EM Fundamentals and Antenna Selection Guide

Antennas in Practice
EM fundamentals and antenna selection
Alan Robert Clark
Andr´e P C Fourie
Version 1.4, December 23, 2002

ii
Titles in this series:
Antennas in Practice: EM fundamentals and antenna selection
(ISBN 0-620-27619-3)
Wireless Technology Overview: Modulation, access methods,
standards and systems (ISBN 0-620-27620-7)
Wireless Installation Engineering: Link planning, EMC, site
planning, lightning and grounding (ISBN 0-620-27621-5)
Copyright c 2001 by Alan Robert Clark and
Poynting Innovations (Pty) Ltd.
33 Thora Crescent
Wynberg
Johannesburg
South Africa.
www.poynting.co.za
Typesetting, graphics and design by Alan Robert Clark.
Published by Poynting Innovations (Pty) Ltd.
This book is set in 10pt Computer Modern Roman with a 12 pt leading by LATEX 2ε.
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior written
permission of Poynting Innovations (Pty) Ltd. Printed in South Africa.
ISBN 0-620-27619-3

Contents
1 Electromagnetics 1
1.1 Transmission line theory . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Characteristic impedance & Velocity of propagation . . . 3
Two-wire line . . . . . . . . . . . . . . . . . . . . . . . . . 4
One conductor over ground plane . . . . . . . . . . . . . . 5
Twisted Pair . . . . . . . . . . . . . . . . . . . . . . . . . 5
Coaxial line . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . . 6
Slotline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Impedance transformation . . . . . . . . . . . . . . . . . . 6
1.1.4 Standing Waves, Impedance Matching and Power Transfer 8
1.2 The Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Frequency and wavelength . . . . . . . . . . . . . . . . . . 10
1.3.2 Characteristic impedance & Velocity of propagation . . . 11
1.3.3 EM waves in free space . . . . . . . . . . . . . . . . . . . 12
1.3.4 Reflection from the Earth’s Surface . . . . . . . . . . . . . 14
1.3.5 EM waves in a conductor . . . . . . . . . . . . . . . . . . 16
2 Antenna Fundamentals 17
2.1 Directivity, Gain and Pattern . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Solid angles . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.2 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Isotropic source . . . . . . . . . . . . . . . . . . . . . . . . 18
Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.4 Radiation pattern . . . . . . . . . . . . . . . . . . . . . . 20
Directivity estimation from beamwidth . . . . . . . . . . . 22
2.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Effective aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Free-space link equation and system calulations . . . . . . . . . . 25
3 Matching Techniques 29
3.1 Balun action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Unbalance and its effect . . . . . . . . . . . . . . . . . . . 29
iii

iv CONTENTS
3.1.2 Balanced and unbalanced lines—a deﬁnition . . . . . . . . 30
3.2 Impedance Transformation . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Angle Correction . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Common baluns/balun transfomers . . . . . . . . . . . . . . . . . 33
3.3.1 Sleeve Balun . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Half-wave balun . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.3 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.4 LC Networks . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.5 Resistive Networks . . . . . . . . . . . . . . . . . . . . . . 34
3.3.6 Quarter wavelength transformer . . . . . . . . . . . . . . 35
3.3.7 Stub matching . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.8 Shifting the Feedpoint . . . . . . . . . . . . . . . . . . . . 36
3.3.9 Ferrite loading . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Impedance versus Gain Bandwidth . . . . . . . . . . . . . . . . . 38
4 Simple Linear Antennas 41
4.1 The Ideal Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Radiation resistance . . . . . . . . . . . . . . . . . . . . . 43
4.1.3 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.4 Concept of current moment . . . . . . . . . . . . . . . . . 44
4.2 The Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Radiation resistance . . . . . . . . . . . . . . . . . . . . . 45
4.2.3 Reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.4 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 The Short Monopole . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.1 Input impedance . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 The Half Wave Dipole . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.1 Radiation pattern . . . . . . . . . . . . . . . . . . . . . . 48
4.4.2 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.3 Input impedance . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 The Folded Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Dipoles Above a Ground Plane . . . . . . . . . . . . . . . . . . . 52
4.7 Mutual Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Arrays and Reﬂector Antennas 57
5.1 Array Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.1 Isotropic arrays . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.2 Pattern multiplication . . . . . . . . . . . . . . . . . . . . 58
5.1.3 Binomial arrays . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1.4 Uniform arrays . . . . . . . . . . . . . . . . . . . . . . . . 60
Beamwidth . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Dipole Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.1 The Franklin array . . . . . . . . . . . . . . . . . . . . . . 65
5.2.2 Series fed collinear array . . . . . . . . . . . . . . . . . . . 65
5.2.3 Collinear folded dipoles on masts . . . . . . . . . . . . . . 67
5.3 Yagi-Uda array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 Pattern formation and gain considerations . . . . . . . . . 68

CONTENTS v
5.3.2 Impedance and matching . . . . . . . . . . . . . . . . . . 69
5.3.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Element correction . . . . . . . . . . . . . . . . . . . . . . 69
Stacking and Spacing . . . . . . . . . . . . . . . . . . . . 70
5.4 Log Periodic Dipole Array . . . . . . . . . . . . . . . . . . . . . . 71
5.4.1 Design procedure . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.2 Feeding LPDA’s . . . . . . . . . . . . . . . . . . . . . . . 75
5.5 Loop Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5.1 The small loop . . . . . . . . . . . . . . . . . . . . . . . . 76
Radiation pattern . . . . . . . . . . . . . . . . . . . . . . 77
Input impedance . . . . . . . . . . . . . . . . . . . . . . . 77
5.6 Helical Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6.1 Normal-mode . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6.2 Axial mode . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Original Kraus design . . . . . . . . . . . . . . . . . . . . 81
King and Wong design . . . . . . . . . . . . . . . . . . . . 81
5.7 Patch antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.8 Phased arrays and Multi-beam “Smart Antennas” . . . . . . . . 87
5.9 Flat reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.10 Corner Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.11 Parabolic Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . 91
A Smith Chart 95

List of Figures
1.1 A transmission line connects a generator to a load. . . . . . . . . 1
1.2 2-wire; Coax; µstrip; waveguide; fibre; RF; 50Hz! . . . . . . . . . 2
1.3 A “Lumpy” model of the TxLn, discretizing the distributed pa-
rameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Twisted-Pair geometry . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Impedance values plotted on the Smith Chart . . . . . . . . . . . 9
1.6 Electromagnetic Wave in free-space . . . . . . . . . . . . . . . . . 13
1.7 Radio Horizon due to earth curvature. . . . . . . . . . . . . . . . 13
1.8 Geometry of Interference between Direct Path and Reflected Waves 14
1.9 Interference pattern field strength contours for h/λ = 1.44 . . . . 15
2.1 Standard coordinate system. . . . . . . . . . . . . . . . . . . . . . 17
2.2 concept of a solid angle . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Three-dimensional pattern of a 10 director Yagi-Uda array . . . . 21
2.4 xy plane cut of the 10 director Yagi-Uda . . . . . . . . . . . . . . 21
2.5 Rectanglar radiation patterns of the 10-director Yagi-Uda in the
two principle planes . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Reciprocity concepts . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Polarization possibilities—wave out of page . . . . . . . . . . . . 23
2.8 Aperture efficiency of a parabolic dish . . . . . . . . . . . . . . . 25
2.9 Effective aperture of dipole physical aperture . . . . . . . . . 25
2.10 Apertures must not overlap! . . . . . . . . . . . . . . . . . . . . . 25
2.11 Point-to-point link parameters . . . . . . . . . . . . . . . . . . . 26
3.1 The reason for unbalanced currents . . . . . . . . . . . . . . . . . 29
3.2 Balanced and Unbalanced Transmission lines . . . . . . . . . . . 30
3.3 Power loss in dB versus VSWR on the line. . . . . . . . . . . . . 31
3.4 Increase in Line Loss because of High VSWR . . . . . . . . . . . 32
3.5 VSWR at Input to Transmission Line versus VSWR at Antenna 32
3.6 The Bazooka or Sleeve balun. . . . . . . . . . . . . . . . . . . . . 33
3.7 A HalfWave 4:1 transformer balun. . . . . . . . . . . . . . . . . . 33
3.8 Transmission-line transformer. . . . . . . . . . . . . . . . . . . . 34
3.9 L-Match network. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.10 Quarter wavelength transformer . . . . . . . . . . . . . . . . . . . 35
3.11 Single Matching Stub . . . . . . . . . . . . . . . . . . . . . . . . 36
3.12 Smith Chart of Single Stub Tuner Arrangement . . . . . . . . . . 37
3.13 T and Gamma Matching Sections . . . . . . . . . . . . . . . . . . 37
3.14 Gain bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
vii

viii LIST OF FIGURES
3.15 Impedance bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 The Ideal Dipole in Relation to the Coordinate System . . . . . . 42
4.2 Pattern of an Ideal Dipole Antenna . . . . . . . . . . . . . . . . . 43
4.3 Current Distribution on a Short Dipole Antenna . . . . . . . . . 45
4.4 The equivalent circuit of a short dipole antenna . . . . . . . . . . 46
4.5 Tuning out dipole capacitive reactance with series inductance . . 46
4.6 Short monopole antenna . . . . . . . . . . . . . . . . . . . . . . . 47
4.7 A Half wave dipole and its assumed current distribution . . . . . 48
4.8 Half wave dipole, short dipole and isotrope patterns . . . . . . . 49
4.9 Shortening factors for different thickness half wave dipoles . . . . 50
4.10 Half Wave Folded Dipole Antenna . . . . . . . . . . . . . . . . . 50
4.11 VSWR bandwidth of a dipole and folded dipole. . . . . . . . . . 51
4.12 Impedance transformation using different thickness elements . . . 52
4.13 Triply Folded Dipoles . . . . . . . . . . . . . . . . . . . . . . . . 52
4.14 A horizontal dipole above a ground plane . . . . . . . . . . . . . 53
4.15 Two dipoles in eschelon . . . . . . . . . . . . . . . . . . . . . . . 54
4.16 Mutual impedance between two parallel half wave dipoles placed
side by side—as a function of their separation, d . . . . . . . . . 55
4.17 Change in resistance of a half wave dipole due to coupling to its
image at different heights above a ground plane . . . . . . . . . . 56
4.18 Mutual impedance between two collinear dipoles . . . . . . . . . 56
5.1 Two Isotropic point sources, separated by d . . . . . . . . . . . . 57
5.2 Two Isotropic Sources separated by λ/2 . . . . . . . . . . . . . . 58
5.3 Pattern multiplication. . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4 Binomial array pattern . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Uniform linear array of isotropic sources. . . . . . . . . . . . . . . 60
5.6 Uniform Isotropic Broadside array . . . . . . . . . . . . . . . . . 61
5.7 Eight In-Phase Isotropic sources, dB vs Linear scales. . . . . . . 61
5.8 Uniform Isotropic Endfire Array . . . . . . . . . . . . . . . . . . 62
5.9 Array pattern of 2 isometric sources 10λ apart, and the element
pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.10 Two in-phase dipoles 10λ apart. . . . . . . . . . . . . . . . . . . 64
5.11 SuperNEC run of the 10λ Interferometer . . . . . . . . . . . . . 65
5.12 The Franklin array . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.13 A Series Fed Four Element Collinear Array . . . . . . . . . . . . 66
5.14 Distortion to folded dipole azimuth pattern in presence of a mast 67
5.15 Yagi-Uda array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.16 Multiplication factor for different diameter to wavelength ratios
of director and reflectors . . . . . . . . . . . . . . . . . . . . . . . 70
5.17 Graph showing the length to be added to parasitic elements to
compensate for the effect of the supporting boom . . . . . . . . . 71
5.18 The log-periodic dipole array . . . . . . . . . . . . . . . . . . . . 72
5.19 Constant directivity contours (dBi) . . . . . . . . . . . . . . . . . 73
5.20 Characteristics for feeder impedances of 100, 250 and 400Ω and
dipoles with L/D ratios of 177, 500 and 1000 . . . . . . . . . . . 74
5.21 Coaxial Connection to LPDA’s . . . . . . . . . . . . . . . . . . . 76
5.22 The small circular loop and the equivalent square loop . . . . . . 76
5.23 The Radiation Pattern of a Short Dipole and a Small Loop . . . 78

