WILEY ANTENNAS User Manual

ANTENNAS

FROM THEORY TO PRACTICE

Yi Huang

University of Liverpool, UK

Kevin Boyle

NXP Semiconductors, UK

A John Wiley and Sons, Ltd, Publication

ANTENNAS

ANTENNAS

FROM THEORY TO PRACTICE

Yi Huang

University of Liverpool, UK

Kevin Boyle

NXP Semiconductors, UK

A John Wiley and Sons, Ltd, Publication

This edition first published 2008



C

2008 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Huang, Yi.

Antennas : from theory to practice / Yi Huang, Kevin Boyle.

p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-51028-5 (cloth)

1. Antennas (Electronics) I. Boyle, Kevin. II. Title.
TK7871.6.H79 2008



621.382

4—dc22 2008013164

A catalogue record for this book is available from the British Library.

ISBN 978-0-470-51028-5 (HB)

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India.
Printed in Singapore by Markono Print Media Pte Ltd, Singapore.

Contents

Preface xi

Acronyms and Constants xiii

1 Introduction 1

1.1 A Short History of Antennas 1

1.2 Radio Systems and Antennas 4

1.3 Necessary Mathematics 6

1.3.1 Complex Numbers 6

1.3.2 Vectors and Vector Operation 7

1.3.3 Coordinates 10

1.4 Basics of Electromagnetics 11

1.4.1 The Electric Field 12

1.4.2 The Magnetic Field 15

1.4.3 Maxwell’s Equations 16

1.4.4 Boundary Conditions 19

1.5 Summary 21
References 21
Problems 21

2 Circuit Concepts and Transmission Lines 23

2.1 Circuit Concepts 23

2.1.1 Lumped and Distributed Element Systems 25

2.2 Transmission Line Theory 25

2.2.1 Transmission Line Model 25

2.2.2 Solutions and Analysis 28

2.2.3 Terminated Transmission Line 32

2.3 The Smith Chart and Impedance Matching 41

2.3.1 The Smith Chart 41

2.3.2 Impedance Matching 44

2.3.3 The Quality Factor and Bandwidth 51

2.4 Various Transmission Lines 55

2.4.1 Two-wire Transmission Line 56

2.4.2 Coaxial Cable 57

2.4.3 Microstrip Line 60

vi Contents

2.4.4 Stripline 63

2.4.5 Coplanar Waveguide (CPW) 66

2.4.6 Waveguide 68

2.5 Connectors 70

2.6 Summary 74
References 74
Problems 74

3 Field Concepts and Radio Waves 77

3.1 Wave Equation and Solutions 77

3.1.1 Discussion on Wave Solutions 79

3.2 The Plane Wave, Intrinsic Impedance and Polarization 80

3.2.1 The Plane Wave and Intrinsic Impedance 80

3.2.2 Polarization 82

3.3 Radio Wave Propagation Mechanisms 83

3.3.1 Reflection and Transmission 83

3.3.2 Diffraction and Huygens’s Principle 91

3.3.3 Scattering 92

3.4 Radio Wave Propagation Characteristics in Media 93

3.4.1 Media Classification and Attenuation 93

3.5 Radio Wave Propagation Models 97

3.5.1 Free Space Model 97

3.5.2 Two-ray Model/Plane Earth Model 98

3.5.3 Multipath Models 99

3.6 Comparison of Circuit Concepts and Field Concepts 101

3.6.1 Skin Depth 101

3.7 Summary 104
References 104
Problems 104

4 Antenna Basics 107

4.1 Antennas to Radio Waves 107

4.1.1 Near Field and Far Field 108

4.1.2 Antenna Parameters from the Field Point of View 112

4.2 Antennas to Transmission Lines 122

4.2.1 Antenna Parameters from the Circuit Point of View 122

4.3 Summary 125
References 126
Problems 126

5 Popular Antennas 129

5.1 Wire-Type Antennas 129

5.1.1 Dipoles 129

5.1.2 Monopoles and Image Theory 137

5.1.3 Loops and the Duality Principle 141

5.1.4 Helical Antennas 147

Contents vii

5.1.5 Yagi–Uda Antennas 152

5.1.6 Log-Periodic Antennas and Frequency-Independent Antennas 157

5.2 Aperture-Type Antennas 163

5.2.1 Fourier Transforms and the Radiated Field 163

5.2.2 Horn Antennas 169

5.2.3 Reflector and Lens Antennas 175

5.2.4 Slot Antennas and Babinet’s Principle 180

5.2.5 Microstrip Antennas 184

5.3 Antenna Arrays 191

5.3.1 Basic Concept 192

5.3.2 Isotropic Linear Arrays 193

5.3.3 Pattern Multiplication Principle 199

5.3.4 Element Mutual Coupling 200

5.4 Some Practical Considerations 203

5.4.1 Transmitting and Receiving Antennas: Reciprocity 203

5.4.2 Baluns and Impedance Matching 205

5.4.3 Antenna Polarization 206

5.4.4 Radomes, Housings and Supporting Structures 208

5.5 Summary 211
References 211
Problems 212

6 Computer-Aided Antenna Design and Analysis 215

6.1 Introduction 215

6.2 Computational Electromagnetics for Antennas 217

6.2.1 Method of Moments (MoM) 218

6.2.2 Finite Element Method (FEM) 228

6.2.3 Finite-Difference Time Domain (FDTD) Method 229

6.2.4 Transmission Line Modeling (TLM) Method 230

6.2.5 Comparison of Numerical Methods 230

6.2.6 High-Frequency Methods 232

6.3 Examples of Computer-Aided Design and Analysis 233

6.3.1 Wire-type Antenna Design and Analysis 233

6.3.2 General Antenna Design and Analysis 243

