Antennas in practice - EM fundamentals and antenna selection

Antennas in Practice

EM fundamentals and antenna selection

Alan Robert Clark

Andre 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, siteplanning, lightning and grounding (ISBN 0-620-27621-5)

Copyright c© 2001 by Alan Robert Clark andPoynting Innovations (Pty) Ltd.33 Thora CrescentWynbergJohannesburgSouth 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 LATEX2ε.

All rights reserved. No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording, or otherwise, without the prior writtenpermission of Poynting Innovations (Pty) Ltd. Printed in South Africa.

ISBN 0-620-27619-3

Contents

1 Electromagnetics 11.1 Transmission line theory . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Characteristic impedance & Velocity of propagation . . . 3

Two-wire line . . . . . . . . . . . . . . . . . . . . . . . . . 4One conductor over ground plane . . . . . . . . . . . . . . 5Twisted Pair . . . . . . . . . . . . . . . . . . . . . . . . . 5Coaxial line . . . . . . . . . . . . . . . . . . . . . . . . . . 5Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . . 6Slotline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.3 Impedance transformation . . . . . . . . . . . . . . . . . . 61.1.4 Standing Waves, Impedance Matching and Power Transfer 8

1.2 The Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Frequency and wavelength . . . . . . . . . . . . . . . . . . 101.3.2 Characteristic impedance & Velocity of propagation . . . 111.3.3 EM waves in free space . . . . . . . . . . . . . . . . . . . 121.3.4 Reflection from the Earth’s Surface . . . . . . . . . . . . . 141.3.5 EM waves in a conductor . . . . . . . . . . . . . . . . . . 16

2 Antenna Fundamentals 172.1 Directivity, Gain and Pattern . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Solid angles . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Isotropic source . . . . . . . . . . . . . . . . . . . . . . . . 18Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.4 Radiation pattern . . . . . . . . . . . . . . . . . . . . . . 20

Directivity estimation from beamwidth . . . . . . . . . . . 222.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Effective aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Free-space link equation and system calulations . . . . . . . . . . 25

3 Matching Techniques 293.1 Balun action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Unbalance and its effect . . . . . . . . . . . . . . . . . . . 29

iii

iv CONTENTS

3.1.2 Balanced and unbalanced lines—a definition . . . . . . . . 303.2 Impedance Transformation . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Angle Correction . . . . . . . . . . . . . . . . . . . . . . . 323.3 Common baluns/balun transfomers . . . . . . . . . . . . . . . . . 33

3.3.1 Sleeve Balun . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.2 Half-wave balun . . . . . . . . . . . . . . . . . . . . . . . 333.3.3 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.4 LC Networks . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.5 Resistive Networks . . . . . . . . . . . . . . . . . . . . . . 343.3.6 Quarter wavelength transformer . . . . . . . . . . . . . . 353.3.7 Stub matching . . . . . . . . . . . . . . . . . . . . . . . . 363.3.8 Shifting the Feedpoint . . . . . . . . . . . . . . . . . . . . 363.3.9 Ferrite loading . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Impedance versus Gain Bandwidth . . . . . . . . . . . . . . . . . 38

4 Simple Linear Antennas 414.1 The Ideal Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 Radiation resistance . . . . . . . . . . . . . . . . . . . . . 434.1.3 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1.4 Concept of current moment . . . . . . . . . . . . . . . . . 44

4.2 The Short Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Radiation resistance . . . . . . . . . . . . . . . . . . . . . 454.2.3 Reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.4 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 The Short Monopole . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Input impedance . . . . . . . . . . . . . . . . . . . . . . . 47

4.4 The Half Wave Dipole . . . . . . . . . . . . . . . . . . . . . . . . 484.4.1 Radiation pattern . . . . . . . . . . . . . . . . . . . . . . 484.4.2 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4.3 Input impedance . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 The Folded Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . 504.6 Dipoles Above a Ground Plane . . . . . . . . . . . . . . . . . . . 524.7 Mutual Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Arrays and Reflector Antennas 575.1 Array Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.1 Isotropic arrays . . . . . . . . . . . . . . . . . . . . . . . . 575.1.2 Pattern multiplication . . . . . . . . . . . . . . . . . . . . 585.1.3 Binomial arrays . . . . . . . . . . . . . . . . . . . . . . . . 595.1.4 Uniform arrays . . . . . . . . . . . . . . . . . . . . . . . . 60

Beamwidth . . . . . . . . . . . . . . . . . . . . . . . . . . 62Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Dipole Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2.1 The Franklin array . . . . . . . . . . . . . . . . . . . . . . 655.2.2 Series fed collinear array . . . . . . . . . . . . . . . . . . . 655.2.3 Collinear folded dipoles on masts . . . . . . . . . . . . . . 67

5.3 Yagi-Uda array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3.1 Pattern formation and gain considerations . . . . . . . . . 68

CONTENTS v

5.3.2 Impedance and matching . . . . . . . . . . . . . . . . . . 695.3.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Element correction . . . . . . . . . . . . . . . . . . . . . . 69Stacking and Spacing . . . . . . . . . . . . . . . . . . . . 70

5.4 Log Periodic Dipole Array . . . . . . . . . . . . . . . . . . . . . . 715.4.1 Design procedure . . . . . . . . . . . . . . . . . . . . . . . 755.4.2 Feeding LPDA’s . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 Loop Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.5.1 The small loop . . . . . . . . . . . . . . . . . . . . . . . . 76

Radiation pattern . . . . . . . . . . . . . . . . . . . . . . 77Input impedance . . . . . . . . . . . . . . . . . . . . . . . 77

5.6 Helical Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.6.1 Normal-mode . . . . . . . . . . . . . . . . . . . . . . . . . 785.6.2 Axial mode . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Original Kraus design . . . . . . . . . . . . . . . . . . . . 81King and Wong design . . . . . . . . . . . . . . . . . . . . 81

5.7 Patch antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.8 Phased arrays and Multi-beam “Smart Antennas” . . . . . . . . 875.9 Flat reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.10 Corner Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.11 Parabolic Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . 91

A Smith Chart 95

vi CONTENTS

List of Figures

1.1 A transmission line connects a generator to a load. . . . . . . . . 11.2 2-wire; Coax; µstrip; waveguide; fibre; RF; 50Hz! . . . . . . . . . 21.3 A “Lumpy” model of the TxLn, discretizing the distributed pa-

rameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Twisted-Pair geometry . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Impedance values plotted on the Smith Chart . . . . . . . . . . . 91.6 Electromagnetic Wave in free-space . . . . . . . . . . . . . . . . . 131.7 Radio Horizon due to earth curvature. . . . . . . . . . . . . . . . 131.8 Geometry of Interference between Direct Path and Reflected Waves 141.9 Interference pattern field strength contours for h/λ = 1.44 . . . . 15

2.1 Standard coordinate system. . . . . . . . . . . . . . . . . . . . . . 172.2 concept of a solid angle . . . . . . . . . . . . . . . . . . . . . . . 182.3 Three-dimensional pattern of a 10 director Yagi-Uda array . . . . 212.4 xy plane cut of the 10 director Yagi-Uda . . . . . . . . . . . . . . 212.5 Rectanglar radiation patterns of the 10-director Yagi-Uda in the

two principle planes . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 Reciprocity concepts . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 Polarization possibilities—wave out of page . . . . . . . . . . . . 232.8 Aperture efficiency of a parabolic dish . . . . . . . . . . . . . . . 252.9 Effective aperture of dipole À physical aperture . . . . . . . . . 252.10 Apertures must not overlap! . . . . . . . . . . . . . . . . . . . . . 252.11 Point-to-point link parameters . . . . . . . . . . . . . . . . . . . 26

3.1 The reason for unbalanced currents . . . . . . . . . . . . . . . . . 293.2 Balanced and Unbalanced Transmission lines . . . . . . . . . . . 303.3 Power loss in dB versus VSWR on the line. . . . . . . . . . . . . 313.4 Increase in Line Loss because of High VSWR . . . . . . . . . . . 323.5 VSWR at Input to Transmission Line versus VSWR at Antenna 323.6 The Bazooka or Sleeve balun. . . . . . . . . . . . . . . . . . . . . 333.7 A HalfWave 4:1 transformer balun. . . . . . . . . . . . . . . . . . 333.8 Transmission-line transformer. . . . . . . . . . . . . . . . . . . . 343.9 L-Match network. . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.10 Quarter wavelength transformer . . . . . . . . . . . . . . . . . . . 353.11 Single Matching Stub . . . . . . . . . . . . . . . . . . . . . . . . 363.12 Smith Chart of Single Stub Tuner Arrangement . . . . . . . . . . 373.13 T and Gamma Matching Sections . . . . . . . . . . . . . . . . . . 373.14 Gain bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vii

viii LIST OF FIGURES

3.15 Impedance bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 The Ideal Dipole in Relation to the Coordinate System . . . . . . 424.2 Pattern of an Ideal Dipole Antenna . . . . . . . . . . . . . . . . . 434.3 Current Distribution on a Short Dipole Antenna . . . . . . . . . 454.4 The equivalent circuit of a short dipole antenna . . . . . . . . . . 464.5 Tuning out dipole capacitive reactance with series inductance . . 464.6 Short monopole antenna . . . . . . . . . . . . . . . . . . . . . . . 474.7 A Half wave dipole and its assumed current distribution . . . . . 484.8 Half wave dipole, short dipole and isotrope patterns . . . . . . . 494.9 Shortening factors for different thickness half wave dipoles . . . . 504.10 Half Wave Folded Dipole Antenna . . . . . . . . . . . . . . . . . 504.11 VSWR bandwidth of a dipole and folded dipole. . . . . . . . . . 514.12 Impedance transformation using different thickness elements . . . 524.13 Triply Folded Dipoles . . . . . . . . . . . . . . . . . . . . . . . . 524.14 A horizontal dipole above a ground plane . . . . . . . . . . . . . 534.15 Two dipoles in eschelon . . . . . . . . . . . . . . . . . . . . . . . 544.16 Mutual impedance between two parallel half wave dipoles placed

side by side—as a function of their separation, d . . . . . . . . . 554.17 Change in resistance of a half wave dipole due to coupling to its

image at different heights above a ground plane . . . . . . . . . . 564.18 Mutual impedance between two collinear dipoles . . . . . . . . . 56

5.1 Two Isotropic point sources, separated by d . . . . . . . . . . . . 575.2 Two Isotropic Sources separated by λ/2 . . . . . . . . . . . . . . 585.3 Pattern multiplication. . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Binomial array pattern . . . . . . . . . . . . . . . . . . . . . . . . 595.5 Uniform linear array of isotropic sources. . . . . . . . . . . . . . . 605.6 Uniform Isotropic Broadside array . . . . . . . . . . . . . . . . . 615.7 Eight In-Phase Isotropic sources, dB vs Linear scales. . . . . . . 615.8 Uniform Isotropic Endfire Array . . . . . . . . . . . . . . . . . . 625.9 Array pattern of 2 isometric sources 10λ apart, and the element

pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.10 Two in-phase dipoles 10λ apart. . . . . . . . . . . . . . . . . . . 645.11 SuperNEC run of the 10λ Interferometer . . . . . . . . . . . . . 655.12 The Franklin array . . . . . . . . . . . . . . . . . . . . . . . . . . 665.13 A Series Fed Four Element Collinear Array . . . . . . . . . . . . 665.14 Distortion to folded dipole azimuth pattern in presence of a mast 675.15 Yagi-Uda array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.16 Multiplication factor for different diameter to wavelength ratios

of director and reflectors . . . . . . . . . . . . . . . . . . . . . . . 705.17 Graph showing the length to be added to parasitic elements to

compensate for the effect of the supporting boom . . . . . . . . . 715.18 The log-periodic dipole array . . . . . . . . . . . . . . . . . . . . 725.19 Constant directivity contours (dBi) . . . . . . . . . . . . . . . . . 735.20 Characteristics for feeder impedances of 100, 250 and 400Ω and

dipoles with L/D ratios of 177, 500 and 1000 . . . . . . . . . . . 745.21 Coaxial Connection to LPDA’s . . . . . . . . . . . . . . . . . . . 765.22 The small circular loop and the equivalent square loop . . . . . . 765.23 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 . . . . 795.25 Axial Mode Helix Antenna and its Typical Pattern. . . . . . . . 805.26 A λ/2 × λ patch (of copper) on a dielectric slab, over a ground

plane. The patch is fed by coax through the dielectric, halfwayalong the longest edge. . . . . . . . . . . . . . . . . . . . . . . . . 83

5.27 Geometry of square patch, as simulated in SuperNEC . . . . . . 845.28 3D pattern of the square patch. . . . . . . . . . . . . . . . . . . . 855.29 2D cut through the maximum gain of the square patch . . . . . . 855.30 Geometry and 3D pattern of a 4-square patch array. . . . . . . . 865.31 2D cut through the maximum gain of the 4-patch array. . . . . . 865.32 Power Splitter followed by phase modification. . . . . . . . . . . 875.33 Binary phase shifter, based on transmission line segments. . . . . 885.34 Electrically achieved DownTilt by progressive phasing. . . . . . . 885.35 A Corporate feed network—equal amplitude and phase. . . . . . 895.36 Gain vs separation for a 0.75λ plate . . . . . . . . . . . . . . . . 905.37 Impedance vs separation for a 0.75λ plate . . . . . . . . . . . . . 905.38 The SuperNEC rendition of a corner reflector. . . . . . . . . . . 915.39 SuperNEC predicted xy-plane radiation pattern of a corner re-

flector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.40 (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 formssince 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 representsa total re-evaluation, re-design and re-implementation. Much (older) materialhas 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 knownby the layman, and are promising easy, wireless “connectivity” at ever increasingrates. Reality is a little different, and is dependant on a practical understandingof the antenna issues involved in these emerging technologies.

Although a fair amount of background theory is covered, its goal is to providea framework for understanding practical antennas that are useful. Many designissues are covered, but in many cases the “cookbook” designs offered in this bookare good-enough starting points, but still nonoptimal designs, only achievableby simulation, and testing.

As a result, the book places a fair amount of emphasis on antenna simulationsoftware, such as SuperNEC. Ordinarily, if this book is run as a CEE course, itis accompanied by a SuperNEC Simulation Workshop, a hands-on introductionto 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

xii Preface

Chapter 1

Electromagnetics

1.1 Transmission line theory

TRANSMISSION lines connect generators to loads as shown in fig 1.1. Inthe RF world, in the transmitting case, this is viewed as connecting the

transmitter to the antenna, and in the receiving case as connecting the antennato 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. ByKirchhoff’s Voltage law, there is no change in voltage or current along the lengthof the “connection”.

