lec 3 design issues for mems 2
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HVI CIN T(VI H THNG)MICROSYSTEMS
MICRO-ELECTRO-MECHANICAL SYSTEMS
MICROMACHINING
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VI H THNG
MEMS
Mt th gii rng m v quyn r
(An fascinating and wide world)
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III. Thit k trong MEMS
1. Mu/ Introduction2. Cc h qu khi thu nh kch thc /
Scaling issues for MEMS
3. M hnh ha v m phng/Modeling and Simulation
4. Thit k qui trnh ch to/Process integration
5. Kt lun/Conclusions
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3. M hnh ha v m phng
Php bin i Lagrange S dng h cc phng trnh vi phn Lagrange trong h ta chung(generalized coordinates) qk (k= 1,2,n) m t tng thi ca h
c inTa chung (qi) x Q V hoc
ng nng (T)
Thnng (U)
Hm suy haoRaleigh (D)
Cng (W) F.x T. V.Q i.
2
2
1xM&
2
2
1&M 2
2
1QL&
22
2
1
2
1= &CCV
2
2
1Kx 2
2
1K
C
Q2
2
1L
2
2
1
22
1xC&
22
1&C 221
QR& R
2
2
1&
Slng cc ta c lp m t hng hc gi l bc t do ca
h (degree of freedom - DOF)
Xy dng m hnh tng qut(concept to first design)
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Phng trnh Lagrange tng qut
3. M hnh ha v m phng
Php bin i Lagrange
i
N
i iii
dqq
U
q
T
q
T
dt
dUTd
=
+
=+1
)(&&
Trong h ta chung: T l hm caqi v vn tc ,U l hm ca qi,iq&
0=+
iii q
UqT
qT
dtd
&&
i vi hbo ton khng thuc trng lc th:
Khi h thc hin cng bng lc bnh thng Qi, c: =
==+N
i
iidqQdWUTd1
)(
iiii
Qq
U
q
T
q
T
dt
d
=
+
&& Phng trnh Lagrange:
Xy dng m hnh tng qut(concept to first design)
Tnh lut bo ton nng lng, c: d(T+U) = 0
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3. M hnh ha v m phng
Php bin i LagrangeLc ma st (khng th- nonpotential force)+ Lc ma st trong h ta Descartes: xCF &=
Hm suy hao: = =
=n
r
n
s
srrs qqCD1 12
1 &&
Lc ma st trong h ta chung:i
iq
DQ
=
i
iiii
Qq
D
q
U
q
T
q
T
dt
d=
+
+
&&
Phng trnh Lagrange c lc khng th:
Xy dng m hnh tng quat(concept to first design)
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3. M hnh ha v m phng
Php bin i Lagrangeiu kin rng buc (constrain equation)
=
+=+=
+
+
M
l i
li
con
ii
iiii q
gQQQ
q
D
q
U
q
T
q
T
dt
d
1&&
Phng trnh Lagrange i vi cc lc khng thuc trng th
+ Trng hp sta n v sbc tdo N khng nhnhau, tc l: n > N
iu kin rng buc: gl (q1, q2, , qN) = 0 (l = 1, 2,,M)
+ iu kin c hphng trnh gii c xc nh mt h: n M = N
=
=M
l i
lcon
i q
gQ
1
Suy hao do lc ging
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Php bin i LagrangeV d: xy dng m hnh ca cm bin gia tc
Hdao ng c, 1-DOF, ngoi lc F c thllc qun tnh, lc tnh in hoc lc in t
2
2
1xMT &=
2
2
1KxU=
2
21 xCD &=
xFW = .
