-
微波工程講義
Chapter 7 Two-port network7.1 Impedance parameters
definition, examples7.2 Admittance parameters
definition, examples7.3 Hybrid parameters
definition, examples7.4 Transmission parameters
definition, examples7.5 Conversion of the impedance, admittance,
chain, and hybrid parameters7.6 Scattering parameters
definition, characteristics, examples7.7 Conversion from
impedance, admittance, chain, and hybrid parameters
to scattering parameters or vice versa7.8 Chain scattering
parameters
7-1
-
微波工程講義
7.1 Impedance parametersBasics1.
7-2
[ ] [ ][ ] [ ] [ ]
2port at impedanceinput circuit -open:
dance transimpeforwardcircuit -open:
dance transimpereversecircuit -open:
1port at impedanceinput circuit -open:
,
response:,source:,
02
222
01
221
02
112
01
111
2221212
2121111
2
1
2221
1211
2
1
1
2
1
2
=
=
=
=
=
=
=
=
+=+=
=
=
I
I
I
I
IVZ
IVZ
IVZ
IVZ
IZIZVIZIZV
II
ZZZZ
VV
VIIZV
linearnetworkV1
I1
port 1 V2
I2
port 2
referenceplane 1
referenceplane 2
-
1/9/2003 2
How to Determine Z Parameters?
1 11 1 12 2
2 21 1 22 2
V Z I Z IV Z I Z I
= ⋅ + ⋅= ⋅ + ⋅
2 0I =
2 2
1 211 21
1 10 0I I
V VZ ZI I= =
= =
1 0I =
1 1
2 122 12
2 20 0I I
V VZ ZI I= =
= =
Two Port Network
I1=0
O.C.
Z22 I2
V2 V1 Reciprocity:
12 21Z Z=
Lecture 4
Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
微波工程講義7-3
Discussion1. Ex.7.1
V1
I1
V2
I2
6Ω[ ]
====
===Ω==
=
=
6666
,66
66
,6
11
1
01
221
2
2
02
1122211
2
1
ZII
IVZ
II
IVZZZ
I
I
2. Ex.7.2
V1
I1
V2
I2
12Ω3Ω
[ ]
0,0
30012
,3,12
01
221
02
112
22211
21
====
===
== II IVZ
IVZ
ZZZ
V1
I1
V2
I212Ω
6Ω
3Ω
3. Ex.7.3[ ] [ ] [ ]
66
,66
Z96618
,9,18
1
1
01
221
2
2
02
112
2132211
21
======
+=
===
==II
IVZ
II
IVZ
ZZZZ
II
4. Z12=Z21 → reciprocal circuitZ12=Z21, Z11=Z22→ symmetrical and
reciprocal circuit
5. Useful for series circuits analysis.
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1/9/2003 10
Series Connection of Two-Port Networks
V1
ZB
I2
ZA
I1
V2
V1A
V1B
V2A
V2B
I1 I2
[ ] [ ] [ ]A BZ Z Z= +
Lecture 4
ELG4105: Microwave Circuits © S. Loyka, Winter 2003
-
1/9/2003 5
Example: T-Network
2
2
111
1 0
221
1 0
A CI
CI
VZ Z ZI
VZ ZI
=
=
= = +
= =
1
12 21
222
2 0
C
B CI
Z Z Z
VZ Z ZI =
= = = = +
I1 I2
V2 V1
ZA ZB
ZC
1 0I =
2 0I =
Lecture 4
Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
微波工程講義7-4
6. Ex. 7.4
V1
I1
V2
I2
Zo,γ
l
[ ]
