Antennas in Practice

EM fundamentals and antenna selection

Alan Robert Clark Andr´eP C Fourie

Version 1.4, December 23, 2002 ii Titles in this series:

Antennas in Practice: EM fundamentals and antenna selection (ISBN 0-620-27619-3) Wireless Technology Overview: Modulation, access methods, standards and systems (ISBN 0-620-27620-7) Wireless Installation Engineering: Link planning, EMC, site planning, lightning and grounding (ISBN 0-620-27621-5)

Copyright °c 2001 by Alan Robert Clark and Poynting Innovations (Pty) Ltd. 33 Thora Crescent Wynberg Johannesburg South Africa. www.poynting.co.za Typesetting, graphics and design by Alan Robert Clark. Published by Poynting Innovations (Pty) Ltd.

This book is set in 10pt Computer Modern Roman with a 12 pt leading by LATEX 2ε. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of Poynting Innovations (Pty) Ltd. Printed in South Africa. ISBN 0-620-27619-3 Contents

1 Electromagnetics 1 1.1 Transmission line theory ...... 1 1.1.1 Impedance ...... 3 1.1.2 Characteristic impedance & Velocity of propagation . . . 3 Two-wire line ...... 4 One conductor over ground plane ...... 5 Twisted Pair ...... 5 Coaxial line ...... 5 Microstrip Line ...... 6 Slotline ...... 6 1.1.3 Impedance transformation ...... 6 1.1.4 Standing Waves, Impedance Matching and Power Transfer 8 1.2 The Smith Chart ...... 9 1.3 Field Theory ...... 10 1.3.1 Frequency and wavelength ...... 10 1.3.2 Characteristic impedance & Velocity of propagation . . . 11 1.3.3 EM waves in free space ...... 12 1.3.4 Reflection from the Earth’s Surface ...... 14 1.3.5 EM waves in a conductor ...... 16

2 Antenna Fundamentals 17 2.1 Directivity, Gain and Pattern ...... 17 2.1.1 Solid angles ...... 17 2.1.2 Directivity ...... 18 Isotropic source ...... 18 Decibels ...... 19 2.1.3 Efficiency ...... 19 2.1.4 Radiation pattern ...... 20 Directivity estimation from beamwidth ...... 22 2.2 Reciprocity ...... 22 2.3 Polarization ...... 23 2.4 Effective aperture ...... 24 2.5 Free-space link equation and system calulations ...... 25

3 Matching Techniques 29 3.1 Balun action ...... 29 3.1.1 Unbalance and its effect ...... 29

iii iv CONTENTS

3.1.2 Balanced and unbalanced lines—a definition ...... 30 3.2 Impedance Transformation ...... 30 3.2.1 Angle Correction ...... 32 3.3 Common baluns/balun transfomers ...... 33 3.3.1 Sleeve Balun ...... 33 3.3.2 Half-wave balun ...... 33 3.3.3 Transformers ...... 34 3.3.4 LC Networks ...... 34 3.3.5 Resistive Networks ...... 34 3.3.6 Quarter wavelength transformer ...... 35 3.3.7 Stub matching ...... 36 3.3.8 Shifting the Feedpoint ...... 36 3.3.9 Ferrite loading ...... 38 3.4 Impedance versus Gain Bandwidth ...... 38

4 Simple Linear Antennas 41 4.1 The Ideal Dipole ...... 41 4.1.1 Fields ...... 41 4.1.2 Radiation resistance ...... 43 4.1.3 Directivity ...... 43 4.1.4 Concept of current moment ...... 44 4.2 The Short Dipole ...... 44 4.2.1 Fields ...... 45 4.2.2 Radiation resistance ...... 45 4.2.3 Reactance ...... 45 4.2.4 Directivity ...... 46 4.3 The Short Monopole ...... 47 4.3.1 Input impedance ...... 47 4.4 The Half Wave Dipole ...... 48 4.4.1 Radiation pattern ...... 48 4.4.2 Directivity ...... 48 4.4.3 Input impedance ...... 49 4.5 The Folded Dipole ...... 50 4.6 Dipoles Above a Ground Plane ...... 52 4.7 Mutual Impedance ...... 53

5 Arrays and Reflector Antennas 57 5.1 Array Theory ...... 57 5.1.1 Isotropic arrays ...... 57 5.1.2 Pattern multiplication ...... 58 5.1.3 Binomial arrays ...... 59 5.1.4 Uniform arrays ...... 60 Beamwidth ...... 62 Interferometer ...... 63 5.2 Dipole Arrays ...... 64 5.2.1 The Franklin array ...... 65 5.2.2 Series fed collinear array ...... 65 5.2.3 Collinear folded dipoles on masts ...... 67 5.3 Yagi-Uda array ...... 67 5.3.1 Pattern formation and gain considerations ...... 68 CONTENTS v

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

A Smith Chart 95 vi CONTENTS List of Figures

1.1 A transmission line connects a generator to a load...... 1 1.2 2-wire; Coax; µstrip; waveguide; fibre; RF; 50Hz! ...... 2 1.3 A “Lumpy” model of the TxLn, discretizing the distributed pa- rameters...... 3 1.4 Twisted-Pair geometry ...... 5 1.5 Impedance values plotted on the Smith Chart ...... 9 1.6 Electromagnetic Wave in free-space ...... 13 1.7 Radio Horizon due to earth curvature...... 13 1.8 Geometry of Interference between Direct Path and Reflected Waves 14 1.9 Interference pattern field strength contours for h/λ = 1.44 . . . . 15

2.1 Standard coordinate system...... 17 2.2 concept of a solid angle ...... 18 2.3 Three-dimensional pattern of a 10 director Yagi-Uda array . . . . 21 2.4 xy plane cut of the 10 director Yagi-Uda ...... 21 2.5 Rectanglar radiation patterns of the 10-director Yagi-Uda in the two principle planes ...... 22 2.6 Reciprocity concepts ...... 23 2.7 Polarization possibilities—wave out of page ...... 23 2.8 Aperture efficiency of a parabolic dish ...... 25 2.9 Effective aperture of dipole À physical aperture ...... 25 2.10 Apertures must not overlap! ...... 25 2.11 Point-to-point link parameters ...... 26

3.1 The reason for unbalanced currents ...... 29 3.2 Balanced and Unbalanced Transmission lines ...... 30 3.3 Power loss in dB versus VSWR on the line...... 31 3.4 Increase in Line Loss because of High VSWR ...... 32 3.5 VSWR at Input to Transmission Line versus VSWR at Antenna 32 3.6 The Bazooka or Sleeve balun...... 33 3.7 A HalfWave 4:1 transformer balun...... 33 3.8 Transmission-line transformer...... 34 3.9 L-Match network...... 35 3.10 Quarter wavelength transformer ...... 35 3.11 Single Matching Stub ...... 36 3.12 Smith Chart of Single Stub Tuner Arrangement ...... 37 3.13 T and Gamma Matching Sections ...... 37 3.14 Gain bandwidth ...... 39

vii viii LIST OF FIGURES

3.15 Impedance bandwidth ...... 39

4.1 The Ideal Dipole in Relation to the Coordinate System ...... 42 4.2 Pattern of an Ideal Dipole Antenna ...... 43 4.3 Current Distribution on a Short Dipole Antenna ...... 45 4.4 The equivalent circuit of a short dipole antenna ...... 46 4.5 Tuning out dipole capacitive reactance with series inductance . . 46 4.6 Short monopole antenna ...... 47 4.7 A Half wave dipole and its assumed current distribution . . . . . 48 4.8 Half wave dipole, short dipole and isotrope patterns ...... 49 4.9 Shortening factors for different thickness half wave dipoles . . . . 50 4.10 Half Wave Folded Dipole Antenna ...... 50 4.11 VSWR bandwidth of a dipole and folded dipole...... 51 4.12 Impedance transformation using different thickness elements . . . 52 4.13 Triply Folded Dipoles ...... 52 4.14 A horizontal dipole above a ground plane ...... 53 4.15 Two dipoles in eschelon ...... 54 4.16 Mutual impedance between two parallel half wave dipoles placed side by side—as a function of their separation, d ...... 55 4.17 Change in resistance of a half wave dipole due to coupling to its image at different heights above a ground plane ...... 56 4.18 Mutual impedance between two collinear dipoles ...... 56

5.1 Two Isotropic point sources, separated by d ...... 57 5.2 Two Isotropic Sources separated by λ/2 ...... 58 5.3 Pattern multiplication...... 58 5.4 Binomial array pattern ...... 59 5.5 Uniform linear array of isotropic sources...... 60 5.6 Uniform Isotropic Broadside array ...... 61 5.7 Eight In-Phase Isotropic sources, dB vs Linear scales...... 61 5.8 Uniform Isotropic Endfire Array ...... 62 5.9 Array pattern of 2 isometric sources 10λ apart, and the element pattern...... 63 5.10 Two in-phase dipoles 10λ apart...... 64 5.11 SuperNEC run of the 10λ Interferometer ...... 65 5.12 The Franklin array ...... 66 5.13 A Series Fed Four Element Collinear Array ...... 66 5.14 Distortion to folded dipole azimuth pattern in presence of a mast 67 5.15 Yagi-Uda array ...... 68 5.16 Multiplication factor for different diameter to wavelength ratios of director and reflectors ...... 70 5.17 Graph showing the length to be added to parasitic elements to compensate for the effect of the supporting boom ...... 71 5.18 The log-periodic dipole array ...... 72 5.19 Constant directivity contours (dBi) ...... 73 5.20 Characteristics for feeder impedances of 100, 250 and 400Ω and dipoles with L/D ratios of 177, 500 and 1000 ...... 74 5.21 Coaxial Connection to LPDA’s ...... 76 5.22 The small circular loop and the equivalent square loop ...... 76 5.23 The Radiation Pattern of a Short Dipole and a Small Loop . . . 78 LIST OF FIGURES ix

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

Preface

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

xi xii Preface Chapter 1

Electromagnetics

1.1 Transmission line theory

RANSMISSION lines connect generators to loads as shown in fig 1.1. In Tthe RF world, in the transmitting case, this is viewed as connecting the transmitter to the antenna, and in the receiving case as connecting the antenna to the receiver.

RGen

VGen ZLoad

Generator Transmission Line Load

TxLn Figure 1.1: A transmission line connects a generator to a load.

From a standard circuits analysis perspective, the transmission line simply con- sists of connecting two parts of the circuit, and does not change anything. By Kirchhoff’s Voltage law, there is no change in voltage or current along the length of the “connection”.

As a rule of thumb, as soon as the “connection” length between the parts of the circuit exceeds a fiftieth of a wavelength (λ/50) then ordinary circuit theory breaks down, and the “connection” becomes a transmission line. The length of the line in terms of the wavelength of the operating frequency adds a finite time-lag between the start and end of the line, and this causes the voltage along the line to change in terms of magnitude and phase as a function of the distance down the line. Naturally, the current also changes as a function of the distance, hence the ratio of the voltage and current (impedance) also changes.

Recall that the free-space wavelength, λ, is simply given by the usefull approx-

1 2 Electromagnetics

imation: 300 λ(m) = f(MHz)

Thus, the use of Transmission line theory as opposed to circuit theory approxi- mations becomes important for transmission lines longer than: 120km for 50Hz power; 2km for ordinary telephone connections; 600mm for 10Mb/s Ethernet; 120mm for a PC Board bus track at 50MHz; 6mm for an on-chip interconnect in a 1GHz PIII.

¾ ²r D ²eff ²0 d=2a

²r = ²eff d D

w ¾ ²r ²eff h ²0

²r = ²eff

b

a

η2

η1

765kV !

txlnexa Figure 1.2: 2-wire; Coax; µstrip; waveguide; fibre; RF; 50Hz!

Transmission line theory is thus a superset of classical circuit theory. In this 1.1 Transmission line theory 3 text we assume a Uniform transmission line, ie one whose properties do not change along the length of the line. A transmission line is anything that transfers power from one point to another, whether picowatts or gigawatts, as shown in fig 1.2.

1.1.1 Impedance

A resistance resists current flow. An impedance impedes current flow. ie They are the same concept, though generally, a resistance is purely real, whereas an impedance has a reactive (energy storage) component. Thus, an impedance Z:

Z = R + jX where X = jωL or X = 1/(jωC); ω being the radian frequency: ω = 2πf (f in It is important to note Hertz), L and C being the inductance and capacitance. that the reactive part of the impedance is a func- tion of frequency. 1.1.2 Characteristic impedance & Velocity of propagation

A transmission line can be viewed simply as a waveguide. It simply provides the boundary conditions that shape the electric and magnetic fields in the medium between the conductors. Of course, there is a direct relationship to the voltage and current on the line too. In circuit terms, the distributed capacitance and inductance etc of the line can be collected in lumped models. The model of the transmission line is then an infinite set of these circuit sections. Many versions of the model exist, but I shall use the standard (Kraus & Fleisch 1999) model, as shown in fig 1.3.

R L

Input G C Output

d` Lumped Figure 1.3: A “Lumpy” model of the TxLn, discretizing the distributed param- eters.

For a given length of transmission line, we can hence lump the series resistance R [Ω/m] and inductance, L [H/m] together; and the shunt conductance G [0/m] and capacitance C [F/m]. These terms are per-unit length, and do not change from one section of the transmission line to another (uniform transmission line). Hence we define a characteristic impedance, Z0, as the ratio of the series to the shunt components; in the lossless (or high frequency) case, R and G are negligeable: s à r ! R + jωL L Z = = Lossless 0 G + jωC C 4 Electromagnetics

The concept of characteristic impedance has nothing to do with loss, but it characterises a transmission line. Further, since the inductance and capacitance between the two conductors largely depends on the geometry and the medium properties, the characteristic impedance is largely dependant on these factors. The Characteristic Impedance of a line is also referred to as the Surge Impedance. If a voltage is applied at one end of a long transmission line, a current will flow regardless of the load at the other end (the surge hasn’t got to the load yet!). The ratio of the voltage and the current of the surge is the surge, or characteristic, impedance of the transmission line. The velocity with which the wave moves down the (lossless) transmission line is also dependant on the material properties of the medium: 1 v = √ m/s LC

Since these factors depend on both the geometry and the material properties, there is a need in many cases to define an effective permittivity, ²r(eff), since the fields pass partially through air and through solid dielectric. If the entire volume around the conductors is solid dielectric then ²r(eff) = ²r. In the case of a low- loss foam dielectric, or similar dielectric-and-air combinations, an approximation must be used.

