Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Antennas in Practice
EM fundamentals and antenna selection
Alan Robert Clark
Andr´e P C Fourie
Version 1.4, December 23, 2...
ii
Titles in this series:
Antennas in Practice: EM fundamentals and antenna selection
(ISBN 0-620-27619-3)
Wireless Techno...
Contents
1 Electromagnetics 1
1.1 Transmission line theory . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Impedance...
iv CONTENTS
3.1.2 Balanced and unbalanced lines—a definition . . . . . . . . 30
3.2 Impedance Transformation . . . . . . . ...
CONTENTS v
5.3.2 Impedance and matching . . . . . . . . . . . . . . . . . . 69
5.3.3 Design . . . . . . . . . . . . . . . ...
vi CONTENTS
List of Figures
1.1 A transmission line connects a generator to a load. . . . . . . . . 1
1.2 2-wire; Coax; µstrip; wavegu...
viii LIST OF FIGURES
3.15 Impedance bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 The Ideal Dipole in R...
LIST OF FIGURES ix
5.24 The normal-mode helix antenna and its radiation pattern . . . . 79
5.25 Axial Mode Helix Antenna a...
Preface
This book “Antennas in Practice” has been in existence in a multitude of forms
since about 1989. It has been run a...
xii Preface
Chapter 1
Electromagnetics
1.1 Transmission line theory
TRANSMISSION lines connect generators to loads as shown in fig 1.1....
2 Electromagnetics
imation:
λ(m) =
300
f(MHz)
Thus, the use of Transmission line theory as opposed to circuit theory appro...
1.1 Transmission line theory 3
text we assume a Uniform transmission line, ie one whose properties do not
change along the...
4 Electromagnetics
The concept of characteristic impedance has nothing to do with loss, but it
characterises a transmissio...
1.1 Transmission line theory 5
One conductor over ground plane
One conductor at a height, h, above a (theoretically) infini...
6 Electromagnetics
Wadell (1991, pg53–65) presents formulae for various offset coaxial, and strip-
in-coax situations.
Micr...
1.1 Transmission line theory 7
line. Under sinusoidal, steady state conditions a transmission line transforms
the load imp...
8 Electromagnetics
The velocity factor of a line can be measured by using the quarter-wave trans-
former principle—if the ...
1.2 The Smith Chart 9
1.2 The Smith Chart
For antenna analysis the most convenient way to represent and manipulate
impedan...
10 Electromagnetics
The Voltage Standing Wave Ratio, VSWR, can be read directly off the Chart
as it is simply the intercept...
1.3 Field Theory 11
• Achievable frequencies are limited by electronics (for oscillators)
• Above 3000 GHz, we get into op...
12 Electromagnetics
Just as the characteristic impedance depends on the geometry and the properties
of the medium, so the ...
1.3 Field Theory 13
x
z
y
Ex
Hy
E(φ)
EMWave
Figure 1.6: Electromagnetic Wave in free-space
In general, EM waves of a frequ...
14 Electromagnetics
The classic means of long distance communication is by means of reflecting
waves from the ionosphere. T...
1.3 Field Theory 15
Clearly there will now be a difference in the path lengths to some distant point
P. At certain elevatio...
16 Electromagnetics
1.3.5 EM waves in a conductor
If we solve Maxwell’s equations for EM wave propagation in a conductor, ...
Chapter 2
Antenna Fundamentals
2.1 Directivity, Gain and Pattern
THE USUAL COORDINATE SYSTEM in 3-dimensional space, the c...
18 Antenna Fundamentals
A radian is defined by saying that there are 2π radians in a circle. As a result,
the arclength rθ ...
2.1 Directivity, Gain and Pattern 19
where the power transmitted by both antennas is the same.
Decibels
Directivity is usu...
20 Antenna Fundamentals
For antennas with negligible losses these two values are approximately equal,
but many antennas wh...
2.1 Directivity, Gain and Pattern 21
Figure 2.3: Three-dimensional pattern of a 10 director Yagi-Uda array
300240
180
120 ...
22 Antenna Fundamentals
−180 −120 −60 0 60 120 180
−20
−10
0
10
15
φ
Gain(dBi)
Radiation Pattern (Azimuth)
(a) xy plane
−9...
2.3 Polarization 23
Va Energy
Flow
Ib
Ia Energy
Flow
Vb
Va causes Ib ; Vb causes Ia
If Vb is made = Va, then Ia will be = ...
24 Antenna Fundamentals
in the vertically polarized case, only ey is present and can be expressed as:
ey = e2 sin(ωt − βz)...
2.5 Free-space link equation and system calulations 25
r
Ap = πr2
Ae ApertureEff
Figure 2.8: Aperture efficiency of a parabo...
26 Antenna Fundamentals
Tx
Gt
Pt
Rx
Gr
Pr
r
LinkEqn
Figure 2.11: Point-to-point link parameters
centrates the transmitted ...
2.5 Free-space link equation and system calulations 27
satellite receiver is only 8pW, how much power is required to be tr...
28 Antenna Fundamentals
Chapter 3
Matching Techniques
3.1 Balun action
3.1.1 Unbalance and its effect
AT RF FREQUENCIES recall that from section 1....
30 Matching Techniques
The job of the BALance to UNbalance converter (Balun) is to prevent the
currents flowing on the outs...
3.2 Impedance Transformation 31
are usually more severe. To illustrate this point the following equation gives the
power r...
32 Matching Techniques
10-1
100
101
10-1 100 101
Additional Loss in dB when Matched
AdditionallossindB
SWR=2
SWR=4
SWR=7
S...
3.3 Common baluns/balun transfomers 33
This is most often accomplished before impedance transformation. Some match-
ing te...
34 Matching Techniques
dipoles. Naturally, the current is halved, hence the impedance (voltage over
current) has a factor ...
3.3 Common baluns/balun transfomers 35
CMatch
LMatch
LResonate
CAnt.Reactance
RRadiation
LMatch
Figure 3.9: L-Match networ...
36 Matching Techniques
3.3.7 Stub matching
This type of matching can be used to obtain a narrow-band match to any
impedanc...
3.3 Common baluns/balun transfomers 37
0 0.2 0.5 1 2 5
0.2
0.5
0.2
0.5
1
2
1
2
∞
Towards
Generator
Towards
Load
P2
P1
d1
d...
38 Matching Techniques
the real part does not always occur as the imaginary part crosses zero,
causing some problems.
• Th...
3.4 Impedance versus Gain Bandwidth 39
200 250 300 350
−10
−5
0
5
10
15
Freq (MHz)
Gain(dBi)
Gain at θ=90, φ=0
Line1
Figur...
40 Matching Techniques
Chapter 4
Simple Linear Antennas
THE SIMPLEST FORM OF ANTENNA , and the form in most common
use is the linear antenna—the ...
42 Simple Linear Antennas
I0
x
y
z
Eθ
Hφ
IdealCoord
Figure 4.1: The Ideal Dipole in Relation to the Coordinate System
Ther...
4.1 The Ideal Dipole 43
θ
dipole
y
z
Doughnut
Figure 4.2: Pattern of an Ideal Dipole Antenna
4.1.2 Radiation resistance
Th...
44 Simple Linear Antennas
The current to an ideal dipole with 1 W input power is given by I0 = 2
Rr
.
Using (4.3) for Rr i...
4.2 The Short Dipole 45
Iin
ShortDip
Figure 4.3: Current Distribution on a Short Dipole Antenna
4.2.1 Fields
The current m...
46 Simple Linear Antennas
C R0
Rr
TxLn Short Dipole
ShortCct
Figure 4.4: The equivalent circuit of a short dipole antenna
...
4.3 The Short Monopole 47
4.3 The Short Monopole
I0
h
ShortMono
Ground Plane
Figure 4.6: Short monopole antenna
When a gro...
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 d...
4.4 The Half Wave Dipole 49
θ = 0◦
(Dipole axis)
Isotrope
Short Dipole
Half Wave Dipole
HalfPat
90◦
Figure 4.8: Half wave ...
50 Simple Linear Antennas
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
101 102 103 104
Multiplyingfactor,K
Ratio of ...
4.5 The Folded Dipole 51
• Larger bandwidth than normal dipoles
• Larger input resistance than a normal dipole
• Can offer ...
52 Simple Linear Antennas
d
2r1 2r2
FDThick
Figure 4.12: Impedance transformation using different thickness elements
The im...
4.7 Mutual Impedance 53
2h
P1
S1
S2
θ
θ
Direct-wave path
Reflected-wave path
Reflecting Surface
ReflRay
Path Difference = 2h ...
54 Simple Linear Antennas
λ/2
d
2Dip
Figure 4.15: Two dipoles in eschelon
sidering two parallel half wave dipoles side-by-...
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Antennas in practice - EM fundamentals and antenna selection
Upcoming SlideShare
Loading in …5
×