+ ng nng ca h:
+ Thnng ca h:
+ Suy gim NL ca h:
+ Cng h thc hin:
Phng trnh vi phn dao ng bc 2: 0=++ FKxxCxM &&&
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
M hnh kt khi (lumped-model) Linh kin MEMS trong thgii thc
C kch thc 3 chiu
C nguyn l hot ng tun theo cc nh lut Vt l
M hnh kt khi cho thit k- s dng l thuyt mch :
n gin ha mt hphc tp (c, nhit, ha) bng cc phn t
mch in M hnh ha tng tc gia cc dng nng lng (energy domain)mt cch hiu qu
M hnh ha tnh cht Tnh v ng ca h m khng cn phi xy
dng v th
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
M hnh kt khi (lumped-model)Phn t kt khi Vt thn l c thtrao i NL vi ccvt thkhc
Tc trao i nng lng: nng lng/thi gianPAB = r1
2 PBA = r22 (r1 v r2 l cc sthc, PAB > 0 v PBA > 0)
Mng nng lng gia A v B
P = PAB - PBA = r12 - r22 = (r1+r2)(r1-r2) NL gia A v B lun c vitdi dng tch ca 2 sthc,
Bin NL lin hp Cp i lng m tch ca chng bng tch tc trao i NL mng
gia 2 phn t
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
M hnh kt khi (lumped-model)Nng lng (generalized power variable) Tc ng (effort): e(t)
Bin i (flow):f(t)
Bin lin hp: e(t).f(t) = NL
Xung lng v chuyn v (momentum and displacement)=t
t
dttetp
0
)()(Xung lng
Chuyn v =t
t
dttftq
0
)()(
e(t).q(t) = NL hay f(t).p(t) = NL
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
M hnh kt khi (lumped-model)
Tc ng Bin i Xung lc Dchchuyn
Hchc Lc(F)
Vn tc(v)
Xung lng(p)
V tr(x)
Mch in Th(V) Dng(I) in tch(q)
Cht lu p sut(P)
Dng chy khi(Q)
p lc()
Th tch(V)
NL lin hp tng ng cc dng NL c th khc nhau
Xy dng m hnh tng qut(concept to first design)
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2 bin cho mi port:
3. M hnh ha v m phng
M hnh kt khi (lumped-model)Phn t mt cng (1-port element) Cng (port): cp li vo/ra ca mt phn tmch in cho php dng i vo v i ra.
Ngang qua (across) Xuyn qua (through)
in Tc ng = thBin i = dng
C Tc ng = lc Bin i = vn tc
Cht lu Tc ng = p sut Bin i = dng chy khi
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Ngun (source): phn t kchhot cung cp NL cho cc phnt khc khi e(t).f(t)
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3. M hnh ha v m phng
Trong chuyn ng (dao ng) c, m hnh in trng vai tr l lmb gim chn (damper) dp tt dao ng (do ma st, nht cht lu).
Xy dng m hnh tng qut(concept to first design)
f (I, v hoc Q)
e (V,F hocP) e.f > 0 gc I v III NL sbtiu th (disipated/absorbed)
e = Rf
Th hin mi quan h tuyn tnh gia tcng v bin i:
M hnh kt khi (lumped-model)M hnh in tr (generalized resistor)
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3. M hnh ha v m phng
e (V,F hocP)
q (q,x hoc V) Th hin mi quan h gia tc ng vdch chuyn: e = (q)
Khi t c tng tc 0 cchuyn v0 t tch trNL
NL lu tr: ==11
1
00
)(qq
q dqqedqW
NL tng ng (co-energy):
==1
0
* )(.)(e
qdeqWqeeW
Xy dng m hnh tng qut(concept to first design)
M hnh kt khi (lumped-model)M hnh T (generalized capacitor)
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3. M hnh ha v m phng
QC= CV
g
AC
=
C
QQW
2)(
2
= )(2
)(2
*QW
CVQW ==
e (V)
q (Q)
Q = CV
Q ca t l dng in
M hnh T p dng cho tin phng
Theo L Hook: F = kx
NL lu tr trong l xo: 10
1 2
1)()(
1
kxdxxFxW
x
==
x ca l xo l dng c
M hnh T p dng cho dao ng c
Xy dng m hnh tng qut(concept to first design)
M hnh kt khi (lumped-model)
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3. M hnh ha v m phng
Th hin mi quan h gia bin i vxung lng
f = (p)
V d:p = mvm
pp = )( m
ppW
2)(
21
1 =
211
*
2
1)( mvpW =
=1
0
1 )()(p
dpppW
)(.)(* pWpfpW =
NL lu tr:
NL tng ng (co-energy):
Trong Ccm hnh cun cm ng vai tr l khi lng (gia trng) dochuyn ng qun tnh
Xy dng m hnh tng qut(concept to first design)
M hnh kt khi (lumped-model)M hnh cun cm (generalized inductor)
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3. M hnh ha v m phng
S mch in
Mc ni tip (serires): p dng cho trng
hp cng chia sbin dng chy (flow) vdch chuyn
Mc song song (paralell): p dng cho trnghp cng chia sbin tc ng (effort)
nh lut Kirschhoff p dng cho dng ti nt i: 0= iI p dng vi thca mch kn: 0= iV
-F + ek+ em+ eb= 0
Xy dng m hnh tng qut(concept to first design)
M hnh kt khi (lumped-model)Gn kt cc phn t (circuit connection)
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3. M hnh ha v m phng
Php bin i Laplace (Lapalace transform) Phng php ton hc c dng ph bin phn tch hoc m hnhha hng hc.