LIST OF FIGURES ix
5.24 The normal-mode helix antenna and its radiation pattern . . . . 79
5.25 Axial Mode Helix Antenna and its Typical Pattern. . . . . . . . 80
5.26 A λ/2 × λ patch (of copper) on a dielectric slab, over a ground
plane. The patch is fed by coax through the dielectric, halfway
along the longest edge. . . . . . . . . . . . . . . . . . . . . . . . . 83
5.27 Geometry of square patch, as simulated in SuperNEC . . . . . . 84
5.28 3D pattern of the square patch. . . . . . . . . . . . . . . . . . . . 85
5.29 2D cut through the maximum gain of the square patch . . . . . . 85
5.30 Geometry and 3D pattern of a 4-square patch array. . . . . . . . 86
5.31 2D cut through the maximum gain of the 4-patch array. . . . . . 86
5.32 Power Splitter followed by phase modification. . . . . . . . . . . 87
5.33 Binary phase shifter, based on transmission line segments. . . . . 88
5.34 Electrically achieved DownTilt by progressive phasing. . . . . . . 88
5.35 A Corporate feed network—equal amplitude and phase. . . . . . 89
5.36 Gain vs separation for a 0.75λ plate . . . . . . . . . . . . . . . . 90
5.37 Impedance vs separation for a 0.75λ plate . . . . . . . . . . . . . 90
5.38 The SuperNEC rendition of a corner reflector. . . . . . . . . . . 91
5.39 SuperNEC predicted xy-plane radiation pattern of a corner re-
flector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.40 (a)Ordinary parabolic dish with feed blockage; (b) an offset feed;
(c) Cassegrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Preface
This book “Antennas in Practice” has been in existence in a multitude of forms
since about 1989. It has been run as a Continuing Engineering Education (CEE)
course only sporadically in those years.
It has been re-vamped on several occasions, mainly reflecting changing type-
setting and graphics capabilities, but this (more formal) incarnation represents
a total re-evaluation, re-design and re-implementation. Much (older) material
has been excised, and a lot of new material has been researched and included.
Wireless technology has really moved out of the esoteric and into the common-
place arena. Technologies like HiperLAN, Bluetooth, WAP, etc are well known
by the layman, and are promising easy, wireless “connectivity” at ever increasing
rates. Reality is a little different, and is dependant on a practical understanding
of the antenna issues involved in these emerging technologies.
Although a fair amount of background theory is covered, its goal is to provide
a framework for understanding practical antennas that are useful. Many design
issues are covered, but in many cases the “cookbook” designs offered in this book
are good-enough starting points, but still nonoptimal designs, only achievable
by simulation, and testing.
As a result, the book places a fair amount of emphasis on antenna simulation
software, such as SuperNEC. Ordinarily, if this book is run as a CEE course, it
is accompanied by a SuperNEC Simulation Workshop, a hands-on introduction
to SuperNEC. It is only through “playing” with simulation software that a gut-
feel is attained for many of the issues at stake in antenna design.
xi

Chapter 1
Electromagnetics
1.1 Transmission line theory
TRANSMISSION lines connect generators to loads as shown in fig 1.1. In
the RF world, in the transmitting case, this is viewed as connecting the
transmitter to the antenna, and in the receiving case as connecting the antenna
to the receiver.
VGen
RGen
Generator Transmission Line
ZLoad
Load
TxLn
Figure 1.1: A transmission line connects a generator to a load.
From a standard circuits analysis perspective, the transmission line simply con-
sists of connecting two parts of the circuit, and does not change anything. By
Kirchhoff’s Voltage law, there is no change in voltage or current along the length
of the “connection”.
As a rule of thumb, as soon as the “connection” length between the parts of the
circuit exceeds a fiftieth of a wavelength (λ/50) then ordinary circuit theory
breaks down, and the “connection” becomes a transmission line. The length
of the line in terms of the wavelength of the operating frequency adds a finite
time-lag between the start and end of the line, and this causes the voltage along
the line to change in terms of magnitude and phase as a function of the distance
down the line. Naturally, the current also changes as a function of the distance,
hence the ratio of the voltage and current (impedance) also changes.
Recall that the free-space wavelength, λ, is simply given by the usefull approx-
1

2 Electromagnetics
imation:
λ(m) =
300
f(MHz)
Thus, the use of Transmission line theory as opposed to circuit theory approxi-
mations becomes important for transmission lines longer than: 120km for 50Hz
power; 2km for ordinary telephone connections; 600mm for 10Mb/s Ethernet;
120mm for a PC Board bus track at 50MHz; 6mm for an on-chip interconnect
in a 1GHz PIII.
d=2a
D
r
0
eff
d D
r = eff
w
h
r
0
eff
a
b
r = eff
η2
η1
765kV !
txlnexa
Figure 1.2: 2-wire; Coax; µstrip; waveguide; fibre; RF; 50Hz!
Transmission line theory is thus a superset of classical circuit theory. In this

1.1 Transmission line theory 3
text we assume a Uniform transmission line, ie one whose properties do not
change along the length of the line.
A transmission line is anything that transfers power from one point to another,
whether picowatts or gigawatts, as shown in fig 1.2.
1.1.1 Impedance
A resistance resists current flow. An impedance impedes current flow. ie They
are the same concept, though generally, a resistance is purely real, whereas an
impedance has a reactive (energy storage) component. Thus, an impedance Z:
Z = R + jX
where X = jωL or X = 1/(jωC); ω being the radian frequency: ω = 2πf (f in It is important to note
that the reactive part of
the impedance is a func-
tion of frequency.
Hertz), L and C being the inductance and capacitance.
1.1.2 Characteristic impedance & Velocity of propagation
A transmission line can be viewed simply as a waveguide. It simply provides the
boundary conditions that shape the electric and magnetic fields in the medium
between the conductors. Of course, there is a direct relationship to the voltage
and current on the line too.
In circuit terms, the distributed capacitance and inductance etc of the line can
be collected in lumped models. The model of the transmission line is then an
infinite set of these circuit sections. Many versions of the model exist, but I
shall use the standard (Kraus & Fleisch 1999) model, as shown in fig 1.3.
CG
LR
d
Input Output
Lumped
Figure 1.3: A “Lumpy” model of the TxLn, discretizing the distributed param-
eters.
For a given length of transmission line, we can hence lump the series resistance
R [Ω/m] and inductance, L [H/m] together; and the shunt conductance G [0/m]
and capacitance C [F/m]. These terms are per-unit length, and do not change
from one section of the transmission line to another (uniform transmission line).
Hence we define a characteristic impedance, Z0, as the ratio of the series to
the shunt components; in the lossless (or high frequency) case, R and G are
negligeable:
Z0 =
R + jωL
G + jωC
=
L
C
Lossless

4 Electromagnetics
The concept of characteristic impedance has nothing to do with loss, but it
characterises a transmission line. Further, since the inductance and capacitance
between the two conductors largely depends on the geometry and the medium
properties, the characteristic impedance is largely dependant on these factors.
The Characteristic Impedance of a line is also referred to as the Surge Impedance.
If a voltage is applied at one end of a long transmission line, a current will flow
regardless of the load at the other end (the surge hasn’t got to the load yet!). The
ratio of the voltage and the current of the surge is the surge, or characteristic,
impedance of the transmission line.
The velocity with which the wave moves down the (lossless) transmission line
is also dependant on the material properties of the medium:
v =
1
√
LC
m/s
Since these factors depend on both the geometry and the material properties,
there is a need in many cases to define an effective permittivity, r(eff), since the
fields pass partially through air and through solid dielectric. If the entire volume
around the conductors is solid dielectric then r(eff) = r. In the case of a low-
loss foam dielectric, or similar dielectric-and-air combinations, an approximation
must be used.
Two-wire line
Conceptually, the simplest type of transmission line. Popular examples are the
300Ω “FM Tape” and the standard unshielded twisted pair (UTP) of networking
fame. The relationship of both conductors to ground is the same, making it a
balanced line. Referring to fig 1.2 for the geometrical definitions, the rigorous
case is given by Wadell (1991, pg66):
Z0 =
µ0µr
π2
0 r(eff)
cosh−1 D
d
Kraus (1992, pg158,499)
derives a similar result
using:
cosh−1
x =
ln(x + x2 − 1)
Usually, the spacing D is much greater than the radius a = d/2 and a simplified
equation is used:
Z0 =
µ0µr
π2
0 r(eff)
ln
D
a
D a
Since magnetic materials are never used, the simplified equation is usually sim-
plified further to the useful:
µ0
0
≈ 120π
Z0 =
120
√
r(eff)
ln
D
a
Typical values of Z0 range from 200 to 800Ω.

One conductor over ground plane
One conductor at a height, h, above a (theoretically) infinite groundplane looks
rather similar to a two-wire line by image theory, but is halved.
Z0 =
60
√
r(eff)
ln
2h
a
Twisted Pair
r
d
D
twisted
Figure 1.4: Twisted-Pair geometry
Twisted Pair is a very common form of transmission line, as all external noise
generated is common to both wires. A differential receiver stage then gets rid
of most of the induced noise. Common in telephony, Ethernet networks etc.
Again, the field lines cross through air and dielectric, making a closed-form
solution difficult. Wadell (1991, pg 68) presents Lefferson (1971)’s empirical
closed-form solution:
Z0 =
120
√
r(eff)
cosh−1 D
d
where: θ is normally between 20
and 45◦
. Less than 20◦
,
the twists are too loose
for uniformity, greater
than 50◦
breaks the
wires!
r(eff) = 1 + q( r − 1)
q = 0.25 + 0.0004θ2
θ = tan−1
(TπD)
T = Twists per length (same units as D).
Coaxial line
Coaxial line has the advantage of shielding the inner conductor, and hence has
less radiation (and reception). Since the relationship of the two conductors to
ground is different, coax is an unbalanced line. Also all field lines pass through
the dielectric, hence r(eff) = r. It is also common to refer to the ratio of the
radii a = d/2; b = D/2:
Z0 =
60
√
r
ln
D
d
or: Z0 =
60
√
r
ln
b
a