Two-wire line

Conceptually, the simplest type of transmission line. Popular examples are the 300Ω “FM Tape” and the standard unshielded twisted pair (UTP) of networking fame. The relationship of both conductors to ground is the same, making it a balanced line. Referring to fig 1.2 for the geometrical definitions, the rigorous case is given by Wadell (1991, pg66): r µ ¶ µ0µr −1 D Z0 = 2 cosh π ²0²r(eff) d Kraus (1992, pg158,499) derives a similar result Usually, the spacing D is much greater than the radius a = d/2 and a simplified using: equation is used: Ã −1 ! r µ ¶ cosh x = µ0µr D p Z0 = 2 ln D À a ln(x + x2 − 1) π ²0²r(eff) a

Since magnetic materials are never used, the simplified equation is usually sim- plified further to the useful: µr ¶ µ µ ¶ 0 ≈ 120π 120 D ²0 Z0 = √ ln ²r(eff) a

Typical values of Z0 range from 200 to 800Ω. 1.1 Transmission line theory 5

One conductor over ground plane

One conductor at a height, h, above a (theoretically) infinite groundplane looks rather similar to a two-wire line by image theory, but is halved. µ ¶ 60 2h Z0 = √ ln ²r(eff) a

Twisted Pair

d

²r

D

twisted Figure 1.4: Twisted-Pair geometry

Twisted Pair is a very common form of transmission line, as all external noise generated is common to both wires. A differential receiver stage then gets rid of most of the induced noise. Common in telephony, Ethernet networks etc. Again, the field lines cross through air and dielectric, making a closed-form solution difficult. Wadell (1991, pg 68) presents Lefferson (1971)’s empirical closed-form solution: µ ¶ 120 −1 D Z0 = √ cosh ²r(eff) d where: θ is normally between 20 and 45◦. Less than 20◦, ²r(eff) = 1 + q(²r − 1) the twists are too loose q = 0.25 + 0.0004θ2 for uniformity, greater ◦ θ = tan−1(T πD) than 50 breaks the wires! T = Twists per length (same units as D).

Coaxial line

Coaxial line has the advantage of shielding the inner conductor, and hence has less radiation (and reception). Since the relationship of the two conductors to ground is different, coax is an unbalanced line. Also all field lines pass through the dielectric, hence ²r(eff) = ²r. It is also common to refer to the ratio of the radii a = d/2; b = D/2: µ ¶ µ ¶ 60 D 60 b Z0 = √ ln or: Z0 = √ ln ²r d ²r a 6 Electromagnetics

Wadell (1991, pg53–65) presents formulae for various offset coaxial, and strip- in-coax situations.

Microstrip Line

Very popular since this is simply printed on a circuit board. Generally used at the higher frequencies. One of the problems here is that the fringing flux plays a very important role in establishing the exact Z0 value. Unfortunately the fringing flux flows through a combination of air and dielectric, hence again the need for an effective ². Gardiol (1984) has developed expressions for the cases where w/h ≤ 1 (large percentage of fringing flux) and for cases where w/h > 1 (less percentage of fringing flux). For w/h ≤ 1: "µ ¶ # 1 1 h −1/2 ³ w ´2 ² ≈ (² + 1) + (² − 1) 1 + 12 + 0.04 1 − r(eff) 2 r 2 r w h

µ ¶ 60 h w Z0 ≈ √ ln 8 + ²r(eff) w 4h

For w/h > 1: µ ¶ 1 1 h −1/2 ² ≈ (² + 1) + (² − 1) 1 + 12 r(eff) 2 r 2 r w 120π hw ³w ´i Z0 ≈ √ + 1.393 + 0.667 ln + 1.444 ²r(eff) h h

Kraus & Fleisch (1999, pg 132) also has an approximate formula (which in earlier editions he qualified as being applicable for w ≥ 2h, but this is often not the case in microstrip lines of interest: 120π Z0 ≈ √ ²r[(w/h) + 2]

Slotline

A slotline has no groundplane, but uses adjacent tracks as the transmission line. The fields are distributed in an elliptical waveguide within and without the dielectric. Closed-form expressions for Z0 span several pages of Gupta, Garg & Bahl (1979, 213–216).

1.1.3 Impedance transformation

Since the voltage and current change down the transmission line, react with the load and reflect back to the source, it appears as though the impedance (ratio of the total voltage to the total current) changes continually down the 1.1 Transmission line theory 7 line. Under sinusoidal, steady state conditions a transmission line transforms the load impedance ZL connected to its output to a different impedance at the line input Zin as follows (lossless case): · ¸ ZL + jZ0 tan β` Zin = Z0 Z0 + jZL tan β`

2π where β = and ` is the length of the line (electrical length). λ This equation is generally known as the Transmission line equation since it fully specifies what happens on the line. It is an unwieldy equation, however, and not generally useful. A graphical technique like the Smith Chart (section 1.2 on page 9) effectively embodies this equation in an easy-to-use manner, without the need for the equation as such. Since the velocity with which waves travel on a transmission line is lower than Note that the velocity on that of light, the physical length of the cable is always shorter than the electrical a line can never be higher length: than the speed of light! Physical line length = VF × (electrical length) where the velocity factor, VF = √1 and is quoted by the manufacturers. ²r Some important simplifications of the Transmission line equation are:

1. ZL = Z0 This condition results in: Zin = ZL = Z0, regardless of line length, or frequency. This is the matched case. λ 2. ` = . Z = Z regardless of characteristic impedance. This is the 2 in L halfwave case.

2 λ Z0 3. ` = “Quarter wave transformer” case. Zin = 4 ZL This configuration is useful since it can transform one load impedance to a different one if a line with the correct impedance can be found. 4. Open or short circuited lines.

Zin(oc) = −jZ0 cot β` for an open circuited line

Zin(sc) = jZ0 tan β` for a short circuited line In both these cases the impedance is purely reactive and if the lines in question are less than a quarter wave it is clear that such lines could be used to “manufacture” capacitive (open circuit case) or inductive (short circuit case) reactances. It should be remembered however that the capac- itance or inductance of such a line would itself be frequency dependent. The open and short circuit cases provide a convenient way to measure the characteristic impedance of a line, since combing them yields: p Z0 = Zin(oc)Zin(sc) 8 Electromagnetics

The velocity factor of a line can be measured by using the quarter-wave trans- former principle—if the load end is open circuited, ZL = ∞, hence Zin =0! The method is then to take an open-circuited line and mesure the input impedance, increasing the frequency until the input impedance drops to a minimum. The line is then at an electrical quarter-wavelength, so

` VF = phys λ/4

1.1.4 Standing Waves, Impedance Matching and Power Transfer

The Maximum Power Transfer theorem determines when maximum power will be transferred to a load or extracted from a source: An impedance connected to two terminals of a network will ab- sorb maximum power from the network when the real part of the impedance is equal to the real part of the impedance as seen looking back into the network from the terminals and when the reactance (if any) is of opposite sign. Stating this in a simplified form appropriate for transmission lines (which nor- mally have a real impedance) The impedance of devices connected to the two ends of a transmis- sion line should have the same input and output resistances as that of the characteristic impedance of the line. When this condition is not satisfied standing waves are set up on the line and not Note that the sub- all of the available power is transferred through the system. A more practical maximum power transfer problem which occurs is that such mismatch conditions often damage the output is not due to losses in electronics of a transmitter if excessive standing waves occur (or in well designed the system, but rather to transmitters the protection circuitry automatically reduces power output). power being reflected. The extent to which power is reflected from the load is dependant on how “bad” the load mismatch is to the line characteristic impedance and is expressed in the voltage reflection coefficient, ρ:

Z − Z ρ = L 0 ZL + Z0

The mismatch in impedance is also often stated in terms of the voltage standing wave ratio (VSWR) on the transmission line. µ ¶ 1 + |ρ| V VSWR = = max 1 − |ρ| Vmin

Many texts and measur- On a Smith Chart the VSWR value can be read off directly without performing ing instruments refer to a any of these tedious manipulations. reflection coefficient as a capitalized gamma: Γ. 1.2 The Smith Chart 9

1.2 The Smith Chart

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

1

0.5 2

0.26+j0.6

0.2

0.19+j0 0.6 5.3

0 0.2 0.5 1 2 5 ∞

−0.2

0.6−j1.4 −0.5 −2 Smith .

−1.4 −1 0.16λ

Figure 1.5: Impedance values plotted on the Smith Chart 10 Electromagnetics

The Voltage Standing Wave Ratio, VSWR, can be read directly off the Chart as it is simply the intercept on the real axis as indicated by the circle centered in the centre of the chart drawn for VSWR of 5.3:1. The constant VSWR circle in fact describes all the possible values of the input impedance of the line as a function of distance down the line. As we move down the line, we travel down a constant VSWR circle, assuming that the line is lossless, and hence get no nearer to a better match. In the case of a lossy line, the magnitude of the reflection coefficient decreases, and instead of a circle, a spiral is described, spiralling in to the centre of the chart, improving the match. ie A lossy line improves the aparrent VSWR at the input to the line, but of course this is at the expense of loss in the cable. The addition of an inductive component to the plotted value on the other hand, would clearly move the point on the constant resistance line (0.6 normalized in our example) and it is immediately clear that more reactance in this example would improve the impedance match since it would force the point closer to the centre of the chart, without the loss of power.

1.3 Field Theory

1.3.1 Frequency and wavelength

As seen previously, the wavelength in free-space is given as

300 λ(m) = f(MHz)

In the part of the Electromagnetic spectrum of interest to us, there are tradi- tional (but otherwise meaningless!) designations of radio “bands” Frequency Wavelength Designation 3–30 Hz 100–10 Mm ELF (Extra Low Frequency) 30–300 Hz 10–1 Mm SLF (Super Low Frequency) 300–3000 Hz 1 Mm–100 km ULF (Ultra Low Frequency) 3–30 kHz 100–10 km VLF (Very Low Frequency) 30–300 kHz 10–1 km LF (Low Frequency) 300–3000 kHz 1000–100 m MF (Medium Frequency) 3–30 MHz 100–10 m HF (High Frequency) 30–300 MHz 10–1 m VHF (Very High Frequency) 300–3000 MHz 1000–100 mm UHF (Ultra High Frequency) 3–30 GHz 100–10 mm SHF (Super High Frequerncy) 30–300 GHz 10–1 mm EHF (Extra High Frequency) 300–3000 GHz 1000–100µm Submillimeter/InfraRed Observations: • Small things (electrically speaking) do not radiate well. Hence from 30MHz to 30GHz is the most commercially used band. • 3–30MHz bounces off ionosphere (so-called Shortwave) and is used for old-fashioned over-the-horizon comms. 1.3 Field Theory 11

• Achievable frequencies are limited by electronics (for oscillators) • Above 3000 GHz, we get into optics. • Ionizing radiation can cause a molecule to change (x-rays and higher). Anything lower in frequency can only cause heating. Humans can take about 100W of heat! The microwave (> 1GHz) region is broken into Radar bands. The most common designations are the IEEE ones, although there are “newer” designations, which I have never seen used. Frequency Wavelength IEEE Designation 1–2 GHz 300–150 mm L 2–4 GHz 150–75 mm S 4–8 GHz 75–37.5 mm C (5m (big) dish) 8–12 GHz 37.5–25 mm X 12–18 GHz 25–16.7 mm Ku (1m (small) dish) 18–26 GHz 16.7–11.5 mm K 26–40 GHz 11.5–7.5 mm Ka 40–300 GHz 7.5–1 mm mm Spectrum Usage: FM Radio 88–108 MHz ClassicfM 102.7 MHz TV 123 200 MHz MNET 615 MHz GSM MTN & Vodacom 900 MHz Cell C 1800 MHz DECT 1880–1900 MHz GPS 1.23 & 1.58 GHz WLan 1.8 GHz Industrial ISM 906–928MHz Scientific ISM 2.4–2.5 GHz Medical ISM 5.8–5.9 GHz µwave oven (K5–513) 2.45 GHz DSTV Ku band (11.7 & 12.3GHz)

1.3.2 Characteristic impedance & Velocity of propagation

In the case of a bounded medium like a transmission line, the geometry and the medium played a part in determining the characteristic impedance. For a propagating Transverse Electromagnetic (TEM) wave in an unbounded medium, the characteristic impedance it sees is: s jωµ0µr Z0 = σ + jω²0²r

Since free space is non-conducting, and the relative quantities are unity: Some texts use eta: η, r and others zeta: ζ to µ0 specifically refer to the Z0(free space) = η = = 377Ω or: η ≈ 120π ²0 Z0 of free-space. 12 Electromagnetics

Just as the characteristic impedance depends on the geometry and the properties of the medium, so the velocity of EM radiation also depends strongly on the medium properties. For example, sound travels at 330m/s in air, but 1500m/s in water. In a non-conducting medium, the velocity of propagation is: 1 v = √ µ0µr²0²r

where ²0 and µ0 refer to the electric permittivty and magnetic permeability of free space; the r subscripts refer to the quantities of the material relative to h t o re sa e i c Ntta the value of µ0 that of free space. SinceNotethat is by definition, and ²0 −7 is actually derived from µ0 ≡ 4π × 10 [H/m] −12 the measured speed of ²0 = 8.85 × 10 [F/m] light. (Kraus 1988, Back Cover) Thus, the speed of light in a vacuum or air, c, is given by: 1 c = √ = 3 × 108m/s = 300km/s µ0²0 Most often, we are not interested in the absolute value of the speed of prop- agation, but in the ratio of the speed to that of light in free-space, known as the Velocity Factor, VF= v/c. This leads to the useful formula for EM wave velocity in non-magnetic media as:

c 1 v = √ or: VF = √ ²r ²r

Since most plastics have a relative permittivity of about 2.3, the reduction in speed is about 0.66, hence the shortening factor for coaxial cable calculations.

1.3.3 EM waves in free space

An EM wave travels in free-space and in most transmission lines as a Transverse EM wave (TEM). This implies that the direction of propagation is at 90◦ to both the Electric and Magnetic wave, which, in turn are at 90◦, as shown in fig 1.6 Since the E field is analogous to voltage and the H field to current in the circuits sense, it is easily seen that the equivalent relationships to Ohms law etc exist in TEM waves in free-space: E (V = I × R)... E = H × η or: H = 120π In a similar fashion, the power relationships hold (power density in EM):

E2 S = EH = = H2 × 120π W/m2 120π where E and H are RMS values. 1.3 Field Theory 13

x Hy z

Ex E(φ)

EMWave y

Figure 1.6: Electromagnetic Wave in free-space

In general, EM waves of a frequency above 30MHz do not bend around the earth, and propagation occurs only within Line-of-Sight (LOS). The earth’s curvature thus limits the radio horizon achievable for a certain height of transmitter, as shown in fig 1.7

d Antenna h Horizon Line

h Earth Shadow Region

radhor

Figure 1.7: Radio Horizon due to earth curvature.

