ECE145a / 218a Siggpnal Flow Graphs€¦ · Siggpnal Flow Graphs Mark Rodwell University of...

class notes, M. Rodwell, copyrighted 2009

ECE145a / 218a Signal Flow Graphsg p

Mark RodwellUniversity of California, Santa Barbara y

[email protected] 805-893-3244, 805-893-3262 fax


Signal Flow Graphs theorysystem control :Mason

if

aSaSb +=

:belowasRepresent )parameters-S :(example

equations of System

2221212

2121111

aSaSbaSaSb

+=+=

:nodes as drepresente Variables

branchesenteringofsumvariableofValue =

branches. of weight times nodes connecting of valuesof sum

branchesenteringofsum variableof Value=


Representation of Generator & Load

000 ZVZVTZV

VVTV

sgens

sgens

Γ+=→

Γ+=−+

−+

11 baa sgen Γ+=→

22 :further ba LΓ= 22 L

:tionRepresenta :tionRepresenta


2nd Example: Cascaded Amplifiers

Circuit

tionRepresenta

ωτjeT −= eT =

represents visually andcompactly graphflow signal Thesystem. thedescribing equationsmany the


Why Use Signal Flow Graphs ?

l iii i ii lL dequationslinear ofset a of tionrepresenta theOrganizes -

: theorysystem control in usedheavily most graphsflow Signal

* RulesGain sMason'* throughsolutionefficient Provides-analysis.inintuition visual Lends-

Graphs" Flow Signal of s PropertieSomeheoryFeedback t" : MasonS.J. −

thtt ltM t

1953. Sept. 1141, p. 41, IRE, Proc.

theory.systemcontrol on ts Many tex:or


Manipulating Signal Flow Graphs

i l il :onsmanipulati Elementary


Reducing a Feedback Loop

:feedback withSystem

1122 xTx =

22112122

2121

xTTaTxxTax+=→

+=

TTT

aTT

Tx

xTTaTx

2112

122

22112122

1−=→

+=

2112


Mason's Gain Rule

on"transmissi"/ Define == abT?find wedoHow T

twice.nodeanythroughgonotdoes which to from routeany as patha Define baPi

twice.nodeany throughgonot does

of )(product theas t coefficien loopa Define 312312 TTTLi

loop.closedany aroundtscoefficien ion transmissthe


Mason's Gain Rule

[ ] [ ][ ] [ ]L

LLL

∑∑∑∑∑∑∑

+−+−+−+−+−+−

==)3()2()1(1

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

)2()2(2

)1()1(1

LLLLLPLLP

abT

tscoefficienloopallofsum)1( =∑L

:Where

tscoefficienloop all of sum )1( =∑L

(1)

hhdhi hloopsfor tscoefficien loop all of sum )1(P

L =∑1pathnot touchdowhich P

loopsorder -second all of sum )2( =∑L

loops. touching-nonofpair any oftscoefficientheofproduct theis looporder -second A

hhdhi hlddllf)2( )1( PL∑ .pathnot touchdo whichloopsorder -second all of sum )2( 1)1( PL =∑

etc.


Analysis of Simple Amplifier

genabT / Find 2=

[ ])2()1(1

)2()1(1

)1()1(1

+−+−

=∑∑∑∑

LLLLP

TL

1

)2()1(1

221112212211

21

ΓΓ+ΓΓ−Γ−Γ−=

+− ∑∑

SSSSSSS

LL

lslsls

!

∑ )1(L ∑ )2(L

!easy =


Input Reflection Coefficient

0ZZini

−=Γ

0ZZinin +Γ

ation RepresentOverall


Input Reflection Coefficient

:inputat wavesreflected andincident between ipRelationsh

[ ]L

LLin S

SSSSabT

Γ−Γ+Γ−

=Γ==22

12212211

1

1

11

LLin S

SSSΓ−

Γ+=Γ22

122111 1 L22

upondependsimpedance

Input ⎬⎫

⎨⎧

.0unlessffi ifl i

impedance load

upondependstcoefficien reflection

Input

2112 =⎭⎬⎫

⎩⎨⎧

⎭⎬

⎩⎨

SStcoefficien reflection 2112⎭⎬

⎩⎨


Output Reflection Coefficient

:outputat wavesreflected andincident between ipRelationsh

[ ]S

SSout S

SSSSabT

Γ−Γ+Γ−

=Γ==11

12211122

2

2

11

SSout S

SSSΓ−

Γ+=Γ11

122122 1 S11

upondependsimpedance

Output ⎬⎫

⎨⎧

.0unlessffi ifl i

impedance source

upondependstcoefficien reflection

Output

2112 =⎭⎬⎫

⎩⎨⎧

⎭⎬

⎩⎨

SStcoefficien reflection 2112⎭⎬

⎩⎨


Implication for Impedance MatchingSSSS

0or0eitherthen0If SSSS

SSout S

SSSΓ−

Γ+=Γ11

122122 1L

Lin SSSSΓ−

Γ+=Γ22

122111 1

.*unilateral*isamplifier the,0 If .directions both in signals passcannot amplifier thecase,either In

0.or 0either then, 0 If

2112

21122112

SS

SSSS

=

===

. and have amplifiers Unilateral

p

22out 11in

2112

SS =Γ=Γ

ng.input tuni thedisturb matchoutput the tuningdoesnor ing,output tunthedisturbnot doesmatchinput the tuningcase, thisIn

difficult. more is design :minimuma at tuningeInteractive.interactiv are ingoutput tun andinput ),0( amplifiers bilateral In 2112SS

→≠

possible.not is matching that find will welarge,ly sufficient is If 2112SS


Origin of Nonzero S12S21

:Tsemitter BJ-common in coupling Reverse:FETssource-common in coupling Reverse

cb

gd

CC

,:BJTsbase-common in coupling Reverse , : FETsgate-common in coupling Reverse

ceb

dsg

CLCL

only arise some ,parasitics device are theseof Some

,p g ceb

.terminalsdevicenear thedesignct interconnepoor from

gainstablemaximumdeviceincreases(usually)andstabilityamplifier increases )(low isolation reverse High 12S

gain.stablemaximumdeviceincreases (usually) and


Available Source Power

{ }gengenGAV ZVP Re42

, ⋅=

0/abb

aZVTb gensgensgen

Γ+

Γ+=

0 loada into launched amplitude wave theis that note

ZZbabb

L

s

genssgen

=

Γ+=

0L

** i.e. load matched-conjugateconnect :Now

SLSL ZZ Γ=Γ=


Available Source Power

22

22 ΓΓ

[ ] [ ]2

22

222

2

1PF d

11 Power Reverse

b

bbS

ss

S

Ls

Γ−

Γ=

Γ−

Γ=

[ ]2

22

1PowerAvailablePowerLoad

1 PowerForward

b

bS

s

==

Γ−=

02

2

2

todeliveredpowertheiswhere

1PowerAvailable PowerLoad

ZZbb

P

b

Ls

AVG

Ss

==

Γ−==

02 todeliveredpower theis where1

ZZbP LsS

AVGΓ−

ECE145a / 218a Siggpnal Flow Graphs€¦ · Siggpnal Flow Graphs Mark Rodwell University of...

Documents

Transcript of ECE145a / 218a Siggpnal Flow Graphs€¦ · Siggpnal Flow Graphs Mark Rodwell University of...