=====
==−
=−
==
==−+=
−
+==
=−=
=+=→
=Γ
−−
−
=
−
−
−
−
=
−
−−
ljZ
ljZ
ljZ
ljZ
Zlj
ZZZlj
ZZZ
Zl
Zee
ZeV
ZV
eVIVZ
Zl
ZeeeeZ
eVZV
eVVIVZ
IeV
ZV
I
eVVeVVV
V
oo
oo
oo
ollo
o
lin
o
in
lin
I
oll
ll
o
o
lin
o
in
linin
I
o
lin
o
in
lin
linin
openin
ββ
ββββ
γ
γ
γγγ
γ
γγ
γγ
γ
γ
γ
γγ
tansin
sintan,sin
,tan
line lossless
sinh2
Z
2
tanhZ
0,Z
2,
1open :2port , source:port1
21122211
12201
221
222
2
01
111
2
2
1
22
1
2
2
-
微波工程講義7-5
7.2 Admittance parametersBasics1.
[ ] [ ][ ] [ ] [ ]
2port at admittanceinput circuit -short:
ttance transadmiforwardcircuit -short:
ttance transadmireversecircuit -short:
1port at admittanceinput circuit -short:
,
response:,source:,
02
222
01
221
02
112
01
111
2221212
2121111
2
1
2221
1211
2
1
1
2
1
2
=
=
=
=
=
=
=
=
+=+=
=
=
V
V
V
V
VIY
VIY
VIY
VIY
VYVYIVYVYI
VV
YYYY
II
IVVYI
linearnetworkV1
I1
port 1 V2
I2
port 2
referenceplane 1
referenceplane 2
-
1/9/2003 3
Y Parameters
[ ] [ ] [ ]V Z I= ⋅ [ ] [ ] [ ]1I Z V−= ⋅ [ ] [ ] 1Y Z −=
1 11 1 12 2
2 21 1 22 2
I Y V Y VI Y V Y V
= ⋅ + ⋅= ⋅ + ⋅
2 0V =
2 2
1 211 21
1 10 0V V
I IY YV V= =
= =
1 0V =
1 1
2 122 12
2 20 0V V
I IY YV V= =
= =
Two Port Network
I1 S.C. Y22 I2
V2 V1=0
Lecture 4
Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
微波工程講義7-6
Discussion1. Ex.7.5
V1
I1
V2
I2
0.05S[ ]
212
2
02
112
12211
05.005.0
05.005.005.005.0
,05.0
1
YVV
VIY
YYY
V
=−=−==
−−
===
=
V1
I1
V2
I20.1S
0.025S
0.2S
2. Ex.7.6
[ ]
−−
=
=−=→−=−=
=+
===
=
=++
+=
=++
+=
=
0769.00615.00615.00692.0
0615.00615.01.0
615.0025.01.0
V:0.1S across voltage,0769.0I
short 1port ,V source:
0769.0025.02.01.0
)025.01.0(2.0
0692.0025.02.01.0
)025.02.0(1.0
2
211221
22
m22222
202
112
22
11
1
Y
YYVVI
VIVVY
VIY
Y
Y
m
V
-
微波工程講義7-7
V1
I1
V2
I20.1S
0.025S
0.2S
0.05S3. Ex.7.7
[ ] [ ] [ ]213
21122221
22
22
2222222
202
112
22
11
1269.01115.01115.01192.0
1115.01115.0)05.00615.0(
0615.00769.00.0250.1
0.1:0.1Srough current th
0769.0:0.2Srough current th
05.005.01269.0
05.0:0.05Srough current th,1269.0I
short 1port ,V source:
1269.0025.02.01.0
)025.01.0(2.005.0
1192.0025.02.01.0
)025.02.0(1.005.0
1
YYY
YYVVVI
VV
VII
IIIVVY
VIY
Y
Y
n
n
V
+=
−−
=
=−=→−=+−=
=+
=−
=+
===
=
=++
++=
=++
++=
=
4. Useful for parallel circuits analysis.
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1/9/2003 11
Parallel Connection of Two-Port Networks
V1
YB
I2
YA
I1
V2
I1A I2A
I2A I2A [ ] [ ] [ ]A BY Y Y= +
Lecture 4
ELG4105: Microwave Circuits © S. Loyka, Winter 2003
-
1/9/2003 6
Example: Pi-Network
2
2
111
1 0
221
1 0
A CV
CV
IY Y YV
IY YV
=
=
= = +
= = −
1
12 21
222
2 0
C
B CV
Y Y Y
IY Y YV =
= = − = = +
YA YB
YC I1 I2
V2V1
1 0V =
2 0V =
Lecture 4
Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
微波工程講義7-8
5. Ex.7.8
V1
I1
V2
I2
Zo,γ
l
[ ]
=====
=−=−
−=−
−==
==−+=
−
+==
−=+=
=−=→
−=Γ
−−
−
=
−
−
−
−
=
−−
−
ljZlZj
lZj
ljZYlZ
jYYljZ
YY
YlZeeZeVV
eZV
VIY
YlZe
eZeVV
eVZV
VIY
eZV
IeV
ZV
I
VeVVVV
oo
oo
oo
oll
ol
inin
l
o
in
V
ol
l
ol
inin
o
lin
o
in
V
l
o
in
o
lin
o
in
linin
shortin
ββ
ββββ
γ
γ
γγγ
γ
γ
γ
γ
γ
γγ
γ
tan1
sin
sintan1
,sin
,tan1 line lossless
sinh121
2
tanh1
111Z
2,
Z
0,
1short :2port , source:port1
21122211
12201
221
222
2
2
2
01
111
2
2
1
22
1
2
2
-
微波工程講義7-9
7.3 Hybrid parametersBasics1.
2port at admittanceinput circuit -open:
gaincurrent forwardcircuit -short:
gain voltagereversecircuit -open:
1port at admittanceinput circuit -short:
responses:, sources,:,
,
02
222
01
221
02
112
01
111
2121
2221212
2121111
2
1
2221
1211
2
1
1
2
1
2
=
=
=
=
=
=
=
=
+=+=
=
I
V
V
V
VIh
IIh
VVh
VIh
IVVIVhIhIVhIhV
VI
hhhh
IV
linearnetworkV1
I1
port 1 V2
I2
port 2
referenceplane 1
referenceplane 2
-
微波工程講義7-10
Discussion1. Useful for transistor circuits analysis.2.
Ex.7.9
V1
I1
V2
I212Ω
6Ω
3Ω
32
366,
33
96
91
631
14636312
01
221
2
2
02
112
02
222
01
111
21
1
1
−=+
−=====
=+
==
=+×+==
==
=
=
VI
I
V
IIh
II
VVh
VIh
IVh
-
微波工程講義7-11
7.4 Transmission (ABCD, chain) parametersBasics1.
gaincurrent reversecircuit -short:
ttance transadmireversecircuit -open:
dance transimpereversecircuit -short:
gain voltagereversecircuit -open:
,
02
1
02
1
02
1
02
1
221
221
2
2
1
1
2
2
2
2
=
=
=
=
−=
=
−=
=
−=−=
−
=
V
I
V
I
IID
VIC
IVB
VVA
DICVIBIAVV
IV
DCBA
IV
linearnetworkV1
I1
port 1 V2
I2
port 2
referenceplane 1
referenceplane 2
-
1/9/2003 7
ABCD Parameters
1 2 2
1 2 2
V A V B II C V D I
= ⋅ + ⋅= ⋅ + ⋅
1 2
1 2
V VA BI IC D
= ⋅
Two Port Network
I1
V1 V2
I2 !!!
