Title: Ch 08 Transmission Lines
1Ch 08 Transmission Lines
2Transverse Electromagnetic Waves (TEM)
- TEM waves propagate in the nonconductor
(dielectric) that separate the two conductors - For a transverse wave, the direction of
propagation is perpendicular to the direction of
(charge) displacement - EM wave is produced by the acceleration of an
electric charge
3Transverse Electromagnetic Waves
In free space
z
Direction of Propagation
y
Magnetic Field
Electric Field
x
4EM waves
- In a conductor, current voltage are always
accompanied by an electric (E) and magnetic field
(H) in the nearby space - E H fields are perpendicular to each other at
90o angles - EM waves that travel along a transmission line
from source to load are called incident waves
and those travel back reflected waves
5Types of Transmission Lines
- Balanced two wires, twisted, untwisted,
shielded, unshielded, open wire one conductor
carries the signal and the other is the return - Unbalanced lines (where one conductor is
grounded) e.g. concentric or coaxial cable. - Transmission lines for microwave use e.g.
striplines, microstrips, and waveguides.
6Equivalent Circuit of a Transmission Line
- Characteristics of a T L (uniformly distributed)
are determined by - Electrical proprieties wire conductivity and
insulator dielectric. - Physical properties wire diameter and conductor
spacing - Primary electric constants
- Series DC resistance R Inductance L
- Shunt Capacitance C Conductance G
7Equivalent Circuit of a Transmission Line Contd
- The primary constants are uniformly distributed
throughout the length of the line, hence called
distributed parameters. - For simplification, distributed parameters are
put up together per a given unit length to form
an - "electrical artificial model"
8Transmission Line Equivalent Circuit
L
L
L
L
R
R
Zo
Zo
C
C
C
C
G
G
Lossless Line
Lossy Line
9Notes on Transmission Line
- Characteristics of a line is determined by its
primary electrical constants or distributed
parameters R (?/m), L (H/m), C (F/m), and G
(S/m). - Characteristic impedance, Zo, is defined as the
input impedance of an infinite line or that of a
finite line terminated with a load impedance,
ZL Zo.
10Matched lines
- The i/p impedance of an infinitely long line at
radio frequency is resistive and equal to Zo - For maximum power transfer from source to load,
the T L must be terminated with the load equal to
the characteristic impedance of the line, also
called matched TL - E/M waves in matched lines travel down the line
without reflection - The ratio of voltage to current at any point
along the line is equal to Z o
11Formulas for Some Lines
For parallel two-wire line
D
m momr e eoer mo 4px10-7 H/m eo 8.854
pF/m
d
For co-axial cable
D
d
12Transmission-Line Wave Propagation
Electromagnetic waves travel at lt c in a
transmission line because of the dielectric
separating the conductors. The velocity of
propagation is given by
m/s
Velocity factor, VF, is defined as
13Transverse Electromagnetic Waves
In free space
z
Direction of Propagation
y
Magnetic Field
Electric Field
x
14Propagation Constant
- Propagation constant, ?, determines the variation
of V or I with distance along the line V
Vse-x? I Ise-x?, where VS, and IS are the
voltage and current at the source end, and x
distance from source. - ? ? j?, where ? attenuation coefficient (
0 for lossless line), and ? phase shift
coefficient 2?/? (rad./m)
15Incident Reflected Waves
- For an infinitely long line or a line terminated
with a matched load, no incident power is
reflected. The line is called a flat or
nonresonant line. - For a finite line with no matching termination,
part or all of the incident voltage and current
will be reflected.
16Reflection Coefficient
The reflection coefficient is defined as
It can also be shown that
Note that when ZL Zo, ? 0 when ZL 0, ?
-1 and when ZL open circuit, ? 1.
17Standing Waves
Vmax Ei Er
Voltage
Vmin Ei - Er
l 2
With a mismatched line, the incident and
reflected waves set up an interference pattern on
the line known as a standing wave. The standing
wave ratio is
18Other Formulas
When the load is purely resistive (whichever
gives an SWR gt 1)
Return Loss, RL Fraction of power reflected
?2, or -20 log ? dB So, Pr ?2Pi
Mismatched Loss, ML Fraction of
power transmitted/absorbed 1 - ?2 or -10
log(1-?2) dB So, Pt Pi (1 - ?2) Pi - Pr
19Time-Domain Reflectometry
d
ZL
Transmission Line
Oscilloscope
Pulse or Step Generator
TDR is a practical technique for determining
the length of the line, the way it is terminated,
and the type and location of any impedance
discontinuities. The distance to the
discontinuity is d vt/2, where t elapsed
time of returned reflection.
20Transmission-Line Input Impedance
The input impedance at a distance l from the load
is
When the load is a short circuit, Zi jZo tan
(?l).
For 0 lt l lt ?/4, shorted line is inductive.
For l ?/4, shorted line a parallel resonant
circuit.
For ?/4 lt l lt ?/2, shorted line is capacitive.
21T-L Input Impedance (contd)
- When the load is an open circuit,
Zi -jZo cot (?l) - For 0 lt l lt ?/4, open circuited line is
capacitive. - For l ?/4, open-line series resonant circuit.
- For ?/4 lt l lt ?/2, open-line is inductive.
- A ?/4 line with characteristic impedance, Zo,
can be used as a matching transformer between a
resistive load, ZL, and a line with
characteristic impedance, Zo, by choosing
22Transmission Line Summary
or
is equivalent to
l gt ?/4
l lt ?/4
is equivalent to
or
l gt ?/4
l lt ?/4
?/4
Zo
ZL
Zo
l ?/4
?/4-section Matching Transformer
23The Smith Chart
- The Smith chart is a graphical aid to solving
transmission-line impedance problems. - The coordinates on the chart are based on the
intersection of two sets of orthogonal circles. - One set represents the normalized resistive
component, r ( R/Zo), and the other the
normalized reactive component, jx ( jX/Zo).
24Smith Chart Basics
j0.7
r 0
z1 1j0.7
z1
r 2
j0
?
z2
z2 2-j1.4
r 1
-j1.4
25Applications of The Smith Chart
- Applications to be discussed in this course
- Find SWR, ???, RL
- Find YL
- Find Zi of a shorted or open line of length l
- Find Zi of a line terminated with ZL
- Find distance to Vmax and Vmin from ZL
- Solution for quarter-wave transformer matching
- Solution for parallel single-stub matching