Electrical Engineering 3 ELECTROMAGNETICS: Transmission Lines Dr' P'J'S' Ewen

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Electrical Engineering 3ELECTROMAGNETICS
Transmission Lines
Dr. P.J.S. Ewen
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• ELECTROMAGNETICS TRANSMISSION LINES
• Arrangements are same as for Prof. Murray's part
  of the module
• Lectures Tuesdays 10.00 10.50 LT 1
• Thursdays 12.10 13.00 LT
  100 (J Black)
• Fridays 15.00 15.50
  LT G10 (Darwin)
• Tutorials Tuesdays 15.00 15.50 CR4

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• SYLLABUS
• Part 1 - Introduction and Basics
• Lecture Topics
• 1. General definition
• Practical definition
• Types of transmission line TE, TM, TEM
  modes
• TEM wave equation - equivalent circuit
  approach
• 2. The "Telegrapher's Equations"
• Solution for lossless transmission lines
  F(tx/v)
• Simplest case of F(tx/v)
• 3. Direction of travel of cos/sin (?t ßx)
  waves
• Phase velocity of a wave on a transmission
  line
• General transmission line attenuation

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• Part 2 - Characteristic Impedance and Reflections
• Lecture Topics
• 4. Current and voltage on a transmission
  line
• Characteristic impedance, ZO
  
• Characteristic impedance of lossless
  lines
• Characteristic impedance of general
  lines
• Infinitely long transmission lines
  
• Reflections on transmission lines
• 5. Transmission line with change of ZO
  voltage
• reflection coefficient
• Voltage reflection coefficient at an
  arbitrary
• distance l from the load ZL
• 6. Impedances of terminated lines
• Voltage Standing Wave Ratio (VSWR)
  
• Voltage Standing Wave measurement

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• Part 3 - The Smith Chart and its Applications
• Lecture Topics
• 7. Introduction to the Smith Chart
• Principle of operation
• Construction of the Smith Chart
• Key points on the Smith Chart
• 8. Using Smith Chart with load and line
  combinations
• Smith Chart and general transmission lines
  
• Effect of variation in frequency
• Smith Chart and VSWR
• Using the Smith Chart and VSWR to find ZL
• 9. Adding components using a Smith Chart
  
• Matching with Smith Chart and series
  components
• Admittance using a Smith Chart
• Single Stub Matching

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• Recommended Text
• J.D. Kraus and D.A. Fleisch,
• "Electromagnetics with Applications",
  
• McGraw-Hill, 1999. (5th Edition)
• This is a comprehensive text covering most
  of
• the material in the Electromagnetics module.
  It is
• also a recommended text for the 4th year
  module
• on RF Engineering.

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• Handout on Transmission Lines
• Lecture notes for all the lectures
• Lecture summaries
• Tutorial sheets A - D
• Lecture examples
• Formula sheet (same as for exam)
• Tutorial solutions will be distributed at
  tutorials.
• The "slides" will be available on the web - click
  
• on PJSE on the Electrical Engineering 3 page

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• Under certain circumstances all these can be
• regarded as transmission
  lines

Co-ax cable
Pair of wires
PCB tracks
IC interconnects
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• GENERAL DEFINITION
• A transmission line can be defined as a device
  for
• propagating or guiding energy from one point to
• another. The propagation
• of energy is for one of two
• general reasons

1. Power transfer (e.g. for lighting, heating,
performing work) - examples are mains
electricity, microwave guides in a microwave
oven, a fibre-optic illuminator.
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• 2. Information transfer examples are
  telephone, radio, and fibre-optic links (in each
  case the energy propagating down the transmission
  line is modulated in some way).

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CE amplifier circuit

• Because signals
• cannot travel
• faster than the
• speed of light, if
• the voltage at A
• changes it will take
• a finite time for the
• information to
• reach B during
• that time the
• voltages at A and
• B will be different.

