passive devices (3 port network)
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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
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 ,
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
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
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
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
Impedance Matrix Z13 = Z31 = Z24 = Z42
0,0 4213
03
113
3
IIIII
VZ
kkI
1111
1111
0
0
0
3
o
e
o
e
jjjj
jjjj
I
I
I
I
eeee
eeee
I
3
3
1
333
3
sin2
2,
2
0
,
.0
,
,0
,0
IZZ
j
Iee
ZZIZIZIZIZ
VVVVV
ee
III
ee
IIIIeeIeeI
IIeeIeeI
IIII
eIeIeIeI
IeIeIeIeI
IIII
IIII
oooe
jjoeoe
oooeooeoeeoe
ooee
jjoejjoejj
ojj
e
oejj
ojj
e
ooee
jo
je
jo
je
jo
je
jo
je
oeoe
oeoe
sin203
113
3
oooe
I
ZZj
I
VZ
kk
Impedance Matrix Z14 = Z41 = Z32 = Z23
0,0 3214
04
114
4
IIIII
VZ
kkI
1111
1111
0
0
0
4
o
e
o
e
jjjj
jjjj
I
I
I
I
eeee
eeee
I
4
4
1
444
4
sin2
2,
2
0
,
.
,0
,0
,0
IZZ
j
Iee
ZZIZIZIZIZ
VVVVV
ee
III
ee
IIIIeeIeeI
IIeeIeeI
IIII
IeIeIeIeI
eIeIeIeI
IIII
IIII
oooe
jjoooe
ooooooeoeeoe
ooee
jjoejjoejj
ojj
e
oejj
ojj
e
ooee
jo
je
jo
je
jo
je
jo
je
oeoe
oeoe
sin204
114
3
oooe
I
ZZj
I
VZ
kk
tantansinsin
tantansinsin
sinsintantan
sinsintantan
2
4
3
2
1
4
3
2
1
I
I
I
I
ZZZZZZZZ
ZZZZZZZZ
ZZZZZZZZ
ZZZZZZZZ
j
V
V
V
V
oooeoooeoooeoooe
oooeoooeoooeoooe
oooeoooeoooeoooe
oooeoooeoooeoooe
LLLet oe
ooo
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L
4-Port Network: Coupled Line Coupler
Single Section Coupled Line Coupler
o
o
o
oo
ZIV
ZIV
ZIV
ZIEVVZIE
44
33
22
111111
,
,
,
Boundary Condition at each Port
kokekkokek IIIVVV ,
oeine
oein
eine
ooe
oeooeein
ZZ
VI
ZZ
ZVV
jZZ
jZZZZ
,1
,
,1
,
,
tan
tan
einZ ,oinZ ,
ooine
ooin
oino
ooo
ooooooin
ZZ
VI
ZZ
ZVV
jZZ
jZZZZ
,1
,
,1
,
,
tan
tan
ooinein
ooineino
ooinein
oeinoinooinein
oe
oein
ZZZ
ZZZZ
ZZZ
ZZZZZZ
II
VV
I
VZ
22
2
,,
2,,
,,
,,,,
11
11
1
1
inZ
2,,
1
oeinein
oin
ZZZ
ZZ matching:# portfor
2,, oeinein ZZZfor
0
)1(
)1(
tan
11
222
222
2
22
22
,,
S
ZtZZjZtZ
tZZjZtZZ
ZZ Z
tZZjZtZZZ
tZZjZtZZZZZ
tjZZtjZZ
tjZZtjZZZZ
tjZZ
tjZZZ
tjZZ
tjZZZZZ
t
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oooeooo
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oooeooooeooooe
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ooo
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VjtC
jCtV
ZZZCZZ
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tjZZZ
tZZZZjV
tZZjZ
tZZjV
tZZjZ
tjZZ
tZZjZ
tjZZV
ZZ
Z
ZZ
ZV
VVVVV
oooeooooe
oooe
oooeo
oooeoooe
oooeo
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oooeo
oeo
ooin
oin
oein
ein
oeoe
23
2
,
,
,
,
11333
1
/21,
/2
/
2
2 2
tan1
tan2
3
jC
jC
V
V
, 222 oe VVVfor
z0zLz
cos2tan
tancos2
)(
tancos2
)(
,
1
1
e
oeo
oeoe
oeooeo
e
e
oeooeo
e
zj
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zje
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V
jZZ
ZZV
jZZZZ
V
VLzV
zjZZZZ
zV
eZZ
ZZeV
eVeVzV
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oeo
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ZZz
0
tan2costancos
1
costan
1
,
,
1
2
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o
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o
e
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ZVV
ZZ
Z
jZZ
Z
V
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Z
ZZ
ZZVVVV
tZZjZ
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tjZZZ
tZjZZtZjZZ
tZjZZ
tZjZZZtZjZZ
tZjZZ
ZtZZjZ
tjZZZZ
tZZjZ
tjZZZZ
ZZ
Z
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o
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oeoooeoe
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2
2
,
,
z0zLz
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tancos2
)(
tancos2
)(
,
1
1
o
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oo
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e
zj
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ooozjo
zjo
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V
jZZ
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jZZZZ
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zjZZZZ
zV
eZZ
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eVeVzV
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ooo
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ZZz
0
tZZjZ
tjZZ
tZZjZZ
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tZjZZZtZjZZ
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tjZZZZ
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Z
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2
2
,
,
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1
costan
1
,
,
1
2
oooeo
o
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ZZjZ
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Z
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V
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0
sincos1
1
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sincos2
2
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224
2
2
2222
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VVV
VjC
C
VjZZZ
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VZZjZ
Z
ZZjZ
ZVVVVV
0
1sincos1
1
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tan
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,
4
2
2
22
2
3
V
V
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C
V
V
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V
V
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