of

Antennas in practice - EM fundamentals and antenna selection Slide 1 Antennas in practice - EM fundamentals and antenna selection Slide 2 Antennas in practice - EM fundamentals and antenna selection Slide 3 Antennas in practice - EM fundamentals and antenna selection Slide 4 Antennas in practice - EM fundamentals and antenna selection Slide 5 Antennas in practice - EM fundamentals and antenna selection Slide 6 Antennas in practice - EM fundamentals and antenna selection Slide 7 Antennas in practice - EM fundamentals and antenna selection Slide 8 Antennas in practice - EM fundamentals and antenna selection Slide 9 Antennas in practice - EM fundamentals and antenna selection Slide 10 Antennas in practice - EM fundamentals and antenna selection Slide 11 Antennas in practice - EM fundamentals and antenna selection Slide 12 Antennas in practice - EM fundamentals and antenna selection Slide 13 Antennas in practice - EM fundamentals and antenna selection Slide 14 Antennas in practice - EM fundamentals and antenna selection Slide 15 Antennas in practice - EM fundamentals and antenna selection Slide 16 Antennas in practice - EM fundamentals and antenna selection Slide 17 Antennas in practice - EM fundamentals and antenna selection Slide 18 Antennas in practice - EM fundamentals and antenna selection Slide 19 Antennas in practice - EM fundamentals and antenna selection Slide 20 Antennas in practice - EM fundamentals and antenna selection Slide 21 Antennas in practice - EM fundamentals and antenna selection Slide 22 Antennas in practice - EM fundamentals and antenna selection Slide 23 Antennas in practice - EM fundamentals and antenna selection Slide 24 Antennas in practice - EM fundamentals and antenna selection Slide 25 Antennas in practice - EM fundamentals and antenna selection Slide 26 Antennas in practice - EM fundamentals and antenna selection Slide 27 Antennas in practice - EM fundamentals and antenna selection Slide 28 Antennas in practice - EM fundamentals and antenna selection Slide 29 Antennas in practice - EM fundamentals and antenna selection Slide 30 Antennas in practice - EM fundamentals and antenna selection Slide 31 Antennas in practice - EM fundamentals and antenna selection Slide 32 Antennas in practice - EM fundamentals and antenna selection Slide 33 Antennas in practice - EM fundamentals and antenna selection Slide 34 Antennas in practice - EM fundamentals and antenna selection Slide 35 Antennas in practice - EM fundamentals and antenna selection Slide 36 Antennas in practice - EM fundamentals and antenna selection Slide 37 Antennas in practice - EM fundamentals and antenna selection Slide 38 Antennas in practice - EM fundamentals and antenna selection Slide 39 Antennas in practice - EM fundamentals and antenna selection Slide 40 Antennas in practice - EM fundamentals and antenna selection Slide 41 Antennas in practice - EM fundamentals and antenna selection Slide 42 Antennas in practice - EM fundamentals and antenna selection Slide 43 Antennas in practice - EM fundamentals and antenna selection Slide 44 Antennas in practice - EM fundamentals and antenna selection Slide 45 Antennas in practice - EM fundamentals and antenna selection Slide 46 Antennas in practice - EM fundamentals and antenna selection Slide 47 Antennas in practice - EM fundamentals and antenna selection Slide 48 Antennas in practice - EM fundamentals and antenna selection Slide 49 Antennas in practice - EM fundamentals and antenna selection Slide 50 Antennas in practice - EM fundamentals and antenna selection Slide 51 Antennas in practice - EM fundamentals and antenna selection Slide 52 Antennas in practice - EM fundamentals and antenna selection Slide 53 Antennas in practice - EM fundamentals and antenna selection Slide 54 Antennas in practice - EM fundamentals and antenna selection Slide 55 Antennas in practice - EM fundamentals and antenna selection Slide 56 Antennas in practice - EM fundamentals and antenna selection Slide 57 Antennas in practice - EM fundamentals and antenna selection Slide 58 Antennas in practice - EM fundamentals and antenna selection Slide 59 Antennas in practice - EM fundamentals and antenna selection Slide 60 Antennas in practice - EM fundamentals and antenna selection Slide 61 Antennas in practice - EM fundamentals and antenna selection Slide 62 Antennas in practice - EM fundamentals and antenna selection Slide 63 Antennas in practice - EM fundamentals and antenna selection Slide 64 Antennas in practice - EM fundamentals and antenna selection Slide 65 Antennas in practice - EM fundamentals and antenna selection Slide 66 Antennas in practice - EM fundamentals and antenna selection Slide 67 Antennas in practice - EM fundamentals and antenna selection Slide 68 Antennas in practice - EM fundamentals and antenna selection Slide 69 Antennas in practice - EM fundamentals and antenna selection Slide 70 Antennas in practice - EM fundamentals and antenna selection Slide 71 Antennas in practice - EM fundamentals and antenna selection Slide 72 Antennas in practice - EM fundamentals and antenna selection Slide 73 Antennas in practice - EM fundamentals and antenna selection Slide 74 Antennas in practice - EM fundamentals and antenna selection Slide 75 Antennas in practice - EM fundamentals and antenna selection Slide 76 Antennas in practice - EM fundamentals and antenna selection Slide 77 Antennas in practice - EM fundamentals and antenna selection Slide 78 Antennas in practice - EM fundamentals and antenna selection Slide 79 Antennas in practice - EM fundamentals and antenna selection Slide 80 Antennas in practice - EM fundamentals and antenna selection Slide 81 Antennas in practice - EM fundamentals and antenna selection Slide 82 Antennas in practice - EM fundamentals and antenna selection Slide 83 Antennas in practice - EM fundamentals and antenna selection Slide 84 Antennas in practice - EM fundamentals and antenna selection Slide 85 Antennas in practice - EM fundamentals and antenna selection Slide 86 Antennas in practice - EM fundamentals and antenna selection Slide 87 Antennas in practice - EM fundamentals and antenna selection Slide 88 Antennas in practice - EM fundamentals and antenna selection Slide 89 Antennas in practice - EM fundamentals and antenna selection Slide 90 Antennas in practice - EM fundamentals and antenna selection Slide 91 Antennas in practice - EM fundamentals and antenna selection Slide 92 Antennas in practice - EM fundamentals and antenna selection Slide 93 Antennas in practice - EM fundamentals and antenna selection Slide 94 Antennas in practice - EM fundamentals and antenna selection Slide 95 Antennas in practice - EM fundamentals and antenna selection Slide 96 Antennas in practice - EM fundamentals and antenna selection Slide 97 Antennas in practice - EM fundamentals and antenna selection Slide 98 Antennas in practice - EM fundamentals and antenna selection Slide 99 Antennas in practice - EM fundamentals and antenna selection Slide 100 Antennas in practice - EM fundamentals and antenna selection Slide 101 Antennas in practice - EM fundamentals and antenna selection Slide 102 Antennas in practice - EM fundamentals and antenna selection Slide 103 Antennas in practice - EM fundamentals and antenna selection Slide 104 Antennas in practice - EM fundamentals and antenna selection Slide 105 Antennas in practice - EM fundamentals and antenna selection Slide 106 Antennas in practice - EM fundamentals and antenna selection Slide 107 Antennas in practice - EM fundamentals and antenna selection Slide 108 Antennas in practice - EM fundamentals and antenna selection Slide 109 Antennas in practice - EM fundamentals and antenna selection Slide 110 Antennas in practice - EM fundamentals and antenna selection Slide 111 Antennas in practice - EM fundamentals and antenna selection Slide 112 Antennas in practice - EM fundamentals and antenna selection Slide 113
Upcoming SlideShare
What to Upload to SlideShare
Next
Download to read offline and view in fullscreen.

54 Likes

Share

Download to read offline

Antennas in practice - EM fundamentals and antenna selection

Download to read offline

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 revamped on several occasions, mainly reflecting changing typesetting 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 commonplace 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 dependent 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 SimulationWorkshop, 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.