Php bin i t vng ph thuc thi gian (time-domain) trong thng
tin a vo (inputs) v ly ra (outputs) l hm ph thuc thi gian sang vngtn s (frequency-domain) ngha l cc thng tin ni trn trthnh hm catn s gc phc (hay rad/s).
Bin i php vi phn (hoc tch phn) thnh php nhn (chia) vi s phng trnh a thc d dng gii s dng php bin i Laplace
nghch chuyn ngc v qu trnh ph thuc thi gian.
nh ngha: bin i Lapalce ca hm ph thuc thi gian, f(t), l hm
F(s), c dng: ==0
)(}{)( dtetxxsX stsL (s l sphc)
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Php bin i Laplace (Lapalace transform)Cc tnh cht Tuyn tnh (linear)
(t) = .x(t) + y(t)(t) l t hp tuyn tnh cax(t) v y(t):
W(s) = .X(s) + Y(s) Bin i Laplace ca (t): Vi phn (differentiation)
)0()(}{)( xssXxtx s = && LBin i Laplace ca )(tx&
Hm n v bc thang (Heaviside unit-step function )
=
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3. M hnh ha v m phng
Php bin i Laplace (Lapalace transform)Hm truyn (transfer function)
)}({
)}({
)(
)()(
tx
ty
sX
sYsH
s
s
L
L==
H lin tc: x(t): hm li vo, y(t): hm li ra
H ri rc:
)(
)()(
zX
zYzH =
Biu din ton hc (php nh x tuyn tnh ca bin i Laplace) th hinmi quan h gia li vo v li ra (ph thuc thi gian) di dng hm tn s:
Y(s) = H(s)X(s)
H(s): hm ph thuc tn s
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Khi gia trng di chuyn (moving mass)
Gii phtr xc nh X(s)
)()()( txmtmatf &&==L 2 Newton:
Bin i Laplace caf(t) }{)()( xmsFtf s &&L=
)]0()0()([
)}0()]0()([{
)]0(}{[)(
2 xsxsXsm
xxssXsm
xxsmsF s
&
&
&&
=
==
== L
00)0( vxx == &&x(0): v tr ban u v l vn tc ban u
p dng nh l vi phn, c:
Xy dng m hnh tng qut(concept to first design)
Php bin i Laplace (Lapalace transform)ng dng cho cc m hnh
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3. M hnh ha v m phng
Nuf(t) = 0, c: 22)0()0()0()0(
)( s
v
s
x
s
x
s
xsX +=+=
&
p dng nh l unit step function v tr khi gia trngx(t):
x(t) =x(0).u(t) + v(0).t.u(t)
v tr ban ux(0) = x0, v v(0) = 0 x(t) = x0 vit 0 vt ng yn
Khi gia trng di chuyn (moving mass)
v(0) kt hp vi thi gian vt Ckhng ngng vi vn tc ban u
phtr vi phn m tC ca vtphtr i s tuyn tnh
Xy dng m hnh tng qut(concept to first design)
Php bin i Laplace (Lapalace transform)ng dng cho cc m hnh
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3. M hnh ha v m phng
Bdao ng gia trng-l xo (mass-spring)
Bin i Laplace
)()0()0()(
)()}0()]0()([{
)()]0(}{[
}{}{
}{0
2 skXxmmsxsXms
skXxxssXsm
skXxxsm
xkxm
kxxm
s
ss
s
+=
=+=
=+=
=+=
=+=
&
&
&&
&&
&&
L
LL
L
Xy dng m hnh tng qut(concept to first design)
Php bin i Laplace (Lapalace transform)ng dng cho cc m hnh
)()()( txmtmatfm &&== L 2 Newton cho m:
L Hook cho k: )()( tkxtfk =
L 3 Newton: 0)()( =+ tftf km 0)()( =+ tkxtxm&&
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Php bin i Laplace (Lapalace transform)ng dng cho cc m hnh Bdao ng gia trng-l xo (mass-spring)
3. M hnh ha v m phng
00)0( vxx == &&x(0): v tr ban u v l vn tc ban u
00
200
/)(
js
r
js
r
mks
vsxsX
++
=+
+=
mk/0= rj
reRv
jxr
=+=
00
0
220
20
20
20
2
xvRr
+=
=
00
01tanx
vr
v tr khi gia trngx(t):
+=
==+=
00
010
0
20
20
20 tancos
}{2)( 000
x
vt
xv
rereerretx tjtjtj
v(0) = 0 x(t) = x0cos(0t)
x(0) = 0 x(t) = (x0/0)sin(0t) vv(t) = v0cos(0t)
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Php bin i Laplace (Lapalace transform)
Bin trng thi:
x1: v tr
x2 = dx1/dt: vn tc
dx2/dt: gia tc
u vo (input) = tc ng
u ra (output) = dch chuyn
12 xx &=
F
m
x
m
bx
m
kx
1212 +=& (L 2 Newton cho h)
ng dng cho cc m hnh Bdao ng l xo-gia trng-gim chn (mass-spring-damper)
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Php bin i Laplace (Lapalace transform)ng dng cho cc m hnhF
mx
x
m
b
m
kx
x
+
=
1010
2
1
2
1
&
&
Fx
x
x
x
+
=
00
1001
2
1
2
1
-x: bin trng thi (ma trn ct) th hin v tr, vn tc,
-U: thng tin u vo (ma trn ct) th hin lc
-y: thng tin u ra (ma trn ct) th hin gia tc,
-A, B, C, D cc i lng thi gian, c ngha lin kt c h.
