Transcript Document

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 Maxwell’s equations for the same problem.
• Maxwell’s 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 Maxwell’s equations for
problems would be hopelessly difficult.
1
4.1 Impedance and Equivalent Voltages and
Currents
Equivalent Voltages and Currents
• The voltage of the + conductor relative to the –

conductor
V   E  dl

I

C
H  dl
V
Z0 
I
• After having defined and determined a voltage,
current, and characteristic impedance, we can proceed
to apply the circuit theory for transmission lines to
characterize this line as a circuit element.
2
Figure 4.1 (p. 163)
Electric and magnetic field lines for an arbitrary two-conductor
TEM line.
3
TE10
E y ( x, y , z ) 
j a
A sin
x
e  j  z  Ae y ( x, y )e  j  z

a
j a
 x  j z
H x ( x, y , z ) 
A sin
e
 Ahx ( x, y )e  j  z

a
Figure 4.2 (p. 163)
Electric field lines for the TE10 mode of a rectangular waveguide.
4
V
 j a

A sin
x
a
e  j  z  dy
y
• There is no “correct” voltage in the sense of being
unique. V ( x  a / 2,0  y  b)  V ( x  0,0  y  b)
• There are many ways to define equivalent voltage,
current, and impedance for waveguides.
– V&I are defined only for a particular waveguide mode.
– The equivalent V&I should be defined so that their product
gives the power flow of the mode.
– V/I for a single traveling wave should be equal to Z0 of the
line. This impedance may be chosen arbitrarily, but is
usually selected as equal to the wave impedance of the line.
5
• For an arbitrarily waveguide mode, the transverse
fields
  j z
Et ( x, y, z )  e ( x, y )( A e
  j z
H t ( x, y, z )  h ( x, y )( A e

A e

j z
A e
e ( x, y )   j  z
)
(V e
 V e j  z )
C1
j z
h ( x, y )   j  z
)
(I e
 V e j z )
C2
where e and h are the transverse field variations of the
mode. Since Et & Ht are related by Zw,
zˆ  e ( x, y)
h ( x, y) 
Zw
• Defining equivalent voltage and current waves as
V ( z )  V  e j  z  V  e j  z ,
  j z
I ( z)  I e

I e
j z
V V
with Z 0    
I
I
6
• The complex power flow for the incident wave
1  2
P  |A |
2

V  I 
ˆ 
S e  h  zds
2C1C2


ˆ
e

h
 zds

S
• Since we want this power to be (1/2)V+I+*,
ˆ
C1C2   e  h   zds
S
where the surface integration is over the cross section
of the waveguide.
V  V  C1
Z0     
I
I
C2
• If it is desired to have Z0 = Zw,
C1
 Z w ( ZTE or ZTM )
C2
7
• For higher order modes,
Vn  j n z Vn j n z
Et ( x, y, z )   (
e

e )en ( x, y )
C1n
n 1 C1n
N
I n  j n z I n j n z
H t ( x, y , z )   (
e

e )hn ( x, y )
C2 n
n 1 C2 n
N
• Ex 4.1
8
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 ( Zw  Et / Ht  1/ Yw ): 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 ( Z0  1/ Y0  L / C ): the ratio of
V/I for a traveling wave on a transmission line. Z0 for TEM
wave is unique. TE and TM waves are not unique.
9
Ex 4.2
Figure 4.3 (p. 167)
Geometry of a partially filled waveguide and its
transmission line equivalent for Example 4.2.
10
Figure 4.4 (p. 168)
An arbitrary one-port network.
11
• The complex power delivered to this network is:
1
P
2

S
E  H   ds  Pl  2 j (Wm  We )
where Pl is real and represents the average power
dissipated by the network.
• If we define real transverse modal fields, e and h,
over the terminal plane of the network such that
Et ( x, y, z )  V ( z )e ( x, y )e  j  z
H t ( x, y , z )  I ( z ) h ( x, y )e  j  z
with a normalization S
P
e  h   ds  1
1
1 


