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Electrical Engineering 3ELECTROMAGNETICS

Transmission Lines

Dr. P.J.S. Ewen

- 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

- 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

- 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

- 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

- 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.

- 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

(No Transcript)

- Under certain circumstances all these can be
- regarded as transmission

lines

Co-ax cable

Pair of wires

PCB tracks

IC interconnects

- 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.

- 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).

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.

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?

VA VB

- 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)

- 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

- 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

Microstrip line

cross section

dielectric

conductors

conductor

dielectric

Waveguide

rectangular waveguides

cross section

Optical fibre

- 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

- 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.

- 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.)

- 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

- 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.

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.

- Example 1.3 - Impedance of an infinite lossless

transmission line.

Determine an expression for the impedance of an

infinite, lossless transmission line.

- 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.

- 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

- Some different types of transmission lines