As a rule of thumb, as soon as the “connection” length between the parts of thecircuit exceeds a fiftieth of a wavelength (λ/50) then ordinary circuit theorybreaks down, and the “connection” becomes a transmission line. The lengthof the line in terms of the wavelength of the operating frequency adds a finitetime-lag between the start and end of the line, and this causes the voltage alongthe line to change in terms of magnitude and phase as a function of the distancedown 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 50Hzpower; 2km for ordinary telephone connections; 600mm for 10Mb/s Ethernet;120mm for a PC Board bus track at 50MHz; 6mm for an on-chip interconnectin a 1GHz PIII.

d=2a

Dεr

ε0

εeff

d Dεr = εeff

w

hεr

ε0

εeff

a

b

εr = εeff

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 notchange 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 Theyare the same concept, though generally, a resistance is purely real, whereas animpedance 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 notethat the reactive part ofthe 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 theboundary conditions that shape the electric and magnetic fields in the mediumbetween the conductors. Of course, there is a direct relationship to the voltageand current on the line too.

In circuit terms, the distributed capacitance and inductance etc of the line canbe collected in lumped models. The model of the transmission line is then aninfinite set of these circuit sections. Many versions of the model exist, but Ishall 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 resistanceR [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 changefrom one section of the transmission line to another (uniform transmission line).Hence we define a characteristic impedance, Z0, as the ratio of the series tothe shunt components; in the lossless (or high frequency) case, R and G arenegligeable:

Z0 =

√R + jωL

G + jωC

(=

√L

CLossless

)

4 Electromagnetics

The concept of characteristic impedance has nothing to do with loss, but itcharacterises a transmission line. Further, since the inductance and capacitancebetween the two conductors largely depends on the geometry and the mediumproperties, 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 flowregardless of the load at the other end (the surge hasn’t got to the load yet!). Theratio 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 lineis 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, εr(eff), since thefields pass partially through air and through solid dielectric. If the entire volumearound the conductors is solid dielectric then εr(eff) = εr. In the case of a low-loss foam dielectric, or similar dielectric-and-air combinations, an approximationmust be used.

Two-wire line

Conceptually, the simplest type of transmission line. Popular examples are the300Ω “FM Tape” and the standard unshielded twisted pair (UTP) of networkingfame. The relationship of both conductors to ground is the same, making it abalanced line. Referring to fig 1.2 for the geometrical definitions, the rigorouscase is given by Wadell (1991, pg66):

Z0 =√

µ0µr

π2ε0εr(eff)cosh−1

(D

d

)

Kraus (1992, pg158,499)derives a similar resultusing:

(cosh−1 x =

ln(x +√

x2 − 1)

)

Usually, the spacing D is much greater than the radius a = d/2 and a simplifiedequation is used:

Z0 =√

µ0µr

π2ε0εr(eff)ln

(D

a

)D À a

Since magnetic materials are never used, the simplified equation is usually sim-plified further to the useful:

(√µ0

ε0≈ 120π

)Z0 =

120√εr(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 looksrather similar to a two-wire line by image theory, but is halved.

Z0 =60√εr(eff)

ln(

2ha

)

Twisted Pair

εr

d

D

twisted

Figure 1.4: Twisted-Pair geometry

Twisted Pair is a very common form of transmission line, as all external noisegenerated is common to both wires. A differential receiver stage then gets ridof most of the induced noise. Common in telephony, Ethernet networks etc.Again, the field lines cross through air and dielectric, making a closed-formsolution difficult. Wadell (1991, pg 68) presents Lefferson (1971)’s empiricalclosed-form solution:

Z0 =120√εr(eff)

cosh−1

(D

d

)

where: θ is normally between 20and 45. Less than 20,the twists are too loosefor uniformity, greaterthan 50 breaks thewires!

εr(eff) = 1 + q(εr − 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 hasless radiation (and reception). Since the relationship of the two conductors toground is different, coax is an unbalanced line. Also all field lines pass throughthe dielectric, hence εr(eff) = εr. It is also common to refer to the ratio of theradii a = d/2; b = D/2:

Z0 =60√εr

ln(

D

d

)or: Z0 =

60√εr

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 atthe higher frequencies. One of the problems here is that the fringing flux playsa very important role in establishing the exact Z0 value. Unfortunately thefringing flux flows through a combination of air and dielectric, hence again theneed for an effective ε. Gardiol (1984) has developed expressions for the caseswhere w/h ≤ 1 (large percentage of fringing flux) and for cases where w/h > 1(less percentage of fringing flux).

For w/h ≤ 1:

εr(eff) ≈12(εr + 1) +

12(εr − 1)

[(1 + 12

h

w

)−1/2

+ 0.04(1− w

h

)2]

Z0 ≈ 60√εr(eff)

ln(

8h

w+

w

4h

)

For w/h > 1:

εr(eff) ≈12(εr + 1) +

12(εr − 1)

(1 + 12

h

w

)−1/2

Z0 ≈ 120π√εr(eff)

[w

h+ 1.393 + 0.667 ln

(w

h+ 1.444

)]

Kraus & Fleisch (1999, pg 132) also has an approximate formula (which inearlier editions he qualified as being applicable for w ≥ 2h, but this is often notthe case in microstrip lines of interest:

Z0 ≈ 120π√εr[(w/h) + 2]

Slotline

A slotline has no groundplane, but uses adjacent tracks as the transmissionline. The fields are distributed in an elliptical waveguide within and withoutthe 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 withthe 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 transformsthe load impedance ZL connected to its output to a different impedance at theline input Zin as follows (lossless case):

Zin = Z0

[ZL + jZ0 tan β`

Z0 + jZL tan β`

]

where β =2π

λand ` is the length of the line (electrical length).

This equation is generally known as the Transmission line equation since it fullyspecifies what happens on the line. It is an unwieldy equation, however, andnot generally useful. A graphical technique like the Smith Chart (section 1.2on page 9) effectively embodies this equation in an easy-to-use manner, withoutthe need for the equation as such.

Since the velocity with which waves travel on a transmission line is lower than Note that the velocity ona line can never be higherthan the speed of light!

that of light, the physical length of the cable is always shorter than the electricallength:

Physical line length = VF× (electrical length)

where the velocity factor, VF = 1√εr

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 linelength, 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 =

Z20

ZL

This configuration is useful since it can transform one load impedance toa 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 tanβ` for a short circuited line

In both these cases the impedance is purely reactive and if the lines inquestion are less than a quarter wave it is clear that such lines could beused to “manufacture” capacitive (open circuit case) or inductive (shortcircuit 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 thecharacteristic 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! Themethod is then to take an open-circuited line and mesure the input impedance,increasing the frequency until the input impedance drops to a minimum. Theline is then at an electrical quarter-wavelength, so

VF =`phys

λ/4

1.1.4 Standing Waves, Impedance Matching and PowerTransfer

The Maximum Power Transfer theorem determines when maximum power willbe 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 theimpedance is equal to the real part of the impedance as seen lookingback 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 thatof the characteristic impedance of the line.

When this condition is not satisfied standing waves are set up on the line and notall of the available power is transferred through the system. A more practicalNote that the sub-

maximum power transferis not due to losses inthe system, but rather topower being reflected.

problem which occurs is that such mismatch conditions often damage the outputelectronics of a transmitter if excessive standing waves occur (or in well designedtransmitters 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 inthe voltage reflection coefficient, ρ:

ρ =ZL − Z0

ZL + Z0

The mismatch in impedance is also often stated in terms of the voltage standingwave 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 areflection coefficient as acapitalized 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 manipulateimpedances and transmission lines is on the Smith Chart (Smith 1939). Inaddition to solving the transmission line equation presented in section 1.1.3 onpage 6, the chart is a visualization tool, enabling design decisions that are notpossible by simply studying the theory behind the equations. Appendix A onpage 95 contains a fully fledged Smith Chart, and this can also be downloadedfrom 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 ina value of 0.6− j1.4 and then plotted on the chart shown in fig 1.5 in the lowerhemisphere.

This transmission line chart has a lot of useful features when manipulating The Smith chart is, infact, simply a polar plotof the (complex) reflec-tion coefficient, ρ, over-laid with lines of constantresistance and reactance.

impedances. If the impedance above is that of an antenna and it is connectedto a transmission line of 50Ω (say of quarter wavelength) then the input valueto this line can be simply read off the chart by rotating the plotted value aroundthe centre of the chart by a quarter wavelength (180 on the chart), yielding0.26 + j0.6, or 13 + j30Ω, denormalized. The distance (in electrical length)travelled down the line is indicated on the outside of the chart. When travellingfrom the load to the generator, a clockwise direction is followed.

On the other hand, if the transmission line is 0.16 wavelengths long, the antennaimpedance 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

5.3 0.6

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 Chartas it is simply the intercept on the real axis as indicated by the circle centeredin the centre of the chart drawn for VSWR of 5.3:1. The constant VSWR circlein fact describes all the possible values of the input impedance of the line as afunction of distance down the line.

As we move down the line, we travel down a constant VSWR circle, assumingthat the line is lossless, and hence get no nearer to a better match. In the case ofa lossy line, the magnitude of the reflection coefficient decreases, and instead ofa circle, a spiral is described, spiralling in to the centre of the chart, improvingthe match. ie A lossy line improves the aparrent VSWR at the input to theline, 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 inour example) and it is immediately clear that more reactance in this examplewould improve the impedance match since it would force the point closer to thecentre 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 Designation3–30 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 from30MHz to 30GHz is the most commercially used band.

• 3–30MHz bounces off ionosphere (so-called Shortwave) and is used forold-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 takeabout 100W of heat!

The microwave (> 1GHz) region is broken into Radar bands. The most commondesignations are the IEEE ones, although there are “newer” designations, whichI have never seen used.

Frequency Wavelength IEEE Designation1–2 GHz 300–150 mm L2–4 GHz 150–75 mm S4–8 GHz 75–37.5 mm C (5m (big) dish)8–12 GHz 37.5–25 mm X12–18 GHz 25–16.7 mm Ku (1m (small) dish)18–26 GHz 16.7–11.5 mm K26–40 GHz 11.5–7.5 mm Ka40–300 GHz 7.5–1 mm mm

Spectrum Usage:

FM Radio 88–108 MHzClassicfM 102.7 MHzTV 123 200 MHzMNET 615 MHzGSM MTN & Vodacom 900 MHzCell C 1800 MHzDECT 1880–1900 MHzGPS 1.23 & 1.58 GHzWLan 1.8 GHzIndustrial ISM 906–928MHzScientific ISM 2.4–2.5 GHzMedical ISM 5.8–5.9 GHzµwave oven (K5–513) 2.45 GHzDSTV 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 andthe medium played a part in determining the characteristic impedance. For apropagating Transverse Electromagnetic (TEM) wave in an unbounded medium,the characteristic impedance it sees is:

Z0 =

√jωµ0µr

σ + jωε0εr

Since free space is non-conducting, and the relative quantities are unity: Some texts use eta: η,and others zeta: ζ tospecifically refer to theZ0 of free-space.

Z0(free space) = η =√

µ0

ε0= 377Ω or: η ≈ 120π

12 Electromagnetics

Just as the characteristic impedance depends on the geometry and the propertiesof the medium, so the velocity of EM radiation also depends strongly on themedium properties. For example, sound travels at 330m/s in air, but 1500m/sin water.

In a non-conducting medium, the velocity of propagation is:

v =1√

µ0µrε0εr

where ε0 and µ0 refer to the electric permittivty and magnetic permeability offree space; the r subscripts refer to the quantities of the material relative tothat of free space. SinceNote that the value of µ0

is by definition, and ε0is actually derived fromthe measured speed oflight. (Kraus 1988, BackCover)

µ0 ≡ 4π × 10−7 [H/m]

ε0 = 8.85× 10−12 [F/m]

Thus, the speed of light in a vacuum or air, c, is given by:

c =1√µ0ε0

= 3× 108m/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 asthe Velocity Factor, VF= v/c. This leads to the useful formula for EM wavevelocity in non-magnetic media as:

v =c√εr

or: VF =1√εr

Since most plastics have a relative permittivity of about 2.3, the reduction inspeed 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 TransverseEM wave (TEM). This implies that the direction of propagation is at 90 toboth the Electric and Magnetic wave, which, in turn are at 90, as shown infig 1.6

Since the E field is analogous to voltage and the H field to current in the circuitssense, it is easily seen that the equivalent relationships to Ohms law etc exist inTEM 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

xz

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 curvaturethus limits the radio horizon achievable for a certain height of transmitter, asshown 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 thetangential point is on it,by Pythagorous, we get19.5km!

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 illustratedabove using either ground wave propagation or reflection from some atmosphericfeature such as the ionosphere, meteors, the troposphere or even an artificialsatellite (although in that case the signal is usually retransmitted).

Ground wave propagation occurs when vertically polarized waves are launchedand these are guided by the earth surface and thus follow the curvature of theearth. They are only practical at lower frequencies where the attenuation thesewaves undergo as result of earth losses are reasonably low. At the very lowfrequency end (round 30 kHz) the wavelength is so large that the waves flow inthe waveguide formed by the earth surface on the one hand and the ionosphereon 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 reflectingwaves from the ionosphere. This layer1 of gas that is ionized by the sun liesat a distance of approximately 80 to 400 km above the earth surface. Onlywaves below a certain critical frequency are reflected (by a process of successiverefraction) by this layer. This so called critical frequency is typically in the HFband (3–30 MHz) and varies strongly with time of day and other variables. Atfrequencies much below this frequency the radio-waves undergo attenuation (orabsorption) and communication using this principle is also impractical.

Other methods of obtaining beyond the horizon communication are the ratherexotic techniques such as meteor scatter propagation: where waves are reflectedoff the ionized trails left by meteors entering the atmosphere; or troposphericscatter: which uses discontinuities in the atmosphere to cause bending of wavesand 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-seafibre 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 reflectorof radio waves. This situation is illustrated in figure 1.8.

2h

P1

S1

S2

θ

θ

Direct-wave path

Reflected-wave path

Reflecting Surface

ReflRayPath 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 equallywell in all directions, the pattern would be modified by the reflected wave. Bythe method of images the situation above is similar to that which exists if amirror 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 pointP . At certain elevation angles θ the path difference would be such that thetwo waves are in phase and thus interfere constructively and for others theinterference 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 thenbe given by:

E = |E0| sin(

2πh sin θ

λ

)

Application of this formula to the particular value of h/λ = 1.44 results in thefield pattern shown in figure 1.9.

This condition is not always advantageous since an antenna that may have hada maximum towards θ = 0 would now have a null in the same direction. Theonly way to improve the situation would be to either make the antenna higherand thus force the angle of the first maximum lower or increase the frequencyand thus ensure an increased h/λ ratio.