6 Electromagnetics
Wadell (1991, pg53–65) presents formulae for various offset coaxial, and strip-
in-coax situations.
Microstrip Line
Very popular since this is simply printed on a circuit board. Generally used at
the higher frequencies. One of the problems here is that the fringing flux plays
a very important role in establishing the exact Z0 value. Unfortunately the
fringing flux flows through a combination of air and dielectric, hence again the
need for an effective . Gardiol (1984) has developed expressions for the cases
where w/h ≤ 1 (large percentage of fringing flux) and for cases where w/h > 1
(less percentage of fringing flux).
For w/h ≤ 1:
r(eff) ≈
1
2
( r + 1) +
1
2
( r − 1) 1 + 12
h
w
−1/2
+ 0.04 1 −
w
h
2
Z0 ≈
60
√
r(eff)
ln 8
h
w
+
w
4h
For w/h > 1:
r(eff) ≈
1
2
( r + 1) +
1
2
( r − 1) 1 + 12
h
w
−1/2
Z0 ≈
120π
√
r(eff)
w
h
+ 1.393 + 0.667 ln
w
h
+ 1.444
Kraus & Fleisch (1999, pg 132) also has an approximate formula (which in
earlier editions he qualified as being applicable for w ≥ 2h, but this is often not
the case in microstrip lines of interest:
Z0 ≈
120π
√
r[(w/h) + 2]
Slotline
A slotline has no groundplane, but uses adjacent tracks as the transmission
line. The fields are distributed in an elliptical waveguide within and without
the dielectric. Closed-form expressions for Z0 span several pages of Gupta, Garg
& Bahl (1979, 213–216).
1.1.3 Impedance transformation
Since the voltage and current change down the transmission line, react with
the load and reflect back to the source, it appears as though the impedance
(ratio of the total voltage to the total current) changes continually down the

line. Under sinusoidal, steady state conditions a transmission line transforms
the load impedance ZL connected to its output to a different impedance at the
line input Zin as follows (lossless case):
Zin = Z0
ZL + jZ0 tan β
Z0 + jZL tan β
where β =
2π
λ
and is the length of the line (electrical length).
This equation is generally known as the Transmission line equation since it fully
specifies what happens on the line. It is an unwieldy equation, however, and
not generally useful. A graphical technique like the Smith Chart (section 1.2
on page 9) effectively embodies this equation in an easy-to-use manner, without
the need for the equation as such.
Since the velocity with which waves travel on a transmission line is lower than Note that the velocity on
a line can never be higher
than the speed of light!
that of light, the physical length of the cable is always shorter than the electrical
length:
Physical line length = VF × (electrical length)
where the velocity factor, VF = 1√
r
and is quoted by the manufacturers.
Some important simplifications of the Transmission line equation are:
1. ZL = Z0 This condition results in: Zin = ZL = Z0, regardless of line
length, or frequency. This is the matched case.
2. =
λ
2
. Zin = ZL regardless of characteristic impedance. This is the
halfwave case.
3. =
λ
4
“Quarter wave transformer” case. Zin =
Z2
0
ZL
This configuration is useful since it can transform one load impedance to
a different one if a line with the correct impedance can be found.
4. Open or short circuited lines.
Zin(oc) = −jZ0 cot β for an open circuited line
Zin(sc) = jZ0 tan β for a short circuited line
In both these cases the impedance is purely reactive and if the lines in
question are less than a quarter wave it is clear that such lines could be
used to “manufacture” capacitive (open circuit case) or inductive (short
circuit case) reactances. It should be remembered however that the capac-
itance or inductance of such a line would itself be frequency dependent.
The open and short circuit cases provide a convenient way to measure the
characteristic impedance of a line, since combing them yields:
Z0 = Zin(oc)Zin(sc)

8 Electromagnetics
The velocity factor of a line can be measured by using the quarter-wave trans-
former principle—if the load end is open circuited, ZL = ∞, hence Zin =0! The
method is then to take an open-circuited line and mesure the input impedance,
increasing the frequency until the input impedance drops to a minimum. The
line is then at an electrical quarter-wavelength, so
VF =
phys
λ/4
1.1.4 Standing Waves, Impedance Matching and Power
Transfer
The Maximum Power Transfer theorem determines when maximum power will
be transferred to a load or extracted from a source:
An impedance connected to two terminals of a network will ab-
sorb maximum power from the network when the real part of the
impedance is equal to the real part of the impedance as seen looking
back into the network from the terminals and when the reactance
(if any) is of opposite sign.
Stating this in a simplified form appropriate for transmission lines (which nor-
mally have a real impedance)
The impedance of devices connected to the two ends of a transmis-
sion line should have the same input and output resistances as that
of the characteristic impedance of the line.
When this condition is not satisfied standing waves are set up on the line and not
all of the available power is transferred through the system. A more practicalNote that the sub-
maximum power transfer
is not due to losses in
the system, but rather to
power being reflected.
problem which occurs is that such mismatch conditions often damage the output
electronics of a transmitter if excessive standing waves occur (or in well designed
transmitters the protection circuitry automatically reduces power output).
The extent to which power is reflected from the load is dependant on how “bad”
the load mismatch is to the line characteristic impedance and is expressed in
the voltage reflection coefficient, ρ:
ρ =
ZL − Z0
ZL + Z0
The mismatch in impedance is also often stated in terms of the voltage standing
wave ratio (VSWR) on the transmission line.
VSWR =
1 + |ρ|
1 − |ρ|
=
Vmax
Vmin
On a Smith Chart the VSWR value can be read off directly without performingMany texts and measur-
ing instruments refer to a
reflection coefficient as a
capitalized gamma: Γ.
any of these tedious manipulations.

1.2 The Smith Chart 9
1.2 The Smith Chart
For antenna analysis the most convenient way to represent and manipulate
impedances and transmission lines is on the Smith Chart (Smith 1939). In
addition to solving the transmission line equation presented in section 1.1.3 on
page 6, the chart is a visualization tool, enabling design decisions that are not
possible by simply studying the theory behind the equations. Appendix A on
page 95 contains a fully fledged Smith Chart, and this can also be downloaded
from Clark (2001).
On the Smith chart, the impedance is first normalized to some convenient value
(often 50Ω) by dividing both real and imaginary components by the normaliza-
tion factor.
The value 30 − j70Ω is plotted on the Smith Chart by normalizing resulting in
a value of 0.6 − j1.4 and then plotted on the chart shown in fig 1.5 in the lower
hemisphere.
This transmission line chart has a lot of useful features when manipulating The Smith chart is, in
fact, simply a polar plot
of the (complex) reflec-
tion coefficient, ρ, over-
laid with lines of constant
resistance and reactance.
impedances. If the impedance above is that of an antenna and it is connected
to a transmission line of 50Ω (say of quarter wavelength) then the input value
to this line can be simply read off the chart by rotating the plotted value around
the centre of the chart by a quarter wavelength (180◦
on the chart), yielding
0.26 + j0.6, or 13 + j30Ω, denormalized. The distance (in electrical length)
travelled down the line is indicated on the outside of the chart. When travelling
from the load to the generator, a clockwise direction is followed.
On the other hand, if the transmission line is 0.16 wavelengths long, the antenna
impedance would be transformed to a purely real value of 0.19 (or about 10Ω).
−1.4
0 0.2 0.5 1 2 5 ∞
0.2
−0.2
0.5
−0.5
1
−1
2
−2
5.30.6
0.6−j1.4
0.16λ
Smith
0.26+j0.6
0.19+j0
.
Figure 1.5: Impedance values plotted on the Smith Chart

10 Electromagnetics
The Voltage Standing Wave Ratio, VSWR, can be read directly off the Chart
as it is simply the intercept on the real axis as indicated by the circle centered
in the centre of the chart drawn for VSWR of 5.3:1. The constant VSWR circle
in fact describes all the possible values of the input impedance of the line as a
function of distance down the line.
As we move down the line, we travel down a constant VSWR circle, assuming
that the line is lossless, and hence get no nearer to a better match. In the case of
a lossy line, the magnitude of the reflection coefficient decreases, and instead of
a circle, a spiral is described, spiralling in to the centre of the chart, improving
the match. ie A lossy line improves the aparrent VSWR at the input to the
line, but of course this is at the expense of loss in the cable.
The addition of an inductive component to the plotted value on the other hand,
would clearly move the point on the constant resistance line (0.6 normalized in
our example) and it is immediately clear that more reactance in this example
would improve the impedance match since it would force the point closer to the
centre of the chart, without the loss of power.
1.3 Field Theory
1.3.1 Frequency and wavelength
As seen previously, the wavelength in free-space is given as
λ(m) =
300
f(MHz)
In the part of the Electromagnetic spectrum of interest to us, there are tradi-
tional (but otherwise meaningless!) designations of radio “bands”
Frequency Wavelength Designation
3–30 Hz 100–10 Mm ELF (Extra Low Frequency)
30–300 Hz 10–1 Mm SLF (Super Low Frequency)
300–3000 Hz 1 Mm–100 km ULF (Ultra Low Frequency)
3–30 kHz 100–10 km VLF (Very Low Frequency)
30–300 kHz 10–1 km LF (Low Frequency)
300–3000 kHz 1000–100 m MF (Medium Frequency)
3–30 MHz 100–10 m HF (High Frequency)
30–300 MHz 10–1 m VHF (Very High Frequency)
300–3000 MHz 1000–100 mm UHF (Ultra High Frequency)
3–30 GHz 100–10 mm SHF (Super High Frequerncy)
30–300 GHz 10–1 mm EHF (Extra High Frequency)
300–3000 GHz 1000–100µm Submillimeter/InfraRed
Observations:
• Small things (electrically speaking) do not radiate well. Hence from
30MHz to 30GHz is the most commercially used band.
• 3–30MHz bounces off ionosphere (so-called Shortwave) and is used for
old-fashioned over-the-horizon comms.

1.3 Field Theory 11
• Achievable frequencies are limited by electronics (for oscillators)
• Above 3000 GHz, we get into optics.
• Ionizing radiation can cause a molecule to change (x-rays and higher).
Anything lower in frequency can only cause heating. Humans can take
about 100W of heat!
The microwave (> 1GHz) region is broken into Radar bands. The most common
designations are the IEEE ones, although there are “newer” designations, which
I have never seen used.
Frequency Wavelength IEEE Designation
1–2 GHz 300–150 mm L
2–4 GHz 150–75 mm S
4–8 GHz 75–37.5 mm C (5m (big) dish)
8–12 GHz 37.5–25 mm X
12–18 GHz 25–16.7 mm Ku (1m (small) dish)
18–26 GHz 16.7–11.5 mm K
26–40 GHz 11.5–7.5 mm Ka
40–300 GHz 7.5–1 mm mm
Spectrum Usage:
FM Radio 88–108 MHz
ClassicfM 102.7 MHz
TV 123 200 MHz
MNET 615 MHz
GSM MTN & Vodacom 900 MHz
Cell C 1800 MHz
DECT 1880–1900 MHz
GPS 1.23 & 1.58 GHz
WLan 1.8 GHz
Industrial ISM 906–928MHz
Scientiﬁc ISM 2.4–2.5 GHz
Medical ISM 5.8–5.9 GHz
µwave oven (K5–513) 2.45 GHz
DSTV Ku band (11.7 & 12.3GHz)
1.3.2 Characteristic impedance & Velocity of propagation
In the case of a bounded medium like a transmission line, the geometry and
the medium played a part in determining the characteristic impedance. For a
propagating Transverse Electromagnetic (TEM) wave in an unbounded medium,
the characteristic impedance it sees is:
Z0 =
jωµ0µr
σ + jω 0 r
Since free space is non-conducting, and the relative quantities are unity: Some texts use eta: η,
and others zeta: ζ to
speciﬁcally refer to the
Z0 of free-space.
Z0(free space) = η =
µ0
0
= 377Ω or: η ≈ 120π

