Notes 14 - Network analysis

```ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 14
Network Analysis
Multiport Networks
1
Multiport Networks
A general circuit can be represented by a multi-port network, where the “ports”
are defined as access terminals at which we can define voltages and currents.
Note: Equal and opposite currents are
assumed on the two wires of a port.
Examples:
1)
One-port network
I1
+

V1
I1
+
V1
R
-
-
2) Two-port network
I2
I2
I1
+
V1
+
V2
-
I1
-

+
V1
-
+
V2
-
2
Multiport Networks (cont.)
IN
3) N-port Network
+ VN -

I1
Im
V1 +-
To represent multi-port networks we use:
+
- Vm
I2
-
Z (impedance) parameters
h (hybrid) parameters
ABCD parameters
Not easily
measurable at
high frequency
-
S (scattering) parameters
Measurable at high frequency
V2 +
-

I3



+ V 3
3
Poynting Theorem (Phasor Domain)

V

2
1
EJ
i*
 dV   S  nˆ dS
S
1
     c E
2
V 
2

1
2 j      H
4
V 
1
   H
2
2

1
4
2

 dV

2

 c E  dV

The last term is the VARS
consumed by the region.
The notation < > denotes
time-average.
Ps  P f  Pd  j  2    Wm  We

4
Self Impedance
Consider a general one-port network
S
I1
V1
+
nˆ
E,H
-
V
Complex power delivered to network:
Pin 

1
2
1
2
  E  H   nˆ
S
V1 I 1
d s  Pd  j 2   W m  W e 
*
Average power
dissipated in [W]
Pd  Pd
Average electric
energy (in [J])
stored inside V
W e  We
Average magnetic energy
(in [J]) stored inside V
W m  Wm
5
Define Self Impedance (Zin)
1
Z in 
V1

I1
V1 I 1
I1
*
2
 2
1
2
 R in  jX in 
V1 I 1
I1
*

2
X in 
1
I1
2
2
Pd  j 2   W m  W e 
1
2
R in 
Pin
I1
2
2 Pd
I1
2
S
I1
4  (W m  W e )
I1
2
V1
+
-
nˆ
E,H
V
6
Self Impedance (cont.)
We can show that for physically realizable networks the following apply:
book for a proof.
V1      V1   
*
 Z in      Z in   
*
I1     I1   
*

R in    is a n e v e n fu n ctio n o f 
Z in     R in     jX in   
X in    is a n o d d fu n ctio n o f 
Note: Frequency is usually defined as a positive quantity. However, we consider
the analytic continuation of the functions into the complex frequency plane.
7
Two-Port Networks
Consider a general 2-port linear network:
I2
I1
V1
+
-
1
+
2
-
V2
In terms of Z-parameters, we have (from superposition)
Impedance (Z) matrix
V1  Z 11 I 1  Z 12 I 2
V 2  Z 21 I 1  Z 2 2 I 2
 V1   Z 11
   
V 2   Z 21
Z 12   I 1 
 
Z 22   I 2 

V    Z  I 
8
Elements of Z-Matrix: Z-Parameters
(open-circuit parameters )
V1  Z 11 I 1  Z 12 I 2
V 2  Z 21 I 1  Z 22 I 2
Port 2 open circuited
Z 11 
Z 21 
Port 1 open circuited
Z ij 
V1
I1
Vi
Ij
Ik 0
Z 12 
k j
I2 0
V2
I1
Z 22 
I2 0
V1
-
I2
I1  0
V2
I2
I1  0
I2
I1
+
V1
1
2
+
-
V2
9
Z-Parameters (cont.)
N-port network

Z ij 
Ij
Vi
Ij
Ik 0
k j
Vi +
-




We inject a current into port j and measure the voltage (with an ideal
voltmeter) at port i. All ports are open-circuited except j.
10
Z-Parameters (cont.)
Z-parameters are convenient for series connected networks.