In cases where radiation at some angle is required this reflection results in anunexpected bonus, however. The maximum value of the E-field in the directionof the maxima is twice the value of the original antenna without reflection. Thisimplies that in that particular direction the power density would be increasedby 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 usedproperly!

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, weget a different result to the free-space case considered above. The Electric fieldmagnitude actually decays exponentially inside the conductor. We define theSkin depth, δ, as the point where the field inside the (good) conductor, Ey, hasbecome 1/e of its original value, E0:

δ =√

2ωµσ

=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 fieldmagnitude. Comparing the skin depth (or depth of penetration) δ for silver,copper & cast iron at 50 Hz and 1GHz:

Relevant parameters: σAg = 6.1 × 1070/m σCu = 5.7 × 1070/m (Both haveµr = 1) σFe = 1060/m and µrFe = 5000

δ50Hz δ1GHz

Ag 9.1mm 2µmCu 9.4mm 2.1µmFe 1mm 0.23µm

The decay is obviously logarithmic: at one further skin depth, the field hasdeacyed a further 37%. ie 9.4mm of Copper to get to 37% of the original signalstrength. Another 9.4mm of copper will take that signal strength to 37% ofthat value, ie 13.7% of the original value. A useful table is then the percentageoverall decrease as a function of the number of skin depths:

δ′s % of Orig.1 372 13.73 5.14 1.95 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 barif 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, butat RF, the current den-sity concentrates in thetwo outer skins.

after 10µm, there is no appreciable field.

This means that the outside of a coaxial braid is a completely separateconductor 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 liketo describe the points in the spherical co-ordinate system as (r, θ, φ), as shownin 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, wherer is the distance from the origin, θ the angle from the zenith (not the elevationangle) 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 a3-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 are4π steradians in a sphere. As a result, an area a on the surface of the sphere ofradius r is said to subtend the solid angle ω. Note that the shape of the area isnot specified. shown in fig 2.2

x = rθ

r

θ

θ = xr radians

z

y

x

Ω

Area=A

Ω = Ar 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 powerin all possible (3-dimensional) angles. By definition, an isotropic antenna has adirectivity, d, of 1, and is infinitely small. Note that an isotropic antenna is animpossibility and is used only as a reference!

Generally, however, an antenna has greater directivity in some directions thanother 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 transmitsPrad watts, and we enclose the source with a sphere of radius, r, and hencesurface area, A = 4πr2, then the power density, S at the surface of the sphereis 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, butthat most equations re-quire it in linear form.

referred to an isotropic source (the Sisotropic in the above), the unit is denoteddBi for dB’s above isotropic. Decibels are very useful in power ratio’s becauseof the large range of values experienced because of the inverse square law. Forexample, if power P1 is 1000 times greater than P2, the ratio expressed in dBsis:

dB = 10 log10

(P1

P2

)

which yields P1 as being 30dB greater than P2. If P1 were a million times thepower of P2, it would be 60dB greater. It can be seen that dB’s are simply moreconvenient 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 ofa squared quantity is simply twice the log of the non-squared quantity, for apower component like voltage, E-field etc:

dB = 20 log10

(E2

E1

)

It is sometimes convenient to represent a signal with respect to a standardreference—eg the absolute power level to a milliwatt of power is denoted dBm:

dBm = 10 log10

(power1mW

)

similarly for a voltage relative to a microvolt:

dBµV = 20 log10

(voltage1µV

)

2.1.3 Efficiency

Directivity, D, is often loosely referred to as Gain, G. However, the term gain Note that “Gain” is notused in the same sense asamplifier “gain”. An an-tenna is passive. It sim-ply denotes an increasedpower density in one di-rection, at the expense ofother directions.

refers to the input power as opposed to the radiated power—the difference beingthe 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 arenot 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 andφ but are often used todescribe these values inthe maximum directions.

more theoretical than practical value.

In practice the gain of an antenna is important since it increases the powerdensity in the direction of the main beam of the antenna. A 100 W transmitterwith a 13dB gain antenna produces the same power density at a distant pointas a 1 kW transmitter with a 3 dB gain antenna. Clearly the former case wouldhave a larger area of lower power density in other directions and the extentto which antenna gain would improve communications would depend on theintended coverage.

In a broadcast scenario for instance, omnidirectional radiation may be requiredin the plane of the earth. The only improvement in gain of broadcast antennasis by minimizing the radiation upwards (at angles of θ smaller than 90). Forpoint-to-point links on the other hand the power can be concentrated as far aspossible in the desired direction.

Hence, the power density S at a distance r away from an antenna with a gainG is:

S =GPin

4πr2=

DPrad

4πr2W/m2

2.1.4 Radiation pattern

An antenna’s radiation pattern is a measure of how it radiates in 3-dimensionalspace. 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 antennaweakens predictably with distance by 1/r2, but not predictably by angle—thisis 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 plotcan 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 radiationpattern of a 10-director Yagi-Uda array is shown in fig 2.3. An antenna oftenhas 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, itcan rarely show detail ac-curately.

the main lobe, like a collar). It can also be seen that the next sidelobe fires upand down, but that there is a null along the y-axis. It is far more common totake a 2D cut through the principle planes of the antenna, as shown in the xyplane cut in fig 2.4.

Fig 2.4 is shown in a polar representation, which is most useful for visualizationpurposes. 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 thefirst sidelobe levels (14.3dB in azimuth; 8.77dB inelevation) 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 pointof 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 peakpower is a relative measurement, and hence an eassier one.


−180 −120 −60 0 60 120 180−20

−10

0

10

15

φ

Gai

n (d

Bi)


(a) xy plane

−90 −30 30 90 150 210 270−20

−10

0

10

15

θ

Gai

n (d

Bi)

Radiation Pattern (Elevation)

(b) yz plane

Figure 2.5: Rectanglar radiation patterns of the 10-director Yagi-Uda in thetwo principle planes

Directivity estimation from beamwidth

Note again that if the “beam” equally covered a whole sphere of 4π squareradians, or steradians, ie an isotropic source, the directivity is 1 by definition.4π square radians =

4π(360/2π)2 square de-grees = 41 253, usuallyrounded 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

θHPφHP

The above approximation produces about 1dB too much gain. the reasoningis that the area represented is square, whereas in reality, the beam is generallyround. hence a common adjustment (Kraus 1988, pg100) is:

D ≈ 36 000θHPφHP

even in this case, however, if fairly significant side- and back-lobes exist, theprediction 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 gives13.5dBi 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 currentto flow at the terminals of antenna B, then an equal current (in magnitude andphase) will occur at the terminals of antenna A if the same voltage is appliedto the terminals of antenna B.

In short, all characteristics of an antenna apply equally well in transmit andreceive mode (radiation pattern (reception pattern), input/output impedanceetc). shown in fig 2.6.

2.3 Polarization 23

Va EnergyFlow

Ib

Ia EnergyFlow

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 EMwaves. In the case of a dipole or a Yagi-Uda array, the E-field is linearlypolarized in the direction of the dipole. Most often, a linear polarization will bespecified 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 receivingantenna vertically polar-ized, there is (ideally) noreceived signal.

polarization of the incident wave and the antenna. eg A cellphone base stationantenna is vertically polarized, and although the mobile has a linearly polarizedantenna, it is seldom held perfectly upright, incurring a loss in received power.

yLinear

x

E2

yElliptical

x

E2

E1

E

yCircular

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 onlyin the y direction (vertical). the maximum value e2 is in the y direction. β isthe 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 aleft-hand or a right-handed fashion, giving rise to lhcp and rhcp. a rhcp antennawill not (ideally) receive a lhcp wave. the e-field is continuously rotating, andthe 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, themajor axis magnitude tothe minor axis magni-tude defines the axial ra-tio. An infinite axial ra-tio implies linear polar-ization, whereas an AR of1 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 willreceive a power Pr [W] and deliver it to a load connected to its terminals. Thisgives 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 berelated 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 actualarea 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

AeApertureEff

Figure 2.8: Aperture efficiency of a parabolic dish

Ae

λ/2

ApertureDipole

Figure 2.9: Effective aperture of dipole À physical aperture

is 1.11m2! (740× larger).

Note that the aperture has meaning—if two antennas are used to capture moreenergy from the wave, the apertures cannot be allowed to overlap as shown infig 2.10

ApertureSteal

Figure 2.10: Apertures must not overlap!

In the case of linear antennas, this is a fair problem, as the apertures easilyoverlap, and the overall capture area will be reduced. With dish antennas, sinceAe < 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 certaindirection 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 theeffective radiated power (ERP) of an antenna in a particular direction as:ERP is the power that

an isotrope would haveto produce in all di-rections to achieve thesame effect in this par-ticular direction. (alsopedantically called EIRP,I for Isotropic, but rarelyused).

ERP = GtPt

at a distance, r, away from the transmitting antenna, the power density, S isgiven 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 =Grλ

2

4π

hence the power received by an antenna in a freespace point-to-point link is:

Pr =GtGrPtλ

2

(4πr)2[W]

note that multipath reflections with constructive and destructive interferenceand atmospheric absorption change this value!

The link equation can obviously also be conveniently expressed in dB’s:The 32.45 factor comesfrom +20 log(300) forfrequency conversion(20 since squared),−20 log(4π), and−20 log(1000) for thekm 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 fis in MHz.

Example: Geostationary satellite comms

A geostationary satellite must be in the Clarke orbit at 36 000 km in orderto 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 transmittedby the earth station? Receiver gain is 20dB, earth station gain is 30dB, at afrequency of 5GHz.

Converting to linear gains:

Gr = 20dB = 100; Gt = 30dB = 1000

λ =3005000

= 60mm

hence the power we are required to transmit is:

Pt =Pr(4πr)2

GtGrλ2=

8pW(4π × 3.6× 107m)2

100 · 1000 · (0.06)2= 4547W

So 4.5kW is needed to get 8pW to the satellite. Note that the link equationignores atmospheric absorption or any other losses along the path, and purelyassumes 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 onlya 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 DCperspective, we view the braid as “one conductor”.

If we feed an antenna using a coaxial cable as shown in fig 3.1, the antennaradiation 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-talesign of external current isa change in a measure-ment parameter when thecable is grabbed by ahand.

the inside and outside currents meet at the antenna junction, this implies thata larger current can flow on one dipole arm than on the other. The antennapattern is distorted, and the input impedance is changed.

29

30 Matching Techniques

The job of the BALance to UNbalance converter (Balun) is to prevent thecurrents flowing on the outside of the braid by presenting a high impedance tothem.

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 thata coaxial line is not—the field strength is much higher on the inner conductorthan the outer conductor.

3.2 Impedance Transformation

The Balun action is to be distinguished from the Transformer action. A circuitthat does both functions should be called a Balun-Transformer, but in practiceall 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 tothe line, and the generator in order that no reflected power exists—maximumpower transfer.

A transmitter cannot deliver full power to an unmatched load, but this is notthe 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 of2:1)

Impedance matching is important both for transmission and reception. It ismore 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 thepower reduction as result of a mismatch in terms of VSWR:

Power lost in transfer = 10 log

(1−

(VSWR− 1VSWR + 1

)2)

dB

Thus a VSWR of 2 : 1 results in a power reduction of only 0.5 dB. Even a VSWRas high as 5 : 1 only causes a reduction of 2.5 dB. The power reduction due to themismatch condition itself is thus not all that significant, but some transmitterswill start reducing power output to protect the driving stage electronics at suchlow values as 1.5 : 1 or 2 : 1 (Or simply blow up if no power reduction protectionis in place). The power lost (in dB) versus VSWR is illustrated in figure 3.3.

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

VSWR

Pow

er lo

st in

dB

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 manufacturerof the lines in terms of dB/30m (ie dB/100ft!). It should be noted that themanufacturer specification assumes matched conditions. The graph in figure 3.4indicates 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 isclearly obtained at theexpense of efficiency,indicating the need toavoid long transmis-sion lines when doingmeasurements of VSWR.

the VSWR at the antenna. A curve quantifying this characteristic is given infigure 3.5


10-1

100

101

10-1 100 101

Additional Loss in dB when Matched

Add

ition

al lo

ss in

dB

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

SWR

atA

nten

na

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 ahigh (theoretically infinite) impedance. Thus the simplest balun is a short-circuited quarter-wave sleeve around the coaxial feeder cable, where the sleeveis short-circuited to the coaxial feeder cable braid a quarter wavelength awayfrom the feed. Currents wanting to flow in the outside of the braid now see ahigh 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 torefer to the sleeve balunas a “Bazooka balun”!

It is rather narrow-band however, and care must be taken to ensure that it isthe 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 differentVF).

3.3.2 Half-wave balun

λ/2HalfWaveBalun

Figure 3.7: A HalfWave 4:1 transformer balun.

In addition to a balun action, due to the halfwave section, there is also a 4:1impedance transformation as shown in fig 3.7.

The impedance transformation is because after the λ/2 transmission line, thevoltage is exactly out of phase, hence twice the voltage is applied between the


dipoles. Naturally, the current is halved, hence the impedance (voltage overcurrent) 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 thata 2:1 voltage transformer is a 4:1 impedance transformer). Transformers arebroadband, but tend to be lossy. The selection of the ferrite core is importantas 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. Theyare usually bifilar(4:1) or trifilar(9:1) wound, and the main coupling mechanismis that of a transmission line. They are effectively autotransformers and areusually wound on a frequency-band specific ferrite core. They are quite broad-For the HF band, the

Philips 4C6 material isexcellent.

band achieving bandwidths of up to 15:1. The transmission line transformer (ifconnected correctly) also has a Balance to Unbalance conversion. If the inputGnd is reversed, NO Balun action.

3.3.4 LC Networks

Physical inductors and capacitors can also be used to resonate the antennaand then transform the impedance. One drawback is that inductors are lossy.Both “pi” and “T” networks are commmon. One popular match is known asthe “L-match”, shown in fig 3.9. In the L-match, the LC transformer networkis simply an extension of the resonating inductor for the antenna. Thereforepopular 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 1.1.3on page 6):

Z0 =√

ZLZS

In matching one impedance to another, if a transmission line can be found witha 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. Atlow frequencies the cable length may be inconvenient, at high frequencies amicrostrip line is used.

