ECE145a / 218a Bilateral Tuned Amplifier Design: Stability · Bilateral Tuned Amplifier Design:...
Transcript of ECE145a / 218a Bilateral Tuned Amplifier Design: Stability · Bilateral Tuned Amplifier Design:...
class notes, M. Rodwell, copyrighted 2009
ECE145a / 218a Bilateral Tuned Amplifier Design: Stability
Mark Rodwell
University of California, Santa Barbara
[email protected] 805-893-3244, 805-893-3262 fax
class notes, M. Rodwell, copyrighted 2009
Stability
? 1 angreater th of magnitude theIs
viewpoint(S) lection Ref
?part real negativea have )( Does
viewpointImpedance
? plane-s of halfright in lieany Do
.)(/)( in poles Find
analysis. nodalby circuit Analyze
:eoryNetwork th
? plane-s of halfright in lieany Do
poles. loop-closed ofpart real Find
margin. phase find :methodsNyquist & Bode
theorysystem control Classical
: versionsequivalentmany :theoryStability
input. zero houtput wit zero-non :yInstabilit
in
in
genout
jZ
sVsV
class notes, M. Rodwell, copyrighted 2009
Stability: LaPlace Transform / Eigenvalue Method
...)exp()exp()exp()(
:response Impulse
)...)(()(
)...)(()(
...1
...1)(
)(etc)(
)(or
)(
)(or
)(
)(
function.transfer limit signal-small in etc.) (circuits, system Physical
332211
321
32122
21
2
211
tsktsktskth
ssssss
ssssssc
sasa
sbsbcsH
sHsI
sV
sa
sb
sV
sV
ppp
ppp
zzz
in
in
gen
out
system. unstable
limit.ut grow witho will)exp( then
)positive( plane-s of halfright in lie polesany If
:complex )(generally are Poles
tsk
js
pii
pi
pipipi
class notes, M. Rodwell, copyrighted 2009
Unstable system if any pole has positive real part
system. stable
decays. )exp(
negative ;
ts
js
pi
pipipipi
system. unstable
grows. )exp(
positive ;
ts
js
pi
pipipipi
class notes, M. Rodwell, copyrighted 2009
Stability from Network Viewpoint
:Impedances
stIetI )(
stVetV )(sCZ /1
sLZ
RZ
conditions initial zero assumes This
class notes, M. Rodwell, copyrighted 2009
Network Theory: One-Port Potential Instability.
)( part to real negativea
having deviceport -1a Consider
jZin
system. combined the
ofstability consider to
load external anconnect We
class notes, M. Rodwell, copyrighted 2009
Network Theory: One-Port Potential Instability.
unstable 0} Re{ negative )(
stable 0} Re{ positive )(
1
22
01)( 0/1
01
/1
1or (stable) 0 Hence
0/1
:analysis Nodal
2,1
2,1
2
2,1
2
psin
psin
sinsinp
sinppinps
pinps
in
in
in
s
in
sRR
sRR
LCL
RR
L
RRs
CRRsLCsLsRCsR
LsRCsRV
sLR
V
sCR
V
class notes, M. Rodwell, copyrighted 2009
Ideas: Unconditional stability, Potential instability
unstable 0} Re{ negative )(
stable 0} Re{ positive )(
2,1
2,1
psin
psin
sRR
sRR
stabley potentiall is device
system unstable an produces of values(positive) some negative, is If
stable.nally unconditio is device
system unstable an produces of value(positive) no positive, is If
sin
sin
RR
RR
class notes, M. Rodwell, copyrighted 2009
Unconditional stability, Potential instability
: viewpointimpedance From
s.frequencie allfor 0)(Re
:if stablenally unconditio isport -one A
jZin
s.frequencie allfor 0)(Re
:if stablenally unconditio isport -onea ely,Alternativ
jYin
unstable.y potentiall is device thethen
s,frequencie somefor 0)(ReIf jZin
class notes, M. Rodwell, copyrighted 2009
Potential instability: Relfection Viewpoint
: viewpointreflection From
s.frequencie allfor 1)(
:if stablenally unconditio isport -one A
)(
)()(
0
0
j
ZjZ
ZjZj
in
in
inin
unstable.y potentiall is device thethen
s,frequencie somefor 1)( If jin
class notes, M. Rodwell, copyrighted 2009
Stability of a Two-Port: Look at in
. and
:freedom of degrees twoare There
LS
:port-1a asit ofThink
L
LinS
SSS
22
122111
1
1 that such tscoefficien reflection load allfor
1)( if stablenally unconditio is device The
LL
in j
?1consider not wedoWhy L
class notes, M. Rodwell, copyrighted 2009
Stability of a Two-Port: Look at out Instead.
