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EE 41139 Microwave Techniques 1
Lecture 3
Impedance and Equivalent Voltages and Currents for Non-TEM Lines
Impedance Properties of One-Port Networks
Impedance, Admittance and Scattering Matrices
Signal Flow Graphs
EE 41139 Microwave Techniques 2
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
show how circuit and network concepts can be extended to handle many microwave analysis and design problems of practical interest
EE 41139 Microwave Techniques 3
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
equivalent voltage and current can be defined uniquely for TEM-type lines (require two conductors) but not so for non-TEM lines
EE 41139 Microwave Techniques 4
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
for non-TEM lines, voltage and current are only defined for a particular waveguide mode, V is related to Et and I to Ht where t denotes the transverse component
EE 41139 Microwave Techniques 5
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
the product of the equivalent V and I should yield the power flow of the mode
V/I for a single traveling wave should be equal to the characteristic impedance of the line or can be normalized to 1
EE 41139 Microwave Techniques 6
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
for an arbitrary waveguide mode with a +ve and -ve (in z) traveling waves
E x y z e x y A e A ee x y
CV e V e
H x y z h x y A e A eh x y
CI e I e
tj z j z j z j z
tj z j z j z j z
( , , ) ( , )( )( , )
( )
( , , ) ( , )( )( , )
( )
1
2
EE 41139 Microwave Techniques 7
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
we can write the voltage and current of an equivalent transmission line as
V z V e V ej z j z( )
I z I e I ej z j z( )
EE 41139 Microwave Techniques 8
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
i.e., we are only interested in certain quantities and these quantities can be derived using circuit and network theory
EE 41139 Microwave Techniques 9
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
the incident power is given by
Where
PV I
C Ce h zds V Iinc
S
1
2
1
21 2
**
C C e h zdsS
1 2*
EE 41139 Microwave Techniques 10
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
the characteristic impedance of the equivalent transmission line is
ZV
I
V
Iand
V
C
I
Co
1 2
ZC
Co 1
2
EE 41139 Microwave Techniques 11
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
the wave impedance is given by
hz e
Zw
EE 41139 Microwave Techniques 12
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
if we choose the characteristic impedance of the line equal to that of the wave impedance, i.e.,
which can be either TE or TM modes
Z ZC
CZo w w 1
2
EE 41139 Microwave Techniques 13
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
therefore, if one can measure the voltage and current for each mode, the field in the waveguide can be determined as sum of the field for each mode
EE 41139 Microwave Techniques 14
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
E x y zV
Ce
V
Ce e x y
H x y zI
Ce
I
Ce h x y
tN
n
n
j z n
n
j zn
tN
n
n
j z n
n
j zn
n n
n n
( , , ) ( ) ( , )
( , , ) ( ) ( , )
1 1 1
1 2 2
EE 41139 Microwave Techniques 15
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
we reiterate that there are various types of impedance
intrinsic impedance
depends on material parameters but is equal to the wave impedance of a plane wave in a homogeneous medium
EE 41139 Microwave Techniques 16
Impedance and Equivalent Voltages and Currents for Non-
TEM Lineswave impedance of a particular type of wave, namely, TEM, TE and TM
depends on frequency, materials and boundary conditions