12 Electromagnetics
Just as the characteristic impedance depends on the geometry and the properties
of the medium, so the velocity of EM radiation also depends strongly on the
medium properties. For example, sound travels at 330m/s in air, but 1500m/s
in water.
In a non-conducting medium, the velocity of propagation is:
v =
1
√
µ0µr 0 r
where 0 and µ0 refer to the electric permittivty and magnetic permeability of
free space; the r subscripts refer to the quantities of the material relative to
that of free space. SinceNote that the value of µ0
is by definition, and 0
is actually derived from
the measured speed of
light. (Kraus 1988, Back
Cover)
µ0 ≡ 4π × 10−7
[H/m]
0 = 8.85 × 10−12
[F/m]
Thus, the speed of light in a vacuum or air, c, is given by:
c =
1
√
µ0 0
= 3 × 108
m/s = 300km/s
Most often, we are not interested in the absolute value of the speed of prop-
agation, but in the ratio of the speed to that of light in free-space, known as
the Velocity Factor, VF= v/c. This leads to the useful formula for EM wave
velocity in non-magnetic media as:
v =
c
√
r
or: VF =
1
√
r
Since most plastics have a relative permittivity of about 2.3, the reduction in
speed is about 0.66, hence the shortening factor for coaxial cable calculations.
1.3.3 EM waves in free space
An EM wave travels in free-space and in most transmission lines as a Transverse
EM wave (TEM). This implies that the direction of propagation is at 90◦
to
both the Electric and Magnetic wave, which, in turn are at 90◦
, as shown in
fig 1.6
Since the E field is analogous to voltage and the H field to current in the circuits
sense, it is easily seen that the equivalent relationships to Ohms law etc exist in
TEM waves in free-space:
(V = I × R)... E = H × η or: H =
E
120π
In a similar fashion, the power relationships hold (power density in EM):
S = EH =
E2
120π
= H2
× 120π W/m2
where E and H are RMS values.

1.3 Field Theory 13
x
z
y
Ex
Hy
E(φ)
EMWave
Figure 1.6: Electromagnetic Wave in free-space
In general, EM waves of a frequency above 30MHz do not bend around the earth,
and propagation occurs only within Line-of-Sight (LOS). The earth’s curvature
thus limits the radio horizon achievable for a certain height of transmitter, as
shown in fig 1.7
dh
h
Earth Shadow Region
Horizon Line
Antenna
radhor
Figure 1.7: Radio Horizon due to earth curvature.
An empirical formula to calculate the radio horizon is: Since the mean Earth Ra-
dius is 6370km, and the
tangential point is on it,
by Pythagorous, we get
19.5km!
dh(km) = 4 h(m)
Thus a 30m mast will only provide a 22km Line-of-Sight range.
Beyond the horizon waves propagate for longer distances than that illustrated
above using either ground wave propagation or reflection from some atmospheric
feature such as the ionosphere, meteors, the troposphere or even an artificial
satellite (although in that case the signal is usually retransmitted).
Ground wave propagation occurs when vertically polarized waves are launched
and these are guided by the earth surface and thus follow the curvature of the
earth. They are only practical at lower frequencies where the attenuation these
waves undergo as result of earth losses are reasonably low. At the very low
frequency end (round 30 kHz) the wavelength is so large that the waves flow in
the waveguide formed by the earth surface on the one hand and the ionosphere
on the other. This could also be considered as a type of ground wave.

14 Electromagnetics
The classic means of long distance communication is by means of reflecting
waves from the ionosphere. This layer1
of gas that is ionized by the sun lies
at a distance of approximately 80 to 400 km above the earth surface. Only
waves below a certain critical frequency are reflected (by a process of successive
refraction) by this layer. This so called critical frequency is typically in the HF
band (3–30 MHz) and varies strongly with time of day and other variables. At
frequencies much below this frequency the radio-waves undergo attenuation (or
absorption) and communication using this principle is also impractical.
Other methods of obtaining beyond the horizon communication are the rather
exotic techniques such as meteor scatter propagation: where waves are reflected
off the ionized trails left by meteors entering the atmosphere; or tropospheric
scatter: which uses discontinuities in the atmosphere to cause bending of waves
and thus long distance propagation.
Most over-the-horizon communication, of course, occurs at microwave frequen-
cies over a satellite relay link at high data volumes. (Discounting under-the-sea
fibre optics for this course!)
1.3.4 Reflection from the Earth’s Surface
In addition to its role as a obstacle, the earth’s surface also acts as a reflector
of radio waves. This situation is illustrated in figure 1.8.
2h
P1
S1
S2
θ
θ
Direct-wave path
Reflected-wave path
Reflecting Surface
ReflRay
Path Difference = 2h sin θ
Figure 1.8: Geometry of Interference between Direct Path and Reflected Waves
It is clear that if S1 is an isotropic source and would normally radiate equally
well in all directions, the pattern would be modified by the reflected wave. By
the method of images the situation above is similar to that which exists if a
mirror image source S2 was positioned at distance h below the reflecting plane.
1Actually, several identifiable layers at differing heights

1.3 Field Theory 15
Clearly there will now be a difference in the path lengths to some distant point
P. At certain elevation angles θ the path difference would be such that the
two waves are in phase and thus interfere constructively and for others the
interference would be destructive and result in a null in the radiation pattern.
If the field due to a single source is termed E0 then the total field would then
be given by:
E = |E0| sin
2πh sin θ
λ
Application of this formula to the particular value of h/λ = 1.44 results in the
field pattern shown in figure 1.9.
This condition is not always advantageous since an antenna that may have had
a maximum towards θ = 0◦
would now have a null in the same direction. The
only way to improve the situation would be to either make the antenna higher
and thus force the angle of the first maximum lower or increase the frequency
and thus ensure an increased h/λ ratio.
In cases where radiation at some angle is required this reflection results in an
unexpected bonus, however. The maximum value of the E-field in the direction
of the maxima is twice the value of the original antenna without reflection. This
implies that in that particular direction the power density would be increased
by a factor of four (power density is proportional to the square of the E field).
Earth reflection can thus be used to gain a 6 dB bonus in antenna gain if used
properly!
0.2
0.4
0.6
0.8
1
30
60
90
0
Interfere
.
Figure 1.9: Interference pattern field strength contours for h/λ = 1.44

16 Electromagnetics
1.3.5 EM waves in a conductor
If we solve Maxwell’s equations for EM wave propagation in a conductor, we
get a different result to the free-space case considered above. The Electric field
magnitude actually decays exponentially inside the conductor. We define the
Skin depth, δ, as the point where the field inside the (good) conductor, Ey, has
become 1/e of its original value, E0:
δ =
2
ωµσ
=
1
√
πfµσ
Note that δ has units of distance. If we look at what happens at one skin depth,
δ, into the conductor, the field is given as: Ey = E0e−x/δ
, then
|Ey| =
|E0|
e
≈ 37%|E0|
or the field inside the conductor at one skin depth is 37% of the original field
magnitude. Comparing the skin depth (or depth of penetration) δ for silver,
copper & cast iron at 50 Hz and 1GHz:
Relevant parameters: σAg = 6.1 × 107
0/m σCu = 5.7 × 107
0/m (Both have
µr = 1) σFe = 106
0/m and µrFe = 5000
δ50Hz δ1GHz
Ag 9.1mm 2µm
Cu 9.4mm 2.1µm
Fe 1mm 0.23µm
The decay is obviously logarithmic: at one further skin depth, the field has
deacyed a further 37%. ie 9.4mm of Copper to get to 37% of the original signal
strength. Another 9.4mm of copper will take that signal strength to 37% of
that value, ie 13.7% of the original value. A useful table is then the percentage
overall decrease as a function of the number of skin depths:
δ s % of Orig.
1 37
2 13.7
3 5.1
4 1.9
5 0.7
After 5 skin depths, the field level is below 1%.
A number of consequences follow:
• Busbars in substations are hollow. It is silly to supply a solid copper bar
if there is no field and current flow in it’s middle!
• At cellphone frequencies, the skin depth of copper is about 2µm. CertainlyAt DC, the current den-
sity is uniform through-
out the cross-section, but
at RF, the current den-
sity concentrates in the
two outer skins.
after 10µm, there is no appreciable field.
This means that the outside of a coaxial braid is a completely separate
conductor from the inside of the coaxial braid from an RF perspective!!

Chapter 2
Antenna Fundamentals
2.1 Directivity, Gain and Pattern
THE USUAL COORDINATE SYSTEM in 3-dimensional space, the carte-
sian co-ordinate system of points being described in terms of (x, y, z) co-
ordinates does not lend itself to the electromagnetic world. Generally, we like
to describe the points in the spherical co-ordinate system as (r, θ, φ), as shown
in ﬁg 2.1.
z
y
x
P(x, y, z)&P(r, θ, φ)
r
φ
θ
Coord
Figure 2.1: Standard coordinate system.
The spherical co-ordinate system is mainly described in terms of angles, where
r is the distance from the origin, θ the angle from the zenith (not the elevation
angle) and φ is the projected angle from the x-axis, in the xy plane.
2.1.1 Solid angles
Clearly, when dealing with 3-dimensional radiation patterns, some idea of a
3-dimensional angle needs to be employed.
17

18 Antenna Fundamentals
A radian is defined by saying that there are 2π radians in a circle. As a result,
the arclength rθ of a circle of radius r is said to subtend the angle θ.
A steradian (or square radian) is a solid angle defined by saying that there are
4π steradians in a sphere. As a result, an area a on the surface of the sphere of
radius r is said to subtend the solid angle ω. Note that the shape of the area is
not specified. shown in fig 2.2
x = rθ
r
θ
θ = x
r radians
z
y
x
Ω
Area=A
Ω = A
r steradians
steradian
Figure 2.2: concept of a solid angle
2.1.2 Directivity
Isotropic source
An isotropic antenna, is as its name implies, one which radiates the same power
in all possible (3-dimensional) angles. By definition, an isotropic antenna has a
directivity, d, of 1, and is infinitely small. Note that an isotropic antenna is an
impossibility and is used only as a reference!
Generally, however, an antenna has greater directivity in some directions than
other directions. This is at the expense of directivity in other areas.
In free-space the EM wave is not guided, as it would be in a transmission line,
and can hypothetically radiate in all directions. If an isotropic source transmits
Prad watts, and we enclose the source with a sphere of radius, r, and hence
surface area, A = 4πr2
, then the power density, S at the surface of the sphere
is given as:
S =
Prad
A
=
Prad
4πr2
[W/m2
]
hence the power reduces inversely proportional to the square of the distance.
Directivity, D, is then defined as:
D =
S(θ, φ)max
Sisotropic