A
B
 V1   V1   V1 
    A   B
V 2  V 2  V 2 
+
A
A
B
B
  Z   I    Z   I 

A
B
  Z    Z 
I 
 I1 
  Z    Z   
I2 

  Z
A
A
B
I1A
I1
V1A
I 2A
+
-
1
A
2
+
-
I2
V2A
+
V2
V1
I 2B
I1B

-
V1B
 I1 
 Z   
I2 
+
-
1
B
2
+
-
V2B
-
B
 V1   Z 11  Z 11
   A
B
V
Z

Z
 2   21
21
A
B
A
B
Z 12  Z 12   I 1 

A
B  
Z 22  Z 22   I 2 
S e rie s

I1  I1  I1
A
B
I2  I2  I2
A
B
11
Consider a 2-port network:
I2
I1
V1
+
1
-
+
2
-
I 1  Y11V1  Y12V 2
matrix
I 2  Y21V1  Y 22V 2
or
 I 1   Y11
  
 I 2   Y 21
Yij 
V2
Y12   V1 
     I    Y V
Y 2 2  V 2 
Ii
Vj

Short-circuit parameters
Vk  0 k  j
12
Y-Parameters (cont.)
N-port network

Yij 
Vj
Ii
Vj
+Vk  0 k  j
Ii




We apply a voltage across port j and measure the current (with an ideal
current meter) at port i. All ports are short-circuited except j.
13
Y-parameters are
convenient for
parallel connected
networks.
I1
+
V1
-
I1A
V1A
+
-
1
A
2
I1B
B
1
 Y11  Y11
  A
B
Y

Y
21
 21
A
B
V
Y12  Y12   V1 

A
B  
Y 22  Y 22  V 2 
A
I2
+ A
- V2
V2
+
-
I 2B
+
B
1 -
 I1   I   I 
    A   B
 I2   I2   I2 
A
1
I 2A
1
B
+ B
- V2
2
B
P a ra lle l
 V1  V1  V1
A
B
V2  V2  V2
A
B
14
Relation between [Z] and [Y] matrices:
V    Z  I 
 I   Y V 
Hence
V    Z   Y V  

  Z Y   V 
 Z Y   U   Identity M atrix
Therefore Y    Z 
1
15
Reciprocal Networks
If a network does not contain non-reciprocal devices or materials* (i.e.
ferrites, or active devices), then the network is “reciprocal.”
 Z ij  Z
ji
  Z  a n d Y
Y

ij
 Y ji 
Note: The inverse of a
symmetric matrix is
symmetric.
a re sym m e tric
* A reciprocal material is one that has reciprocal permittivity and permeability tensors.
A reciprocal device is one that is made from reciprocal materials
Example of a nonreciprocal material: a biased ferrite
(This is very useful for making isolators and circulators.)
16
Reciprocal Materials
D   E
B   Η
 D x    xx

 
D y    yx


 D z    zx

 xy
 xz   E x 
 yy
 yz  E y
 zy
 
 
 zz   E z 
 B x    xx
  
B  
 y   yx
 B z    zx
 xy
 yy
 zy
 xz    x 

 yz    y 
 
 zz    z 
Reciprocal:  ij   ji ,  ij   ji
Ferrite:
 

  0  j

 0
j

0
0

0

1 
 is not symmetric!
17
Reciprocal Networks (cont.)
We can show that the equivalent circuits for reciprocal 2-port networks are:
Z 22  Z 21
Z11  Z 21
T-equivalent
Z 21
Y21
Pi-equivalent
Y11  Y21
Y22  Y21
18
ABCD-Parameters
There are defined only for 2-port networks.
