ECE145a / 218a Siggpnal Flow Graphs - UC Santa · PDF fileThe signal flow graph compactly and...
Transcript of ECE145a / 218a Siggpnal Flow Graphs - UC Santa · PDF fileThe signal flow graph compactly and...
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
class notes, M. Rodwell, copyrighted 2009
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=
class notes, M. Rodwell, copyrighted 2009
Representation of Generator & Load
000 ZVZVTZV
VVTV
sgens
sgens
Γ+=→
Γ+=−+
−+
11 baa sgen Γ+=→
22 :further ba LΓ= 22 L
:tionRepresenta :tionRepresenta
class notes, M. Rodwell, copyrighted 2009
2nd Example: Cascaded Amplifiers
Circuit
tionRepresenta
ωτjeT −= eT =
represents visually andcompactly graphflow signal Thesystem. thedescribing equationsmany the
class notes, M. Rodwell, copyrighted 2009
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
class notes, M. Rodwell, copyrighted 2009
Manipulating Signal Flow Graphs
i l il :onsmanipulati Elementary
class notes, M. Rodwell, copyrighted 2009
Reducing a Feedback Loop
:feedback withSystem
1122 xTx =
22112122
2121
xTTaTxxTax+=→
+=
TTT
aTT
Tx
xTTaTx
2112
122
22112122
1−=→
+=
2112
class notes, M. Rodwell, copyrighted 2009
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
class notes, M. Rodwell, copyrighted 2009
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.
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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 =
class notes, M. Rodwell, copyrighted 2009
Input Reflection Coefficient
0ZZini
−=Γ
0ZZinin +Γ
ation RepresentOverall
class notes, M. Rodwell, copyrighted 2009
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⎭⎬
⎩⎨
class notes, M. Rodwell, copyrighted 2009
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⎭⎬
⎩⎨
class notes, M. Rodwell, copyrighted 2009
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
class notes, M. Rodwell, copyrighted 2009
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
class notes, M. Rodwell, copyrighted 2009
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=
class notes, M. Rodwell, copyrighted 2009
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−