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Ch 08 Transmission Lines

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... 4 line with characteristic impedance, Zo', can be used as a matching transformer ... One set represents the normalized resistive component, r (= R/Zo), and ... – PowerPoint PPT presentation

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Title: Ch 08 Transmission Lines


1
Ch 08 Transmission Lines
2
Transverse 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

3
Transverse Electromagnetic Waves
In free space
z
Direction of Propagation
y
Magnetic Field
Electric Field
x
4
EM 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

5
Types 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.

6
Equivalent 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

7
Equivalent 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"

8
Transmission Line Equivalent Circuit
L
L
L
L
R
R
Zo
Zo
C
C
C
C
G
G
Lossless Line
Lossy Line
9
Notes 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.

10
Matched 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

11
Formulas 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
12
Transmission-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
13
Transverse Electromagnetic Waves
In free space
z
Direction of Propagation
y
Magnetic Field
Electric Field
x
14
Propagation 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)

15
Incident 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.

16
Reflection 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.
17
Standing 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
18
Other 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
19
Time-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.
20
Transmission-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.
21
T-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

22
Transmission 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

23
The 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).

24
Smith Chart Basics
j0.7
r 0
z1 1j0.7
z1
r 2
j0
?
z2
z2 2-j1.4
r 1
-j1.4
25
Applications 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
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