sinh (tm thi) dng duy tr cc lp tip theo v s c xa b lm l(gii phng) vi cu trc cn ch to (c gii phng hoc treo). Cng ngh ny c thc hin ln u cho cc IC v ngdng cho ch to cc vi cu trc trong thp nin 1980. Trn b mt ca mt mng silicon, cc lp mng ca vt liu cu trcv hy sinh c kt ta v to cu trc. Sau , vt liu hy sinh c xa b v to ra mt cu trc hay thit b vi c hc.Hnh 14.3 minh ha mt dy qui trnh c bn ca cng ngh ch to vi gia cng b mt.

Thng thng, lp hy sinh l ioxit silic (SiO2), ioxit silic c kch thch bng photpho, hoc silic nitrit (Si3N4). Cclp cu trc sau c hnh thnh vi polysilicon, v xa b lp hy sinh. C th, sau qu trnh gia cng vi cu trc v vithit b b mt (cc vi my), mng silicon c th l khi t c n mn hnh thnh cc l pha di cu thnh b mtcho php khong rng hn trong chuyn ng cn thit cho thit b. C th thc hin n mn t bng cch dng flohyric vaxit flohyric m, kali hyroxit, etylen-iamin-pyrocatecol, tetramethylam-monium dydroxide, hoc sodium hydroxide.Cng ngh vi gia cng b mt c dng ch to cc vi my dng quay [6]. V d, c th dng polysilicon kch thchmnh bng photpho ch to cc roto v stato, v dng silic nitrit nh l vt liu cu trc thu c s cch ly in. Titdin ngang ca vi ng c khng rnh c gia cng trn nn silicon vi stato polysilicon vi cun kt ta, roto polysiliconvi nam chm c kt ta, v trc c minh ha trn hnh 14.4. Vi ng c c iu khin bng cch dng cc IC dnng/nhn bit v iu khin/x l. ch to vi ng c v IC trn mt chip 1 mt hoc 2 mt (c hiu sut hn hn), ngita dng cc cng ngh v qui trnh ch to ng dng v cc vn v tnh tng thch s c xc nh v gii quyt. Ccqui trnh vi gia cng b mt c tch hp vi cng ngh CMOS (chng hn, cc vt liu ng dng, quang khc, n mn, vcc k thut khc). ch to MEMS tch hp, c th p dng cc k thut tin, pha trn v hu CMOS/vi gia cng [1-3].

Cc cng ngh LIGA v gn LIGA

Cn thit phi pht trin cc k thut ch to cho php to cc vi cu trc c t l tng quan cao. Qui trnh LIGA

(Lithography-Galvanoforming-Molding) thch hp cho vic to cc vi cu trc 3D ti hng centimt chiu cao vi t l tngquan (chiu su i vi kch thc bn) ln hn 100 [2,7,8]. Cng ngh LIGA da trn quang khc tia X m bo bc sngngn hn (t hn 10 A dn n cc hiu ng nhiu x khng ng k) v su tp trung ln hn so vi quang khc thngthng. Kh nng ch to cc vi cu trc v vi thit b trong khong centimt l c bit quan trng trong cc c cu chp

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S tay C in t

hnh v cc ng dng ng c do cc thng s k thut s dng lc v m men tiu th t cc vi thit b, v do mt lc vm men gii hn, nh thit kt i mt vi s cn thit phi tng kch thc c cu chp hnh.

14.4 C s in t v m hnh MEMS

Phn loi, tng hp cu trc v ti u ha MEMS c trnh by trong phn 14.2. Vic phn loi v ti u ha datrn s suy xt v tng hp h thng in t, phn tch lc chuyn ng manht, thit k hnh hc v tp ca MEMS, v ti

u ha cc i lng khc. Ngi ta phn loi cc vi thit b dng chuyn ng tnh tin v quay (hng tm hoc quanhtrc) bng cch dng cc h thng endless (ng), open-ended(m), v tch hp.

Mc tiu ca chng ta l tip cn v gii quyt mt phm vi rng cc vn thc t bt gp trong thit k phi tuyn, mhnh ha, phn tch, iu khin v ti u ha cc vi cu trc v vi thit b chuyn ng vi mch dn ng/nhn bit ciu khin bi cc IC cho MEMS hiu sut cao. Nghin cu MEMS, tm quan trng c t vo:

Thit k MEMS hiu sut cao thng qua vic ngh ra cc vi thit b chuyn ng mi vi cc vi thit b nng lngpht x, mch dn ng/nhn bit c micro, v cc IC iu khin/x l tn hiu,

Ti u ha v phn tch cc thit b chuyn ng quay v tnh tin,

Pht trin cc IC x l v iu khin tn hiu vi hiu sut cao cho cc vi thit b mi,

Pht trin m hnh ton hc vi cp n gin ha v gi nh nh nht trong min thi gian,

Thit k cc thut ton iu khin bn vng ti u,

Thit k cc h thng thng minh thng qua s t thch nghi, t t chc, kin thc tin ha, to quyt nh, v trthng minh,

Pht trin phn mm v phn cng tin tin t c cp thng minh, tch hp, hiu qu, v hiu sut caonht.

Trong phn ny, mc tiu ca chng ta l thc hin m hnh ha, phn tch v m phng phi tuyn. t c mc tiuny, chng ta p dng m hnh tng hp MEMS, pht trin cc m hnh ton hc phi tuyn m hnh ha ng lc hc int-c hc phc tp, thc hin ti u ha, thit k cc h iu khin vng kn, v thc thi phn tch d liu tp trung trongmin thi gian.

m hnh ha cc thit b chuyn ng in t, ngi ta thng dng th nng vc t t A v th nng v hng inV gii cc phng trnh vi phn ring phn

22

2

A A

A Vt tms me ms

- + + = -

ur

dng phn tch phn t hu hn. y, m, s v e ln lt l thm, dn, v hng s in mi.

Tuy nhin, thit k MEMS in t cng nh thc hin phn tch v ti u ha in t-c hc, cn gii cc phngtrnh vi phn trong min thi gian. Thc t, cc hin tng c bn khng th m hnh, phn tch v nh gi mt cch y bng cch dng phn tch phn t hu hn cho cc m hnh v li gii trng thi n nh. Cn thit phi pht trin cc cngc m hnh ha cho php tng thm tnh in t v c hc phi tuyn trong mt li m hnh in t-c hc n t cphn tch c trung thc cao vi nh gi kh nng v d bo kt qu.

Cc nguyn l hot ng ca MEMS da trn cc nguyn l in t. Mt m hnh in t y nhn c di dng 5vc t in t trng. C th l 3 vc t in trng v 2 vc t t trng. Cc vc t in trng l cng in trng E, mt thng lng in D , v mt dng in J . Cc vc t t trng l cng t trng H v mt t trng B. Cc phng trnh sai phn cho vi thit b chuyn ng c in c ly t cc phng trnh Maxwell, phng trnh cu trc

(ph tr), v c hc c in.Cc phng trnh vi phn ring phn Maxwell trong min E v H dng im l

( , , , )( , , , )

( , , , ) ( , , , )( , , , ) ( , , , ) ( , , , )

( , , , )( , , , )

( , , , ) 0

v

H x y z tE x y z t

t

E x y z t E x y z t H x y z t J x y z t E x y z t

t t

x y z tE x y z t

H x y z t

m

e e s

r

e

= -

= + = +

=

=

ur

ur uruur ur ur

ur

uur

y e l hng s in mi, l thm, s l dn, v vr l mt in tch khi.Cc phng trnh cu trc (ph tr) thu c t hng s in mi e , tenx thm m, v dn s . C th l:

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C cu chp hnh

D Ee= hoc D E Pe= +

B Hm= hoc ( )B H Mm= +

Es= hoc vJ vr=r

Phng trnh Maxwell c th gii c bng cch dng cc iu kin rng buc trn cc vc t trng. Trong mi trng2 vng, chng ta c

2 1 2 1 2 1 2 1( ) 0, ( ) , ( ) , ( ) 0N N s N s Na E E a H H J a D D a B Br - = - = - = - =r r r r

y,sJ l vc t mt dng in b mt, Na l vc t chun php ca b mt ti ranh gii t vng 2 vo trong vng 1, v

sr l mt in tch b mt.

Cc quan h cu trc miu t mi trng c th tch hp vi cc phng trnh Maxwell gn vi cc trng nhm tm ra 2phng trnh vi phn ring phn. Dng cc mt in trng E v t trng H m hnh cc trng in t trongMEMS, ta c

2 22

2 2

22

2

( ) ( )

( ) ( )

J D E EE E E

t t t t H H

H H Ht t

m m ms me

ms me

= - = - - = - -

= - = - -

r r r

r rr r r

Cp phng trnh sng ng nht v khng ng nht di y

22

2

22

20

vE EEt t

H HH

t t

rms me

e

ms me

- - =

- - =

r

r rr

tng ng vi 4 phng trnh Maxwell v quan h cu trc. Vi mt s trng hp, 2 phng trnh ny c th gii c lp.Cn phi nhn mnh rng khng th dng cc iu kin rng buc ch cho E v H , v nh vy, vn ny lun khng n

gin vi 2 vc t in t trng. Bi th, cn dng th nng v hng in v vc t t. Gi th nng vc t t l A v thnng v hng in l V, chng ta c

A A B H E V

tm

= = = - -

r r r r

in t trng l vi phn ca th nng. Dng phng trnh Lorentz

VA

t

= -

r

phng trnh sng th nng vc t khng ng nht cn c gii l

22

2

A AA Vt tms me ms

- + + = -

r

m hnh cc thit b chuyn ng, cn dng cc phng trnh c hc, v nh lut 2 Newton thng c p dng thu c phng trnh chuyn ng.