Z
Z
1)
2)
3)
[ ] ?ABCD −
[ ] ?ABCD −
[ ] ?ABCD −
Lecture 4
ELG4105: Microwave Circuits © S. Loyka, Winter 2003
-
1/9/2003 12
Cascade Connection of Two-Port Networks
V1A V2A V1B V2B ABCDB ABCDA
I1A I2A I2BI1B
[ ] [ ] [ ]A BABCD ABCD ABCD= +
Lecture 4
ELG4105: Microwave Circuits © S. Loyka, Winter 2003
-
微波工程講義7-12
V1
I1
V2
I2
1Ω
Discussion1. Useful for cascade circuits analysis.2. Ex.7.10
===
−=
=−
===
==
==
1011
,0,1
1,1
02
1
02
1
02
1
02
1
22
22
IV
VI
VIC
IVB
IID
VVA
3. Ex.7.11
===
−=
=−
===
==
==
101
,,0
1,1
02
1
02
1
02
1
02
1
22
22
jwjw
VIC
IVB
IID
VVA
IV
VI
V1
I1
V2
I21Ω
jwS
1Ω
V1
I1
V2
I2
jwS
4. Ex.7.12
+++
=
+
+=
jwjwjwjw
jwjw
jwjw
jw
121
1011
111
111
101
1011
5. AD-BC=1→ reciprocal circuitA=D, AD-BC=1→ symmetrical and
reciprocal circuit
-
微波工程講義7-13
6. Ex.7.13
V1
I1
V2
I2
Zo,γ
l
=−
===+
==
=−==+=
===
=+=+
==−=−
=
−=+==−=
=−
=−
=
−
−
=−
−
=
−−−
==
−
−
−
−
−
−
−
−
−−
−
==
llZj
ljZl
lZeV
eVZV
VICl
eVeVV
VVA
IeV
ZV
IeVVeVVV
ΓVIC
VVA
lee
eZV
eVZV
DlZeeZ
eZV
eVVB
eZVIeV
ZVIVeVVV
-ΓIID
IVB
o
o
ol
in
o
lin
o
in
Il
in
linin
I
o
lin
o
inlin
linin
II
l
l
l
o
in
o
lin
o
in
ol
l
ol
o
in
linin
l
o
in
o
lin
o
inlinin
VV
ββ
ββ
γγ
γγ
γ
γ
γ
γ
γγγ
γ
γ
γ
γ
γ
γ
γ
γ
γγ
γ
cossinh1sincos
line lossless
sinh12
Z,cosh
2
0,Z
,2,
1open :2port 1,port :source,,
cosh2
12
Z,sinh
21
2
2,
Z,0,
1short :2port 1,port :source,,
2
02
12
02
1
2
2
122
1
02
1
02
1
2
2
22
2
2
122
1
02
1
02
1
22
22
22
-
1/9/2003 8
ABCD Parameters of TL I1 I2
V2V1 ,l β
0 l
( )
( )
1 10
2
20
1,
1
j l j l
j l j l
V V V I V VZ
V V e V e
I V e V eZ
+ − + −
+ − β − β
+ − β − β
= + = −
= + = −
2 1 1 0
12 1
0
cos sin
sin cos
V V l jI Z lVI j l I lZ
= β + β = β + β
[ ] 00
cos sinsin cos
l jZ lABCD
jY l lβ β
= β β
Lecture 4
ELG4105: Microwave Circuits © S. Loyka, Winter 2003
-
1/9/2003 9
Transformation Between Different Sets of Parameters
• Any set of parameters can be transformed into any other set of
parameters
Lecture 4
ELG4105: Microwave Circuits © S. Loyka, Winter 2003
R. L
udw
ig a
nd P
. Bre
tchk
o, R
F C
ircui
t Des
ign:
The
ory
and
Appl
icat
ions
, Pre
ntic
e H
all
-
1/16/2003 1
S-Parameters• Why S (scattering) parameters?• Z, Y and ABCD
parameters: O.C. or S.C. terminations
– very difficult at microwave frequencies• O.C. & S.C. :
standing waves make measurements
difficult and can destroy elements• S-parameters: defined in
terms of incident/reflected
waves• Easy to measure at microwaves: matched terminations
Lecture 5
Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
微波工程講義7-14
7.5 Conversion of the impedance, admittance, chain, and hybrid
parameterssee p.267, Table 7.1
7.6 Scattering parametersBasics1. Z, Y, H, and ABCD parameters
require an open or short circuit at
port. It is difficult or impossible to determine the parameters
of a network at radio and microwave frequencies.
2.
linearnetwork
b1
a1
port 1
a2
b2
port 2
referenceplane 1
referenceplane 2
ibia
aSaSbaSaSb
aa
SSSS
bb
i
i
port at wave(power) reflected:port at wave(power)incident :
,2221212
2121111
2
1
2221
1211
2
1
+=+=
=
-
1/16/2003 3
Definition of S-Parameters
2 2
1 1
1 111
1 0 1 0
2 222
2 0 2 0
a V
a V
b VSa V
b VSa V
+
+
−
+= =
−
+= =
= = =
= = =
reflected wave at port 1
incident wave at port 1
reflected power at port 2
incident power at port 2
2 2
1 1
2 221
1 0 1 0
1 112
2 0 2 0
a V
a V
b VSa V
b VSa V
+
+
−
+= =
−
+= =
= = =
= = =
transmitted power at port 2
incident power at port 1
transmitted power at port 1
incident power at port 2
Two-
port
netw
ork
Inpu
t po
rtO
utpu
t po
rt
Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
微波工程講義7-15
3. Measurable parameters for radio frequency and microwave
frequency
match aby d terminate1port with 2port at t coefficien
reflection:
match aby d terminate1port with 1port to2port fromt coefficienon
transmissi:
match aby d terminate2port with 2port to1port fromt coefficienon
transmissi:
match aby d terminate2port with 1port at t coefficien
reflection:
02
222
02
112
01
221
01
111
1
1
2
2
=
=
=
=
=
=
=
=
a
a
a
a
abS
abS
abS
abS
linearnetwork
b1
a1
port 1
a2
b2
port 2
referenceplane 1
referenceplane 2
-
微波工程講義7-16
4. shifting the reference planes
linearnetwork
b1
a1
port 1
a2
b2
port 2
referenceplane 1
referenceplane 2
b'1
a'1
referenceplane 1'
a'2
b'2
referenceplane 2'
port 1' port 2'
l1 l2
[ ]
=
====
====
====
−+−
+−−
−
=
+−
=
+−
=
−
=
−−−−
221
211
2
1
21
1
21
2
1
2
2211
222
)(21
)(12
211
222
02
222
)(12
02
112
)(21
01
221
211
0'1
111
22221111
'
','
','''
',',','
ljllj
lljlj
lj
a
llj
a
llj
a
lj
a
ljljljlj
eSeSeSeS
S
eSabSeS
abS
eSabSeS
abS
ebbeaaebbeaa
ββ
ββ
ββ
ββ
ββββ
-
微波工程講義7-17
Discussion1.
