passive devices (3 port network) s-matrix of 3 port networks (1) all ports are matched and network...
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![Page 1: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/1.jpg)
Passive Devices (3 port network)
S-matrix of 3 port networks
333231
232221
131211
SSS
SSS
SSS
S
(1) All ports are matched and network is reciprocal
0
0
0
2313
2312
1312
SS
SS
SS
S
(2) can be lossless?
? * USS T
10.0 ,1
1 0 ,0 ,1
1 0 ,0 ,1
2
23
2
1313*12
2
23
2
13
2
12*2312
*23
2
23
2
12
2
12*1323
*13
2
13
2
12
SSSSSS
SSletSSSS
SSletSSSS
3 Port Network can not be lossless, reciprocal, and match at all port
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Matched, lossless, and nonreciprocal 3 port network
Circulator
0
0
0
3231
2321
1312
SS
SS
SS
S
00
00
00
1,0)2(
00
00
00
1,0)1(
.0 ,1
,0 ,1
,0 ,1
3
2
1
3
2
1
312312133221
313221312312
13*12
2
32
2
31
21*23
2
23
2
21
23*31
2
13
2
12
j
j
j
j
j
j
e
e
e
S
SSSSSS
e
e
e
S
SSSSSS
SSSS
SSSS
SSSS
![Page 3: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/3.jpg)
Reciprocal, matched, and lossy 3 port network
Reciprocal, lossless, and only 2 ports matched
0
0
332313
2312
1312
SSS
SS
SS
S
1.0 ,1
1 0 ,0 ,1
1 0 ,0 ,1
2
3333*2313
*12
2
33
2
23
2
13
2
12*2313
*3312
*23
2
23
2
12
2
12*1323
*13
2
13
2
12
SSSSSSSS
SSletSSSSSS
SSletSSSS
00
00
00
j
j
j
e
e
e
S Port 3 is isolated
0
0
0
2313
2312
1312
SS
SS
SS
S Lossy power divider
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Passive Devices (4 port network)S-matrix of reciprocal and matched
network
0
0
0
0
342414
342313
242312
141312
SSS
SSS
SSS
SSS
S
For lossless, S-matrix must be unitary matrix
0
0
0
0:)3,2(
,0:(2,4) ,1:)4,4(
,0:(1,3) ,1:)3,3(
,0:(3,4) ,1:)2,2(
,0:(1,2) ,1:)1,1(
2314
2
34
2
122323*3412
*143434
*1423
*1212
2
13
2
24*1423
*2413
*14
*1324
*1423
*13
*24
34*2413
*12
23*3412
*14
2
34
2
24
2
14
34*1423
*12
2
34
2
23
2
13
23*2413
*14
2
24
2
23
2
12
24*1423
*13
2
14
2
13
2
12
SS
SSSSSSSSSSSSS
SSSSSSSSSSSSS
SSSS
SSSSSSS
SSSSSSS
SSSSSSS
SSSSSSS
S
![Page 5: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/5.jpg)
00
00
00
00
00
00
00
00
,,0 ,0 ,2/ ,0
)2( )1(
)12(
0
,,,
1
1,1
,1,1
2413341224133412
34241312
)()(34
*2413
*12
24133412
22
22
34
2
12
2
34
2
24
2
24
2
12
22
24
2
13
2
24
2
12
2
13
2
12
34241312
24133412
BA
BA
AB
AB
S
BjA
BjA
jAB
jAB
S
andletandlet
CouplerricAntisymmetCouplerSymmetric
n
ABeABeSSSS
AeSAeSBeSBeS
BA
BSSSSSS
ASSSSSS
jj
jjjj
![Page 6: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/6.jpg)
dB
dB log20log10
dB log20log10
dB log20log10
144
1
144
3
3
1
CDI
SP
PIIsolation
S
A
P
PDyDirectivit
AP
PCCoupling
00
00
00
00
BjA
BjA
jAB
jAB
S
Hybrid Coupler(3 dB Coupler)
0110
1001
1001
0110
2
1
010
100
001
010
2
1
S
j
j
j
j
S
couplerhybridricAntisymmetcouplerhybridSymmetric
BA
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Even-Odd Mode Circuit Analysis
