Electrical Engineering 3

ELECTROMAGNETICS: Transmission Lines

Dr. P.J.S. Ewen

ELECTROMAGNETICS – TRANSMISSION LINES

Arrangements are same as for Prof. Murray's part of the course:

Lectures: Tuesdays 9.00 – 9.50 LT 2 Thursdays 15.10 – 16.00 LT 2 Fridays 15.10 – 16.00 LT 2

Tutorials: Tuesdays 15.10 – 16.00 CR10

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(t±x/v) Simplest case of F(t±x/v)

3. Direction of travel of cos/sin (ωt ± βx) 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

Part 1 - Introduction and Basics

Lecture 1• Define what is meant by a

transmission line• Look at different types of line

Lecture 2• Equations governing the

current and voltage on a transmission line

• Solution to these equations for the simplest case – current and voltage propagate as waves

I or V

For a forward For a forward travellingtravelling wave:wave:

V = VV = V11 e e jjtt eexx = V= V11 ee x x e e j(j(tt--xx))

amplitude factor amplitude factor phase factor phase factor

time variation time variation

VoltageVoltageVV1 1

xx

Distance, xLecture 3• Properties of waves on a

transmission line• General transmission line -

attenuation

− Reflectedvoltage wave

+x -x

− Forward voltage wave

Part 2 - Characteristic Impedance and Reflections

Lecture 4• Characteristic impedance, ZO

• Reflections on transmission lines

Lecture 5• Voltage reflection coefficient

Lecture 6• Impedances of terminated lines

Vr / Vi = r

Zin ≠ Zo + ZL Zo ZL

Lecture 7• Introduction to the Smith Chart

• Principle of operationLecture 8• Using the Smith Chart to solve

various transmission line problems

Lecture 9• Designing matching circuits

using the Smith Chart

Part 3 - The Smith Chart and its Applications

Source Transmission line (Zo) Load

Matchingcircuit ornetwork

ZL ZoForward power only

Zo

Recommended Text:

J.D. Kraus and D.A. Fleisch, "Electromagnetics with Applications", McGraw-Hill.

This is a comprehensive text covering most of the material in the Electromagnetics course. It is also a recommended text for the 4th year course 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 (and are available on LEARN).

The PowerPoint "slides" are available on the web …

… go to the“Electromagnetics 3: Signal Transmission” page on Learnand click on “Teaching materials – Transmission Lines”

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

Co-ax cablePair of wires

PCB tracks IC interconnects

GENERAL DEFINITION

A transmission line can be defined as a device forpropagating or guiding energy from one point to another. The propagation of energy is for one of twogeneral 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).

Because signalscannot 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.

Vs

Ci

Vcc

Co

CERER2

R1 RC

Vo

xA Bx

CE amplifier circuit

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?

A

G

BA

G

Generating station Transformer

600 km

Vm

0.81Vm

VA

Time0.002s

VA VB

GeneratorTransformer

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 2π radians. So the phase change over a distance l is just 360º l / λ (or 2π l / λ radians)

λ

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, l , of the line:

l

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:

Radio link with antennas

2-wire line (dc)

2-wire line (ac)

Coaxial line (dc, ac, rf)

Microstrip line (rf)

Rectangularwaveguide (rf)

Opticalfibre (light)

Cross section

Microstrip line

cross section

conductor

dielectric

dielectricconductors

Waveguide

cross section

rectangular waveguides

Optical fibre

MODES OF PROPAGATIONThe energy propagating down a transmission line propagates as an electromagnetic wave. Different patterns of E and H fields are possible for these waves. Each pattern constitutes a “mode of propagation”:

MODES OF PROPAGATION

These modes 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

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.

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.

TEM WAVE EQUATIONThe 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 alumped 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, Can inductance per unit length, Land 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 >> a:

a = wire radius d = wire spacingεo = permittivity of free spaceμo = permeability of free space

FOR COAXIAL CABLE:

a = radius of inner conductorb = inner radius of outer conductorε = permittivity of dielectric in cableμ = permeability of dielectric in cable

Cd a

F m

L d a H m

o

o

ln( / )/

ln( / ) /

Cb a

F m

L b a H m

2

2

ln( / )/

ln( / ) /

TEM WAVE EQUATIONThe 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 alumped 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, Can inductance per unit length, Land a (leakage) conductance per unit length, G.

(Note that R represents the resistance of both conductors in the line.)

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:

Δx

RΔx LΔx

CΔx GΔx

To make up the whole line, repeat the equivalent circuit a sufficient number of times.

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”.

PRIMARY LINE CONSTANTS

Dx

RDx

LDx CD

xGDx

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, l , of the line:

l

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

Dx

RDx

LDx CD

xGDx