Passive Devices (3 port network)
S-matrix of 3 port networks
 S11 S12
S   S21 S22
 S31 S32
S13 
S 23 
S33 
(1) All ports are matched and network is reciprocal
(2) can be lossless?  S T S *  U  ?
0
S   S12
 S13
S12  S13  1, S13* S 23  0,  let S13*  0  S12  1
2
2
2
*
*
S12  S 23  1, S 23
S12  0,  let S 23
 0  S12  1
2
2
2
S13  S 23  1, S12* S13  0.  S13  S 23  0  1
2
2
2
2
3 Port Network can not be
lossless, reciprocal, and match at all port
S12
0
S 23
S13 
S 23 
0 
Matched, lossless, and nonreciprocal 3 port network
Circulator
*
S12  S13  1, S 31
S 23  0,
2
2
*
S 21  S 23  1, S 23
S 21  0,
2
2
S 31  S 32  1, S12* S13  0.
2
2
(1) S12  S 23  S 31  0, S 21  S 32  S 31  1
 0
0
e  j1 


 S   e  j 2
0
0 
 j 3
 0
e
0 

(2) S 21  S 32  S13  0, S12  S 23  S 31  1
 0

 S    0
 e  j 3

e  j1
0
0
0 

e  j 2 
0 
 0 S12
S   S21 0
 S31 S32
S13 
S 23 
0 
Reciprocal, lossless, and only 2 ports matched
0
S   S12
 S13
S12  S13  1, S13* S 23  0,  let S13*  0  S12  1
2
2
2
*
*
*
S12  S 23  1, S 23
S12  S33
S13  0,  let S 23
 0  S12  1
2
2
2
*
S13  S 23  S33  1, S12* S13  S 23
S33  0.  S33  1
2
2
 0
S   e  j
 0

2
e  j
0
0
2
0 

0 
e  j 
Port 3 is isolated
Reciprocal, matched, and lossy 3 port network
0
S   S12
 S13
S12
0
S 23
S13 
S 23 
0 
Lossy power divider
S12
0
S 23
S13 
S 23 
S33 
Passive Devices (4 port network)
S-matrix of reciprocal and matched network
0
S
S    12
 S13

 S14
S12
S13
0
S 23
S 23
0
S 24
S34
S14 
S 24 
S34 

0
S 
For lossless, S-matrix must be unitary matrix
(1,1) : S12  S13  S14  1, (1,2) : S13* S 23  S14* S 24  0,
2
2
2
*
(2,2) : S12  S 23  S 24  1, (3,4) : S14* S13  S 24
S 23  0,
2
2
2
(3,3) : S13  S 23  S34  1, (1,3) : S12* S 23  S14* S34  0,
2
2
2
*
(4,4) : S14  S 24  S34  1, (2,4) : S14* S12  S34
S 23  0,
2

S
2
2
 
  S S
*
(2,3) : S12* S13  S 24
S34  0

 S

S
*
*
S 24
S13* S 23  S14* S 24  S13* S14* S13  S 24
S 23  S14* S 24  S13
S12
*
12
S 23  S14* S34
 S14  S 23  0
34
*
S  S34
S 23
*
14 12
23
12
2
2
 S34
2
2
 0
 0
S12  S13  1, S12  S 24  1  S13  S 24  A2 ,
2
2
2
2
2
2
S12  S 24  1, S 24  S34  1  S12  S 34  B 2
2
2
2
2
2
2
A2  B 2  1
S12  Be  j12 , S34  Be  j34 , S13  Ae  j13 , S 24  Ae  j 24
*
S12* S13  S 24
S34  ABe j (12 13 )  ABe j ( 24  34 )  0
 12  13   24   34   (2n  1)
(1) Symmetric Coupler
(2) Antisymmetric Coupler
let 12   34  0, and 13   24   / 2,
let 12   34  0, and 13  0,  24   ,
0
B
S   
 jA

0
0 B A 0 
 B 0 0  A

S   
A 0 0 B 


0  A B 0 
B
jA
0
0
0
0
jA
B
0
jA
B

0
0
B
S   
 jA

0
B
jA
0
0
0
0
jA
B
0
jA
B

0
Coupling  C  10 log
P1
 20 log A dB
P3
Directivit y  D  10 log
Isolation  I  10 log
P3
A
 20 log
dB
P4
S14
P1
 20 log S14 dB
P4
I  D  C dB
Hybrid Coupler(3 dB Coupler)
A B
Symmetric hybrid coupler
Antisymmetric hybrid coupler
0 1 j 0 