An empirical formula to calculate the radio horizon is: Since the mean Earth Ra- p dius is 6370km, and the dh(km) = 4 h(m) tangential point is on it, by Pythagorous, we get Thus a 30m mast will only provide a 22km Line-of-Sight range. 19.5km! Beyond the horizon waves propagate for longer distances than that illustrated above using either ground wave propagation or reflection from some atmospheric feature such as the ionosphere, meteors, the troposphere or even an artificial satellite (although in that case the signal is usually retransmitted). Ground wave propagation occurs when vertically polarized waves are launched and these are guided by the earth surface and thus follow the curvature of the earth. They are only practical at lower frequencies where the attenuation these waves undergo as result of earth losses are reasonably low. At the very low frequency end (round 30 kHz) the wavelength is so large that the waves flow in the waveguide formed by the earth surface on the one hand and the ionosphere on the other. This could also be considered as a type of ground wave. 14 Electromagnetics

The classic means of long distance communication is by means of reflecting waves from the ionosphere. This layer1 of gas that is ionized by the sun lies at a distance of approximately 80 to 400 km above the earth surface. Only waves below a certain critical frequency are reflected (by a process of successive refraction) by this layer. This so called critical frequency is typically in the HF band (3–30 MHz) and varies strongly with time of day and other variables. At frequencies much below this frequency the radio-waves undergo attenuation (or absorption) and communication using this principle is also impractical. Other methods of obtaining beyond the horizon communication are the rather exotic techniques such as meteor scatter propagation: where waves are reflected off the ionized trails left by meteors entering the atmosphere; or tropospheric scatter: which uses discontinuities in the atmosphere to cause bending of waves and thus long distance propagation. Most over-the-horizon communication, of course, occurs at microwave frequen- cies over a satellite relay link at high data volumes. (Discounting under-the-sea fibre optics for this course!)

1.3.4 Reflection from the Earth’s Surface

In addition to its role as a obstacle, the earth’s surface also acts as a reflector of radio waves. This situation is illustrated in figure 1.8.

Direct-wave path

θ S1

Reflected-wave path

2h θ Reflecting Surface

P1

Path Difference = 2h sin θ S2 ReflRay

Figure 1.8: Geometry of Interference between Direct Path and Reflected Waves

It is clear that if S1 is an isotropic source and would normally radiate equally well in all directions, the pattern would be modified by the reflected wave. By the method of images the situation above is similar to that which exists if a mirror image source S2 was positioned at distance h below the reflecting plane.

1Actually, several identifiable layers at differing heights 1.3 Field Theory 15

Clearly there will now be a difference in the path lengths to some distant point P . At certain elevation angles θ the path difference would be such that the two waves are in phase and thus interfere constructively and for others the interference would be destructive and result in a null in the radiation pattern. If the field due to a single source is termed E0 then the total field would then be given by: µ ¶ 2πh sin θ E = |E | sin 0 λ

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

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

In cases where radiation at some angle is required this reflection results in an unexpected bonus, however. The maximum value of the E-field in the direction of the maxima is twice the value of the original antenna without reflection. This implies that in that particular direction the power density would be increased by a factor of four (power density is proportional to the square of the E field). Earth reflection can thus be used to gain a 6 dB bonus in antenna gain if used properly!

90

1 60

0.8

0.6

30

0.4

0.2

0

. Interfere

Figure 1.9: Interference pattern field strength contours for h/λ = 1.44 16 Electromagnetics

1.3.5 EM waves in a conductor

If we solve Maxwell’s equations for EM wave propagation in a conductor, we get a different result to the free-space case considered above. The Electric field magnitude actually decays exponentially inside the conductor. We define the Skin depth, δ, as the point where the field inside the (good) conductor, Ey, has become 1/e of its original value, E0: r 2 1 δ = = √ ωµσ πfµσ

Note that δ has units of distance. If we look at what happens at one skin depth, −x/δ δ, into the conductor, the field is given as: Ey = E0e , then

|E | |E | = 0 ≈ 37%|E | y e 0 or the field inside the conductor at one skin depth is 37% of the original field magnitude. Comparing the skin depth (or depth of penetration) δ for silver, copper & cast iron at 50 Hz and 1GHz: 7 7 Relevant parameters: σAg = 6.1 × 10 0/m σCu = 5.7 × 10 0/m (Both have 6 µr = 1) σFe = 10 0/m and µrFe = 5000

δ50Hz δ1GHz Ag 9.1mm 2µm Cu 9.4mm 2.1µm Fe 1mm 0.23µm The decay is obviously logarithmic: at one further skin depth, the field has deacyed a further 37%. ie 9.4mm of Copper to get to 37% of the original signal strength. Another 9.4mm of copper will take that signal strength to 37% of that value, ie 13.7% of the original value. A useful table is then the percentage overall decrease as a function of the number of skin depths: δ0s % of Orig. 1 37 2 13.7 3 5.1 4 1.9 5 0.7 After 5 skin depths, the field level is below 1%. A number of consequences follow: • Busbars in substations are hollow. It is silly to supply a solid copper bar if there is no field and current flow in it’s middle! At DC, the current den- • At cellphone frequencies, the skin depth of copper is about 2µm. Certainly sity is uniform through- after 10µm, there is no appreciable field. out the cross-section, but This means that the outside of a coaxial braid is a completely separate at RF, the current den- conductor from the inside of the coaxial braid from an RF perspective!! sity concentrates in the two outer skins. Chapter 2

Antenna Fundamentals

2.1 Directivity, Gain and Pattern

HE USUAL COORDINATE SYSTEM in 3-dimensional space, the carte- Tsian co-ordinate system of points being described in terms of (x, y, z) co- ordinates does not lend itself to the electromagnetic world. Generally, we like to describe the points in the spherical co-ordinate system as (r, θ, φ), as shown in fig 2.1. z

P (x, y, z)&P (r, θ, φ) θ

r y

φ Coord

x Figure 2.1: Standard coordinate system.

The spherical co-ordinate system is mainly described in terms of angles, where r is the distance from the origin, θ the angle from the zenith (not the elevation angle) and φ is the projected angle from the x-axis, in the xy plane.

2.1.1 Solid angles

Clearly, when dealing with 3-dimensional radiation patterns, some idea of a 3-dimensional angle needs to be employed.

17 18 Antenna Fundamentals

A radian is defined by saying that there are 2π radians in a circle. As a result, the arclength rθ of a circle of radius r is said to subtend the angle θ. A steradian (or square radian) is a solid angle defined by saying that there are 4π steradians in a sphere. As a result, an area a on the surface of the sphere of radius r is said to subtend the solid angle ω. Note that the shape of the area is not specified. shown in fig 2.2 z

Area=A

Ω

θ x = rθ y r

x

x A θ = r radians Ω = r steradians

steradian Figure 2.2: concept of a solid angle

2.1.2 Directivity

Isotropic source

An isotropic antenna, is as its name implies, one which radiates the same power in all possible (3-dimensional) angles. By definition, an isotropic antenna has a directivity, d, of 1, and is infinitely small. Note that an isotropic antenna is an impossibility and is used only as a reference! Generally, however, an antenna has greater directivity in some directions than other directions. This is at the expense of directivity in other areas. In free-space the EM wave is not guided, as it would be in a transmission line, and can hypothetically radiate in all directions. If an isotropic source transmits Prad watts, and we enclose the source with a sphere of radius, r, and hence surface area, A = 4πr2, then the power density, S at the surface of the sphere is given as: P S = rad A P = rad [W/m2] 4πr2 hence the power reduces inversely proportional to the square of the distance. Directivity, D, is then defined as: S(θ, φ) D = 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- referred to an isotropic source (the Sisotropic in the above), the unit is denoted ways quoted in dB’s, but dBi for dB’s above isotropic. Decibels are very useful in power ratio’s because that most equations re- of the large range of values experienced because of the inverse square law. For quire it in linear form. example, if power P1 is 1000 times greater than P2, the ratio expressed in dBs is: µ ¶ P1 dB = 10 log10 P2 which yields P1 as being 30dB greater than P2. If P1 were a million times the power of P2, it would be 60dB greater. It can be seen that dB’s are simply more convenient for expressing power ratio’s than linear quantities. One can also use dB’s in field-related quantities that yield power when squared— from Ohms law in free space, we see that S = E2/120π. Since the log of a squared quantity is simply twice the log of the non-squared quantity, for a power component like voltage, E-field etc: µ ¶ E2 dB = 20 log10 E1

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

³power´ dBm = 10 log 10 1mW similarly for a voltage relative to a microvolt: µ ¶ voltage dBµV = 20 log 10 1µV

2.1.3 Efficiency

Directivity, D, is often loosely referred to as Gain, G. However, the term gain Note that “Gain” is not refers to the input power as opposed to the radiated power—the difference being used in the same sense as the power lost in the ohmic losses in the antenna. The antenna efficiency is then: amplifier “gain”. An an- tenna is passive. It sim- P η = rad ply denotes an increased Pin power density in one di- rection, at the expense of and hence: other directions. G = ηD 20 Antenna Fundamentals

For antennas with negligible losses these two values are approximately equal, but many antennas which are small in terms of wavelength or broadband are not efficient and the two values can be quite different. In general the gain is of importance for calculating power levels at various sites. Both Gain and Directiv- The directivity is more a indication of the pattern of an ideal radiator and is of ity are functions of θ and more theoretical than practical value. φ but are often used to In practice the gain of an antenna is important since it increases the power describe these values in density in the direction of the main beam of the antenna. A 100 W transmitter the maximum directions. with a 13dB gain antenna produces the same power density at a distant point as a 1 kW transmitter with a 3 dB gain antenna. Clearly the former case would have a larger area of lower power density in other directions and the extent to which antenna gain would improve communications would depend on the intended coverage. In a broadcast scenario for instance, omnidirectional radiation may be required in the plane of the earth. The only improvement in gain of broadcast antennas is by minimizing the radiation upwards (at angles of θ smaller than 90◦). For point-to-point links on the other hand the power can be concentrated as far as possible in the desired direction. Hence, the power density S at a distance r away from an antenna with a gain G is: GP DP S = in = rad W/m2 4πr2 4πr2

2.1.4 Radiation pattern

An antenna’s radiation pattern is a measure of how it radiates in 3-dimensional space. Generally, a radiation pattern refers to one that is measured in the far- field (see later). As a result, it is a plot of transmitted power versus angle. It has nothing to do with distance. Note that the signal from an antenna weakens predictably with distance by 1/r2, but not predictably by angle—this is described by the radiation pattern. The pattern can show power density(W/m2) versus angle, radiation intensity (W/sr) versus angle, E-field (V/m) or H-field (A/m) versus angle. The plot can be rectangular or polar. Generally, however, it is a plot of power density, expressed in dBi. A representation of a SuperNEC (Nitch 2001) three dimensional radiation pattern of a 10-director Yagi-Uda array is shown in fig 2.3. An antenna often has a main beam, with some sidelobes and backlobes, which may not be desired. Although a 3d pattern The main lobe can be clearly seen, as can the first sidelobe (all the way around may look impressive, it the main lobe, like a collar). It can also be seen that the next sidelobe fires up can rarely show detail ac- and down, but that there is a null along the y-axis. It is far more common to curately. take a 2D cut through the principle planes of the antenna, as shown in the xy plane cut in fig 2.4. Fig 2.4 is shown in a polar representation, which is most useful for visualization It is easier to get a feel purposes. To read off values, however, a rectangular form is preferred as shown for the front-to-back ra- tio, (14.3 dB), and the first sidelobe levels (14.3 dB in azimuth; 8.77dB in elevation) from the rect- angular plots. 2.1 Directivity, Gain and Pattern 21

Figure 2.3: Three-dimensional pattern of a 10 director Yagi-Uda array

Radiation Pattern (Azimuth) 10 dBi 120 60 0

−10

−20

−30

180 0

° Structure: θ=90

240 300

Figure 2.4: xy plane cut of the 10 director Yagi-Uda

in fig 2.5.

The rectangular plots also show the half power beamwidth (HPBW) (-3dB point of the main beam) and the beamwidth between first nulls (BWFN) quite clearly, and these quantities are often useful to determine the absolute gain of the an- tenna, which is difficult to measure accurately, whereas a 3dB drop from peak power is a relative measurement, and hence an eassier one. 22 Antenna Fundamentals

Radiation Pattern (Azimuth) Radiation Pattern (Elevation) 15 15

10 10

0 0 Gain (dBi) Gain (dBi)

−10 −10

−20 −20 −180 −120 −60 0 60 120 180 −90 −30 30 90 150 210 270 φ θ

(a) xy plane (b) yz plane

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

Directivity estimation from beamwidth

Note again that if the “beam” equally covered a whole sphere of 4π square 4π square radians = radians, or steradians, ie an isotropic source, the directivity is 1 by definition. 4π(360/2π)2 square de- This gives rise to an approximation which is often used (Kraus & Fleisch 1999, grees = 41 253, usually pg 255): rounded off to 41 000. 4π 4π 41 000 D = ≈ ≈ ◦ ◦ ΩA θHPφHP θHPφHP The above approximation produces about 1dB too much gain. the reasoning is that the area represented is square, whereas in reality, the beam is generally round. hence a common adjustment (Kraus 1988, pg100) is:

36 000 D ≈ ◦ ◦ θHPφHP

even in this case, however, if fairly significant side- and back-lobes exist, the prediction is optimistic. In the above example of the 10-director Yagi-Uda array, we get HPBW’s of 42 and 38 degrees, which according to the above formula gives 13.5dBi gain, whereas SuperNEC gives 12.7dBi.

2.2 Reciprocity

The reciprocity theorem stated by Lord Rayleigh, and generalised by Carson (1929), states that if a voltage is applied to antenna A which causes a current to flow at the terminals of antenna B, then an equal current (in magnitude and phase) will occur at the terminals of antenna A if the same voltage is applied to the terminals of antenna B. In short, all characteristics of an antenna apply equally well in transmit and receive mode (radiation pattern (reception pattern), input/output impedance etc). shown in fig 2.6. 2.3 Polarization 23

Ib Vb

Va Energy Ia Energy Flow Flow

reciprocity

Va causes Ib ; Vb causes Ia If Vb is made = Va, then Ia will be = Ib Figure 2.6: Reciprocity concepts

However this does (obviously) not hold for the near fields—it is a far-field phe- nomenon. ie far-field patterns are identical, but near-field patterns are different.