22,
2
2
,,,,,,
22
,,,,,,
,
,
,
,
,,
,,
:iport the todeliveredpower
2)
*Re(
21)*Re(
21
2)
*Re(
21)*Re(
21
)(22
1222
)(22
1222
2
2)(1
iiid
ioi
iref
oi
irefirefirefirefiref
ioi
iin
oi
iiniiniiniiniavs
ioioi
i
oi
ioii
oi
irefi
ioioi
i
oi
ioii
oi
iini
ioiiiref
ioiiiin
irefiinoi
i
irefiini
baP
bZ
VZ
VVIVP
aZ
VZV
VIVP
IZZV
ZIZV
Z
Vb
IZZV
ZIZV
ZV
a
IZVV
IZVV
VVZ
I
VVV
−=
====
====
−=−
=≡
+=+
=≡
−=
+=
→
−=
+=
linearnetwork
Vref,1
Vin,1
Vin,2
Vref,2
linearnetworkV1
I1
port 1 V2
I2
port 2
referenceplane 1
referenceplane 2
02,
2,
02
222
02,
1,
02
112
01,
2,
01
221
01,
1,
01
11121
1,1
1,12,22,2
,, if
==
======
==
======→=
in
ininin
Vin
ref
a
Vin
ref
aVin
ref
aVin
ref
aoo
VV
abS
VV
abS
VV
abS
VV
abSZZ
-
微波工程講義7-18
2.
linearnetwork
b1
a1
a2
b2Zs
Vs1
Z1, Γ1 Z2, Γ2
ZL
s
s
L
L
L
LL
s
ss
SSSS
aaSS
ab
SSSS
aaSS
ab
SS
aa
baS
aaS
abaSaSb
SS
aa
ba
aaSS
abaSaSb
Γ−Γ
+=+==Γ
Γ−Γ
+=+==Γ
ΓΓ−
=→=Γ+=→+=
ΓΓ−
=→=Γ+=→+=
11
211222
2
12122
2
22
22
211211
1
21211
1
11
21
22
2
1
2
222
2
121
2
22221212
12
11
1
2
1
1
1
21211
1
12121111
1
1
1,
1,
-
1/16/2003 4
Measurement of S-ParametersMeasurement Setup
DUT
0LZ Z=
1 1 2 211 1 21
1 11 1,b V b VS S
a aV V
− −
+ += = = Γ = =
Forward voltage gain: 2211
2
G
VSV
=
Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
微波工程講義7-19
3. Ex.7.14 2,1,2,1,2211 ,50,10015.0,3012,401.0,010 refino
VVZIVIV →=°∠=°∠=°∠=°∠=
2,1,2,1,
,,
,,,2
,2
refrefinin
ioiiiref
ioiiiin
VVVV
IZVVIZVV
→
−=
+=
4. Ex.7.15
12
1,
1,1,
1,
1
1,
2
1,
2,21
2211
22
222
2
02,
02,
02,02,
SZZ
ZZZ
ZZZZZ
ZZZ
VVV
VZZ
ZV
VV
VV
S
ZZZ
ZZZZZZSS
o
o
o
o
o
o
o
o
Vin
refin
V
in
o
o
VinVin
ref
ooo
oo
in
in
inin
=+
=++
+=
++
=+
===
+=
++−+
==
=
=
==
ZZO ZO
5. S12=S21 → reciprocal circuitS12=S21, S11=S22→ symmetrical and
reciprocal circuit
-
微波工程講義7-20
6. Lossless circuitEx.7.15
≠=
=∑ kjkj
SS ikij for 0for 1*
jXZO ZO
0
22
2222
222
22
2
14
44
22
22
22
*2122
*1112
*2221
*1211
*
22
2
22
22
222
11*
1221
2211
=
+−++
+−−
+=+
+−−
++
+−+=+
=
=+
++
=+=
+=
+==
+=
+==
∑
∑
≠
=
o
o
ooo
o
oo
o
o
o
oik
kjij
o
o
oik
kjij
o
o
o
o
oo
ZjXZ
ZjXjX
ZjXjX
ZjXZ
SSSS
ZjXjX
ZjXZ
ZjXZ
ZjXjXSSSS
SS
ZXZ
ZXXSSSS
ZjXZ
ZZZSS
ZjXjX
ZZZSS
-
微波工程講義7-21
YZO ZO
7. Ex.7.16
12111,
1,1,
1,
1
1,
2
1,
2,21
1
1
1
122111
22
1
211
11
,
02,02,02,02,
SYY
YS
VVV
VV
VV
VV
S
YYY
YYYY
YY
YYSSYYY
o
o
Vin
refin
VinVinVin
ref
oo
o
o
oo
inininin
=+
=+=+
====
+−=
+−
=+
−==+=
====
8. Ex.7.17
12211
1,
1,1,
1,
1
1,
2
1,
2,21
2
2
2
2
222
2
2
2
11
1211/
11,
11
02,02,02,02,
Snn
nS
VVV
nVnV
VV
VV
S
nn
ZnZ
ZnZ
Snn
ZZnZZnS
inininin Vin
refin
VinVinVin
ref
oo
oo
oo
oo
=+
=+
=+
====
+−=
+
−=
+−=
+−
=
====
n:1
V2
1I 2I
1VZO ZO
-
微波工程講義7-22
9. Ex.7.18
Zo,γ
l
a2b2
b1
a1
121,
1,
1,
2,21
2211
02,02,
0
SeVeV
VV
S
ZZZZSS
l
Vin
lin
Vin
ref
oo
oo
inin
====
=+−
==
−−
==
γγ
10. Ex.7.19
-j25
j50
ΖΟ=50 ΖΟ=50
121,
1
1,
2
1,
2,21
222
11
1
9067.0)306030401(
301020103010
2010
11774.0503010503010,3010
255050)25()5050(
11774.0503010503010
30102010502550
)25(5050
02,
02,02,
Sjj
jj
VjjV
VV
VV
S
jjSj
jjjjZ
jjS
jjjjjjZ
in
inin
V
inVinVin
ref =°−∠=++−+
+−=+
−
===
°−∠=+−−−=−=
−+−×+=
°∠=++−+=
+=−+=−−×+=
=
==
-
1/16/2003 5
Example: T-Network I1 I2
V2V1
R1 R2
R3
1 2 3
0
25 , 10075
R R RZ
= = Ω = Ω= Ω