2.1 Directivity, Gain and Pattern 19
where the power transmitted by both antennas is the same.
Decibels
Directivity is usually expressed in decibels (DdB = 10 log10 D), and since it is Directivity is almost al-
ways quoted in dB’s, but
that most equations re-
quire it in linear form.
referred to an isotropic source (the Sisotropic in the above), the unit is denoted
dBi for dB’s above isotropic. Decibels are very useful in power ratio’s because
of the large range of values experienced because of the inverse square law. For
example, if power P1 is 1000 times greater than P2, the ratio expressed in dBs
is:
dB = 10 log10
P1
P2
which yields P1 as being 30dB greater than P2. If P1 were a million times the
power of P2, it would be 60dB greater. It can be seen that dB’s are simply more
convenient for expressing power ratio’s than linear quantities.
One can also use dB’s in field-related quantities that yield power when squared—
from Ohms law in free space, we see that S = E2
/120π. Since the log of
a squared quantity is simply twice the log of the non-squared quantity, for a
power component like voltage, E-field etc:
dB = 20 log10
E2
E1
It is sometimes convenient to represent a signal with respect to a standard
reference—eg the absolute power level to a milliwatt of power is denoted dBm:
dBm = 10 log10
power
1mW
similarly for a voltage relative to a microvolt:
dBµV = 20 log10
voltage
1µV
2.1.3 Efficiency
Directivity, D, is often loosely referred to as Gain, G. However, the term gain Note that “Gain” is not
used in the same sense as
amplifier “gain”. An an-
tenna is passive. It sim-
ply denotes an increased
power density in one di-
rection, at the expense of
other directions.
refers to the input power as opposed to the radiated power—the difference being
the power lost in the ohmic losses in the antenna. The antenna efficiency is then:
η =
Prad
Pin
and hence:
G = ηD

For antennas with negligible losses these two values are approximately equal,
but many antennas which are small in terms of wavelength or broadband are
not efficient and the two values can be quite different.
In general the gain is of importance for calculating power levels at various sites.
The directivity is more a indication of the pattern of an ideal radiator and is ofBoth Gain and Directiv-
ity are functions of θ and
φ but are often used to
describe these values in
the maximum directions.
more theoretical than practical value.
In practice the gain of an antenna is important since it increases the power
density in the direction of the main beam of the antenna. A 100 W transmitter
with a 13dB gain antenna produces the same power density at a distant point
as a 1 kW transmitter with a 3 dB gain antenna. Clearly the former case would
have a larger area of lower power density in other directions and the extent
to which antenna gain would improve communications would depend on the
intended coverage.
In a broadcast scenario for instance, omnidirectional radiation may be required
in the plane of the earth. The only improvement in gain of broadcast antennas
is by minimizing the radiation upwards (at angles of θ smaller than 90◦
). For
point-to-point links on the other hand the power can be concentrated as far as
possible in the desired direction.
Hence, the power density S at a distance r away from an antenna with a gain
G is:
S =
GPin
4πr2
=
DPrad
4πr2
W/m2
2.1.4 Radiation pattern
An antenna’s radiation pattern is a measure of how it radiates in 3-dimensional
space. Generally, a radiation pattern refers to one that is measured in the far-
field (see later). As a result, it is a plot of transmitted power versus angle.
It has nothing to do with distance. Note that the signal from an antenna
weakens predictably with distance by 1/r2
, but not predictably by angle—this
is described by the radiation pattern.
The pattern can show power density(W/m2
) versus angle, radiation intensity
(W/sr) versus angle, E-field (V/m) or H-field (A/m) versus angle. The plot
can be rectangular or polar. Generally, however, it is a plot of power density,
expressed in dBi.
A representation of a SuperNEC (Nitch 2001) three dimensional radiation
pattern of a 10-director Yagi-Uda array is shown in fig 2.3. An antenna often
has a main beam, with some sidelobes and backlobes, which may not be desired.
The main lobe can be clearly seen, as can the first sidelobe (all the way aroundAlthough a 3d pattern
may look impressive, it
can rarely show detail ac-
curately.
the main lobe, like a collar). It can also be seen that the next sidelobe fires up
and down, but that there is a null along the y-axis. It is far more common to
take a 2D cut through the principle planes of the antenna, as shown in the xy
plane cut in fig 2.4.
Fig 2.4 is shown in a polar representation, which is most useful for visualization
purposes. To read off values, however, a rectangular form is preferred as shownIt is easier to get a feel
for the front-to-back ra-
tio, (14.3 dB), and the
first sidelobe levels (14.3
dB in azimuth; 8.77dB in
elevation) from the rect-
angular plots.

2.1 Directivity, Gain and Pattern 21
Figure 2.3: Three-dimensional pattern of a 10 director Yagi-Uda array
300240
180
120 60
0
10 dBi
0
−10
−20
−30
Radiation Pattern (Azimuth)
Structure: θ=90°
Figure 2.4: xy plane cut of the 10 director Yagi-Uda
in fig 2.5.
The rectangular plots also show the half power beamwidth (HPBW) (-3dB point
of the main beam) and the beamwidth between first nulls (BWFN) quite clearly,
and these quantities are often useful to determine the absolute gain of the an-
tenna, which is difficult to measure accurately, whereas a 3dB drop from peak
power is a relative measurement, and hence an eassier one.

−180 −120 −60 0 60 120 180
−20
−10
0
10
15
φ
Gain(dBi)
Radiation Pattern (Azimuth)
(a) xy plane
−90 −30 30 90 150 210 270
−20
−10
0
10
15
θ
Gain(dBi)
Radiation Pattern (Elevation)
(b) yz plane
Figure 2.5: Rectanglar radiation patterns of the 10-director Yagi-Uda in the
two principle planes
Directivity estimation from beamwidth
Note again that if the “beam” equally covered a whole sphere of 4π square
radians, or steradians, ie an isotropic source, the directivity is 1 by definition.4π square radians =
4π(360/2π)2
square de-
grees = 41 253, usually
rounded off to 41 000.
This gives rise to an approximation which is often used (Kraus & Fleisch 1999,
pg 255):
D =
4π
ΩA
≈
4π
θHPφHP
≈
41 000
θ◦
HPφ◦
HP
The above approximation produces about 1dB too much gain. the reasoning
is that the area represented is square, whereas in reality, the beam is generally
round. hence a common adjustment (Kraus 1988, pg100) is:
D ≈
36 000
θ◦
HPφ◦
HP
even in this case, however, if fairly significant side- and back-lobes exist, the
prediction is optimistic. In the above example of the 10-director Yagi-Uda array,
we get HPBW’s of 42 and 38 degrees, which according to the above formula gives
13.5dBi gain, whereas SuperNEC gives 12.7dBi.
2.2 Reciprocity
The reciprocity theorem stated by Lord Rayleigh, and generalised by Carson
(1929), states that if a voltage is applied to antenna A which causes a current
to flow at the terminals of antenna B, then an equal current (in magnitude and
phase) will occur at the terminals of antenna A if the same voltage is applied
to the terminals of antenna B.
In short, all characteristics of an antenna apply equally well in transmit and
receive mode (radiation pattern (reception pattern), input/output impedance
etc). shown in fig 2.6.

2.3 Polarization 23
Va Energy
Flow
Ib
Ia Energy
Flow
Vb
Va causes Ib ; Vb causes Ia
If Vb is made = Va, then Ia will be = Ib
reciprocity
Figure 2.6: Reciprocity concepts
However this does (obviously) not hold for the near fields—it is a far-field phe-
nomenon. ie far-field patterns are identical, but near-field patterns are different.
2.3 Polarization
Polarization generally refers to the direction of the E-field vector in the far-
field. note that the H-field is at 90◦
to the E-field and is present in all EM
waves. In the case of a dipole or a Yagi-Uda array, the E-field is linearly
polarized in the direction of the dipole. Most often, a linear polarization will be
specified as horizontally polarized (most TV antennas) or vertically polarized
(most broadcast radio signals) (reference is the earth).
The amount of power received varies as the cosine of the angle between the Note that if the sig-
nal is horizontally polar-
ized, and the receiving
antenna vertically polar-
ized, there is (ideally) no
received signal.
polarization of the incident wave and the antenna. eg A cellphone base station
antenna is vertically polarized, and although the mobile has a linearly polarized
antenna, it is seldom held perfectly upright, incurring a loss in received power.
y
Linear
x
E2
y
Elliptical
x
E2
E1
E
y
Circular
x
E2
E1
E
Polarization
Figure 2.7: Polarization possibilities—wave out of page

in the vertically polarized case, only ey is present and can be expressed as:
ey = e2 sin(ωt − βz)
ie a time varying and distance varying (out of the page) component, but only
in the y direction (vertical). the maximum value e2 is in the y direction. β is
the wave number, given as:
β =
2π
λ
the e-field can be made to progressively change direction in a circular fashion
(eg with a helical antenna, popular on satellites). the helix can be wound in a
left-hand or a right-handed fashion, giving rise to lhcp and rhcp. a rhcp antenna
will not (ideally) receive a lhcp wave. the e-field is continuously rotating, and
the peak x-component is equal to the peak y-component of the field.
Generalizing to elliptical polarization, the x and y components are not equal.In the general case of el-
liptical polarization, the
major axis magnitude to
the minor axis magni-
tude defines the axial ra-
tio. An infinite axial ra-
tio implies linear polar-
ization, whereas an AR of
1 implies circular polar-
ization.
The field components are thus:
Ex = E1 sin(ωt − βz)
Ey = E2 sin(ωt − βz + δ)
where δ is the time phase angle by which Ey leads Ex.
2.4 Effective aperture
If an antenna is immersed in a field with a power density of S [W/m2
], it will
receive a power Pr [W] and deliver it to a load connected to its terminals. This
gives rise to the concept of an effective aperture, Ae [m2
].
Ae =
Pr
S
In general (Kraus & Fleisch 1999, pg 258), the aperture of an antenna can be
related to the antenna gain:
Ae =
Gλ2
4π
ie when the gain is large the aperture is large.
Note that the effective aperture of a dish antenna may not be as big as the actual
area of the dish (see fig 2.8), leading to the concept of an aperture efficiency,
εap:
εap =
Ae
Ap
where Ap is the physical size of the antenna aperture. (Mainly caused by im-
perfect parabolicness—witness the hubble telescope)
As an example as shown in figure 2.9, a half-wave dipole for classicfm (102.7mhz)
made of 1mm diameter rod, has a physical aperture of 1.46m×0.001=0.0015m2
.
since a half-wave dipole has a gain of 2.16dBi=1.64 linear, the effective aperture

2.5 Free-space link equation and system calulations 25
r
Ap = πr2
Ae ApertureEff
Figure 2.8: Aperture efficiency of a parabolic dish
Ae
λ/2
ApertureDipole
Figure 2.9: Effective aperture of dipole physical aperture
is 1.11m2
! (740× larger).
Note that the aperture has meaning—if two antennas are used to capture more
energy from the wave, the apertures cannot be allowed to overlap as shown in
fig 2.10
ApertureSteal
Figure 2.10: Apertures must not overlap!
In the case of linear antennas, this is a fair problem, as the apertures easily
overlap, and the overall capture area will be reduced. With dish antennas, since
Ae < Ap, it is not possible to have an overlapping aperture!
2.5 Free-space link equation and system calula-
tions
The gain of an antenna is defined as the maximum power density in a certain
direction as compared to a hypothetical isotrope. Since the antenna gain con-