Alan Robert Clark
Andre P C Fourie
Version 1.4, December 23, 2002

Related Books

Free with a 30 day trial from Scribd

See all

Related Audiobooks

Free with a 30 day trial from Scribd

See all

Antennas in practice - EM fundamentals and antenna selection

  1. 1. Antennas in Practice EM fundamentals and antenna selection Alan Robert Clark Andr´e P C Fourie Version 1.4, December 23, 2002
  2. 2. 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
  3. 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
  4. 4. 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
  5. 5. 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
  6. 6. vi CONTENTS
  7. 7. List of Figures 1.1 A transmission line connects a generator to a load. . . . . . . . . 1 1.2 2-wire; Coax; µstrip; waveguide; fibre; RF; 50Hz! . . . . . . . . . 2 1.3 A “Lumpy” model of the TxLn, discretizing the distributed pa- rameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Twisted-Pair geometry . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Impedance values plotted on the Smith Chart . . . . . . . . . . . 9 1.6 Electromagnetic Wave in free-space . . . . . . . . . . . . . . . . . 13 1.7 Radio Horizon due to earth curvature. . . . . . . . . . . . . . . . 13 1.8 Geometry of Interference between Direct Path and Reflected Waves 14 1.9 Interference pattern field strength contours for h/λ = 1.44 . . . . 15 2.1 Standard coordinate system. . . . . . . . . . . . . . . . . . . . . . 17 2.2 concept of a solid angle . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Three-dimensional pattern of a 10 director Yagi-Uda array . . . . 21 2.4 xy plane cut of the 10 director Yagi-Uda . . . . . . . . . . . . . . 21 2.5 Rectanglar radiation patterns of the 10-director Yagi-Uda in the two principle planes . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Reciprocity concepts . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Polarization possibilities—wave out of page . . . . . . . . . . . . 23 2.8 Aperture efficiency of a parabolic dish . . . . . . . . . . . . . . . 25 2.9 Effective aperture of dipole physical aperture . . . . . . . . . 25 2.10 Apertures must not overlap! . . . . . . . . . . . . . . . . . . . . . 25 2.11 Point-to-point link parameters . . . . . . . . . . . . . . . . . . . 26 3.1 The reason for unbalanced currents . . . . . . . . . . . . . . . . . 29 3.2 Balanced and Unbalanced Transmission lines . . . . . . . . . . . 30 3.3 Power loss in dB versus VSWR on the line. . . . . . . . . . . . . 31 3.4 Increase in Line Loss because of High VSWR . . . . . . . . . . . 32 3.5 VSWR at Input to Transmission Line versus VSWR at Antenna 32 3.6 The Bazooka or Sleeve balun. . . . . . . . . . . . . . . . . . . . . 33 3.7 A HalfWave 4:1 transformer balun. . . . . . . . . . . . . . . . . . 33 3.8 Transmission-line transformer. . . . . . . . . . . . . . . . . . . . 34 3.9 L-Match network. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.10 Quarter wavelength transformer . . . . . . . . . . . . . . . . . . . 35 3.11 Single Matching Stub . . . . . . . . . . . . . . . . . . . . . . . . 36 3.12 Smith Chart of Single Stub Tuner Arrangement . . . . . . . . . . 37 3.13 T and Gamma Matching Sections . . . . . . . . . . . . . . . . . . 37 3.14 Gain bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 vii
  8. 8. 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
  9. 9. 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
  10. 10. 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
  11. 11. xii Preface
  12. 12. Chapter 1 Electromagnetics 1.1 Transmission line theory TRANSMISSION lines connect generators to loads as shown in fig 1.1. In the RF world, in the transmitting case, this is viewed as connecting the transmitter to the antenna, and in the receiving case as connecting the antenna to the receiver. VGen RGen Generator Transmission Line ZLoad Load TxLn Figure 1.1: A transmission line connects a generator to a load. From a standard circuits analysis perspective, the transmission line simply con- sists of connecting two parts of the circuit, and does not change anything. By Kirchhoff’s Voltage law, there is no change in voltage or current along the length of the “connection”. As a rule of thumb, as soon as the “connection” length between the parts of the circuit exceeds a fiftieth of a wavelength (λ/50) then ordinary circuit theory breaks down, and the “connection” becomes a transmission line. The length of the line in terms of the wavelength of the operating frequency adds a finite time-lag between the start and end of the line, and this causes the voltage along the line to change in terms of magnitude and phase as a function of the distance down the line. Naturally, the current also changes as a function of the distance, hence the ratio of the voltage and current (impedance) also changes. Recall that the free-space wavelength, λ, is simply given by the usefull approx- 1
  13. 13. 2 Electromagnetics imation: λ(m) = 300 f(MHz) Thus, the use of Transmission line theory as opposed to circuit theory approxi- mations becomes important for transmission lines longer than: 120km for 50Hz power; 2km for ordinary telephone connections; 600mm for 10Mb/s Ethernet; 120mm for a PC Board bus track at 50MHz; 6mm for an on-chip interconnect in a 1GHz PIII. d=2a D r 0 eff d D r = eff w h r 0 eff a b r = eff η2 η1 765kV ! txlnexa Figure 1.2: 2-wire; Coax; µstrip; waveguide; fibre; RF; 50Hz! Transmission line theory is thus a superset of classical circuit theory. In this
  14. 14. 1.1 Transmission line theory 3 text we assume a Uniform transmission line, ie one whose properties do not change along the length of the line. A transmission line is anything that transfers power from one point to another, whether picowatts or gigawatts, as shown in fig 1.2. 1.1.1 Impedance A resistance resists current flow. An impedance impedes current flow. ie They are the same concept, though generally, a resistance is purely real, whereas an impedance has a reactive (energy storage) component. Thus, an impedance Z: Z = R + jX where X = jωL or X = 1/(jωC); ω being the radian frequency: ω = 2πf (f in It is important to note that the reactive part of the impedance is a func- tion of frequency. Hertz), L and C being the inductance and capacitance. 1.1.2 Characteristic impedance & Velocity of propagation A transmission line can be viewed simply as a waveguide. It simply provides the boundary conditions that shape the electric and magnetic fields in the medium between the conductors. Of course, there is a direct relationship to the voltage and current on the line too. In circuit terms, the distributed capacitance and inductance etc of the line can be collected in lumped models. The model of the transmission line is then an infinite set of these circuit sections. Many versions of the model exist, but I shall use the standard (Kraus & Fleisch 1999) model, as shown in fig 1.3. CG LR d Input Output Lumped Figure 1.3: A “Lumpy” model of the TxLn, discretizing the distributed param- eters. For a given length of transmission line, we can hence lump the series resistance R [Ω/m] and inductance, L [H/m] together; and the shunt conductance G [0/m] and capacitance C [F/m]. These terms are per-unit length, and do not change from one section of the transmission line to another (uniform transmission line). Hence we define a characteristic impedance, Z0, as the ratio of the series to the shunt components; in the lossless (or high frequency) case, R and G are negligeable: Z0 = R + jωL G + jωC = L C Lossless
  15. 15. 4 Electromagnetics The concept of characteristic impedance has nothing to do with loss, but it characterises a transmission line. Further, since the inductance and capacitance between the two conductors largely depends on the geometry and the medium properties, the characteristic impedance is largely dependant on these factors. The Characteristic Impedance of a line is also referred to as the Surge Impedance. If a voltage is applied at one end of a long transmission line, a current will flow regardless of the load at the other end (the surge hasn’t got to the load yet!). The ratio of the voltage and the current of the surge is the surge, or characteristic, impedance of the transmission line. The velocity with which the wave moves down the (lossless) transmission line is also dependant on the material properties of the medium: v = 1 √ LC m/s Since these factors depend on both the geometry and the material properties, there is a need in many cases to define an effective permittivity, r(eff), since the fields pass partially through air and through solid dielectric. If the entire volume around the conductors is solid dielectric then r(eff) = r. In the case of a low- loss foam dielectric, or similar dielectric-and-air combinations, an approximation must be used. Two-wire line Conceptually, the simplest type of transmission line. Popular examples are the 300Ω “FM Tape” and the standard unshielded twisted pair (UTP) of networking fame. The relationship of both conductors to ground is the same, making it a balanced line. Referring to fig 1.2 for the geometrical definitions, the rigorous case is given by Wadell (1991, pg66): Z0 = µ0µr π2 0 r(eff) cosh−1 D d Kraus (1992, pg158,499) derives a similar result using: cosh−1 x = ln(x + x2 − 1) Usually, the spacing D is much greater than the radius a = d/2 and a simplified equation is used: Z0 = µ0µr π2 0 r(eff) ln D a D a Since magnetic materials are never used, the simplified equation is usually sim- plified further to the useful: µ0 0 ≈ 120π Z0 = 120 √ r(eff) ln D a Typical values of Z0 range from 200 to 800Ω.
  16. 16. 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. Z0 = 60 √ r(eff) ln 2h a Twisted Pair r d D twisted Figure 1.4: Twisted-Pair geometry Twisted Pair is a very common form of transmission line, as all external noise generated is common to both wires. A differential receiver stage then gets rid of most of the induced noise. Common in telephony, Ethernet networks etc. Again, the field lines cross through air and dielectric, making a closed-form solution difficult. Wadell (1991, pg 68) presents Lefferson (1971)’s empirical closed-form solution: Z0 = 120 √ r(eff) cosh−1 D d where: θ is normally between 20 and 45◦ . Less than 20◦ , the twists are too loose for uniformity, greater than 50◦ breaks the wires! r(eff) = 1 + q( r − 1) q = 0.25 + 0.0004θ2 θ = tan−1 (TπD) T = Twists per length (same units as D). Coaxial line Coaxial line has the advantage of shielding the inner conductor, and hence has less radiation (and reception). Since the relationship of the two conductors to ground is different, coax is an unbalanced line. Also all field lines pass through the dielectric, hence r(eff) = r. It is also common to refer to the ratio of the radii a = d/2; b = D/2: Z0 = 60 √ r ln D d or: Z0 = 60 √ r ln b a
  17. 17. 6 Electromagnetics Wadell (1991, pg53–65) presents formulae for various offset coaxial, and strip- in-coax situations. Microstrip Line Very popular since this is simply printed on a circuit board. Generally used at the higher frequencies. One of the problems here is that the fringing flux plays a very important role in establishing the exact Z0 value. Unfortunately the fringing flux flows through a combination of air and dielectric, hence again the need for an effective . Gardiol (1984) has developed expressions for the cases where w/h ≤ 1 (large percentage of fringing flux) and for cases where w/h > 1 (less percentage of fringing flux). For w/h ≤ 1: r(eff) ≈ 1 2 ( r + 1) + 1 2 ( r − 1) 1 + 12 h w −1/2 + 0.04 1 − w h 2 Z0 ≈ 60 √ r(eff) ln 8 h w + w 4h For w/h > 1: r(eff) ≈ 1 2 ( r + 1) + 1 2 ( r − 1) 1 + 12 h w −1/2 Z0 ≈ 120π √ r(eff) w h + 1.393 + 0.667 ln w h + 1.444 Kraus & Fleisch (1999, pg 132) also has an approximate formula (which in earlier editions he qualified as being applicable for w ≥ 2h, but this is often not the case in microstrip lines of interest: Z0 ≈ 120π √ r[(w/h) + 2] Slotline A slotline has no groundplane, but uses adjacent tracks as the transmission line. The fields are distributed in an elliptical waveguide within and without the dielectric. Closed-form expressions for Z0 span several pages of Gupta, Garg & Bahl (1979, 213–216). 1.1.3 Impedance transformation Since the voltage and current change down the transmission line, react with the load and reflect back to the source, it appears as though the impedance (ratio of the total voltage to the total current) changes continually down the
  18. 18. 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): Zin = Z0 ZL + jZ0 tan β Z0 + jZL tan β where β = 2π λ and is the length of the line (electrical length). This equation is generally known as the Transmission line equation since it fully specifies what happens on the line. It is an unwieldy equation, however, and not generally useful. A graphical technique like the Smith Chart (section 1.2 on page 9) effectively embodies this equation in an easy-to-use manner, without the need for the equation as such. Since the velocity with which waves travel on a transmission line is lower than Note that the velocity on a line can never be higher than the speed of light! that of light, the physical length of the cable is always shorter than the electrical length: Physical line length = VF × (electrical length) where the velocity factor, VF = 1√ r and is quoted by the manufacturers. Some important simplifications of the Transmission line equation are: 1. ZL = Z0 This condition results in: Zin = ZL = Z0, regardless of line length, or frequency. This is the matched case. 2. = λ 2 . Zin = ZL regardless of characteristic impedance. This is the halfwave case. 3. = λ 4 “Quarter wave transformer” case. Zin = Z2 0 ZL This configuration is useful since it can transform one load impedance to a different one if a line with the correct impedance can be found. 4. Open or short circuited lines. Zin(oc) = −jZ0 cot β for an open circuited line Zin(sc) = jZ0 tan β for a short circuited line In both these cases the impedance is purely reactive and if the lines in question are less than a quarter wave it is clear that such lines could be used to “manufacture” capacitive (open circuit case) or inductive (short circuit case) reactances. It should be remembered however that the capac- itance or inductance of such a line would itself be frequency dependent. The open and short circuit cases provide a convenient way to measure the characteristic impedance of a line, since combing them yields: Z0 = Zin(oc)Zin(sc)
  19. 19. 8 Electromagnetics The velocity factor of a line can be measured by using the quarter-wave trans- former principle—if the load end is open circuited, ZL = ∞, hence Zin =0! The method is then to take an open-circuited line and mesure the input impedance, increasing the frequency until the input impedance drops to a minimum. The line is then at an electrical quarter-wavelength, so VF = phys λ/4 1.1.4 Standing Waves, Impedance Matching and Power Transfer The Maximum Power Transfer theorem determines when maximum power will be transferred to a load or extracted from a source: An impedance connected to two terminals of a network will ab- sorb maximum power from the network when the real part of the impedance is equal to the real part of the impedance as seen looking back into the network from the terminals and when the reactance (if any) is of opposite sign. Stating this in a simplified form appropriate for transmission lines (which nor- mally have a real impedance) The impedance of devices connected to the two ends of a transmis- sion line should have the same input and output resistances as that of the characteristic impedance of the line. When this condition is not satisfied standing waves are set up on the line and not all of the available power is transferred through the system. A more practicalNote that the sub- maximum power transfer is not due to losses in the system, but rather to power being reflected. problem which occurs is that such mismatch conditions often damage the output electronics of a transmitter if excessive standing waves occur (or in well designed transmitters the protection circuitry automatically reduces power output). The extent to which power is reflected from the load is dependant on how “bad” the load mismatch is to the line characteristic impedance and is expressed in the voltage reflection coefficient, ρ: ρ = ZL − Z0 ZL + Z0 The mismatch in impedance is also often stated in terms of the voltage standing wave ratio (VSWR) on the transmission line. VSWR = 1 + |ρ| 1 − |ρ| = Vmax Vmin On a Smith Chart the VSWR value can be read off directly without performingMany texts and measur- ing instruments refer to a reflection coefficient as a capitalized gamma: Γ. any of these tedious manipulations.
