Notes 14 - Network analysis
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Transcript 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
Y (admittance) 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
EJ
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:
Please see the Pozar
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
Admittance (Y) Parameters
Consider a 2-port network:
I2
I1
V1
+
1
-
+
2
-
I 1 Y11V1 Y12V 2
Admittance
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
Admittance (Y) Parameters
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
Admittance (Y) Parameters
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
Cascaded Networks
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
easy to use with cascaded
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
j2s L
j2s 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
j2s z
L
1
e
L
j2s L
Hence, we have
V1 0
e
e
L
j2s L
V
2
0
V1 0
e
j s L
1
L
e
j2s 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
j2s L
1 L
V1 0
1 S 11
1 S 11 1 L e j L
s
a2 0
1 Le
j2sL
so
1 S 11 1 L e j L
s
S 21
1 Le
j2sL
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
50 [] loads.
2) Find the input impedance looking into port 1 when port 2 is terminated in a 75 []
load and port 3 is terminated in a 50 [] load.
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
For cascaded 2-port networks:
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
cascaded set of
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