S-Parameters ???
Zin
R1 R2
R3 Z0
S11 , S21( )1 3 2 0 75inZ R R R Z= + + = Ω
11 220inS S= Γ = =
( )( )
3 2 0021
2 0 1 3 2 0
12
R R ZZSR Z R R R Z
+= =
+ + +
12 21S S=
Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter
2003
-
The Scattering MatrixEXAMPLE
Find the S-parameters of the 3 dB attenuator circuit.
SolS11 can be found as the reflection coefficient seen at port 1
when port 2 is terminated in a matched loads (Note that Zo= 50Ω and
Zin(2) =50Ω)
But, , so . Because of the symmetry of the circuit, .
Ω=++++= 50)5056.88.141(/)]5056.8(8.141[56.8)1(inZ
Zin(2)Zin(1)
011 =S
22 0S =
11 121 1
21 222 2
S SV VS SV V
− +
− +
⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
-
The Scattering Matrix
S21 can be found by applying an incident wave at port 1, V1+ ,
and measuring the outcoming wave at port 2, V2-. This is equivalent
to the transmission coefficient from port 1 to port 2 when port 2
is matched,
From the fact that S11=S22=0, we know that V1- =0 when port 2 is
matched to 50Ω. This also implies that V2+=0.In this case we then
have that and .
Zin(2)Zin(1)
11 121 1
21 222 2
S SV VS SV V
− +
− +
⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
22 VV =−
11 VV =+
-
The Scattering Matrix
So by applying a voltage V1 at port 1 ( ) and the voltage cross
the 50 load resistor at port 2 is ( )
where .Thus, .If the input power is , then the output power
is
which is one-self (-3 dB) of the input power.
11 VV =+
)56.588.141(/)56.58(8.14144.41 +=707.02112 == SS
0
2
1 2/ ZV+
0
2
1
2
10
2
2102
1210
2
2 4/2/2/||2/ ZVVZSZVSZV+++− ===
Zin(2)Zin(1)
11 121 1
21 222 2
S SV VS SV V
− +
− +
⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
2 2V V− =
-
The Scattering MatrixEXAMPLE EXAMPLE
A two-port net work is measured and the following scattering
matrix is obtained:
a) determine whether the network is reciprocal or lossless. b)
If port two is terminated with a matched load, what will be the
return
loss at port 1?c) If a short-circuit is placed on port 2, what
will be the resulting return
loss at port 1?SolSola) Since [S] is not symmetry, the net work
is reciprocal.
To be lossless, the [S] parameters must satisfy (4.53).
Since
Thus, the network is not lossless.
0.15 0 0.85 45[ ]
0.85 45 0.2 0
o o
o oS
⎡ ⎤∠ ∠−= ⎢ ⎥∠ ∠⎣ ⎦
2 2 2 211 22| | | | 0.15 0.85 0.745 1S S+ = + = ≠
-
The Scattering Matrixb) When port 2 is terminated with matched
load, the reflection coefficient at port 1 is Γ = S11 = 0.15,
Thus,
When port 2 is terminated with a short circuit, we have .
Thus,
The second equation gives
Substituting into the first equation we have
20log | | 20 log(0.15) 16.5RL dB= − Γ = − =
−+ −= 22 VV
(0.85 45 )(0.85 45 )0.15 0.4521 0.2
o o∠− ∠= − = −
+
-
The Scattering MatrixSo the return loss is
NOTENOTE
The reflection coefficient looking into port n is not equal to
Snnunless all other ports are matched.
Similarly, the transmission coefficient from port m to port n is
not equal to Snm, unless all other ports are matched.
The parameters of a network are properties only of the network
itself (assuming the network is linear), and are defined under the
condition that all ports are matched.