V 1   A
  
 I1  C
I1
B  V 2 
 '
D   I2 
V1


I 2'
1
2
V
 2
I2   I2
'
A
V1
V2
B 
'
I2 0
V1
V2
D 
'
I2
I1
C 
V2  0
'
I2 0
I1
'
I2
V2  0
19
I1
I1A
V1

V1A
I


1 A 2
V

A'
2
A
2
I
B
1
B
1
V
I

1 B 2

A
V 2 
V 1  V 1 
A
    A    A B C D   A ' 
 I1   I1 
 I 2 
B'
2
V

B
2
I 2'

V2
A
V 1 
A
  A B C D   B 
 I1 
B
V 2 
A
B
  A B C D   A B C D   ' 
B
 I 2 
B
V1 
  
 I1 
 ABCD

AB
A nice property of the
ABCD matrix is that it is
networks: you simply
multiply the ABCD
matrices together.
V 2 
 '
 I
 2
20
Scattering Parameters
 At high frequencies, Z, Y, h & ABCD parameters are difficult
(if not impossible) to measure.
o V and I are not uniquely defined
o Even if defined, V and I are extremely difficult
to measure (particularly I).
o Required open and short-circuit conditions are
often difficult to achieve.
 Scattering (S) parameters are often the best
representation for multi-port networks at high frequency.
21
Scattering Parameters (cont.)
S-parameters are defined
assuming transmission lines are connected to each port.
a1
b1
Z 01 ,  1
1
Z 02 ,  2
2
z1
a2
b2
z2
Local coordinates
On each transmission line:
Vi  zi   Vi 0 e

I i  zi  
Vi

  i zi
 zi 
Z 0i

 Vi 0 e

Vi

  i zi
 zi 
 Vi

 zi   Vi   zi 
i  1, 2
Z 0i
Incom ing w ave function  a i
 z i   Vi   z i 
Z 0i
O utgoing w ave function  bi
 z i   Vi   z i 
Z 0i
22
For a One-Port Network
L
L
0
 
V1  0 
V1


Z 01
a1
Z 01
Z 01
b1
b1  0 
a1  0 
l1

b1  0    L a 1  0 
 S 11 a 1  0 
For a one-port network,
S11 is defined to be the
same as L.
 S11
Incom ing w ave function  a i
 z i   Vi   z i 
Z 0i
O utgoing w ave function  bi
 z i   Vi   z i 
Z 0i
23
For a Two-Port Network
a1
Z 01 ,  1
1
Z 02 ,  2
2
b1
z1
b2
z2
b1  0   S 11 a 1  0   S 12 a 2  0 
b2  0   S 21 a 1  0   S 22 a 2  0 
 b1  0    S 11
 
 
 b2  0    S 21
a2
Scattering
matrix
S 12   a 1  0  
   b    S  a 

S 22   a 2  0  
24
Scattering Parameters
b1  0   S 11 a 1  0   S 12 a 2  0 
b2  0   S 21 a 1  0   S 22 a 2  0 
S11 
S12 
S 21 
S 22 
b1  0 
a1  0 
Output is
matched
a2  0
b1  0 
a2 0
a1  0
b2  0 
a1  0 
a2  0
b2  0 
a2 0
Input is
matched
Output is
matched
Input is
matched
a1  0
input reflection coef.
w/ output matched
reverse transmission coef.
w/ input matched
forward transmission coef.
w/ output matched
output reflection coef.
w/ input matched
25
Scattering Parameters (cont.)
For a general multiport network:
S ij 
All ports except j are semi-infinite (or matched)
bi  0 
a j 0
ak  0 k  j
N-port network
Semi-infinite

aj
Port i
bi

Port j



26
Scattering Parameters (cont.)
Illustration of a three-port network
a1
b1
2
1
3
a2
b2
a3
b3
27
Scattering Parameters (cont.)
For reciprocal networks, the S-matrix is symmetric.
 S ij  S ji
i j
Note: If all lines entering the network have the same characteristic impedance, then
S ij
0
 
V j 0
V
Vi


k
0 k  j
28
Scattering Parameters (cont.)