Ordinarily there is no balun action. However, if a “series” connection of twocoaxial cables is used as the transformer, there is balun action since both linesare shielded. A “series” connection uses two pieces of coax, with the two braidsconnected 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 anyimpedance. 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, andonly varies the length ofthe 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 simplyadded. To achieve this the normalized antenna impedance is plotted (P1) androtated 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 thenecessary amount of susceptance (inverse of reactance) using the open or shortcircuited stub. This value can be found by starting at the outer circle on thechart at the right hand end (admittance of infinity) for short circuited case orthe left hand side (admittance of zero) for open circuit stub and finding therequired 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-bandmatch. 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 difficultto analyze their performance but the following general trends referring to the Tmatch 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 60percent of the total antenna length.

• Higher values of input impedance can be realized when the antenna isshortened to cancel the inductive reactance of the matching section.

Some flexibility can be obtained by inserting variable capacitors in series withthese sections at the feed. As a first approximation values of about 7 pF permeter of wavelength can be used.

3.3.9 Ferrite loading

Wrapping the coax/TxLn round a ferrite toroid/clamping ferrite beads aroundthe TxLn. (eg VGA cables). This simply increases the impedance to externalcurrent flow.

3.4 Impedance versus Gain Bandwidth

The goal of matching is to make the antenna present an acceptable impedanceto the transmitter over the bandwidth that is necessary. However it is senselessto achieve a broad bandwidth match if the antenna does not maintain its gainover that bandwidth. Hence we can speak of an impedance bandwidth and again 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 producesa gain bandwith as shown in 3.14, and an impedance bandwith (normalizingto 200Ω) as shown in fig 3.15. It is clear that some work needs to be done toimprove the match so that the impedance bandwidth falls in line with the gainbandwidth.

3.4 Impedance versus Gain Bandwidth 39

200 250 300 350−10

−5

0

5

10

15

Freq (MHz)

Gai

n (d

Bi)

Gain at θ=90, φ=0

Line1

Figure 3.14: Gain bandwidth

200 250 300 3501

1.5

2

2.5

3

3.5

4

4.5

5

Freq (MHz)

VS

WR

Line1

Figure 3.15: Impedance bandwidth


Chapter 4

Simple Linear Antennas

THE SIMPLEST FORM OF ANTENNA , and the form in most commonuse 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 idealcase and thence deduceresults for more complexexamples.

derstand as a large number of other antennas are analyzed using the equationsthat are quite easily developed for this antenna. Examples of these are theshort dipole, loop antennas, travelling wave antennas and some arrays. Theradiation pattern of any wire construction on which the currents are known canalso be readily determined by considering the structure to consist of connectedideal dipoles and adding the pattern contribution due to each to form the fullpattern. Many computer analysis codes rely on this approach.

The ideal dipole is defined as a linear wire antenna with length very small withrespect to the wavelength and a uniform current distribution. For convenience,this antenna is positioned at the centre of the coordinate system and aligned inthe 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 easyto 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 antennahas an E-field in the θ direction, Eθ, and a φ-directed H-field, Hφ only. Theexpression 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`

λrjej(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. Consideringit factor by factor:

• 60π is the constant or magnitude

• Io is the (constant) current magnitude. An increase in this value resultsin a corresponding increase in the field

• `λ is the electrical length of the antenna and again an increase in this ratiowill imply a larger field. Changes in this ratio should only be made suchthat the assumption of small electrical length still holds (0.1λ maximum).

• jej(2πf−βr) is the phase factor. This factor is relatively unimportant unlessthis antenna is combined with another and the total pattern becomes anaddition of the fields where phase plays an important role.

• sin θ is the pattern factor. This is the only factor indicating variation withrespect to the spherical coordinate system angles. Since none of the factorscontain a φ-term this antenna has constant pattern characteristics in theazimuth direction. The resulting pattern has the familiar “doughnut”Another way of putting

this is that the antennahas 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 fielddistributions. Such distributions are always a function of excitation, geometryin terms of wavelength and θ and φ angles. The relative pattern of the antennacan be drawn using only the sin θ term and regarding the rest as a normalizingfactor. Where absolute field strengths are required the total equation shouldclearly 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 distributionis known. Using circuit concepts, the radiation resistance Rr is given by:

Rr =2Pt

I20

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 thefact that I0 is the peakcurrent and not the RMSvalue.

density over a surface surrounding the antenna. Clearly if the power densitiesin all directions have been accounted for, the total power is found. The powerdensity 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 obtainedand 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 of1 W to the antenna. Since the reference used is always the isotrope, the powerthat 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 =√

2Rr

.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 therelationship:

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 idealdipole 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 ofthe antenna and the current (assumed constant over the whole antenna). Thecurrent moment M for an ideal dipole is therefore the area under the currentdistribution:

M = I0`

The power density and power transmitted is proportional to the current momentsquared — ie:

E ∝ M

P ∝ M2

4.2 The Short Dipole

The short dipole antenna is a practically realizable antenna which is assumedto have a triangular current distribution when shorter than about a tenth of awavelength, as shown in figure 4.3.

Since it can be shown that the current distribution on thin linear radiatorsis sinusoidal, the two small parts of a sinusoid starting at either tip of theshort dipole is well approximated by two straight lines and hence a triangulardistribution.

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`

λrsin θ

4.2.2 Radiation resistance

The power transmitted by the short dipole is proportional to the square of thecurrent moment (ie a quarter):

Pt(short dipole) =Pt(ideal dipole)

4

since Pt = I2R the radiation resistance of the short dipole would be a quarterof 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 theseshort antennas is a verystrong function of the an-tenna thickness and (ob-viously) length.

and is not as easily calculated as the radiation resistance. Reactance valuescan be measured for a specific antenna—and tables (King & Harrison 1969)are available for different thickness antennas. The equivalent circuit of a shortdipole antenna can be given as in figure 4.4


C R0

RrTxLn 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 beincluded when that value is significant in relation to the radiation resistance Rr.

This antenna thus presents a serious problem when power has to be deliveredto it. The capacitive reactance (X = −1/2πfC) is typically a few hundredohms which is a large mismatch condition. Matching is usually accomplishedby 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 thusresonating the antenna, as shown in figure 4.5.

L C R0

RrTxLn 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 explainsthe inherent difficulty in transferring power to small antennas:

• The coil will have some loss resistance which is very often large comparedto radiation resistance (which is often a fraction of an ohm) resulting invery low efficiency.

• To decrease coil losses the Q of the inductor should be increased but thiscauses a reduced operating bandwidth and a more sensitive antenna, alsoincreasing the circulating currents and hence the voltages associated withthem.

• If the decrease in bandwidth can be tolerated, the resultant real (resonant)impedance would approximate the very low radiation resistance and thisstill 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 thepattern is exactly the same as that of the ideal dipole. The directivity (gain) ofa 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 trueof any antenna/imagecombination) the ground-plane behaviour can bededuced from that of thefree space equivalent.

terms of image theory. The antenna/image combination has the same radiationpattern 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 sameinput power as the dipole, the monopole radiates twice as much powerinto the upper hemisphere.

The power radiated is halved and the radiation resistance is half that of a shortdipole when expressed in terms of `. For monopoles, the length of the antennaabove the ground h = `/2 is clearly more relevant than ` and in terms of thisthe radiation resistance is:

Rr = 40π2

(h

λ

)2

All the power is radiated in the upper hemisphere which implies double powerdensity in all directions in comparison to short—or ideal dipoles with the samepower input. The directivity of this antenna would thus also be double that ofthe previous two antennas: As usual, increased gain

is at the expense of de-creased gain elsewhere—under the ground planein 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 halfthat of the equivalent short dipole. The same applies to the capacitive reactanceof 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 wavedipole with an assumed sinusoidal current distribution as shown in figure 4.7can also be found by considering the antenna to be made up of small idealdipoles. 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 bedetermined.

be assumed to be constant and that the path lengths to a distant point P candiffer 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 aswas shown by Schelkunoff (1941) and others. For more complex structures (andthick 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 theantenna wires. Once this is known it is a relatively straightforward task tocalculate impedance and radiation pattern of the antenna.

4.4.1 Radiation pattern

Using the sinusoidal current assumption, the magnitude of the electric fielddistribution around the dipole can be determined as (noting that `/λ = λ/2):

E =60I

r· cos

(π2 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 shortdipole. 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)

IsotropeShort 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 ofthe half wave dipole and that of the short dipole. This initially does not makesense since a short dipole can be very much smaller than a dipole and hencecheaper and more practical. The primary reason for the popularity of the halfwave dipole is its large and resonant input impedance—which was the problemwith the short dipole.

Similarly, the directivity of a quarter wave monopole—which is the image theoryequivalent 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. Veryoften antenna gain and directivity is quoted relative to a half wave dipole sincethis is a physically realizable antenna unlike the isotrope. The gain can thus bedirectly measured by comparing the signal strength received from a half wavedipole to that of the test antenna. When gain is quoted relative to a dipoleit should be clearly stated and often this is done by using the notation dBd It is always important to

ascertain which of thesetwo references are usedwhen gain is specifiedor quoted since manysources 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 halfwave 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

Mul

tiply

ing

fact

or, 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 foreasy transfer of power and virtually lossless antennas when good conductors areused. Efficiencies are typically 99% or higher and losses can thus be neglected.The impedance band-

width of thin dipoleantennas as defined bythe VSWR 2 : 1 limi-tation is typically 5%of the centre frequency.For thicker antennas(small length to diameterratios) this bandwidthcan be larger.

This may be untrue in cases of extremely thin wires or high frequencies (>1000MHz).

4.5 The Folded Dipole

It is very seldom that folded dipoles of other values than half wave length (orslightly less to achieve resonance) are used. The term folded dipole would thusbe used to denote such an antenna unless otherwise stated. A typical FoldedDipole 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 boomsince this is a zero volt-age point

This antenna’s characteristics are again easily understood using the currentmoment technique. Clearly the currents in each arm are sinusoidally distributedas in a half wave dipole and are in the same direction. The current moment ofthis 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 VSWR2 : 1 limit) of 10 to 12%. This increase relative to an half-wave dipole can beexplained by noting that the antenna can be considered to be a superpositionof a normal dipole and a short circuit transmission line. When the frequency islowered the dipole exhibits a capacitive reactance whereas the transmission linestarts to be inductive. These two factors initially cancel each other causing theincrease in bandwidth. A comparison of the VSWR bandwidths of an ideallymatched dipole and folded dipole is shown in figure 4.11.

200 250 300 350 400 450 5001

1.5

2

2.5

3

3.5

4

4.5

5

freq (MHz)

VS

WR

dipbw

DipoleFolded Dipole

Figure 4.11: VSWR bandwidth of a dipole and folded dipole.

A very interesting fact about these antennas is that the input resistance canbe increased or decreased by making the two elements of different diameters asshown 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)2Rin(half wave dipole)

where a =log(d/r1)log(d/r2)

The impedance can also be manipulated by using more than two elements, asshown 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 = n2Rin(half wave dipole)

4.6 Dipoles Above a Ground Plane

So far only monopole antennas which are dependent on the ground for theiroperation have been discussed. It is interesting to observe the changes thatoccur for horizontal dipoles above a ground plane as shown in figure 4.14. Afew general cases will be considered here to indicate the effects qualitatively.Since image theory is

used to determine groundplane effects, it impliesthat a dipole aboveground can be consideredto be a two elementarray (see section 5.1 onpage 57).

4.7 Mutual Impedance 53

2h

P1

S1

S2

θ

θ

Direct-wave path

Reflected-wave path

Reflecting Surface

ReflRayPath 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 constructiveand destructive interference will take place—depending on the path and hencephase difference. When constructive interference occurs, the two waves will addin phase and double the E-field result as compared to the free space pattern. Inpower 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 atthe expense of reduced gain at other angles where the interference would bedestructive, resulting in a null. However, with intelligent use it can be put togreat effect.

One of the main disadvantages of antennas close to the ground is that thecurrent 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 theground plane

can be a severe problem for ground-to-ground communications and the onlyreal remedy is to mount the antenna sufficiently high so that this effect can beneglected.

4.7 Mutual Impedance

When two dipoles (or in fact any two antennas) are close enough to each otherto cause appreciable currents to flow on the one antenna as result of radiation bythe other they are said to be mutually coupled. The radiation patterns of suchan intentional or accidental combination can easily be determined by addingthe field contributions from each vectorially. Another important consequence isthat the input impedance of the antennas is often altered considerably.

The same statement clearly applies to antennas close to a reflecting plane sinceinteraction 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 hasa 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 andZ22. Due to the interaction between the two dipoles their input impedancesin the configuration shown is different. This can be calculated by defining themutual impedance between the two dipoles, Z12 as the ratio of the voltage atthe terminals of the second antenna V2 as a result of a terminal current appliedto the first dipole I1. That is:

Z12 =V2

I1Ω

Curves of the mutual impedance for the geometry shown in figure 4.15 is givenin figure 4.16.

The following equations can be written to find the currents on the antennas andhence 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 antennasand thus using the above two equations, the resultant currents I1 and I2 arefound. The input impedance to dipole 1 would then be conventionally definedas:

Zin =V1

I1

This value will generally be different to the free space self impedance of theantenna Z11.

4.7 Mutual Impedance 55

-40

-20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Spacing in terms of wavelength

Impe

danc

e (O

hms)

Mutual ResistanceMutual Reactance

ddvalre

Figure 4.16: Mutual impedance between two parallel half wave dipoles placedside by side—as a function of their separation, d

This theory can be extended to include further dipoles, where the equations arethen arranged in matrix form and solved using any of the standard techniques.This is usually the method used by computer programs specifically written to If the one dipole in this

arrangement is a para-sitic element like thosefound in Yagi antennas,the equations still applyand V2 is simply set tozero.

analyze Yagi and log periodic arrays. As was said before: once the currentdistribution on an antenna is known all other parameters can be calculated—not only the input impedance as was shown above.

From the curves in figure 4.16, it is clear that the mutual impedance is largewhen antennas are close together and decreases as they are moved furtherapart—as one would expect. It is clear that the coupling becomes quite smallfor separations of more than about one wavelength, and this is a useful rule-of-thumb. This gives a general indication of the distances to conducting bodieswhich can start to effect antenna performance.

Once again these effects can be used to advantage if they are taken into account—since one can sometimes actually improve the match to a antenna by choosingthe optimum separation. It also indicates the importance of considering the Remember that the sepa-

ration in terms of wave-length is important, oneshould not be fooled byphysical dimensions.

effects of surroundings on antenna performance.

For cases where the antenna is close to a reflecting plane, the separation isclearly twice the distance from the surface if these curves are to be applied tosuch cases. The effect on the input resistance of a half-wave dipole above aground plane is shown in figure 4.17.

Again a feel for the distances where coupling becomes significant is obtainedfrom this curve. Note that radiation resistance drops to zero when a dipoleis brought very close to a ground plane. Seen from a different point of viewthis makes sense—since at close separation the currents in the antenna and theimage are oppositely directed as in a transmission line and thus no nett radiationresults.