Tx
Gt
Pt
Rx
Gr
Pr
r
LinkEqn
Figure 2.11: Point-to-point link parameters
centrates the transmitted power into a particular direction, we can define the
effective radiated power (ERP) of an antenna in a particular direction as:ERP is the power that
an isotrope would have
to produce in all di-
rections to achieve the
same effect in this par-
ticular direction. (also
pedantically called EIRP,
I for Isotropic, but rarely
used).
ERP = GtPt
at a distance, r, away from the transmitting antenna, the power density, S is
given as:
S(r) =
GtPt
4πr2
the power received, Pr, by the receiving antenna is then Pr = SAer where Aer
is the effective aperture of the receiving antenna, given by
Aer =
Grλ2
4π
hence the power received by an antenna in a freespace point-to-point link is:
Pr =
GtGrPtλ2
(4πr)2
[W]
note that multipath reflections with constructive and destructive interference
and atmospheric absorption change this value!
The link equation can obviously also be conveniently expressed in dB’s:The 32.45 factor comes
from +20 log(300) for
frequency conversion
(20 since squared),
−20 log(4π), and
−20 log(1000) for the
km conversion.
Pr = Pt + Gt + Gr − 32.45 − 20 log10 r − 20 log10 f
where Pr and Pt are expressed in dBm, Gt and Gr are in dBi, r is in km, and f
is in MHz.
Example: Geostationary satellite comms
A geostationary satellite must be in the Clarke orbit at 36 000 km in order
to appear stationary above a point on the earth. If the power required at the

2.5 Free-space link equation and system calulations 27
satellite receiver is only 8pW, how much power is required to be transmitted
by the earth station? Receiver gain is 20dB, earth station gain is 30dB, at a
frequency of 5GHz.
Converting to linear gains:
Gr = 20 dB = 100; Gt = 30 dB = 1000
λ =
300
5000
= 60mm
hence the power we are required to transmit is:
Pt =
Pr(4πr)2
GtGrλ2
=
8pW(4π × 3.6 × 107
m)2
100 · 1000 · (0.06)2
= 4547W
So 4.5kW is needed to get 8pW to the satellite. Note that the link equation
ignores atmospheric absorption or any other losses along the path, and purely
assumes line-of-sight (LOS) situations.
Actually, satellites transmit at hundreds of watts, not kW!
The dB version is thus:
Pt = Pr − Gt − Gr + 32.45 + 20 log r + 20 log f
= −80.9 − 20 − 30 + 32.45 + 91.12 + 73.98
= 66.58dBm = 4556kW

Chapter 3
Matching Techniques
3.1 Balun action
3.1.1 Unbalance and its effect
AT RF FREQUENCIES recall that from section 1.3.5, the skin-depth is only
a few microns. As a direct result, the inside of the coax braid is quite a
seperate conductor than the outside of the braid, even though, from a DC
perspective, we view the braid as “one conductor”.
If we feed an antenna using a coaxial cable as shown in fig 3.1, the antenna
radiation can induce a current flow on the outside of the braid, i3, quite inde-
pendant of the current flow on the inside of the braid due to the transmitter,
i2.
i1i2i3
i1i2 + i3
BalunReason
Figure 3.1: The reason for unbalanced currents
Since on the inside of the braid, balanced currents must flow i1 = i2, and both An absolutely tell-tale
sign of external current is
a change in a measure-
ment parameter when the
cable is grabbed by a
hand.
the inside and outside currents meet at the antenna junction, this implies that
a larger current can flow on one dipole arm than on the other. The antenna
pattern is distorted, and the input impedance is changed.
29

30 Matching Techniques
The job of the BALance to UNbalance converter (Balun) is to prevent the
currents flowing on the outside of the braid by presenting a high impedance to
them.
3.1.2 Balanced and unbalanced lines—a definition
Balanced
UnBalanced
UnBalanced
BalLine
Tr Ln Cross Sections
Figure 3.2: Balanced and Unbalanced Transmission lines
Essentially, a balanced line is one in which symmetry ensures that equal fields,
currents, voltages exist along it in the equal and opposite sense!
It is clear that a two-wire line of equal diameter conductors is balanced, but that
a coaxial line is not—the field strength is much higher on the inner conductor
than the outer conductor.
3.2 Impedance Transformation
The Balun action is to be distinguished from the Transformer action. A circuit
that does both functions should be called a Balun-Transformer, but in practice
all are simply called Baluns, whether they transform, or just do a Balance-to-
Unbalance conversion (sometimes even when they only transform.
Impedance transformation has the goal of matching the antenna impedance to
the line, and the generator in order that no reflected power exists—maximum
power transfer.
A transmitter cannot deliver full power to an unmatched load, but this is not
the only consideration:
• High VSWR means high V &I, hence txln losses.
• High V means flashover/dielectric breakdown.
• High I means hotspots/copper melting.
• Output electronics of the transmitter can be damaged, or more likely, the
automatic power reduction circuitry kicks in. (Typically at a VSWR of
2:1)
Impedance matching is important both for transmission and reception. It is
more critical, however, for the transmitting case and the VSWR specifications

3.2 Impedance Transformation 31
are usually more severe. To illustrate this point the following equation gives the
power reduction as result of a mismatch in terms of VSWR:
Power lost in transfer = 10 log 1 −
VSWR − 1
VSWR + 1
2
dB
Thus a VSWR of 2 : 1 results in a power reduction of only 0.5 dB. Even a VSWR
as high as 5 : 1 only causes a reduction of 2.5 dB. The power reduction due to the
mismatch condition itself is thus not all that significant, but some transmitters
will start reducing power output to protect the driving stage electronics at such
low values as 1.5 : 1 or 2 : 1 (Or simply blow up if no power reduction protection
is in place). The power lost (in dB) versus VSWR is illustrated in figure 3.3.
1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
VSWR
PowerlostindB
pwrvswr
Figure 3.3: Power loss in dB versus VSWR on the line.
The additional line losses with a higher VSWR is also of concern and is a func-
tion of the normal line losses, which are usually specified by the manufacturer
of the lines in terms of dB/30m (ie dB/100ft!). It should be noted that the
manufacturer specification assumes matched conditions. The graph in figure 3.4
indicates the additional loss under mismatch conditions (ARRL 1988, pg3-12).
Losses in the line will improve the VSWR at the input end as compared to This improvement is
clearly obtained at the
expense of efficiency,
indicating the need to
avoid long transmis-
sion lines when doing
measurements of VSWR.
the VSWR at the antenna. A curve quantifying this characteristic is given in
figure 3.5

10-1
100
101
10-1 100 101
Additional Loss in dB when Matched
AdditionallossindB
SWR=2
SWR=4
SWR=7
SWR=20
AddLoss
Figure 3.4: Increase in Line Loss because of High VSWR
1
5
10
50
100
1 2 3 5 10
SWR at TransmitterSWRChange
0dB Loss
1dB Loss
2dB Loss
3dB Loss
SWRatAntenna
Figure 3.5: VSWR at Input to Transmission Line versus VSWR at Antenna
3.2.1 Angle Correction
Remember that a low-loss or lossless transmission line has a purely real char-
acteristic impedance. For matching to occur, the antenna has to be resonated.

3.3 Common baluns/balun transfomers 33
This is most often accomplished before impedance transformation. Some match-
ing techniques include angle correction, some don’t.
3.3 Common baluns/balun transfomers
3.3.1 Sleeve Balun
Recall that a short-circuited quarter-wavelength transmission line presents a
high (theoretically infinite) impedance. Thus the simplest balun is a short-
circuited quarter-wave sleeve around the coaxial feeder cable, where the sleeve
is short-circuited to the coaxial feeder cable braid a quarter wavelength away
from the feed. Currents wanting to flow in the outside of the braid now see a
high impedance, as shown in fig 3.6.
λ/4
Bazooka
Figure 3.6: The Bazooka or Sleeve balun.
A sleeve balun has no transformer action, but simply prevents the current flow. The Americans tend to
refer to the sleeve balun
as a “Bazooka balun”!
It is rather narrow-band however, and care must be taken to ensure that it is
the correct length in terms of the Velocity Factor of the coax outer insulation,
which is normally a different material than the inner dielectric (hence a different
VF).
3.3.2 Half-wave balun
λ/2
HalfWaveBalun
Figure 3.7: A HalfWave 4:1 transformer balun.
In addition to a balun action, due to the halfwave section, there is also a 4:1
impedance transformation as shown in fig 3.7.
The impedance transformation is because after the λ/2 transmission line, the
voltage is exactly out of phase, hence twice the voltage is applied between the

dipoles. Naturally, the current is halved, hence the impedance (voltage over
current) has a factor of four.
Obviously, the half-wave balun is narrow-band too.
3.3.3 Transformers
Ordinary transformers can be used as impedance transformers (remember that
a 2:1 voltage transformer is a 4:1 impedance transformer). Transformers are
broadband, but tend to be lossy. The selection of the ferrite core is important
as they are frequency selective.
A very useful variety is the Transmission Line Transformer shown in fig 3.8.
TxLn 9:1 Ant
TLTrans
Figure 3.8: Transmission-line transformer.
Sevick (1990) is an excellent reference on transmission-line transformers. They
are usually bifilar(4:1) or trifilar(9:1) wound, and the main coupling mechanism
is that of a transmission line. They are effectively autotransformers and are
usually wound on a frequency-band specific ferrite core. They are quite broad-For the HF band, the
Philips 4C6 material is
excellent.
band achieving bandwidths of up to 15:1. The transmission line transformer (if
connected correctly) also has a Balance to Unbalance conversion. If the input
Gnd is reversed, NO Balun action.
3.3.4 LC Networks
Physical inductors and capacitors can also be used to resonate the antenna
and then transform the impedance. One drawback is that inductors are lossy.
Both “pi” and “T” networks are commmon. One popular match is known as
the “L-match”, shown in fig 3.9. In the L-match, the LC transformer network
is simply an extension of the resonating inductor for the antenna. Therefore
popular when the antenna is short in terms of wavelength, eg mobile HF whips.
LC networks do not have a Balun action, and are naturally narrow-band.
3.3.5 Resistive Networks
Resistive networks (pi and T) are also used, and are broadband, but very lossy.
They are used only in extreme cases, and have no balun action.

CMatch
LMatch
LResonate
CAnt.Reactance
RRadiation
LMatch
Figure 3.9: L-Match network.
3.3.6 Quarter wavelength transformer
ZLZin
x = λ/4 quarter
Z0
Figure 3.10: Quarter wavelength transformer
For a transmission line of a quarter-wavelength long, recall that (section 1.1.3
on page 6):
Z0 = ZLZS
In matching one impedance to another, if a transmission line can be found with
a Z0 of the geometric mean of the two impedances, a match can be made.
The match is fairly narrow-band, and requires a practically available Z0. At
low frequencies the cable length may be inconvenient, at high frequencies a
microstrip line is used.
Ordinarily there is no balun action. However, if a “series” connection of two
coaxial cables is used as the transformer, there is balun action since both lines
are shielded. A “series” connection uses two pieces of coax, with the two braids
connected at both ends, but the two centres are used as the transmission line.
The characteristic impedance of the series connected line is half the character-
istic impedance of the coax.