Dng mt in tch khi vr , ta tm c lc Lorentz lin h cc hin tng in t v c hc

( )v vF E v B E J Br r= + = + r

C th tm lc in t bng cch dng phng php tenx ng sut Maxwell. Khi nim ny dng mt tch phn khi thu c nng lng tch tr, v xc nh ng sut ti tt c cc im ca mt b mt ranh gii. Tng cc ng sut cc b scho lc mng li. C th, ng sut in t l

1( )v

v s

F E J B dv T dsabrm

= + =

r r r r rt

Tenx nng lng ng sut in t (tenx ng sut Maxwell th 2) l

7


8/27

S tay C in t

0

0

0

0

x y z

x z y

y z x

z y x

E E E

E B BT

E B B

E B B

ab

- -

= - - - -

r r rt

r r r

r r r

Ni chung, m men in t trong vi cu trc chuyn ng c tm bng cch dng in t trng. C th, tenx ngsut in t c cho nh sau

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

12

12

12

1

2 12

12

E Ms s s

j j

j j

j j

j j

j j

j j

T T T

E D E D E D E D

E D E D E D E D

E D E D E D E D

B H B H B H B H

B H B H B H B H

B H B H B H B H

= +

-

= - -

-

+ - -

Vi cc h ta Cartesian, tr v cu dng pht trin m hnh ton hc, chng ta c

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3

, , , , , ,

, , , , ,

, , , , , ,

, , , , ,

, , , , , ,

, , ,

x y z x y z

x y z x y z

r z r z

r z r z

E E E E E E D D D D D D

H H H H H H B B B H B B


H H H H H H B B B B B B


H H H H H H B B

q q

q q

r q q r q f

r q f r

= = = = = =

= = = = = =

= = = = = =

= = = = = =

= = = = = =

= = = = 1 2 3, ,B B B Bq f= =

C th dng mi trng Matlab gii cc phng trnh Maxwell.

Trong cc vi thit b chuyn ng, nh thit k phn tch cc c cu sinh ra m men hoc lc.

nh lut 2 Newton vi cc chuyn ng quay v tnh tin l

1,

1,

r rr

d dT

dt J dt dv dx

F vdt m dt

w qwS

S

= =

= =

r

r

y rw v rq ln lt l vn tc v dch chuyn gc, v vx ln lt l vn tc v dch chuyn di, FS l lc mng li,J l m men qun tnh tng ng, v m l khi lng.

14.5 M hnh ton hc ca MEMS

Cc vn m hnh v iu khin MEMS rt quan trng trong nhiu ng dng. Mt m hnh ton hc l mt m t tonhc (di dng hm s hoc phng trnh) ca MEMS tch hp cc vi thit b chuyn ng (cc c cu chp hnh v cmbin c micro), vi thit b nng lng pht x, mch dn ng/nhn bit c micro, cc IC iu khin/x l tn hiu. Mc chca vic pht trin m hnh l hiu v lnh hi v hin tng cng nh phn tch hnh vi gn nhau.

m hnh MEMS, cc phng php phn tch tin tin c dng gii chnh xc cc hin tng, hiu ng, v qutrnh vt l c phc tp cao. S cn thit ca phn tch chnh xc cao, thut ton c hiu nng tnh ton, v vic gim thigian m phng tng ln ng k vi cc vi thit b phc tp, kh khn trong vic ng dng cc phng trnh Maxwell gii quyt vn . Nh minh ha trong phn trc, cc hin tng in t v bin i nng lng phi tuyn c m tbi cc phng trnh vi phn ring phn. Vic dng cc phng trnh Maxwell p ng s cn thit cho cc kh nng phntch d liu tp trung vi s d bo kt qu trong ton b min m phng cng nh c bit cn thit cho m phng v phntch MEMS hiu sut cao. Ngoi ra, ngi ta cn s dng cc phng php m phng v phn tch khc. C th dng m8


9/27

C cu chp hnh

hnh ton hc gp c m t bng cc phng trnh vi phn ton phn. Qui trnh m hnh ton hc v pht trin m hnhnh sau.

Bc u tin l xy dng cng thc cho vn cn m hnh:

kho st v phn tch MEM nh khi nim a phn cp, pht trin cc cp h thng con vo-ra nhiu bin, chnghn, cc vi cu trc chuyn ng (cc c cu chp hnh v cm bin c micro), cc vi thit b nng lng pht tn,mch c micro, cc IC, b iu khin, cc thit b vo/ra;

hiu v lnh hi cu trc MEMS v cu hnh h thng;

thu thp d liu v thng tin;

pht trin cc cp bin vo-ra, nhn dng iu khin, nhiu, u ra, tham chiu (lnh), trng thi c lp v phthuc v cc bin hot ng cng nh cc s kin;

to cc gii thit chnh xc, n gin vn lm cho MEMS cn nghin cu tr nn d kim sot v mt tonhc (l s l tng ha hin tng vt l, cc m hnh ton hc khng bao gi chnh xc tuyt i v vic cc mhnh ton hc n gin ha thc t cho php nh thit k thc hin mt s phn tch trit v to ra cc d onchnh xc cho hot ng ca h thng)

Bc th 2 l a ra cc phng trnh lin h cc bin v s kin:

nh ngha v nh r cc nh lut c bn (Kirchhoff, Lagrange, Maxwell, Newton, v cc nh lut khc) cdng xy dng cc phng trnh chuyn ng. Cc m hnh ton hc ca cc h thng con c micro in t,in, v c hc c th tm c v gia c to ra cc m hnh ton hc ca MEMS thng qua cc bin v s kin

nh ngha. a ra cc m hnh ton hc

Bc th 3 l m phng, phn tch v ph chun:

nhn dng cc phng php gii tch v s c dng phn tch v m phng;

gii bng phng php gii tch v/hoc s cc phng trnh ton hc (v d, cc phng trnh vi phn hoc saiphn, cc phng trnh phi tuyn, v.v..)

dng cc bin (c o hoc quan st) v s kin thng tin, tng hp cc hm tng hp v khng tng hp;

kim tra cc kt qu thng qua vic so snh ton din gia li gii (cc tp nh x u vo-trng thi-u ra- skin ca m hnh) vi cc d liu thc nghim (cc tp nh x u vo-trng thi-u ra- s kin thc nghim);

tnh ton cc hm tng hp v khng tng hp;

nghin cu d liu gii tch v s da vo d liu v du hiu thc nghim mi.Nu khng m bo s tng hp vi chnh xc cho trc, cn phi ci tin m hnh ton hc ca MEMS v nh thit

k phi bt u li chu trnh.

L thuyt in t v c hc c in hnh thnh c s cho s pht trin m hnh ton hc cho MEMS. Ngi ta ch rarng MEMS c th c m hnh bng cc phng trnh Maxwell v cc phng trnh chuyn ng c hc xon. Tuy nhin,t m hnh, phn tch, thit k, iu khin, v m phng, c th hnh thnh v s dng cc m hnh ton hc c cho bi ccphng trnh vi phn ton phn.

Xem xt vi cu trc dng quay (nam chm thi, cun dng, v vi slnit) trong mt t trng khng i, xem hnh 14.5.Vi cu trc s quay nu c m men in t. Cn nghin cu in t trng tm m men in t.

M men hng thng m men t m vi B , v

T m B= ur

HNH 14.5 Quay theo chiu kim ng h ca vi cu trc chuyn ng

Cho mt vi cu trc c ng knh ngoi rD , cng t Q. Khi , m men t l rm QD= , v lc c tnh lF QB= .

M men in t l

9


10/27

S tay C in t

12 sin sin sin

2 r rT F D QD mBa a a= = =

Dng vc t n v theo hng m men t ma , ta thu c

m mrT m B a m B QD a B= = = ur r r

Vi mt cun dng c din tchA, ta tnh c m men

m mT m B a m B iAa B= = = ur r r

Vi mt solenoid cNvng, ta thu c

m mT m B a m B iANa B= = = ur r r

T m men in t tm c, dng nh lut 2 Newton, ta c

1 1( ),r rL r

d dT T T

dt J J dt

w qwS= = - =

r ur ur

y LT l m men ti.

C th dng cc lc in ng (emf) v t ng (mmf) khi pht trin m hnh.