1 1 0 0 j 
S  
2  j 0 0 1


0 j 1 0 
0 1

1 1 0
S  
2 1 0

0  1
0
0  1
0 1

1 0
1
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
1
1
1
1
e  o , B2  Te  To ,
2
2
2
2
1
1
1
1
B3  Te  To , B4  e  o ,
2
2
2
2
B1 
3-Port Network: Power Divider
(1) Reciprocal and Lossless
-> some ports mismatched
Lossless Power Divider: Pin  P1 P2
P1
Port #1
Input Admittance at Input Port Yin
P in
1
1
Yin  jB  
Z1 Z 2
for Input Port Matching, B=0 and
1
1
1
Yin  

Z1 Z 2 Z o
P2
Ex) 1:2 divider (with B=0 & Port #1 matched: S11  0)
P2  2 P1  Z1  2 Z 2
 P1 : P2  Z 2 : Z1
1
1
3
1



 Z1  3Z o  150, Z 2  75
Z1 Z 2 Z1 Z o
Reflection at Port #2: 1 
Reflection at Port #3: 2 
50 75  150
50 75  150
50 150  75
50 150  75
1 Vo2
1 Vo2
1 Vo2
Pin 
, P1 
, P2 
2 Zo
2 Z1
2 Z2
 0.666
S 22
 0.333
S 33
(2)Resistive Divider
– All ports are matched
- output ports are not isolated
Zo  Zo
Z

   Zo   o  Zo   Zo
3  3
 3

2Z o
2
3
V  V1
 V1
Zo
2Z
3
 o
3
3
Zo
3
1
V2  V3  V
 V  V1
Z
4
2
Zo  o
3
1
 S 21  S31 
2
0 1 1 
1
S   1 0 1
2
1 1 0
Z in 
1 V12
1 V22 1 V12 1
Pin 
, P2  P3 

 Pin
2 Zo
2 Zo 8 Zo 4
 P2  P3 
1
Pin  Pin
2
S23  0
Wilkinson Power Divider( equal power division)
Resistive divider -all ports are matched
-out-ports are isolated
P2
Pin
Pin  P2  P3
Microstrip
Wilkinson Power Divider
P3
Equivalent Circuit
Wilkinson Power Divider
Vg 2  Vg 3  2
even mode
Vg 2  Vg 3  2
odd mode
Equivalent Circuit in normalized and symmetric form
Wilkinson Power Divider: even mode
Z ine
Z2
Z 
 Z  2 for matching ( S 22,e  0, S33,e  0)
2
e
in
Voltage on transmission line
V (x )
V ( x )  V  e  j x  V  e j x  V  ( e  j  x   e j  x )
V ( x  0)  V  (1  )  V2  1,  V  
1
1 
V ( x   4)  V1  V  ( j  j)  jV  (  1)  j
2 2
2 2
V
1
 S13,e  1   j
  j 0.707
V2
2
 S 23,o  0 ( port open)
V1   j
S12,e
S32,e
1
2

 1
 1
Wilkinson Power Divider: odd mode
Z ino
r
 r  2 for matching ( S 22,o  0, S33,o  0)
2
V1  0  port 1 is shorted
Z ino 
 S12,o  S13,o 
V1
0
V2
S 32,o  S 23,o  0 ( port short )
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
Z in
Z in
 2
Z 
2
2
1
Z in  Z in Z in  1  in  0  S11  0
in
Wilkinson Power Divider: S-matrix
 0  j  j
S    j 0 0  
 j 0
0 
•Symmetric
•Not Unitary
•All ports are matched
•Reciprocal
•Lossy
•Output ports are isolated
Wilkinson Power Divider: unequal power division
Let K 2 
P3
,
P2
Z o3  Z o
1 K 2
,
3
K
P2


Pin
Z o 2  K 2 Z o3  Z o K 1  K 2 ,
1

R  Z o  K  .
K

Wilkinson Power Divider: N-way Divider
P3
4-Port Network: Quadrature Hybrid
3 dB Directional Coupler with Quadrature Phase Difference
between two output ports
0 j 1 0 