2.3 Polarization

Polarization generally refers to the direction of the E-field vector in the far- field. note that the H-field is at 90◦ to the E-field and is present in all EM waves. In the case of a dipole or a Yagi-Uda array, the E-field is linearly polarized in the direction of the dipole. Most often, a linear polarization will be specified as horizontally polarized (most TV antennas) or vertically polarized (most broadcast radio signals) (reference is the earth). The amount of power received varies as the cosine of the angle between the Note that if the sig- polarization of the incident wave and the antenna. eg A cellphone base station nal is horizontally polar- antenna is vertically polarized, and although the mobile has a linearly polarized ized, and the receiving antenna, it is seldom held perfectly upright, incurring a loss in received power. antenna vertically polar- ized, there is (ideally) no Linear Elliptical Circular received signal. y y y

E2 E2 E2 E E

x x x E1 E1

Polarization Figure 2.7: Polarization possibilities—wave out of page 24 Antenna Fundamentals

in the vertically polarized case, only ey is present and can be expressed as:

ey = e2 sin(ωt − βz)

ie a time varying and distance varying (out of the page) component, but only in the y direction (vertical). the maximum value e2 is in the y direction. β is the wave number, given as: 2π β = λ the e-field can be made to progressively change direction in a circular fashion (eg with a helical antenna, popular on satellites). the helix can be wound in a left-hand or a right-handed fashion, giving rise to lhcp and rhcp. a rhcp antenna will not (ideally) receive a lhcp wave. the e-field is continuously rotating, and the peak x-component is equal to the peak y-component of the field. In the general case of el- Generalizing to elliptical polarization, the x and y components are not equal. liptical polarization, the The field components are thus: major axis magnitude to the minor axis magni- Ex = E1 sin(ωt − βz) tude defines the axial ra- tio. An infinite axial ra- Ey = E2 sin(ωt − βz + δ) tio implies linear polar- where δ is the time phase angle by which Ey leads Ex. ization, whereas an AR of 1 implies circular polar- ization. 2.4 Effective aperture

If an antenna is immersed in a field with a power density of S [W/m2], it will receive a power Pr [W] and deliver it to a load connected to its terminals. This 2 gives rise to the concept of an effective aperture, Ae [m ]. P A = r e S In general (Kraus & Fleisch 1999, pg 258), the aperture of an antenna can be related to the antenna gain: Gλ2 A = e 4π ie when the gain is large the aperture is large. Note that the effective aperture of a dish antenna may not be as big as the actual area of the dish (see fig 2.8), leading to the concept of an aperture efficiency, εap: Ae εap = 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

2 Ap = πr

r

Ae ApertureEff 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 more energy from the wave, the apertures cannot be allowed to overlap as shown in fig 2.10

ApertureSteal Figure 2.10: Apertures must not overlap!

In the case of linear antennas, this is a fair problem, as the apertures easily overlap, and the overall capture area will be reduced. With dish antennas, since Ae < Ap, it is not possible to have an overlapping aperture!

2.5 Free-space link equation and system calula- tions

The gain of an antenna is defined as the maximum power density in a certain direction as compared to a hypothetical isotrope. Since the antenna gain con- 26 Antenna Fundamentals

Gt Gr

Pr Pt

Tx Rx r

LinkEqn

Figure 2.11: Point-to-point link parameters

centrates the transmitted power into a particular direction, we can define the ERP is the power that effective radiated power (ERP) of an antenna in a particular direction as: an isotrope would have to produce in all di- ERP = GtPt rections to achieve the at a distance, r, away from the transmitting antenna, the power density, S is same effect in this par- given as: ticular direction. (also G P S(r) = t t pedantically called EIRP, 4πr2 I for Isotropic, but rarely the power received, Pr, by the receiving antenna is then Pr = SAer where Aer used). is the effective aperture of the receiving antenna, given by G λ2 A = r er 4π hence the power received by an antenna in a freespace point-to-point link is:

G G P λ2 P = t r t [W] r (4πr)2

note that multipath reflections with constructive and destructive interference and atmospheric absorption change this value! The 32.45 factor comes The link equation can obviously also be conveniently expressed in dB’s: from +20 log(300) for frequency conversion Pr = Pt + Gt + Gr − 32.45 − 20 log10 r − 20 log10 f (20 since squared), where P and P are expressed in dBm, G and G are in dBi, r is in km, and f −20 log(4π), and r t t r is in MHz. −20 log(1000) for the km conversion. Example: Geostationary satellite comms

A geostationary satellite must be in the Clarke orbit at 36 000 km in order to appear stationary above a point on the earth. If the power required at the 2.5 Free-space link equation and system calulations 27 satellite receiver is only 8pW, how much power is required to be transmitted by the earth station? Receiver gain is 20dB, earth station gain is 30dB, at a frequency of 5GHz. Converting to linear gains:

Gr = 20 dB = 100; Gt = 30 dB = 1000 300 λ = = 60mm 5000 hence the power we are required to transmit is:

2 7 2 Pr(4πr) 8pW(4π × 3.6 × 10 m) Pt = 2 = 2 = 4547W GtGrλ 100 · 1000 · (0.06) So 4.5kW is needed to get 8pW to the satellite. Note that the link equation ignores atmospheric absorption or any other losses along the path, and purely assumes line-of-sight (LOS) situations. Actually, satellites transmit at hundreds of watts, not kW! The dB version is thus:

Pt = Pr − Gt − Gr + 32.45 + 20 log r + 20 log f = −80.9 − 20 − 30 + 32.45 + 91.12 + 73.98 = 66.58dBm = 4556kW 28 Antenna Fundamentals Chapter 3

Matching Techniques

3.1 Balun action

3.1.1 Unbalance and its effect

T RF FREQUENCIES recall that from section 1.3.5, the skin-depth is only Aa few microns. As a direct result, the inside of the coax braid is quite a seperate conductor than the outside of the braid, even though, from a DC perspective, we view the braid as “one conductor”. If we feed an antenna using a coaxial cable as shown in fig 3.1, the antenna radiation can induce a current flow on the outside of the braid, i3, quite inde- pendant of the current flow on the inside of the braid due to the transmitter, i2.

i2 + i3 i1

i3 i2 i1

BalunReason Figure 3.1: The reason for unbalanced currents

Since on the inside of the braid, balanced currents must flow i1 = i2, and both An absolutely tell-tale the inside and outside currents meet at the antenna junction, this implies that sign of external current is a larger current can flow on one dipole arm than on the other. The antenna a change in a measure- pattern is distorted, and the input impedance is changed. ment parameter when the cable is grabbed by a hand. 29 30 Matching Techniques

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

3.1.2 Balanced and unbalanced lines—a definition

Tr Ln Cross Sections Balanced

UnBalanced

UnBalanced

BalLine

Figure 3.2: Balanced and Unbalanced Transmission lines

Essentially, a balanced line is one in which symmetry ensures that equal fields, currents, voltages exist along it in the equal and opposite sense! It is clear that a two-wire line of equal diameter conductors is balanced, but that a coaxial line is not—the field strength is much higher on the inner conductor than the outer conductor.

3.2 Impedance Transformation

The Balun action is to be distinguished from the Transformer action. A circuit that does both functions should be called a Balun-Transformer, but in practice all are simply called Baluns, whether they transform, or just do a Balance-to- Unbalance conversion (sometimes even when they only transform. Impedance transformation has the goal of matching the antenna impedance to the line, and the generator in order that no reflected power exists—maximum power transfer. A transmitter cannot deliver full power to an unmatched load, but this is not the only consideration: • High VSWR means high V &I, hence txln losses. • High V means flashover/dielectric breakdown. • High I means hotspots/copper melting. • Output electronics of the transmitter can be damaged, or more likely, the automatic power reduction circuitry kicks in. (Typically at a VSWR of 2:1) Impedance matching is important both for transmission and reception. It is more critical, however, for the transmitting case and the VSWR specifications 3.2 Impedance Transformation 31 are usually more severe. To illustrate this point the following equation gives the power reduction as result of a mismatch in terms of VSWR:

à µ ¶ ! VSWR − 1 2 Power lost in transfer = 10 log 1 − dB VSWR + 1

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

5

4.5

4

3.5

3

2.5

Power lost in dB 2

1.5

1

0.5

0 pwrvswr 1 2 3 4 5 6 7 8 9 10 VSWR

Figure 3.3: Power loss in dB versus VSWR on the line.

The additional line losses with a higher VSWR is also of concern and is a func- tion of the normal line losses, which are usually specified by the manufacturer of the lines in terms of dB/30m (ie dB/100ft!). It should be noted that the manufacturer specification assumes matched conditions. The graph in figure 3.4 indicates the additional loss under mismatch conditions (ARRL 1988, pg3-12).

Losses in the line will improve the VSWR at the input end as compared to This improvement is the VSWR at the antenna. A curve quantifying this characteristic is given in clearly obtained at the figure 3.5 expense of efficiency, indicating the need to avoid long transmis- sion lines when doing measurements of VSWR. 32 Matching Techniques

101 SWR=20

SWR=7

SWR=4

100

Additional loss in dB SWR=2

10-1 10-1 100 101

AddLoss Additional Loss in dB when Matched

Figure 3.4: Increase in Line Loss because of High VSWR 100

50 3dB Loss 2dB Loss

1dB Loss 10 SWR at Antenna 5 0dB Loss

1 1 2 3 5 10

SWRChange SWR at Transmitter

Figure 3.5: VSWR at Input to Transmission Line versus VSWR at Antenna

3.2.1 Angle Correction

Remember that a low-loss or lossless transmission line has a purely real char- acteristic impedance. For matching to occur, the antenna has to be resonated. 3.3 Common baluns/balun transfomers 33

This is most often accomplished before impedance transformation. Some match- ing techniques include angle correction, some don’t.

3.3 Common baluns/balun transfomers

3.3.1 Sleeve Balun

Recall that a short-circuited quarter-wavelength transmission line presents a high (theoretically infinite) impedance. Thus the simplest balun is a short- circuited quarter-wave sleeve around the coaxial feeder cable, where the sleeve is short-circuited to the coaxial feeder cable braid a quarter wavelength away from the feed. Currents wanting to flow in the outside of the braid now see a high impedance, as shown in fig 3.6.

λ/4

Bazooka

Figure 3.6: The Bazooka or Sleeve balun.

A sleeve balun has no transformer action, but simply prevents the current flow. The Americans tend to It is rather narrow-band however, and care must be taken to ensure that it is refer to the sleeve balun the correct length in terms of the Velocity Factor of the coax outer insulation, as a “Bazooka balun”! which is normally a different material than the inner dielectric (hence a different VF).

3.3.2 Half-wave balun

λ/2

HalfWaveBalun Figure 3.7: A HalfWave 4:1 transformer balun.

In addition to a balun action, due to the halfwave section, there is also a 4:1 impedance transformation as shown in fig 3.7. The impedance transformation is because after the λ/2 transmission line, the voltage is exactly out of phase, hence twice the voltage is applied between the 34 Matching Techniques

dipoles. Naturally, the current is halved, hence the impedance (voltage over current) has a factor of four. Obviously, the half-wave balun is narrow-band too.

3.3.3 Transformers

Ordinary transformers can be used as impedance transformers (remember that a 2:1 voltage transformer is a 4:1 impedance transformer). Transformers are broadband, but tend to be lossy. The selection of the ferrite core is important as they are frequency selective. A very useful variety is the Transmission Line Transformer shown in fig 3.8.

TxLn 9:1 Ant

TLTrans Figure 3.8: Transmission-line transformer.

Sevick (1990) is an excellent reference on transmission-line transformers. They are usually bifilar(4:1) or trifilar(9:1) wound, and the main coupling mechanism is that of a transmission line. They are effectively autotransformers and are For the HF band, the usually wound on a frequency-band specific ferrite core. They are quite broad- Philips 4C6 material is band achieving bandwidths of up to 15:1. The transmission line transformer (if excellent. connected correctly) also has a Balance to Unbalance conversion. If the input Gnd is reversed, NO Balun action.

3.3.4 LC Networks

Physical inductors and capacitors can also be used to resonate the antenna and then transform the impedance. One drawback is that inductors are lossy. Both “pi” and “T” networks are commmon. One popular match is known as the “L-match”, shown in fig 3.9. In the L-match, the LC transformer network is simply an extension of the resonating inductor for the antenna. Therefore popular when the antenna is short in terms of wavelength, eg mobile HF whips. LC networks do not have a Balun action, and are naturally narrow-band.

3.3.5 Resistive Networks

Resistive networks (pi and T) are also used, and are broadband, but very lossy. They are used only in extreme cases, and have no balun action. 3.3 Common baluns/balun transfomers 35

RRadiation

CAnt.Reactance

LResonate

LMatch CMatch

LMatch Figure 3.9: L-Match network.

3.3.6 Quarter wavelength transformer

Zin Z0 ZL

x = λ/4 quarter

Figure 3.10: Quarter wavelength transformer

For a transmission line of a quarter-wavelength long, recall that (section 1.1.3 on page 6): p Z0 = ZLZS

In matching one impedance to another, if a transmission line can be found with a Z0 of the geometric mean of the two impedances, a match can be made.

The match is fairly narrow-band, and requires a practically available Z0. At low frequencies the cable length may be inconvenient, at high frequencies a microstrip line is used.

Ordinarily there is no balun action. However, if a “series” connection of two coaxial cables is used as the transformer, there is balun action since both lines are shielded. A “series” connection uses two pieces of coax, with the two braids connected at both ends, but the two centres are used as the transmission line. The characteristic impedance of the series connected line is half the character- istic impedance of the coax. 36 Matching Techniques

3.3.7 Stub matching

This type of matching can be used to obtain a narrow-band match to any impedance. The general arrangement is shown in figure 3.11.

d1

Z0 Z0 ZL

Z0

Stub

d2

Figure 3.11: Single Matching Stub

Both the lengths d1 and d2 need to be adjustable which can present some diffi- If the cable is coax, vary- culty. This type of matching is most clearly illustrated on the Smith Chart and ing d1 can prove im- is shown in figure 3.12. possible. The Double- Due to the fact that stubs are always connected in parallel it is easier to use ad- Stub match uses fixed dis- mittance, rather than impedance notation where parallel components are simply tances from the load, and added. To achieve this the normalized antenna impedance is plotted (P ) and only varies the length of 1 rotated halfway round the chart to obtain the corresponding admittance value the stubs (P2).

Once an admittance has been converted by length d1 to lie on the circle indicated (Rn = 1) a match can always be achieved by simply adding or subtracting the necessary amount of susceptance (inverse of reactance) using the open or short circuited stub. This value can be found by starting at the outer circle on the chart at the right hand end (admittance of infinity) for short circuited case or the left hand side (admittance of zero) for open circuit stub and finding the required value of susceptance and reading of the length of the required stub (d2) from the Smith Chart.

3.3.8 Shifting the Feedpoint

Changes can be made to the feed geometry in order to effect a narrow-band match. The T and Gamma matches are well suited to feed folded dipole anten- nas. They are illustrated in fig 3.13 These two types of matching sections are very similar in behaviour. It is difficult to analyze their performance but the following general trends referring to the T match have been observed: 3.3 Common baluns/balun transfomers 37

d1

Towards 1 Generator 1.8 0.5 2

P2

P3 0.2

0 0.2 0.5 1 2 5 ∞

0.2

d P1 2

0.5 2 Towards 1 1.8 Load

SmithStub

Figure 3.12: Smith Chart of Single Stub Tuner Arrangement

` A

B

Balanced

C

TGamma Unbalanced

Figure 3.13: T and Gamma Matching Sections

• The input impedance increases as A is made larger. The desired value of 38 Matching Techniques

the real part does not always occur as the imaginary part crosses zero, causing some problems. • The maximum impedance values occur in the region where A is 40 to 60 percent of the total antenna length. • Higher values of input impedance can be realized when the antenna is shortened to cancel the inductive reactance of the matching section. Some flexibility can be obtained by inserting variable capacitors in series with these sections at the feed. As a first approximation values of about 7 pF per meter of wavelength can be used.