Ta c

1 1( )

s

Bemf E dl v B dl ds

t

= = -

r r rur r r

v

l s s

Dmmf H dl J ds ds

t

= = +

ruur ur r r

Vi thit k s b, khi cn chnh xc va , c th dng cc nh lut Faraday hoc Lenz cho lc in ng didng thay i t trng theo thi gian. C th,

r

rr r

ddemf

dt t dt t

q y y y y y w

q q

= - = - - = - -

yt

y

l s hng bin i.

Lin h thng lng tng l

1

4 s pNy p= F

y SN l s vng v PF l thng lng trn mt cc.

Vi cc vi my dng t p hng tm, ta c

2

s

p in st e

iNR L

P g

mF =

y i l dng in trn vi cun pha (cung cp bi IC), instR l bn knh stato bn trong,L l t cm,Pl s cc, v eg l

khe h tng ng bao gm khe h khng kh v dy hng tm ca nam chm vnh cu.

Gi s vng trn mt pha l SN , lc in ng l

coss riN

mmf PP

q=

Biu thc n gin ha cho lc in ng vi vi my khng chi qut dng t p hng tm l

12 ag s s r r

T PB i N L D=

y agB l mt thng lng khe h khng kh, ( )/ 2 coseg S e r B iN Pg Pm q= , si l dng in tng, rL l chiu di hot

ng (chiu di trc roto), v rD l ng knh roto pha ngoi.

10


11/27

C cu chp hnh

Cc vi my khng chi qut dng t p hng trc c th c thit k v ch to. M men in t c cho nh sau

2ax ag s s aT k B i N D=

y axk l h s phi tuyn c tm nh dy dn hot ng v chiu di nam chm vnh cu mng mng; v aD l chiu ditng ng l mt hm s ca cun dy v t p nam chm vnh cu.

V d 14.5.1: M hnh ton hc ca vi bin i tnh tin

Hnh 14.6 minh ha mt vi cu trc dch chuyn n gin vi mt b phn ng yn v mt vi cu trc c kh nngchuyn ng tnh tin (pittng) c gia cng bng qui trnh x l l lin tc. Cun dy c in bng cng ngh vi giacng/CMOS.

Chng ta p dng nh lut 2 Newton nghin cu ng lc hc. nh lut Newton pht biu rng gia tc ca mt vt tl vi lc mng li. Vc t tng ca tt c cc lc c tm nh sau

22

1 22( ) ( ) ( )v s s e

d x dx F t m B k x k x F t

dt dt = + + + +

HNH 14.6 S vi chuyn i vi vi cu trc chuyn ng tnh tin

yx l dch chuyn ca vi cu trc tnh tin (pittng), m l khi lng ca pittng dch chuyn, vB l h s ma st nht,

1sk v 2sk l cng (l xo c th lm t polysilicon), v ( )eF t l lc t c tm t ng nng lng eW .

Cc lc cng v phc hi khng t l trc tip vi dch chuyn, v cc lc ny khc nhau c 2 pha ca v tr cn bng.

Lc cng/phc hi ca l xo polysilicon c biu din nh sau ( )21 2s sk x k x + .

Gi s rng h t l tuyn tnh, ng nng lng c biu din nh sau

21( , ) ( )2c

W i x L x i=

Tip theo

2 ( )1

( , ) 2edL x

F i x i dx= t cm tm c nh sau

220( )

2 ( 2 )f f g

f g g f f f

N A ANL x

A l A x d

mm

m= =

+ + +

y f v g ln lt l t tr ca vt liu st t v khe h khng kh, fA v gA l cc tit din ngang lin kt, v fl v

( )2x d+ ln lt l cc chiu di ca vt liu t v khe h khng kh.

Nh vy

2 2 22 0

2[ 2 ( 2 )]

N A AgdL f f

dx A l A x d g f f f

m m

m= -

+ +

Dng nh lut Kirchhoff, phng trnh in p cho vi mch pha l

11


12/27

S tay C in t

a

du ri

dt

y= +

y lin kt thng lng y c cho nh sau ( )L x iy = .

Chng ta thu c

( )( )a

dL xdi dx u ri L x i

dt dx dt = + +

v nh vy

2 2 20

2

2 1

( ) ( )[ 2 ( 2 )] ( )f f g

a

g f f f

N A Adi ri iv u

dt L x L x A l A x d L x

mm

m= - + +

+ +

Thm phng trnh sau vo phng trnh vi phn

22

1 22( ) ( ) ( )v s s e

d x dx F t m B k x k x F t

dt dt = + + + +

tm c 3 phng trnh vi phn phi tuyn cho vi thit b chuyn ng tnh tin cn nghin cu

2 20 0

2 2 20 2 2

1 22

[ 2 ( 2 )] 2 2 ( 2 )

2 ( 2 )

1( )

[ 2 ( 2 )]

g f f f f f g f f f

af f g g f f f f f g

f f g vs s

g f f f

r A l A x d A A l A x d dii iv u

dt N A A A l A x d N A A

dxv

dt

N A A Bdvi k x k x v

dt m A l A x d m m

m m m

mm m mm

mm

m

+ + + += - + +

+ +

=

= - + -+ +

V d 14.5.2: M hnh ton hc ca vi ng c t tr ng b s cp

Xem xt mt vi ng c t tr 1 pha c gia cng trc tip bng cc cng ngh CMOS, LIGA, gn-LIGA truyn thng.Vt liu st t c dng gia cng cc stato v roto c micro, v c th kt ta cun dy trn stato, xem hnh 14.7.

Cc trc t vung gc v thngc c nh vi vi ng c khi n quay vi vn tc gc rw . Cc trc t ny quay vi

vn tc gc w. Gi s iu kin u bng khng. Nh vy, dch chuyn gc ca roto rq v dch chuyn gc ca trc tvung gc q bng nhau, v

0 0

( ) ( )t t

r rt t

d dq q w t t w t t = = =

T tr do t ha m l mt hm ca dch chuyn gc roto rq . Vi s vng dy SN , t tr do t ha l

2

( )( )

Sm r

m r

NL q

q=

g

T tr do t ha thay i 2 ln trong 1 vng quay roto v c gi tr nh nht v ln nht, v

2 2

min maxmax min 1 3 5

0, ,2 ,... , , ...2 2 2

,( ) ( )r r

S S

m m

m r m r

N N

L Lq p p q p p p

q q= =

= =

HNH 14.7 ng c t tr n pha c microo vi vi cu trc (vi ng c) chuyn ng quay

12


13/27

C cu chp hnh

HNH 14.8 T tr do t ha ( )m rL q

Gi s rng s thay i ny l mt hm hnh sin ca dch chuyn gc roto. Khi o,

( ) cos2mm r m r L L Lq q= - V

y mL l gi tr trung bnh ca t tr do t ha v mLD l mt na bin bin thin hnh sin ca t tr do t ha.

th ca ( )m rL q c thy trn hnh 14.8.

M men in t trong cc ng c t tr 1 pha c tnh nh biu thc ca ng nng lng ( ),e as r W i q . T

( ) ( ) 21, cos22e as r ls m m r as

W i L L L iq qD= + - , ta tm c

2122( , ) [ ( cos2 )] sin2

mc as r as ls m r e m as r

r r

W i i L L LT L i

q qq

q q

+ -= = =

V

V

M men in t khng c cc ng c t tr ng b nu IC nui dng hoc in p 1 chiu trong cun dy ca ngc bi v 2as sin2e m rT L i qD= . Nh vy, khng th p dng cc thut ton iu khin truyn thng, v cn phi nghin cu

cc phng php mi da trn cc c tnh in t. Gi tr trung bnh ca Te khng bng khng nu dng in l hm ca rq . minh ha, chng ta gi s dng in di y c nui trong cun dy ca ng c

Re( sin2 )as M r i i q=

Khi , m men in t l

2 2 2sin2 (Re sin2 ) sin2 0e m as r m M r r T L i L iq q q= = V V

v

2 2

0

1 1sin2

4eav m as r r m MT L i d L i

p

q qp

= = V V

C th tm m hnh ton hc ca ng c t tr 1 pha c micro bng cch dng nh lut Kirchhoff v nh lut 2 Newton

asas s as

du r i

dt

y= + (phng trnh mch)

2

2r

e m r L dT B T Jdt

qw- - = (phng trnh c hc-xon)

T ( )cos2as ls m m r asL L L iy qD= + - , ta thu c 3 phng trnh vi phn phi tuyn bc nht. C th nh sau

2

2 1sin2

cos2 cos2 cos2

1( sin2

as s mas as r r as

m m mls m r ls m r ls m r

rm as r m r L

rr

di r Li i u

dt L L L L L L L L L

dL i B T

dt Jd

dt

w qq q q

wq w

qw

= - ++ - + - + -

= - -

=

V

V V V

V

V d 14.5.3: M hnh ton hc ca cc vi ng c bc nam chm vnh cu 2 phaVi cc vi ng c bc nam chm vnh cu 2 pha, ta c

13


14/27

S tay C in t

asas s as

bsbs s bs

du r i

dtd

u r idt

y

y

= +

= +

y cc lin kt t thng l as asas as asbs bs asmL i L iy y= + + v bs bsas as bsbs bs bsmL i L iy y= + + .

y, asu v bsu l cc in p pha trong cc vi cun dy stato as v bs; asi v bsi l cc cc dng in trong cc vi cun

dy stato; asy v bsy l cc lin kt t thng stato, rs l in tr ca cun dy stato;Lasas,Lasbs,Lbsas,Lbsbs l cc t tr chung.