  20. 20. 1.2 The Smith Chart 9 1.2 The Smith Chart For antenna analysis the most convenient way to represent and manipulate impedances and transmission lines is on the Smith Chart (Smith 1939). In addition to solving the transmission line equation presented in section 1.1.3 on page 6, the chart is a visualization tool, enabling design decisions that are not possible by simply studying the theory behind the equations. Appendix A on page 95 contains a fully fledged Smith Chart, and this can also be downloaded from Clark (2001). On the Smith chart, the impedance is first normalized to some convenient value (often 50Ω) by dividing both real and imaginary components by the normaliza- tion factor. The value 30 − j70Ω is plotted on the Smith Chart by normalizing resulting in a value of 0.6 − j1.4 and then plotted on the chart shown in fig 1.5 in the lower hemisphere. This transmission line chart has a lot of useful features when manipulating The Smith chart is, in fact, simply a polar plot of the (complex) reflec- tion coefficient, ρ, over- laid with lines of constant resistance and reactance. impedances. If the impedance above is that of an antenna and it is connected to a transmission line of 50Ω (say of quarter wavelength) then the input value to this line can be simply read off the chart by rotating the plotted value around the centre of the chart by a quarter wavelength (180◦ on the chart), yielding 0.26 + j0.6, or 13 + j30Ω, denormalized. The distance (in electrical length) travelled down the line is indicated on the outside of the chart. When travelling from the load to the generator, a clockwise direction is followed. On the other hand, if the transmission line is 0.16 wavelengths long, the antenna impedance would be transformed to a purely real value of 0.19 (or about 10Ω). −1.4 0 0.2 0.5 1 2 5 ∞ 0.2 −0.2 0.5 −0.5 1 −1 2 −2 5.30.6 0.6−j1.4 0.16λ Smith 0.26+j0.6 0.19+j0 . Figure 1.5: Impedance values plotted on the Smith Chart
  21. 21. 10 Electromagnetics The Voltage Standing Wave Ratio, VSWR, can be read directly off the Chart as it is simply the intercept on the real axis as indicated by the circle centered in the centre of the chart drawn for VSWR of 5.3:1. The constant VSWR circle in fact describes all the possible values of the input impedance of the line as a function of distance down the line. As we move down the line, we travel down a constant VSWR circle, assuming that the line is lossless, and hence get no nearer to a better match. In the case of a lossy line, the magnitude of the reflection coefficient decreases, and instead of a circle, a spiral is described, spiralling in to the centre of the chart, improving the match. ie A lossy line improves the aparrent VSWR at the input to the line, but of course this is at the expense of loss in the cable. The addition of an inductive component to the plotted value on the other hand, would clearly move the point on the constant resistance line (0.6 normalized in our example) and it is immediately clear that more reactance in this example would improve the impedance match since it would force the point closer to the centre of the chart, without the loss of power. 1.3 Field Theory 1.3.1 Frequency and wavelength As seen previously, the wavelength in free-space is given as λ(m) = 300 f(MHz) In the part of the Electromagnetic spectrum of interest to us, there are tradi- tional (but otherwise meaningless!) designations of radio “bands” Frequency Wavelength Designation 3–30 Hz 100–10 Mm ELF (Extra Low Frequency) 30–300 Hz 10–1 Mm SLF (Super Low Frequency) 300–3000 Hz 1 Mm–100 km ULF (Ultra Low Frequency) 3–30 kHz 100–10 km VLF (Very Low Frequency) 30–300 kHz 10–1 km LF (Low Frequency) 300–3000 kHz 1000–100 m MF (Medium Frequency) 3–30 MHz 100–10 m HF (High Frequency) 30–300 MHz 10–1 m VHF (Very High Frequency) 300–3000 MHz 1000–100 mm UHF (Ultra High Frequency) 3–30 GHz 100–10 mm SHF (Super High Frequerncy) 30–300 GHz 10–1 mm EHF (Extra High Frequency) 300–3000 GHz 1000–100µm Submillimeter/InfraRed Observations: • Small things (electrically speaking) do not radiate well. Hence from 30MHz to 30GHz is the most commercially used band. • 3–30MHz bounces off ionosphere (so-called Shortwave) and is used for old-fashioned over-the-horizon comms.
  22. 22. 1.3 Field Theory 11 • Achievable frequencies are limited by electronics (for oscillators) • Above 3000 GHz, we get into optics. • Ionizing radiation can cause a molecule to change (x-rays and higher). Anything lower in frequency can only cause heating. Humans can take about 100W of heat! The microwave (> 1GHz) region is broken into Radar bands. The most common designations are the IEEE ones, although there are “newer” designations, which I have never seen used. Frequency Wavelength IEEE Designation 1–2 GHz 300–150 mm L 2–4 GHz 150–75 mm S 4–8 GHz 75–37.5 mm C (5m (big) dish) 8–12 GHz 37.5–25 mm X 12–18 GHz 25–16.7 mm Ku (1m (small) dish) 18–26 GHz 16.7–11.5 mm K 26–40 GHz 11.5–7.5 mm Ka 40–300 GHz 7.5–1 mm mm Spectrum Usage: FM Radio 88–108 MHz ClassicfM 102.7 MHz TV 123 200 MHz MNET 615 MHz GSM MTN & Vodacom 900 MHz Cell C 1800 MHz DECT 1880–1900 MHz GPS 1.23 & 1.58 GHz WLan 1.8 GHz Industrial ISM 906–928MHz Scientific ISM 2.4–2.5 GHz Medical ISM 5.8–5.9 GHz µwave oven (K5–513) 2.45 GHz DSTV Ku band (11.7 & 12.3GHz) 1.3.2 Characteristic impedance & Velocity of propagation In the case of a bounded medium like a transmission line, the geometry and the medium played a part in determining the characteristic impedance. For a propagating Transverse Electromagnetic (TEM) wave in an unbounded medium, the characteristic impedance it sees is: Z0 = jωµ0µr σ + jω 0 r Since free space is non-conducting, and the relative quantities are unity: Some texts use eta: η, and others zeta: ζ to specifically refer to the Z0 of free-space. Z0(free space) = η = µ0 0 = 377Ω or: η ≈ 120π
  23. 23. 12 Electromagnetics Just as the characteristic impedance depends on the geometry and the properties of the medium, so the velocity of EM radiation also depends strongly on the medium properties. For example, sound travels at 330m/s in air, but 1500m/s in water. In a non-conducting medium, the velocity of propagation is: v = 1 √ µ0µr 0 r where 0 and µ0 refer to the electric permittivty and magnetic permeability of free space; the r subscripts refer to the quantities of the material relative to that of free space. SinceNote that the value of µ0 is by definition, and 0 is actually derived from the measured speed of light. (Kraus 1988, Back Cover) µ0 ≡ 4π × 10−7 [H/m] 0 = 8.85 × 10−12 [F/m] Thus, the speed of light in a vacuum or air, c, is given by: c = 1 √ µ0 0 = 3 × 108 m/s = 300km/s Most often, we are not interested in the absolute value of the speed of prop- agation, but in the ratio of the speed to that of light in free-space, known as the Velocity Factor, VF= v/c. This leads to the useful formula for EM wave velocity in non-magnetic media as: v = c √ r or: VF = 1 √ r Since most plastics have a relative permittivity of about 2.3, the reduction in speed is about 0.66, hence the shortening factor for coaxial cable calculations. 1.3.3 EM waves in free space An EM wave travels in free-space and in most transmission lines as a Transverse EM wave (TEM). This implies that the direction of propagation is at 90◦ to both the Electric and Magnetic wave, which, in turn are at 90◦ , as shown in fig 1.6 Since the E field is analogous to voltage and the H field to current in the circuits sense, it is easily seen that the equivalent relationships to Ohms law etc exist in TEM waves in free-space: (V = I × R)... E = H × η or: H = E 120π In a similar fashion, the power relationships hold (power density in EM): S = EH = E2 120π = H2 × 120π W/m2 where E and H are RMS values.
  24. 24. 1.3 Field Theory 13 x z y Ex Hy E(φ) EMWave Figure 1.6: Electromagnetic Wave in free-space In general, EM waves of a frequency above 30MHz do not bend around the earth, and propagation occurs only within Line-of-Sight (LOS). The earth’s curvature thus limits the radio horizon achievable for a certain height of transmitter, as shown in fig 1.7 dh h Earth Shadow Region Horizon Line Antenna radhor Figure 1.7: Radio Horizon due to earth curvature. An empirical formula to calculate the radio horizon is: Since the mean Earth Ra- dius is 6370km, and the tangential point is on it, by Pythagorous, we get 19.5km! dh(km) = 4 h(m) Thus a 30m mast will only provide a 22km Line-of-Sight range. Beyond the horizon waves propagate for longer distances than that illustrated above using either ground wave propagation or reflection from some atmospheric feature such as the ionosphere, meteors, the troposphere or even an artificial satellite (although in that case the signal is usually retransmitted). Ground wave propagation occurs when vertically polarized waves are launched and these are guided by the earth surface and thus follow the curvature of the earth. They are only practical at lower frequencies where the attenuation these waves undergo as result of earth losses are reasonably low. At the very low frequency end (round 30 kHz) the wavelength is so large that the waves flow in the waveguide formed by the earth surface on the one hand and the ionosphere on the other. This could also be considered as a type of ground wave.
  25. 25. 14 Electromagnetics The classic means of long distance communication is by means of reflecting waves from the ionosphere. This layer1 of gas that is ionized by the sun lies at a distance of approximately 80 to 400 km above the earth surface. Only waves below a certain critical frequency are reflected (by a process of successive refraction) by this layer. This so called critical frequency is typically in the HF band (3–30 MHz) and varies strongly with time of day and other variables. At frequencies much below this frequency the radio-waves undergo attenuation (or absorption) and communication using this principle is also impractical. Other methods of obtaining beyond the horizon communication are the rather exotic techniques such as meteor scatter propagation: where waves are reflected off the ionized trails left by meteors entering the atmosphere; or tropospheric scatter: which uses discontinuities in the atmosphere to cause bending of waves and thus long distance propagation. Most over-the-horizon communication, of course, occurs at microwave frequen- cies over a satellite relay link at high data volumes. (Discounting under-the-sea fibre optics for this course!) 1.3.4 Reflection from the Earth’s Surface In addition to its role as a obstacle, the earth’s surface also acts as a reflector of radio waves. This situation is illustrated in figure 1.8. 2h P1 S1 S2 θ θ Direct-wave path Reflected-wave path Reflecting Surface ReflRay Path Difference = 2h sin θ Figure 1.8: Geometry of Interference between Direct Path and Reflected Waves It is clear that if S1 is an isotropic source and would normally radiate equally well in all directions, the pattern would be modified by the reflected wave. By the method of images the situation above is similar to that which exists if a mirror image source S2 was positioned at distance h below the reflecting plane. 1Actually, several identifiable layers at differing heights
  26. 26. 1.3 Field Theory 15 Clearly there will now be a difference in the path lengths to some distant point P. At certain elevation angles θ the path difference would be such that the two waves are in phase and thus interfere constructively and for others the interference would be destructive and result in a null in the radiation pattern. If the field due to a single source is termed E0 then the total field would then be given by: E = |E0| sin 2πh sin θ λ Application of this formula to the particular value of h/λ = 1.44 results in the field pattern shown in figure 1.9. This condition is not always advantageous since an antenna that may have had a maximum towards θ = 0◦ would now have a null in the same direction. The only way to improve the situation would be to either make the antenna higher and thus force the angle of the first maximum lower or increase the frequency and thus ensure an increased h/λ ratio. In cases where radiation at some angle is required this reflection results in an unexpected bonus, however. The maximum value of the E-field in the direction of the maxima is twice the value of the original antenna without reflection. This implies that in that particular direction the power density would be increased by a factor of four (power density is proportional to the square of the E field). Earth reflection can thus be used to gain a 6 dB bonus in antenna gain if used properly! 0.2 0.4 0.6 0.8 1 30 60 90 0 Interfere . Figure 1.9: Interference pattern field strength contours for h/λ = 1.44
  27. 27. 16 Electromagnetics 1.3.5 EM waves in a conductor If we solve Maxwell’s equations for EM wave propagation in a conductor, we get a different result to the free-space case considered above. The Electric field magnitude actually decays exponentially inside the conductor. We define the Skin depth, δ, as the point where the field inside the (good) conductor, Ey, has become 1/e of its original value, E0: δ = 2 ωµσ = 1 √ πfµσ Note that δ has units of distance. If we look at what happens at one skin depth, δ, into the conductor, the field is given as: Ey = E0e−x/δ , then |Ey| = |E0| e ≈ 37%|E0| or the field inside the conductor at one skin depth is 37% of the original field magnitude. Comparing the skin depth (or depth of penetration) δ for silver, copper & cast iron at 50 Hz and 1GHz: Relevant parameters: σAg = 6.1 × 107 0/m σCu = 5.7 × 107 0/m (Both have µr = 1) σFe = 106 0/m and µrFe = 5000 δ50Hz δ1GHz Ag 9.1mm 2µm Cu 9.4mm 2.1µm Fe 1mm 0.23µm The decay is obviously logarithmic: at one further skin depth, the field has deacyed a further 37%. ie 9.4mm of Copper to get to 37% of the original signal strength. Another 9.4mm of copper will take that signal strength to 37% of that value, ie 13.7% of the original value. A useful table is then the percentage overall decrease as a function of the number of skin depths: δ s % of Orig. 1 37 2 13.7 3 5.1 4 1.9 5 0.7 After 5 skin depths, the field level is below 1%. A number of consequences follow: • Busbars in substations are hollow. It is silly to supply a solid copper bar if there is no field and current flow in it’s middle! • At cellphone frequencies, the skin depth of copper is about 2µm. CertainlyAt DC, the current den- sity is uniform through- out the cross-section, but at RF, the current den- sity concentrates in the two outer skins. after 10µm, there is no appreciable field. This means that the outside of a coaxial braid is a completely separate conductor from the inside of the coaxial braid from an RF perspective!!