Why are the wave functions (a and b) defined as they are?
a1
Z 01 ,  1
1
Z 02 ,  2
2
b1
z1
Pi

0 
a2
b2
z2
1


R e V i  0  I i

2
*
1 Vi
 0   
2

0
2
(assuming lossless lines)
Z 0i
Note:
a i  0   Vi

0
1
 Pi


2
0
Z 0i
ai  0 
2
29
Scattering Parameters (cont.)
Similarly,
Pi

0 
1 Vi

2
0
2

Z 0i
1
2
bi  0 
2
Also,
Vi
Vi
 Pi
Pi



  li   V i   0 
e

  li   V i   0 
e
  li 
  li 

1
2

1
2
a i   li 
2
bi   l i 
2
  i li
  i li

1
2

1
2
ai  0  e
2
bi  0  e
2
 2  i li
 2  i li
30
Example
Find the S parameters for a series impedance Z.
a1
Z0
b1
V1
Z



V

2
z1
Z0
a2
b2
z2
Note that two different coordinate systems are being used here!
31
Example (cont.)
Semi-infinite
a1
V1
Z0
b1


Z

V

2
Z in
Z0
b2
z2
z1
S11 Calculation:
S 11 

b1  0 
a1  0 
0
 
V1  0 
V1
a2  0
S 11 

Z
Z  2Z0

Z in  Z 0
Z in  Z 0

a2  0
Z
Z
 Z0   Z0
 Z0   Z0
By symmetry:
S 22  S 11
32
Example (cont.)
S21 Calculation:
Semi-infinite
a1
Z0
b1
V1


Z

V

2
Z in
b2  0 
a1  0 

a2  0  V2
a2  0
0

V1  0 
a


V1

b2
z2
z1
S 21 
Z0
V2
 0   a1  0 
2
0
 0   V2  0 
 Z0 
V 2  0   V1  0  

 Z  Z0 
V1  0   a 1 Z 0  1  S 11 
Z0

 V2
 0   V 2  0   a1
 Z0 
Z 0  1  S 11  

Z

Z
0 

33
Example (cont.)
Semi-infinite
a1
Z0
V1
b1


Z
S 21 
2
Z in
z1
a1  0 

V

Z0
b2
z2
 Z0 
Z 0  1  S 11  

Z

Z
0


a1  0 
Z0
 Z0  
  Z0   2Z  2Z0   Z0 
Z
  1  S 11  

1

 




Z

Z
Z

2
Z
Z

Z
Z

2
Z
Z

Z
0 
0 
0 
0 
0 



Hence
S 21 
2 Z0
Z  2 Z0
S 12  S 21
34
Example
Find the S parameters for a length L of transmission line.
L
Z0s ,  s
Z0
z1
Z0
z
z2
Note that three different coordinate systems are being used here!
35
Example (cont.)
L 
S11 Calculation:
L
L
V1  0 
Z0
+
Z0s ,  s
-
+
V  z
-
z
z1
S 11 
Z in
a2  0
b1
a1


a2  0
Z in
Z in
a2  0
a2  0
 Z0
 Z0
 Z 0  jZ 0 s tan  s L 
Z 0s
 Z 0 s  jZ 0 tan  s L 
+
-
V2  0 
z2
Z 0  Z 0s
Z 0  Z 0s
Z0
Semi-infinite
 S 22 ( by sym m etry )
 Z 0s
1  
1  
e
L
e
L
 j2s L
 j2s L


36
Example (cont.)
L
L
V1  0 
Z0
+
Z0s ,  s
-
+
V  z
-
z
z1
+
-
V2  0 
Z0
z2
Hence
Z 0s
S 11  S 22 
Z 0s
N o te : If
 Z 0  jZ 0 s tan  s L 
 Z0
Z

jZ
tan

L
 0s

0
s
 Z 0  jZ 0 s tan  s L 
 Z0
 Z 0 s  jZ 0 tan  s L 
Z 0s  Z 0 
Z in
a2  0
 Z0 
S 11  S 22  0
37
Example (cont.)