. and
:freedom of degrees twoare There
LS
:port-1a asit ofThink
1 that such tscoefficien reflection load allfor
1)( if stablenally unconditio is device The
SS
out j
?1consider not wedoWhy S
S
SoutS
SSS
11
122122
1
class notes, M. Rodwell, copyrighted 2009
Input Stability Circle
S
SoutS
SSS
11
122122
1
plane-S
1 give which of values outS
22 so 0 SoutS
class notes, M. Rodwell, copyrighted 2009
Is the inside or the outside of the circle stable ?
S
SoutS
SSS
11
122122
1plane-S
1 give which of values outS
22 so 0 SoutS
stable. ischart Smith theofcenter the then, 1 If 22 S
unstable. ischart Smith theofcenter the then, 1 If 22 S
class notes, M. Rodwell, copyrighted 2009
Output Stability Circle
plane-L
1 give which of values inL
11 so 0 SinL
L
LinS
SSS
22
122111
1
class notes, M. Rodwell, copyrighted 2009
Is the inside or the outside of the circle stable ?
stable. ischart Smith theofcenter the then, 1 If 11 S
unstable. ischart Smith theofcenter the then, 1 If 11 S
plane-L
1 give which of values outL
11 so 0 SinL
L
LinS
SSS
22
122111
1
class notes, M. Rodwell, copyrighted 2009
Potentially Unstable Amplifier
circlestability Source
1 out
122 S
plane-S
circlestability Load
1 in
111 S
plane-L
s.frequencie allat must test frequency; specific oneat a test is This
class notes, M. Rodwell, copyrighted 2009
Unconditionally stable Amplifier
circlestability Source
1 out
plane-S
circlestability Load
1 in
plane-L
s.frequencie allat must test frequency; specific oneat a test is This
122 S
111 S
class notes, M. Rodwell, copyrighted 2009
Is this possible ?????
circlestability Source
1 out
plane-S
circlestability Load
1 in
plane-L
clearly... think and Stop
122 S
111 S
class notes, M. Rodwell, copyrighted 2009
We need check only one stability circle
n.oscillatio cause can and of ncombinatio no then
,1 for which no is thereIf
LS
inL
n.oscillatio cause can and of ncombinatio no then
,1 for which no is thereIf
LS
outS
other. must the so thenest,stablity t thepasses circlestablity one If
class notes, M. Rodwell, copyrighted 2009
"Safe" and "Unsafe" Impedances
circlestability Source
1 out
122 S
plane-S
S
SoutS
SSS
11
122122
1
. wrong withgiven oscillatemight 1, gives region shaded thein inside
.any withoscillatecannot 1, gives region shaded theoutside
LoutS
LoutS
class notes, M. Rodwell, copyrighted 2009
"Safe" and "Unsafe" Impedances
circlestability Load
1 in
111 S
plane-L
. wrong withgiven oscillatemight 1, gives region shaded thein inside
.any withoscillatecannot 1, gives region shaded theoutside
SinL
SinL
L
LinS
SSS
22
122111
1
class notes, M. Rodwell, copyrighted 2009
Stable Interfaces to a Potentially Unstable Device
circlestability Source
122 S
circlestability Load
111 S
S
L
stable. be llcircuit wi thezones,danger theoutside lies or either If LS
class notes, M. Rodwell, copyrighted 2009
Stability Factors
L
LinS
SSS
22
122111
1
S
SoutS
SSS
11
122122
1
. stablenally unconditio isnetwork the,1 for which no is thereIf inL
2
21122211
2
22
2
111
1
nd
21122211
2112
2
21122211
2
22
2
11
1 2b)
wherepositive, be B measurestability that theiscondition 2 ealternativAn
1. 2a)
and
2
1 )1
where1,Kfactor stability Rollet theif stablenally unconditio isport -2A
SSSSSSB
SSSS
SS
SSSSS-S-K
. stablenally unconditio isnetwork the,1 for which no is thereIf outS