C th tm vn tc v dch chuyn gc in bng cch dng s rng roto RT,

r rm

r rm

RT

RT

w w

q q

=

=

y rw v rmw ln lt l cc vn tc gc in v roto, v rq v rmq ln lt l cc dch chuyn gc in v roto.

Cc lin kt thng lng l cc hm ca s rng roto RT, v bin ca cc lin kt thng lng sinh ra bi nam chmvnh cu my . C th,

cos( )asm m rmRT y y q= v sin( )bsm m rmRT y y q=

t cm ca cun dy stato lmss asas bsbs lsL L L L L= = = +

Cc vi cun dy stato c dch chuyn mt gc 90 in. Nh vy, cc t tr chung gia cc vi cun dy stato bngkhng, 0asbs bsasL L= = .

Khi , ta c

cos( )asm ss as m rmL i RT y y q= + v sin( )bs ss bs m rmL i RT y y q= +

Lu n cc phng trnh mch, ta c

[ cos( )sin( )

[ sin( ) cos( )

ss as m rm asas s as s as ss m rm rm

ss bs m rm bsbs s bs s bs ss m rm rm

d L i RT diu r i r i L RT RT

dt dt

d L i RT diu r i r i L RT RT

dt dt

y q y w q

y q y w q

+= + = + -

+= + = + +

Do , ta thu c

1sin( )

1cos( )

as s mas rm rm as

ss ss ss

bs s mbs rm rm bs

ss ss ss

di r RT i RT u

dt L L L

di r RT i RT u

dt L L L

yw q

yw q

= - + +

= - - +

Dng nh lut 2 Newton, ta c

1( )rm e m rm L

rmrm

dT B T

dt Jd

dt

ww

qw

= - -

=

C th tm c biu thc cho m men in t cho cc vi ng c bc nam chm vnh cu. Lu ti mi lin h ngnng lng

2 21( ) cos( ) sin( )2c ss as ss bs m as rm m bs rm PM

W L i L i i RT i RT W y q y q= + + + +

ta tm c m men in t

[ sin( ) cos( )]ce m as rm bs rmrm

WT RT i RT i RT y q q

q

= = - -

Do , s pht trin tc thi ca cc dng in pha ias v ibs, vn tc gc roto rmw , v dch chuyn gc roto rmq c lyt cc phng trnh vi phn di y

14


15/27

C cu chp hnh

1sin( )

1cos( )

1[ sin( ) cos( )]

as s mas rm rm as

ss ss ss

bs s mbs rm rm bs

ss ss ss

rm m mas rm bs rm rm L

rmrm

di r RT i RT u

dt L L L

di r RT i RT u

dt L L L

d RT Bi RT i RT T

dt J J J

ddt

yw q

yw q

w yq q w

q w

= - + +

= - - +

= - - - -

=

Bn phng trnh vi phn phi tuyn ny c vit li di dng khng gian trng thi nh sau

0 0 0 sin( )

0 0 0 cos( )

0 0 0 [ sin(

0 0 1 0

as s mrm rm

ss ssasbs s m

bs rm rmss ss

rmrmm m

asrm

rm

di r RTRT

dt L Lidi r RT

i RTdt L Ld

B RTi RTdt

J Jd

dt

yw q

yw q

wwy

qq

- - - = + - -

) cos( )]

0

rm bs rmi RTq q

-

10 0

010 1

0 00

0 0

ss

as

Lss bs

L

uTL u

J

+ -

Vic phn tch phng trnh m men

[ sin( ) cos( )]e m as rm bs rmT RT i RT i RT y q q= - -

dn n kt lun rng biu thc cho mt tp hnh sin dng in 2 pha cn bng l2 sin( )as M rmi i RT q= - v 2 cos( )bs M rmi i RT q=

Nu nhng dng in pha ny c nui, m men in t l mt hm ca bin dng in iM v

2e m MT RT iy=

Cc dng in pha cn c nui l cc hm ca dch chuyn gc roto. Gi s rng cc t tr l nh khng ng k, ta ccc in p pha di y

2 sin( )as M rmu u RT q= - v 2 cos( )bs M rmu u RT q=

V d 14.5.4: M hnh ton hc ca vi ng c ng b nam chm vnh cu 2 pha

Xem xt cc vi ng c ng b nam chm vnh cu 2 pha. Dng nh lut in p Kirchhoff, ta c

asas s as

bsbs s bs

du r i

dtd

u r idt

y

y

= +

= +

y cc lin kt thng lng c biu din nh sau as asas as asbs bs asmL i L iy y= + + v bs bsas as bsbs bs bsmL i L iy y= + + .

Cc lin kt thng lng l cc hm tun hon ca dch chuyn gc (v tr roto), v t

sinasm m rm y y q= v cosbsm m rm y y q= -

t cm ca cc vng dy stato c tnh nh sau

mss asas bsbs lsL L L L L= = = +

15


16/27

S tay C in t

Cc cun dy stato c dch chuyn 90 in, v do , cc t tr chung gia cc cun dy stato l 0asbs bsasL L= = .Nh vy, ta c

sinas ss as m rmL i y y q= + v cosbs ss bs m rmL i y y q= -

Do , ta tm c

( sin )cos

( cos )sin

ss as m rm asas s as s as ss m rm rm

ss bs m rm bsbs s bs s bs ss m rm rm

d L i diu r i r i L

dt dt d L i di

u r i r i Ldt dt

y q y w q

y q y w q

+= + = + +

-= + = + -

Dng nh lut 2 Newton

2

2rm

e m rm L

dT B T J

dt

qw- - =

ta c

1( )rm e m rm L

rm

rm

dT B T

dt Jd

dt

ww

qw

= - -

=

Chng ta c th thu c biu thc m men in t cho ng c nam chm vnh cu bng cch dng ng nng lng

2 21( ) sin cos2c ss as ss bs m as rm m bs rm PM

W L i L i i i W y q y q= + + - +

Khi , ta c

( cos sin )2

c rme as rm bs rm

rm

W PT i i

yq q

q

= = +

Thm cc qu trnh chuyn tip mch vo ng lc hc c hc-xon. ta tm c m hnh ton hc ca vi ng c namchm vnh cu 2 pha dng sau

1cos

1sin

1( cos sin )

2

as s mas rm rm as

ss ss ss

bs s mbs rm rm bs

ss ss ss


rmrm

di r i udt L L L

di ri u

dt L L L

d P Bi i T

dt J J Jd

dt

y w q

yw q

w yq q w

qw

= - - +

= - + +

= + - -

=

HNH 14.9 Dng sng ca mmf khe h khng kh v dng in pha

Vi cc ng c 2 pha (i vi cc phn b cun dy hnh sin v cc dng song mmf hnh sin), m men in t c biudin nh sau

16


17/27

C cu chp hnh

( cos sin )2

me as rm bs rm

PT i i

yq q= +

Nh vy, m bo trng thi cn bng, cn nui

2 cosas M rmi i q= v 2 sinbs M rmi i q=

cc i m men in t. Tm li, ta thu c

2 2( cos sin ) 2 (cos sin )2 2 2

m m me as rm bs rm M rm rm M

P P PT i i i i y y y q q q q= + = + =

Hnh 14.9 v cc dng sng khe h khng kh v dng in pha.

14.6 iu khin MEMS

M hnh ton hc ca MEMS c pht trin vi cc phc tp tnh ton khc nhau. Cn nhn mnh l ngoi cc mhnh thit b chuyn ng c micro, nn xc nh ng lc hc nhanh ca IC. Do phc tp tnh ton ca cc m hnh ICy , s khng thc t ca cc phng trnh pht trin, v ng lc hc rt nhanh, c th m hnh ng lc hc IC dngphng trnh vi phn rt gn hay nh ng lc hc khng m hnh. Vi MEMS, m hnh dng cc phng trnh vi phntuyn tnh v phi tuyn

min max

min max

( ) , ,( ) ( , , , ) ( , , ) , ( )z p

x t Ax Bu u u u y Hx x t F t x r z B t x p u u u u y H x

= + == + =

&&

c th thit k cc thut ton khc nhau.

y, cc vc t trng thi, iu khin, u ra, v tham chiu (lnh) c nh ngha l x, u,y, v r; cc trng thi thams khng r rng (nh cc h s bin i theo thi gian, ng lc hc khng m hnh, nhng thay i khng xc nh, v.v..)dng cc vc tz vp m hnh ha.

Cc ma trn h s lA,B, vH. Cc trng nh x lm mn m hnh tuyn tnh c nh ngha l ( ).zF , ( ).pB v ( ).H .

