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


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


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


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


Stability from Network Viewpoint

:Impedances

stIetI )(

stVetV )(sCZ /1

sLZ

RZ

conditions initial zero assumes This


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


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


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


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


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


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


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


Input Stability Circle

S

SoutS

SSS

11

122122

1

plane-S

1 give which of values outS

22 so 0 SoutS


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


Output Stability Circle

plane-L

1 give which of values inL

11 so 0 SinL

L

LinS

SSS

22

122111

1


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


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


Unconditionally stable Amplifier


1 out

plane-S


1 in

plane-L

s.frequencie allat must test frequency; specific oneat a test is This

122 S

111 S


Is this possible ?????


1 out

plane-S


1 in

plane-L

clearly... think and Stop

122 S

111 S


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


"Safe" and "Unsafe" Impedances


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


"Safe" and "Unsafe" Impedances


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


Stable Interfaces to a Potentially Unstable Device


122 S


111 S

S

L

stable. be llcircuit wi thezones,danger theoutside lies or either If LS


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

ECE145a / 218a Bilateral Tuned Amplifier Design: Stability · Bilateral Tuned Amplifier Design:...

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