•If circuit can be decomposed into the superposition of an even-mode excitation and odd-mode excitation
•Adding two sets( even & odd) of excitations produces the original excitation
,2
1
2
1,
2
1
2
1
,2
1
2
1,
2
1
2
1
43
21
oeoe
oeoe
BTTB
TTBB
![Page 8: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/8.jpg)
3-Port Network: Power Divider
(1)Reciprocal and Lossless
-> some ports mismatched
Lossless Power Divider: 21 PPPin
inYInput Admittance at Input Port
21
11
ZZjBYin
for Input Port Matching, B=0 and
oin ZZZ
Y111
21
inP
1P
2P
Port #1Port
#2
Port #3
1221
2
2
21
2
1
2
::
2
1,
2
1,
2
1
ZZPP
Z
VP
Z
VP
Z
VP oo
o
oin
Ex) 1:2 divider (with B=0 & Port #1 matched: )
75,15031311
22
21121
2112
ZZZZZZZ
ZZPP
oo
666.01507550
15075501
Reflection at Port #2:
333.07515050
75150502
Reflection at Port #3:
22S
33S
011 S
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(2)Resistive Divider – All ports are matched
- output ports are not isolated
inin
inooo
in
oo
o
oo
o
ooo
ooo
in
PPPP
PZ
V
Z
VPP
Z
VP
S
SS
VVZZ
ZVVV
VZZ
Z
VV
ZZZ
ZZZ
Z
2
1
4
1
8
1
2
1,
2
1
011
101
110
2
1
2
1
2
1
4
3
3
3
2
32
3
32
333
32
21
22
32
21
3121
132
11
023 S
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Wilkinson Power Divider( equal power division)
Resistive divider -all ports are matched
-out-ports are isolated
Microstrip Wilkinson Power Divider
Equivalent Circuit
inP
2P
3P32 PPPin
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Wilkinson Power Divider
even mode
odd mode
232 gg VV
232 gg VV
Equivalent Circuit in normalized and symmetric form
![Page 12: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/12.jpg)
Wilkinson Power Divider: even mode
einZ
)0,0( 22 ,33,22
2
eeein SSmatchingforZ
ZZ
Voltage on transmission line)(xV
) ( 0
707.02
1
22
22
2
1
1
1)1()()4(
1
1,1)1()0(
)()(
,23,32
2
11312
1
1
2
openportS S
jjV
VS S
jV
jjVjjVVxV
VVVxV
eeVeVeVxV
oe
,e,e
xjxjxjxj
![Page 13: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/13.jpg)
Wilkinson Power Divider: odd mode
) ( 0
0
1 0
)0,0( 22
,23,32
2
1,13,12
1
,33,22
shortportSS
V
VSS
shortedisportV
SSmatchingforrr
Z
oo
oo
oooin
oinZ
All power from Port #2 is delivered to r/2 resister, and all dissipated
No power is delivered to Port #1
![Page 14: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/14.jpg)
Wilkinson Power Divider: input port
inZ
inZ
001
21
2
11
2
SZZZ
Z
inininin
in
Wilkinson Power Divider: S-matrix
00
00
0
j
j
jj
S
•All ports are matched•Reciprocal•Lossy•Output ports are isolated
•Symmetric•Not Unitary
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Wilkinson Power Divider: unequal power division
inP
2P
3P
.1
,1
,1
,
23
22
3
2
3
2
32
KKZR
KKZZKZ
K
KZZ
P
PKLet
o
ooo
oo
Wilkinson Power Divider: N-way Divider
![Page 16: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/16.jpg)
4-Port Network: Quadrature Hybrid
3 dB Directional Coupler with Quadrature Phase Difference between two output ports
Reciprocal, Lossless, and all ports are matched.
