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Chapter 4. Microwave Network Analysis
It is much easier to apply the simple and intuitive idea
of circuit analysis to a microwave problem than it is
to solve Maxwells equations for the same problem. Maxwells equations for a given problem is complete,
it gives the E & H fields at all points in space.
Usually we are interested in only the V & I at a set of
terminals, the power flow through a device, or some
other type of global quantity.
A field analysis using Maxwells equations forproblems would be hopelessly difficult.
2
4.1 Impedance and Equivalent Voltages andCurrentsEquivalent Voltages and Currents
The voltage of the + conductor relative to theconductor
After having defined and determined a voltage,current, and characteristic impedance, we can proceedto apply the circuit theory for transmission lines to
characterize this line as a circuit element.
0
C
V E dl
I H dl
VZ
I
3
Figure 4.1 (p. 163)Electric and magnetic field lines for an arbitrary two-conductor
TEM line.
4
Figure 4.2 (p. 163)Electric field lines for the TE10mode of a rectangular waveguide.
10
( , , ) sin ( , )
( , , ) sin ( , )
j z j z
y y
j z j z
x x
TE
j a xE x y z A e Ae x y e
a
j a xH x y z A e Ah x y e
a
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The Concept of Impedance
Various types of impedance
Intrinsic impedance ( ) of the medium: depends
on the material parameters of the medium, and is equal to
the wave impedance for plane waves.
Wave impedance ( ): a characteristic of
the particular type of wave. TEM, TM and TE waves each
have different wave impedances which may depend on the
type of the line or guide, the material, and the operating
frequency.
Characteristic impedance ( ): the ratio ofV/I for a traveling wave on a transmission line. Z0for TEM
wave is unique. TE and TM waves are not unique.
/
/ 1/w t t wZ E H Y
0 01/ /Z Y L C
10
Geometry of a partially filled waveguide
Geometry of a partially filled waveguide and itstransmission line equivalent.
Reflection coefficient
11
An arbitrary one-port network.
The complex power delivered to this network is:
wherePlis real and represents the average power dissipated by the
network
12 ( )
2 l m e
SP E H ds P j W W
12
If we define real transverse modal fields, eand h,
over the terminal plane of the network such that
with a normalization
The input impedance is
If the network is lossless, thenPl= 0 andR= 0. Then
Zinis purely imaginary, with a reactance
( , , ) ( ) ( , )
( , , ) ( ) ( , )
j z
t
j z
t
E x y z V z e x y e
H x y z I z h x y e
, ,
1S
e h ds
e1 1
2 2SP VI e h ds VI
e
2 2 21 12 2
2 ( )l m ein
P j W WV VI P Z R jX
I I I I
2
4 ( )m eW WXI
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Even and Odd Properties of Z() and ()
Consider the driving point impedance, Z(), at the
input port of an electrical network.V() = I()
Z().
Since v(t) must be real v(t) = v*(t),
Re{V()} is even in , Im{V()} is odd in . I()
holds the same as V().
1( ) ( )2
j tv t V e d
( ) ( ) ( )
( ) ( )
j t j t j tV e d V e d V e d
V V
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )V Z I Z I V Z I
14
The reflection coefficient at the input port
0 0
0 0
0 0
0 0
2 2
( ) ( ) ( )( )
( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
Z Z R Z jX
Z Z Z Z jX
R Z jX R Z jX
Z Z jX Z Z jX
15
Impedance and Admittance Matrices
At the nthterminal plane, the total voltage and current
is whenz= 0.
The impedance matrix
Similarly,
where
,n n n n n nV V V I I I
V Z I
I Y V
11 12 1
121
1
N
N NN
Y Y Y
YY Z
Y Y
16
An arbitrary N-port
microwave network.