Reciprocal, Lossless, and all ports are matched.
1  j 0 0 1
S  
2 1 0 0 j 


0
1
j
0


Even-mode
 A B  1 0  0
C D    j 1 

 
j 2
j / 2  1 0 1   1 j 




2  j  1
0   j 1
A  B  C  D  1  j  j  1 / 2

0
A  B  C  D  1  j  j  1 / 2
2
2
1
1  j 
Te 


A  B  C  D  1  j  j  1 / 2
2
e 
Odd-mode
 A B   1 0  0
C D    j 1 

 
j 2
j / 2   1 0 1 1 j 




2  j 1
0   j 1
A  B  C  D 1  j  j  1 / 2

0
A  B  C  D 1  j  j  1 / 2
2
2
1
1  j 
To 


A  B  C  D 1  j  j  1 / 2
2
e 
e  o
S11 
 0,
2
T T
1
S 21  e o   j
,
2
2
T T
1
S31  e o  
,
2
2
e  o
S 41 
0
2
0 j 1 0 


1  j 0 0 1
S  
2 1 0 0 j 


0 1 j 0 
Coupled Line Theory
Propagating Modes on Symmetric Coupled Line
ground
Even Mode
Odd Mode
Even Mode
Line Characteristics for even mode propagation
•Capacitance per unit length w/o dielectrics:
•Inductance per unit length:
Cea  C11a  C22a
L  o o / Cea
•Capacitance per unit length with dielectrics: Ced  C11d  C22d   eff Cea
e
•Characteristic Impedance:
Z o  L Ced
•Propagation Constant:    LCe   o o eff
e
e
Odd Mode
Line Characteristics for even mode propagation
•Capacitance per unit length w/o dielectrics: Coa  C11a  2C12a  C22a  2C12a
•Inductance per unit length:
L  o o / Coa
•Capacitance per unit length with dielectrics:
•Characteristic Impedance:
•Propagation Constant:
Z o  L Ced
Cod  C11d  2C12d
o
 C22d  2C12d   eff
Coa
 o   LCo   o o effo
(1)Voltage and Current Waves of Even Mode Excitation
Z
  j e z
1e
v1 ( z )  V e
e
,

oe

1e
V e
j e z
i1 ( z )  I1e e  j z  I1e e j
e
  j e z
2e
v2 ( z )  V e

e

2e
V1e  j e z V1e j e z

e

e
Z oe
Z oe
z
V e
j e z
i2 ( z )  I 2e e  j z  I 2e e j
e
V1e  V2e , V1e  V2e
e
z
V2e  j e z V2e j e z

e

e
Z oe
Z oe
(1)Voltage and Current Waves of Odd Mode Excitation
Z
  j o z
1o
v1 ( z )  V e
  j o z
1o
i1 ( z )  I e
v2 ( z )  V e

1o
V e

1o
I e
  j o z
2o

o
,

oo
j o z
j o z

2o
V e
j o z
i2 ( z )  I 2o e  j z  I 2o e j
o
V1o  j o z V1o j o z

e

e
Z oo
Z oo
o
z
V1o  V2o , V1o  V2o
V2o  j o z V2o j o z

e

e
Z oo
Z oo
Voltage and Current Waves on Coupled Lines
v2 ( z ), i2 ( z )
Z
oe

,  e , Z oo ,  o

v1 ( z), i1 ( z)
(1)Voltage and Current Waves on Line #1
v1 ( z )  Ve e  j z  Ve e j z  Vo e  j z  Vo e j
e
  j e z
e
i1 ( z )  I e
e

e
I e
j e z
o
  j o z
o
I e

o
I e
o
z
j o z
Ve  j e z Ve j e z Vo  j o z Vo j o z

e

e

e

e
Z oe
Z oe
Z oo
Z oo
(2)Voltage and Current Waves on Line #2
v2 ( z )  Ve e  j z  Ve e j z  Vo e  j z  Vo e j
e
e
o
i2 ( z )  I e e  j z  I e e j z  I o e  j z  I o e j
e
e
o
o
o
z
z
Ve  j e z Ve j e z Vo  j o z Vo j o z

e

e

e

e
Z oe
Z oe
Z oo
Z oo
 I1   1
 I   e  j e L
 2  
I 3   1
    j e L
 I 4  e
e  j
1
o
e j
L
1
 e  j
e
L
e
L
1
o
e j
L
  I e 
o
 
e j L   I o 
 1   I e 
o 

 e j L   I o 
1
 Ve 
o
e
o
 
e  j L e j L e j L  Vo 
1
1
 1  Ve 
o
e
o 

 e  j L e j L  e j L  Vo 
0
1
1
1   Z oe 0
o
e
o


0
e  j L e j L e j L   0 Z oo
0  Z oe
1
1
 1  0
o
e
o 
0
0
 e  j L e j L  e j L   0
V1   1
V  e  j e L
 2  
V3   1
    j e L
V4  e
 1
 e  j e L