Cn nhn mnh rng iu khin c gii hn. V d, dng h s s dng Dd lm tn hiu iu khin, ta c 0 1Dd hoc 1 1Dd- + . Dng cc IC bn gc do kh nng tt hn, v 1 1Dd- + . Nh vy, ta c 1 1u- + . Tuynhin, ni chung, min maxu u u .

iu khin t l - tch phn vi phn

Nhiu MEMS c th c iu khin bi cc b iu khin t l - tch phn vi phn (PID) cng vi cc gii hn iukhin, nh thy trong [9]

max

min

max

min

2 1 2 12 12 1 2 12 1

min max0 0 0

( ) , ,

,

uu

j jj

uu pj ij dj

j j j

deu t e edt

dt

k e k e dt k e u u uz s a

m gb

+ +++ ++

= = =

=

= + +

sat

sat &

y pjk , ijk v djk l cc ma trn h s khuch i phn hi t l, tch phn, v vi phn; , b , s , m, a v g l cc s

nguyn khng m.Trong cc b iu khin PID phi tuyn, ngi ta dng sai s bm. C th

( ) ( ) ( )e t r t y t = -

Cc b iu khin gii hn tuyn tnh c th c thit k trc tip. V d, t 0V b s m= = = = , ta c lut iu khinPI phi tuyn nh sau:

( )maxmin 0 0( ) ( )uu p iu t k e t k etdt = + sat

17


18/27

S tay C in t

Cc b iu khin vi s m rng phn hi trng thi c th c tng hp nh saumax

min

max

min

2 1 2 12 12 1 2 12 1

min max0 0 0

( ) ( , )

( , )( ) ,

uu

j jj

uu pj ij dj

j j j

u t e x

V e xk e k e dt k e G t B u u ue

x

z s am gb

+ +++ ++

= = =

=

= + + +

sat

sat &

y ( ),V e x l hm s tha mn cc yu cu chung cho cp Lyapunov [9], v d, dng cc iu kin cho n nh.

R rng rng cc nh x phn hi phi tuyn c a ra v c th tng hp v s dng hm khng ton phng ( ),V e x thu c thut ton iu khin v cc h s khuch i phn hi.

iu khin bm

iu khin bm c thit k cho cc h thng c khuch i bng cc bin trng thi v ng lc hc tham chiu. Cth, t

( ) , ( ) ( ) ( ) ( ) ( )ref x t Ax Bu x t r t y t r t Hx t = + = - = -& &

ta tm c

00( ) , , , , ,

100refx A B

x t A x B u N r y Hx x A B Nx HS S S S S S S S S

= + + = = = = = -

&

Cc tiu phim hm c trng ton phng

0

1( )

2

tfT T

t J x Qx u Gu dt S S= +

ta tm c lut iu khin dng iu kin cn thit bc nht cho vic ti u. C th, ta c

1 1

0

T

TBV V

u G B Gx x

- -S

S S

= - = -

y, Q l ma trn h s c nh bn xc nh dng, G l ma trn trng s c nh dng.

Li gii ca phng trnh Hamilton-Jacobi

11 12 2

T T

T TV V V V x Qx Ax B G Bt x x x

-S S S S S

S S S

- = + -

tha mn bi hm hi tip ton phng12

TV x K x = . y, K l ma trn i xng, c tm c tm bng cch gii

phng trnh vi phn phi tuyn

1 , ( )T T T T f fK Q A K K A K B G B K K t K -

S S S S- = + + - =&

B iu khin c cho nh sau

1 1

0

T

TB

u G B K x G K x - -S S S

= - = -

T ( ) ( )refx t e t =& , ta c

( ) ( )ref x t e t dt = Do , ta thu c lut iu khin tch phn

1( )

( ) 0 ( )

T x tB

u t G K e t dt

-

= -

Trong thut ton iu khin ny, vc t sai s c b sung trong phn hi trng thi.

18


19/27

C cu chp hnh

Nh minh ha, cc gii hn c p cho iu khin, v min maxu u u . Nh vy, phi thit k cc b iu khin cgii hn. Dng phim hm c trng ton phng [9]

( )0

1tanft T

t J x Qx G udu dt -S S= +

vi ma trn h s c nh bn xc nh dng Q v ma trn xc nh dng G, ta tm c

1 1 11

( ) ( )( ) tanh , 1 10 0( ) ( )

T T

x t x t B Bu t G K sat G K ue t dt e t dt

- + --

= - - -

B iu khin ny thu c vi gi thit rng li gii ca phng trnh vi phn ring phn c th xp x bng hm hi tipton phng

12

TV x K x S S=

yKl ma trn i xng.

iu khin ti u thi gian

Mt b iu khin ti u thi gian c th c thit k bng cch dng phim hm

0

1( )

2

ft T

t x Qx dtS S=

Ta c phng trnh Hamilton-Jacobi

1 1

1min ( )

2

T

T

u

V V x Qx Ax B u

t xS S S S

- S

- = + +

b iu khin dng rle c tm nh sau

sgn , 1 1TV

u B ux

S

S

= - -

Thut ton iu khin ti u khng th thc hin trong thc t do hin tng rung. Do , thng dng cc lut iukhin dng r le vi vng cht

sgn , 1 1T

deadzone

Vu B u

xS

S

= - -

iu khin dng trt

Lut iu khin dng trt chuyn mm c thy trong [9]. Thut ton chuyn mm dng trt cung cp c trng tthn, v loi b c hiu ng rung.

thit k cc b iu khin, chng ta m hnh ng lc hc trng thi v sai s nh sau:

( ) , 1 1

( ) ( )

x t Ax Bu u

e t Nr t HAx HBu

= + -

= - -

&

& &

Bn sao trt mn l

0

01

{( , , ) ( , , ) 0}

{( , , ) ( , , )} 0m

jj

M t x e R X E t x e

t x e R X E t x e

u

u

=

= =

= =

B mt chuyn phi tuyn thay i theo thi gian l ( ) ( ), , , , 0xet x e K t x euu = = . Lut iu khin chuyn mm c chonh sau

( , , ) ( ), 1 1, 0u t x e G u Gf u= - -

y ( ).f l hm gii tch thc lin tc ca lp C e ( 1e ), v d, tanh v erf.

19


20/27

S tay C in t

iu khin c rng buc ca MEMS phi tuyn: phng php Hamilton-Jacobi

Ti u ha c rng buc ca MEMS l mt ch rt c quan tm trong thc t. Dng l thuyt Hamilton-Jacobi, cth tng hp cc b iu khin c gii hn cho h thng lin tc c m hnh nh sau

MEMS MEMS MEMS 2 1 MEMS

MEMS MEMSmin max 0 0

( ) ( ) ( ) ,

, ( )

ws s x t F x B x u y Hx

u u u x t x

+= + =

=

&

y, MEMSx X l vc t trng thi; u U l vc t cc u vo iu khin; y Y l u ra o c; ( ).sF , ( ).sB v( ).H l cc nh x lm mn; ( )0 0sF = , ( )0 0sB = , v ( )0 0H = ; v w l s nguyn khng m.

thit k b iu khin bm, ta b sung ng lc hc MEMS

( )MEMS MEMS MEMS 2 1 MEMS

MEMS MEMSmin max 0 0

( ) ( ) ( ) ,

, ( )

w

s sx t F x B x u y H x

u u u x t x

+= + =

=

&

vi ng lc hc ngoi sinh ( ) ( )MEMSref x t Nr y Nr H x = - = -& .

Dng vc t trng thi b sung

MEMS

ref

xx X

x

=

ta thu c

2 1min max 0 0( ) ( , ) ( ) , , ( ) ,

0( ) ( )( , ) , ( )

0( )

MEMS

w

ref

MEMS MEMSs s

MEMS

xx t F x r B x u u u u x t x x

x

F x B xF x r r B x

NH x

+ = + = =

= + = -

&

Tp iu khin c th chp nhn Ugm hm o c Lebesgue ( ).u v mt b iu khin gii hn c th c thit kvi cc tp iu khin rng buc

min max{ , 1,...., }m

i i iU u u u u i m= =

Chng ta nh x cc gii hn iu khin vo mt hm lin tc gi tr vc t, Lipschitz hon ton, n nh, kh tch v b

gii hn ( )1Ce eF . Mc tiu ca chng ta l thit k bng gii tch b iu khin phn hi trng thi gii hn chp

nhn c dng ng nh ( )u x= F . Hu ht F thng thng l cc hm n nh, kh tch, kh vi lin tc i s v siuvit (m, hyperbolic, logarit, lng gic). V d, hm kh tch n nh l tanh vi min ( ),- + nh x sang cc gii hniu khin. Hm ny c hm nghch o tng ng l tanh-1.

Hm tn tht cn cc tiu ha c cho nh sau

0 0

1 1 2[ ( ) ( )] [ ( ) (2 1) ( ( )) ( ) ]T w x u x t tW x W u dt W x w u G diag u du dt

- -= + = + + F y 1 m mG - l ma trn ng cho xc nh dng.