3.3.7 Stub matching
This type of matching can be used to obtain a narrow-band match to any
impedance. The general arrangement is shown in figure 3.11.
ZLZ0
Z0
Z0
d1
d2
Stub
Figure 3.11: Single Matching Stub
Both the lengths d1 and d2 need to be adjustable which can present some diffi-
culty. This type of matching is most clearly illustrated on the Smith Chart andIf the cable is coax, vary-
ing d1 can prove im-
possible. The Double-
Stub match uses fixed dis-
tances from the load, and
only varies the length of
the stubs
is shown in figure 3.12.
Due to the fact that stubs are always connected in parallel it is easier to use ad-
mittance, rather than impedance notation where parallel components are simply
added. To achieve this the normalized antenna impedance is plotted (P1) and
rotated halfway round the chart to obtain the corresponding admittance value
(P2).
Once an admittance has been converted by length d1 to lie on the circle indicated
(Rn = 1) a match can always be achieved by simply adding or subtracting the
necessary amount of susceptance (inverse of reactance) using the open or short
circuited stub. This value can be found by starting at the outer circle on the
chart at the right hand end (admittance of infinity) for short circuited case or
the left hand side (admittance of zero) for open circuit stub and finding the
required value of susceptance and reading of the length of the required stub
(d2) from the Smith Chart.
3.3.8 Shifting the Feedpoint
Changes can be made to the feed geometry in order to effect a narrow-band
match. The T and Gamma matches are well suited to feed folded dipole anten-
nas. They are illustrated in fig 3.13
These two types of matching sections are very similar in behaviour. It is difficult
to analyze their performance but the following general trends referring to the T
match have been observed:

0 0.2 0.5 1 2 5
0.2
0.5
0.2
0.5
1
2
1
2
∞
Towards
Generator
Towards
Load
P2
P1
d1
d2
1.8
1.8
P3
SmithStub
Figure 3.12: Smith Chart of Single Stub Tuner Arrangement
C
B
A
Unbalanced
Balanced
TGamma
Figure 3.13: T and Gamma Matching Sections
• The input impedance increases as A is made larger. The desired value of

the real part does not always occur as the imaginary part crosses zero,
causing some problems.
• The maximum impedance values occur in the region where A is 40 to 60
percent of the total antenna length.
• Higher values of input impedance can be realized when the antenna is
shortened to cancel the inductive reactance of the matching section.
Some flexibility can be obtained by inserting variable capacitors in series with
these sections at the feed. As a first approximation values of about 7 pF per
meter of wavelength can be used.
3.3.9 Ferrite loading
Wrapping the coax/TxLn round a ferrite toroid/clamping ferrite beads around
the TxLn. (eg VGA cables). This simply increases the impedance to external
current flow.
3.4 Impedance versus Gain Bandwidth
The goal of matching is to make the antenna present an acceptable impedance
to the transmitter over the bandwidth that is necessary. However it is senseless
to achieve a broad bandwidth match if the antenna does not maintain its gain
over that bandwidth. Hence we can speak of an impedance bandwidth and a
gain bandwidth. Impedance bandwidth is usually taken as the 2:1 VSWR level,
whereas gain bandwidth is usually taken as 3dB down on the peak gain.
As an example, a SuperNEC run on a 10-director Yagi-Uda array produces
a gain bandwith as shown in 3.14, and an impedance bandwith (normalizing
to 200Ω) as shown in fig 3.15. It is clear that some work needs to be done to
improve the match so that the impedance bandwidth falls in line with the gain
bandwidth.

3.4 Impedance versus Gain Bandwidth 39
200 250 300 350
−10
−5
0
5
10
15
Freq (MHz)
Gain(dBi)
Gain at θ=90, φ=0
Line1
Figure 3.14: Gain bandwidth
200 250 300 350
1
1.5
2
2.5
3
3.5
4
4.5
5
Freq (MHz)
VSWR
Line1
Figure 3.15: Impedance bandwidth

Chapter 4
Simple Linear Antennas
THE SIMPLEST FORM OF ANTENNA , and the form in most common
use is the linear antenna—the dipole. Used on its own, or in arrays, it is
the most recognisable antenna.
4.1 The Ideal Dipole
The ideal dipole must be one of the most useful theoretical antennas to un- Though it may seem im-
practical, it is often eas-
iest to consider an ideal
case and thence deduce
results for more complex
examples.
derstand as a large number of other antennas are analyzed using the equations
that are quite easily developed for this antenna. Examples of these are the
short dipole, loop antennas, travelling wave antennas and some arrays. The
radiation pattern of any wire construction on which the currents are known can
also be readily determined by considering the structure to consist of connected
ideal dipoles and adding the pattern contribution due to each to form the full
pattern. Many computer analysis codes rely on this approach.
The ideal dipole is defined as a linear wire antenna with length very small with
respect to the wavelength and a uniform current distribution. For convenience,
this antenna is positioned at the centre of the coordinate system and aligned in
the z-direction, as shown in figure 4.1.
4.1.1 Fields
Using Maxwell’s equations and the simplicity of this geometry it is very easy
to find the fields due to the constant current I (Kraus & Fleisch 1999, pg278).
When such an analysis is performed it is found that the far field of the antenna
has an E-field in the θ direction, Eθ, and a φ-directed H-field, Hφ only. The
expression for the E-field will be given but the H-field can clearly be found by
“Ohm’s Law of Free Space” as discussed in section 1.3.3.
Eθ =
60πI0
λr
jej(2πf−βr)
sin θ (4.1)
41

42 Simple Linear Antennas
I0
x
y
z
Eθ
Hφ
IdealCoord
Figure 4.1: The Ideal Dipole in Relation to the Coordinate System
There are a number of important points relating to this expression. Considering
it factor by factor:
• 60π is the constant or magnitude
• Io is the (constant) current magnitude. An increase in this value results
in a corresponding increase in the field
• λ is the electrical length of the antenna and again an increase in this ratio
will imply a larger field. Changes in this ratio should only be made such
that the assumption of small electrical length still holds (0.1λ maximum).
• jej(2πf−βr)
is the phase factor. This factor is relatively unimportant unless
this antenna is combined with another and the total pattern becomes an
addition of the fields where phase plays an important role.
• sin θ is the pattern factor. This is the only factor indicating variation with
respect to the spherical coordinate system angles. Since none of the factors
contain a φ-term this antenna has constant pattern characteristics in the
azimuth direction. The resulting pattern has the familiar “doughnut”Another way of putting
this is that the antenna
has omnidirectional az-
imuthal coverage.
shape as illustrated in figure 4.2
The form of equation (4.1) is common to the expressions for most antenna field
distributions. Such distributions are always a function of excitation, geometry
in terms of wavelength and θ and φ angles. The relative pattern of the antenna
can be drawn using only the sin θ term and regarding the rest as a normalizing
factor. Where absolute field strengths are required the total equation should
clearly be used.

4.1 The Ideal Dipole 43
θ
dipole
y
z
Doughnut
Figure 4.2: Pattern of an Ideal Dipole Antenna
4.1.2 Radiation resistance
The radiation resistance of the antenna can be found once the ﬁeld distribution
is known. Using circuit concepts, the radiation resistance Rr is given by:
Rr =
2Pt
I2
0
Ω (4.2)
The total power transmitted Pt is found by integrating (adding) the power The factor of two is in-
troduced as result of the
fact that I0 is the peak
current and not the RMS
value.
density over a surface surrounding the antenna. Clearly if the power densities
in all directions have been accounted for, the total power is found. The power
density in any direction can be found using the expression discussed before:
Pd =
E2
2(120π)
Performing this integration, an expression for total power radiated is obtained
and using 4.2 the radiation resistance is found as:
Rr = 80π2
λ
2
Ω (4.3)
This value is clearly always small since the ratio of antenna length to wavelength
( /λ) was assumed to be small (≤0.1) at the outset of the analysis.
4.1.3 Directivity
The directivity of the ideal dipole is calculated by assuming an input power of
1 W to the antenna. Since the reference used is always the isotrope, the power
that it would radiate, given the same input power is simply Pd (isotrope) = 1
4πr2 .

The current to an ideal dipole with 1 W input power is given by I0 = 2
Rr
.
Using (4.3) for Rr in the expression above results in:
I0 =
2
80π2( /λ)2
(4.4)
Substituting (4.4) into (4.1) the E-field can be found in the maximum direction
(θ = 90◦
). The power density in this direction, Pd (ideal dipole) is found by the
relationship:
Pd =
E2
2(120π)
=
(60π)2
2 2
(λr)2 80π2 ( /λ)2 2(377)
The directivity by definition is the ratio, which becomes:
D =
Pd (ideal dipole)
Pd (isotrope)
= 1.5(= 1.76dBi)
4.1.4 Concept of current moment
An important concept which allows the use of the results achieved for the ideal
dipole above to other antennas is that of current moment. By inspection of
(4.1) it is clear that the E-field is proportional to the product of the length of
the antenna and the current (assumed constant over the whole antenna). The
current moment M for an ideal dipole is therefore the area under the current
distribution:
M = I0
The power density and power transmitted is proportional to the current moment
squared — ie:
E ∝ M
P ∝ M2
4.2 The Short Dipole
The short dipole antenna is a practically realizable antenna which is assumed
to have a triangular current distribution when shorter than about a tenth of a
wavelength, as shown in figure 4.3.
Since it can be shown that the current distribution on thin linear radiators
is sinusoidal, the two small parts of a sinusoid starting at either tip of the
short dipole is well approximated by two straight lines and hence a triangular
distribution.

4.2 The Short Dipole 45
Iin
ShortDip
Figure 4.3: Current Distribution on a Short Dipole Antenna
4.2.1 Fields
The current moment of the short dipole in terms of the feedpoint current Iin is:
M =
Iin
2
The E-field from the antenna is thus half the E-field found for the ideal dipole
(disregarding the phase terms which would be the same) i.e.
E =
30πIin
λr
sin θ
4.2.2 Radiation resistance
The power transmitted by the short dipole is proportional to the square of the
current moment (ie a quarter):
Pt(short dipole) =
Pt(ideal dipole)
4
since Pt = I2
R the radiation resistance of the short dipole would be a quarter
of that of the ideal dipole
Rr(short dipole) = 20π2
λ
2
Ω
4.2.3 Reactance
The reactance of a short dipole is always capacitive and usually quite large The reactance of these
short antennas is a very
strong function of the an-
tenna thickness and (ob-
viously) length.
and is not as easily calculated as the radiation resistance. Reactance values
can be measured for a specific antenna—and tables (King & Harrison 1969)
are available for different thickness antennas. The equivalent circuit of a short
dipole antenna can be given as in figure 4.4

C R0
Rr
TxLn Short Dipole
ShortCct
Figure 4.4: The equivalent circuit of a short dipole antenna
The R0 value indicated in figure 4.4 refers to the loss resistance and should be
included when that value is significant in relation to the radiation resistance Rr.
This antenna thus presents a serious problem when power has to be delivered
to it. The capacitive reactance (X = −1/2πfC) is typically a few hundred
ohms which is a large mismatch condition. Matching is usually accomplished
by placing an inductor in series with the feed line which has a positive reac-
tance (X = 2πfL) that is equal in magnitude to the capacitive reactance thus
resonating the antenna, as shown in figure 4.5.
L C R0
Rr
TxLn Short Dipole
L/2L/2
Tuning
Figure 4.5: Tuning out dipole capacitive reactance with series inductance
This is an improvement but a few “catch-22” problems still exist which explains
the inherent difficulty in transferring power to small antennas:
• The coil will have some loss resistance which is very often large compared
to radiation resistance (which is often a fraction of an ohm) resulting in
very low efficiency.
• To decrease coil losses the Q of the inductor should be increased but this
causes a reduced operating bandwidth and a more sensitive antenna, also
increasing the circulating currents and hence the voltages associated with
them.
• If the decrease in bandwidth can be tolerated, the resultant real (resonant)
impedance would approximate the very low radiation resistance and this
still presents a matching problem.
4.2.4 Directivity
It is clear from the sin θ factor in the E-field expression that the shape of the
pattern is exactly the same as that of the ideal dipole. The directivity (gain) of
a short dipole is therefore equal to the gain of the ideal dipole:
D (short dipole) = 1.5

4.3 The Short Monopole 47
4.3 The Short Monopole
I0
h
ShortMono
Ground Plane
Figure 4.6: Short monopole antenna
When a ground plane is present as in ﬁgure 4.6 antennas can be analyzed in Once image theory is ap-
plied (and this is true
of any antenna/image
combination) the ground-
plane behaviour can be
deduced from that of the
free space equivalent.
terms of image theory. The antenna/image combination has the same radiation
pattern as the short dipole. The two major diﬀerences between the two are:
• the monopole current moment is half that of the dipole
• the monopole radiates no power in the lower hemisphere—for the same
input power as the dipole, the monopole radiates twice as much power
into the upper hemisphere.
The power radiated is halved and the radiation resistance is half that of a short
dipole when expressed in terms of . For monopoles, the length of the antenna
above the ground h = /2 is clearly more relevant than and in terms of this
the radiation resistance is:
Rr = 40π2 h
λ
2
All the power is radiated in the upper hemisphere which implies double power
density in all directions in comparison to short—or ideal dipoles with the same
power input. The directivity of this antenna would thus also be double that of
the previous two antennas: As usual, increased gain
is at the expense of de-
creased gain elsewhere—
under the ground plane
in this case!
D (short monopole) = 2 (1.5) = 3
4.3.1 Input impedance
It was shown above that the radiation resistance of the short monopole is half
that of the equivalent short dipole. The same applies to the capacitive reactance
of the antenna.

