Ch 08 Transmission Lines

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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

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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

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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"

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

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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

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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
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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
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Transverse Electromagnetic Waves
In free space
z
Direction of Propagation
y
Magnetic Field
Electric Field
x
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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)

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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.

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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.
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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
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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
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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.
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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.
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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

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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

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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).

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