  28. 28. Chapter 2 Antenna Fundamentals 2.1 Directivity, Gain and Pattern THE USUAL COORDINATE SYSTEM in 3-dimensional space, the carte- sian co-ordinate system of points being described in terms of (x, y, z) co- ordinates does not lend itself to the electromagnetic world. Generally, we like to describe the points in the spherical co-ordinate system as (r, θ, φ), as shown in fig 2.1. z y x P(x, y, z)&P(r, θ, φ) r φ θ Coord Figure 2.1: Standard coordinate system. The spherical co-ordinate system is mainly described in terms of angles, where r is the distance from the origin, θ the angle from the zenith (not the elevation angle) and φ is the projected angle from the x-axis, in the xy plane. 2.1.1 Solid angles Clearly, when dealing with 3-dimensional radiation patterns, some idea of a 3-dimensional angle needs to be employed. 17
  29. 29. 18 Antenna Fundamentals A radian is defined by saying that there are 2π radians in a circle. As a result, the arclength rθ of a circle of radius r is said to subtend the angle θ. A steradian (or square radian) is a solid angle defined by saying that there are 4π steradians in a sphere. As a result, an area a on the surface of the sphere of radius r is said to subtend the solid angle ω. Note that the shape of the area is not specified. shown in fig 2.2 x = rθ r θ θ = x r radians z y x Ω Area=A Ω = A r steradians steradian Figure 2.2: concept of a solid angle 2.1.2 Directivity Isotropic source An isotropic antenna, is as its name implies, one which radiates the same power in all possible (3-dimensional) angles. By definition, an isotropic antenna has a directivity, d, of 1, and is infinitely small. Note that an isotropic antenna is an impossibility and is used only as a reference! Generally, however, an antenna has greater directivity in some directions than other directions. This is at the expense of directivity in other areas. In free-space the EM wave is not guided, as it would be in a transmission line, and can hypothetically radiate in all directions. If an isotropic source transmits Prad watts, and we enclose the source with a sphere of radius, r, and hence surface area, A = 4πr2 , then the power density, S at the surface of the sphere is given as: S = Prad A = Prad 4πr2 [W/m2 ] hence the power reduces inversely proportional to the square of the distance. Directivity, D, is then defined as: D = S(θ, φ)max Sisotropic
  30. 30. 2.1 Directivity, Gain and Pattern 19 where the power transmitted by both antennas is the same. Decibels Directivity is usually expressed in decibels (DdB = 10 log10 D), and since it is Directivity is almost al- ways quoted in dB’s, but that most equations re- quire it in linear form. referred to an isotropic source (the Sisotropic in the above), the unit is denoted dBi for dB’s above isotropic. Decibels are very useful in power ratio’s because of the large range of values experienced because of the inverse square law. For example, if power P1 is 1000 times greater than P2, the ratio expressed in dBs is: dB = 10 log10 P1 P2 which yields P1 as being 30dB greater than P2. If P1 were a million times the power of P2, it would be 60dB greater. It can be seen that dB’s are simply more convenient for expressing power ratio’s than linear quantities. One can also use dB’s in field-related quantities that yield power when squared— from Ohms law in free space, we see that S = E2 /120π. Since the log of a squared quantity is simply twice the log of the non-squared quantity, for a power component like voltage, E-field etc: dB = 20 log10 E2 E1 It is sometimes convenient to represent a signal with respect to a standard reference—eg the absolute power level to a milliwatt of power is denoted dBm: dBm = 10 log10 power 1mW similarly for a voltage relative to a microvolt: dBµV = 20 log10 voltage 1µV 2.1.3 Efficiency Directivity, D, is often loosely referred to as Gain, G. However, the term gain Note that “Gain” is not used in the same sense as amplifier “gain”. An an- tenna is passive. It sim- ply denotes an increased power density in one di- rection, at the expense of other directions. refers to the input power as opposed to the radiated power—the difference being the power lost in the ohmic losses in the antenna. The antenna efficiency is then: η = Prad Pin and hence: G = ηD
  31. 31. 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. The directivity is more a indication of the pattern of an ideal radiator and is ofBoth Gain and Directiv- ity are functions of θ and φ but are often used to describe these values in the maximum directions. more theoretical than practical value. In practice the gain of an antenna is important since it increases the power density in the direction of the main beam of the antenna. A 100 W transmitter with a 13dB gain antenna produces the same power density at a distant point as a 1 kW transmitter with a 3 dB gain antenna. Clearly the former case would have a larger area of lower power density in other directions and the extent to which antenna gain would improve communications would depend on the intended coverage. In a broadcast scenario for instance, omnidirectional radiation may be required in the plane of the earth. The only improvement in gain of broadcast antennas is by minimizing the radiation upwards (at angles of θ smaller than 90◦ ). For point-to-point links on the other hand the power can be concentrated as far as possible in the desired direction. Hence, the power density S at a distance r away from an antenna with a gain G is: S = GPin 4πr2 = DPrad 4πr2 W/m2 2.1.4 Radiation pattern An antenna’s radiation pattern is a measure of how it radiates in 3-dimensional space. Generally, a radiation pattern refers to one that is measured in the far- field (see later). As a result, it is a plot of transmitted power versus angle. It has nothing to do with distance. Note that the signal from an antenna weakens predictably with distance by 1/r2 , but not predictably by angle—this is described by the radiation pattern. The pattern can show power density(W/m2 ) versus angle, radiation intensity (W/sr) versus angle, E-field (V/m) or H-field (A/m) versus angle. The plot can be rectangular or polar. Generally, however, it is a plot of power density, expressed in dBi. A representation of a SuperNEC (Nitch 2001) three dimensional radiation pattern of a 10-director Yagi-Uda array is shown in fig 2.3. An antenna often has a main beam, with some sidelobes and backlobes, which may not be desired. The main lobe can be clearly seen, as can the first sidelobe (all the way aroundAlthough a 3d pattern may look impressive, it can rarely show detail ac- curately. the main lobe, like a collar). It can also be seen that the next sidelobe fires up and down, but that there is a null along the y-axis. It is far more common to take a 2D cut through the principle planes of the antenna, as shown in the xy plane cut in fig 2.4. Fig 2.4 is shown in a polar representation, which is most useful for visualization purposes. To read off values, however, a rectangular form is preferred as shownIt is easier to get a feel for the front-to-back ra- tio, (14.3 dB), and the first sidelobe levels (14.3 dB in azimuth; 8.77dB in elevation) from the rect- angular plots.
  32. 32. 2.1 Directivity, Gain and Pattern 21 Figure 2.3: Three-dimensional pattern of a 10 director Yagi-Uda array 300240 180 120 60 0 10 dBi 0 −10 −20 −30 Radiation Pattern (Azimuth) Structure: θ=90° Figure 2.4: xy plane cut of the 10 director Yagi-Uda in fig 2.5. The rectangular plots also show the half power beamwidth (HPBW) (-3dB point of the main beam) and the beamwidth between first nulls (BWFN) quite clearly, and these quantities are often useful to determine the absolute gain of the an- tenna, which is difficult to measure accurately, whereas a 3dB drop from peak power is a relative measurement, and hence an eassier one.
  33. 33. 22 Antenna Fundamentals −180 −120 −60 0 60 120 180 −20 −10 0 10 15 φ Gain(dBi) Radiation Pattern (Azimuth) (a) xy plane −90 −30 30 90 150 210 270 −20 −10 0 10 15 θ Gain(dBi) Radiation Pattern (Elevation) (b) yz plane Figure 2.5: Rectanglar radiation patterns of the 10-director Yagi-Uda in the two principle planes Directivity estimation from beamwidth Note again that if the “beam” equally covered a whole sphere of 4π square radians, or steradians, ie an isotropic source, the directivity is 1 by definition.4π square radians = 4π(360/2π)2 square de- grees = 41 253, usually rounded off to 41 000. This gives rise to an approximation which is often used (Kraus & Fleisch 1999, pg 255): D = 4π ΩA ≈ 4π θHPφHP ≈ 41 000 θ◦ HPφ◦ HP The above approximation produces about 1dB too much gain. the reasoning is that the area represented is square, whereas in reality, the beam is generally round. hence a common adjustment (Kraus 1988, pg100) is: D ≈ 36 000 θ◦ HPφ◦ HP even in this case, however, if fairly significant side- and back-lobes exist, the prediction is optimistic. In the above example of the 10-director Yagi-Uda array, we get HPBW’s of 42 and 38 degrees, which according to the above formula gives 13.5dBi gain, whereas SuperNEC gives 12.7dBi. 2.2 Reciprocity The reciprocity theorem stated by Lord Rayleigh, and generalised by Carson (1929), states that if a voltage is applied to antenna A which causes a current to flow at the terminals of antenna B, then an equal current (in magnitude and phase) will occur at the terminals of antenna A if the same voltage is applied to the terminals of antenna B. In short, all characteristics of an antenna apply equally well in transmit and receive mode (radiation pattern (reception pattern), input/output impedance etc). shown in fig 2.6.
  34. 34. 2.3 Polarization 23 Va Energy Flow Ib Ia Energy Flow Vb Va causes Ib ; Vb causes Ia If Vb is made = Va, then Ia will be = Ib reciprocity Figure 2.6: Reciprocity concepts However this does (obviously) not hold for the near fields—it is a far-field phe- nomenon. ie far-field patterns are identical, but near-field patterns are different. 2.3 Polarization Polarization generally refers to the direction of the E-field vector in the far- field. note that the H-field is at 90◦ to the E-field and is present in all EM waves. In the case of a dipole or a Yagi-Uda array, the E-field is linearly polarized in the direction of the dipole. Most often, a linear polarization will be specified as horizontally polarized (most TV antennas) or vertically polarized (most broadcast radio signals) (reference is the earth). The amount of power received varies as the cosine of the angle between the Note that if the sig- nal is horizontally polar- ized, and the receiving antenna vertically polar- ized, there is (ideally) no received signal. polarization of the incident wave and the antenna. eg A cellphone base station antenna is vertically polarized, and although the mobile has a linearly polarized antenna, it is seldom held perfectly upright, incurring a loss in received power. y Linear x E2 y Elliptical x E2 E1 E y Circular x E2 E1 E Polarization Figure 2.7: Polarization possibilities—wave out of page
  35. 35. 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: β = 2π λ the e-field can be made to progressively change direction in a circular fashion (eg with a helical antenna, popular on satellites). the helix can be wound in a left-hand or a right-handed fashion, giving rise to lhcp and rhcp. a rhcp antenna will not (ideally) receive a lhcp wave. the e-field is continuously rotating, and the peak x-component is equal to the peak y-component of the field. Generalizing to elliptical polarization, the x and y components are not equal.In the general case of el- liptical polarization, the major axis magnitude to the minor axis magni- tude defines the axial ra- tio. An infinite axial ra- tio implies linear polar- ization, whereas an AR of 1 implies circular polar- ization. The field components are thus: Ex = E1 sin(ωt − βz) Ey = E2 sin(ωt − βz + δ) where δ is the time phase angle by which Ey leads Ex. 2.4 Effective aperture If an antenna is immersed in a field with a power density of S [W/m2 ], it will receive a power Pr [W] and deliver it to a load connected to its terminals. This gives rise to the concept of an effective aperture, Ae [m2 ]. Ae = Pr S In general (Kraus & Fleisch 1999, pg 258), the aperture of an antenna can be related to the antenna gain: Ae = Gλ2 4π ie when the gain is large the aperture is large. Note that the effective aperture of a dish antenna may not be as big as the actual area of the dish (see fig 2.8), leading to the concept of an aperture efficiency, εap: εap = Ae Ap where Ap is the physical size of the antenna aperture. (Mainly caused by im- perfect parabolicness—witness the hubble telescope) As an example as shown in figure 2.9, a half-wave dipole for classicfm (102.7mhz) made of 1mm diameter rod, has a physical aperture of 1.46m×0.001=0.0015m2 . since a half-wave dipole has a gain of 2.16dBi=1.64 linear, the effective aperture
  36. 36. 2.5 Free-space link equation and system calulations 25 r Ap = πr2 Ae ApertureEff Figure 2.8: Aperture efficiency of a parabolic dish Ae λ/2 ApertureDipole Figure 2.9: Effective aperture of dipole physical aperture is 1.11m2 ! (740× larger). Note that the aperture has meaning—if two antennas are used to capture more energy from the wave, the apertures cannot be allowed to overlap as shown in fig 2.10 ApertureSteal Figure 2.10: Apertures must not overlap! In the case of linear antennas, this is a fair problem, as the apertures easily overlap, and the overall capture area will be reduced. With dish antennas, since Ae < Ap, it is not possible to have an overlapping aperture! 2.5 Free-space link equation and system calula- tions The gain of an antenna is defined as the maximum power density in a certain direction as compared to a hypothetical isotrope. Since the antenna gain con-
  37. 37. 26 Antenna Fundamentals Tx Gt Pt Rx Gr Pr r LinkEqn Figure 2.11: Point-to-point link parameters centrates the transmitted power into a particular direction, we can define the effective radiated power (ERP) of an antenna in a particular direction as:ERP is the power that an isotrope would have to produce in all di- rections to achieve the same effect in this par- ticular direction. (also pedantically called EIRP, I for Isotropic, but rarely used). ERP = GtPt at a distance, r, away from the transmitting antenna, the power density, S is given as: S(r) = GtPt 4πr2 the power received, Pr, by the receiving antenna is then Pr = SAer where Aer is the effective aperture of the receiving antenna, given by Aer = Grλ2 4π hence the power received by an antenna in a freespace point-to-point link is: Pr = GtGrPtλ2 (4πr)2 [W] note that multipath reflections with constructive and destructive interference and atmospheric absorption change this value! The link equation can obviously also be conveniently expressed in dB’s:The 32.45 factor comes from +20 log(300) for frequency conversion (20 since squared), −20 log(4π), and −20 log(1000) for the km conversion. Pr = Pt + Gt + Gr − 32.45 − 20 log10 r − 20 log10 f where Pr and Pt are expressed in dBm, Gt and Gr are in dBi, r is in km, and f is in MHz. Example: Geostationary satellite comms A geostationary satellite must be in the Clarke orbit at 36 000 km in order to appear stationary above a point on the earth. If the power required at the
  38. 38. 2.5 Free-space link equation and system calulations 27 satellite receiver is only 8pW, how much power is required to be transmitted by the earth station? Receiver gain is 20dB, earth station gain is 30dB, at a frequency of 5GHz. Converting to linear gains: Gr = 20 dB = 100; Gt = 30 dB = 1000 λ = 300 5000 = 60mm hence the power we are required to transmit is: Pt = Pr(4πr)2 GtGrλ2 = 8pW(4π × 3.6 × 107 m)2 100 · 1000 · (0.06)2 = 4547W So 4.5kW is needed to get 8pW to the satellite. Note that the link equation ignores atmospheric absorption or any other losses along the path, and purely assumes line-of-sight (LOS) situations. Actually, satellites transmit at hundreds of watts, not kW! The dB version is thus: Pt = Pr − Gt − Gr + 32.45 + 20 log r + 20 log f = −80.9 − 20 − 30 + 32.45 + 91.12 + 73.98 = 66.58dBm = 4556kW