3.3.9 Ferrite loading

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

3.4 Impedance versus Gain Bandwidth

The goal of matching is to make the antenna present an acceptable impedance to the transmitter over the bandwidth that is necessary. However it is senseless to achieve a broad bandwidth match if the antenna does not maintain its gain over that bandwidth. Hence we can speak of an impedance bandwidth and a gain bandwidth. Impedance bandwidth is usually taken as the 2:1 VSWR level, whereas gain bandwidth is usually taken as 3dB down on the peak gain. As an example, a SuperNEC run on a 10-director Yagi-Uda array produces a gain bandwith as shown in 3.14, and an impedance bandwith (normalizing to 200Ω) as shown in fig 3.15. It is clear that some work needs to be done to improve the match so that the impedance bandwidth falls in line with the gain bandwidth. 3.4 Impedance versus Gain Bandwidth 39

Gain at θ=90, φ=0 15 Line1

10

5 Gain (dBi) 0

−5

−10 200 250 300 350 Freq (MHz)

Figure 3.14: Gain bandwidth

5 Line1

4.5

4

3.5

3 VSWR

2.5

2

1.5

1 200 250 300 350 Freq (MHz)

Figure 3.15: Impedance bandwidth 40 Matching Techniques Chapter 4

Simple Linear Antennas

HE SIMPLEST FORM OF ANTENNA , and the form in most common Tuse 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- derstand as a large number of other antennas are analyzed using the equations practical, it is often eas- that are quite easily developed for this antenna. Examples of these are the iest to consider an ideal short dipole, loop antennas, travelling wave antennas and some arrays. The case and thence deduce radiation pattern of any wire construction on which the currents are known can results for more complex also be readily determined by considering the structure to consist of connected examples. ideal dipoles and adding the pattern contribution due to each to form the full pattern. Many computer analysis codes rely on this approach. The ideal dipole is defined as a linear wire antenna with length very small with respect to the wavelength and a uniform current distribution. For convenience, this antenna is positioned at the centre of the coordinate system and aligned in the z-direction, as shown in figure 4.1.

4.1.1 Fields

Using Maxwell’s equations and the simplicity of this geometry it is very easy to find the fields due to the constant current I (Kraus & Fleisch 1999, pg278). When such an analysis is performed it is found that the far field of the antenna has an E-field in the θ direction, Eθ, and a φ-directed H-field, Hφ only. The expression for the E-field will be given but the H-field can clearly be found by “Ohm’s Law of Free Space” as discussed in section 1.3.3.

60πI ` E = 0 jej(2πf−βr) sin θ (4.1) θ λr

41 42 Simple Linear Antennas

z

` I0 Hφ

Eθ y

IdealCoord x

Figure 4.1: The Ideal Dipole in Relation to the Coordinate System

There are a number of important points relating to this expression. Considering it factor by factor:

• 60π is the constant or magnitude

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

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

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

• sin θ is the pattern factor. This is the only factor indicating variation with respect to the spherical coordinate system angles. Since none of the factors contain a φ-term this antenna has constant pattern characteristics in the Another way of putting azimuth direction. The resulting pattern has the familiar “doughnut” this is that the antenna shape as illustrated in figure 4.2 has omnidirectional az- imuthal coverage. The form of equation (4.1) is common to the expressions for most antenna field distributions. Such distributions are always a function of excitation, geometry in terms of wavelength and θ and φ angles. The relative pattern of the antenna can be drawn using only the sin θ term and regarding the rest as a normalizing factor. Where absolute field strengths are required the total equation should clearly be used. 4.1 The Ideal Dipole 43

z

θ

y

dipole Doughnut

Figure 4.2: Pattern of an Ideal Dipole Antenna

4.1.2 Radiation resistance

The radiation resistance of the antenna can be found once the field distribution is known. Using circuit concepts, the radiation resistance Rr is given by:

2Pt Rr = 2 Ω (4.2) I0

The total power transmitted Pt is found by integrating (adding) the power The factor of two is in- density over a surface surrounding the antenna. Clearly if the power densities troduced as result of the in all directions have been accounted for, the total power is found. The power fact that I0 is the peak density in any direction can be found using the expression discussed before: current and not the RMS value. E2 P = d 2(120π)

Performing this integration, an expression for total power radiated is obtained and using 4.2 the radiation resistance is found as: µ ¶ ` 2 R = 80π2 Ω (4.3) r λ

This value is clearly always small since the ratio of antenna length to wavelength (`/λ) was assumed to be small (≤0.1) at the outset of the analysis.

4.1.3 Directivity

The directivity of the ideal dipole is calculated by assuming an input power of 1 W to the antenna. Since the reference used is always the isotrope, the power 1 that it would radiate, given the same input power is simply Pd (isotrope) = 4πr2 . 44 Simple Linear Antennas

q 2 The current to an ideal dipole with 1 W input power is given by I0 = . Rr Using (4.3) for Rr in the expression above results in: s 2 I = (4.4) 0 80π2(`/λ)2

Substituting (4.4) into (4.1) the E-field can be found in the maximum direction ◦ (θ = 90 ). The power density in this direction, Pd (ideal dipole) is found by the relationship: E2 P = d 2(120π) (60π)2 2 `2 = (λr)2 80π2 (`/λ)2 2(377)

The directivity by definition is the ratio, which becomes:

P D = d (ideal dipole) = 1.5(= 1.76dBi) Pd (isotrope)

4.1.4 Concept of current moment

An important concept which allows the use of the results achieved for the ideal dipole above to other antennas is that of current moment. By inspection of (4.1) it is clear that the E-field is proportional to the product of the length of the antenna and the current (assumed constant over the whole antenna). The current moment M for an ideal dipole is therefore the area under the current distribution:

M = I0`

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

E ∝ M P ∝ M 2

4.2 The Short Dipole

The short dipole antenna is a practically realizable antenna which is assumed to have a triangular current distribution when shorter than about a tenth of a wavelength, as shown in figure 4.3. Since it can be shown that the current distribution on thin linear radiators is sinusoidal, the two small parts of a sinusoid starting at either tip of the short dipole is well approximated by two straight lines and hence a triangular distribution. 4.2 The Short Dipole 45

Iin

`

ShortDip

Figure 4.3: Current Distribution on a Short Dipole Antenna

4.2.1 Fields

The current moment of the short dipole in terms of the feedpoint current Iin is: I ` M = in 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.

30πI ` E = in sin θ λr

4.2.2 Radiation resistance

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

P (ideal dipole) P (short dipole) = t t 4

2 since Pt = I R the radiation resistance of the short dipole would be a quarter of that of the ideal dipole µ ¶ ` 2 R (short dipole) = 20π2 Ω r λ

4.2.3 Reactance

The reactance of a short dipole is always capacitive and usually quite large The reactance of these and is not as easily calculated as the radiation resistance. Reactance values short antennas is a very can be measured for a specific antenna—and tables (King & Harrison 1969) strong function of the an- are available for different thickness antennas. The equivalent circuit of a short tenna thickness and (ob- dipole antenna can be given as in figure 4.4 viously) length. 46 Simple Linear Antennas

C R0

Rr TxLn Short Dipole ShortCct

Figure 4.4: The equivalent circuit of a short dipole antenna

The R0 value indicated in figure 4.4 refers to the loss resistance and should be included when that value is significant in relation to the radiation resistance Rr. This antenna thus presents a serious problem when power has to be delivered to it. The capacitive reactance (X = −1/2πfC) is typically a few hundred ohms which is a large mismatch condition. Matching is usually accomplished by placing an inductor in series with the feed line which has a positive reac- tance (X = 2πfL) that is equal in magnitude to the capacitive reactance thus resonating the antenna, as shown in figure 4.5.

C L R0

Rr L/2 L/2 TxLn Short Dipole Tuning

Figure 4.5: Tuning out dipole capacitive reactance with series inductance

This is an improvement but a few “catch-22” problems still exist which explains the inherent difficulty in transferring power to small antennas: • The coil will have some loss resistance which is very often large compared to radiation resistance (which is often a fraction of an ohm) resulting in very low efficiency. • To decrease coil losses the Q of the inductor should be increased but this causes a reduced operating bandwidth and a more sensitive antenna, also increasing the circulating currents and hence the voltages associated with them. • If the decrease in bandwidth can be tolerated, the resultant real (resonant) impedance would approximate the very low radiation resistance and this still presents a matching problem.

4.2.4 Directivity

It is clear from the sin θ factor in the E-field expression that the shape of the pattern is exactly the same as that of the ideal dipole. The directivity (gain) of a short dipole is therefore equal to the gain of the ideal dipole:

D (short dipole) = 1.5 4.3 The Short Monopole 47

4.3 The Short Monopole

h

I0 ` Ground Plane

ShortMono

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- terms of image theory. The antenna/image combination has the same radiation plied (and this is true pattern as the short dipole. The two major differences between the two are: of any antenna/image combination) the ground- • the monopole current moment is half that of the dipole plane behaviour can be • the monopole radiates no power in the lower hemisphere—for the same deduced from that of the input power as the dipole, the monopole radiates twice as much power free space equivalent. into the upper hemisphere. The power radiated is halved and the radiation resistance is half that of a short dipole when expressed in terms of `. For monopoles, the length of the antenna above the ground h = `/2 is clearly more relevant than ` and in terms of this the radiation resistance is: µ ¶ h 2 R = 40π2 r λ

All the power is radiated in the upper hemisphere which implies double power density in all directions in comparison to short—or ideal dipoles with the same power input. The directivity of this antenna would thus also be double that of the previous two antennas: As usual, increased gain is at the expense of de- D (short monopole) = 2 (1.5) = 3 creased gain elsewhere— under the ground plane in this case! 4.3.1 Input impedance

It was shown above that the radiation resistance of the short monopole is half that of the equivalent short dipole. The same applies to the capacitive reactance of the antenna. 48 Simple Linear Antennas

4.4 The Half Wave Dipole

λ/2

I0

HalfDip

Figure 4.7: A Half wave dipole and its assumed current distribution

Although the derivation will not be performed here, the fields from a half wave dipole with an assumed sinusoidal current distribution as shown in figure 4.7 can also be found by considering the antenna to be made up of small ideal It is interesting to note dipoles. The only difference in this case is that the phase of the current can not that the current distribu- be assumed to be constant and that the path lengths to a distant point P can tion must be known be- differ from the different locations on the antenna. fore the various parame- In the above cases, the current distributions were assumed to be sinusoidal mak- ters of an antenna can be ing analysis possible. This assumption is quite valid for thin linear radiators as determined. was shown by Schelkunoff (1941) and others. For more complex structures (and thick dipoles) the current distribution may be more difficult to determine. Com- putational techniques such as the Method of Moments, embodied in SuperNEC, are therefore primarily concerned with the determination of the current on the antenna wires. Once this is known it is a relatively straightforward task to calculate impedance and radiation pattern of the antenna.

4.4.1 Radiation pattern

Using the sinusoidal current assumption, the magnitude of the electric field distribution around the dipole can be determined as (noting that `/λ = λ/2): ¡ ¢ 60I cos π cos θ E = · 2 r sin θ

4.4.2 Directivity

The pattern of this antenna relative to that of a short dipole is shown in figure 4.8

The directivity of this antenna is clearly not much larger than that of the short dipole. The accurate value is:

D (half wave dipole) = 1.64

this is equivalent to 2.16 dBi (relative to isotropic). 4.4 The Half Wave Dipole 49

θ = 0◦ (Dipole axis) Isotrope Short Dipole

90◦

Half Wave Dipole

HalfPat

Figure 4.8: Half wave dipole, short dipole and isotrope patterns

It is immediately clear that there is not a large difference between the gain of the half wave dipole and that of the short dipole. This initially does not make sense since a short dipole can be very much smaller than a dipole and hence cheaper and more practical. The primary reason for the popularity of the half wave dipole is its large and resonant input impedance—which was the problem with the short dipole. Similarly, the directivity of a quarter wave monopole—which is the image theory equivalent of a half wave dipole—can be found as:

D (quarter wave monopole) = 2.16 + 3 = 5.16 dBi

The notation dBi is quite important and has been assumed until now. Very often antenna gain and directivity is quoted relative to a half wave dipole since this is a physically realizable antenna unlike the isotrope. The gain can thus be directly measured by comparing the signal strength received from a half wave dipole to that of the test antenna. When gain is quoted relative to a dipole it should be clearly stated and often this is done by using the notation dBd It is always important to (decibels relative to dipole). The conversion between the two is evident: ascertain which of these two references are used dBi = dBd + 2.16 when gain is specified or quoted since many sources do not distin- guish between the two— 4.4.3 Input impedance not an ignorable differ- ence! By analysis, the input impedance for thin half wave dipoles is:

Zin = 73 + j43 Ω

This antenna is thus slightly longer than the length required for resonance. When a thin antenna is shortened by about 2% resonance can be obtained. Figure 4.9 indicates the shortening required for various length to diameter ratios. As before, the quarter wave monopole has half the input impedance of the half wave dipole.

Zin (quarter wave monopole) = 36.5 + j21 Ω 50 Simple Linear Antennas

1

0.99

0.98

0.97

0.96

0.95

0.94 Multiplying factor, K 0.93

0.92

0.91

0.9 101 102 103 104

Ratio of half-wave length to diameter Short

Figure 4.9: Shortening factors for different thickness half wave dipoles

The relatively large values of radiation resistance of these antennas makes for easy transfer of power and virtually lossless antennas when good conductors are The impedance band- used. Efficiencies are typically 99% or higher and losses can thus be neglected. width of thin dipole This may be untrue in cases of extremely thin wires or high frequencies (>1000 antennas as defined by MHz). the VSWR 2 : 1 limi- tation is typically 5% of the centre frequency. 4.5 The Folded Dipole For thicker antennas (small length to diameter ratios) this bandwidth It is very seldom that folded dipoles of other values than half wave length (or can be larger. slightly less to achieve resonance) are used. The term folded dipole would thus be used to denote such an antenna unless otherwise stated. A typical Folded Dipole is shown in figure 4.10. Folded dipoles are often used instead of normal

λ/2

s ¿ λ

FD

Figure 4.10: Half Wave Folded Dipole Antenna

dipoles for the following reasons: • Mechanically easier to manipulate and more sturdy 4.5 The Folded Dipole 51

• Larger bandwidth than normal dipoles • Larger input resistance than a normal dipole • Can offer a direct DC path to ground —for lightning protection. The element centre op- posite the feed can be This antenna’s characteristics are again easily understood using the current “shorted” to the boom moment technique. Clearly the currents in each arm are sinusoidally distributed since this is a zero volt- as in a half wave dipole and are in the same direction. The current moment of age point this antenna is thus double that of the normal dipole. This implies:

Rin = 4(70) = 280 Ω

D (folded dipole) = D (half wave dipole) = 2.16dBi

E (folded dipole) = 2E (half wave dipole) Folded dipoles typically have an impedance bandwidth (defined by the VSWR 2 : 1 limit) of 10 to 12%. This increase relative to an half-wave dipole can be explained by noting that the antenna can be considered to be a superposition of a normal dipole and a short circuit transmission line. When the frequency is lowered the dipole exhibits a capacitive reactance whereas the transmission line starts to be inductive. These two factors initially cancel each other causing the increase in bandwidth. A comparison of the VSWR bandwidths of an ideally matched dipole and folded dipole is shown in figure 4.11.