S21 Calculation:
L 
L
L
Z0
V1  0 
+
Z0s ,  s
-
0

V1  0 

b2
a1
-

a2  0
V2
V1  0   V1

V2  0 
Z 0  Z 0s
Z0
Semi-infinite
z2
z
z1
S 21 
+
+
V  z
-
Z 0  Z 0s
Z0
Z0
a2  0
 0   1  S 11 
Hence, for the denominator of the S21 equation we have
V1

0

V1  0 
1  S 11
We now try to put the
numerator of the S21
equation in terms of V1 (0).
38
Example (cont.)
L
L
V1  0 
Z0
+
Z0s ,  s
-

-
V2  0 
Z0
z2
z
z1
V2
+
+
V  z
-
 0   V 2  0   V  0   V   0  1   L 
Next, use
V z  V

 0  e  j
 V1  0   V   L   V
V

0 
s
1  
z

 0  e  j
 j s L
1   L e
s
 j2s z
L

1  
e
L
 j2s L

Hence, we have
V1  0 
e
e
L
 j2s L

V

2
0 
V1  0 
e
 j s L
1  
L
e
 j2s L

1   L 
39
Example (cont.)
L
L
V1  0 
Z0
+
-
Z0s ,  s
+
V  z
-
z
z1
+
-
V2  0 
Z0
z2

V2
0 
V1  0 
e
Therefore, we have
V1
0
 
V1  0 

S 21
 j s L
V2

1   L e
0

 j2s L

1   L 
V1  0 
1  S 11
1  S 11  1   L  e  j  L
s

a2  0
1   Le
 j2sL
so
1  S 11  1   L  e  j  L
s
S 21 
1   Le
 j2sL
 S 12 by sym m etry
40
Example (cont.)
L
Special cases:
a)
Z0s ,  s
Z 0 s  Z 0  S 11  S 22  0,
S 21  S 12  e
b) L 
g
2
 Z in
e
 j s L
 j s L
 s L 
a2  0
 Z0
 1

L  0
2  g
g 2
 0
 S     j s L
e
e
 j s L
0




 S 11  S 22  0
0
S   
 1
 1

0 
S 21   1
41
Example
Find the S parameters for a step-impedance discontinuity.
S 11 
S 22 
Z 02  Z 01
Z 02  Z 01
Z 01  Z 02
Z 02  Z 01
Z 01
Z 02
  S 11
V2
S 21 
b2  0 
a1  0 

a2  0

0
Z 02
V1

0
Z 01
a2  0
42
Example (cont.)
Semi-infinite
S21 Calculation:
+
V1
Because of continuity of the voltage
across the junction, we have:

V2
0 a
2 0
 V2  0 
V2
S 21  

a2  0
0
V1
Z 02
V1

 V1  0 

0
Z 01

 V1
a2  0
 0  1 

1  S 11  1 
Z 02
V1

+
-
Z 02
V2
 0  1  S 11 
S11 
0
Z 01
a2  0
-
Z 01

a2  0
Z 02  Z 01
Z 02  Z 01
2 Z 02
Z 02  Z 01
so
S 21  1  S 11 
Z 01
Z 02
Hence
S 21  S 12  2
Z 01 Z 02
Z 01  Z 02
43
Properties of the S Matrix
For reciprocal networks, the S-matrix is symmetric.
 S   S 
T
N o te :
 A  B   U 
If
For lossless networks, the S-matrix is unitary.
 S  S 
T
Equivalently,
*
  S   S   U 
*
T
th e n
 B  A   U 
Identity matrix
Notation:  S    S    S 
†
S 
T*
 S 
1
so
H
S 
†
 S 
1
N-port network
N
T a k e ( i , j ) e le m e n t 

k 1
N
S
T
ik
S
*
kj


k 1
S ki S kj   ij
*
1 ; i  j
 ij  
0 ; i  j
44
T*
Properties of the S Matrix (cont.)