VI
e

h

ds

VI

S
2
2
12
• The input impedance is
Pl  2 j (Wm  We )
V VI 
P
Z in  R  jX   2 

2
2
1
1
I
I
I
I
2
2
• If the network is lossless, then Pl = 0 and R = 0. Then
Zin is purely imaginary, with a reactance
X
4(Wm  We )
I
2
13
Even and Odd Properties of Z(ω) and Γ(ω)
• Consider the driving point impedance, Z(ω), at the
input port of an electrical network.  V(ω) = I(ω)
Z(ω). v(t )  1  V ( )e jt d
2


• Since v(t) must be real v(t) = v*(t),




V ( )e d   V ( )e
jt


 jt

d   V  ( )e jt d

V ( )  V  ( )
 Re{V(ω)} is even in ω, Im{V(ω)} is odd in ω. I(ω)
holds the same as V(ω).
V  ()  Z  () I  ()  Z  () I ()  V ()  Z () I ()
14
• The reflection coefficient at the input port
Z ( )  Z 0 R( )  Z 0  jX ( )
( ) 

Z ( )  Z 0 Z ( )  Z 0  jX ( )
( ) 
R( )  Z 0  jX ( ) R( )  Z 0  jX ( )

  ( )
Z ( )  Z 0  jX ( ) Z ( )  Z 0  jX ( )
( )  ( ) ( )  ( )( )  ( )
2
2
15
4.2 Impedance and Admittance Matrices
• At the nth terminal plane, the total voltage and current
is Vn  Vn  Vn , In  In  In as seen from (4.8) when
z = 0.
• The impedance matrix
V    Z  I 
• Similarly,  I   Y V 
where
 Y11 Y12
Y
Y    21

YN 1
Y1N 

   Z 1


YNN 
16
Figure 4.5 (p.
169)
An arbitrary N-port
microwave network.
17
• Zij can be defined as
Vi
Zij 
Ij
I k  0 for k  j
In words, Zij can be found by driving port j with the
current Ij, open-circuiting all other ports (so Ik=0 for
k≠j), and measuring the open-circuit voltage at port i.
• Zii: input impedance seen looking into port i when all
other ports are open.
• Zij: transfer impedance between ports i and j when all
other ports are open.
• Similarly,
Ii
Yij 
Vj
Vk 0 for k  j
18
Reciprocal Networks
• Let Fig. 4.5 to be reciprocal (no active device, ferrites,
or plasmas), with short circuits placed at all terminal
planes except those of ports 1 and 2.
• Let Ea, Ha and Eb, Hb be the fields anywhere in the
network due to 2 independent sources, a and b,
located somewhere in the network.
• From the reciprocity theorem,

S
Ea  H b  ds 

S
Eb  H a  ds
19
• The fields due to sources a and b at the terminal
planes t1 and t2: E1a  V1a e1 , H1a  I1a h1
(4.31)
E1b  V1b e1 ,
H1b  I1b h1
E2 a  V2 a e2 ,
H 2 a  I 2 a h2
E2b  V2b e2 ,
H 2b  I 2b h2
where e1, h1 and e2, h2 are the transverse modal fields
of port 1 and 2. Therefore,
V1a I1b  V1b I1a  S e1  h1  ds  V2a I 2b  V2b I 2a  S e2  h2  ds  0
where S1, S2 are the cross-sectional areas at the
terminal planes of ports 1 and 2.
• Comparing (4.31) to (4.6), C1 = C2 = 1 for each port,
so that
from (4.10).
e  h  ds  e  h  ds
1

S1
1
2
1

S2
2
2
20
• This leads to V1a I1b V1b I1a   V2a I2b V2b I2a   0
• For 2 port, I1  Y11V1  Y12V2 , I 2  Y21V1  Y22V2
V1a Y11V1b  Y12V2b   V1b Y11V1a  Y12V2 a  
V2 a Y21V1b  Y22V2b   V2b Y21V1a  Y22V2 a   0
V1aV2b V1bV2a Y12  Y21   0
Y12  Y21
Generally, Yij  Yji
21
Lossless Networks
• Consider a reciprocal lossless N-port network.
• If the network is lossless, Re{Pav} = 0.
1
1
1 t
t
t


Pav  [V ] [ I ]  [ Z ][ I ] [ I ]  [ I ] [ Z ][ I ]
2
2
2
1
 ( I1Z11 I1  I1Z12 I 2  I 2 Z 21 I1  )
2
1 N N
  I m Z mn I n
2 n 1 m 1
• Since the Ins are independent, only nth current is taken.