010
001
100
010
2
1
j
j
j
j
S
![Page 17: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/17.jpg)
Even-mode
1
1
2
1
1
01
02
2/0
1
01
j
j
jj
j
jDC
BA
j
jjDCBAT
jj
jj
DCBA
DCBA
e
e
12
1
2/11
22
02/11
2/11
![Page 18: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/18.jpg)
Odd-mode
1
1
2
1
1
01
02
2/0
1
01
j
j
jj
j
jDC
BA
j
jjDCBAT
jj
jj
DCBA
DCBA
o
e
12
1
2/11
22
02/11
2/11
![Page 19: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/19.jpg)
02
,2
1
2
,2
1
2
,02
41
31
21
11
oe
oe
oe
oe
S
TTS
jTT
S
S
010
001
100
010
2
1
j
j
j
j
S
![Page 20: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/20.jpg)
Coupled Line Theory
Propagating Modes on Symmetric Coupled Line
Even Mode Odd Mode
ground
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Even Mode
Line Characteristics for even mode propagation
•Capacitance per unit length w/o dielectrics:
•Inductance per unit length:
•Capacitance per unit length with dielectrics:
•Characteristic Impedance:
•Propagation Constant:
aaea CCC 2211
eaoo CL /
eaeeffdded CCCC 2211
edo CLZ eeffooe
e LC
![Page 22: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/22.jpg)
Odd Mode
Line Characteristics for even mode propagation
•Capacitance per unit length w/o dielectrics:
•Inductance per unit length:
•Capacitance per unit length with dielectrics:
•Characteristic Impedance:
•Propagation Constant:
aaaaoa CCCCC 12221211 22
oaoo CL /
oaoeffdd
ddod
CCC
CCC
1222
1211
2
2
edo CLZ oeffooo
o LC
![Page 23: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/23.jpg)
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(1)Voltage and Current Waves of Even Mode Excitation
eoeZ ,
eeee VVVV 2121 ,
zj
oe
ezj
oe
ezje
zje
zje
zje
eeee
ee
eZ
Ve
Z
VeIeIzi
eVeVzv
11111
111
)(
)(
zj
oe
ezj
oe
ezje
zje
zje
zje
eeee
ee
eZ
Ve
Z
VeIeIzi
eVeVzv
22222
222
)(
)(
![Page 25: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/25.jpg)
(1)Voltage and Current Waves of Odd Mode Excitation
oooZ ,
zj
oo
ozj
oo
ozjo
zjo
zjo
zjo
oooo
oo
eZ
Ve
Z
VeIeIzi
eVeVzv
11111
111
)(
)(
zj
oo
ozj
oo
ozjo
zjo
zjo
zjo
oooo
oo
eZ
Ve
Z
VeIeIzi
eVeVzv
22222
222
)(
)(
oooo VVVV 2121 ,
![Page 26: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/26.jpg)
Voltage and Current Waves on Coupled Lines
(1)Voltage and Current Waves on Line #1
zj
oo
ozj
oo
ozj
oe
ezj
oe
e
zjo
zjo
zje
zje
zjo
zjo
zje
zje
ooee
ooee
ooee
eZ
Ve
Z
Ve
Z
Ve
Z
V
eIeIeIeIzi
eVeVeVeVzv
)(
)(
1
1
(2)Voltage and Current Waves on Line #2
zj
oo
ozj
oo
ozj
oe
ezj
oe
e
zjo
zjo
zje
zje
zjo
zjo
zje
zje
ooee
ooee
ooee
eZ
Ve
Z
Ve
Z
Ve
Z
V
eIeIeIeIzi
eVeVeVeVzv
)(
)(
2
2
ooo
eoe ZZ ,,,
)(),( 11 zizv
)(),( 22 zizv
![Page 27: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/27.jpg)
o
e
o
e
Ljoo
Ljoe
Ljoo
Ljoe
oooeoooe
Ljoe
Ljoe
Ljoo
Ljoe
oooeoooe
o
e
o
e
oo
oe
oo
oe
LjLjLjLj
LjLjLjLj
o
e
o
e
LjLjLjLj
LjLjLjLj
o
e
o
e
LjLjLjLj
LjLjLjLj
I
I
I
I
eZeZeZeZ
ZZZZ
eZeZeZeZ
ZZZZ
I
I
I
I
Z
Z
Z
Z
eeee
eeee
V
V
V
V
eeee
eeee
V
V
V
V
I
I
I
I
eeee
eeee
I
I
I
I
oeoe
oeoe
oeoe
oeoe
oeoe
oeoe
oeoe
oeoe
000
000
000
000
1111
1111
1111
1111
1111
1111
4
3
2
1
4
3
2
1
![Page 28: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/28.jpg)
Impedance Matrix Z11 = Z22=Z33= Z44
0,0 4321
01
111
1
IIIII
VZ
kkI
&
1111
1111
0
0
01
jLjLjeo
o
e
o
e
LjLjLjLj
LjLjLjLj
eeeβlet β
I
I
I
I
eeee