Example:
 S 11
 S    S 21
 S 31
U n ita ry 
S 12
S 22
S 32
S 13 

S 23

S 33 
S 11 S 11  S 21 S 21  S 31 S 31  1
*
S 12 S
*
12
*
 S 22 S
*
22
*
 S 32 S
*
32
1
S 13 S 13  S 23 S 23  S 33 S 33  1
*
S 11 S
*
12
*
 S 21 S
*
22
*
 S 31 S
*
32
 0
S 11 S 13  S 21 S 23  S 31 S 33  0
*
*
The column vectors form
an orthogonal set.
*
The rows also form
orthogonal sets (see the
note on the previous slide).
S 12 S 13  S 22 S 23  S 32 S 33  0
*
*
*
45
Example

 0

j
S    
 50  
2

j
 2
j
2
0
0
j

2


0


0 

1
S50
2
3
Not unitary  Not lossless
(For example, column 2 doted with the conjugate of column three is not zero.
1) Find the input impedance looking into port 1 when ports 2 and 3 are terminated in
2) Find the input impedance looking into port 1 when port 2 is terminated in a 75 []
46
Example (cont.)
1 If ports 2 and 3 are terminated in 50 [Ω]: (a2 = a3 = 0)
b1  S 11 a1  S 12 a 2  S 13 a 3
  in 1 
in1
a1
b1
b1
a1
 S 11  0

2
1
S50
3
Z in 1  50 [  ]
a2
b2
a3
b3
2  0
3  0
47
Example (cont.)
2) If port 2 is terminated in 75 [Ω] and port 3 in 50 [Ω]:
 2 
in1
a1
b1
a2
b2

75  50
75  50
2
1
S50
3

1
5
a2
b2
a3
b3
2 
1
5
3  0
48
Example (cont.)

 0
 b1  
  j
b 
 2  2
 b3  
j
 2
j
2
0
0
  in 1 
j

2


0


0 

b1
a1
 a1 
 
a
 2
 a 3 
in1
 S 11  S 12
a2
a1
 S 13
a1
b1
a3
2
1
S50
3
a2
b2
a3
b3
2 
1
5
3  0
a 
a 
b 2 / a1  S 21  S 22  2   S 23  3 
 a1 
 a1 
a1

b 
1
  j  1   j 
 S 12   2 2   S 12   2 S 21   


 

a
5
10
2
2







1 
a 2   2 b2
 Z in 1
 1   in
1
 50 
1 
in 1


  44.55 [  ]


49
Transfer (T) Matrix
1
A
1
a
b1A
2
A
a
b2A
T12   b2 
 
T 22   a 2 
 b2 
 T   
a2 
1
2
b
B
a1B
T Matrix:
 a 1   T11
  
 b1   T 21
B
1
A
2
T 
 1
S
21
 
 S 11
S
 21
b2B
a2B
 S 22


S 21

S S 
S 12  11 22
S 21 
  T 21
 T
22
S   
2

T12
 T11 
T 22

1 
T 22 

T12 

T 22 
(Derivation omitted)
50
Transfer (T) Matrix (cont.)
 a1 
 b2 
A
  A    T   A 
 b1 
a2 
A
B ut
A
The T matrix of a
networks is the
product of the T
matrices.
b 
a 
 A   B 
a2 
 b1 
A
2
B
1
 a1 
 a1 
A
  A    T   B 
 b1 
 b1 
A
B
 a1 
 b2 
A
B
 A    T   T   B 
 b1 
a2 
A
Hence
B
T

AB


51
Conversion Between Parameters
52
Example
Derive Sij from the Z parameters.
(The result is given inside row 1, column 2, of the previous table.)
