Download abcd hybrid matrix scattering

Microwave Engineering
Cheng-Hsing Hsu
Department of Electrical Engineering
National United University
Outline
1. Transmission Line Theory
2. Transmission Lines and Waveguides
General Solutions for TEM, TE, and TM waves ; Parallel Plate waveguide ; Rectangular Waveguide ; Coaxial
Line ; Stripline ; Microstrip
3. Microwave Network Analysis
Impedance and Equivalent Voltages and Currents ; Impedance and Admittance Matrices ; The Scattering
Matrix ; ABCD Matrix ; Signal Flow Graphs ; Discontinuties and Model Analysis
4. Impedance Matching and Tuning
Matching with Lumped Elements ; Single-Stub Tuning ; Double-Stub Tuning ; The Quarter-Wave Transformer ;
The Theory of Small Reflections
5. Microwave Resonators
Series and Parallel Resonant Circuits ; Transmission Line Resonators ; Rectangular Waveguide Cavities
Dielectric Resonators
6. Power Dividers and Directional Couplers
Basic Properties of Dividers and Couplers ; The T-Junction Power Divider ; The Wilkinson Power Divider ;
Coupled Line Directional Couplers ; 180o hybrid
7. Microwave Filters
Periodic Structure ; Filter Design by the Insertion Loss Method ; Filter Transformations ; Filter Implementation ;
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3. Microwave Network Analysis
Impedance and Equivalent Voltages and Currents
Impedance and Admittance Matrices
The Scattering Matrix
ABCD Matrix
Signal Flow Graphs
Discontinuties and Model Analysis
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Circuits operating at low frequencies, for which the circuit dimensions are small relative to the
wavelength, can be treated as an interconnection of lumped passive or active components. In this
situation the circuit dimensions are small enough so that there is negligible phase change from one
point in the circuit to another. In addition, the fields can be considered as TEM fields supported by
two or more conductors. -> this leads to a quasi-static type of solution to Maxwell’
s equations, and
to the kirchhoff voltage and current laws and impedance concepts of circuit theory. But these
techniques can not be directly applied to microwave circuits..
=> Because the solution to Maxwell’
s equation for a given problem is complete, it gives the
electric and magnetic fields at all points in space. But we are interested in only the voltage or
current at a set of terminals, the power flow through a device. Another reason for using circuit or
network analysis is that it is then very easy to modify the original problem, or combine several
elements together and find the response, without having to analyze in detail the behavior of each
element in combination with its neighbors. A field analysis using Maxwell’
s equations for such
problems would be hopelessly difficult.
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Impedance and equivalent voltages and currents
Equivalent voltages and currents
At microwave frequencies the measurement of voltage or current is difficult (impossible), unless a
clearly defined terminal pair (TEM type lines such as coaxial cable, microstrip, or stripline) is
available. => not exist for non-TEM lines such as rectangular, circular, or surface waveguide
The voltage, V, of the conductor relative to the - conductor can be found as

V 
E
d

I 
H
d  (Ampere's law)

C
V
 Z o  (charactersitic impedance)
I
charactersitic impedance (assuming we know the propagation constant for line)
apply the circuit theory for transmission lines developed to characterize this line as a circuit element.
more difficult for waveguides
For a rectangular waveguide, TE10 mode
ja
x
A sin e jz Ae y 
x, y 
e jz

a
ja
x
Hx
x, y , z 

A sin e jz Ahx 
x, y 
e jz

a
Ey 
x, y , z 

Electric field lines for the TE10 mode
of a rectangular waveguide.
Electric and magnetic field lines for
an arbitrary two-conductor TEM line.
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Applied to electric field
ja
x
V
A sin e jz 
dy
y