eeee
I
oe
oeoe
oeoe
12
2
1
2
21
21
122
22
22
1
21
1
12,
12
111
011
,
.0
,0
,0
,
IZZ
e
eIZIZIZIZ
VVVVV
e
eIII
e
IIIIeIeI
IIeIeI
eIIeII
eIeIeIeI
eIeIeIeI
IIII
IIIII
oooej
j
ooooooeoeeoe
ooee
j
j
oejoej
oj
e
oej
oj
e
joo
jee
jo
je
jo
je
jo
je
jo
je
oeoe
oeoe
cot2
01
111
1
oooe
I
ZZj
I
VZ
kk
![Page 29: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/29.jpg)
Impedance Matrix Z12=Z21=Z34=Z43
0,0 4312
02
112
2
IIIII
VZ
kkI
1111
1111
0
0
0
2
o
e
o
e
jjjj
jjjj
I
I
I
I
eeee
eeeeI
22
2
1
2
22
22
222
22
22
2
21
1
12,
12
111
011
,
.0
,0
,
,0
IZZ
e
eIZIZIZIZ
VVVVV
e
eIII
e
IIIIeIeI
IIeIeI
eIIeII
eIeIeIeI
eIeIeIeI
IIIII
IIII
oooej
j
ooooooeoeeoe
ooee
j
j
oejoej
oj
e
oej
oj
e
joo
jee
jo
je
jo
je
jo
je
jo
je
oeoe
oeoe
cot2
02
112
2
oooe
I
ZZj
I
VZ
kk
![Page 30: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/30.jpg)
Impedance Matrix Z13 = Z31 = Z24 = Z42
0,0 4213
03
113
3
IIIII
VZ
kkI
1111
1111
0
0
0
3
o
e
o
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2
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3
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![Page 31: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/31.jpg)
Impedance Matrix Z14 = Z41 = Z32 = Z23
0,0 3214
04
114
4
IIIII
VZ
kkI
1111
1111
0
0
0
4
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e
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sin204
114
3
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ZZj
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kk
![Page 32: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/32.jpg)
tantansinsin
tantansinsin
sinsintantan
sinsintantan
2
4
3
2
1
4
3
2
1
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![Page 33: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/33.jpg)
4-Port Network: Coupled Line Coupler
![Page 34: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/34.jpg)
Single Section Coupled Line Coupler
o
o
o
oo
ZIV
ZIV
ZIV
ZIEVVZIE
44
33
22
111111
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,
,
Boundary Condition at each Port
![Page 35: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/35.jpg)
kokekkokek IIIVVV ,
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VI
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![Page 36: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/36.jpg)
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![Page 37: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/37.jpg)
, 222 oe VVVfor
z0zLz
cos2tan
tancos2
)(
tancos2
)(
,
1
1
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![Page 38: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/38.jpg)
z0zLz
cos2tan
tancos2
)(
tancos2
)(
,
1
1
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![Page 39: Passive Devices (3 port network) S-matrix of 3 port networks (1) All ports are matched and network is reciprocal (2) can be lossless? 3 Port Network can](https://reader035.vdocuments.mx/reader035/viewer/2022062314/56649f205503460f94c38e70/html5/thumbnails/39.jpg)
0
sincos1
1
sincos/2
/2
sincos2
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tan2cos22
224
2
2
2222
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4
2
2
22
2
3
V
V
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fat o
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