20log | | 20 log(0.452) 6.9RL dB= − Γ = − =
-
微波工程講義7-23
7.7 Conversion from impedance, admittance, chain, and hybrid
parameters to scattering parameters o.r vice versa
Discussion1. See p.288, Table 7.42. Derivation ABCD→S
DCZZADVVCZZVAVV
abS
DCZZADCZZA
DVVCZZVAVDVVCZZVAV
abS
ZV
ZIZVb
ZDVVCZZVAV
ZIDZVCZIAV
ZIZV
b
ZDVVCZZVAV
ZIDZVCZIAV
ZIZVa
IZV
ZZZZIZV
bZIZV
aIV
DCBA
IV
ooooa
oo
oo
oo
oo
a
oo
o
o
oo
o
oo
o
o
o
oo
o
oo
o
o
o
ooooi
ioiii
oi
ioiii
+++=
+++==
+++−−+
=+++−−+
==
=−
=
−−+=
+−−=
−=
+++=
−+−=
+=
−=
==−
=+
=
−
=
=
=
/B2
/B2
/B/B
/B/B
222
22
22/B
22B
22
22/B
22B
22
2,port at load matched )1(
,22
,22
,
2222
2
01
221
2222
2222
01
111
2222
22222222111
22222222111
22
212
2
1
1
2
2
-
微波工程講義7-24
����
�
�
����
�
�
++++−+−
+++
+++∆
+++−−+
=��
���
�
+++∆=
+++∆
==
++++−+−
=+++−−+
==
=−
=
−−+∆
=+−−
∆=
−=
+++∆
=−+−
∆=
+=
−=
==−
=+
=��
���
�
−��
���
�
∆=�
�
���
�
=
=
DCZZADCZZA
DCZZA
DCZZADCZZADCZZA
SSSS
DCZZAAVVCZZVDVV
abS
DCZZADCZZA
AVVCZZVDVAVVCZZVDV
abS
ZV
ZIZV
b
ZAVVCZZVDV
ZIAZVCZIDV
ZIZV
b
ZAVVCZZVDV
ZIAZVCZIDV
ZIZV
a
IZV
ZZZZIZVb
ZIZVa
IV
ACBD
IV
oo
oo
oo
oooo
oo
ooooa
oo
oo
oo
oo
a
oo
o
o
oo
o
oo
o
o
o
oo
o
oo
o
o
o
ooooi
ioiii
oi
ioiii
/B/B
/B2
/B2
/B/B
/B2
/B2
/B/B
/B/B
222
22
22/B1
22B1
22
22/B1
22B1
22
1,port at load matched )2(
,22
,22
,1
2221
1211
1111
1
02
112
1111
1111
02
222
1111
11111111222
11111111222
11
211
1
2
2
1
1
-
微波工程講義7-25
7.8 Chain scattering (scattering transfer, T-)
parametersBasics1.