Phng trnh trng thi tng qut BUAxx +=&
DUCxy +=
(1)
(2)
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Php bin i Laplace (Lapalace transform)ng dng cho cc m hnh
Bin i Laplace vi phng trnh (1): )()()0()( sBUsAXxssX +=
Bin i Laplace th hin p ng ca h:[ ])()0()()( 1 sBUxAsIsX +=
p ng iu kin ban u (cha c li vo): )0()()( 1xAsIsXzir=
)()()( 1 sBUAsIsXzsr=p ng tc ng li vo:
iu kin c nghim: 0)det( AsI
Bi ton tr ring: tn ti gi tr s sao cho: 0)det( =AsI
Xy dng m hnh tng qut(concept to first design)
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p li ra:i vi h l xo/gia trng/gim chn: D = 0
)()()()()()( 1 sUsHsBUAsICsCXsY zsrzsr ===
3. M hnh ha v m phng
Php bin i Laplace (Lapalace transform)ng dng cho cc m hnh
BAsICsH 1)()( =Hm truyn:
Bin i Laplace vi phng trnh (2):
[ ] )()()(
)()()(
sDUsXsXC
sDUsCXsY
zsrzir ++=
=+=
H(s) l ma trn c: s hng = sbin trng thi, s ct l bin li vo
=
++
++=
))((
1))((
111
)(
21
21
ssss
s
m
ssssm
kbsms
skbsmssH
2
0
2
2
2,1 22
=
=
=m
k
m
b
m
bs
Xy dng m hnh tng qut(concept to first design)
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3. M hnh ha v m phng
Cc phng php m phng
3 phng php phbincng dng cho MEMS
Sai phn hu hn (finite differantial method - FDM)
Phn t hu hn (finite element method - FEM)
Th tch hu hn (finite volume method - FVM)
cim Bini cc phng trnh vi phn thnh phng trnhi s xp x
Chia nh cu trc thnh mng li (grid hay mesh)
Gii h cc phng trnhi s
Cc bi ton chc cu trcc m t bng cc phng trnh o hmring mt slng ln cc phng trnh cn c gii tm nghim chomi im trn ton b m hnh (hay mt min) phc tp kh thc hin viphng php gii tch thng thng cc phng php gii s gn ngc s dng vi cng c l my tnh thng qua mt ngn ng lp trnh hocmt phn mm ng dng.
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FDM
FEM
FVM
3. M hnh ha v m phng
Cc phng php m phng
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Phng php FDM
Xut pht t cch xcnh khong cch gia 2 im = php trphpsai phn
nh ngha php lyo hm
+=
)()(lim)('
0PFPPFPF
Nu hu hn:
+
=)()(
)(' PFPPF
PF
3. M hnh ha v m phng
Cc phng php m phng
Sai phn gia: F(P)=- F(P-1/2P) - F(P-1/2P)P P + P -
Sai phn tin: F(P) = F(P+P) - F(P)
Sai phn li: F(P) = F(P) - F(P-P)P P+PP-P
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3. M hnh ha v m phng
Cc phng php m phngPhng php FDM
Bi tonng: bini hin (explicit method)
Bi tonng: binin (implicit method)
S dng php sai phn tinthiimtj
2
11
1 2
h
uuu
k
uu n
j
n
j
n
j
n
j
n
j
+
+ +=
nj
nj
nj
nj ruruuru 111 )21( ++ ++= Vi 2/ hkr=
S dng php sai phn lithiimtj+1
2
11
11
11
1 2
h
uuu
k
uu i
j
i
j
i
j
n
j
n
j
+
+
+ +=
11
11
1)21( +++
+ ++=+ iji
j
i
j
n
j ruruuur Vi2/ hkr=
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3. M hnh ha v m phng
Cc phng php m phng
Phng php FDM
S dng php sai phn giathiimt(j+1)/2
++
+=
+
+
++
+
+
2 112
1
1
11
1
1 22
21 h
uuu
h
uuu
k
uu n
j
n
j
n
j
n
j
n
j
n
j
n
j
n
j
Vi 2/ hkr=
)()()22()22( 111111
1 ++
++
+ ++++=+ njn
j
n
j
n
j
n
j
n
j uuruururur
Bi tonng: bini Crank-Nicolson
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Phng php FDM
Cc bc thc hin
Xy dng thut ton.