Re I n Z nn I n  I n Re Z nn   0
2
ReZnn   0
22
• Take Im and In only  Re  I n I m  I m I n  Z mn   0
• Since (InIm*+ ImIn*) is purely real, Re{Zmn} = 0,.
• Therefore, Re{Zmn} = 0 for any m, n.
Ex 4.3
23
4.3 The Scattering Matrix
• The scattering matrix relates the voltage waves
incident on the ports to those reflected from the ports.
• The scattering parameters can be calculated using
network analysis technique. Otherwise, they can be
measured directly with a vector network analyzer.
• Once the scattering matrix is known, conversion to
other matrices can be performed.
• Consider the N-port network in Fig. 4.5.
24
V1   S11
  
V2    S 21
  
  
VN   S N 1
or
S12
S1N  V1 
 
 V2 
 
 
S NN  VN 
V     S  V  
Vi 
Sij  
Vj
Vk 0 for k  j
• Sii the reflection coefficient seen looking into port i
when all other ports are terminated in matched loads,
• Sij the transmission coefficient from port j to port i
when all other ports are terminated in matched loads.
25
Figure 4.7 (p. 175)
A photograph of the HewlettPackard HP8510B Network
Analyzer. This test instrument is
used to measure the scattering
parameters (magnitude and
phase) of a one- or two-port
microwave network from 0.05
GHz to 26.5 GHz. Built-in
microprocessors provide error
correction, a high degree of
accuracy, and a wide choice of
display formats. This analyzer
can also perform a fast Fourier
transform of the frequency
domain data to provide a time
domain response of the network
under test.
Courtesy of Agilent
Technologies.
26
Ex.4, Evaluation of Scattering Parameters
Figure 4.8 (p. 176)
A matched 3B attenuator with a 50 Ω Characteristic impedance
27
• Show how [S]  [Z] or [Y]. Assume Z0n are all
identical, for convenience Z0n = 1.
Vn  Vn  Vn , I n  I n  I n  Vn Vn
[ Z ][ I ]  [ Z ][V  ]  [ Z ][V  ]  [V ]  [V  ]  [V  ]
([ Z ]  [U ])[V  ]  ([ Z ]  [U ])[V  ]
where
1 0
0 1
[U ]  


0
0




1
Therefore, [S ]  [V  ][V  ]1  ([Z ]  [U ])1 ([Z ]  [U ])
• For a one-port network,
z11  1
S11 
z11  1
28
• To find [Z], [Z ][S ]  [U ][S ]  [Z ]  [U ]
[ Z ]  ([U ]  [ S ])([U ]  [ S ]) 1
Reciprocal Networks and Lossless Networks
• As in Sec. 4.2, the [Z] and [Y] are symmetric for
reciprocal networks, and purely imaginary for
lossless networks.
1