Hm kh tch c trng ( ).xW v ( ).uW l cc hm gi tr thc, xc nh dng v kh tch kh vi lin tc. Dng cc tnhcht ca hm F , ta chng minh c hm nghch o 1-F l kh tch. V vy, tn ti tch phn

1 1 2( ( )) ( )T wu G diag u du- -F

V d

Xem xt mt h ng lc phi tuyn

3 min max,dx ax bu u u udt = +

Vi

20


21/27

C cu chp hnh

1 1 2( ) (2 1) ( ( )) ( )T wuW u w u G diag u du- -= + F

ta c hm kh tch xc nh dng

1 1 2 3 1 2 11 1 1( ) 3 tanh tanh ln(1 ),3 6 3u

W u uG u du u u u G- - - -= = + - =

Ni chung, nu dng mt hm tang hyperbol nh x hiu ng bo ha, cho trng hp mt u vo, ta c

2 12 1 2 1 1

2 2( ) (2 1) tanh tanh

ww w

u

u u uW u w u du u k du

k k k u

+- + -= + = -

-

Cc iu kin cn thit hm iu khin ( ).u m bo cc tiu ti hm Hamilton

1 1 2 2 1( )( ) (2 1) ( ( )) ( ) [ ( , ) ( ) ]T

T w wx

V xH W x w u G diag u du F x r B x u

x

- - += + + F + +

l: iu kin cn thit bc nht 1n ,

0H

u

=

v iu kin cn thit bc hai 2n ,

2

0T

H

u u

>

Hm hi tip xc nh dng ( ). , , 1V V Ck k , l

0 0 0( ) inf ( , ) inf ( , ()) 0u U

V x J x u J x

= = F

Phng trnh Hamilton-Jacobi-Bellman c cho nh sau

1 1 2 2 1( )min ( ) (2 1) ( ( )) ( ) [ ( , ) ( ) ]T

T w wx

u U

V xVW x w u G diag u du F x r B x u

t x

- - +

- = + + F + +

C th xy dng b iu khin bng cch tm gi tr iu khin t vic cc tiu ti hm khng ton phng. iu kincn thit bc nht ( 1n ) dn chng ta ti mt lut iu khin gii hn c th chp nhn c. C th,

( )( ) ,T

V xu GB x u U

x

= - F

iu kin cn thit bc hai cho ti u ( 2n ) c p ng do ma trn 1G - xc nh dng. Nh vy, thit k c mtng c vin iu khin lin tc, gii tch thc, gii hn v duy nht.

Nu tn ti mt hm thch hp ( )V x tha mn phng trnh Hamilton-Jacobi, h thng ng thu c l n nh thtrong cc tp trng thi danh ngha X v iu khin U, v bm bn vng c m bo trong tp li v compact

( ), , ,o oXY X U R E . Tc l, tn ti mt min n nh bt bin

0 0

0 0 0

{ , : ( ) ( , ) ( ), ( ) ( , ) ( ) ( ),

( , ), [ , ), ( , , )}

c b x u e r y

c b

S x e x t x t u e t e t r y

x X X U t t e E E R Y

= + + +

" " "

v iu khin ( ). ,u u U dn sai s bm n tp

0 0 0, 0

0 0 0

( ) { : , ( ), , , [ , )

( ) ( , ) , 0, ( , , ), [ , 0}

bE

be

S e e E x X X U r R y Y t t

e t e t e E E R Y t t

d

d d

=

+ " "

y xQ v cQ l cc hm KL; v uQ , rQ v yQ l cc hm K.

C th tm li gii phng trnh hm bng cch dng cc hm hi tip khng ton phng. thu c ( ).V , hm tntht c c lng ti cc gi tr trng thi v iu khin cho php. Cc phim hm phi tuyn v tuyn tnh nhn cc gi trcui cng, v gi tr tn tht ton phng nh nht c cho bi cc dng chui ly tha [9]. Tc l,

2( 1)

2 1min 0

0

( ) , 0,1,2,..., 0,1, 2,...i

i

J v x

ghg h g

+ +

+

=

= = =

21


22/27

S tay C in t

Li gii ca phng trnh vi phn ring phn c tha mn bng mt hm hi tip xc nh dng kh vi lin tc

1 12 1 2 1

0

2 1( )

2( 1)

Ti i

i

i

V x x K x i

g ghg gg

g

+ + + ++ +

=

+ = + +

y ma trnKi c tnh bng cch gii phng trnh Hamilton-Jacobi.

Hm hi tip ton phng trong ( )1

2

T

o

V x x K x = c tnh bng cch t 0h g= = . C th dng hm ny ch khi nh

thit k c th b qua cc thnh phn bc cao trong khai trin chui Taylor. Dng 1h = v 0g = , ta thu c

2 20 1

1 1( ) ( )

2 4T TV x x K x x K x = +

trong khi vi 4h = v 1g = , ta c hm sau

2/ 3 2/ 3 4/ 3 4/ 3 5/ 3 5/ 3 2 20 1 2 3 4

3 1 3 3 1( ) ( ) ( ) ( ) ( )

4 2 8 10 4T T T T T V x x K x x K x x K x x K x x K x = + + + +

B iu khin gii hn phi tuyn c cho nh sau

12 1 2 1

0

2 11

2 12

2 1

2 11

2 1

( ) ( ) ( ) ( ) ,

0 0 0

0 0 0

( )

0 0 0

0 0 0

i i

T

ii

i

i

i

i

c

i

c

u GB x diag x t K t x t

x

x

diag x t

x

x

g ghg g

g

g

g

g

g

g

g

g

g

g

- + ++ +

=

-+

-+

-+

-+

-

-+

= - F

=

L

L

M M O M M

L

M

Nu cc ma trnKi l ma trn ng cho, chng ta c thut ton iu khin nh sau:

2 12 1

0

( )i

Ti

i

u GB x K x h

g

++

=

= - F

iu khin rng buc ca MEMS khng xc nh phi tuyn: phng php Lyapunov

Trn mc [ , )ot , ta coi ng lc hc ca MEMS c m hnh nh sau

min max 0 0( ) ( , , , ) ( , , ) , ( ), , ( )z px t F t x r z B t x p u y H x u u u x t x = + = =&

y 0t l thi gian; x X l vc t khng gian trng thi; u U l vc t u vo iu khin gii hn; r R v

y Y l cc vc t tham chiu v u ra o c; z Z v p P l cc phn c th tham s, cc hm ( ).z v ( ).p lLebesgue o c v bit trong gii hn; Zv P l cc tp compact khng rng bit; v ( ).zF , ( ).pB v ( ).H l cctrng nh x lm mn.

Chng ta s lp cng thc v gii bi ton iu khin chuyn ng bng cch tng hp cc b iu khin bn vng mbo tnh bm n nh v bn vng. Mc tiu ca chng ta l nhm thit k cc lut iu khin lm n nh bn vng cc h

phi tuyn vi cc tham s khng xc nh v hng sai s bm ( ) ( ) ( ) ,e t r t y t e E= - ti n tp compact mt cch bn

vng.Vi MEMS c m hnh bi cc phng trnh vi phn phi tuyn vi bin thin tham s, bm bn vng ca vc t ura o c y Y cn c hon thnh i vi vc t u vo tham chiu gii hn ng dng o c r R .

ng lc hc khng xc nh v danh ngha c nh x bi ( ).F , ( ).B v ( ).X . Nh vy, tin trnh h thng c mt nh sau

min max 0 0( ) ( , , ) ( , ) ( , , , , ), ( ) , ( )zx t F t x r B t x u t x u z p y H x u u u x t x = + + X = =&

Tn ti mt qui tc l ( ), , , ,t x u z pX v ( ) ( ), , , , ,t x u z p t x rX , y ( ).r l hm o c Lebesgue lin tc. Mc tiu

ca chng ta l gii quyt bi ton iu khin chuyn ng, v cn tng hp cc b iu khin bm dng vc t sai s bm v

22


23/27

C cu chp hnh

cc bin trng thi. Hn na, m bo tnh bn vng v m rng bin n nh, tng kh nng ng lc hc, v nhmtha mn cc yu cu khc, cc hm Lyapunov khng ton phng ( ), ,V t e x s c dng trong phn tch v thit k n nhca cc lun iu khin bm bn vng.

Gi s rng l mt tp iu khin chp nhn c Ubao gm c hm o c Lebesgue ( ).u . C th dng l thuytHamilton-Jacobi tm cc lut iu khin, v vic ti thiu ha cc phim hm c trng khng ton phng dn n biu khin gii hn.

t ( ), ,u t e x = F , ta thu c mt tp cc b iu khin chp nhn c. p dng phn hi sai s v trng thi, ta xcnh mt h b iu khin bm nh sau

( , , ) ( , , )1( ) ( , , ) ( ) ( ) ( , ) ( ) ( , ) ,T TE E X

V t e x V t e x du x t e x x G t B t x G t B t x s

s e x dt

= W F = - W F + =

y ( ).W l hm phi tuyn; ( ).EG v ( ).XG l cc hm ma trn ng cho xc nh trn [ , )ot ; ( ).EB l hm ma trn; v( ).V l hm lin tc, kh vi, v gii tch thc.