5

4.5

4

3.5

Dipole 3 Folded Dipole VSWR

2.5

2

1.5

1 dipbw 200 250 300 350 400 450 500 freq (MHz)

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

A very interesting fact about these antennas is that the input resistance can be increased or decreased by making the two elements of different diameters as shown in figure 4.12. 52 Simple Linear Antennas

d

2r1 2r2

FDThick

Figure 4.12: Impedance transformation using different thickness elements

The impedance transformation ratio is

2 Rin(folded dipole) = (1 + a) Rin(half wave dipole)

log(d/r ) where a = 1 log(d/r2)

The impedance can also be manipulated by using more than two elements, as shown in figure 4.13. The impedance step-up ratio under these conditions (where

λ/2

FDMany

Figure 4.13: Triply Folded Dipoles

n is the number of elements) is:

2 Rin = n Rin(half wave dipole)

4.6 Dipoles Above a Ground Plane

So far only monopole antennas which are dependent on the ground for their operation have been discussed. It is interesting to observe the changes that occur for horizontal dipoles above a ground plane as shown in figure 4.14. A Since image theory is few general cases will be considered here to indicate the effects qualitatively. used to determine ground plane effects, it implies that a dipole above ground can be considered to be a two element array (see section 5.1 on page 57). 4.7 Mutual Impedance 53

Direct-wave path

θ S1

Reflected-wave path

2h θ Reflecting Surface

P1

Path Difference = 2h sin θ S2 ReflRay

Figure 4.14: A horizontal dipole above a ground plane

There are are two rays with different path lengths to point P and constructive and destructive interference will take place—depending on the path and hence phase difference. When constructive interference occurs, the two waves will add in phase and double the E-field result as compared to the free space pattern. In power density terms this would be an increase of 4 times, or 6 dB! This apparent “free” gain obtained by placing an antenna above a ground plane is clearly at the expense of reduced gain at other angles where the interference would be destructive, resulting in a null. However, with intelligent use it can be put to great effect. One of the main disadvantages of antennas close to the ground is that the current in the antenna and that in the image are clearly in opposite directions. At grazing angles they will always be out of phase—and cause a null. This Grazing angles are those can be a severe problem for ground-to-ground communications and the only that are close to the real remedy is to mount the antenna sufficiently high so that this effect can be ground plane neglected.

4.7 Mutual Impedance

When two dipoles (or in fact any two antennas) are close enough to each other to cause appreciable currents to flow on the one antenna as result of radiation by the other they are said to be mutually coupled. The radiation patterns of such an intentional or accidental combination can easily be determined by adding the field contributions from each vectorially. Another important consequence is that the input impedance of the antennas is often altered considerably. The same statement clearly applies to antennas close to a reflecting plane since interaction with the image results. This effect is graphically illustrated by con- 54 Simple Linear Antennas

λ/2

d

2Dip

Figure 4.15: Two dipoles in eschelon

sidering two parallel half wave dipoles side-by-side. Each of these antennas has a self impedance which is the impedance seen without the other antenna present. For the two dipoles in this example, their self impedances are called Z11 and Z22. Due to the interaction between the two dipoles their input impedances in the configuration shown is different. This can be calculated by defining the mutual impedance between the two dipoles, Z12 as the ratio of the voltage at the terminals of the second antenna V2 as a result of a terminal current applied to the first dipole I1. That is:

V2 Z12 = Ω I1 Curves of the mutual impedance for the geometry shown in figure 4.15 is given in figure 4.16. The following equations can be written to find the currents on the antennas and hence their impedance (Balanis 1982, sec 8.6,pg 412):

V1 = I1Z11 + I2Z12

V2 = I1Z12 + I2Z22

Usually some voltage is assumed as an excitation (V1 & V2) for the two antennas and thus using the above two equations, the resultant currents I1 and I2 are found. The input impedance to dipole 1 would then be conventionally defined as: V1 Zin = I1 This value will generally be different to the free space self impedance of the antenna Z11. 4.7 Mutual Impedance 55

80

60

Mutual Resistance 40 Mutual Reactance

20

Impedance (Ohms) 0

-20

-40 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Spacing in terms of wavelength ddvalre

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

This theory can be extended to include further dipoles, where the equations are then 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 analyze Yagi and log periodic arrays. As was said before: once the current arrangement is a para- distribution on an antenna is known all other parameters can be calculated— sitic element like those not only the input impedance as was shown above. found in Yagi antennas, the equations still apply From the curves in figure 4.16, it is clear that the mutual impedance is large and V is simply set to when antennas are close together and decreases as they are moved further 2 zero. apart—as one would expect. It is clear that the coupling becomes quite small for 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 bodies which 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 choosing the optimum separation. It also indicates the importance of considering the Remember that the sepa- effects of surroundings on antenna performance. ration in terms of wave- length is important, one For cases where the antenna is close to a reflecting plane, the separation is should not be fooled by clearly twice the distance from the surface if these curves are to be applied to physical dimensions. such cases. The effect on the input resistance of a half-wave dipole above a ground plane is shown in figure 4.17. Again a feel for the distances where coupling becomes significant is obtained from this curve. Note that radiation resistance drops to zero when a dipole is brought very close to a ground plane. Seen from a different point of view this makes sense—since at close separation the currents in the antenna and the image are oppositely directed as in a transmission line and thus no nett radiation results. 56 Simple Linear Antennas

100

90

80

70

60 R(Ω) 50

40

30

20

10

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RinMut Height above Ground, λ

Figure 4.17: Change in resistance of a half wave dipole due to coupling to its image 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 less sensitive to coupling considerations.

30

25

20 R21 X21 15

Ω 10

5

0

-5

-10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Spacing, λ

Figure 4.18: Mutual impedance between two collinear dipoles Chapter 5

Arrays and Reflector Antennas

5.1 Array Theory

UCH OF ANTENNA THEORY consists of correctly adding field contri- Mbutions 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 and phase must occur.

5.1.1 Isotropic arrays

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

θ 1 2 θ = 0◦ d/2 d/2

d cos θ TwoIso

Figure 5.1: Two Isotropic point sources, separated by d

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

jψ/2 −jψ/2 E = E2e + E1e where ψ = βd cos θ = (2πd/λ) cos θ is the phase-angle difference between the fields 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, ³π ´ E = E cos cos θ 0 2 which is shown in figure 5.2.

h ³π ´i E = E cos cos θ 0 2

1 2 θ = 0◦

d = λ/2 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 the multiplication of the element field pattern and the array field pattern. h ³π ´i E = E cos cos θ 0 2

E = k sin α α

1 2 θ = 0◦

h ³π ´i E = E cos cos θ × k sin α 0 2

PatMult d = λ/2

Figure 5.3: Pattern multiplication.

Two short dipoles placed in eschelon λ/2 apart. The element pattern is k sin α, a figure¡ of eight¢ perpendicular to the dipoles. From the above, the array factor π is cos 2 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 the Note that in this case, the overlap is very small. element and array pat- tern to have strength to- gether. 5.1.3 Binomial arrays

If we take the two element isotropic array above, the (normalized) pattern is ³π ´ E = cos cos θ 2

If we place another identical array one λ/2 away, we get a three element array with relative current magnitudes of 1:2:1. By applying pattern multiplication, the pattern of this array is ³π ´ E = cos2 cos θ 2

h ³π ´i E = E cos cos θ 0 2

h ³π ´i E = E cos6 cos θ 0 2

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 source magnitudes given by Pascal’s triangle, the pattern of which is shown in fig 5.4. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Clearly, the pattern multiplies each time, and we get that the pattern of an array of n sources is: ³π ´ E = cosn−1 cos θ 2

This array has no minor lobes, but its directivity is less than that of an array of the same size with equal amplitude sources. In general, most arrays fall in between these two extremes (binomial/uniform). 60 Arrays and Reflector Antennas

5.1.4 Uniform arrays

θ = 90◦

2πd ψ = cos θ λ θ ψ

θ = 0 1 2 3 4 5 d d d d 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 given by: ³ ´ jψ j2ψ j3ψ j(n−1)ψ E = E0 1 + e + e + e + ··· + e (5.1)

where ψ = βd cos θ + δ, δ being the progressive phase difference between the sources. (Phase reference is source 1) Since (5.1) is an infinite Multiplying (5.1) by ejψ and subtracting (5.1) from the result yields: series, we adopt the usual jψ jnψ geometric series method. (1 − e ) = E0(1 − e ) ie: µ ¶ 1 − ejnψ E = E 0 1 − ejψ

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

sin(nψ/2) E = (5.2) sin(ψ/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 field are of interest—broadside and end-fire arrays. Broadside (as in a naval galleon :-) fires its maximum at θ = 90◦ as shown in fig 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 fed in-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

90 Radiation Pattern (Elevation) 10 120 60 10 dBi 120 60 8 0 6 150 30 −10 4 −20 2 −30 180 0 180 0

210 330

240 300

−120 −60 270

(a) “Ordinary” dB scale (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 5.7 which doesn’t look anything like the above until you plot on a linear scale shown next to it. Hence the usefulness of the dB scale. Endfire has its maximum at θ = 0◦, hence ψ = 0 = βd cos 0◦ + δ, hence for max endfire field, 2π δ = −βd = − d λ As an example, if the sources a spaced a quarter wavelength apart, 2π λ π δ = − · = − = −90◦ λ 4 2 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 phase shift of π. Again, note the lobing. 62 Arrays and Reflector Antennas

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 the proviso that sin(ψ/2) cannot also be zero!) We are interested only in the first null, and this occurs at: nψ/2 = ±π, or 2π ψ = ± n (= βd cos θ0 + δ), where θ0 is the angle of the first null. Hence the first null occurs at ·µ ¶ ¸ 2π λ θ = cos−1 ± − δ 0 n 2πd

For the Broadside case, we are interested in the beamwidth at θ = 90◦, hence we use the complementary angle γ = 90 − θ. Recall that for broadside, δ = 0, so that the first broadside null is given by: µ ¶ λ γ = sin−1 ± Broadside 0 nd

If the array is large, nd À λ and the argument to the arcsin is small, (for small angles θ ≈ sin θ): 1 1 γ = = 0 nd/λ L/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: 2 114.6◦ BWFN = 2γ ≈ [rad] = 0 L/λ L/λ

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

57.3◦ HPBW = L/λ

2π For the Endfire case, recall that δ = − λ d, hence the first null occurs at · ¸ λ θ = cos−1 ± + 1 Endfire 0 nd 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 µ ¶ θ λ 1 − 2 sin2 0 = ± + 1 2 nd

Hence µ ¶ r θ λ sin 0 = ∓ ≈ θ /2 2 2nd 0

As before, using L ≈ nd, the first null angle is: s 2 θ = 0 L/λ s s 2 2 BWFN = 2θ = 2 [rad] = 114.6◦ 0 L/λ L/λ and hence s 2 HPBW = 57.3◦ L/λ

Interferometer

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

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

Hence the E field pattern is given as

E = 2 sin θ cos ψ/2 64 Arrays and Reflector Antennas

where ψ = βd cos θ µ ¶ πd = 2 sin θ cos cos θ λ

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 1 γ = sin−1 0 2d/λ

Assuming the distance between the two elements is large, we can approximate: 57.3◦ BWFN = 2γ = 0 d/λ

Hence the further apart Across continent, or over large separations, the time-base is critical to ensure in- they are, the smaller the phase feeding. Radio-astronomers have even used the yearly cycle of the earth beam is between first around the sun (with very accurate timing) to resolve distant astronomical nulls, and the better the artefacts. A SuperNEC simulation of the 10λ separation interferometer is resolution. shown in fig 5.11.

5.2 Dipole Arrays

From array theory, it is clear that a collinear array of dipoles is a uniform broadside 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 array should always be in phase. These arrays are useful due to the omnidirectional 5.2 Dipole Arrays 65

Radiation Pattern (Elevation) 0 10 dBi

0

−10 60 −60 −20

−30

° Structure: φ=90

120 −120

180

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 still produce 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 a wavelength spacing, and is shown in fig 5.12. The quarter wave phasing section reverses the current phase by 180◦ and ensures that all three dipoles are in phase. The gain of this arrangement has been measured at 4 to 5 dBi. Often, the phase reversal section is a loading coil, which has a sufficient amount of 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 wavelength spacing. The value of 0.7 wavelength spacing results in one wavelength electrical length (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 mounted in a fibreglass radome to give them mechanical rigidity. Unfortunately there is 66 Arrays and Reflector Antennas

λ/4 λ/2

λ/2

λ/2

Franklin

Figure 5.12: The Franklin array

Coax Inner Conductor Coax Outer (Braid) Dipole 4 Additional Sleeve 0.7λ

Break in Coax Braid. Dipole 3

λ/2 Dipole 2

Dipole 1

CoaxArray

Figure 5.13: A Series Fed Four Element Collinear Array

no DC path to earth, and lightning protection is a problem with this array in o t f ia o d to s Teerthe cell net- South African conditions.Theyear work first rolled out, in June or so, these ar- rays were used all around Gauteng, 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 of dipoles in free space. The effects of a mast will distort the azimuth pattern somewhat however. Folded dipoles are usually mounted about a quarter of a wavelength from the mast 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, mounted λ/4 away from a mast of λ/8 diameter is shown in figure 5.14. The gain in front 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 mounted symmetrically on opposite sides of the mast.