• From V   1 V  I 
V  V  I 
n
n
2
n
n
1
[V ]  ([ Z ]  [U ])[ I ]
2

2
n
n
1
[V ]  ([ Z ]  [U ])[ I ]
2

[V  ]  ([ Z ]  [U ])([ Z ]  [U ]) 1[V  ]
[ S ]  ([ Z ]  [U ])([ Z ]  [U ]) 1

[ S ]  ([ Z ]  [U ])
t
1
 ([Z ]  [U ])
t
t
29
• If the network is reciprocal, [Z]t = [Z].
[ S ]t  ([ Z ]  [U ]) 1 ([ Z ]  [U ])
[ S ]  [ S ]t
• If the network is lossless, no real power delivers to
the network.
1
1
Re{[V ]t [ I ]}  Re{([V  ]t  [V  ]t )([V  ]  [V  ] )}
2
2
1
 Re{([V  ]t [V  ]  [V  ]t [V  ]  [V  ]t [V  ]  [V  ]t [V  ] )}
2
1  t   1  t  
 [V ] [V ]  [V ] [V ]  0
2
2
Pav 
[V  ]t [V  ]  [V  ]t [V  ]
 ([ S ][V  ])t ([ S ][V  ])
 [V  ]t [ S ]t [ S ][V  ]
30
• For nonzero [V+], [S]t[S]*=[U], or [S]*={[S]t}-1. 
Unitary matrix  N

S
S
 ki kj  ij , for all i, j
k 1
• If i = j,
• If i ≠ j,
where  ij  1 if i  j,  ij  0 if i  j .
N

S
S
 ki ki  1
k 1
N

S
S
 ki kj  0.
k 1
• Ex 4.5 Application of Scattering Parameters
• The S parameters of a network are properties only of
the network itself (assuming the network in linear),
and are defined under the condition that all ports are
matched.
31
A Shift in Reference Planes
Figure 4.9 (p. 181)
Shifting reference planes for an N-port network.
32
• [S]: the scattering matrix at zn = 0 plane.
• [S']: the scattering matrix at zn = ln plane.
[V  ]  [ S ][V  ],
[V  ]  [ S ][V  ]

 jn

  jn


Vn  Vn e ,Vn  Vn e
e j1


 0

e  j1
0 






 V    S  
 0
e j N 

e  j1
e  j1
0 



V    
 S  
 j N 
 0
 0
e



0 




V
  
e  j N 
0 




V
  
 j N 
e

33
e j1

 S   
 0

e j1
0 


 S  
 0
e j N 

0 


e j N 
34
Generalized Scattering Parameters
Figure 4.10 (p. 181)
An N-port network with different characteristic impedances.
35
an  Vn / Z0n , bn  Vn / Z0n
Vn  Vn  Vn  Z 0 n (an  bn )


1
In 
Vn  Vn  Z 0 n (an  bn )
Z0n




1
1
2
2

Pn  Re Vn I n  Re an  bn  bn an  bnan
2
2
1
1 2
2
 an  bn
2
2

36
• The generalized scattering matrix can be used to
relate the incident and reflected waves,
b   S a
bi
Sij 
aj
Vi _
Sij  
Vj
ak  0 for k  j
Vk 0 for k  j
37
Figure on page 183
38
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 2ports.
39
V1  AV2  BI 2
I1  CV2  DI 2
V1   A1
 I   C
 1  1
B1  V2 
D1   I 2 
V2   A2
 I   C
 2  2
B2  V3 



D2   I 3 
V1   A
 I   C
 1 
V1   A1
 I   C
 1  1
B  V2 
D   I 2 
B1   A2


D1  C2
B2  V3 



D2   I 3 
Ex. 4.6 Evaluation of ABCD Parameters
40
Relation to Impedance Matrix
• From the Z parameters with -I2 ,
V1  I1Z11  I 2 Z12
I1  I1Z 21  I 2 Z 22
A
B
C
D
V1
V2
V1
I2
I1
V2
I1
I2