a
The voltage depends on the position x ,as well as the length of the integration contour along the y
direction, and is quite different from the different position ( x=0, a/2).  wrong to obtain voltage,
and the current is also the same situation. (for non-TEM lines)
There are many ways to define equivalent voltage, current, and impedance for waveguide.
 Voltage and current are defined only for a particular waveguide mode, and are defined so
that the voltage is proportional to the transverse electric field, and the current is proportional to
the transverse magnetic field.
 In order to be used in a manner similar to voltages and currents of circuit theory, the
equivalent voltages and currents should be defined so that their product gives the power flow of
the mode.
The ratio of the voltage to the current for a single traveling wave should be equal to the
characteristic impedance of the line. This impedance may be chosen arbitrary, but is usually
selected as equal to the wave impedance of the line, or else normalized to unity.
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For an arbitrary waveguide mode with both positively and negatively traveling waves,
the transverse fields can be writtern as
e
x, y   jz
Et 
x, y, z e
x, y Ae jz Ae jz 
V e
V e jz
C1

h
x, y   jz
Ht 
x, y, z h
x, y Ae jz Ae jz 
I e
I e jz
C2







where e and h are the transverse field variations of the mode,
and A, A- are the field amplitudes of the traveling waves.
zˆ
e
x, y 
Since Et and H t are related by the wave impedance, Z w , h
x, y 
Zw
Defines equivalent voltage and current waves as
V 
z V e jz V e jz
I 
z I e jz I e jz
with V /I  V /I  Z o
The proportionality constants for this relationship are C1 V /A V - /A- and C2 I /A I - /Acan be determined from the remaining two conditions for power and impedance.
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1 2
V I *
*
The complex power flow for the incident wave is given by P  A 
e h 
zˆ
ds 

2
2C1C2*
S

e h


*

zˆ
ds
S
1
this power to be equal to V I *  C1C2* 
e h* 
zˆ
ds

2
S
where S : the surface integration is over the cross section of the waveguide
V  V  C1
The characteristic impedance is Z o     
since V  C1 A and I  C2 A
I
I
C2
If it is desired to have Z o Z w , the wave impedance ( Z TM or ZTE ) of the mode, then
C1
C
Z w 
Z TM or Z TE 
 normalize the characteristic impedance to unity Z o 1  1 1
C2
C2
A general field in a waveguide
N
Vn jn z Vn jn z 
I n jn z I n jn z 
Et 
x, y, z 
 e 
en 
x, y ; H t 
x, y, z 
 e 
hn 
x, y 
C e

C e

C
C
n
1  1n
n
1  2n
1n
2n


N
where Vn and I n are wquivalent voltages and currents for the nth mode ;
C1n and C2 n are the proportionality constants for rach mode
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The concept of Impedance
The concept of impedance forms as important link between field thoery and transmission line or circuit theory
Summarize the various types of impedance we have used so far and their notation:
•=(/
)1/2=intrinsic impedance of the medium. dependent only on the materials parameters of
the medium, and equal to the wave impedance for plane waves.
•Zw=Et/Ht=1/Yw=wave impedance is a characteristic of the particular type of wave. TEM, TM, and
TE waves each have different wave impedance (ZTEM, ZTM, ZTE) depend on the type of line or
guide, the material, and the operating frequency.
•Zo=1/Yo=(L/C)1/2=characteristic impedance is the ratio of voltage to current for a traveling wave
on a transmission line. (the characteristic impedance of TEM wave is uniquely, the characteristic
impedance of TM, TE waves are in various ways due to have a uniquely TE and TM waves.
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Ex. Consider a rectangular waveguide with a = 2.286 cm and b = 1.016 cm (X-band guide), air
filled for z < 0 and rexolite filled (dielectric constant = 2.54) for z > 0, as shown in Figure. If the
operating frequency is 10 GHz, use an equivalent transmission line model to compute the reflection
coefficient of a TE10 wave incident on the interface from z < 0.
<Sol> The propagation constants in the air (z 0) and the dielectric (z 0) regions are


a  k   158.0 m -1 , b  r ko2   304.1 m -1 where ko 209.4 m -1
a 
a 
Treat the problem as the reflection of an incident voltage wave at the junction of two infinite transmission lines.
2
2
2
o
the equivalent characteristic impedances for the two lines are
k 
209.4 