  39. 39. 28 Antenna Fundamentals
  40. 40. Chapter 3 Matching Techniques 3.1 Balun action 3.1.1 Unbalance and its effect AT RF FREQUENCIES recall that from section 1.3.5, the skin-depth is only a few microns. As a direct result, the inside of the coax braid is quite a seperate conductor than the outside of the braid, even though, from a DC perspective, we view the braid as “one conductor”. If we feed an antenna using a coaxial cable as shown in fig 3.1, the antenna radiation can induce a current flow on the outside of the braid, i3, quite inde- pendant of the current flow on the inside of the braid due to the transmitter, i2. i1i2i3 i1i2 + i3 BalunReason Figure 3.1: The reason for unbalanced currents Since on the inside of the braid, balanced currents must flow i1 = i2, and both An absolutely tell-tale sign of external current is a change in a measure- ment parameter when the cable is grabbed by a hand. the inside and outside currents meet at the antenna junction, this implies that a larger current can flow on one dipole arm than on the other. The antenna pattern is distorted, and the input impedance is changed. 29
  41. 41. 30 Matching Techniques The job of the BALance to UNbalance converter (Balun) is to prevent the currents flowing on the outside of the braid by presenting a high impedance to them. 3.1.2 Balanced and unbalanced lines—a definition Balanced UnBalanced UnBalanced BalLine Tr Ln Cross Sections Figure 3.2: Balanced and Unbalanced Transmission lines Essentially, a balanced line is one in which symmetry ensures that equal fields, currents, voltages exist along it in the equal and opposite sense! It is clear that a two-wire line of equal diameter conductors is balanced, but that a coaxial line is not—the field strength is much higher on the inner conductor than the outer conductor. 3.2 Impedance Transformation The Balun action is to be distinguished from the Transformer action. A circuit that does both functions should be called a Balun-Transformer, but in practice all are simply called Baluns, whether they transform, or just do a Balance-to- Unbalance conversion (sometimes even when they only transform. Impedance transformation has the goal of matching the antenna impedance to the line, and the generator in order that no reflected power exists—maximum power transfer. A transmitter cannot deliver full power to an unmatched load, but this is not the only consideration: • High VSWR means high V &I, hence txln losses. • High V means flashover/dielectric breakdown. • High I means hotspots/copper melting. • Output electronics of the transmitter can be damaged, or more likely, the automatic power reduction circuitry kicks in. (Typically at a VSWR of 2:1) Impedance matching is important both for transmission and reception. It is more critical, however, for the transmitting case and the VSWR specifications
  42. 42. 3.2 Impedance Transformation 31 are usually more severe. To illustrate this point the following equation gives the power reduction as result of a mismatch in terms of VSWR: Power lost in transfer = 10 log 1 − VSWR − 1 VSWR + 1 2 dB Thus a VSWR of 2 : 1 results in a power reduction of only 0.5 dB. Even a VSWR as high as 5 : 1 only causes a reduction of 2.5 dB. The power reduction due to the mismatch condition itself is thus not all that significant, but some transmitters will start reducing power output to protect the driving stage electronics at such low values as 1.5 : 1 or 2 : 1 (Or simply blow up if no power reduction protection is in place). The power lost (in dB) versus VSWR is illustrated in figure 3.3. 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 VSWR PowerlostindB pwrvswr Figure 3.3: Power loss in dB versus VSWR on the line. The additional line losses with a higher VSWR is also of concern and is a func- tion of the normal line losses, which are usually specified by the manufacturer of the lines in terms of dB/30m (ie dB/100ft!). It should be noted that the manufacturer specification assumes matched conditions. The graph in figure 3.4 indicates the additional loss under mismatch conditions (ARRL 1988, pg3-12). Losses in the line will improve the VSWR at the input end as compared to This improvement is clearly obtained at the expense of efficiency, indicating the need to avoid long transmis- sion lines when doing measurements of VSWR. the VSWR at the antenna. A curve quantifying this characteristic is given in figure 3.5
  43. 43. 32 Matching Techniques 10-1 100 101 10-1 100 101 Additional Loss in dB when Matched AdditionallossindB SWR=2 SWR=4 SWR=7 SWR=20 AddLoss Figure 3.4: Increase in Line Loss because of High VSWR 1 5 10 50 100 1 2 3 5 10 SWR at TransmitterSWRChange 0dB Loss 1dB Loss 2dB Loss 3dB Loss SWRatAntenna Figure 3.5: VSWR at Input to Transmission Line versus VSWR at Antenna 3.2.1 Angle Correction Remember that a low-loss or lossless transmission line has a purely real char- acteristic impedance. For matching to occur, the antenna has to be resonated.
  44. 44. 3.3 Common baluns/balun transfomers 33 This is most often accomplished before impedance transformation. Some match- ing techniques include angle correction, some don’t. 3.3 Common baluns/balun transfomers 3.3.1 Sleeve Balun Recall that a short-circuited quarter-wavelength transmission line presents a high (theoretically infinite) impedance. Thus the simplest balun is a short- circuited quarter-wave sleeve around the coaxial feeder cable, where the sleeve is short-circuited to the coaxial feeder cable braid a quarter wavelength away from the feed. Currents wanting to flow in the outside of the braid now see a high impedance, as shown in fig 3.6. λ/4 Bazooka Figure 3.6: The Bazooka or Sleeve balun. A sleeve balun has no transformer action, but simply prevents the current flow. The Americans tend to refer to the sleeve balun as a “Bazooka balun”! It is rather narrow-band however, and care must be taken to ensure that it is the correct length in terms of the Velocity Factor of the coax outer insulation, which is normally a different material than the inner dielectric (hence a different VF). 3.3.2 Half-wave balun λ/2 HalfWaveBalun Figure 3.7: A HalfWave 4:1 transformer balun. In addition to a balun action, due to the halfwave section, there is also a 4:1 impedance transformation as shown in fig 3.7. The impedance transformation is because after the λ/2 transmission line, the voltage is exactly out of phase, hence twice the voltage is applied between the
  45. 45. 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 usually wound on a frequency-band specific ferrite core. They are quite broad-For the HF band, the Philips 4C6 material is excellent. band achieving bandwidths of up to 15:1. The transmission line transformer (if connected correctly) also has a Balance to Unbalance conversion. If the input Gnd is reversed, NO Balun action. 3.3.4 LC Networks Physical inductors and capacitors can also be used to resonate the antenna and then transform the impedance. One drawback is that inductors are lossy. Both “pi” and “T” networks are commmon. One popular match is known as the “L-match”, shown in fig 3.9. In the L-match, the LC transformer network is simply an extension of the resonating inductor for the antenna. Therefore popular when the antenna is short in terms of wavelength, eg mobile HF whips. LC networks do not have a Balun action, and are naturally narrow-band. 3.3.5 Resistive Networks Resistive networks (pi and T) are also used, and are broadband, but very lossy. They are used only in extreme cases, and have no balun action.
  46. 46. 3.3 Common baluns/balun transfomers 35 CMatch LMatch LResonate CAnt.Reactance RRadiation LMatch Figure 3.9: L-Match network. 3.3.6 Quarter wavelength transformer ZLZin x = λ/4 quarter Z0 Figure 3.10: Quarter wavelength transformer For a transmission line of a quarter-wavelength long, recall that (section 1.1.3 on page 6): Z0 = ZLZS In matching one impedance to another, if a transmission line can be found with a Z0 of the geometric mean of the two impedances, a match can be made. The match is fairly narrow-band, and requires a practically available Z0. At low frequencies the cable length may be inconvenient, at high frequencies a microstrip line is used. Ordinarily there is no balun action. However, if a “series” connection of two coaxial cables is used as the transformer, there is balun action since both lines are shielded. A “series” connection uses two pieces of coax, with the two braids connected at both ends, but the two centres are used as the transmission line. The characteristic impedance of the series connected line is half the character- istic impedance of the coax.
  47. 47. 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. ZLZ0 Z0 Z0 d1 d2 Stub Figure 3.11: Single Matching Stub Both the lengths d1 and d2 need to be adjustable which can present some diffi- culty. This type of matching is most clearly illustrated on the Smith Chart andIf the cable is coax, vary- ing d1 can prove im- possible. The Double- Stub match uses fixed dis- tances from the load, and only varies the length of the stubs is shown in figure 3.12. Due to the fact that stubs are always connected in parallel it is easier to use ad- mittance, rather than impedance notation where parallel components are simply added. To achieve this the normalized antenna impedance is plotted (P1) and rotated halfway round the chart to obtain the corresponding admittance value (P2). Once an admittance has been converted by length d1 to lie on the circle indicated (Rn = 1) a match can always be achieved by simply adding or subtracting the necessary amount of susceptance (inverse of reactance) using the open or short circuited stub. This value can be found by starting at the outer circle on the chart at the right hand end (admittance of infinity) for short circuited case or the left hand side (admittance of zero) for open circuit stub and finding the required value of susceptance and reading of the length of the required stub (d2) from the Smith Chart. 3.3.8 Shifting the Feedpoint Changes can be made to the feed geometry in order to effect a narrow-band match. The T and Gamma matches are well suited to feed folded dipole anten- nas. They are illustrated in fig 3.13 These two types of matching sections are very similar in behaviour. It is difficult to analyze their performance but the following general trends referring to the T match have been observed:
  48. 48. 3.3 Common baluns/balun transfomers 37 0 0.2 0.5 1 2 5 0.2 0.5 0.2 0.5 1 2 1 2 ∞ Towards Generator Towards Load P2 P1 d1 d2 1.8 1.8 P3 SmithStub Figure 3.12: Smith Chart of Single Stub Tuner Arrangement C B A Unbalanced Balanced TGamma Figure 3.13: T and Gamma Matching Sections • The input impedance increases as A is made larger. The desired value of
  49. 49. 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.
  50. 50. 3.4 Impedance versus Gain Bandwidth 39 200 250 300 350 −10 −5 0 5 10 15 Freq (MHz) Gain(dBi) Gain at θ=90, φ=0 Line1 Figure 3.14: Gain bandwidth 200 250 300 350 1 1.5 2 2.5 3 3.5 4 4.5 5 Freq (MHz) VSWR Line1 Figure 3.15: Impedance bandwidth
  51. 51. 40 Matching Techniques
  52. 52. Chapter 4 Simple Linear Antennas THE SIMPLEST FORM OF ANTENNA , and the form in most common use is the linear antenna—the dipole. Used on its own, or in arrays, it is the most recognisable antenna. 4.1 The Ideal Dipole The ideal dipole must be one of the most useful theoretical antennas to un- Though it may seem im- practical, it is often eas- iest to consider an ideal case and thence deduce results for more complex examples. derstand as a large number of other antennas are analyzed using the equations that are quite easily developed for this antenna. Examples of these are the short dipole, loop antennas, travelling wave antennas and some arrays. The radiation pattern of any wire construction on which the currents are known can also be readily determined by considering the structure to consist of connected ideal dipoles and adding the pattern contribution due to each to form the full pattern. Many computer analysis codes rely on this approach. The ideal dipole is defined as a linear wire antenna with length very small with respect to the wavelength and a uniform current distribution. For convenience, this antenna is positioned at the centre of the coordinate system and aligned in the z-direction, as shown in figure 4.1. 4.1.1 Fields Using Maxwell’s equations and the simplicity of this geometry it is very easy to find the fields due to the constant current I (Kraus & Fleisch 1999, pg278). When such an analysis is performed it is found that the far field of the antenna has an E-field in the θ direction, Eθ, and a φ-directed H-field, Hφ only. The expression for the E-field will be given but the H-field can clearly be found by “Ohm’s Law of Free Space” as discussed in section 1.3.3. Eθ = 60πI0 λr jej(2πf−βr) sin θ (4.1) 41
  53. 53. 42 Simple Linear Antennas I0 x y z Eθ Hφ IdealCoord Figure 4.1: The Ideal Dipole in Relation to the Coordinate System There are a number of important points relating to this expression. Considering it factor by factor: • 60π is the constant or magnitude • Io is the (constant) current magnitude. An increase in this value results in a corresponding increase in the field • λ is the electrical length of the antenna and again an increase in this ratio will imply a larger field. Changes in this ratio should only be made such that the assumption of small electrical length still holds (0.1λ maximum). • jej(2πf−βr) is the phase factor. This factor is relatively unimportant unless this antenna is combined with another and the total pattern becomes an addition of the fields where phase plays an important role. • sin θ is the pattern factor. This is the only factor indicating variation with respect to the spherical coordinate system angles. Since none of the factors contain a φ-term this antenna has constant pattern characteristics in the azimuth direction. The resulting pattern has the familiar “doughnut”Another way of putting this is that the antenna has omnidirectional az- imuthal coverage. shape as illustrated in figure 4.2 The form of equation (4.1) is common to the expressions for most antenna field distributions. Such distributions are always a function of excitation, geometry in terms of wavelength and θ and φ angles. The relative pattern of the antenna can be drawn using only the sin θ term and regarding the rest as a normalizing factor. Where absolute field strengths are required the total equation should clearly be used.
  54. 54. 4.1 The Ideal Dipole 43 θ dipole y z Doughnut Figure 4.2: Pattern of an Ideal Dipole Antenna 4.1.2 Radiation resistance The radiation resistance of the antenna can be found once the field distribution is known. Using circuit concepts, the radiation resistance Rr is given by: Rr = 2Pt I2 0 Ω (4.2) The total power transmitted Pt is found by integrating (adding) the power The factor of two is in- troduced as result of the fact that I0 is the peak current and not the RMS value. density over a surface surrounding the antenna. Clearly if the power densities in all directions have been accounted for, the total power is found. The power density in any direction can be found using the expression discussed before: Pd = E2 2(120π) Performing this integration, an expression for total power radiated is obtained and using 4.2 the radiation resistance is found as: Rr = 80π2 λ 2 Ω (4.3) This value is clearly always small since the ratio of antenna length to wavelength ( /λ) was assumed to be small (≤0.1) at the outset of the analysis. 4.1.3 Directivity The directivity of the ideal dipole is calculated by assuming an input power of 1 W to the antenna. Since the reference used is always the isotrope, the power that it would radiate, given the same input power is simply Pd (isotrope) = 1 4πr2 .