ZE
Hwt
t
EE 41139 Microwave Techniques 17
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
characteristic impedance
unique for TEM waves, non-unique for TE and TM
I
VZo
EE 41139 Microwave Techniques 18
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
we will calculate the wave impedance of the TE10 mode waveguide
now consider the field equations for the TE10 rectangular waveguide
mode, the field components are
EE 41139 Microwave Techniques 19
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
ak,eAeA
a
xsin
akjH,0H
a
xsineAeAeAeA
a
xsin
akjE,0E
czj
10zj
102c
xy
zjzjzj10
zj102
cyx
EE 41139 Microwave Techniques 20
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
a
xsineAeA
Z
1H zjzj
TEy
10
k
H
EZ
x
yTE10
EE 41139 Microwave Techniques 21
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
compare with the transmission line equations and the incident power
zj
o
zj
o
zjzj
eZ
Ve
Z
V)z(I
eVeV)z(V
EE 41139 Microwave Techniques 22
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
*21
2*
TE
2
S
*xy CCA
2
1IV
2
1
Z4
AabdsHE
2
1P
10
oTE2
1 ZZC
C
I
V10
2
ab
Z
1C,
2
abC
10TE21
EE 41139 Microwave Techniques 23
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
therefore, we can relate the field and circuit parameters for the TE10 waveguide mode
we can also the transverse resonance technique to look at the wavenumber of the TE10 mode in the y direction (height of the waveguide)
EE 41139 Microwave Techniques 24
Impedance and Equivalent Voltages and Currents for Non-
TEM Linesthe waveguide can be regarded as a transmission line with certain characteristic impedance and is shorted at both ends
y = 0
y = y’
y = b
Z
EE 41139 Microwave Techniques 25
Impedance and Equivalent Voltages and Currents for Non-
TEM Linesusing the transmission line equation to transfer the short circuit to y =y'
according to the transverse resonance technique, we have impedance looking downward +
impedance looking upward = 0
0)'yb(ktanjZ'yktanjZ yy
EE 41139 Microwave Techniques 26
Impedance and Equivalent Voltages and Currents for Non-
TEM Lines
for the above equation to be true for any value of y', ky must be zero
this is correct as ky = n/b where n = 0 for TE10 mode
0)'yb(ktanjZ'yktanjZ yy
EE 41139 Microwave Techniques 27
Impedance Properties of One-Port Networks
consider the arbitrary one-port network shown here
+-V
I
Zinone-portnetwork
nE, H
S
P E H ds P j W Ws
l m e 1
22* ( )
EE 41139 Microwave Techniques 28
Impedance Properties of One-Port Networks
assume that
if then
E x y z V z e x y etj z( , , ) ( ) ( , )
H x y z I z h x y etj z( , , ) ( ) ( , )
e h dss
1 P VI e h ds VIs
1
2
1
2* *
EE 41139 Microwave Techniques 29
Impedance Properties of One-Port Networks
the input impedance
for a lossless network, R = 0, therefore
ZV
I
VI
II
P
I
P j W W
IR jXin
l m e
*
* | |
( )
| |
2 2 42 2
XW W
I
m e4
2( )
| |
EE 41139 Microwave Techniques 30
Impedance Properties of One-Port Networks
the input impedance is purely imaginary
the reactance is positive for an inductive load (Wm > We) and is negative for a capacitive load
EE 41139 Microwave Techniques 31
Impedance Properties of One-Port Networks
Foster s reactance theorem: the rate of change of the reactance for the lossless one-port network with frequency is
X W W
I
e m42
( )
| |
EE 41139 Microwave Techniques 32
Impedance Properties of One-Port Networks
from Maxwells equations
and the vector identity
E j H xH j E ,
( ) ( ) ( )A B xA B xB A
EE 41139 Microwave Techniques 33
Impedance Properties of One-Port Networks
( ) ( | | | | )
( ) ( | | | | )
( ) ( ) ( )
* *
* *
* *
EH E
H j E H
EH E
H dv j E H dv
EH E
H n ds j W W
v v
es