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A matched 3B attenuator with a 50 Characteristic impedance
Evaluation of Scattering Parameters
26
Show how [S][Z] or [Y]. AssumeZ0nare all
identical, for convenienceZ0n= 1.
where
Therefore,
For a one-port network,
,n n n n n n n nV V V I I I V V
[ ][ ] [ ][ ] [ ][ ] [ ] [ ] [ ]
([ ] [ ])[ ] ([ ] [ ])[ ]
Z I Z V Z V V V V
Z U V Z U V
1 0 0
0 1[ ]
0 1
U
1 1[ ] [ ][ ] ([ ] [ ]) ([ ] [ ])S V V Z U Z U
1111
11
1
1
zS
z
27
To find [Z],
Reciprocal Networks and Lossless Networks
As in Sec. 4.2, the [Z] and [Y] are symmetric forreciprocal networks, and purely imaginary for
lossless networks.
From
1
[ ][ ] [ ][ ] [ ] [ ]
[ ] ([ ] [ ])([ ] [ ])
Z S U S Z U
Z U S U S
1
2
1[ ] ([ ] [ ])[ ]
2
n n nV V I
V Z U I
1
2
1[ ] ([ ] [ ])[ ]
2
n n nV V I
V Z U I
1
1
1
[ ] ([ ] [ ])([ ] [ ]) [ ]
[ ] ([ ] [ ])([ ] [ ])
[ ] ([ ] [ ]) ([ ] [ ])t
t t
V Z U Z U V
S Z U Z U
S Z U Z U
28
If the network is reciprocal, [Z]t= [Z].
If the network is lossless, no real power delivers to
the network.
1[ ] ([ ] [ ]) ([ ] [ ])
[ ] [ ]
t
t
S Z U Z U
S S
1 1Re{[ ] [ ] } Re{([ ] [ ] )([ ] [ ] )}2 2
1Re{([ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] )}
2
1 1[ ] [ ] [ ] [ ] 0
2 2
t t t
av
t t t t
t t
P V I V V V V
V V V V V V V V
V V V V
[ ] [ ] [ ] [ ]([ ][ ]) ([ ][ ])
[ ] [ ] [ ] [ ]
t t
t
t t
V V V V S V S V
V S S V
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Generalized Scattering Parameters
Figure 4.10 (p. 181)AnN-port network with different characteristic impedances.
34
0 0/ , /n n n n n na V Z b V Z
0
0
0
( )
1( )
n n n n n n
n n n n n n
n
V V V Z a b
I V V Z a bZ
2 2
2 2
1 1Re Re2 2
1 1
2 2
n n n n n n n n n
n n
P V I a b b a b a
a b
35
The generalized scattering matrix can be used to
relate the incident and reflected waves,
b S a
0 fork
iij
j a k j
bS
a
_
0 fork
iij
j V k j
VS
V
36
Figure on page 183
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4.4 The Transmission (ABCD) Matrix
The ABCD matrix of the cascade connection of 2 or
more 2-port networks can be easily found by
multiplying the ABCD matrices of the individual 2-
ports.
38
1 2 2
1 2 2
V AV BI
I CV DI
1 2
1 2
V VA B
I C D I
1 21 1
1 11 2
V VA B
C DI I
32 2 2
2 22 3
VV A B
C DI I
31 1 1 2 2
1 1 2 21 3
VV A B A B
C D C DI I
Ex. 4.6 Evaluation of ABCD Parameters
39
Relation to Impedance Matrix
From the Z parameters with -I2,
1 1 11 2 12
1 1 21 2 22
V I Z I Z
I I Z I Z
2
2 2 2
2
2
1 1 1111 21
2 1 210
1 1 11 2 12 1 1 22 11 22 12 2111 12 11 12
2 2 2 1 21 210 0 0
1 121
2 1 210
1 2 2222 21
2 20
/
1/
/
I
V V V
I
V
V I ZA Z Z
V I Z
V I Z I Z I I Z Z Z Z Z B Z Z Z Z
I I I I Z Z
I IC ZV I Z
I I ZD Z Z
I I
40
If the network is reciprocal,Z12=Z21, andAD-BC=1.
Equivalent Circuits for 2-port Networks
Table 4-2
A transition between a coaxial line and a microstrip
line. Because of the physical discontinuity in thetransition from a coaxial line to a microstrip line,
electric and/or magnetic energy can be stored in the
vicinity of the junction, leading to reactive effects.
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Figure 4.12 (p.
188)A coax-to-microstrip
transition and equivalent
circuit representations.
(a) Geometry of the
transition. (b)
Representation of the
transition by a black
box.