Chng ta cng thit k hm Lyapunov. Vn ny l mt vn ti hn v cha ng nhiu kh khn. C th dng ccng c vin cho hm Lyapunov. Tuy nhin, vi cc h phi tuyn khng xc nh, cc hm khng ton phng ( ), ,V t e x chophp thc hin kh nng y ca l thuyt da trn Lyapunov v a chng ta ti cc nh x phn hi phi tuyn cn thu c cc i tng thit k i lp. Ta c h cc ng c vin cho hm Lyapunov di y:

1 11 1 2 1 2 12 1 2 1

0 0

2 1 2 1( , , ) ( ) ( )2( 1) 2( 1) i

T T

i ii i

Ei X

i i

V t e x e K t e x K t x i i

g gb b hV g gb bb gb g

+ + + ++ + + + + ++ +

= =

+ + = = + + + + +

y ( ).EiK v ( ).XiK l cc ma trn i xng; z, b , , v g l cc s nguyn khng m; 0,1,2,...z= ; 0,1,2,...b = ;0,1,2,...h = ; v 0,1,2,...g =

Dng ton phng ni ting ca ( ), ,V t e x c tm bng cch t 0 z b h g= = = = , v ta c

0 0

1 1( , , ) ( ) ( )

2 2T T

E XV t e x e K t e x K t x = +

Bng cch dng 1z= , 1b = , 1h = , v 1g = , ta thu c mt ng c vin khng ton phng:

2 2 2 20 1 0 1

1 1 1 1( , , ) ( ) ( ) ( ) ( )2 4 2 4

T T

T TE E X X V t e x e K t e e K t e x K t x e K t x = + + +

Ta thu c lun iu khin bm sau:

2 1 2 1

0

12 1 2 1

0

2 11

2 12

2 1

1( ) ( ) ( , ) ( ) ( ) ( )

( ) ( , ) ( ) ( ) ( )

0 0 0

0 0 0

( )

i i

TE E Ei

i

i i

T X Xi

i

i

i

i

u x G t B t x diag e t K t e t s

G t B t x diag x t K t x t

e

e

diag e t

b bVb b

g gVg g

b

b

b

b

b

b

- ++ +

=

- + ++ +

=

-+

-

+

-+

= - W F +

=

L

L

M M

2 11

2 1

0 0 0

0 0 0

i

b

i

b

e

e

b

b

b

b

-+

-

-+

O M M

L

M

v

23


24/27

S tay C in t

2 11

2 12

2 1

2 1

1

2 1

0 0 0

0 0 0

( )

0 0 0

0 0 0

i

i

i

i

n

i

n

x

x

diag x t

x

x

g

g

g

g

g

g

g

g

g

g

-+

-+

-

+

-+

--

+

=

L

L

M M O M M

L

M

Nu cc ma trn EiK v XiK l ma trn ng cho, ta c

2 12 12 12 1

0 0

1( ) ( ) ( , ) ( ) ( ) ( ) ( , ) ( ) ( )

ii

T TE E Ei X Xi

i i

u x G t B t x K t e t G t B t x K t x t s

hVgb

++++

= =

= - W F +

Mt h thng khng xc nh ng l n nh th trong ( ), , ,oX X U Z P v vic m bo bm bn vng trong tp li v

compact ( ), ,oE E Y R nn vi cc u vo tham chiu r R v cc tham s khng xc nh trong Zv P tn ti mt hm

dng ( )1C k k : ( ).V cng nh cc hm dng K : ( )1 .Xr , ( )2 .Xr , ( )1 .Er , ( )2 .Er v cc hm dng K: ( )3 .Xr , ( )3 .Er ,nh cc iu kin di y

1 1 2 2

3 3

( ) ( ( , , ) ( ) ( )

( , , )( ) ( )

X E X E

X E

x e V t e x x e

dV t e xx e

dt

r r r r

r r

+ +

- - -

c m bo trong min n nh bt bin S, v ( ), , , , ,o oXE X E U R Z P S .

Cc iu kin cho vic gii bi ton iu khin bn vng c cho. Qua vic tnh o hm ca ( ), ,V t e x , c th tm

c cc h s cha bit ca ( ), ,V t e x . Tc l, thu c cc ma trn ( ).EiK v ( ).XiK . Vn ny c gii bng cchdng khi nim bt ng thc phi tuyn [9].

V d 14.6.1: iu khin vi ng c bc nam chm vnh cu 2 phaMEMS hiu sut cao vi cc vi ng c bc nam chm vnh cu c thit k v ch to. Cc b iu khin cn thit

c thit k iu khin vi ng c bc nam chm vnh cu, v iu chnh vn tc v dch chuyn gc quay bng cchthay i cng in p tc ng hoc dng in nui trong cun dy stato (xem v d 14.5.3). Dch chuyn roto c ohoc quan st p cc in p vo cc cun pha. gii quyt vn iu khin chuyn ng, cn thit k b iu khin.C th thy rng cn trin khai cc thut ton iu khin mi cc i m men. Trn thc t, khng th dng cc b iukhin truyn thng

1 T Vu G Bx

- = -

v 1 TV

u G Bx

- = - F

Dng khi nim ng nng lng, ta tm c biu thc cho m men in t c cho nh sau

[ sin( ) cos( )]e m as rm bs rmT RT i RT i RT y q q= - -

v nh vy, phi nui dng in pha di dng hm sin hoc cosin ca dch chuyn roto.

M hnh ton hc ca vi ng c bc nam chm vnh cu c tm trong v d 14.5.3 nh sau

1sin( )

1cos( )

1[ sin( ) cos( )]

as s mas rm rm as

ss ss ss

bs s mbs rm rm bs

ss ss ss


rm

rm

di r RT i RT u

dt L L L

di r RT i RT u

dt L L L

d RT Bi RT i RT T

dt J J Jd

dt

yw q

yw q

w yq q w

qw

= - + +

= - - +

= - - - -

=

24


25/27

C cu chp hnh

in tr roto l mt hm ca nhit do sut in tr l ( )1 20o oT o Trr r a= + - . Nh vy, ( ) min max.s s sr r r .

nhy ca cc nam chm vnh cu (cc mng mng) tng theo chiu tng ca nhit . Cc tham s ca h thng servo khc

cng thay i; c th, ( ) min max.ss ss ssL L L v

( )min max.ss ss ssB B B

.

Phng trnh chuyn ng tm c di dng vc t nh sau:

min max

0 0

( ) ( , , , , ) ( ) ,

( ) , , ,

z p

as

asbs

rmbsrm

rm

x t F t x r d z B p u u u u

i

ui x t x x u y

uq

w

q

= +

= = = =

&

y, x X v u U ln lt l cc vc t trng thi v iu khin, r R v y Y ln lt l tham chiu v u rao c, d D l nhiu, Ld T= , v z Z cng p P l phn khng xc nh tham s cha bit v gii hn.

Mc tiu ca chng ta l thit k iu khin gii hn ( ).u trong tp rng buc

2 2min max min max{ : , 0, 0}U u u u u u u= < >

Mt lut iu khin c th chp nhn m bo mt in p 2 pha cn bng cho cc cun ab v m bo sn phm c mmen in t ln nht c tng hp nh sau

sin( ) 0

0 cos( )

( , , ) ( , , ) ( , , )1( ) ( ) ( )

as rm

bs rm

T T T X e e i e

u RTu

u RT

V t x e V t x e V t x eG t B G t B G t B

x e s e

q

q

- = = F + +

y e E l sai s bm o c, ( ) ( ) ( )e t r t y t = - ; ( ).F l hm gii hn (erf, sat, tanh), v ( ) max, .U VF F , maxV l

in p t l; ( ).xG , ( ).cG , v ( ).iG c gii hn v i xng, 0xG > , 0cG > v 0iG > ; v ( ).V l hm gii tch thc,

kh vi lin tc trn ( )1C k k .