M ha thut ton bng cch s dng cc ngn ngbin dch (C, pascal,Fortran) hoc cc phn mmng dng (MatLAB, Mathematica) vitchng trnh tnh ton cho yu cu ca bi ton (lp trnh).
a bi ton v dng khng th nguyn.
Thc hin gii bi ton v phn tch kt qu.
3. M hnh ha v m phng
Cc phng php m phng
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3. M hnh ha v m phng
Cc phng php m phng10
4
+1-1
i+1
i-1
i
2 3
5 11
12
678
9 1 0
(i,j)
Mng
2
501
2
02 2
+
= www
x
w
2
703
2
02 2
+
= www
y
w
Xy dng thut ton thng qua php sai phn
1 2 1
2
1
1
Phng php FDMng dng FDM m phng lch (un cong)mng mng ca cm bin p sut Phng trnh lch mng khi c p sut
),(24
4
22
4
4
4
yxPy
wD
yx
wD
x
wD =++
Phng trnh lch mng khng th nguyn
124
4
22
4
4
4
=++y
w
yx
w
x
w
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3. M hnh ha v m phng
Cc phng php m phngPhng php FDMng dng FDM m phng lch (un cong)mng mng ca cm bin p sut
1 1
1
1
8 820
8
8
2 2
22
1 4 6 4 1
1
6
4
4
1
2 24
2
2
1 1
11
4
115019
4
04 464
++
= wwwww
x
w
4
654703812
22
04 2222
++++=
wwwwwwwww
yx
w
4
1270310
40
4 464
++= wwwww
y
w
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3. M hnh ha v m phng
Cc phng php m phngPhng php FDMng dng FDM m phng lch (un cong) mngmng ca cm bin p sut
a x + a x + ... + a x = b
a x + a x + ... + a x = b
...
a x + a x + ... + a x = b
11 1 12 2 1N N 1
21 1 22 2 2N N 2
N1 1 N2 2 NN N N
=
NNNN
N
N
aaa
aaa
aaa
A
,...,,
...
,...,,
,...,,
21
22221
11211
=
Nx
x
x
X
...
2
1
=
Nb
b
b
B
...
2
1
Phng trnhi s tuyn tnh:
Hay: AX = BVi:
X = B.A-1
Gii bi ton xcnhn s:
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3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
tng chia mt cu trc phc tp thnh nhng khi nh (phn t) c
cu trcn gin v d dng khng ch.Hnh trn = hnh tam gic, c din tch:
Din tch hnh trn:
ii RS = sin21 2
Element
Si
22
1
2sin
2
1R
NNRSS
N
i
iN
== =
khi N
Mt cu trccc trng bi ma trn h s cng (stifffness). Mi phn t, ma trn h s cng lin kt c cu trc bng phngtrnh cbn.
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3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
Khi mt cu trcc chia (mesh) thnh cc phn t, ma trn cng ca
ton b cu trc c mi thnh phn l cng ca mi phn t tngng.
Nu cu trc chu tc dng ca mt ngoi lc F phng trnh quan hgia F v dch chuyn (chuyn v) vi h s t l l h s cng.
{F} = {k}.{x} {x}= {k}-1
.{F}
Ta, kch thc, sutn hi Young, t s Poisson hay khi lng ring(density) l cc thng s u vo gii phng trnh ny.
FEM l phng php ri rc ha mt min phc tp thnh cc min conn gin, c cng tnh cht, gi l cc phn t (element) m c m tbngcc phng trnh ton, th hin hnh vi ca n vi mt ti xc nh. Tng hptt c cc hnh vi trong m hnh s cho ta hnh vi chung ca c h vt l.