2222211
2122111
2
2
2221
1211
1
1 ,aTbTbaTbTa
ab
TTTT
ba
+=+=
=
Discussion1. See p.289 Table 7.5 for T- and S-parameters
conversion
21
21122211
21
2211211222211222
02
1
02
22122
21
22
02
21112212211
1
21
11
121
111
02
121
21121
1
02
111
12121111
2
2
1
2221
1211
2
1
,
,0
01,port at load matched )2(
,1
,02,port at load matched )1(
11
1
22
SSSSS
SSSSSSTST
ab
abTT
SS
abTTaTbT
a
SS
aSaS
bbT
SaSa
baT
aSbaSba
aa
SSSS
bb
aa
a
aa
−−=
−=−==+
−=−==+
=
======
===
=
==
=
==
Hw #6(due 2 weeks)2, 6, 11,21
-
微波工程講義
Chapter 9 Signal-flow graphs and applications9.1 Definitions and
manipulation of signal-flow graphs
definition, reduction rules9.2 Signal-flow graph representation
of a voltage source9.3 Signal-flow graph representation of a
passive single-port device9.4 Power gain equations
transducer power gain, operating power gain, available power
gain
9-1
-
微波工程講義
9.1 Definitions and manipulation of signal-flow graphsBasics1.
Definitions: signal-flow graph: representation of a linear
systemnode (input and output nodes) : representation of a
variablebranch: representation of direction and relation between
nodespath: a continuous succession of branches traversed in the
same
directionloop: a path originates and ends at the same node
without
encountering other nodes more than once along its traverse
9-2
-
微波工程講義9-3
2. Reduction rules(1) Rule 1
(2) Rule 2
Discussion1. Scattering parameters
linearnetwork
b1
a1
port 1
a2
b2
port 2
referenceplane 1
referenceplane 2
2221212
2121111
2
1
2221
1211
2
1 ,aSaSbaSaSb
aa
SSSS
bb
+=+=
=
a1
b1
b2
a2
S11 S22
S12
S21
node
branch
5s 2 10s
5s
3 5s+3
-
微波工程講義9-4
(3) Rule 3
sabasbbsbab21
44224−
=→=−→+=
42s
s214
−
(4) Rule 4
C1C2
C3
C4
C2
C1
C3
C4C4
C4(5) Rule 5
C1
C2 C4C3
C1C2
C1
C1
C4
C3
-
微波工程講義9-5
3. Mason’s gain rule:
pointany at t touch don' that loopsorder first 3 ofproduct :gain
looporder 3rdpointany at t touch don' that loopsorder first 2
ofproduct :gain looporder 2nd
nodeany at Ppath not touch do that gains looporder -first all of
sum:1
nodeany at Ppath not touch do that gains looporder -second all
of sum:2
nodeany at Ppath not touch do that gains looporder -first all of
sum:1
gains looporder -second all of sum:2
gains looporder -first all of sum:1
....3211
.....3211
....3211path forwardith theofgain :
....)(function transfer
2)2(
1)1(
1)1(
)2()2()2(2
)1()1()1(1
332211
∑
∑
∑
∑
∑
∑∑ ∑
∑∑ ∑
∑∑ ∑
+−+=∆
+−+=∆
+−+=∆
∆+∆+∆+∆
=
)L(
)L(
)L(
)L(
)L(
)L()L()L(-
)L()L()L(-
)L()L()L(-P
PPPsT
i
-
微波工程講義9-6
4. Ex. 9.5 find the transfer function
2121
23212121
2
1
3
2
1
1)1()1(
)(
251
3
loops 2
141)4(
1111
6161)2)(1(
3132
11
111
paths forward 3
LLLLLPLLLLPPsH
ssL
sL
ssP
Psss
ss
sP
+−−−++−−+
=
+−=+
−=
+−=×−×
+××=
=××=++
=××+
××+
××=
1 1
-31 1
6-4
-53
11+s 2+s
s
P1
P2 P3
-
微波工程講義9-7
9.2 Signal-flow graph representation of a voltage source
Es
Zs
Γsas
bG 1 bs
Γsasbs
bs
as
1
1Vs
Is
ssGs
o
ins
so
os
o
s
so
o
o
refs
insso
sos
so
orefs
refso
sins
o
srefsinss
o
refsins
refsinssrefsinsssss
abbZ
VZZZZ
ZE
ZZZ
Z
V
VZZZZ
EZZ
ZV
VZZV
ZZVVZ
ZVV
VVZIIVZIE
Γ+=→
+−
++
=→
+−
−+
=→
++−=++−
−=
+++−=+−=
222
)1()1(
)()(
,,
,,
,,,,,,
,,,,
-
微波工程講義9-8
9.3 Signal-flow graph representation of a passive single-port
device
ZLΓL
aLaL
ΓLbLbL
1
1VL
IL
LLL
o
inL
oL
oL
o
refL
inLoL
oLrefL
refLinL
Lo
refLinLLrefLinLLLL
abZ
VZZZZ
Z
V
VZZZZ
V
VV
ZZ
VVZIIZIV
Γ=→
+−
=→
+−
=→
+=
−=+==
22
)(
,,
,,
,,
,,,,
-
微波工程講義9-9
Discussion1. Ex.9.6 find Γin
Γin two-portnetwork ZL
a1
b1
b2
a2
S11 S22
S12
S21
ΓL
L
L
L
LLin
L
L
L
LLL
LLLLLLin
SSSS
SSSSS
LPLP
SLSSP
SP
SSSSSSSS
SSSSSSSSSS
Γ−Γ
+=Γ−
Γ+Γ−=
−+−
=Γ
Γ=
Γ==
Γ−Γ
+=+Γ+Γ+=
+ΓΓΓ+ΓΓ+Γ+=Γ
22
122111
22
12212211
1
211
221
12212
111
22
12211122122111
22221221221221122111
11)1(
1)1(
loop 1
paths forward 2
rule sMason')2(1
...)1(
.... )1(
Γin
-
微波工程講義9-10
2. Ex.9.7 find ΓoutΓouttwo-portnetwork ZL
Zs
Vsa1
b1
b2
a2
S11 S22
S12
S21
Γs Γout
s
s
s
ssout
s
s
s
sss
ssLsssout
SSS
SS
SSSSL
PLPSL
SSPSP
SSSSSSSS
SSSSSSSSSS
Γ−Γ
+=Γ−
Γ+Γ−=
−+−
=Γ
Γ=
Γ==
Γ−Γ
+=+Γ+Γ+=
+ΓΓΓ+ΓΓ+Γ+=Γ
11
122122
11
12211122
1
211
111
12212
221
11
12211111122122
11111221111221122122
11)1(
1)1(
loop 1
paths forward 2
rule sMason')2(1
...)1(
.... )1(
-
微波工程講義9-11
3. Ex.9.8 find Pd:power delivered from source, PL:power
delivered to the load, Pavs:maximum power available from source
bs
as
ΓL
bL
aL
Γs
bG 1 1
1
source load
2
22
2
2s
s
222222
2222
L
2
L
2
L
22
LL
L
1)1(
1
:conditionmatch conjugate
)1()1(
)1()1(111
1)
1(1
1
s
Gs
Gdavs
L
LsLLLLL
LLsLs
G
s
LG
s
Gssd
s
LGG
s
G
ss
Gss
s
Gs
sLsGsLLGsLGssGs
bbPP
babaP
PbbbbabP
bbbbba
bb
bbabbbabb
Ls Γ−=Γ−
Γ−==
Γ=Γ
Γ−=Γ−=−=
=Γ−=Γ−ΓΓ−
=ΓΓ−
Γ−
ΓΓ−=−=
ΓΓ−Γ
=−ΓΓ−Γ
=Γ−
=
ΓΓ−=
→
ΓΓ+=ΓΓ+=Γ+=Γ+=
∗Γ=Γ
∗
Pd PL
-
微波工程講義9-12
4. Ex.9.9 find b3/bs
3-portnetwork
b1
a1
a2
b2Zs
Vs ZL
ZD a3b3
a1
b1
b2
a2
S11 S22
S12
S21Γs ΓL
bG
b3 a3
S31
ΓD
S33
S32 S31 S23
321
827143323121
81
2221
22113
13318
32237
3221136
1223315
12214
223
222
111
32212
311
)3(
)2(
...)1(1,1
)3()2()1(1
loop 8
paths forward 2
LLLL
LLLLLLLLLLLLL
LLLS
LLLPP
bb
SSLSSLSSSLSSSLSSL
SLSLSL
SSPSP
L
s
Ds
DL
DLs
DLs
Ls
D
L
s
L
=
+++++=
++==∆Γ−=∆
−+−∆+∆
=
ΓΓ=ΓΓ=
ΓΓΓ=ΓΓΓ=
ΓΓ=Γ=Γ=Γ=
Γ==
∑
∑
∑
∑ ∑ ∑bs
as
-
微波工程講義9-13
9.4 Power gain equationsBasics
[ S ]Vs
Zs
ZL
Γs Γin Γout ΓL
** ,
),,(gain power transducer
),(gain power available
),(gain power operating
outLSinLavninavs
LSavs
LT
Savs
avnA
Lin
LP
PPPP
SPPG
SPPG
SPPG
Γ=ΓΓ=Γ==
ΓΓ≡
Γ≡
Γ≡
Pavs PavnPin PL
-
微波工程講義
Discussion1.