Vi , , , ,oX X u U r R d D z Z v p P , ta thu c tp tin trnh trng thi X. Tp trng thi u ra l

0

0 0 0

( , , , , , )

{( , ) : , , , , , , [ , )}

XY X U R D X P

x y X Y x X u U r R d D z Z p P t t

=

=

v c th tm mt nh x u vo-tham chiu. Mc tiu ca chng ta l tm b iu khin gii hn nh sai s bm( ) [ ). : ,oe t E vi oE E ly trong tp ng danh ngha

10 0 0 0

0 0 0,

( ) { : , ( , , , , , ), [ , )

( ) ( , ) , 0, ( , , , ), [ )}

e

e r d y

S e e E x X X U R D X P t t

e t t e r d y e E E R D Y t t

d

r r r r d d

=

+ + + + " "

y, ( ).cr l hm KL; ( ).rr , ( ).dr v ( ).yr l cc hm K.Mt min n nh bt bin dng c tm cho h thng vng kn vi o ox X , o oe E , u U , r R , d D v

p P . C th

4 10

0 0 0

0 0

{ , : ( ) ( , ) ,

( , , , , , ), [ , ), ( ) ( , )

, ( , , , ), [ , )},

s x r d

e r

d y

S x e x t t x r d

x X X U R D Z P t t e t t e r

d y e E E R D Y t t

r r r d

r r

r r d

= + + +

" " +

+ + + + " "

y ( ).xr l hm KL.

nghin cu tnh bn vng, bm, v loi b nhiu, ta xem xt mt tp li trng thi

0 0 0 0 0 0

0

( , , , , , , ) {( , ) : , , ,, , , , [ , )}

XE X E U R D Z P x e X E x X e E u Ur R d D z Z p P t t

=

25


26/27

S tay C in t

S bm bn vng, tnh n nh, v loi b nhiu c m bo nu s XE S . Tp chp nhn c sS c tm bngcch dng l thuyt n nh Lyapunov [9], v

{

}

4 10 0 0 0 0

1 2 3 4 5 6

0 0 0

, : , , , , , ,

( , , )( , , ) , ,

( , , , , , ), ( , , , ), [ , )

sS x e x X e E u U r R d D z Z p P

dV t x e x e V t x e x e x e

dt

x X X U R P Z P e E E R D Y t t

r r r r r r

=

+ + - -

" " "

y ( )1 .r , ( )2 .r , ( )3 .r v ( )4 .r l cc hm K ; v ( )5 .r v ( )6 .r l cc hm K.

Nu trong XE tn ti mt hm Lyaponov dng C k : ( ), ,V t x e sao cho vi mi o ox X , o oe E , u U , r R ,

d D v p P trn [ ),ot iu kin cho tnh n nh ( )1s

1 2 3 4( , , ) x e V t x e x er r r r + +

v bt ng thc

5 6

( , , )dV t x ex e

dtr r - -

l iu kin cho n nh 2s , th1. li gii ( ) [ ). : ,o x t X cho h vng ng l gii hn v n nh th,

2. s hi t ca vc t sai s ( ) [ ). : ,oe t E ti eS c tha mn trong XE ,

3. XE li v compact, v s XE S.

Tc l, nu m bo tiu chun ( )1s v ( )2s , ta c s XE S .

Dng ng c vin cho Lyapunov khng ton phng

1) 1)1 12 1 2 1 2 1 2 1

0 0

1) 12 1 2 1

0

2 1 2 1( , , ) ( ) ( )

2( 1) 2( 1)

2 1 ( )2( 1)

T Tj jj j

xj ej

j j

Tj j

ij

i

V t x e x K t x e K t ej j

e K t ej

g bg bh Vg g b b

m ms m m

g b

g b

mm

+ + + ++ + + ++ + + +

= =

+ + + ++ +

=

+ + = + + + + +

+ + + +

ta thu c b iu khin gii hn nh sau

12 1 2 1

0

2 12 12 12 1

0 0

sin( ) 0( ) ( )

0 cos( )

1( ) ( ) ( ) ( )

j jas rm

T x xj

bs jrm

jj

T Te e ej i e ij

j j

u RTu G t B diag x K t x

u RT

G t B K t e G t B K t es

g ghg g

V smb

q

q

- + ++ +

=

++++

= =

- = = F -

+ +

y, ( ).xjK l cc hm ma trn cha bit, v ( ).ejK v ( ).ijK l cc h s cha bit; 0,1,2,...h = ; 0,1,2,...g = ;

0,1,2,...z= ; 0,1,2,...b = ; 0,1,2,...s = v 0,1,2,...m=Nu gi thit rng oX , oE ,R,D,Z, vPc th chp nhn c, bi ton bm bn vng c th gii c trong XE. Tc

l, iu khin gii tch thc gii hn ( ).u m bo tnh n nh bn vng v dn sai s bm ti eS . Hn na, tnh n nhc m bo, vic loi b nhiu c thc hin, kh nng bm u vo-u ra danh ngha c th t c.

p dng b iu khin c thit k, ta cc i ha m men in t trong vi ng c bc vnh cu. iu ny c th thyd dng bng cch dng biu thc cho m men in t, tp in p hnh sin 2 pha cn bng (p dng cho cc in p asu v

bsu ), cng nh ng nht thc lng gic 2 2sin cos 1a a+ = .

C th thit k b iu khin dng sai s bm. C th, ta c

2 12 1 2 12 1

0 0

sin( ) 0 1( ) ( ) ( ) ( )0 cos( )

as

bs

jjrm T Te e ei i e ij

j jrm

uu u

RT G t B K t e G t B K t eRT s

V smbq

q

++ ++

= =

=

- = F + -

26


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C cu chp hnh

Thit k, thc thi, v kim tra thc nghim b iu khin c trnh by trong [9].

14.7 Kt lun

Chng ny trnh by tnh trng hin nay, cng b cc kt qu mi nht, nghin cu cc m hnh mi trong tng hp,m hnh ha, phn tch, m phng, iu khin v ti u ha MEMS hiu sut cao. Cc kt qu thu c ny ng dng ccphng php m hnh ha, phn tch, tng hp, iu khin, v ti u ha phi tuyn cho php t c s nh gi kh nng

v d bo kt qu. MEMS mi c pht minh. ng dng ca dng hnh hc ng c tm, cu, torroidal, nn, tr, vkhng i xng, cng nh cc h thng in t endless, open-ended, v tch hp, cho php ta phn loi MEMS. tng nyc bit hu ch trong vic nghin cu MEMS c cng nh trong vic tng hp MEMS hiu sut cao mi nht. V d, cth p dng h thng in t tch hp v hnh dng khng i xng (khng thng thng). Ti u ha c th c thc hin,v m hnh phn loi nh l im bt u t cc cu hnh tin tin c th c tng hp v th hin trc tip. Dng hnh hccc thit b chuyn ng v cc h in t (ng vai tr trung tm) lin quan vi nhau. S tng hp cu trc v ti u haMEMS c hnh thnh v th hin bng cch dng cc tng mi. M hnh phn loi MEMS dn n phn tch nhlng. Trn thc t, dng cc nh lut c bn ca in t v c hc (nh cc phng trnh Maxwell, Kirchhoff v Newton),c th bt ngun v ng dng cc phng trnh vi phn m hnh cc hin tng v hiu ng in t v c hc nhm thuc phn tch kh nng vi vic d bo kt qu. Cc m hnh ton hc cho MEMS c tm ra. Vic dng cc m hnhton hc ny cho phn tch v ti u ha c thc hin, v cc thut ton iu khin phi tuyn c thit k. Cc ctrng v hin tng in t c tch hp trong phn tch, m hnh ha, tng hp, v ti u. C th thy, t c cp hot ng danh ngha, MEMS hiu sut cao mi cn c tng hp ha, v cc m phng phi tuyn ng lc hc m t cao

phi c thc hin. Cc kt qu bo co c cc ng dng trc tip ti vic phn tch v thit k MEMS hiu sut cao. Cth pht minh, tng hp ha, nh ngha, v thit k cc MEMS khc nhau, v nghin cu mt s cc vn tn ti lin quanti in t v tnh thay i hnh hc. Cc kt qu chun ny cho php lp cng thc li v ci tin cc vn c bit quantrng trong l thuyt MEMS, v gii quyt mt s vn rt phc tp trong thit k v ti u ha vi mc tiu cui cng tng hp ha MEMS hiu sut cao mi, m men ln, mt nhiu.

Ti liu tham kho

[1] Lyshevski, S. E., Nano- and Micro-Electromechanical Systems: Fundamentals of Nano- and Micro-Engineering, CRCPress, Boca Raton, FL, 2000.

[2] Madou, M., Fundamentals of Microfabrication, CRC Press, Boca Raton, FL, 1997.

[3] Campbell, S. A., The Science and Engineering of Microelectronic Fabrication, Oxford University Press, New York,2001.

[4] Lyshevski, S. E., Electromechanical Systems, Electric Machines, and Applied Mechatronics, CRC Press, Boca Raton,FL, 1999.

[5] Lyshevski, S. E. and Lyshevski, M. A., Analysis, dynamics, and control of micro-electromechanical systems, Proc.American Control Conference, Chicago, IL, pp. 30913095, 2000.

[6] Mehregany, M. and Tai, Y. C., Surface micromachined mechanisms and micro-motors, J. Micromechanics andMicroengineering, vol. 1, pp. 7385, 1992.

[7] Becker, E. W., Ehrfeld, W., Hagmann, P., Maner, A., and Mynchmeyer, D., Fabrication of microstructures with highaspect ratios and great structural heights by synchrotron radiation lithography, galvanoformung, and plastic molding(LIGA process), Microelectronic Engineering, vol. 4, pp. 3556, 1986.

[8] Guckel, H., Christenson, T. R., Skrobis, K. J., Klein, J., and Karnowsky, M., Design and testing of planar magneticmicromotors fabricated by deep X-ray lithography and electroplating, Technical Digest of International Conference

on Solid-State Sensors and Actuators, Transducers 93, Yokohama, Japan, pp. 6064, 1993.[9] Lyshevski, S. E., Control Systems Theory with Engineering Applications, Birkhuser, Boston, MA, 2001.

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