Passive Devices (3 port network) S-matrix of 3 port networks 33 32 31 23 22 21 13 12 11 S S S S S S S S S S (1) All ports are matched and network is reciprocal 0 0 0 23 13 23 12 13 12 S S S S S S S (2) can be lossless? ? * U S S T 1 0 . 0 , 1 1 0 , 0 , 1 1 0 , 0 , 1 2 23 2 13 13 * 12 2 23 2 13 2 12 * 23 12 * 23 2 23 2 12 2 12 * 13 23 * 13 2 13 2 12 S S S S S S S S let S S S S S S let S S S S 3 Port Network can not be lossless, reciprocal, and match at all port
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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.
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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
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
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
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
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
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
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
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
(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
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
Wilkinson Power Divider
even mode
odd mode
232 gg VV
232 gg VV
Equivalent Circuit in normalized and symmetric form
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
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
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
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
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
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
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
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
Coupled Line Theory
Propagating Modes on Symmetric Coupled Line
Even Mode Odd Mode
ground
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
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
(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
)(
)(
(1)Voltage and Current Waves of Odd Mode Excitation