377 
k 
209.4

377 
Z 0a  0 0 
500.0 ; Z 0d  
259.6
a
158.0
d
304.1
Z 0 Z 0 a
 d
0.316
Z 0d Z 0a
with this results, expressions for the incident, reflected and transmitted waves can be written in terms of fields,
or in terms of equivalent voltages and currents.
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Consider one - port network and drive a general relation between its impedance properties and
electromagnetic energy stored in, and the power dissipated by, the network.
The complex power delivered to this network is
1
P 
E H * 
ds P2 j
Wm We where Pis real and represents
2 S
the average power dissipated by the network and Wm and We represent the stored magnetic and electric energy
If define real transverse modal fields, e and h, over the terminal plane of the network
Et 
x,y,z 
V 
z e x,y 
e -jβj; H t 
x,y,z I 
z h x,y 
e jββ
with a normalization :
e h 
d s 1

S
then above equation can be expressed in terms of the terminal voltage and current :
1
1
P 
VI * e h 
d s  VI *
2 S
2
Then the input impedance is
An arbitrary one-port network.
P 2 j
Wm We 
V VI *
P
Z in R jX   2 

where R of the input impedance
1 2
1 2
I
I
I
I
2
2
is related to dissipated power, while X is related to the net energy stored in the network
If the network is lossless, then P0 and R 0.
4
Wm We 
 Z in is purely imaginary, with a reactance X 
which is positive for an inductive
2
I
load (Wm We ), and negative for a capacitive load (Wm We )
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Impedance and admittance matrices
Using the impedance and/or admittance matrices of circuit theory to relate these terminal or “port”
quantities to each other, and thus to essentially arrive at a matrix description of the network.
Consider an arbitrary N-port microwave network (any type of transmission line or transmission line
equivalent of a single propagating waveguide mode. At a specific point on the nth port, a terminal
plane, tn, is defined along with equivalent voltages and currents for the incident (Vn+, In+) and
reflected (Vn-, In-) waves.
The impedance matrix [Z] of the microwave Network

Vn = Vn+ + VnIn = In+ - In-
[V] = [Z][I] =>
when z = 0
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admittance matrices
Similarly, we can define can admittance matrix [Y]
[Y] = [Z]-1 , both the [Z] and [Y] matrices relate the total port voltages and currents.
V
Z ij  i
Ij
I
Yij  i
Vj
I k 0 for k j
Vk 0 for k j
Driving port j with the current Ij , open-circuiting all other port, and measuring the
open-circuit voltage at port i => Zii is the input impedance seen looking into port i
when all other ports are open-circuited
Driving port j with the voltage Vj , short-circuiting all other port, and measuring the
short-circuit current at port i => Yii is the input admittance seen looking into port i
when all other ports are short-circuited
In general, each Zij or Yij element may be complex. However, many networks are either reciprocal or lossless, or
both. If the network is reciprocal, the impedance and admittance matrices are symmetric. => Zij = Zji and Yij =
Yji ; If the network is lossless, all the Zij or Yij elements are purely imaginary
=> Either of these special cases => can reduce the number of independent quantities or degrees of freedom that
an N-port network.
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Ex. Find the Z parameters of the two-port T-network
Z11 can be found as the input impedance of port 1 when port 2 is open circuited
V
Z 11 1
I1
I 2 0
Z A Z C
Z12 (transfer impedance) can be found measuring the open-circuit voltage
at port 1 when a current I 2 is appled at port 2.
V
Z12  1
I2
I1 0
V
ZC
2
Z C
I 2 Z B Z C
Z11 Z 22 reciprocal
V
Z 22  2
I2
I1 0
Z B Z C
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Reciprocal Networks
Yij = Yji if [Y] is a symmetric matrix, its inverse, [z], is also symmetric
Lossless Networks
Now consider a reciprocal lossless N-port junction; If the network is lossless, then
the net real power delivered to the network must be zero. the elements of the
impedance and admittance matrices must be pure imaginary.
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The scattering matrix
A representation more in accord with direct measurements, and with the ideas of incident, reflected,
and transmitted waves, is given by the scattering matrix. Like the impedance and admittance matrices
for an N-port network, the scattering matrix provides a complete description of the network as seen
at its N ports. => While the impedance and admittance matrices relate the total voltages and
currents at the ports, the scattering matrix relates the voltage waves incident on the ports to those
reflected from the ports. => scattering parameters can be measured directly with a vector network
analyzer.
Vn+ is the amplitude of the voltage wave incident on port n, and Vn- is the amplitude of the voltage wave
reflected from port n. => scattering matrix  [S] matrix
Sij is found by driving port j with an incident wave of voltage Vj+, and measuring the reflected wave amplitude,
Vi-, coming out of port i.=> the incident waves on all ports except the jth port are set to zero, which means that
all ports should be terminated in matched loads to avoid reflections. (Noted Sii)
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Ex. Find the S parameter of the 3dB attenuator circuit shown in Figure.
Vi 
From Sij   V 0 for k j ,
Vj k
S11 can be found as the reflection coefficient seen at port 1 when port 2 is terminated in a matched load (Zo 50)
1
V1
Z in
Z o