Radiation Pattern (Azimuth) 10 dBi 120 60 0

−10

−20

−30

180 0

240 300

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-Uda array. This indicates the popularity of this antenna and not without reason. A Yagi antenna is probably the simplest, cheapest and most effective medium gain antenna available, and is found on every rooftop! The Yagi-Uda antenna, shown in fig 5.15 on the next page was invented in 1926 by Dr. H Yagi and Shintaro Uda (Yagi 1928). Since then numerous reports The Yagi antenna can on this antenna have appeared in the literature—one of the most noteworthy is be analyzed using mu- the study done by Viezbicke (1976) of the U. S. National Bureau of Standards tual coupling theory pre- (NBS). He optimized the gain of a number of Yagi antennas and investigated the sented earlier. 68 Arrays and Reflector Antennas

effect of the boom and element length on the performance of the antenna. The NBS experimental findings were later confirmed during an excellent series of articles 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 Yagi Reflector

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; the other elements are not directly driven, but are parasitic, obtaining their current via mutual coupling. The spacing between elements is approximately a quarter wavelength. The reflector is slightly longer than required for resonance and is thus 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 direction and it behaves like a travelling wave or endfire array. As a general rule of thumb the gain is directly proportional to boom length for well designed Yagis. In other words, a 3 dB gain increase is obtained by doubling the boom length. The number of elements per se is not the determining factor. Generally, the gain bandwidth of Yagi antennas decreases with an increase in gain (length of Yagi). Also, for a given length Yagi antenna the bandwidth will decrease with increased gain and vice versa. The gain bandwidth of a Yagi is usually a function of the reflector length relative to director lengths. If a given design is therefore built with slightly longer reflector and shorter directors gain bandwidth will be larger with slightly lower maximum gain. The comments above apply to impedance bandwidth as well. Experience has ie the Yagi-Uda is typ- shown that the VSWR 2 : 1 bandwidth will correspond more or less to the 1.5 ically impedance band- dB gain bandwidth. width limited, not gain bandwith limited, see fig 3.14 and 3.15 on page 39. 5.3 Yagi-Uda array 69

5.3.2 Impedance and matching

The free space input impedance of the folded dipole is usually considerably reduced from 300Ω due to the mutual coupling to the parasitic elements. The mutual coupling also results in some reactive component being introduced to the input impedance. The reactive part can be eliminated by changing the overall length of the driven dipole to achieve resonance—the driven element length does not influence the antenna performance much and may be manipulated to achieve matching. The resultant resistive input impedance is usually around 200Ω and can be matched to a 50Ω line using a 4 : 1 balun transformer. This value must not be assumed automatically! Measurements or SuperNEC simulations will get the accurate impedance.

5.3.3 Design

The NBS data for Yagi design forms a good base line for design, but are based on unequal director lengths. The difference in gain between these and equal length director 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 specification for all elements results in much simpler antennas with equivalent performance as shown in the table below: Yagi Design Details (All dimensions in wavelengths) Boom length 0.4 0.8 1.2 2.2 3.2 4.2 Reflector 0.482 0.482 0.482 0.482 0.482 0.475 Reflector spacing 0.2 0.2 0.2 0.2 0.2 0.2 No of directors 1 3 4 10 15 13 Director 0.442 0.427 0.424 0.402 0.395 0.401 Director spacing 0.2 0.2 0.25 0.2 0.2 0.308 G(dBd) (Lawson) 7.1 9.2 10.2 12.25 13.4 14.2 Driven (SN) 0.426 0.421 0.417 0.423 0.435 0.434 Rin 8.6 20.4 19.0 43.4 55.6 44.5 Driven FD (SN) 0.389 0.382 0.378 0.382 0.396 0.396 Rin 34.1 76.1 72.1 158.3 202.9 166.2 SuperNEC 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 condition since it does not affect gain operation much. As an example, the SuperNEC determined resonant lengths and input impedances are shown in the above table. Driven (SN) refers to the SuperNEC derived dipole driven lengths, and Driven FD (SN) is the SuperNEC derived folded dipole driven lengths. All diameters are 0.008λ, and the folded dipole separation is 0.05λ. The gain as calculated by SuperNEC is shown in the last line, in dBi. As can be clearly seen, the Lawson gain figures are optimistic for the longer arrays.

Element correction

The element lengths shown above are for a diameter to wavelength ratio of 0.008. For different radius elements the same design data can be modified by 70 Arrays and Reflector Antennas

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

1.1

1.08

1.06 Reflector 1.04 Directors 1.02

1

0.98 Correction factor

0.96

0.94

0.92

0.9 10-3 10-2 10-1

Element diameter to wavelength ratio ElemFactor

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

When the elements are If the elements are mounted through a metal boom, as is often done for mechanical mounted on insulators— support, a further correction factor must be applied to get the correct length. at least an element radius This is due to the slight distortion of the field at the centre of the dipole due away from the boom— to the boom. It is viewed as some additional capacitance and will thus require this effect becomes negli- the length to be increased with increased boom diameter. The curve to do this gible. correction is given in figure 5.17.

Stacking and Spacing

Yagi antennas can be vertically stacked or horizontally spaced to give additional gain. As seen from section 2.4 on page 24, the criterion for determining the spacing is the effective aperture of the antenna. Since the beamwidths are different in the E and H planes, the optimum separation for stacking Yagi antennas is 1.6 wavelengths apart and for spacing side by side it is 2 wavelengths between antenna centres. Either of these two methods produce 2.5 dB gain relative to a single antenna. Clearly four yagis can be both stacked and spaced to produce 5 dB of extra gain. These techniques are attractive since the gain is increased without compromising bandwidth. The antennas clearly have to be driven in phase via a phasing harness 5.4 Log Periodic Dipole Array 71

0.03

0.025

0.02

0.015 Increase in Electrical Length

0.01

0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Diameter to wavelength ratio of Boom BoomFactor

Figure 5.17: Graph showing the length to be added to parasitic elements to compensate 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 obtained using the antenna image when mounted above a conducting plane. This is often possible when mounting antennas on metal roofs or for lower frequencies when the antenna is mounted above the earth.

5.4 Log Periodic Dipole Array

The LPDA, first proposed by Isbell (1960), is a truly frequency-independent antenna 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 HF to microwaves. Carrel (1961) developed a particularly straight-forward design procedure for these antennas which has ensured their success. Carrel disregarded the effects of the characteristic impedance of the antenna boom/transmission line and also the thickness of the elements. Peixeiro (1988) presents a more complete design technique catering for these defects, but which still incorporates a large number of the features introduced by Carrel. The general 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 radiation is toward this end. The lengths of the dipoles and their spacing are varied such that these dimensions bear a constant ratio to each other—regardless of the 72 Arrays and Reflector Antennas

1 2 n − 1 n n + 1 N d n Ln

2α ZT

Diameter Dn

LPDA

Figure 5.18: The log-periodic dipole array

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

L d τ = n+1 = n+1 (5.3) Ln dn

One of the characteristics It is apparent that these conditions cause the ends of the dipoles to trace out an of frequency independent angle, 2α. When the antenna is fed from the small end with a frequency that is antennas is that they can much too low for the short dipoles to resonate, these elements will absorb very be defined in terms of an- little power (hence they will radiate very little power too). The phase of the gles rather than linear di- current is mechanically changed by 180 degrees between these electrically short mensions. elements. The radiation any of these will produce will therefore be cancelled by the out-of-phase radiation of adjacent elements. Once a portion on the antenna has been reached where the dipoles are resonant and electrically further apart these dipoles will absorb most of the energy from the transmission line and radiate it. This part of the antenna is called the active region. If the frequency is increased, this active region will simply move towards the small end of the antenna. This explains the frequency independence of the antenna 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 the active element being shorter than resonance and therefore capacitive—and act as directors. Similarly, elements behind (towards the large end) act as reflectors— giving the antenna a endfire beam towards the small end. Referring again to figure 5.18 the space factor, σ, of the antenna is:

d σ = n (5.4) 2Ln 5.4 Log Periodic Dipole Array 73

The antenna must always be truncated at both ends which is determined by the minimum and maximum frequencies of operation. The truncation coefficients K1 and K2 are defined to determine the values of shortest and longest elements to ensure satisfactory performance:

L1 ≥ K1λmax (5.5)

LN ≤ K2λmin (5.6)

The number of dipoles can then be obtained using ln(L /L ) N = 1 + 1 N 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 improvement to the theory by Peixeiro—and both of the authors’ curves are reproduced in figure 5.19 for the case where Z0 = 100Ω ; Ln/Dn = 177

Improved Pexiero

Original Carrel

CarrPex

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 74 Arrays and Reflector Antennas

Figure 5.20: Characteristics for feeder impedances of 100, 250 and 400Ω and dipoles with L/D ratios of 177, 500 and 1000 5.4 Log Periodic Dipole Array 75

• Ln — the thickness of the elements. Dn 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 constant directivity D, input resistance R and truncation coefficient contours K1; K2 as shown in figure 5.20. • Pairs (τ, σ) are selected taking into account both directivity and input resistance specifications.

• Find the truncation coefficients, K1 and K2 from the same curves based on 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 be calculated using equations 5.3 and 5.4. If constraints are placed on some of the parameters, the procedure is naturally iterative in order to find the best compromise. The following comments may help 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 moderate influence on the LPDA parameters. Often the maximum boom length is one of the constraints imposed in practical examples. This value is not immediately evident from the various design constants but can be found using the following equation: 2σ(L − L ) boom length = 1 N (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—which is usually convenient and prevents the cable from affecting the radiation pattern. This connection has the additional advantage that it acts as a broadband balun as result of the hollow boom forming a sleeve around the outside of the coaxial cable.

1IEE Proceedings, Part H, Vol 135, No 2, pg 100 76 Arrays and Reflector Antennas

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 the simple concepts such as ideal dipoles, isotropic array theory and transmission line 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 square loop with the same area.

I I

d 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 loop will duplicate the performance of its circular equivalent as long as the areas of both are equal: µ ¶ d 2 s2 = π 2 5.5 Loop Antennas 77

This is true in general for different loop configurations such as triangular or multi-sided—as long as they contain some approximately circular area and are not “flat”. A “flat” or narrow rectangle, for instance, looks more like a short circuited 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 four ideal 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 dipoles 1 and 3 do not contribute to radiation in this plane, since they carry opposing currents 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 two element 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 two isotropic sources may be applied to this case. The separation is s and there is a 180◦ phase difference between the two resulting in the E field distribution: µ ¶ βs sin θ E(θ) = 2E sin 0 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 an ideal dipole and the E-field of this antenna is used: 60πIs E = 0 λr This gives the full E-field of the loop as

120π2IA E(θ) = sin θ rλ2 where A = s2, the area of the loop. The radiation pattern of the small loop is similar to that of the ideal and short dipoles 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 was outlined in Chapter 4. 320π4A2 R = Ω rad λ4 78 Arrays and Reflector Antennas

θ = 0◦

Eθ

Eφ

LoopDip

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 increase by N times and the radiation resistance is proportional to the current moment squared. The equation above for a multi-turn loop of N turns is:

320π4A2N 2 R = Ω rad λ4

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

2 Rrad (ferrite loop) = µeff Rrad (air loop)

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

2 L = µ0N A 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 and ideal dipoles. The pattern of such an array is the same as the pattern of the 5.6 Helical Antennas 79

d

` h

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 array factor—and is shown in figure 5.24. The pattern contains both vertical E-field components due to the dipoles; and horizontal components due to the loops. This results in circular polarization when these values are equal. Usually they are not equal resulting in predomi- nantly horizontal or vertical polarization. The ratio between the two types of polarizations is: E(vertical) `λ = E(horizontal) 2πA

If this value is large the antenna pattern is vertically polarized and if it is small the polarization is approximately horizontal. If the ratio is close to unity the polarization is circular. The pattern (intensities) are not affected however. The reason for the popularity of this antenna is due to the fact that much shorter length resonant linear antennas can be produced. This is due to the fact that the velocity of propagation along the helix axis is slow and the current and voltage will thus be in phase before the helix is a quarter wavelength. The reduction in resonant length, k, is given by the formula:

1 k = p 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:

2 Rrad = (25.3h/λ) where h is the height of the monopole helix. This results in slightly lower values of input resistance when compared to the 36Ω input resistance of the quarter wave monopole. 80 Arrays and Reflector Antennas

5.6.2 Axial mode

In 1946, J.D Kraus attended a physics lecture in which a helical structure was used to guide an electron beam in a traveling wave tube. He asked the lecturer about the possibility of using the helix to radiated electromagnetic wave into space, to which the answer was an emphatic NO! Nevertheless, Kraus went home and started to experiment with the structure. As he suspected (or was it to his amazement), the helix showed good promise as an antenna. When the diameter D and the spacing S are large fractions of a wavelength, the operation of the helical antenna changes considerably from the normal-mode behaviour, and it behaves as an endfire array of loops and the pattern therefore has a main beam in the axial direction as shown in figure 5.25.

D

S L α S C = πD

Helix Axis

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 C 0.8 ≤ ≤ 1.15 λ with a circumference of 1 near optimum. The spacing, S must be about λ/4 and 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, whose diameter 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 the helical turn are 180◦ out of phase, hence providing the E-field vector in that plane. Also, referring back to the Yagi-Uda array where the directors are about 0.2λ apart, in order to capacitively “suck” the wave forward, the turns are about 0.21 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 and published 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 a maximum gain given by: µ ¶ µ ¶ C 2 NS G = K G λ λ

where KG is the gain factor, originally 15, but later reduced to 12 by Kraus (1988, pg284) • The Half Power BeamWidth (HPBW) is given by: K λ3/2 HPBW = B√ C NS

where KB is the beam factor, which is about 52, derived from the standard approximation on beamwidths: 41000 G = HPθHPφ

Since the beam is generally circularly symmetric, HPθ =HPφ=HPBW: p 41000/K λ3/2 HPBW = √ G C NS p where 41000/KG = KB, the beam factor. • The input impedance is nearly resistive and is given by: µ ¶ C R = 140 Ω λ

• The beam is circularly polarized.

King and Wong design

King & Wong (1980) performed a study which involved varying the parameters of a uniform helix and measuring the electrical performance of the structure. They found that the expressions derived by Kraus tended to overestimate the performance of the antenna. Their results are summarized (empirically) as follows: µ ¶√ µ ¶ µ ¶√ C N+2−1 NS 0.8 tan(12.5) N/2 G = 8.3 λ λ tan α

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). The beamwidth factor, KB, is therefore between 61 and 70 (compared to 52). Please note: 82 Arrays and Reflector Antennas

C • The revised factors are valid for antennas with 0.8 ≤ λ ≤ 1.2. • King and Wong also note that the Kraus original factors depend on other design 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, C • The peak gain of a helix occurs when λ = 1.155 for N = 5; and for C λ = 1.07 for N = 35. 2 • Since the beam is circular, HPBWθ=HPBWφ=HPBW. The Gain-HPBW product was found to be significantly less than 41 000, and lies in the range of 18 000 to 31 500. They note that: 2 C – G × HPBW = 18 000 for N = 35 and 0.75 ≤ λ ≤ 1.1 2 C – G × HPBW = 31 000 for N = 5 and 0.75 ≤ λ ≤ 1.2 – The smaller the pitch, the larger the gain-beamwidth product. • The gain bandwidth of the helix is presented as:

µ ¶ √4 f 0.91 3 N H ≈ 1.07 fL G/Gpeak

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 3dB point). – Note that the bandwidth decreases as the axial length/ gain/ number of turns increases. – The bandwidth is approximately 42% for a helix of N = 5; and approximately 21% for N = 35. • The impedance bandwidth (2:1 VSWR) is typically 70%. The input impedance of the helix (with C/λ = 1) is about 140Ω, almost purely resistive. However, if the last quarter-turn of the helix is made parallel to the ground plane, it creates a quarter-wave transformer, which allows matching down to 50Ω. Since a frequency-selective device has now been introduced, the 70% impedance bandwidth drops to about 40%. This can be 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, the more 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 of elements 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 lossy dielectrics. At microwave frequencies, therefore, antennas are manufactured on low-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). The patch 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 ground plane. The patch is fed by coax through the dielectric, halfway along the longest edge.