 1
  j e L
e
1
 Z oe
 Z e  j e L
  oe
 Z oe

 j e L
 Z oee
1
1
 Z oe
Z oo
Z ooe  j
1
o
L
 Z ooe  j
o
 Z oee j
 Z oo
 Z oo
e
L
e
L
 Z oe
L
 Z oee j
  I e 
o
 
 Z oee j L   I o 
 Z oo   I e 
o 

 Z ooe j L   I o 
0   I e 
 
0   I o 
0   I e 
 
 Z oo   I o 
Impedance Matrix Z11 = Z22=Z33= Z44
Z11 
V1
I1
 I1   1
 0  e  j e L
 
0  1
    j e L
 0  e
 I1  0, I 2  I 3  I 4  0
0
Ik
k 1
I e  I o  I e  I o  I1 ,

e

o

e
let

o
I  I  I  I  0,
1
e  j
o
1
 e  j
I e   I e e  j 2 , I o   I o e  j 2
I e 1  e  j 2  I o 1  e  j 2  0  I e  I o
I
 j 2

o
 j 2
1
I1
I1  e  j 2


, Ie  Io 
1  I  I 
2 1  e  j 2
2 1  e  j 2

e

o
V1  Ve  Ve  Vo  Vo
1  e  j 2 Z oe  Z oo 
 Z I Z I Z I Z I 
I1
1  e  j 2
2

oe e

oe e

oo o
 Z11 

oo o
V1
I1
j
Ik
k 1
0
Z oe  Z oo  cot 
2
1
e
L
e
L
1
o
e j
L
β o  β e & e  j
I e e  j  I o e  j  I e e j  I o e j  0.

e
e j
L
I e e  j  I o e  j  I e e j  I o e j  0,

 

1  e   I 1  e   I
  I e 
o
 
e j L   I o 
 1   I e 
o 

 e j L   I o 
1
e
L
 e  j
o
L
 e  j
Impedance Matrix Z12=Z21=Z34=Z43
Z12 
V1
I2
0  1
 I   e  j
 2  
0  1
    j
 0  e
 I 2  0, I1  I 3  I 4  0
Ik
0
k 2
1
1
e  j
e j
1
1
 e  j
e j
1   I e 
 
e j   I o 
 1   I e 
 
 e j   I o 
I e  I o  I e  I o  0,
I e  I o  I e  I o  I 2 ,
I e e  j  I o e  j  I e e j  I o e j  0,
I e e  j  I o e  j  I e e j  I o e j  0.
I e   I e e  j 2 , I o   I o e  j 2

 

1  e   I 1  e   I
I e 1  e  j 2  I o 1  e  j 2  0  I e   I o
I

e
 j 2

o
 j 2
2
1 I2
I 2  e  j 2


 I  I 
, Ie  Io 
2 1  e  j 2
2 1  e  j 2

e

o
V1  Ve  Ve  Vo  Vo
1  e  j 2 Z oe  Z oo
 Z I Z I Z I Z I 
I2
1  e  j 2
2

oe e

oe e

oo o

oo o
 Z12 
V1
I2
j
Ik
k 2
0
Z oe  Z oo  cot 
2
Impedance Matrix Z13 = Z31 = Z24 = Z42
Z13
V
 1
I3
0  1
 0  e  j
 
I3   1
    j
 0  e
 I 3  0, I1  I 2  I 4  0
Ik
0
k 3
1
1
e  j
e j
1
1
 e  j
e j
1   I e 
 
e j   I o 
 1   I e 
 
 e j   I o 
I e  I o  I e  I o  0,
I e  I o  I e  I o  0,
I e e  j  I o e  j  I e e j  I o e j  I 3 ,
I e e  j  I o e  j  I e e j  I o e j  0.
I e   I e , I o   I o

e
 
  I e

 I
I e e  j  e j  I o e  j  e j  0  I e  I o
I e
 j
 e j

o
 j
 e j
3
 I e  I o 
I3
I3




I

I
,
o
e
2 e  j  e j 
2 e  j   e j



V1  Ve  Ve  Vo  Vo
 Z oe I e  Z oe I e  Z oo I e  Z oo I o 
 j
Z oe  Z oe
I3
e  j   e j
Z oe  Z oo
I3
2 sin 
 Z13 
V1
I3
 j
Ik
k 3
0
Z oe  Z oo
2 sin 