1
S11   V 0  V 0  
2
V1 2
Z in1 Z o
Z o on port 2
1
Z in
8.56 [141.8
8.56 50 
] / 141.8 8.56 5050, so S11 0 S 22 0 (symmetry)
From S11 S 22 0 V1- 0 when port 2 is terminated in Z o 50 , and that V2 0
V1 V1 and V2 V2
 41.44 
 50 
V2 V2 V1 


0.707V1
41.44 8.56 
50 8.56 
S12 S 21 0.707
2
If the input power is V1 / 2 Z o , then the output power is V2
2
2
2 Z o S 21V1 2 Z o S 21
2
2
2
2 Z o V1 V1 4 Z o
which is one - half(3dB) of the input power
A matched 3B attenuator with a 50 Ω Characteristic impedance
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The [s] matrix can be determined from the [z] or [Y] matrix.
Vn = Vn+ + VnIn = In+ - In- = Vn+ - Vn- (Z0n =1)
=>[Z][I] = [Z][V+] –[Z][V-] = [V] = [V+] + [V-]
which can be rewritten as ([Z]+[U])[V-] = ([Z]-[U])[V+]
where [U] is unit or identity matrix [U]=1
=> [S]=([Z]+[U])-1 ([Z]-[U]) giving the scattering matrix in terms of the impedance matrix.
=> [Z] = ([U]+[S])([U]-[S])-1
Reciprocal Networks
[U] is diagonal, so [U]t=[U] => [Z]t=[Z] => [S] matrix for a reciprocal network is symmetric.
=> [S] = [S]t
Lossless Networks
[S] matrix for a lossless network is unitary.
=> [S]t[S]* = [U] => [S]* = {[S]t}-1
1

*
S
S



ki kj
0
k
1

N
i j
i j
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A shift in Reference Planes
Due to the S parameters relate amplitudes (magnitude and phase) of traveling waves incident on and
reflected from a microwave networks, phase reference planes must be specified for each port of the
network. Now consider the S parameters are transformed when the reference planes are move from
their original locations.
let the new [S’
] is denoted as new scattering matrix (zn=ln), and [S] is denoted as scattering matrix
for the network with this set of terminal planes (zn=0).
 The incident and reflected port voltages : [V-]=[S][V+]
[V’
-]=[S’
][V’
+]
For the theory of traveling waves on lossless transmission lines

+ = V + ejn ; V ’
- -jn
Vn’
n
n = Vn e
n=nln electrical length of the outward shift of the
reference plane of port n
Shifting reference planes for an N-port network.
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Combination above two equations
-2jn S , meaning that the phase of S is shifted by twice the electrical length of the shift
S’
nn = e
nn
nn
in terminal plane n, because the wave travels twice over this length upon incident and reflection.

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Generalized Scattering Parameters
In general case, scattering parameters of networks with the same characteristic impedance for all
ports (50). However, the characteristic impedances of a multi-port network may be different, which
requires a generalization of the scattering parameters as defined up to this point.
Z0n is the (real) characteristic impedance of the nth port ; Vn+ and Vn- represent the incident and
reflected voltage waves at port n.
incident (power) wave : an Vn
Z 0n
; reflected (power) wave : bn Vn-