m
2 2
2 2
4
EE 41139 Microwave Techniques 34
Impedance Properties of One-Port Networks
( ) ( )* *EH E
H ds j W Wes
m
4
( ) ( )* * * *VI
e h V Ieh V
I e h VIe
h ds j W Ws
e m
4
( ) ( )* * * *VI
e h V Ieh V
I e h VIe
h ds j W Ws
e m
4
EE 41139 Microwave Techniques 35
Impedance Properties of One-Port Networks
Note that
and therefore,
=0
E x y z V z e x y e
H x y z I z h x y e
tj z
tj z
( , , ) ( ) ( , )
( , , ) ( ) ( , )
eh e
h
( )* *
sV Ie
hVI
eh ds real
EE 41139 Microwave Techniques 36
Impedance Properties of One-Port Networks
V= j XI for lossless line
( ) ( )
| | ( )
( )
| |,
( )
| |
* * * *
* *
VI
e hV
I e h ds VI V
I
jXII
jX
I jXI
I j W W
X W W
Isimilarly
B W W
V
s
e m
e m e m
2
2 2
4
4 4
EE 41139 Microwave Techniques 37
Impedance Properties of One-Port Networks
poles and zeros must alternate in position as the slope is always positive
zero poleX
EE 41139 Microwave Techniques 38
Even and Odd Properties of Z(w) and G(w)
define the Fourier transform as
note that v(t) must be a real quantity, i.e., v(t) = v*(t), therefore,
or V(-) = V*()
v t V e dj t( ) ( ) 1
2
V e d V e d V e dj t j t j t( ) * ( ) * ( )
EE 41139 Microwave Techniques 39
Even and Odd Properties of Z(w) and G(w)
note that we can only measure V(), we need its complex conjugate to obtain v(t), similar arguments hold for I()
V*(-) = V()=Z()I()=Z*(-)I*(-)=Z*(- Z() =Z*(-
EE 41139 Microwave Techniques 40
Even and Odd Properties of Z(w) and G(w)
therefore, the real part of Z, i.e, R is an even function of while the imaginary part X is an odd function of
the reflection coefficient also has an even real part and an odd imaginary part
EE 41139 Microwave Techniques 41
Even and Odd Properties of Z(w) and G(w)
( )( )
( )
( ) ( )
( ) ( )
( )( )
( )
( ) ( )
( ) ( ), . . , ( ) * ( )
Z Z
Z Z
R jX Z
R jX Z
Z Z
Z Z
R jX Z
R jX Zi e
o
o
o
o
o
o
o
o
( )( )
( )
( ) ( )
( ) ( )
( )( )
( )
( ) ( )
( ) ( ), . . , ( ) * ( )
Z Z
Z Z
R jX Z
R jX Z
Z Z
Z Z
R jX Z
R jX Zi e
o
o
o
o
o
o
o
o
EE 41139 Microwave Techniques 42
Impedance, Admittance and Scattering Matrices
t n
V , I n n
V , I n n
+ +
- -
EE 41139 Microwave Techniques 43
Impedance, Admittance and Scattering Matrices
N-port microwave network, each port has a reference plane tn
at the reference plane of port N, we have
V V V
I I I
n n n
n n n
EE 41139 Microwave Techniques 44
Impedance, Admittance and Scattering Matrices
if we are only interested in knowing the relationship among the voltages and currents at the ports, we can define a impedance matrix Z so that
[ ] [ ][ ]V Z I
EE 41139 Microwave Techniques 45
Impedance, Admittance and Scattering Matrices
the element Zij of the impedance matrix
is given by
similar equations can be written for the admittance matrix
ZV
Iiji
jI k jk
| ,0
EE 41139 Microwave Techniques 46
Impedance, Admittance and Scattering Matrices
for reciprocal network, the impedance (admittance) matrix is symmetric
for lossless network, all matrix elements are purely imaginary
EE 41139 Microwave Techniques 47
Impedance, Admittance and Scattering Matrices
the scattering matrix relate the voltage waves incident on the ports to those reflected from the ports
the scattering parameter is written as
[ ] [ ][ ]V S V
EE 41139 Microwave Techniques 48
Impedance, Admittance and Scattering Matrices
each element is given by
Sij is found by driving port j with an incident wave of voltage Vj
+, and measuring the reflected amplitude coming Vi
-, out of port i
SV
Vij
i
jV k jk
|,0
EE 41139 Microwave Techniques 49
Impedance, Admittance and Scattering Matrices
the incident waves on all ports except the jth port are set to zero which implies that all these ports are terminated with match loads