  55. 55. 44 Simple Linear Antennas The current to an ideal dipole with 1 W input power is given by I0 = 2 Rr . Using (4.3) for Rr in the expression above results in: I0 = 2 80π2( /λ)2 (4.4) Substituting (4.4) into (4.1) the E-field can be found in the maximum direction (θ = 90◦ ). The power density in this direction, Pd (ideal dipole) is found by the relationship: Pd = E2 2(120π) = (60π)2 2 2 (λr)2 80π2 ( /λ)2 2(377) The directivity by definition is the ratio, which becomes: D = Pd (ideal dipole) Pd (isotrope) = 1.5(= 1.76dBi) 4.1.4 Concept of current moment An important concept which allows the use of the results achieved for the ideal dipole above to other antennas is that of current moment. By inspection of (4.1) it is clear that the E-field is proportional to the product of the length of the antenna and the current (assumed constant over the whole antenna). The current moment M for an ideal dipole is therefore the area under the current distribution: M = I0 The power density and power transmitted is proportional to the current moment squared — ie: E ∝ M P ∝ M2 4.2 The Short Dipole The short dipole antenna is a practically realizable antenna which is assumed to have a triangular current distribution when shorter than about a tenth of a wavelength, as shown in figure 4.3. Since it can be shown that the current distribution on thin linear radiators is sinusoidal, the two small parts of a sinusoid starting at either tip of the short dipole is well approximated by two straight lines and hence a triangular distribution.
  56. 56. 4.2 The Short Dipole 45 Iin ShortDip Figure 4.3: Current Distribution on a Short Dipole Antenna 4.2.1 Fields The current moment of the short dipole in terms of the feedpoint current Iin is: M = Iin 2 The E-field from the antenna is thus half the E-field found for the ideal dipole (disregarding the phase terms which would be the same) i.e. E = 30πIin λr sin θ 4.2.2 Radiation resistance The power transmitted by the short dipole is proportional to the square of the current moment (ie a quarter): Pt(short dipole) = Pt(ideal dipole) 4 since Pt = I2 R the radiation resistance of the short dipole would be a quarter of that of the ideal dipole Rr(short dipole) = 20π2 λ 2 Ω 4.2.3 Reactance The reactance of a short dipole is always capacitive and usually quite large The reactance of these short antennas is a very strong function of the an- tenna thickness and (ob- viously) length. and is not as easily calculated as the radiation resistance. Reactance values can be measured for a specific antenna—and tables (King & Harrison 1969) are available for different thickness antennas. The equivalent circuit of a short dipole antenna can be given as in figure 4.4
  57. 57. 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. L C R0 Rr TxLn Short Dipole L/2L/2 Tuning Figure 4.5: Tuning out dipole capacitive reactance with series inductance This is an improvement but a few “catch-22” problems still exist which explains the inherent difficulty in transferring power to small antennas: • The coil will have some loss resistance which is very often large compared to radiation resistance (which is often a fraction of an ohm) resulting in very low efficiency. • To decrease coil losses the Q of the inductor should be increased but this causes a reduced operating bandwidth and a more sensitive antenna, also increasing the circulating currents and hence the voltages associated with them. • If the decrease in bandwidth can be tolerated, the resultant real (resonant) impedance would approximate the very low radiation resistance and this still presents a matching problem. 4.2.4 Directivity It is clear from the sin θ factor in the E-field expression that the shape of the pattern is exactly the same as that of the ideal dipole. The directivity (gain) of a short dipole is therefore equal to the gain of the ideal dipole: D (short dipole) = 1.5
  58. 58. 4.3 The Short Monopole 47 4.3 The Short Monopole I0 h ShortMono Ground Plane Figure 4.6: Short monopole antenna When a ground plane is present as in figure 4.6 antennas can be analyzed in Once image theory is ap- plied (and this is true of any antenna/image combination) the ground- plane behaviour can be deduced from that of the free space equivalent. terms of image theory. The antenna/image combination has the same radiation pattern as the short dipole. The two major differences between the two are: • the monopole current moment is half that of the dipole • the monopole radiates no power in the lower hemisphere—for the same input power as the dipole, the monopole radiates twice as much power into the upper hemisphere. The power radiated is halved and the radiation resistance is half that of a short dipole when expressed in terms of . For monopoles, the length of the antenna above the ground h = /2 is clearly more relevant than and in terms of this the radiation resistance is: Rr = 40π2 h λ 2 All the power is radiated in the upper hemisphere which implies double power density in all directions in comparison to short—or ideal dipoles with the same power input. The directivity of this antenna would thus also be double that of the previous two antennas: As usual, increased gain is at the expense of de- creased gain elsewhere— under the ground plane in this case! D (short monopole) = 2 (1.5) = 3 4.3.1 Input impedance It was shown above that the radiation resistance of the short monopole is half that of the equivalent short dipole. The same applies to the capacitive reactance of the antenna.
  59. 59. 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 dipoles. The only difference in this case is that the phase of the current can notIt is interesting to note that the current distribu- tion must be known be- fore the various parame- ters of an antenna can be determined. be assumed to be constant and that the path lengths to a distant point P can differ from the different locations on the antenna. In the above cases, the current distributions were assumed to be sinusoidal mak- ing analysis possible. This assumption is quite valid for thin linear radiators as was shown by Schelkunoff (1941) and others. For more complex structures (and thick dipoles) the current distribution may be more difficult to determine. Com- putational techniques such as the Method of Moments, embodied in SuperNEC, are therefore primarily concerned with the determination of the current on the antenna wires. Once this is known it is a relatively straightforward task to calculate impedance and radiation pattern of the antenna. 4.4.1 Radiation pattern Using the sinusoidal current assumption, the magnitude of the electric field distribution around the dipole can be determined as (noting that /λ = λ/2): E = 60I r · cos π 2 cos θ sin θ 4.4.2 Directivity The pattern of this antenna relative to that of a short dipole is shown in figure 4.8 The directivity of this antenna is clearly not much larger than that of the short dipole. The accurate value is: D (half wave dipole) = 1.64 this is equivalent to 2.16 dBi (relative to isotropic).
  60. 60. 4.4 The Half Wave Dipole 49 θ = 0◦ (Dipole axis) Isotrope Short Dipole Half Wave Dipole HalfPat 90◦ Figure 4.8: Half wave dipole, short dipole and isotrope patterns It is immediately clear that there is not a large difference between the gain of the half wave dipole and that of the short dipole. This initially does not make sense since a short dipole can be very much smaller than a dipole and hence cheaper and more practical. The primary reason for the popularity of the half wave dipole is its large and resonant input impedance—which was the problem with the short dipole. Similarly, the directivity of a quarter wave monopole—which is the image theory equivalent of a half wave dipole—can be found as: D (quarter wave monopole) = 2.16 + 3 = 5.16 dBi The notation dBi is quite important and has been assumed until now. Very often antenna gain and directivity is quoted relative to a half wave dipole since this is a physically realizable antenna unlike the isotrope. The gain can thus be directly measured by comparing the signal strength received from a half wave dipole to that of the test antenna. When gain is quoted relative to a dipole it should be clearly stated and often this is done by using the notation dBd It is always important to ascertain which of these two references are used when gain is specified or quoted since many sources do not distin- guish between the two— not an ignorable differ- ence! (decibels relative to dipole). The conversion between the two is evident: dBi = dBd + 2.16 4.4.3 Input impedance By analysis, the input impedance for thin half wave dipoles is: Zin = 73 + j43 Ω This antenna is thus slightly longer than the length required for resonance. When a thin antenna is shortened by about 2% resonance can be obtained. Figure 4.9 indicates the shortening required for various length to diameter ratios. As before, the quarter wave monopole has half the input impedance of the half wave dipole. Zin (quarter wave monopole) = 36.5 + j21 Ω
  61. 61. 50 Simple Linear Antennas 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 101 102 103 104 Multiplyingfactor,K Ratio of half-wave length to diameter Short Figure 4.9: Shortening factors for different thickness half wave dipoles The relatively large values of radiation resistance of these antennas makes for easy transfer of power and virtually lossless antennas when good conductors are used. Efficiencies are typically 99% or higher and losses can thus be neglected.The impedance band- width of thin dipole antennas as defined by the VSWR 2 : 1 limi- tation is typically 5% of the centre frequency. For thicker antennas (small length to diameter ratios) this bandwidth can be larger. This may be untrue in cases of extremely thin wires or high frequencies (>1000 MHz). 4.5 The Folded Dipole It is very seldom that folded dipoles of other values than half wave length (or slightly less to achieve resonance) are used. The term folded dipole would thus be used to denote such an antenna unless otherwise stated. A typical Folded Dipole is shown in figure 4.10. Folded dipoles are often used instead of normal λ/2 s λ FD Figure 4.10: Half Wave Folded Dipole Antenna dipoles for the following reasons: • Mechanically easier to manipulate and more sturdy
  62. 62. 4.5 The Folded Dipole 51 • Larger bandwidth than normal dipoles • Larger input resistance than a normal dipole • Can offer a direct DC path to ground —for lightning protection. The element centre op- posite the feed can be “shorted” to the boom since this is a zero volt- age point This antenna’s characteristics are again easily understood using the current moment technique. Clearly the currents in each arm are sinusoidally distributed as in a half wave dipole and are in the same direction. The current moment of this antenna is thus double that of the normal dipole. This implies: Rin = 4(70) = 280 Ω D (folded dipole) = D (half wave dipole) = 2.16dBi E (folded dipole) = 2E (half wave dipole) Folded dipoles typically have an impedance bandwidth (defined by the VSWR 2 : 1 limit) of 10 to 12%. This increase relative to an half-wave dipole can be explained by noting that the antenna can be considered to be a superposition of a normal dipole and a short circuit transmission line. When the frequency is lowered the dipole exhibits a capacitive reactance whereas the transmission line starts to be inductive. These two factors initially cancel each other causing the increase in bandwidth. A comparison of the VSWR bandwidths of an ideally matched dipole and folded dipole is shown in figure 4.11. 200 250 300 350 400 450 500 1 1.5 2 2.5 3 3.5 4 4.5 5 freq (MHz) VSWR dipbw Dipole Folded Dipole Figure 4.11: VSWR bandwidth of a dipole and folded dipole. A very interesting fact about these antennas is that the input resistance can be increased or decreased by making the two elements of different diameters as shown in figure 4.12.
  63. 63. 52 Simple Linear Antennas d 2r1 2r2 FDThick Figure 4.12: Impedance transformation using different thickness elements The impedance transformation ratio is Rin(folded dipole) = (1 + a)2 Rin(half wave dipole) where a = log(d/r1) log(d/r2) The impedance can also be manipulated by using more than two elements, as shown in figure 4.13. The impedance step-up ratio under these conditions (where λ/2 FDMany Figure 4.13: Triply Folded Dipoles n is the number of elements) is: Rin = n2 Rin(half wave dipole) 4.6 Dipoles Above a Ground Plane So far only monopole antennas which are dependent on the ground for their operation have been discussed. It is interesting to observe the changes that occur for horizontal dipoles above a ground plane as shown in figure 4.14. A few general cases will be considered here to indicate the effects qualitatively.Since image theory is used to determine ground plane effects, it implies that a dipole above ground can be considered to be a two element array (see section 5.1 on page 57).
  64. 64. 4.7 Mutual Impedance 53 2h P1 S1 S2 θ θ Direct-wave path Reflected-wave path Reflecting Surface ReflRay Path Difference = 2h sin θ Figure 4.14: A horizontal dipole above a ground plane There are are two rays with different path lengths to point P and constructive and destructive interference will take place—depending on the path and hence phase difference. When constructive interference occurs, the two waves will add in phase and double the E-field result as compared to the free space pattern. In power density terms this would be an increase of 4 times, or 6 dB! This apparent “free” gain obtained by placing an antenna above a ground plane is clearly at the expense of reduced gain at other angles where the interference would be destructive, resulting in a null. However, with intelligent use it can be put to great effect. One of the main disadvantages of antennas close to the ground is that the current in the antenna and that in the image are clearly in opposite directions. At grazing angles they will always be out of phase—and cause a null. This Grazing angles are those that are close to the ground plane can be a severe problem for ground-to-ground communications and the only real remedy is to mount the antenna sufficiently high so that this effect can be neglected. 4.7 Mutual Impedance When two dipoles (or in fact any two antennas) are close enough to each other to cause appreciable currents to flow on the one antenna as result of radiation by the other they are said to be mutually coupled. The radiation patterns of such an intentional or accidental combination can easily be determined by adding the field contributions from each vectorially. Another important consequence is that the input impedance of the antennas is often altered considerably. The same statement clearly applies to antennas close to a reflecting plane since interaction with the image results. This effect is graphically illustrated by con-
  65. 65. 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: Z12 = V2 I1 Ω Curves of the mutual impedance for the geometry shown in figure 4.15 is given in figure 4.16. The following equations can be written to find the currents on the antennas and hence their impedance (Balanis 1982, sec 8.6,pg 412): V1 = I1Z11 + I2Z12 V2 = I1Z12 + I2Z22 Usually some voltage is assumed as an excitation (V1 & V2) for the two antennas and thus using the above two equations, the resultant currents I1 and I2 are found. The input impedance to dipole 1 would then be conventionally defined as: Zin = V1 I1 This value will generally be different to the free space self impedance of the antenna Z11.
  • FadilHamdani2