Z 0n

1
1

Vn VnVn  Z 0 n 
an bn ; I n 
VnVn 
an bn 
Z 0n
Z 0n
Now the average power delivered to the nth port is
1
1
1 2 1 2
2
2
Pn  Re Vn I n*  Re an bn bn an* bn*an  an  bn
2
2
2
2
An N-port network with different characteristic impedances.
the quantity bn an* bn*an is purely imaginary
 





the average power delivered through port n is equal to the power in the incident wave
minus the power in the reflected wave (physically satisfying)
If expressed in terms of Vn and Vn- , the corresponding result would be
dependent on the characteristic impedance of the nth port.
Generalized scattering matrix : [b] [S][a]
b
Sij  i
aj
ak 0 for k j
Vi  Z 0 j
S ij  
V j Z 0i
Vk0 for k j
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The transmission (ABCD) matrix
The Z, Y, and S parameter representations can be used to characterize a microwave network with an
arbitrary number of ports, but in practice many microwave consist of a cascade connection of two or
more two-port networks.
The ABCD matrix is defined for a two-port network in terms of the total voltages and currents to 2x2
transmission.
V1 AV2 BI 2
For two-port network
I1 CV2 DI 2
V  A B 
V2 


1 


I1  
I2 
C D



For cascade connection of two two-port networks
V1  
A

1


I1  
C1

B1 
V2  
V 
A B2 
V3 


; 2  2





I3 
D1 
I2  
I2  
C2 D2 



V 
A B1 
A2 B2 
V3 



1 1




I1  
C1 D1 
C2 D2 
I3 




(a) A two-port network;
(b) a cascade connection of two-port networks.
Relation to impedance Matrix
Knowing the Z parameter of a network -> determine the ABCD matrix
V1 I1Z11 I 2 Z12 ; V2 I1Z 21 I 2 Z 22
V
A 1
V2
I 2 0
IZ
V
 1 11 Z11 Z 21 ; B  1
I2
I1Z 21
I
C1
V2
I 2 0
I
I
 1 1 Z 21 ; D  1
I2
I1Z 21
V2 0
V2 0
I Z I Z
 1 11 2 12
I2
V2 0
Z11
I1
I2
V2 0
Z Z Z12 Z 21
Z12  11 22
Z 21
I Z /Z
 2 22 21 Z 22 Z 21
I2
If the network is reciprocal, then Z 21 Z12  AD-BC 1
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Equivalent Circuits for Two-Port Networks
The special case of a two-port microwave network occurs so frequently in practice that is deserves
further attention. => Discuss the use of the equivalent circuits to represent an arbitrary two-port
network.
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Figure shows a transition between a coaxial line and a microstrip line, and serves as an example of
a two-port network. Terminal planes can be defined at arbitrary points on the two transmission lines.
=>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 which can be obtained by measurements or by theoretical analysis (two-port “
black box”
: Z,
Y, S, or ABCD), suited transition from one type of transmission to another or transmission line
discontinuity.
The arbitrary two-port network can be described in the terms of impedance parameters as
V1 = Z11I1 + Z12 I2
or
;
V2 = Z21I1 + Z22I2
I1 = Y11V1 + Y12V2 ;
I2 = Y21V1 + Y22V2
If the network is reciprocal, then Z12 = Z21 and Y12 = Y21
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].
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Other equivalent circuits can also be used to represent a two-port network such as T and
equivalent circuits.
If the network is reciprocal, there are six degrees of
freedom (the real and imaginary parts of three matrix
elements), so the equivalent circuit should have six
independent parameters
If the network is lossless, which is good approximation
for many practical two-port junctions, some
simplification can be made in the equivalent circuit.
=> This reduces the degrees of freedom for such a
network to three, and implies that the T and equivalent
circuits can be constructed from purely reactive
elements.
Equivalent circuits for a reciprocal two-port network.
(a) T equivalent. (b) π equivalent.
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Signal Flow Graphs
In this section we discuss the signal flow graph, which is an additional technique that is very useful
for the analysis of microwave networks in terms of transmitted and reflected waves.
first discuss the features and the construction of the flow graph itself ; second present a technique
for the reduction, or solution, of the flow graph.
The primary components of a signal flow graph are nodes and branches:
Nodes: Each port, i , of a microwave network has two nodes, ai and bi.
=> 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 a directed path between two nodes, representing signal flow from one node to
another. Every branch has an associated S parameter or reflection coefficient.
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Consider the flow graph of an arbitrary two-port network. The flow graph gives an intuitive graphical
illustration of the network behavior.
a1 incident at port, part going through S11 and out
port 1 as a reflected wave and part transmitted
through S21 to node b2.
If a load with nonzero reflection coefficient is
connected at port 2, this wave (b2 wave) will be
partly reflected and reenter the two-port network at
node a2.
=> part of the wave can be reflected back out port 2
via S22, and part can be transmitted out port 1
through S12.
The signal flow graph representation of a two-port network.
(a) Definition of incident and reflected waves. (b) Signal flow graph.
Two other special networks, a one-port network and a