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3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
Cu trc c to t cc phn t c s gm: mt phng 4 cnh
(quadrilateral plates), mt phng tam gic (triangular plates), khi (solid brickelement), hay thanh mng (beam).
Phn t1 chiu1-D (line) element
(spring, beam, truss, pipe)
Phn t2 chiu
2-D (plane) element
(membrane, plane, shell)
Phn t3 chiu3-D (solid) element
(trng vt l 3 chiu: nhit,chuyn v, ng sut, dng chy)
Nt (node)
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3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
H s n hi - cng (stiffness) ca l xo
L xo (phn t n hi) chu tc dng cangoi lc F dch chuyn (bin dng) on:
)( ij uukkF ==
L 3 Newton cp lc cn bng ti 2 v trca chuyn di - nt (node) i vj:
jiiji kukuuukFf === )(
jiijj kukuuukFf +=== )(
Hay:
=
j
i
j
i
f
f
u
u
kk
kk Fku=
u
Fk=
3 M h h h h
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3. M hnh ha v m phng
Cc phng php m phngPhng php FEMH s cng ca thanh (bar)
Thanh (bar) c diL, thit din ngangA, h sn hiE, dch chuynu(x),bin dng(x) vng sut(x).
LL
uu
dx
du ji =
==
L
EE
==
c
v Mt khc:A
F= === k
L
EAAF vi:
L
EAk=
Thanhng x nh l xo
=
=
= 11
11
L
EA
L
EA
L
EAL
EA
L
EA
kk
kk
k
Phng trnh cn bng phn t:
=
j
i
j
i
f
f
u
u
L
EA
11
11 Ct thj (=1, 2) ca ma trnkthhin lctc dng ln bar duy tr tnh trng bin dng
ti ntj v khng bin dng ti cc nt khc
3 M h h h h
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3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
Hm hnh dng v nng lng bin dng
ji uLxu
Lxxu +
= 1)( Chuyn vu(x) dc theo trc x:
Hm hnh dng: =
1)(iN =
)(jN ViL
x= 10
[ ] Nuuu
NNuNuNuxuj
i
jijjii =
=+== )()()()()()(
B: ma trn (h st l) bin dng-chuyn v
[ ]LLB /1/1=
Khi BuuNdx
d
dx
du=
==
v EBuE == NL bin dng: th nng lu tr trong h bin dng n hi, c gi tr bng cng h thc hintorab in dng.
VLAFWU ====
2
1
2
1.
2
1 Vi thanh c
II C s c in
3 M hnh ha m phng
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II. Cs vcin3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
H s cng tng qut
( ) ( ) udVEBBudVEBuBudVUV
TT
V
TT
V
T
=== 2
1
2
1
2
1
Hm nng lng bin dng tng qut:
Trong: T
vuT
vBT
l cc ma trn hon v (hng ct) Cng thc hin bi 2 lc nt fuufufUW Tjjii 2
1
2
1
2
1=+==
hay ( ) fuudVEBBu TV
TT
2
1
2
1=
( ) kufudVEBB
V
T ==
Ma trn cng phn t: ( )=V
TdVEBBk
Hm nng lng bin dng thu gn kuuU T
2
1=
p dng chung chomi loi phn t
II C s v c in
3 M hnh ha v m phng
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II. Cs vcin3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
Thc hin xy dng m hnh (modeling) i tng cn tnh ton, nhp ccthng s vt liu c trng (Material Properties), xc lp kiu phn t (ElementType), chia nhi tng thnh mt slng cc phn t(Mesh), t cc iukin bin (Boundary Condition) xc lp vng khng chu ti (Constrains) v
vng chu ti (Load), cng nhiu kin ban u (Initial Condition).
Cc bc thc hin
Gii h cc phng trnh i s tuyn tnh (p dng phng php loi trGauss) tm bin chnh l chuyn v{u} v bin ph l bin dng, ng sut,moment
Phn tch v x l kt qu tnh ton.