The antenna functions as an array of slots. The long edges radiate as two vertical slot antennas. It is common to have large arrays of patches to achieve a high gain. Since this 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 given by: r c 2 w = 2fr ²r + 1

²r(eff) The effective dielectric constant (since w À h) is determined by (Balanis 1989): · ¸ ² + 1 ² − 1 h −1/2 ² = r + r 1 + 12 r(eff) 2 2 w

∆` The extension length, ∆`, is found using (Hammerstad 1975): ¡ ¢ ∆` (² + 0.3) w + 0.264 r(eff) h¡ ¢ = 0.412 w h (²r(eff) − 0.258) h + 0.8

Length The patch length, `, is then given by: 1 ` = √ √ − 2∆` 2fr ²r(eff) µ0²0 84 Arrays and Reflector Antennas

Actually, there are so many ways to feed a patch, and so many ways to construct it, that a comprehensive guide cannot be readily established. Essentially, it is required that a set of modes are resonant within it—eg with feedpoints roughly 90◦ in spatial separation, and with a 90◦ phase separation, circular polarization is 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 5.29 on the facing page Patches are simply asking to be arrayed, and fig 5.30 on page 86 shows a four edge-fed square patch array with its 3D pattern; the 2D pattern at max gain is shown in fig 5.31. 5.7 Patch antennas 85

20 3D Radiation Pattern 8.2

15 4.2

10 0.3

−3.7 5 Y X −7.6

0 −11.6

−5 −15.5

−19.5 −10

−23.4 |Gain | Tot −15

−20

20 35 15 30 10 25 5 20 15 0 Figure 5.28: 3D pattern of the square patch.10 −5 5 −10 0 −15 −5 −10 −20 −15

Radiation Pattern (Elevation) 0 10 dBi

0

−10 60 −60 −20

−30 ° Structure: φ=92

120 −120

180

Figure 5.29: 2D cut through the maximum gain of the square patch 86 Arrays and Reflector Antennas

30 Z

3D Radiation Pattern 10.4 20 5.3

0.2 10

−4.9

0 −10.1

−15.2

−10 −20.3

−25.4 −20 Y

−30.5 |Gain | Tot

−30

−30 −20 −10 0 10 −40 −30 20 −20 −10 30 0 Figure 5.30: Geometry and 3D10 pattern of a 4-square patch array. 40 20 30 40

Radiation Pattern (Elevation) 0 10 dBi

0

−10 60 −60 −20

−30 ° Structure: φ=92

120 −120

180

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 steerable beams (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 (911 requirement 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 intentional jammer. • Re-use of spectrum: Instead of an omnidirectional pattern for cellphone coverage, you could have multiple beams to track individual users, and use 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, shown in fig 5.32, which may provide differing amounts of power to each antenna, or equal power. This is possibly followed by a phase shifting section, so that the individual phases of the sources in the array can be changed.

φ

φ

φ Power Splitter φ

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 phase One of the simplest (in theory) phase shifter is a transmission line based binary shifters use transmission one as shown in fig 5.33, where various sections of TxLn are switched into the lines as above for the circuit. lower frequencies, but variable lumped elements For example a four element array, spaced λ/2 apart. If the beam needs to fire at higher frequencies. at 60◦, the E-field must be in phase at 60◦. Hence the incremental phase change between sources ψ must be zero at 60◦. Thus 2πd ψ = cos θ + δ (= 0 at θ = 60◦) λ

Hence we get that δ = −90◦ ie that the phase difference between each source Interestingly, if δ = +90◦, we get a Null at θ = 60◦. 88 Arrays and Reflector Antennas

1×

2×

4×

8×

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 the relative 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

Radiation Pattern (Elevation) 0 10 dBi Broadside ° 0 5 Downtilt ° 10 Downtilt −10 60 −60 −20

−30 DownTilt

120 −120

180

Figure 5.34: Electrically achieved DownTilt by progressive phasing.

careful look at the magnitudes in fig 5.34 will show a slight degradation in peak value (10.81;10.80;10.06) dBi respectively. This is due to the array pattern being multiplied by the element pattern which is a sinusoidal function of theta. The electrical downtilt is achieved by slightly lengthening the cables progressively in the 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 reasonable gain requires a very large number of elements. This implies a high frequency in order for the array to be of reasonable size. At radar frequencies (11GHz), this is 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 the system operates, resulting in a beamshift. Assuming that the exact frequency is not critical, one can get a fair degree of beamshift. Useful in RFID applications where the exact frequency used to excite the passive tags is not critical, but where 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 easily analysed by image theory. The distance between the dipole and plate affects the gain markedly. Kraus (1988, pg546) Supergain conditions hold at very close spacings, and a very impressive gain can be obtained, but at the expense of having a very low input resistance, (close coupling with a negative image), making the antenna difficult to feed. The supergain condition is also very narrowband. A tradeoff distance has to be obtained which provides a decent gain and a decent (feedable) input impedance. 90 Arrays and Reflector Antennas

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

Gain vs separation for a 0.75 by 0.75 plate 9

8.5

8

7.5

7

6.5 Gain (dBi) 6

5.5

5

4.5

4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Separation (Wavelengths)

Figure 5.36: Gain vs separation for a 0.75λ plate

Impedance vs separation for a 0.75 by 0.75 plate 120 Real part Imag Part

100 ) Ω 80

60

40 Real and Imaginary impedance (

20

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Separation (Wavelengths)

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 many shapes and sizes, but the most common form is where the “corner” is defined as being 90◦, and the reflector panels are simply made from vertical rods. At some 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 with each of the two panels being 0.7λ × 0.7λ, with the dipole separation, s, from the corner being 0.3λ. The SuperNEC predicted xy-plane radiation pattern of such a reflector is shown in fig 5.39 If the panels are made larger (1.2λ × 1.2λ, say) the gain improves slightly, but at the expense of the antenna being very large. I have not seen more than about 9dBi 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 feed is placed at the focal point of the parabola, all rays are reflected with the same efective path length, and are hence in phase after leaving the parabola. It is therefore possible to achieve a very high gain with a dish, limited mainly by the constructional deformities of the actual parabola. 92 Arrays and Reflector Antennas

Radiation Pattern (Azimuth) 10 dBi 120 60 0

−10

−20

−30

° Structure: θ=90 180 0

240 300

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 more than 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. Most DSTV antennas A better idea is shown in fig 5.40(b), where the feed is offset. Simply put, the use offset feeds. It shape still follows a parabola, and the rays are still in phase, but the feed is not therefore appears that obscured. they “point” very low on the horizon—but stand Fig 5.40(c) shows a Cassegrain feed. The subreflector, which is not at the fo- fig 5.40 so that the dish cal point of the parabola, is a hyperbola. The combination of hyperbola and in (b) is almost vertical parabola ensures in-phase behaviour of the dish. Note that there is a blockage and see where it “points” of the area behind the sub-reflector again, resulting in a lowered Aperture ef- to! ficiency. The Cassegrain is mainly used in large dishes, as the feed system is more 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, which finally results in a uniform wavefront. 94 Arrays and Reflector Antennas Appendix A

Smith Chart

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

95 The Smith Chart Calculator

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RADIALLY SCALED PARAMETERS SWR TOWARD LOAD —> <— TOWARD GENERATOR Rtn loss [dB] ∞100 40 20 10 5 4 3 2.5 2 1.8 1.6 1.4 1.2 1.1 1 15 10 7 5 4 3 2 1 Atten. [dB] dBS SW Loss coeff Rfl coeff,P ∞ 40 30 20 15 10 8 6 5 4 3 2 1 1 1 1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 ∞

Rfl coeff E/I 0 1 2 3 4 5 6 7 8 9 10 12 14 20 30 ∞ 0 0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15∞ Rfl loss [dB] 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.01 0 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 peak (const P)

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CENTRE 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Trans coeff,E/I

ORIGIN References

ARRL (1988), The ARRL Antenna Book, 14th edn, The American Radio Relay League. Balanis, C. A. (1982), Antenna Theory: Analysis and Design, 2nd edn, John Wiley & 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 uniform helical antennas”, IEEE Transactions on Antennas & Propagation AP- 7, 291. King, R. W. P. & Harrison, C. W. (1969), Antennas and Waves - a Modern Approach, MIT Press, Mass, chapter Appendix 4 - Tables of Impedance and 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, fifth edn, 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, Part H 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 Radio Relay League, Newington. Smith, P. H. (1939), “Transmission line calculator”, Electronics 12, 29. Viezbicke, P. P. (1976), Yagi antenna design, Tech. doc. NBS-TN-688, National Bureau 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, 70 medium, 11 Array transmission lines, 3 binomial, 59 coaxial line, 6 collinear, 64 microstrip line, 6 dipole, 64 one-wire line, 5 Franklin, 65 twisted pair, 5 isotropic, 57 two-wire line, 4 LPDA, see LPDA Circular polarization, 24 pattern multiplication, 58 Coaxial characteristic impedance, 6 series fed collinear, 65 Collinear dipole array, 64 theory, 57 series fed, 65 uniform, 60 Corner reflector, 89 broadside, 60 Corporate feed, 88 endfire, 61 Current moment, 44 Yagi-Uda, see Yagi-Uda array Curvature of earth, 13 Autotransformer, 34 Axial ratio, 24 dB, 19 Axial-mode helix, 80 vs. linear scale, 61 dBd, 49 Balun Decibels, see dB bazooka, 32 half-wave, 33 Depth of penetration, see Skin depth sleeve, 32 Design transmission line transformer, 33 LPDA, 75 Baluns, 30 Yagi-Uda, 69 Bandwidth Dipole impedance, 50, 51 array, 64 vs Q, 46 folded, 50 Bazooka balun, 32 ground interaction, 52 Beam steering, 88 half wavelength, 48 Beamwidth ideal, 41 between first nulls (BWFN), 21 short, 44 half power(HPBW), 21 Directivity, 19 uniform array, 62 folded dipole, 51 Binomial array, 59 half wave dipole, 48 Broadside array, 60 ideal dipole, 43 quarter wave monopole, 49 Cassegrain parabolic, 91 short dipole, 46 Characteristic impedance short monopole, 47 free space, 11 Directivity estimation, 22 measuring, 7 Downtilt, 88

99 100 INDEX

Earth curvature, 13 Input impedance Effect of mast on pattern, 67 folded dipole, 51 Effective aperture, 24 half wave dipole, 49 Efficiency, 19 quarter wave monopole, 49 Electrical downtilt, 88 short monopole, 47 Element correction, 69 small loop, 77 Elliptical polarization, 24 Interferometer, 63 Endfire array, 61 Inverse square law, 18 Equation Isotropic array, 57 free-space wavelength, 2, 10 Isotropic source, 18 power loss ito VSWR, 31 L-match, 34 Ferrite cores, 34 LC networks, 34 Fields Line of sight, 13 ideal dipole, 41 Linear polarization, 23 short dipole, 45 Linear vs. dB scale, 61 Flat reflector antennas, 89 Log periodic dipole array, see LPDA Folded dipole, 50 Loop antenna, 76 Franklin array, 65 input impedance, 77 Free-space small, 76 characteristic impedance, 11 LPDA, 71 Ohm’s law, 12 design procedure, 75 permeability, 12 feeding, 75 permittivity, 12 Mast mounted dipoles, 67 speed of propagation, 12 Matching, 8, 30 wavelength, 2, 10 tuning coil, 46 Frequency independance, 71 Yagi-Uda, 69 Frequency scanning arrays, 89 Maximum power transfer, 8 Front-to-back ratio, 21 Measuring characteristic impedance, 7 Gain, 19 gain, 49 bandwidth, 37 velocity factor, 8 estimation, 22 Medium, 3 isotropic, 18 Method of Moments, see SuperNEC measuring, 49 Microstrip line characteristic impedance, Half wavelength dipole, 48 6 Half-power beamwidth (HPBW), 21 Mutual impedance, 53 Half-wave balun, 33 Normal-mode helix, 78 Helix, 78–82 axial-mode, 80 Offset feed parabolic, 91 King and Wong design, 81 Ohm’s law in free-space, 12 normal-mode, 78 One-wire line characteristic impedance, original Kraus design, 81 5

Ideal dipole, 41 Parabolic reflector, 91 Impedance bandwidth, 37 Patch antennas, 82–84 folded dipole, 51 Pattern multiplication, 58 half wave dipole, 50 Permeability of free space, 12 helix, 82 Permittivity of free space, 12 INDEX 101

Polarization, 23–24 Steradians, 22 axial ratio, 24 Stub matching, 35 circular, 24 SuperNEC, 20, 22, 48, 61, 64, 67, elliptical, 24 84, 90 horizontal, 23 Surge impedance, see Characteris- linear, 23 tic impedance loss, 23 vertical, 23 Table frequency allocations & desig- Q, 46 nations, 10 Quarter wave transformer, 7, 34 microwave bands, 11 skin depths, 16 Radiation pattern, 20 typical spectrum usage, 11 backlobe, 20 Yagi-Uda array design, 69 effect of mast on, 67 Transmission line half wave dipole, 48 additional loss ito VSWR, 31 principle planes, 20 behaviour, 1 sidelobes, 20 characteristic impedance, 3 small loop, 77 equation, 7 Radiation resistance transformer balun, 33 ideal dipole, 43 Tuning coil, 46 short dipole, 45 Twisted pair characteristic impedance, short monopole, 47 5 Radio horizon, 13 Two-wire line characteristic impedance, Reactance 4 short dipole, 45 Reflection coefficient, 8 Uniform array, 60 Rule of thumb beamwidth, 62 coupling separation, 55 Velocity factor, 7, 12 transmission line behaviour, 1 measuring, 8 Yagi gain vs boomlength, 68 plastics, 12 Series connected coax, 35 Velocity of propagation, see Speed of propagation Series fed collinear, 65 VSWR, 8, 10 Short dipole, 44 Short monopole, 47 Waveguide, 3 Skin depth, 16 Wavelength Sleeve balun, 32 in free-space, 2, 10 Small loop, 76 Smith chart, 9 Yagi-Uda download, 9 matching, 69 Solid angle, 22 stacking, 70 Speed of propagation Yagi-Uda array, 67 free-space, 12 design, 69 medium, 12 sound, 12 Z0, see Characteristic impedance transmission line, 4 Spherical coordinate system, 17 Square degrees, 22 Stacking, 70