voltage source.
it is easy matter to solve for the ratio of any
combination of wave amplitudes.
=> using four basic decomposition rules, but the
same results can also be obtained using Mason’
s rule
from control system theory.
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 flow graph.
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Decomposition of Signal Flow Graphs
A signal flow graph can be reduced to a signal branch between two nodes using the four basic
decomposition rules below, to obtain any desired wave amplitude ratio.
Rule 1 (Series Rule):
two branches, common node has only one incoming and one outing wave. => to form
a single branch whose coefficient is the product of the coefficients of the original branches. V3 = S32 V2 = S32 S21 V1
Rule 2 (Parallel Rule):
two branches, from common node to another common node may be combined into a
single branch => whose coefficient is the sum of the coefficients of the original branches.
V2 = Sa V1 + Sb V1 = (Sa + Sb) V1
Rule 3 (Self-Loop Rule):
When a node has a self loop (a branch
that begins and ends on the same node) of coefficient S, the self loop can be
eliminated by multiplying coefficients of the branched feeling that node by
1 / (1-S) => V2 = S21 V1 + S22 V2 , V3 = S32 V2 =>eliminating V2 :
V3 = [S32S21 / (1 –S22)] V1
Rule 4 (Splitting Rule):
A node may be split into two separate
nodes as long as the resulting flow graph contains, once and only once, each
combination of separate (not self loops) input and output branches that
connect to the original node. => V4 = S42 V2 = S21 S42 V1
Decomposition rules. (a) Series rule. (b) Parallel rule.
(c) Self-loop rule. (d) Splitting rule.
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Ex: Use signal graphs to derive expressions for in and out for the two-port network.
b
S S 
in  1 S11  21 12 
a1
1 S 22 
b2
S12 S 21s
out  S 22 
a2
1 S11s
Decompositions of the flow graph 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) Using Rule 3 for the self-loop at node a1.
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Application to TRL Network Analyzer Calibration
For a further application of signal flow graphs, calibration of a network analyzer using the Thru-Reflect Line (TRL)
technique. The general problem is intended to measure the S-parameters of a two-port device at the indicated
reference planes.
=>The primary reference plane for such measurements is generally at some point within the analyzer itself.
 the measurement will include losses and phase delays caused by the effects of the connectors, cables, and
transitions that must be used to connect the device under test (DUT) to the analyzer.
 these effects are lumped together in a two-port error box (a calibration procedure is used to characterize the
error boxes before measurement of the DUT) placed at each port between the actual measurement reference plane
and the desired reference plane for the two-port DUT. => the actual error-corrected S-parameter of the DUT can
be calculated. (Measurement of a one-port network can be considered as a reduced case of the two-port)
The simplest way to calibrate a network analyzer is to use three or more known loads, such as shorts,
opens, and matched loads. => uses three simple connections to allow the error boxes to be characterized
completely. => [I] Thru connection is made by directly connecting port 1 to port 2, as the reference
planes
[II] Reflect connection uses as a load
having a large reflection coefficient, L,
such as a nominal open or short.
[III] Line connection involves connecting
ports 1 and 3 together through a length of
matched transmission line
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Use signal flow graphs to derive the set of equations necessary to find the S-parameter for the error
boxes in the TRL calibration procedure. The Thru, Reflect, and Line connections at the reference
plane for the DUT, and measure the S-parameter for these three cases at the measurement planes.
=>Assume the same characteristic impedance for ports 1 and 2, and the error boxes (characterized by
the S-matrix [S], and alternatively by the ABCD matrix) are reciprocal and identical for both ports.
=> S21=S12 and an inverse relation between the ABCD matrices of the error boxes for ports 1 and 2.
 [T] ; [R] ; [L] matrices
Thru connection
The thru connection and the corresponding signal flow
graph. Observe that we have made use of the fact that
S21=S12 and that the error boxes are identical and
symmetrically arranged.
can be easily reduced using the decomposition rules
to give the measured S-parameters at the measurement
planes in terms of the S-parameters of the error boxes
b
T11  1
a1
b
T12  1
a2
a2 0
S 22 S122
S11 
1 S 222
(1)
a1 0
S2
 12 2
1 S 22
(2)
By symmetry to T22=T11, and by reciprocity to T12=T21
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Reflect connection
Note that this arrangement effectively decouples the two measurement ports, so R12=R21=0
b
=> can be easily reduced to show that R11  1
a1
a2 0
By symmetry to R22=R11
S122 L
S11 
1 S 22 L
(3)
Line connection
A reduction similar to that used for the thru case
b
L11  1
a1
b
L12  1
a2
a2 0
a1 0
S 22 S122 e 2
S11 
2 2
1 S 22
e
(4)
S122 e 2