Phn mmng dng ca FEM ANSYS, NASTRAN, ABAQUS, COSMOS, ALGOR, PATRAN,hyperMESH, Dyna-3D
c vit ch yu bng ngn ng FORTRAN
II C s v c in
3 M hnh ha v m phng
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II. Cs vcin3. M hnh ha v m phng
Cc phng php m phngPhng php FEMng dng FEM trong cc bi ton k thut
II C s v c in
3 M hnh ha v m phng
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II. Cs vcin3. M hnh ha v m phng
Cc phng php m phngPhng php FEM
ng dng FEM (ANSYS) trong MEMSM phng hotng ca cm bin gia tc
M phng hotng ca cm bino pH do
hiung swelling ca polymer hydrogel
M phng hotng ca bnh rng vi c
II C s v c in
3 M hnh ha v m phng
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II. Cs vcin3. M hnh ha v m phng
Cc phng php m phngSo snh FDM v FEM FDM v FEM u c h o li gii trn ton b min tnh ton v khng cn tnhlin tc. FEM mnh v kh nng x l cc bi ton c hnh hc phc tp trongkhi FDM ch gii hn nhngi tng c hnh dngn gin
FEM c la chn ch yu cho cc bi ton cu trc chc trong khiFDM oph hp hn cho cc ton v ng hc cht lu (computational fluiddynamics - CFD).
Cs ton hc ca php gnng trong FEM cho php vic chia nh itng thc hin tnh ton mnh hn FDM. Tuy nhin FDM d thc hin,trong khi FEM i hi qu trnh tnh ton nhiu hn v lu hn sai s trong
cc tnh ton ca FEM nh hn so vi FDM.
C rt nhiu phn mmc vit sn cho FEM, tch hp (nhng) phn x l ha (CAD), km theo cc b th vin cha tngiy cc loi phnt, cc dng bi ton k thut trong thc t nn rt thun li cho ngi s dng kt qu tnh tonc biu din bng s liu, th, hnhnh 2,3 chiu v
c bit lnhng (animation) di dng file ui .avi.
III Thit k trong MEMS
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III. Thit k trong MEMS
1. Mu/ Introduction2. Cc h qu khi thu nh kch thc /
Scaling issues for MEMS
3. M hnh ha v m phng/Modeling and Simulation
4. Thit k qui trnh ch to/Process integration
5. Kt lun/Conclusions
II C s v c in
4 Thit k qui trnh ch to
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II. Cs vcin4. Thit k qui trnh ch to
Thit k qui trnh ch to
Xy dng cc bc cng ngh chi tit cho ch tolinh kin (fabrication layout)
Thit kb MASK (mt n) quang hc
II. C s v c in
4. Thit k qui trnh ch to
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II. Cs vcin4. Thit k qui trnh ch to
Cc nguyn l thit k qui trnh Chn cch thc th hin cu trc linh kin d dng phc tho hnh dnglinh kin cho qui trnh ch to:
- Hnh v 3 chiu bng cc cng c thit k trn my tnh (CAD)
- Hnh v mt ct th hinc cu trcy ca linh kin
La chn cng ngh v cch thc ch to:
- Vi ckhi kh (hoct) hay vi cb mt
- Thc hin to cu trcin trc (front-end process) hay cu trc c trc(back-end process).
- Cn nhc cc bc cng ngh sao cho bom an ton ca cu trc cn chto (do c tnh ca cu trc linh kin MEMS rt mnh mai, db gy hay v).
Lun kt hp gia thit k v kim tra mt qui trnh ch tom botnhn nhng yu t cng ngh lm tng tnh chnh xc ca qui trnh.
Lu n tnh cht khngngu ca b mt phin trc v ngay trongqu trnh ch to (cc bc cng ngh lm thay i phn b b mt phin)
trnh b tr linh kinst mp phin
II. C s v c in
4. Thit k qui trnh ch to
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II. Cs vcin4. Thit k qui trnh ch to
Gii hn ca k thut quang khc (photolithography) Cc chi tit ca linh kin MEMS cnh dng, tonh vn mn hoc
bc lp (lift-off) bng qui trnh quang khc Gii hn quang khc l kh nng to ra kch thcnh nht c th cho linh kin kch thc chi tit(feature size - FS) hay kch thc ti hn (critical
dimension CD), xcnh bng rng vch (linewidth - w) v qui tc thit k khong cch (spacedesign rules - s) .
Vi phm qui tc
khng t c cu trcmong mun
Chi tit trn
MASK quang hc
a < FS
Chi titc
nh dng
nh hng ca cng ngh ch to
II. Cs vcin
4. Thit k qui trnh ch to
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q
Gii hn ca k thut quang khc (photolithography) Hiung quang hc (giao thoa, nhiu x) lm cho chi tit cncnh
dng trong qui trnh quang khc b thay i khu vc vin mp (distortededge effect).