0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

R(Ω)

Height above Ground, λRinMut

Figure 4.17: Change in resistance of a half wave dipole due to coupling to itsimage at different heights above a ground plane

The mutual coupling is also shown for collinear half-wave dipoles in figure 4.18,which shows that the coupling is much smaller—collinear configurations are lesssensitive to coupling considerations.

-10

-5

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

R21X21

Ω

Spacing, λ

Figure 4.18: Mutual impedance between two collinear dipoles

Chapter 5

Arrays and Reflector Antennas

5.1 Array Theory

MUCH OF ANTENNA THEORY consists of correctly adding field contri-butions at a point from all parts of an antenna, or antenna array. Note

that fields must be used, not power, as proper vector addition of magnitude andphase must occur.

5.1.1 Isotropic arrays

Consider two isotropic sources, separated by d, having the same magnitude andphase.

1 2θ = 0

d/2 d/2

θ

d cos θTwoIso

Figure 5.1: Two Isotropic point sources, separated by d

The far E-field is given by (Kraus & Fleisch 1999, pg260):

E = E2ejψ/2 + E1e

−jψ/2

where ψ = βd cos θ = (2πd/λ) cos θ is the phase-angle difference between thefields from the two sources. If E1 = E2 = E0, we get:

E = 2E0 cos(ψ/2)

57

58 Arrays and Reflector Antennas

For the special case of d = λ/2,

E = E0 cos(π

2cos θ

)

which is shown in figure 5.2.

1 2θ = 0

d = λ/2

E = E0

[cos

(π

2cos θ

)]

TwoIsoPat

Figure 5.2: Two Isotropic Sources separated by λ/2

5.1.2 Pattern multiplication

The total field pattern of an array of non-isotropic sources is given by themultiplication of the element field pattern and the array field pattern.

1 2θ = 0

d = λ/2

α

E = E0

[cos

(π

2cos θ

)]

E = k sinα

E = E0

[cos

(π

2cos θ

)]× k sin α

PatMult

Figure 5.3: Pattern multiplication.

Two short dipoles placed in eschelon λ/2 apart. The element pattern is k sinα,a figure of eight perpendicular to the dipoles. From the above, the array factoris cos

(π2 cos θ

), a figure of eight parallel to the dipoles.

5.1 Array Theory 59

By pattern multiplication, we now get four (weak) lobes at 45 as seen in fig 5.3. Ordinarily one wants theelement and array pat-tern to have strength to-gether.

Note that in this case, the overlap is very small.

5.1.3 Binomial arrays

If we take the two element isotropic array above, the (normalized) pattern is

E = cos(π

2cos θ

)

If we place another identical array one λ/2 away, we get a three element arraywith relative current magnitudes of 1:2:1.

By applying pattern multiplication, the pattern of this array is

E = cos2(π

2cos θ

)

E = E0

[cos

(π

2cos θ

)]

E = E0

[cos6

(π

2cos θ

)]

Binomial

Figure 5.4: Binomial array pattern

If this process is repeated, we will have a four source array with relative cur-rent magnitudes of 1:3:3:1. Clearly, continuing the process will provide sourcemagnitudes given by Pascal’s triangle, the pattern of which is shown in fig 5.4.

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

Clearly, the pattern multiplies each time, and we get that the pattern of anarray of n sources is:

E = cosn−1(π

2cos θ

)

This array has no minor lobes, but its directivity is less than that of an arrayof the same size with equal amplitude sources. In general, most arrays fall inbetween these two extremes (binomial/uniform).


5.1.4 Uniform arrays

1

d

2

d

3

d

4

d

5 θ = 0

θ = 90

ψ

ψ =2πd

λcos θ

θ

Uniform

Figure 5.5: Uniform linear array of isotropic sources.

If instead, we have an array of equal amplitude, E0, and spacing, d, (not neces-sarily equal phase as shown in figure 5.5, the far field E-field at angle θ is givenby:

E = E0

(1 + ejψ + ej2ψ + ej3ψ + · · ·+ ej(n−1)ψ

)(5.1)

where ψ = βd cos θ + δ, δ being the progressive phase difference between thesources. (Phase reference is source 1)

Multiplying (5.1) by ejψ and subtracting (5.1) from the result yields:Since (5.1) is an infiniteseries, we adopt the usualgeometric series method. (1− ejψ) = E0(1− ejnψ) ie:

E = E0

(1− ejnψ

1− ejψ

)

This can be manipulated using half-angle expansion, and assuming a new phasereference in the middle of the array, we get:

E =sin(nψ/2)sin(ψ/2)

(5.2)

As ψ → 0, E = nE0, ie the E field of n sources at the same point, as it should!This is the maximum E field attainable. Two special cases of maximum fieldare of interest—broadside and end-fire arrays.

Broadside (as in a naval galleon :-) fires its maximum at θ = 90 as shown infig 5.6. For max field, ψ = 0 = βd cos 90 + δ, hence for max broadside field,

δ = 0

This means that there is no progressive phase shift, ie that all sources are fedin-phase. Note the minor lobing that occurs. (The sources are λ/4 apart.)

5.1 Array Theory 61

n=4 n=8

UniformBroad

Figure 5.6: Uniform Isotropic Broadside array

180

120 60

0

−60−120

10 dBi

0

−10

−20

−30


(a) “Ordinary” dB scale

2

4

6

8

10

30

210

60

240

90

270

120

300

150

330

180 0

(b) Linear Scale

Figure 5.7: Eight In-Phase Isotropic sources, dB vs Linear scales.

It is instructive to compare the results with a SuperNEC run, shown in fig 5.7which doesn’t look anything like the above until you plot on a linear scale shownnext to it. Hence the usefulness of the dB scale.

Endfire has its maximum at θ = 0, hence ψ = 0 = βd cos 0 + δ, hence formax endfire field,

δ = −βd = −2π

λd

As an example, if the sources a spaced a quarter wavelength apart,

δ = −2π

λ· λ

4= −π

2= −90

ie that there needs to be a 90 progressive phase shift between sources.

Fig 5.8 shows an Endfire Array with sources λ/2 apart, with a progressive phaseshift of π. Again, note the lobing.


n=4 n=9 n=12

UniformEnd

Figure 5.8: Uniform Isotropic Endfire Array

Beamwidth

From (5.2) it can be seen that the nulls occur when sin(nψ/2) = 0 (with theproviso that sin(ψ/2) cannot also be zero!)

We are interested only in the first null, and this occurs at: nψ/2 = ±π, orψ = ± 2π

n (= βd cos θ0 + δ), where θ0 is the angle of the first null.

Hence the first null occurs at

θ0 = cos−1

[(±2π

n− δ

)λ

2πd

]

For the Broadside case, we are interested in the beamwidth at θ = 90, hencewe use the complementary angle γ = 90 − θ. Recall that for broadside, δ = 0,so that the first broadside null is given by:

γ0 = sin−1

(± λ

nd

)Broadside

If the array is large, nd À λ and the argument to the arcsin is small, (for smallangles θ ≈ sin θ):

γ0 =1

nd/λ=

1L/λ

where L is the length of the array L = (n− 1)d ≈ nd for a large array.

The BWFN is obviously twice this angle, hence:

BWFN = 2γ0 ≈ 2L/λ

[rad] =114.6

L/λ

For most purposes, we can say that HPBW≈BWFN/2, hence

HPBW =57.3

L/λ

For the Endfire case, recall that δ = − 2πλ d, hence the first null occurs at

θ0 = cos−1

[± λ

nd+ 1

]Endfire

5.1 Array Theory 63

Recognizing that we wish to use the small angle approximation again, (θ ≈ sin θ)we convert to a sin via cos 2α = 1− 2 sin2 α

1− 2 sin2

(θ0

2

)= ± λ

nd+ 1

Hence

sin(

θ0

2

)=

√∓ λ

2nd≈ θ0/2

As before, using L ≈ nd, the first null angle is:

θ0 =

√2

L/λ

BWFN = 2θ0 = 2

√2

L/λ[rad] = 114.6

√2

L/λ

and hence

HPBW = 57.3√

2L/λ

Interferometer

The resolution of (beamwidth) of an antenna can be improved by having a largeraperture, but this is not always possible. A synthetically large aperture can beachieved by array theory (Kraus 1988, 522). Consider a simple two-elementarray consisting of two short dipoles placed 10λ apart and fed in phase, shownin fig 5.10. The element pattern is sin θ and the array factor is cos ψ/2.

Array 10λ Dipole

TenWave

Figure 5.9: Array pattern of 2 isometric sources 10λ apart, and the elementpattern.

Hence the E field pattern is given as

E = 2 sin θ cos ψ/2


where ψ = βd cos θ

= 2 sin θ cos(

πd

λcos θ

)

Fig 5.9 shows the pattern multiplication process.

Interferometer

Figure 5.10: Two in-phase dipoles 10λ apart.

Following similar arguments as before, and using the complementary angle γ,the First Null occurs at

γ0 = sin−1 12d/λ

Assuming the distance between the two elements is large, we can approximate:

BWFN = 2γ0 =57.3

d/λ

Across continent, or over large separations, the time-base is critical to ensure in-Hence the further apartthey are, the smaller thebeam is between firstnulls, and the better theresolution.

phase feeding. Radio-astronomers have even used the yearly cycle of the eartharound the sun (with very accurate timing) to resolve distant astronomicalartefacts. A SuperNEC simulation of the 10λ separation interferometer isshown in fig 5.11.

5.2 Dipole Arrays

From array theory, it is clear that a collinear array of dipoles is a uniformbroadside array. It is almost exclusively used in the broadside mode of oper-ation since the dipoles themselves have maximum directivity in this direction.From the broadside condition this implies that the elements of such an arrayshould always be in phase. These arrays are useful due to the omnidirectional

5.2 Dipole Arrays 65

180

120

60

0

−60

−120

10 dBi

0

−10

−20

−30


Structure: φ=90°

Interferometer

Figure 5.11: SuperNEC run of the 10λ Interferometer

characteristics of the pattern about the array axis. If such an array is there-fore vertically mounted it will give omnidirectional azimuth coverage but stillproduce gain due to concentrating the radiation in the elevation plane.

5.2.1 The Franklin array

This is an ingenious method of arraying three half-wave dipoles with half awavelength spacing, and is shown in fig 5.12.

The quarter wave phasing section reverses the current phase by 180 and ensuresthat all three dipoles are in phase. The gain of this arrangement has beenmeasured at 4 to 5 dBi.

Often, the phase reversal section is a loading coil, which has a sufficient amountof self-capacitance to form a resonant L-C network, which performs the 180

phase shift. Many cellphone car-kits employ this technique.

5.2.2 Series fed collinear array

An example of this structure is shown in figure 5.13.

This arrangement is thus equivalent to a 4 dipole array with a 0.7 wavelengthspacing. The value of 0.7 wavelength spacing results in one wavelength electricallength (0.7/0.66) between the excitation slots and so ensures in-phase operation.The gain of this antenna is about 8.5 dBi. Usually such antennas are mountedin a fibreglass radome to give them mechanical rigidity. Unfortunately there is


Franklin

λ/2

λ/2

λ/2

λ/4

Figure 5.12: The Franklin array

CoaxArray

Coax Outer (Braid)

Additional Sleeve0.7λ

λ/2

Break in Coax Braid.

Dipole 1

Dipole 2

Dipole 3

Dipole 4

Coax Inner Conductor

Figure 5.13: A Series Fed Four Element Collinear Array

no DC path to earth, and lightning protection is a problem with this array inSouth African conditions.The year the cell net-

work first rolled out, inJune or so, these ar-rays were used all aroundGauteng, come Septem-ber, they were replaced!

5.3 Yagi-Uda array 67

5.2.3 Collinear folded dipoles on masts

From array theory, it is fairly easy to calculate the gain of a collinear array ofdipoles in free space. The effects of a mast will distort the azimuth patternsomewhat however.

Folded dipoles are usually mounted about a quarter of a wavelength from themast to yield some gain from the mast reflection and to ensure practical boom-lengths and feed harnesses. A SuperNEC run on a λ/2 folded dipole, mountedλ/4 away from a mast of λ/8 diameter is shown in figure 5.14. The gain infront of the dipole is 5.01dBi, and behind the mast is -3.43dBi. If omnidirec-tional coverage is essential the antennas forming the array should be mountedsymmetrically on opposite sides of the mast.

300240

180

120 60

0

10 dBi

0

−10

−20

−30


Figure 5.14: Distortion to folded dipole azimuth pattern in presence of a mast

5.3 Yagi-Uda array

If you ask a 6 year old child to draw an “aerial” it is likely to be a Yagi-Udaarray. This indicates the popularity of this antenna and not without reason.A Yagi antenna is probably the simplest, cheapest and most effective mediumgain antenna available, and is found on every rooftop!

The Yagi-Uda antenna, shown in fig 5.15 on the next page was invented in 1926by Dr. H Yagi and Shintaro Uda (Yagi 1928). Since then numerous reports The Yagi antenna can

be analyzed using mu-tual coupling theory pre-sented earlier.

on this antenna have appeared in the literature—one of the most noteworthy isthe study done by Viezbicke (1976) of the U. S. National Bureau of Standards(NBS). He optimized the gain of a number of Yagi antennas and investigated the


effect of the boom and element length on the performance of the antenna. TheNBS experimental findings were later confirmed during an excellent series ofarticles on Yagi antenna design by James Lawson in the Ham Radio magazine(1979 - 1980). These articles were later combined in a book by the ARRL(Lawson 1986), which is the best practical Yagi-Uda design text available today.

Main beam

Directors

Driven Element

Reflector

Yagi

Figure 5.15: Yagi-Uda array

5.3.1 Pattern formation and gain considerations

The antenna usually has only one driven element, usually a folded dipole; theother elements are not directly driven, but are parasitic, obtaining their currentvia mutual coupling. The spacing between elements is approximately a quarterwavelength. The reflector is slightly longer than required for resonance and isthus inductive (current phase retarded). The directors are shorter than reso-nance and therefore exhibit a capacitive reactance and hence a phase advance.The overall structure therefore has a progressive phase in the forward directionand it behaves like a travelling wave or endfire array.

As a general rule of thumb the gain is directly proportional to boom length forwell designed Yagis. In other words, a 3 dB gain increase is obtained by doublingthe boom length. The number of elements per se is not the determining factor.

Generally, the gain bandwidth of Yagi antennas decreases with an increase ingain (length of Yagi). Also, for a given length Yagi antenna the bandwidth willdecrease with increased gain and vice versa. The gain bandwidth of a Yagi isusually a function of the reflector length relative to director lengths. If a givendesign is therefore built with slightly longer reflector and shorter directors gainbandwidth will be larger with slightly lower maximum gain.