2 2
1 S 22
e
(5)
By symmetry and reciprocity to L22=L11 and L21=L12
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Now have five equations for five unknowns S11, S12, S22, L, e-l
=> Since equation 3 is the only equation that contains L, we can first solve the other four equations
for the other four unknowns. => equation 2 can be eliminate S12 from equations 1, 4 and 5, and then
S11 can be eliminated from equations 1 and 4.
L12e-2l –L12S222 = T12e-l –T12S222e-l
e-2l (T11-S22T12)-T11S222 = L11(e-2l –S222) –S22T12
=>
e
l
2
2
2
L12
T122 (T11 L11 ) 2  [ L12
T122 (T11 L11 ) 2 ]2 4 L12
T122

2 L12T12
 real and imaginary parts of be positive, or by knowing the phase of L to within 180º
 We multiply equation 2 by S22 and subtract from equation 1 to get T11 = S11 + S22T12, and similarly
multiply equation 5 by S22 and subtract from equation 4 to get L11 = S11 + S22L12e-l
Eliminating S11 from these two equations gives S22 in terms of e-l as S22 = (T11-L11) / (T12-L12e-l)
 S11 = T11-S22T12 and solving equation 2 for S12 gives S122 = T12(1-S222)
 Finally equation 3 can be solved fro L to give L = (R11-S11) / [S122 + S22(R11-S11)]
From the results give the S-parameters for the error boxes, as well as the unknown reflection
coefficient, L, and the propagation factor, e-l.=> this completes the calibration procedure for the
TRL method.
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Since we are now working with a cascade of three two-port networks, it id convenient to use
ABCD parameters. We convert the error box S-parameters to the corresponding ABCD
parameters, and convert the measured S-parameters of the cascade to the corresponding
AmBmCmDm parameters. If we use A’
B’
C’
D’to denote the parameters for the DUT, then
1
A B
A B
Am B m  
A' B' 


m





m 
C
D
C
'
D
'
C
D
C
D








from which we can determine the ABCD parameters for the DUT as
1
Am
A' B'  A B  



m
C
'
D
'
C
D
C

 