Impedance Matrix Z14 = Z41 = Z32 = Z23
Z14
V
 1
I4
0  1
 0   e  j
 
0  1
    j
 I 4  e
 I 4  0, I1  I 2  I 3  0
Ik
0
k 4
1
1
e  j
e j
1
1
 e  j
e j
1   I e 
 
e j   I o 
 1   I e 
 
 e j   I o 
I e  I o  I e  I o  0,
I e  I o  I e  I o  0,
I e e  j  I o e  j  I e e j  I o e j  0,
I e e  j  I o e  j  I e e j  I o e j  I 4 .
I e   I e , I o   I o

e
 
 I e

 I
I e e  j  e j  I o e  j  e j  0  I e   I o
I e
 j
 e j

o
 j
 e j
4
 I e   I o 
I4
I4




I


I
,
o
e
2 e  j   e j
2 e  j  e j




V1  Ve  Ve  Vo  Vo
 Z oe I e  Z oe I e  Z oo I o  Z oo I o 
 j
Z oe  Z oo
I4
2 sin 
Z oe  Z oo
I4
e  j  e j
 Z14 
V1
I4
 j
Ik
k 3
0
Z oe  Z oo
2 sin 
Z
oe

,  e , Z oo ,  o

L
Let  e L   o L  
 Z oe  Z oo
 tan 
V1 
 Z Z
oo
V 
 oe
tan 
 2  j 
V3  2  Z oe  Z oo
 
 sin 
V
 4
 Z Z
oo
 oe
 sin 
Z oe  Z oo

tan 
Z  Z oo
 oe
tan 
Z oe  Z oo
sin 
Z oe  Z oo
sin 
Z oe  Z oo
sin 
Z oe  Z oo
sin 
Z  Z oo
 oe
tan 
Z oe  Z oo

tan 
Z oe  Z oo 
sin   I
Z oe  Z oo   1 
I 
sin    2 
Z  Z oo   I 3 
 oe
tan    I 
Z oe  Z oo   4 


tan  
4-Port Network: Coupled Line Coupler
Single Section Coupled Line Coupler

Boundary Condition at each Port
E1  I1Z o  V1  V1  E1  I1Z o ,
V2   I 2 Z o ,
V3   I 3 Z o ,
V4   I 4 Z o

Z in V  V  V , I  I  I
k
ke
ko
k
ke
ko
Z in ,o
Z in,e
Z in,e  Z oe
V1e  V
Z in 
Z o  jZ oe tan 
Z oe  jZ o tan 
Z in,e
Z in,e  Z o
, I1e 
V
Z in,e  Z o
Z in,o  Z oo
V1o  V
Z o  jZoo tan 
Z oo  jZo tan 
Z in,o
Z in,o  Z o
, I1e 
V
Z in,o  Z o
Z Z
 Z o   Z in,o Z in,e  Z o 
V  V1o
V1
 1e
 in,e in,o
I1
I1e  I1o
Z in,e  Z in,o  2 Z o
 Zo  2
Z in,e Z in,o  Z o2
Z in,e  Z in,o  2 Z o
for port #1 matching:Z in  Z o
 Z in,e Z in,e  Z o2
for Zin,e Zin,e  Z o2
t  tan 
Z in,e Z in,e  Z oe
Z o  jZ oet
Z  jZ oot
Z oo o
Z oe  jZ o t
Z oo  jZ o t
Z o  jZoet Z o  jZoot 
Z oe  jZot Z oo  jZot 
Z o2  Z oe Z oot 2  jZ o Z oe  Z oo  t
 Z oe Z oo
Z oe Z oo  Z o2t 2  jZ o Z oe  Z oo  t
 Z oe Z oo
Z  Z oe Z oo
2
o
Z
2
o
Z o2 (1  t 2 )  jZ o Z oe  Z oo  t
 Z o2
2
2
Z o (1  t )  jZ o Z oe  Z oo  t
 S11  0
V3  V3e  V3o  V1e  V1o
 Z in,e