a1
b1
b2
a2
S11 S22
S21
Γs
bs
as
bG
ΓL
Pavs Pin Pavn PL
source amplifier load
2
2
222
11
221
2
22
2221122
22
22
2221
222
22
22212222
22
21
22
1212
22
222
12
12
1
1
11
)1(1
1
)1(11,
11
)1(
11
)1()1(
)1)(1(1
1
)1()1(
1
Gs
inavs
Gouts
out
outsinsG
insout
outLavn
GinsL
LLLLLL
insL
G
LL
Gins
ininin
ins
GssinsGs
bPP
bS
S
S
Sb
S
SPP
bS
SabaP
SbS
SaSba
babaP
abbbbb
ins
outLoutL
Γ−==
Γ−Γ−=
Γ−
Γ−Γ−=ΓΓ−
ΓΓ−Γ−
Γ−==
ΓΓ−Γ−
Γ−=Γ−=−=
ΓΓ−Γ−=
Γ−==
ΓΓ−
Γ−=Γ−=−=
=ΓΓ−
=→ΓΓ+=
∗
∗∗
Γ=Γ
∗Γ=Γ∗Γ=Γr
r
r
aL
bL
9-14
-
微波工程講義
2.
2211
22221
2222
22221
2211
2221
2222
2221
22
222
11
221
222
22
22212
2
2
11
)1)(1(
11
)1)(1(
)1(1
)1(
)1(1
)1(
11,
)1(1
11
)1(,
1
)1(
outLs
sL
insL
sL
avs
LT
outs
s
avs
avnA
inL
L
in
LP
Gs
inavsGouts
Lavn
GinsL
LLG
ins
inin
S
S
S
SPPG
S
SPP
G
S
SPPG
bPPbS
SPP
bS
SPbP
insoutL
ΓΓ−Γ−
Γ−Γ−=
ΓΓ−Γ−
Γ−Γ−==
Γ−Γ−
Γ−==
Γ−Γ−
Γ−==
Γ−==
Γ−Γ−==
ΓΓ−Γ−
Γ−=
ΓΓ−
Γ−=
∗∗ Γ=ΓΓ=Γ
9-15
-
微波工程講義9-16
222
2212
11
max2211
222
22
21211
211in12
11
11,
1
1
1
1
0
SS
SGSS
GGGS
SS
G
SS
TULs
LoSL
L
s
sTU
−−=→Γ=Γ=
=Γ−
Γ−
Γ−
Γ−=
=Γ→=
∗∗
5. Unilateral transducer power gain GTU
[ S ]Go
Zo Zo
Γs Γin Γout ΓL
Outputmatchingcircuit GL
Inputmatchingcircuit Gs
-
微波工程講義9-17
6. A 800MHz amplifier (Zo=50Ω) with S11=0.45∠150°,
S12=0.01∠-10°, S21=2.0∠10°, S22=0.4∠-150°, Zs=20Ω, ZL=30Ω→ GT, GP,
GA
dBS
SPPG
dBS
SPP
G
dBS
SPPG
SSSS
SSSS
ZZZZ
ZZZZ
Lins
Ls
avs
LT
sout
s
avs
avnA
Lin
L
in
LP
s
sout
L
Lin
oL
oL
os
os
4.7487.511
)1)(1(
7.7855.51)1(
)1(
7.7937.51)1(
)1(
87.150408.01
,32.150455.01
25.0,429.0
222
2
22221
211
2
2221
222
2
2221
11
122122
22
122111
Ls
==Γ−ΓΓ−
Γ−Γ−=≡
==Γ−Γ−
Γ−=≡
==Γ−Γ−
Γ−=≡
°∠=Γ−Γ
+=Γ°∠=Γ−Γ
+=Γ
−=−−
=Γ−=−−
=Γ
-
微波工程講義9-18
dBS
SS
GGGG
S
SPPG
S
SPPG
S
SPPG
SSSSS
TUPAT
Lins
Ls
avs
LT
sout
s
avs
avnA
Lin
L
in
LP
outin
2542.3221
11
111
)1)(1(
1)1(
)1(,
1)1(
)1(
,
,0
222
2212
11max
222
2
22221
211
2
2221
222
2
2221
22L11s
221112
==−−
====
Γ−ΓΓ−
Γ−Γ−=≡
Γ−Γ−
Γ−=≡
Γ−Γ−
Γ−=≡
=Γ=Γ
=Γ=Γ→=∗∗
7. A 2GHz amplifier (Zo=50Ω) with S11=0.97∠-43°, S12=0,
S21=3.39∠140°, S22=0.63∠-32°, Γs =0.97∠43°, ΓL =0.63∠32°, → GT, GP,
GA
Hw #7 (due 2 weeks)5, 10, 14