4.4 The Half Wave Dipole
λ/2
I0
HalfDip
Figure 4.7: A Half wave dipole and its assumed current distribution
Although the derivation will not be performed here, the fields from a half wave
dipole with an assumed sinusoidal current distribution as shown in figure 4.7
can also be found by considering the antenna to be made up of small ideal
dipoles. The only difference in this case is that the phase of the current can notIt is interesting to note
that the current distribu-
tion must be known be-
fore the various parame-
ters of an antenna can be
determined.
be assumed to be constant and that the path lengths to a distant point P can
differ from the different locations on the antenna.
In the above cases, the current distributions were assumed to be sinusoidal mak-
ing analysis possible. This assumption is quite valid for thin linear radiators as
was shown by Schelkunoff (1941) and others. For more complex structures (and
thick dipoles) the current distribution may be more difficult to determine. Com-
putational techniques such as the Method of Moments, embodied in SuperNEC,
are therefore primarily concerned with the determination of the current on the
antenna wires. Once this is known it is a relatively straightforward task to
calculate impedance and radiation pattern of the antenna.
4.4.1 Radiation pattern
Using the sinusoidal current assumption, the magnitude of the electric field
distribution around the dipole can be determined as (noting that /λ = λ/2):
E =
60I
r
·
cos π
2 cos θ
sin θ
4.4.2 Directivity
The pattern of this antenna relative to that of a short dipole is shown in figure 4.8
The directivity of this antenna is clearly not much larger than that of the short
dipole. The accurate value is:
D (half wave dipole) = 1.64
this is equivalent to 2.16 dBi (relative to isotropic).

4.4 The Half Wave Dipole 49
θ = 0◦
(Dipole axis)
Isotrope
Short Dipole
Half Wave Dipole
HalfPat
90◦
Figure 4.8: Half wave dipole, short dipole and isotrope patterns
It is immediately clear that there is not a large difference between the gain of
the half wave dipole and that of the short dipole. This initially does not make
sense since a short dipole can be very much smaller than a dipole and hence
cheaper and more practical. The primary reason for the popularity of the half
wave dipole is its large and resonant input impedance—which was the problem
with the short dipole.
Similarly, the directivity of a quarter wave monopole—which is the image theory
equivalent of a half wave dipole—can be found as:
D (quarter wave monopole) = 2.16 + 3 = 5.16 dBi
The notation dBi is quite important and has been assumed until now. Very
often antenna gain and directivity is quoted relative to a half wave dipole since
this is a physically realizable antenna unlike the isotrope. The gain can thus be
directly measured by comparing the signal strength received from a half wave
dipole to that of the test antenna. When gain is quoted relative to a dipole
it should be clearly stated and often this is done by using the notation dBd It is always important to
ascertain which of these
two references are used
when gain is specified
or quoted since many
sources do not distin-
guish between the two—
not an ignorable differ-
ence!
(decibels relative to dipole). The conversion between the two is evident:
dBi = dBd + 2.16
4.4.3 Input impedance
By analysis, the input impedance for thin half wave dipoles is:
Zin = 73 + j43 Ω
This antenna is thus slightly longer than the length required for resonance.
When a thin antenna is shortened by about 2% resonance can be obtained.
Figure 4.9 indicates the shortening required for various length to diameter ratios.
As before, the quarter wave monopole has half the input impedance of the half
wave dipole.
Zin (quarter wave monopole) = 36.5 + j21 Ω

0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
101 102 103 104
Multiplyingfactor,K
Ratio of half-wave length to diameter Short
Figure 4.9: Shortening factors for different thickness half wave dipoles
The relatively large values of radiation resistance of these antennas makes for
easy transfer of power and virtually lossless antennas when good conductors are
used. Efficiencies are typically 99% or higher and losses can thus be neglected.The impedance band-
width of thin dipole
antennas as defined by
the VSWR 2 : 1 limi-
tation is typically 5%
of the centre frequency.
For thicker antennas
(small length to diameter
ratios) this bandwidth
can be larger.
This may be untrue in cases of extremely thin wires or high frequencies (>1000
MHz).
4.5 The Folded Dipole
It is very seldom that folded dipoles of other values than half wave length (or
slightly less to achieve resonance) are used. The term folded dipole would thus
be used to denote such an antenna unless otherwise stated. A typical Folded
Dipole is shown in figure 4.10. Folded dipoles are often used instead of normal
λ/2
s λ
FD
Figure 4.10: Half Wave Folded Dipole Antenna
dipoles for the following reasons:
• Mechanically easier to manipulate and more sturdy

4.5 The Folded Dipole 51
• Larger bandwidth than normal dipoles
• Larger input resistance than a normal dipole
• Can offer a direct DC path to ground —for lightning protection. The element centre op-
posite the feed can be
“shorted” to the boom
since this is a zero volt-
age point
This antenna’s characteristics are again easily understood using the current
moment technique. Clearly the currents in each arm are sinusoidally distributed
as in a half wave dipole and are in the same direction. The current moment of
this antenna is thus double that of the normal dipole. This implies:
Rin = 4(70) = 280 Ω
D (folded dipole) = D (half wave dipole) = 2.16dBi
E (folded dipole) = 2E (half wave dipole)
Folded dipoles typically have an impedance bandwidth (defined by the VSWR
2 : 1 limit) of 10 to 12%. This increase relative to an half-wave dipole can be
explained by noting that the antenna can be considered to be a superposition
of a normal dipole and a short circuit transmission line. When the frequency is
lowered the dipole exhibits a capacitive reactance whereas the transmission line
starts to be inductive. These two factors initially cancel each other causing the
increase in bandwidth. A comparison of the VSWR bandwidths of an ideally
matched dipole and folded dipole is shown in figure 4.11.
200 250 300 350 400 450 500
1
1.5
2
2.5
3
3.5
4
4.5
5
freq (MHz)
VSWR
dipbw
Dipole
Folded Dipole
Figure 4.11: VSWR bandwidth of a dipole and folded dipole.
A very interesting fact about these antennas is that the input resistance can
be increased or decreased by making the two elements of different diameters as
shown in figure 4.12.

d
2r1 2r2
FDThick
Figure 4.12: Impedance transformation using different thickness elements
The impedance transformation ratio is
Rin(folded dipole) = (1 + a)2
Rin(half wave dipole)
where a =
log(d/r1)
log(d/r2)
The impedance can also be manipulated by using more than two elements, as
shown in figure 4.13. The impedance step-up ratio under these conditions (where
λ/2
FDMany
Figure 4.13: Triply Folded Dipoles
n is the number of elements) is:
Rin = n2
Rin(half wave dipole)
4.6 Dipoles Above a Ground Plane
So far only monopole antennas which are dependent on the ground for their
operation have been discussed. It is interesting to observe the changes that
occur for horizontal dipoles above a ground plane as shown in figure 4.14. A
few general cases will be considered here to indicate the effects qualitatively.Since image theory is
used to determine ground
plane effects, it implies
that a dipole above
ground can be considered
to be a two element
array (see section 5.1 on
page 57).

4.7 Mutual Impedance 53
2h
P1
S1
S2
θ
θ
Direct-wave path
Reflected-wave path
Reflecting Surface
ReflRay
Path Difference = 2h sin θ
Figure 4.14: A horizontal dipole above a ground plane
There are are two rays with different path lengths to point P and constructive
and destructive interference will take place—depending on the path and hence
phase difference. When constructive interference occurs, the two waves will add
in phase and double the E-field result as compared to the free space pattern. In
power density terms this would be an increase of 4 times, or 6 dB! This apparent
“free” gain obtained by placing an antenna above a ground plane is clearly at
the expense of reduced gain at other angles where the interference would be
destructive, resulting in a null. However, with intelligent use it can be put to
great effect.
One of the main disadvantages of antennas close to the ground is that the
current in the antenna and that in the image are clearly in opposite directions.
At grazing angles they will always be out of phase—and cause a null. This Grazing angles are those
that are close to the
ground plane
can be a severe problem for ground-to-ground communications and the only
real remedy is to mount the antenna sufficiently high so that this effect can be
neglected.
4.7 Mutual Impedance
When two dipoles (or in fact any two antennas) are close enough to each other
to cause appreciable currents to flow on the one antenna as result of radiation by
the other they are said to be mutually coupled. The radiation patterns of such
an intentional or accidental combination can easily be determined by adding
the field contributions from each vectorially. Another important consequence is
that the input impedance of the antennas is often altered considerably.
The same statement clearly applies to antennas close to a reflecting plane since
interaction with the image results. This effect is graphically illustrated by con-

λ/2
d
2Dip
Figure 4.15: Two dipoles in eschelon
sidering two parallel half wave dipoles side-by-side. Each of these antennas has
a self impedance which is the impedance seen without the other antenna present.
For the two dipoles in this example, their self impedances are called Z11 and
Z22. Due to the interaction between the two dipoles their input impedances
in the configuration shown is different. This can be calculated by defining the
mutual impedance between the two dipoles, Z12 as the ratio of the voltage at
the terminals of the second antenna V2 as a result of a terminal current applied
to the first dipole I1. That is:
Z12 =
V2
I1
Ω
Curves of the mutual impedance for the geometry shown in figure 4.15 is given
in figure 4.16.
The following equations can be written to find the currents on the antennas and
hence their impedance (Balanis 1982, sec 8.6,pg 412):
V1 = I1Z11 + I2Z12
V2 = I1Z12 + I2Z22
Usually some voltage is assumed as an excitation (V1 & V2) for the two antennas
and thus using the above two equations, the resultant currents I1 and I2 are
found. The input impedance to dipole 1 would then be conventionally defined
as:
Zin =
V1
I1
This value will generally be different to the free space self impedance of the
antenna Z11.

EM Fundamentals and Antenna Selection Guide

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