Transmission Line Analysis
Propagating electric field
Space factor
E X = E0 X cos(t kz )
Time factor
Phase velocity
vp = f =
1
=
c
r
Traveling voltage wave
V ( z , t ) = E0 X
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sin(t kz )
k
194
High frequency implies spatial voltage distribut
From Reflection Coefficient to Load
Impedance (Smith Chart)
Reflection coefficient in phasor form
Z L Z0
0 =
= 0 r + j0i =| 0 | e j L
Z L + Z0
The load reflection
coefficient is identified in
the complex domain
0
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2
Normalized impedance
1 + (d
Special Termination Conditions
Lossless transmission line
Z in ( d ) = Z 0
Z L + jZ 0 tan( d )
Z 0 + jZ L tan( d )
Characteristic impedance
Z0 =
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L
C
2
Input impedance of short circuit transmission line
Voltage:
V (d ) = 2 jV + sin( d )
Curren
RF Behavior of Passive Components
Conventional circuit analysis
R is frequency independent
Ideal inductor: X L = jL
Ideal capacitor: X C = 1
C
Evaluation
Impedance chart
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2
Impedance Chart
(impedance of C & L vs frequency)
ZC=1/(2fC)
ZL=2f
General Transmission Line Equations
Detailed analysis of a differential section
Note: Analysis applies to all types of transmission lines such as
coax cable, two-wire, microstrip, etc.
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2
Kirchhoffs laws on a microscopic level
Over a differen
Single and Multi-Port Networks
basic current and voltage definitions definitions
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2
Impedance and admittance networks
cfw_V = [ Z ]cfw_I
cfw_I = [Y ]cfw_V
cfw_V = [ Z ][Y ]cfw_I
1
[Y ] = [ Z ]
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3
Example Z-representation o
Parallel Connection of R and L Elements
(Smith Chart)
parallel connection of R and L elements
1
yin ( L ) = g j
L LY0
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2
Parallel connection of R and C elements
yin (L ) = g + jZ0 LC
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3
Series connection of R and L elements
LL
Lossless Transmission Line Model
Lossless implies:
R = G = 0!
Line representation
Characteristic impedance:
Z0 =
( R + j L )
(G + j C )
Note: R, L, G, C are given per unit length and depend on geometry
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2
Transmission Line Parameters for diffe
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m.
m
m
n.
H.
m
m.
M
u
m
m
m
m
m
JHPEDHH'CE GR Dhl'fTTnHCE EDIDHDIHATEE
Fig. +35. hnpednme-admiumc: mnveniun an Lb. Smith Chart. RF Emu-1113mm
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1
I.
1.
u
I
H
m
E
m
V
n
R
a.
w
I
I
k
n
m
E.
S
.
u
E
H
u
H
T
s
m
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a
J.
.-
. H
.l
M
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a
r
Impedance Transformation
(Smith Chart)
Reflection coefficient in phasor form
0
Z L Z0
0 =
= 0 r + j0i =| 0 | e j L
Z L + Z0
Z in (d ) / Z 0 = zin = r + jx =
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1 + (d ) 1 + r + ji
=
1 (d ) 1 r ji
2
Generic Smith Chart computation
Normalize load
Sourced and Loaded Transmission Lines
Lossless transmission line with source
Z Z0
= G
ZG + Z0
Z L Z0
=
ZL + Z0
Z in
)
Vin = V (1 + in ) = VG (
Z in + Z G
+
in
Voltage at the beginning of the transmission line is
composed of an incident and reflected comp