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Example 1.1 - Voltage and phase difference
along a transmission line
A remote step-down transformer (B) is
connected by a transmission line 600 km long to
a generating station (A) supplying 50 Hz AC. At
time t 0 the generator is switched into the
line and the voltage at the generator is at its
maximum, Vm. What are the voltage and phase
differences between the ends of the line at the
instant power reaches the transformer?
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VA VB
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• Example 1.2 - Phase difference between the ends
  of a cable.
• Determine the phase difference between the ends
  of
• (a) a 10m length of mains cable for a 50Hz
  electricity supply
• (b) a 10m length of coaxial cable carrying a
  750MHz TV signal

?
N.B. one wavelength corresponds to one complete
cycle or wave, and hence to a phase change of
360º or 2p radians. So the phase change over a
distance l is just 360º ? l /
? (or 2p ? l / ? radians)
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• PRACTICAL DEFINITION

We have to treat a conducting system as a
transmission line if the wavelength of the signal
propagating down the line is less than or
comparable with the length of the line
Associated with transmission lines there may be

• Propagation losses
• Distortion
• Interference due to reflection at the load
• Time delays
• Phase changes

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• Some different types of transmission lines

Cross section
2-wire line (dc)
2-wire line (ac)
Coaxial line (dc, ac, rf)
Microstrip line (rf)
Rectangular waveguide (rf)
Optical fibre (light)
Radio link with antennas
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Microstrip line
cross section
dielectric
conductors
conductor
dielectric
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Waveguide
rectangular waveguides
cross section
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Optical fibre
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• MODES OF PROPAGATION

The energy propagating down a transmission line
propagates as a wave. Different modes of
propagation (i.e. different patterns of E and H
fields) are possible. These fall into two
categories TE TRANSVERSE ELECTRIC TM
TRANSVERSE MAGNETIC

• TEM Modes In the special case
• where E and H are both transverse
• (i.e. at right angles) to the direction
• of energy flow, the mode is termed TEM.
• E and H will also be at right angles to each
  other.
• TEM TRANSVERSE ELECTROMAGNETIC

TE mode
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• The kinds of mode that can propagate down a line
  
• depend on the geometry and materials of the
  line.
• Transmission lines can be classified into 2
  groups
• according to the type of mode that normally
• propagates down them.

• 1. LINES PROPAGATING TEM MODES

• There is no E or H field in the direction of
  propagation.
• twin-wire, coaxial, stripline and
  (approximately) microstrip lines are in this
  group.

2. LINES PROPAGATING TE OR TM MODES

• E or H have components in the direction of energy
  flow.
• waveguides and optical fibres are in this group.

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• TEM WAVE EQUATION

The details of wave propagation on a transmission
line can be deduced from Maxwell's Equations.
However, TEM guided waves on a transmission line
can also be analysed using a lumped equivalent
circuit approach.
EQUIVALENT CIRCUIT APPROACH TO
TRANSMISSION LINE ANALYSIS
Real transmission lines have associated with
them a resistance per unit length, R a
capacitance per unit length, C an inductance per
unit length, L and a (leakage) conductance per
unit length, G. (Note that R represents the
resistance of both conductors in the line.)
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• FOR PARALLEL WIRES

for wires in air and with d gtgt a a wire
radius, d wire spacing
FOR COAXIAL CABLE
a radius of inner conductor b inner radius of
outer conductor
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• EQUIVALENT CIRCUIT FOR A
• TRANSMISSION LINE

The existence of an inductance, capacitance,
resistance and conductance (per unit length)
allows us to represent the transmission line by
an equivalent circuit in which each
infinitessimal length of transmission line is
represented by the same combination of 4
components
?x
To make up the whole line, repeat the equivalent
circuit a sufficient number of times.
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PRIMARY LINE CONSTANTS
C capacitance per unit length (F/m) L
inductance per unit length (H/m) R resistance
per unit length (W/m) G conductance per unit
length (S/m)

• Note-
• R, C, L and G are all expressed per unit length
• R G should be small for a good transmission
  line.
• If R 0 and G 0, the line is termed
  lossless.

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• Example 1.3 - Impedance of an infinite lossless
  transmission line.

Determine an expression for the impedance of an
infinite, lossless transmission line.
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• Summary

• PRACTICAL DEFINITION - Transmission line
  analysis must be used when the wavelength of the
  energy propagating down the line is less than or
  comparable with the length of the line.
• Definition of TE, TM TEM modes - twin-wire,
  coaxial and strip lines only propagate TEM modes.
• Transmission lines have capacitance, inductance
  and resistances associated with them and can be
  represented by an equivalent circuit.

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• C, L, R and G are the primary line constants and
  are
• all expressed PER UNIT LENGTH
• For a LOSSLESS LINE, R 0 and G 0

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• Some different types of transmission lines