    Oct. 29, 2019
  • MONRAI

    Oct. 22, 2019
  • APatro1

    Apr. 16, 2019
  • farqadharoon

    Jul. 31, 2017
  • FadliMukhtarif

    Jul. 17, 2017
  • oshwaish

    Jul. 16, 2017
  • ebiele

    Jul. 16, 2017
  • dylan83

    Jul. 16, 2017
  • russelhossain

    Jul. 14, 2017
  • srinivasvgn

    Jul. 13, 2017
  • daba1987

    Jul. 11, 2017
  • TakehiroKawahara

    Jul. 10, 2017
  • kamalpreet08

    Jul. 10, 2017
  • JoshAranov

    Jul. 10, 2017
  • NilayTailor1

    Jul. 10, 2017
  • MichaelMardinian

    Jul. 9, 2017
  • SergioChiodi

    Jul. 8, 2017
  • CarmelKambire

    Jul. 8, 2017
  • BenJones75

    Jul. 7, 2017
  • diazpomalayajose

    Jul. 6, 2017

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 revamped on several occasions, mainly reflecting changing typesetting 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 commonplace 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 dependent 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 SimulationWorkshop, 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. Alan Robert Clark Andre P C Fourie Version 1.4, December 23, 2002

Views

Total views

15,132

On Slideshare

0

From embeds

0

Number of embeds

6,023

Actions

Downloads

1,312

Shares

0

Comments

0

Likes

54

×