The comments above apply to impedance bandwidth as well. Experience hasshown that the VSWR 2 : 1 bandwidth will correspond more or less to the 1.5ie the Yagi-Uda is typ-

ically impedance band-width limited, not gainbandwith limited, seefig 3.14 and 3.15 onpage 39.

dB gain bandwidth.

5.3 Yagi-Uda array 69

5.3.2 Impedance and matching

The free space input impedance of the folded dipole is usually considerablyreduced from 300Ω due to the mutual coupling to the parasitic elements. Themutual coupling also results in some reactive component being introduced to theinput impedance. The reactive part can be eliminated by changing the overalllength of the driven dipole to achieve resonance—the driven element length doesnot influence the antenna performance much and may be manipulated to achievematching. The resultant resistive input impedance is usually around 200Ω andcan be matched to a 50Ω line using a 4 : 1 balun transformer. This value mustnot be assumed automatically! Measurements or SuperNEC simulations willget the accurate impedance.

5.3.3 Design

The NBS data for Yagi design forms a good base line for design, but are based onunequal director lengths. The difference in gain between these and equal lengthdirector Yagis is negligible for short antennas and about 0.5 dB for long ones.Lawson (1986) showed that using the average length of the NBS specificationfor all elements results in much simpler antennas with equivalent performanceas shown in the table below:

Yagi Design Details (All dimensions in wavelengths)Boom length 0.4 0.8 1.2 2.2 3.2 4.2Reflector 0.482 0.482 0.482 0.482 0.482 0.475Reflector spacing 0.2 0.2 0.2 0.2 0.2 0.2No of directors 1 3 4 10 15 13Director 0.442 0.427 0.424 0.402 0.395 0.401Director spacing 0.2 0.2 0.25 0.2 0.2 0.308G(dBd) (Lawson) 7.1 9.2 10.2 12.25 13.4 14.2Driven (SN) 0.426 0.421 0.417 0.423 0.435 0.434Rin 8.6 20.4 19.0 43.4 55.6 44.5Driven FD (SN) 0.389 0.382 0.378 0.382 0.396 0.396Rin 34.1 76.1 72.1 158.3 202.9 166.2SuperNEC gains(dBi) 9.1 10.5 10.8 12.3 12.7 13.3

The length of the driven element can be chosen for the optimum match conditionsince it does not affect gain operation much. As an example, the SuperNECdetermined resonant lengths and input impedances are shown in the above table.Driven (SN) refers to the SuperNEC derived dipole driven lengths, and DrivenFD (SN) is the SuperNEC derived folded dipole driven lengths. All diametersare 0.008λ, and the folded dipole separation is 0.05λ. The gain as calculatedby SuperNEC is shown in the last line, in dBi. As can be clearly seen, theLawson gain figures are optimistic for the longer arrays.

Element correction

The element lengths shown above are for a diameter to wavelength ratio of0.008. For different radius elements the same design data can be modified by


using a curve with correction factors as determined by the NBS team. Theyused a very complicated process to determine the correct element lengths dueto a clumsy formulation of these curves. Figure 5.16 shows a modified curvederived from their sets of curves that achieves the same effect quite simply.

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

10-3 10-2 10-1

Cor

rect

ion

fact

or

Element diameter to wavelength ratioElemFactor

Reflector

Directors

Figure 5.16: Multiplication factor for different diameter to wavelength ratios ofdirector and reflectors

If the elements are mounted through a metal boom, as is often done for mechanicalWhen the elements aremounted on insulators—at least an element radiusaway from the boom—this effect becomes negli-gible.

support, a further correction factor must be applied to get the correct length.This is due to the slight distortion of the field at the centre of the dipole dueto the boom. It is viewed as some additional capacitance and will thus requirethe length to be increased with increased boom diameter. The curve to do thiscorrection is given in figure 5.17.

Stacking and Spacing

Yagi antennas can be vertically stacked or horizontally spaced to give additionalgain. As seen from section 2.4 on page 24, the criterion for determining thespacing is the effective aperture of the antenna. Since the beamwidths aredifferent in the E and H planes, the optimum separation for stacking Yagiantennas is 1.6 wavelengths apart and for spacing side by side it is 2 wavelengthsbetween antenna centres. Either of these two methods produce 2.5 dB gainrelative to a single antenna. Clearly four yagis can be both stacked and spacedto produce 5 dB of extra gain.

These techniques are attractive since the gain is increased without compromisingbandwidth. The antennas clearly have to be driven in phase via a phasing harness

5.4 Log Periodic Dipole Array 71

0.005

0.01

0.015

0.02

0.025

0.03

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Incr

ease

in E

lect

rica

l Len

gth

Diameter to wavelength ratio of BoomBoomFactor

Figure 5.17: Graph showing the length to be added to parasitic elements tocompensate for the effect of the supporting boom

to achieve the desired effect.

It is also evident that the 6 dB reflection gain mentioned earlier can be obtainedusing the antenna image when mounted above a conducting plane. This is oftenpossible when mounting antennas on metal roofs or for lower frequencies whenthe antenna is mounted above the earth.

5.4 Log Periodic Dipole Array

The LPDA, first proposed by Isbell (1960), is a truly frequency-independentantenna and probably the most popular broadband array. The term true fre-quency independence in this instance implies pattern and input impedance con-stancy. These antennas are used successfully in applications ranging from HFto microwaves. Carrel (1961) developed a particularly straight-forward designprocedure for these antennas which has ensured their success.

Carrel disregarded the effects of the characteristic impedance of the antennaboom/transmission line and also the thickness of the elements. Peixeiro (1988)presents a more complete design technique catering for these defects, but whichstill incorporates a large number of the features introduced by Carrel. Thegeneral form of an LPDA antenna is shown in figure 5.18.

The array is fed at the small end of the structure, and the maximum radiationis toward this end. The lengths of the dipoles and their spacing are varied suchthat these dimensions bear a constant ratio to each other—regardless of the


N

n + 1n

n− 12

1

ZT

dn

LPDA

2α

Ln

Diameter Dn

Figure 5.18: The log-periodic dipole array

position on the antenna (except at the two ends). This design ratio, τ , is oneof the design parameters and the following relationships hold:

τ =Ln+1

Ln=

dn+1

dn(5.3)

It is apparent that these conditions cause the ends of the dipoles to trace out anOne of the characteristicsof frequency independentantennas is that they canbe defined in terms of an-gles rather than linear di-mensions.

angle, 2α. When the antenna is fed from the small end with a frequency that ismuch too low for the short dipoles to resonate, these elements will absorb verylittle power (hence they will radiate very little power too). The phase of thecurrent is mechanically changed by 180 degrees between these electrically shortelements. The radiation any of these will produce will therefore be cancelled bythe out-of-phase radiation of adjacent elements.

Once a portion on the antenna has been reached where the dipoles are resonantand electrically further apart these dipoles will absorb most of the energy fromthe transmission line and radiate it. This part of the antenna is called the activeregion. If the frequency is increased, this active region will simply move towardsthe small end of the antenna. This explains the frequency independence of theantenna for frequencies where the active region is not at one of the two ends.

The directional property of the antenna is due to the elements in front of theactive element being shorter than resonance and therefore capacitive—and act asdirectors. Similarly, elements behind (towards the large end) act as reflectors—giving the antenna a endfire beam towards the small end. Referring again tofigure 5.18 the space factor, σ, of the antenna is:

σ =dn

2Ln(5.4)


The antenna must always be truncated at both ends which is determined by theminimum and maximum frequencies of operation. The truncation coefficientsK1 and K2 are defined to determine the values of shortest and longest elementsto ensure satisfactory performance:

L1 ≥ K1λmax (5.5)

LN ≤ K2λmin (5.6)

The number of dipoles can then be obtained using

N = 1 +ln(L1/LN )

ln(1/τ)

Carrel produced curves to find the (τ, σ) pairs for different values of directivity.These curves were modified using computer techniques, with some improvementto the theory by Peixeiro—and both of the authors’ curves are reproduced infigure 5.19 for the case where Z0 = 100Ω ;Ln/Dn = 177

CarrPex

Original Carrel

Improved Pexiero

Figure 5.19: Constant directivity contours (dBi)

Peixeiro also produced numerous curves showing the relationships between:

• τ and σ — the LPDA design constants

• K1 and K2 — the truncation coefficients

• R — the input resistance

• Z0 — the connecting transmission line characteristic impedance

• D — the directivity of the antenna and


Figure 5.20: Characteristics for feeder impedances of 100, 250 and 400Ω anddipoles with L/D ratios of 177, 500 and 1000


• LnDn

— the thickness of the elements.

For accuracy, these curves are reproduced from his document in figure 5.201.

5.4.1 Design procedure

An easy-to-use design procedure for the LPDA can be based on sets of constantdirectivity D, input resistance R and truncation coefficient contours K1; K2 asshown in figure 5.20.

• Pairs (τ, σ) are selected taking into account both directivity and inputresistance specifications.

• Find the truncation coefficients, K1 and K2 from the same curves basedon the required directivity and selected spacing factor.

• L1 and LN are then calculated by equations 5.5 and 5.6.

• Having chosen suitable dipole diameters the antenna geometry can becalculated using equations 5.3 and 5.4.

If constraints are placed on some of the parameters, the procedure is naturallyiterative in order to find the best compromise. The following comments mayhelp to do so efficiently:

The effects of increasing Z0 are

• a decrease in directivity

• an increase in the input resistance

• a decrease in the truncation coefficients.

On the other hand, the effects of length-to-diameter ratio have a moderateinfluence on the LPDA parameters. Often the maximum boom length is one ofthe constraints imposed in practical examples. This value is not immediatelyevident from the various design constants but can be found using the followingequation:

boom length =2σ(L1 − LN )

(1− τ)

5.4.2 Feeding LPDA’s

A very useful method of feeding a LPDA is illustrated in figure 5.21.

The coaxial cable is passed through the hollow boom from the large end—whichis usually convenient and prevents the cable from affecting the radiation pattern.This connection has the additional advantage that it acts as a broadband balunas result of the hollow boom forming a sleeve around the outside of the coaxialcable.

1IEE Proceedings, Part H, Vol 135, No 2, pg 100


LPDAFeed

Figure 5.21: Coaxial Connection to LPDA’s

5.5 Loop Antennas

The theoretical analysis of loop antennas exploits the dipole and array theory.This again emphasizes the power available to the antenna designer once thesimple concepts such as ideal dipoles, isotropic array theory and transmissionline operation are understood.

5.5.1 The small loop

Figure 5.22 shows a small circular loop and how it can be analyzed as a squareloop with the same area.

d

I

I

I

I s

Loop

Figure 5.22: The small circular loop and the equivalent square loop

In this analysis the term small loop implies that d and s ¿ λ. The square loopwill duplicate the performance of its circular equivalent as long as the areas ofboth are equal:

s2 = π

(d

2

)2

5.5 Loop Antennas 77

This is true in general for different loop configurations such as triangular ormulti-sided—as long as they contain some approximately circular area and arenot “flat”. A “flat” or narrow rectangle, for instance, looks more like a shortcircuited stub than a loop antenna.

Radiation pattern

Since the loop circumference is small it will carry a uniform current with con-stant phase. The square antenna can therefore be considered as made up of fourideal dipoles. The radiation in the xy-plane is clearly isotropic due to the sym-metry of the situation—which is quite clear when the circular loop is considered.The xz-plane is of specific interest and is also shown in figure 5.22. The dipoles1 and 3 do not contribute to radiation in this plane, since they carry opposingcurrents and the path difference to all points in the θ plane is exactly equal.Dipoles 2 and 4 also have equal and out of phase currents, but they form a twoelement array which can be analyzed using the theory in the previous chapter.

The dipoles are isotropic in the xz-plane and the equation for a array of twoisotropic sources may be applied to this case. The separation is s and there isa 180 phase difference between the two resulting in the E field distribution:

E(θ) = 2E0 sin(

βs sin θ

2

)

but s ¿ λ(impliesβ s ¿ 1), thus

E(θ) = E0βs sin θ

This is the equation for the isotropic array and the E0 term refers to the radi-ation due to one of the isotropes. In the case of the loop this value is due to anideal dipole and the E-field of this antenna is used:

E0 =60πIs

λr

This gives the full E-field of the loop as

E(θ) =120π2IA

rλ2sin θ

where A = s2, the area of the loop.

The radiation pattern of the small loop is similar to that of the ideal and shortdipoles as shown in figure 5.23.

Input impedance

From the E-field distribution the power density distribution can be obtained.Integrating this over a full sphere results in an expression for the total trans-mitted power and from this the radiation resistance can be obtained as wasoutlined in Chapter 4.

Rrad =320π4A2

λ4Ω


θ = 0

LoopDip

Eθ

Eφ

Figure 5.23: The Radiation Pattern of a Short Dipole and a Small Loop

When the loop contains N turns instead of one the current moment will increaseby N times and the radiation resistance is proportional to the current momentsquared. The equation above for a multi-turn loop of N turns is:

Rrad =320π4A2N2

λ4Ω

If the loops are made around a ferrite rod the so-called “ferrite loop” resultswith the radiation resistance modified by the ferrite effective permeability, µeff :

Rrad (ferrite loop) = µ2effRrad (air loop)

The radiation resistance of small loops is usually small as was the case for shortdipoles. The reactive part is always inductive and the value can be roughlyestimated using the standard equation for loop inductance:

L = µ0N2A H

the inductive reactance will then be:

X = 2πfL Ω

5.6 Helical Antennas

5.6.1 Normal-mode

This antenna is analyzed by considering it to consist of an array of loops andideal dipoles. The pattern of such an array is the same as the pattern of the

5.6 Helical Antennas 79

h

d

`

Helix Axis

SmallHelix

Figure 5.24: The normal-mode helix antenna and its radiation pattern

individual elements since the array size is small resulting in an isotropic arrayfactor—and is shown in figure 5.24.

The pattern contains both vertical E-field components due to the dipoles; andhorizontal components due to the loops. This results in circular polarizationwhen these values are equal. Usually they are not equal resulting in predomi-nantly horizontal or vertical polarization. The ratio between the two types ofpolarizations is:

E(vertical)E(horizontal)

=`λ

2πA

If this value is large the antenna pattern is vertically polarized and if it is smallthe polarization is approximately horizontal. If the ratio is close to unity thepolarization is circular. The pattern (intensities) are not affected however.