(c) A possible equivalentcircuit for the transition
[6].
42
Figure 4.13 (p. 188)Equivalent circuits for a reciprocal two-port network. (a) T equivalent.
(b) equivalent.
43
4.5 Signal Flow Graphs
Very useful for the features and the construction of
the flow transmitted and reflected waves.
Nodes: Each port, i, of a microwave network has 2nodes, aiand bi. Node aiis identified with a wave
entering port i, while node biis identified with a wave
reflected from port i. The voltage at a node is equal to
the sum of all signals entering that node.
Branches: A branch is directed path between 2 nodes,
representing signal flow from one node to another.Every branch has an associated Sparameter or
reflection coefficient.
44
Figure 4.14 (p. 189)The signal flow graph representation of a two-port network. (a)
Definition of incident and reflected waves. (b) Signal flow graph.
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Figure 4.18 (p. 192)Signal flow path for the two-port network with general source and
load impedances of Figure 4.17.
50
Figure 4.19 (p. 192)
Decompositions of the flow graph of Figure 4.18 to find in=b1/a1and out= b2/a2. (a) Using Rule 4 on node a2. (b) Using
Rule 3 for the self-loop at node b2. (c) Using Rule 4 on node b1. (d)
Using Rule 3 for the self-loop at node a1.
51
Figure 4.20 (p. 193)Block diagram of a network analyzer measurement of a two-port
device.
52
Figure 4.21a (p. 194)Block diagram and signal flow graph for the Thruconnection.
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Figure 4.21b (p. 194)Block diagram and signal flow graph for theReflectconnection.
54
Figure 4.21c (p. 194)Block diagram and signal flow graph for theLineconnection.
55
Figure 4.22 (p. 198)Rectangular waveguide
discontinuities.
56
Some common microstripdiscontinuities. (a) Open-
ended microstrip. (b) Gap
in microstrip. (c) Change in
width.
(d) T-junction. (e) Coax-to-
microstrip junction.
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Figure 4.24 (p. 200)Geometry of anH-plane step (change in width) in rectangular
waveguide.
58
Figure 4.25 (p. 203)Equivalent inductance of an H-plane asymmetric step.
59
Figure on page 204Reference: T.C. Edwards, Foundations for Microwave Circuit Design, Wiley, 1981.
60
Figure 4.26 (p. 205)An infinitely long rectangular waveguide with surface current
densities atz= 0.
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Figure 4.27 (p. 206)An arbitrary electric or magnetic current source in an infinitely
long waveguide.
62
Figure 4.28 (p. 208)A uniform current probe in a rectangular waveguide.
63
Figure 4.29 (p. 210)
Various waveguide and other transmission line configurations usingaperture coupling. (a) Coupling between two waveguides wit an
aperture in the common broad wall. (b) Coupling to a waveguide
cavity via an aperture in a transverse wall. (c) Coupling between
two microstrip lines via an aperture in the common ground plane. (d)
Coupling from a waveguide to a stripline via an aperture.
64
Figure 4.30 (p. 210)Illustrating the development of equivalent electric and magnetic
polarization currents at an aperture in a conducting wall (a) Normal
electric field at a conducting wall. (b) Electric field lines around an
aperture in a conducting wall. (c) Electric field lines around electricpolarization currents normal to a conducting wall. (d) Magnetic field
lines near a conducting wall. (e) Magnetic field lines near an
aperture in a conducting wall. (f)Magnetic field lines near magnetic
image theory to the
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image theory to the
problem of an aperture in
the transverse wall of a
waveguide. (a) Geometry
of a circular aperture in
the transverse wall of a
waveguide. (b) Fieldswith aperture closed. (c)
Fields with aperture open.
(d) Fields with aperture
closed and replaced with
equivalent dipoles.
(e) Fields radiated by
equivalent dipoles forx< 0; wall removed by
image theory.
(f) Fields radiated by
equivalent dipoles forz>
0; all removed by image
66
Figure 4.32 (p. 214)Equivalent circuit of the aperture in a transverse waveguide wall.
67
Figure 4.33 (p. 214)Two parallel waveguides coupled through an aperture in a
common broad wall.