I1Z11
 Z11 / Z 21
I1Z 21

I1Z11  I 2 Z12
I2
I 2 0
V2  0
V2  0

I1
 1/ Z 21
I1Z 21

I 2 Z 22
 Z 22 / Z 21
I2
I 2 0
V2  0
 Z11
I1
I2
 Z12  Z11
V2  0
I1Z 22
Z Z  Z12 Z 21
 Z12  11 22
I1Z 21
Z 21
41
• If the network is reciprocal, Z12=Z21, and AD-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 the
transition 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.
42
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 equivalent
circuit for the transition
[6].
43
Figure 4.13 (p. 188)
Equivalent circuits for a reciprocal two-port network. (a) T equivalent.
(b) π equivalent.
44
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 2
nodes, ai and bi. Node ai is identified with a wave
entering port i, while node bi is 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 S parameter or
reflection coefficient.
45
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.
46
Figure 4.15 (p. 190)
The signal flow graph representations of a one-port network and a
source.
(a) A one-port network and its flow graph. (b) A source and its
47
flow graph.
Decomposition of Signal Flow Graphs
• A signal flow graph can be reduced to a single branch
between 2 nodes using the 4 basic decomposition
rules below, to obtain any desired wave amplitude
ratio.
– Rule 1 (Series Rule): V3 = S32V2 = S32S21V1 .
– Rule 2 (Parallel Rule): V2 = SaV1 + SbV1 = (Sa + Sb)V1.
– Rule 3 (Self-Loop Rule): V2 = S21V1 + S22V2, V3 = S32V2.
 V3 = S32S21V1/(1-S22)
– Rule 4 (Splitting Rule): V4 = S42V2 = S21S42V1.
48
Figure 4.16 (p.
191)
Decomposition rules.
(a) Series rule.
(b) Parallel rule.
(c) Self-loop rule.
(d) Splitting rule.
49
Ex 4.7 Application of Signal Flow Graph
Figure 4.17 (p. 192)
A terminated two-port network.
50
Figure 4.18 (p. 192)
Signal flow path for the two-port network with general source and
51
load impedances of Figure 4.17.
Figure 4.19 (p. 192)
Decompositions of the flow graph of Figure 4.18 to find Γin =
b1/a1 and Γout = 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)52
Using Rule 3 for the self-loop at node a1.
Figure 4.20 (p. 193)
Block diagram of a network analyzer measurement of a two-port
device.
53
Figure 4.21a (p. 194)
Block diagram and signal flow graph for the Thru connection.
54
Figure 4.21b (p. 194)
Block diagram and signal flow graph for the Reflect connection.
55
Figure 4.21c (p. 194)
Block diagram and signal flow graph for the Line connection.
56
Figure 4.22 (p. 198)
Rectangular waveguide
discontinuities.
57
Figure 4.23 (p. 199)
Some common microstrip
discontinuities. (a) Openended microstrip. (b) Gap
in microstrip. (c) Change in
width.
(d) T-junction. (e) Coax-tomicrostrip junction.
58
Figure 4.24 (p. 200)
Geometry of an H-plane step (change in width) in rectangular
waveguide.
59
Figure 4.25 (p. 203)
Equivalent inductance of an H-plane asymmetric step.
60
Figure on page 204
Reference: T.C. Edwards, Foundations for Microwave Circuit Design, Wiley, 1981.
61
Figure 4.26 (p. 205)
An infinitely long rectangular waveguide with surface current
densities at z = 0.
62
Figure 4.27 (p. 206)
An arbitrary electric or magnetic current source in an infinitely
long waveguide.
63
Figure 4.28 (p. 208)
A uniform current probe in a rectangular waveguide.
64
Figure 4.29 (p. 210)
Various waveguide and other transmission line configurations using
aperture 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)
65
Coupling from a waveguide to a stripline via an aperture.
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 electric
polarization currents normal to a conducting wall. (d) Magnetic field
lines near a conducting wall. (e) Magnetic field lines near an
66
aperture in a conducting wall. (f) Magnetic field lines near magnetic
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) Fields
with aperture closed. (c)
Fields with aperture open.
(d) Fields with aperture
closed and replaced with
equivalent dipoles.
(e) Fields radiated by
equivalent dipoles for x
< 0; wall removed by
image theory.
(f) Fields radiated by
equivalent dipoles for z >
0; all removed by image
67
Figure 4.32 (p. 214)
Equivalent circuit of the aperture in a transverse waveguide wall.
68
Figure 4.33 (p. 214)
Two parallel waveguides coupled through an aperture in a
common broad wall.
69