The reason for the popularity of this antenna is due to the fact that muchshorter length resonant linear antennas can be produced. This is due to thefact that the velocity of propagation along the helix axis is slow and the currentand voltage will thus be in phase before the helix is a quarter wavelength. Thereduction in resonant length, k, is given by the formula:

k =1√

1 + 20(nd)2.5(d/λ)1/2

where n is the number of turns per meter.

The radiation resistance of this antenna when it is resonant is given by:

Rrad = (25.3h/λ)2

where h is the height of the monopole helix.

This results in slightly lower values of input resistance when compared to the36Ω input resistance of the quarter wave monopole.


5.6.2 Axial mode

In 1946, J.D Kraus attended a physics lecture in which a helical structure wasused to guide an electron beam in a traveling wave tube. He asked the lecturerabout the possibility of using the helix to radiated electromagnetic wave intospace, to which the answer was an emphatic NO! Nevertheless, Kraus wenthome and started to experiment with the structure. As he suspected (or was itto his amazement), the helix showed good promise as an antenna.

When the diameter D and the spacing S are large fractions of a wavelength, theoperation of the helical antenna changes considerably from the normal-modebehaviour, and it behaves as an endfire array of loops and the pattern thereforehas a main beam in the axial direction as shown in figure 5.25.

SL

C = πDα

Helix Axis

D

S

AxialHelix

Figure 5.25: Axial Mode Helix Antenna and its Typical Pattern.

To excite the axial mode of operation, the circumference of the helix, C(= πD),must be in the range

0.8 ≤ C

λ≤ 1.15

with a circumference of 1 near optimum. The spacing, S must be about λ/4and the pitch angle, α in the range:

12 ≤ α ≤ 14

where α = arctan(S/C)

Most often the antenna is used in conjunction with a ground plane, whosediameter is at least λ/2. The number of helix turns, N , should be more than 4.

Intuitively, the circularly polarization comes about since “opposite” sides of thehelical turn are 180 out of phase, hence providing the E-field vector in thatplane. Also, referring back to the Yagi-Uda array where the directors are about0.2λ apart, in order to capacitively “suck” the wave forward, the turns are about0.21 to 0.25 λ apart (For a C/λ of 1).

5.6 Helical Antennas 81

Original Kraus design

During the years 1948-1949, Kraus empirically studied the helical antenna andpublished the following findings (assuming 0.8 ≤ C/λ ≤ 1.15; 12 ≤ α ≤ 14;and n ≥ 4):

• The radiation pattern of a helix is predominantly cigar shaped and has amaximum gain given by:

G = KG

(C

λ

)2 (NS

λ

)

where KG is the gain factor, originally 15, but later reduced to 12 byKraus (1988, pg284)

• The Half Power BeamWidth (HPBW) is given by:

HPBW =KBλ3/2

C√

NS

where KB is the beam factor, which is about 52, derived from the standardapproximation on beamwidths:

G =41000

HPθHPφ

Since the beam is generally circularly symmetric, HPθ =HPφ=HPBW:

HPBW =

√41000/KGλ3/2

C√

NS

where√

41000/KG = KB , the beam factor.

• The input impedance is nearly resistive and is given by:

R = 140(

C

λ

)Ω

• The beam is circularly polarized.

King and Wong design

King & Wong (1980) performed a study which involved varying the parametersof a uniform helix and measuring the electrical performance of the structure.They found that the expressions derived by Kraus tended to overestimate theperformance of the antenna. Their results are summarized (empirically) asfollows:

G = 8.3(

C

λ

)√N+2−1 (NS

λ

)0.8 (tan(12.5)

tanα

)√N/2

When comparing this result to that published by Kraus, the gain factor KG

is between 4.2 and 7.7 (compared to Kraus’s reduced estimate of 12). Thebeamwidth factor, KB , is therefore between 61 and 70 (compared to 52).

Please note:


• The revised factors are valid for antennas with 0.8 ≤ Cλ ≤ 1.2.

• King and Wong also note that the Kraus original factors depend on otherdesign parameters of the helix and are only constant for helices with ap-proximately 10 turns. The revised factors do not suffer from this limita-tion.

In addition,

• The peak gain of a helix occurs when Cλ = 1.155 for N = 5; and for

Cλ = 1.07 for N = 35.

• Since the beam is circular, HPBWθ=HPBWφ=HPBW. The Gain-HPBW2

product was found to be significantly less than 41 000, and lies in the rangeof 18 000 to 31 500. They note that:

– G×HPBW2 = 18 000 for N = 35 and 0.75 ≤ Cλ ≤ 1.1

– G×HPBW2 = 31 000 for N = 5 and 0.75 ≤ Cλ ≤ 1.2

– The smaller the pitch, the larger the gain-beamwidth product.

• The gain bandwidth of the helix is presented as:

fH

fL≈ 1.07

(0.91

G/Gpeak

) 43√

N

where the subscript L refers to the lower frequency, and H the higher;G/Gpeak is 3dB or 2dB etc according to preference (usually want the 3dBpoint).

– Note that the bandwidth decreases as the axial length/ gain/ numberof turns increases.

– The bandwidth is approximately 42% for a helix of N = 5; andapproximately 21% for N = 35.

• The impedance bandwidth (2:1 VSWR) is typically 70%. The inputimpedance of the helix (with C/λ = 1) is about 140Ω, almost purelyresistive. However, if the last quarter-turn of the helix is made parallelto the ground plane, it creates a quarter-wave transformer, which allowsmatching down to 50Ω. Since a frequency-selective device has now beenintroduced, the 70% impedance bandwidth drops to about 40%. This canbe ameliorated by tapering the matching section in the usual way.

5.7 Patch antennas

At higher frequencies, the convenience of manufacturing antennas on PCBs(Printed Circuit Boards) becomes attractive. The higher the frequency, themore patches can be used, resulting in better gain, and more controllable pa-rameters.

At V/UHF frequencies, it is simply not possible to have a large number ofelements in an array. One problem at the higher frequencies is that the dielectric

5.7 Patch antennas 83

of the PCB has to be carefully controlled, and “standard” PCBs have quite lossydielectrics. At microwave frequencies, therefore, antennas are manufactured onlow-loss substrates.

The simplest patch antenna is a rectangular patch, roughly λ/2 × λ in size,λ being the wavelength in the dielectric (Kraus & Fleisch 1999, pg307). Thepatch is fed in the middle of one of its longest edges, as shown in fig 5.26.

patch

Figure 5.26: A λ/2 × λ patch (of copper) on a dielectric slab, over a groundplane. The patch is fed by coax through the dielectric, halfway along the longestedge.

The antenna functions as an array of slots. The long edges radiate as twovertical slot antennas.

It is common to have large arrays of patches to achieve a high gain. Sincethis kind of array is readily produced using ordinary printed circuit technology(albeit using fancy dielectrics), it is a popular method of producing arrays.

Balanis (1982, pg.730) outlines a design method which produces good results:

Width The patch width, w, which produces good radiation efficiencies is givenby:

w =c

2fr

√2

εr + 1

εr(eff) The effective dielectric constant (since w À h) is determined by (Balanis1989):

εr(eff) =εr + 1

2+

εr − 12

[1 + 12

h

w

]−1/2

∆` The extension length, ∆`, is found using (Hammerstad 1975):

∆`

h= 0.412

(εr(eff) + 0.3)(

wh + 0.264

)

(εr(eff) − 0.258)(

wh + 0.8

)

Length The patch length, `, is then given by:

` =1

2fr√

εr(eff)√

µ0ε0− 2∆`


Actually, there are so many ways to feed a patch, and so many ways to constructit, that a comprehensive guide cannot be readily established. Essentially, it isrequired that a set of modes are resonant within it—eg with feedpoints roughly90 in spatial separation, and with a 90 phase separation, circular polarizationis easily achieved.

The size of groundplane also affects all parameters. An example of an (essen-tially) edge-fed square patch (SuperNEC run) is shown in fig 5.27

Figure 5.27: Geometry of square patch, as simulated in SuperNEC

The 3D and max 2D patterns are shown in figs 5.28 on the next page and 5.29on the facing page

Patches are simply asking to be arrayed, and fig 5.30 on page 86 shows a fouredge-fed square patch array with its 3D pattern; the 2D pattern at max gain isshown in fig 5.31.

5.7 Patch antennas 85

3D Radiation Pattern

|GainTot

|−23.4

−19.5

−15.5

−11.6

−7.6

−3.7

0.3

4.2

8.2

−20−15

−10−5

05

1015

20

−15−10

−50

510

1520

2530

35

−20

−15

−10

−5

0

5

10

15

20

YX

Figure 5.28: 3D pattern of the square patch.

180

120

60

0

−60

−120

10 dBi

0

−10

−20

−30


Structure: φ=92°

Figure 5.29: 2D cut through the maximum gain of the square patch


3D Radiation Pattern

|GainTot

|−30.5

−25.4

−20.3

−15.2

−10.1

−4.9

0.2

5.3

10.4

−40−30

−20−10

010

2030

40

−30

−20

−10

0

10

20

30

40

−30

−20

−10

0

10

20

30

Y

Z

Figure 5.30: Geometry and 3D pattern of a 4-square patch array.

180

120

60

0

−60

−120

10 dBi

0

−10

−20

−30


Structure: φ=92°

Figure 5.31: 2D cut through the maximum gain of the 4-patch array.

5.8 Phased arrays and Multi-beam “Smart Antennas” 87

5.8 Phased arrays and Multi-beam “Smart An-tennas”

It is often important to track a moving target–eg direction finding. For this pur-pose, relatively large antenna arrays can be employed, with electrically steerablebeams (or multiple beams) (Johnson & Jasik 1984, ch.32).

Applications are:

• Classic Direction-Finding: Beacons for aircraft navigation; Illegal trans-mitters; Triangulation of a Cell phone from different adjacent cells (911requirement in USA 2005?); Finding enemy radar :-)

• Classic Interference minimizing: Steer a Null towards an interfering signal–eg multipath (hence delayed) signal; Steer a null towards an intentionaljammer.

• Re-use of spectrum: Instead of an omnidirectional pattern for cellphonecoverage, you could have multiple beams to track individual users, anduse the SAME frequency for both! Also used in the (now largely defunct)satellite mobile phone Low-Earth Orbiting antennas.

To feed an array, the input is usually passed through a power splitter, shownin fig 5.32, which may provide differing amounts of power to each antenna, orequal power. This is possibly followed by a phase shifting section, so that theindividual phases of the sources in the array can be changed.

φ

φ

φ

φ

Pow

erSp

litte

r

PowerSplit

Figure 5.32: Power Splitter followed by phase modification.

Changing the phasing allows the beam to be steered to follow an object etc. Commercial phaseshifters use transmissionlines as above for thelower frequencies, butvariable lumped elementsat higher frequencies.

One of the simplest (in theory) phase shifter is a transmission line based binaryone as shown in fig 5.33, where various sections of TxLn are switched into thecircuit.

For example a four element array, spaced λ/2 apart. If the beam needs to fireat 60, the E-field must be in phase at 60. Hence the incremental phase changebetween sources ψ must be zero at 60. Thus

ψ =2πd

λcos θ + δ (= 0 at θ = 60)

Hence we get that δ = −90 ie that the phase difference between each source Interestingly, if δ =+90, we get a Null atθ = 60.


8×

4×

2×

1×

BinPhase

Figure 5.33: Binary phase shifter, based on transmission line segments.

must be −90. The exercise can be repeated for any angle, and by changing therelative phases of the (uniform amplitude) array, we can steer the beam.

Beam steering in Elevation is often used in cellular systems, as shown in fig 5.34,which shows a standard “eight-stack”, ie 8 λ/2 dipoles, spaced 3/4λ apart. A

180

120

60

0

−60

−120

10 dBi

0

−10

−20

−30


Broadside

5° Downtilt

10° Downtilt

DownTilt

Figure 5.34: Electrically achieved DownTilt by progressive phasing.

careful look at the magnitudes in fig 5.34 will show a slight degradation in peakvalue (10.81;10.80;10.06) dBi respectively. This is due to the array pattern beingmultiplied by the element pattern which is a sinusoidal function of theta. Theelectrical downtilt is achieved by slightly lengthening the cables progressively inthe corporate feed network, shown in fig 5.35.

For the 5 downtilt, θ = 95, so from ψ = βd cos(95) + δ, we get δ =0.41radians = 24. (Typical spacing is 3/4λ).

For the 10 downtilt, δ = 47. This translates directly to additional cable length

5.9 Flat reflectors 89

All 75Ω; nλ/4; n odd

Corporate

Figure 5.35: A Corporate feed network—equal amplitude and phase.

required (NB at 50Ω) to each antenna.

It should be obvious that to have a number of beams, each with a reasonablegain requires a very large number of elements. This implies a high frequency inorder for the array to be of reasonable size. At radar frequencies (11GHz), thisis possible.

Apart from actually switching in different amounts of phase to each element,one can use a fixed phasing network and change the frequency at which thesystem operates, resulting in a beamshift. Assuming that the exact frequency isnot critical, one can get a fair degree of beamshift. Useful in RFID applicationswhere the exact frequency used to excite the passive tags is not critical, butwhere a relatively large area needs to be scanned.

5.9 Flat reflectors

The simplest reflector antenna is a dipole in front of a flat plate, which is easilyanalysed by image theory. The distance between the dipole and plate affectsthe gain markedly. Kraus (1988, pg546)

Supergain conditions hold at very close spacings, and a very impressive gain canbe obtained, but at the expense of having a very low input resistance, (closecoupling with a negative image), making the antenna difficult to feed. Thesupergain condition is also very narrowband.

A tradeoff distance has to be obtained which provides a decent gain and a decent(feedable) input impedance.


A SuperNEC run of a 0.75 λ square plate and a λ/2 dipole shows gain versusdistance in figure 5.36, and the input impedance in figure 5.37.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9Gain vs separation for a 0.75 by 0.75 plate

Gai

n (d

Bi)

Separation (Wavelengths)

Figure 5.36: Gain vs separation for a 0.75λ plate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120Impedance vs separation for a 0.75 by 0.75 plate

Rea

l and

Imag

inar

y im

peda

nce

(Ω)

Separation (Wavelengths)

Real partImag Part

Figure 5.37: Impedance vs separation for a 0.75λ plate

5.10 Corner Reflectors 91

5.10 Corner Reflectors

Kraus (1988, Pg549) first designed a corner reflector. They can come in manyshapes and sizes, but the most common form is where the “corner” is definedas being 90, and the reflector panels are simply made from vertical rods. Atsome distance from the corner, symmetrically placed is a dipole driven element.A SuperNEC run shows the structure in fig 5.38.

Figure 5.38: The SuperNEC rendition of a corner reflector.