Chi tit sau khicnh dng
Dng hnh hctrn MASK quang
nh hng ca cng ngh ch to
II. Cs vcin
4. Thit k qui trnh ch to
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q
Gii hn ca k thut quang khc (photolithography) Gii hn khong cch tiu c (depth focus) ca knh hin vi trn thit b
quang khc Du so MASK (alignment marks) quytnh tnh chnh xc ca qui trnhch tophn gii ca h quang hc trn thit b quang khc cnctnhn khi thc hin thit kb MASk quang hc.
Hin tngng sut ni xut hin khi ph lp cm quang (photoresist) cth lmnh hngn tnh chnh xc ca chi tit cn ch to .
nh hng ca cng ngh ch to
II. Cs vcin
4. Thit k qui trnh ch to
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q
nh hng ca cng ngh ch toGii hn ca k thutn mn (etching) Vt liu bo v (mask) b mt phin thch hpm bo:
- Chungc thi gian di trong qui trnhn mnt.- Chuc nhit cao ca mi tr ng plasma trong qu trnhn mn kh.
Convex corner
Hin tngn mn ngang:
- Notching effect trong k thutn mn kh.
- Undercutting lm cu trc cnn mn b lmgc () trong k thutn mnt
Notching effect
II. Cs vcin
4. Thit k qui trnh ch to
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Thc hin m hnh ha qui trnh ch to da trn nguyn l vt l (physic-based process modeling tool): Ch yu l cc phng php m hnh ha vphng qu trnh to mng mng cng nh n mn cung cp cc thng tin vthng s qu trnh gip xy dngc hnhnh c th ca linh kin s cch to.
Thc hin m hnh ha qui trnh ch to da trn cc bc cng ngh(geometric emulation of MEMS processes): m t hnh dng nhn c saumi bc cng ngh kt ni vi nhau xy dng cu trc hnh hc hon
chnh ca linh kin s c ch to.
Kim tra thit k (design rule checking)
II. Cs vcin
4. Thit k qui trnh ch to
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Kim tra thit k (design rule checking)
Cng c tonho
Hnh hc caMASK (2 D)
nho 3D
nh o 2-D(mt ct dc)
Xc inhqui trnh
II. Cs vcin
4. Thit k qui trnh ch to
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Cc cng c thit k (design tools)
Thit k MASK: Corel Draw, LEdit,
Thit k qui trnh: SUMMiT, MEMCAD, SIMULINK
II. Cs vcin
4. Thit k qui trnh ch to
-
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Qui trnh ch to cm bin p sutpin tr (back-end process)
Ch to cu trc c (mngmng nhy p sut) trc, cutrcin sau.
Ch to theo cng ngh vi ckhi (bulk microfabrication) s dng qui trnh n mn t(wet etching)
Thc hin trn thit bquang khc 1 mt (one-sidealigner)
II. Cs vcin
4. Thit k qui trnh ch to
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Qui trnh ch to cm bin p sutpin tr (back-end process)
Thit kb MASK (corel draw)
II. Cs vcin
4. Thit k qui trnh ch to
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Qui trnh ch to cm bin p sutpin tr (back-end process)
Thit kb MASK (corel draw)
II. Cs vcin
4. Thit k qui trnh ch to
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Qui trnh ch to cm bin p sutpin tr (front-end process)
Ch to cu trcin trc,cu trc c (mng mng nhyp sut) sau.
Ch to theo cng ngh vi c
khi (bulk microfabrication) sdng qui trnhn mn t (dryetching)
Thc hin trn thit bquang khc 2 mt (two-sidealigner)
Si
Buried SiO2
SiO2
Boron doping layer
Al
1
2
3
54
6
7
8
II. Cs vcin
4. Thit k qui trnh ch to
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Qui trnh ch to cm bin p sutpin tr (front-end process)
Thit kb MASK (LEDit)
MASK 1
MASK 2
II. Cs vcin
4. Thit k qui trnh ch to
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Qui trnh ch to cm bin p sutpin tr (front-end process)
Thit kb MASK (LEDit)
MASK 4
MASK 3
5. Kt lun
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Thit k MEMS l cng vic phc tpi hi cc kh nng tduy tru tng v ton hc thng qua thc hin m hnh ha vm phng (tnh ton) ccng x ca linh kin.
Nhng nhu cu ca th trng, kh nng cng ngh kt hpvi tng sng to l cscho vic thc hin thit k mt linhkin MEMS.
Cc nguyn l, nh lut vt l kch thc vi m s ancc hiung khc so vi mc v mi hi cc tnh ton vthit kph hp.
Thit k qui trnh ch to cn tun th cc qui tc thit k vkim tra thit k nhmt to rac mt qui trnh cng nghch to tiu, ph hp cciu kin thc t.
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