for a reciprocal network [S] = [S]t, i.e., the matrix is symmetric
EE 41139 Microwave Techniques 50
Impedance, Admittance and Scattering Matrices
for a lossless network
for all I,j
the scattering parameters can be readily measured by a Network Analyzer
S Skik
Nkj ij
1*
EE 41139 Microwave Techniques 51
A Shift in Reference Planes
the S parameters relate the amplitude of traveling wave incident on and reflected from a microwave network, phase reference planes must be specified for each port of the network
we need to know how the S parameters change when the reference planes are moved
EE 41139 Microwave Techniques 52
A Shift in Reference Planes
port1
z lz l =0=l1
V1+
-V1
V1
V1‘-
‘+
port NlN
EE 41139 Microwave Techniques 53
A Shift in Reference Planes
let the original reference at zl=0, the
incident and reflected port voltages are related by
at the new reference planes at zn=ln
[ ] [ ][ ]V S V
[ ' ] [ ' ][ ' ]V S V
EE 41139 Microwave Techniques 54
A Shift in Reference Planes
for a lossless transmission line
V V e
V V e l
n nj
n nj
n n n
n
n
'
' ,
EE 41139 Microwave Techniques 55
A Shift in Reference Planes
In matrix form, we have
e
e
e
V S
e
e
e
V
j
j
j
j
j
jN N
1
2
1
2
0
0
0
0
[ ] [ ] [ ]' '
[ ] [ ] [ ]' 'V
e
e
e
S
e
e
e
V
j
j
j
j
j
jN N
1
2
1
2
0
0
0
0
EE 41139 Microwave Techniques 56
A Shift in Reference Planes
note that each diagonal term is shifted by twice the electrical length of the shift in the reference plane, i.e., the shift is a round trip shift
[ ' ] [ ]S
e
e
e
S
e
e
e
j
j
j
j
j
jN N
1
2
1
2
0
0
0
0
EE 41139 Microwave Techniques 57
Generalized Scattering Parameters
note that not all the ports are of the same characteristic impedance, let the nth port has a characteristic impedance of Zon
EE 41139 Microwave Techniques 58
Generalized Scattering Parameters
we define a new set of wave amplitude as
an represents an incident wave at the
nth port and bn represents a reflected
wave from that port
a V Z
b V Z
n n on
n n on
/
/
EE 41139 Microwave Techniques 59
Generalized Scattering Parameters
at the reference plane, we have
V V V Z a b
IZ
V VZ
a b
n n n on n n
non
n non
n n
( )
( ) ( )1 1
EE 41139 Microwave Techniques 60
Generalized Scattering Parameters
the average power delivered to the nth port is
the average power delivered through port n is the incident power minus the reflected power
P V I a b b a b a a bn n n n n n n n n n n 1
2
1
2
1
22 2 2 2Re{ } Re{| | | | ( )} (| | | | )* * *
P V I a b b a b a a bn n n n n n n n n n n 1
2
1
2
1
22 2 2 2Re{ } Re{| | | | ( )} (| | | | )* * *
EE 41139 Microwave Techniques 61
Generalized Scattering Parameters
a generalized scattering matrix can be defined with the matrix element given by
or S
b
aiji
ja k jk
| ,0
SV Z
V Zij
i oj
i oiV k jk
| ,0
EE 41139 Microwave Techniques 62
Generalized Scattering Parameters
note that it only depends on the ratio of the characteristic impedances, not the characteristic impedance themselves
EE 41139 Microwave Techniques 63
Signal Flow Graph
the primary components of a signal flow graph are nodes and branches
EE 41139 Microwave Techniques 64
Signal Flow Graph
for a two-port network, we have
[S]port 1 port 2
b1
a1a2
b2
a1
b1
b2
a2
S11
S12
S22
S21
EE 41139 Microwave Techniques 65
Signal Flow Graph
each port has two nodes, node a is identify with a wave entering the port while node b is identify with a wave reflected from the port
nodes a and b are connected by a branch and each branch is associated with a scattering parameter
EE 41139 Microwave Techniques 66
Simplification of Signal Flow Graphs
series rule: two branches, whose common node has only one incoming and one outgoing wave many be combined to form a single branch