After several iterations, my “standard design” is now simply a 90 corner witheach of the two panels being 0.7λ×0.7λ, with the dipole separation, s, from thecorner being 0.3λ.

The SuperNEC predicted xy-plane radiation pattern of such a reflector isshown in fig 5.39

If the panels are made larger (1.2λ× 1.2λ, say) the gain improves slightly, butat the expense of the antenna being very large. I have not seen more than about9dBi in a common corner reflector.

5.11 Parabolic Reflectors

The classic parabolic dish is shown in fig 5.40 on the next page(a). If the feedis placed at the focal point of the parabola, all rays are reflected with the sameefective path length, and are hence in phase after leaving the parabola. It istherefore possible to achieve a very high gain with a dish, limited mainly by theconstructional deformities of the actual parabola.


300240

180

120 60

0

10 dBi

0

−10

−20

−30


Structure: θ=90°

Figure 5.39: SuperNEC predicted xy-plane radiation pattern of a corner re-flector.

(a) (b) (c)

Figure 5.40: (a)Ordinary parabolic dish with feed blockage; (b) an offset feed;(c) Cassegrain

The feed system for the parabola must illuminate the entire dish, but no morethan the dish, otherwise it results in spillover loss. The feed (and its supports)obviously block some energy, resulting in a slight shadow behind the feed.

A better idea is shown in fig 5.40(b), where the feed is offset. Simply put, theMost DSTV antennasuse offset feeds. Ittherefore appears thatthey “point” very low onthe horizon—but standfig 5.40 so that the dishin (b) is almost verticaland see where it “points”to!

shape still follows a parabola, and the rays are still in phase, but the feed is notobscured.

Fig 5.40(c) shows a Cassegrain feed. The subreflector, which is not at the fo-cal point of the parabola, is a hyperbola. The combination of hyperbola andparabola ensures in-phase behaviour of the dish. Note that there is a blockageof the area behind the sub-reflector again, resulting in a lowered Aperture ef-ficiency. The Cassegrain is mainly used in large dishes, as the feed system ismore readily accessible as compared to it being out on supports.

Although a parabolic reflector ensures that the phase of the wavefront is con-stant, there is usually an amplitude taper associated with the wavefront. It is

5.11 Parabolic Reflectors 93

thus usually desirable to design the feed system to have the inverse taper, whichfinally results in a uniform wavefront.


Appendix A

Smith Chart

The next page contains a smith chart. A postscript and Laserjet version can befound at http://ytdp.ee.wits.ac.za/smith.html.

95

0.10.1

0.1

0.20.2

0.2

0.30.3

0.3

0.40.4

0.4

0.50.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

0.9

1.0

1.0

1.0

1.2

1.2

1.2

1.4

1.4

1.4

1.6

1.6

1.6

1.81.8

1.8

2.02.0

2.0

3.03.0

3.0

4.04.0

4.0

5.05.0

5.0

1010

10

2020

20

5050

50

0.20.2

0.20.2

0.40.4

0.40.4

0.60.6

0.60.6

0.80.8

0.80.8

1.01.0

1.01.0

20−20

30−30

40−40

50−50

60−60

70−70

80−8090

−90100

−100

110−110

120−120

130−130

140

−140

150

−150

160

−160

170

−170

180

±

90-9

085

-85

80-8

0

75-7

5

70-7

0

65-6

5

60-6

0

55-5

5

50-5

0

45

-45

40-40

35-35

30-30

25-25

20-20

15-15

10-10

0.04

0.04

0.05

0.05

0.06

0.06

0.070.07

0.08

0.080.09

0.090.1

0.10.11

0.110.12

0.12

0.13

0.13

0.14

0.14

0.15

0.15

0.16

0.16

0.17

0.17

0.18

0.18

0.190.19

0.20.2

0.210.21

0.22

0.220.23

0.230.24

0.24

0.25

0.25

0.26

0.26

0.27

0.27

0.28

0.28

0.29

0.29

0.3

0.3

0.31

0.31

0.320.32

0.330.33

0.340.34

0.350.35

0.36

0.360.37

0.37

0.38

0.38

0.39

0.39

0.4

0.4

0.41

0.41

0.42

0.42

0.43

0.43

0.44

0.44

0.45

0.45

0.46

0.46

0.47

0.47

0.48

0.48

0.49

0.49

0.0

0.0

AN

GLE O

F TRA

NSM

ISSION

CO

EFFICIEN

T IN D

EGR

EES

AN

GLE O

F REFLEC

TION

CO

EFFICIEN

T IN D

EGR

EES

—>

WA

VEL

ENG

THS

TOW

AR

D G

ENER

ATO

R —

><—

WA

VEL

ENG

THS

TOW

AR

D L

OA

D <

—

IND

UC

TIV

E RE

AC

TAN

CE C

OM

PON

ENT (+

jX/Zo), O

R CAPACITIVE SUSCEPTANCE (+jB/Yo)

CAPACITIVE REACTANCE COMPONENT (-

jX/Zo)

, OR IN

DU

CTIVE

SUSC

EPTA

NC

E (-

jB/Y

o)

RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)

RADIALLY SCALED PARAMETERS

TOWARD LOAD —> <— TOWARD GENERATOR1.11.21.41.61.822.5345102040100

SWR 1∞

12345681015203040dBS

1∞

1234571015 Atten. [d

B]

1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 SW Loss

coeff

1 ∞0 1 2 3 4 5 6 7 8 9 10 12 14 20 30

Rtn loss [dB] ∞

0.010.050.10.20.30.40.50.60.70.80.91

Rfl coeff,P0

0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15 Rfl loss

[dB]

∞0

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 SW pea

k (const

P)

0 ∞0.10.20.30.40.50.60.70.80.91

Rfl coeff E/I 0 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Trans c

oeff,P

1

CENTRE1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Tra

ns coeff

,E/I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ORIGIN

The Smith Chart Calculator

References

ARRL (1988), The ARRL Antenna Book, 14th edn, The American Radio RelayLeague.

Balanis, C. A. (1982), Antenna Theory: Analysis and Design, 2nd edn, JohnWiley & Sons.

Balanis, C. A. (1989), Advanced Engineering Electromagnetics, John Wiley &Sons, New York.

Carrel, R. L. (1961), Analysis and Design of the Log periodic Dipole Antenna,PhD thesis, Elec Eng Dept, University of Illinois, Ann Arbor, MI.

Carson, J. R. (1929), “Reciprocal theorems in radio communication”, Proceed-ings of the Institute of Radio Engineers 17(6), 952–956.

Clark, A. R. (2001), “Smith chart downloads in various forms”,http://ytdp.ee.wits.ac.za/smith.html.

Gardiol, F. E. (1984), Introduction to Microwaves, Artech House, Inc, Dedham,MA, USA.

Gupta, K. C., Garg, R. & Bahl, I. J. (1979), Microstrip lines and Slotlines,Artech House, Inc, Dedham, MA, USA.

Hammerstad, E. O. (1975), Equations for microstrip circuit design, in “Proc.Fifth European Micrwave Conf.”, pp. 268–272.

Isbell, D. E. (1960), “Log periodic dipole arrays”, IRE Trans. Antennas Prop-agat. AP-8, 260–267.

Johnson, R. C. & Jasik, H. (1984), Antenna Engineering Handbook, 2nd edn,McGraw Hill.

King, H. E. & Wong, J. L. (1980), “Characteristics of 1 to 8 wavelength uniformhelical antennas”, IEEE Transactions on Antennas & Propagation AP-7, 291.

King, R. W. P. & Harrison, C. W. (1969), Antennas and Waves - a ModernApproach, MIT Press, Mass, chapter Appendix 4 - Tables of Impedanceand Admittance of Electrically Long Antennas - Theory of Wu, pp. 740–757.

Kraus, J. D. (1988), Antennas, second edn, McGraw-Hill.

Kraus, J. D. (1992), Electromagnetics, fourth edn, McGraw-Hill.

97

98 REFERENCES

Kraus, J. D. & Fleisch, D. A. (1999), Electromagnetics: with Applications, fifthedn, WCB/McGraw-Hill.

Lawson, J. L. (1986), Yagi Antenna Design, The American Radio Relay League,Newington, CT. Publication 72.

Lefferson, P. (1971), “Twisted magnet wire transmission line”, IEEE Transac-tions on Parts, Hybrids and Packaging PHP-7(4), 148–154.

Nitch, D. C. (2001), SuperNEC, 1.5 edn.

Peixeiro, C. (1988), “Design of log-periodic antennas”, IEE Proceedings, PartH 135(2), 98–102.

Schelkunoff, S. A. (1941), “Theory of antennas of arbitary size and shape”,Proceedings of the IRE 29(September), 493–521.

Sevick, J. (1990), Transmission line transformers, 2nd edn, American RadioRelay League, Newington.

Smith, P. H. (1939), “Transmission line calculator”, Electronics 12, 29.

Viezbicke, P. P. (1976), Yagi antenna design, Tech. doc. NBS-TN-688, NationalBureau of Standards, Dept. of Commerce, Washington D.C.

Wadell, B. C. (1991), Transmission Line Design Handbook, Artech House, Inc.,Norwood, MA, USA.

Yagi, H. (1928), “Beam transmission of ultra short waves”, IRE Proc. 16, 715–741.

Index

Aperture, 24, 70Array

binomial, 59collinear, 64dipole, 64Franklin, 65isotropic, 57LPDA, see LPDApattern multiplication, 58series fed collinear, 65theory, 57uniform, 60

broadside, 60endfire, 61

Yagi-Uda, see Yagi-Uda arrayAutotransformer, 34Axial ratio, 24Axial-mode helix, 80

Balunbazooka, 32half-wave, 33sleeve, 32transmission line transformer, 33

Baluns, 30Bandwidth

impedance, 50, 51vs Q, 46

Bazooka balun, 32Beam steering, 88Beamwidth

between first nulls (BWFN), 21half power(HPBW), 21uniform array, 62

Binomial array, 59Broadside array, 60

Cassegrain parabolic, 91Characteristic impedance

free space, 11measuring, 7

medium, 11transmission lines, 3

coaxial line, 6microstrip line, 6one-wire line, 5twisted pair, 5two-wire line, 4

Circular polarization, 24Coaxial characteristic impedance, 6Collinear dipole array, 64

series fed, 65Corner reflector, 89Corporate feed, 88Current moment, 44Curvature of earth, 13

dB, 19vs. linear scale, 61

dBd, 49Decibels, see dBDepth of penetration, see Skin depthDesign

LPDA, 75Yagi-Uda, 69

Dipolearray, 64folded, 50ground interaction, 52half wavelength, 48ideal, 41short, 44

Directivity, 19folded dipole, 51half wave dipole, 48ideal dipole, 43quarter wave monopole, 49short dipole, 46short monopole, 47

Directivity estimation, 22Downtilt, 88

99

100 INDEX

Earth curvature, 13Effect of mast on pattern, 67Effective aperture, 24Efficiency, 19Electrical downtilt, 88Element correction, 69Elliptical polarization, 24Endfire array, 61Equation

free-space wavelength, 2, 10power loss ito VSWR, 31

Ferrite cores, 34Fields

ideal dipole, 41short dipole, 45

Flat reflector antennas, 89Folded dipole, 50Franklin array, 65Free-space

characteristic impedance, 11Ohm’s law, 12permeability, 12permittivity, 12speed of propagation, 12wavelength, 2, 10

Frequency independance, 71Frequency scanning arrays, 89Front-to-back ratio, 21

Gain, 19bandwidth, 37estimation, 22isotropic, 18measuring, 49

Half wavelength dipole, 48Half-power beamwidth (HPBW), 21Half-wave balun, 33Helix, 78–82

axial-mode, 80King and Wong design, 81normal-mode, 78original Kraus design, 81

Ideal dipole, 41Impedance bandwidth, 37

folded dipole, 51half wave dipole, 50helix, 82

Input impedancefolded dipole, 51half wave dipole, 49quarter wave monopole, 49short monopole, 47small loop, 77

Interferometer, 63Inverse square law, 18Isotropic array, 57Isotropic source, 18

L-match, 34LC networks, 34Line of sight, 13Linear polarization, 23Linear vs. dB scale, 61Log periodic dipole array, see LPDALoop antenna, 76

input impedance, 77small, 76

LPDA, 71design procedure, 75feeding, 75

Mast mounted dipoles, 67Matching, 8, 30

tuning coil, 46Yagi-Uda, 69

Maximum power transfer, 8Measuring

characteristic impedance, 7gain, 49velocity factor, 8

Medium, 3Method of Moments, see SuperNECMicrostrip line characteristic impedance,

6Mutual impedance, 53

Normal-mode helix, 78

Offset feed parabolic, 91Ohm’s law in free-space, 12One-wire line characteristic impedance,

5

Parabolic reflector, 91Patch antennas, 82–84Pattern multiplication, 58Permeability of free space, 12Permittivity of free space, 12

INDEX 101

Polarization, 23–24axial ratio, 24circular, 24elliptical, 24horizontal, 23linear, 23loss, 23vertical, 23

Q, 46Quarter wave transformer, 7, 34

Radiation pattern, 20backlobe, 20effect of mast on, 67half wave dipole, 48principle planes, 20sidelobes, 20small loop, 77

Radiation resistanceideal dipole, 43short dipole, 45short monopole, 47

Radio horizon, 13Reactance

short dipole, 45Reflection coefficient, 8Rule of thumb

coupling separation, 55transmission line behaviour, 1Yagi gain vs boomlength, 68

Series connected coax, 35Series fed collinear, 65Short dipole, 44Short monopole, 47Skin depth, 16Sleeve balun, 32Small loop, 76Smith chart, 9

download, 9Solid angle, 22Speed of propagation

free-space, 12medium, 12sound, 12transmission line, 4

Spherical coordinate system, 17Square degrees, 22Stacking, 70

Steradians, 22Stub matching, 35SuperNEC, 20, 22, 48, 61, 64, 67,

84, 90Surge impedance, see Characteris-

tic impedance

Tablefrequency allocations & desig-

nations, 10microwave bands, 11skin depths, 16typical spectrum usage, 11Yagi-Uda array design, 69

Transmission lineadditional loss ito VSWR, 31behaviour, 1characteristic impedance, 3equation, 7transformer balun, 33

Tuning coil, 46Twisted pair characteristic impedance,

5Two-wire line characteristic impedance,

4

Uniform array, 60beamwidth, 62

Velocity factor, 7, 12measuring, 8plastics, 12

Velocity of propagation, see Speedof propagation

VSWR, 8, 10

Waveguide, 3Wavelength

in free-space, 2, 10

Yagi-Udamatching, 69stacking, 70

Yagi-Uda array, 67design, 69

Z0, see Characteristic impedance

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