A B
Bm 




C
D
Dm 


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Discontinuities and Model Analysis
By either necessity or design, microwave networks often consist of transmission lines with various
types of transmission line discontinuities.
 Discontinuities are unavoidable result of mechanical or electrical transitions from one medium
to another (e.g.: a junction between two waveguides, or a coax-to-microstrip transition).
 Depending on the type of discontinuity, the equivalent circuit may be a simple shunt or series
element across the line or, in the more general case, a T- or -equivalent circuit may be required.
In addition, the component value of an equivalent circuit depend on the parameters of the line and
the discontinuity, as well as the frequency of operation.
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Figure a to c, thin metallic diaphragms
(or “
irises”
) can be placed in the cross
section of a waveguide to yield equivalent
shunt inductance, capacitance, or a
resonant combination.
Similar discontinuities can be also be
made in circular waveguide
Rectangular waveguide discontinuities.
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Some typical microstrip discontinuities and
transitions are shown in figure.
Similar geometries exist for stripline and other
printed transmission lines such as slotline,
covered microstrip, coplanar waveguide, etc.
Some common microstrip discontinuities. (a) Open-ended microstrip. (b) Gap in microstrip.
(c) Change in width. (d) T-junction. (e) Coax-to-microstrip junction.
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Model Analysis of an H-Plane Step in Rectangular Waveguide
The field analysis of most discontinuity problems is very difficult, and the technique of modal
analysis is relatively straightforward and similar in principle to the reflection/transmission problems.
In addition, modal analysis is a rigorous and versatile technique that can be applied to many coax,
waveguide, and planar transmission line discontinuity problems, and lends itself well to computer
implementation. => we will present the technique of modal analysis by applying it to the problem
of finding the equivalent circuit of an H-plane step (change in width) in rectangular waveguide.
Assume that only the dominant TE10 mode is propagating
in guide 1 (z < 0), and that such a mode is incident on the
junction from z < 0. it is also assumed that no modes are
propagating in guide 2 , although the analysis to follow is
still valid if propagation can occur in guide 2.
The transverse components of the incident TE10 mode can
be written, for z < 0,
E yi sin
x j1a z
e
a
a
1
x
H xi  a sin e j1 z
Z1
a
;
n
Geometry of an H-plane step (change in width) in rectangular waveguide.
where   k  
a 
is the propagation constant of the TEn 0 mode in guide 1(of width a)
2
a
n
k
and Z na  0 a0
n
2
0
is the wave impedance of the TEn0 mode in guide 1.
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Due to the discontinuity at z = 0 there will be reflected and transmitted waves in both guides,
consisting of infinite sets of TEn0 modes in guides 1 and 2. => only the TE10 mode will propagate
in guide 1, but the higher-order modes are also important because they account for stored energy,
localized near z = 0.
=> In addition, there is no y variation introduced by this discontinuity, TEnm modes for m 0 are
not excited, nor are any TM modes. (However, a more general discontinuity may excite such modes.
The reflected modes in guide 1 may then be written, for z 0, as
nx jna z
E An sin
e
a
n
1

r
y
A
nx jna z
H  na sin
e
Z
a
n
1
n

;
r
x
where An is unknown amplitude coefficient of the reflected TEn 0 mode in guide 1.
The reflected coefficient of the incident TE10 mode is then A1.
Similarly, the transmitted modes into guide 2 can be written, for z 0, as
nx jnc z
E Bn sin
e
c
n
1

t
y
B
nx jnc z
H  nc sin
e
Z
c
n
1
n

;
t
x
n
where the propagation constant in guide 2 is nc  k02  
c 
k
and the wave impedance in guide 2 is Z nc  0 c0
n
2
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The transverse field ( E y , H x ) must be continuous for 0  x  c at z 0.
In addition, E y must be zero for c x  a because of the step.
Enforcing these boundary condistions leads to the following equations :
nx

x
nx 
B
sin
for 0 x c,

n
E y sin An sin

(1)
c
n
1
a n 1
a

for c x a,
0

1
x  An
nx
B
nx
H x  a sin  a sin
 nc sin
for 0 x c. (2)
Z1
a n 1 Z n
a
Z
c
n
1
n

The fact that the equivalent circuit of the H - plane step looks inductive is a result of the actual value
of the reflection coefficient, A1 , but we can verify this result by computing the complex power into
the evanescent modes on either side of the discontinuity.
  B*
c
c 
nx 
mx 

*
m
 P 
E H z 0  
z dxdy b E y H x dx b 
Bn sin
 * sin
dx




c
x 0 y 0
x 0
x 0
c
c
n

1
m

1




 Zm

c
b
*
2
bc  Bn
jbc 
2
  c* 
B
nc

n
2 n 1 Z n
2k00 n 1
(3)
Equation 3 shows that the complex power flow into guide 2 is purely inductive. A similar result can be
derived for the evanescent modes in guide 1. (this is left as a problem)
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Microstrip Discontinuity Compensation
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