V2 V3V1
S21 S32
V1 V3S21S32
132213 VSSV
EE 41139 Microwave Techniques 67
Simplification of Signal Flow Graphs
parallel rule: two branches that are parallel may be combined as
EE 41139 Microwave Techniques 68
Simplification of Signal Flow Graphs
parallel rule: two branches that are parallel may be combined as
V 1 V 2
V 1 V 2
Sa
Sb
Sa Sb+
1ba2 V)SS(V
EE 41139 Microwave Techniques 69
Simplification of Signal Flow Graphs
self-loop rule: when a loop has a self-loop, it can be eliminated
V2 V3V1
S21 S32
V1 V3
S21/(1-S22)
S22
V2
S32
VS S
SV3 1
32 21
1 22
EE 41139 Microwave Techniques 70
Simplification of Signal Flow Graphs
splitting rule:a node may be split into two separate nodes
V2 V3V1
S21 S32
V1 V3
S42
V2
S32
V4 S21
S21
S42 V4
142212424 VSSVSV
EE 41139 Microwave Techniques 71
Mason’s rule
independent variable node is the node of an incident wave
dependent variable node is the node of a reflected wave
EE 41139 Microwave Techniques 72
Mason’s rule
path is a series of codirectional branches from an independent node to a dependent node, along which no node is crossed more than once, the value of a path is the product of all the branch coefficients along the path
EE 41139 Microwave Techniques 73
Mason’s rule
first-order loop is the product of branch coefficient encountered in a round trip from a node back to that same node, without crossing the node twice
second-order loop is the product of any two nontouching first-order loop
third-order loop is the product of three nontouching first-order loop
EE 41139 Microwave Techniques 74
Mason’s rule
the Mason’s rule for the ratio T of the wave amplitude of a dependent variable to the wave amplitude of an independent variable is given as
,are the coefficients of the possible paths connecting the independent and dependent variables
TP L L P L
L L L
11 1
221 1 2 1 1
1 1 2 3
[ ( ) ( ) ] [ ( ) ]
( ) ( ) ( )
P P1 2, ,
EE 41139 Microwave Techniques 75
Mason’s rule
are the sums of all the first-order, second-order, … loops
are the sums of all first-order, second-order, … loops that do not touch the first path between the variables
L L( ), ( )1 2
L L( ) , ( )1 21 1
EE 41139 Microwave Techniques 76
Mason’s rule
are the sums of all first-order, second-order, … loops that do not touch the second path between the variables and so on, for all the path between the independent and dependent variables
L L( ) , ( )1 22 2
EE 41139 Microwave Techniques 77
flow graph simplification
find in
in
a1
b1
S11
S21
S22
S12
l
b2
a2
EE 41139 Microwave Techniques 78
flow graph simplification
Splittling rule
in
a1
b1
S11
S21
S22
S12
l
b2
a2
l
EE 41139 Microwave Techniques 79
flow graph simplification
self-loop rule
in
a1
b1
S11
S21
S12
l
b2
a2
1-S22l
EE 41139 Microwave Techniques 80
flow graph simplification
series rule follows by the parallel rule yields
in
l
l
b
aS
S S
S
1
111
21 12
221
EE 41139 Microwave Techniques 81
flow graph simplification
Mason’s rule
Two paths
TP L L P L
L L L
11 1
221 1 2 1 1
1 1 2 3
[ ( ) ( ) ] [ ( ) ]
( ) ( ) ( )
in
a1
b1
S11
S21
S22
S12
l
b2
a2P S Sl2 21 12 P S1 11
EE 41139 Microwave Techniques 82
flow graph simplification
first-order loop
in
a1
b1
S11
S21
S22
S12
l
b2
a2
EE 41139 Microwave Techniques 83
flow graph simplification
there is no second-order loopthe sum of all the first-order loop not touching P1
the sum of all the first-order loop not touching P2 is zero
L S l( )1 22
L S l( )1 122
EE 41139 Microwave Techniques 84
flow graph simplification
Therefore,
this is the same result obtained by the other method
in
l l
l
l
l
S S S S
SS
S S
S
11 22 21 12
2211
21 12
22
1
1 1
( )