Z in,o
V 


Z in,o  Z o 
 Z in,e  Z o




Z o  jZ oet
Z o  jZ oot
V 


 2 Z o  j Z oe  Z oo  t 2 Z o  j Z oe  Z oo  t 
j Z oe  Z oo  t
j Z oe  Z oo / Z oe  Z oo  t
V
V
2 Z o  j Z oe  Z oo  t
2 Z o / Z oe  Z oo   j t
let C 
V3 

Z oe  Z oo
, 1  C 2  2 Z o / Z oe  Z oo 
Z oe  Z oo
jCt
1  C 2  jt
V
V3
jC tan 

V
1  C 2  j tan 
for V2  V2 e  V2o ,
z e z  0  Z o  Z oe
z  L
on line ,
V ( z )  Ve e  jz  Ve e jz
 Z oe Z oo  jZoet 

Z oe 

Z oe  j Z oe Z oo t 
Z in,e


Z in,e  Z o
 Z oe Z oo  jZoet 
  Zo
Z oe 
Z  j Z Z t
oe oo 
 oe
  j z
Z o  Z oe jz 
 Ve 
e

e 


Z

Z
o
oe


2 cos  z
Z o  jZoe tan  z 
 Ve
Z o  Z oe
V ( z   L )  V1e
 Ve
 Ve 
2 cos 
Z o  jZoe tan  
Z o  Z oe
Z o  Z oe
V1e
Z o  jZ oe tan  2 cos 

Z o  Z oe
V2 e  Ve  Ve  Ve 
1


Z o  Z oe

Zo
V1e

Z o  jZoe tan  cos 

Z o  Z oe
z 0








Z oe

Z oe
Z oe

Z oo  j Z oe t

Z oo  j Z oe t  Z oe Z oo

Z oe

Z oe Z oo  jZoet

2 Z oe Z oo  j Z oe  Z oo t
Z in,e
Zo
Zo
1
V
V 
cos  Z o  jZ oe tan  Z in,e  Z o
cos  2 Z o  j Z oe  Z oo  tan 
Z oe  j Z oo t
Z oo  j Z oe t
Z oo  j Z oe t  Z oo
Z o  jZoet
2 Z o  j Z oe  Z oo t




Z oe  j Z oo t


z
z  L
o z  0  
z 0
Z o  Z oo
Z o  Z oo
on line ,
V ( z )  Vo e  jz  Vo e jz
  j z
Z  Z oo jz 
 Vo 
 o
e 
e

Z

Z
o
oo


2 cos  z
Z o  jZoo tan z 
 Ve
Z o  Z oo
V ( z   L)  V1o  Vo

o
V
2 cos 
Z o  jZoo tan  
Z o  Z oo
Z o  Z oo
V1o

Z o  jZ oo tan  2 cos 
 Z oe Z oo  jZoot 

Z oo 

Z oo  j Z oe Z oo t 
Z in,o


Z in,o  Z o
 Z oe Z oo  jZoot 
  Zo
Z oo 
Z  j Z Z t
oe oo 
 oo




Z o  Z oo
V2 o  V  V  V 
1  Z  Z
o
oo

Zo
V1o

Z o  jZoo tan  cos 

o


o

o





Z oo

Z oo
Z oo

Z oe  j Z oo t

Z oe  j Z oo t  Z oe Z oo

Z oo



Z oo  j Z oe t
Z oe  j Z oo t

Z oe  j Z oo t  Z oe


Z oo  j Z oe t
Z oe Z oo  jZoot
2 Z oe Z oo  j Z oe  Z oo t
Z o  jZoot
2 Z o  j Z oe  Z oo t
Z in,o
Zo
Zo
1
V
V 
cos  Z o  jZoo tan  Z in,o  Z o
cos  2 Z o  j Z oe  Z oo  tan 
 V2e  V2o


V2  V2 e  V2 o  2V2 e  2



Zo
V
cos  2 Z o  j Z oe  Z oo  tan 
2Z o
V
2 Z o cos   j Z oe  Z oo  sin 
2 Z o / Z oe  Z oo 
V
2 Z o / Z oe  Z oo  cos   j sin 
1 C 2
1  C cos   j sin 
V4  V2 e  V2 o  0
2
at f o ,  

V
,
2
jC tan 
V3

C
2
V
1  C  j tan 
V2
1 C 2

  j 1 C 2
V
1  C 2 cos   j sin 
V4
0
V
from C 
Z oe  Z o
Z oe  Z oo
and Z o2  Z oe Z oo ,
Z oe  Z oo
1 C
and Z oo  Z o
1 C
1 C
1 C