S11 Calculation:
1
Z 22  Z 21
Z11  Z 21
S11
Z0
2
Z0
Z 21
Semi infinite
S 1 1   in 1 
Z in  Z 0
Z in  Z 0
Z in 
 Z 1 1  Z 2 1    Z 2 1  ||   Z 2 2
 Z 2 1   Z 0 
53
Example (cont.)
1
Z 22  Z 21
Z11  Z 21
S11
Z0
Z 21
2
Z0
Semi infinite
Z in   Z 1 1  Z 2 1    Z 2 1  ||   Z 2 2  Z 2 1   Z 0 
  Z 11  Z 21  

Z 21  Z 22  Z 0  Z 21 
 Z 11  Z 21   Z 22
Z 22  Z 0
 Z 0   Z 21  Z 22  Z 0  Z 21 
Z 22  Z 0
Z 11 Z 22  Z 11 Z 0  Z 21 Z 22  Z 21 Z 0  Z 21 Z 22  Z 21 Z 0  Z 21
2

Z 22  Z 0
Z 11 Z 22  Z 11 Z 0  Z 21
2

Z 22  Z 0
54
Example (cont.)
1
Z 22  Z 21
Z11  Z 21
S11
Z0
2
Z0
Z 21
Semi infinite
Z in 
Z 11  Z 0  Z 22
S11 
so
S11 
  Z 221
Z 22  Z 0
Z in  Z 0
Z in  Z 0
Z 11  Z 0  Z 22
  Z 221  Z 0  Z 0
2
Z 22   Z 21  Z 0  Z 0
 Z 22

Z 22 
Z 11  Z 0 

55
Example (cont.)
1
Z 22  Z 21
Z11  Z 21
S11
Z0
2
Z0
Z 21
Semi infinite
S11 
Z 11  Z 0  Z 22
  Z 221  Z 0  Z 0
2
Z 22   Z 21  Z 0  Z 0
 Z 22
Z 11  Z 0 

Z 11 Z 0  Z 11 Z 22  Z 21  Z 0  Z 0 Z 22
2



Z 22 
2
Z 11 Z 0  Z 11 Z 22  Z 21  Z 0  Z 0 Z 22
2
Z0
Z0
2
  Z 1 1  Z 0   Z 221
2
Z 22   Z 11  Z 0   Z 21
 Z 22

56
Example (cont.)
1
Z 22  Z 21
Z11  Z 21
S11
Z0
2
Z0
Z 21
Semi infinite
S11 
Z0
Z0
  Z 11 
Z 22   Z 11 
 Z 22

  Z 221
2
Z 0   Z 21
Z0
Note: to get S22, simply let Z11  Z22 in the previous result.
S 22 
Z0
Z0
 Z 11   Z 22  Z 0
 Z 11   Z 22 
  Z 221
2
Z 0   Z 21
57
Example (cont.)
S21 Calculation:
S11
V1  0 1
Z11  Z 21
Z0
Vc
2
Z 22  Z 21
V2  0 
Z0
Z 21
Semi infinite
Assume V1   0   1 [ V ]
V1  0   1  S 1 1

S 21  V 2
0
 V2  0 
Use voltage divider equation twice:

 Z 2 1  ||   Z 2 2  Z 2 1   Z 0 
V c  V1  0  
  Z  Z    Z  ||   Z  Z   Z 
11
21
21
22
21
0 



Z0
V2  0   Vc 
 Z  Z   Z
22
21
0








58
Example (cont.)
V1  0 1
S11
Z0
Z11  Z 21
Vc
Z 22  Z 21
2
V2  0 
Z0
Z 21
Semi infinite
Hence
S 21

 Z 21  ||   Z 22  Z 21   Z 0 
  1  S 11  
  Z  Z    Z  ||   Z  Z   Z 
11
21
21
22
21
0 



Z0

 Z  Z   Z
22
21
0




After simplifying, we should get the result in the table.
(You are welcome to check it!)
59
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