6.4 Summary 251
References 251
Problems 252

7 Antenna Manufacturing and Measurements 253

7.1 Antenna Manufacturing 253

7.1.1 Conducting Materials 253

7.1.2 Dielectric Materials 255

7.1.3 New Materials for Antennas 255

7.2 Antenna Measurement Basics 256

7.2.1 Scattering Parameters 256

7.2.2 Network Analyzers 258

viii Contents

7.3 Impedance, S11, VSWR and Return Loss Measurement 261

7.3.1 Can I Measure These Parameters in My Office? 261

7.3.2 Effects of a Small Section of a Transmission Line or a Connector 262

7.3.3 Effects of Packages on Antennas 262

7.4 Radiation Pattern Measurements 263

7.4.1 Far-Field Condition 264

7.4.2 Open-Area Test Sites (OATS) 265

7.4.3 Anechoic Chambers 267

7.4.4 Compact Antenna Test Ranges (CATR) 268

7.4.5 Planar and Cylindrical Near-Field Chambers 270

7.4.6 Spherical Near-Field Chambers 270

7.5 Gain Measurements 272

7.5.1 Comparison with a Standard-Gain Horn 272

7.5.2 Two-Antenna Measurement 272

7.5.3 Three-Antenna Measurement 273

7.6 Miscellaneous Topics 273

7.6.1 Efficiency Measurements 273

7.6.2 Reverberation Chambers 274

7.6.3 Impedance De-embedding Techniques 275

7.6.4 Probe Array in Near-Field Systems 276

7.7 Summary 281
References 281
Problems 282

8 Special Topics 283

8.1 Electrically Small Antennas 283

8.1.1 The Basics and Impedance Bandwidth 283

8.1.2 Antenna Size-Reduction Techniques 299

8.2 Mobile Antennas, Antenna Diversity and Human Body Effects 304

8.2.1 Introduction 304

8.2.2 Mobile Antennas 305

8.2.3 Antenna Diversity 318

8.2.4 User Interaction 325

8.3 Multiband and Ultra-Wideband Antennas 334

8.3.1 Introduction 334

8.3.2 Multiband Antennas 334

8.3.3 Wideband Antennas 337

8.4 RFID Antennas 340

8.4.1 Introduction 340

8.4.2 Near-Field Systems 343

8.4.3 Far-Field Systems 349

8.5 Reconfigurable Antennas 352

8.5.1 Introduction 352

8.5.2 Switching and Variable-Component Technologies 352

8.5.3 Resonant Mode Switching/Tuning 354

Contents ix

8.5.4 Feed Network Switching/Tuning 355

8.5.5 Mechanical Reconfiguration 355

8.6 Summary 356
References 356

Index 357

Preface

As an essential element of a radio system, the antenna has always been an interesting but
difficult subject for radio frequency (RF) engineering students and engineers. Many good
books on antennas have been published over the years and some of them were used as our
major references.

This book is different from other antenna books. It is especially designed for people who
know little about antennas but would like to learn this subject from the very basics to practical
antenna analysis, design and measurement within a relatively short period of time. In order
to gain a comprehensive understanding of antennas, one must know about transmission lines
and radio propagation. At the moment, people often have to read a number of different books,
which may not be well correlated. Thus, it is not the most efficient way to study the subject.
In this book we put all the necessary information about antennas into a single volume and
try to examine antennas from both the circuit point of view and the field point of view. The
book covers the basic transmission line and radio propagation theories, which are then used
to gain a good understanding of antenna basics and theory. Various antennas are examined
and design examples are presented. Particular attention is given to modern computer-aided
antenna design. Both basic and advanced computer software packages are used in examples to
illustrate how they can be used for antenna analysis and design. Antenna measurement theory
and techniques are also addressed. Some special topics on the latest antenna development are
covered in the final chapter.

The material covered in the book is mainly based on a successful short course on antennas
for practising professionals at the University of Oxford and the Antennas module for students
at the University of Liverpool. The book covers important and timely issues involving modern
practical antenna design and theory. Many examples and questions are given in each chapter. It
is an ideal textbook for universityantenna courses, professionaltraining courses and self-study.
It isalso a valuable reference forengineers anddesigners who work with RF engineering, radar
and radio communications.

The book is organized as follows:

Chapter 1:Introduction.The objective of this chapter is tointroduce theconcept of antennas
and review essential mathematics and electromagnetics, especially Maxwell’s equations. Ma­terial properties (permittivity,permeability and conductivity) are discussed and some common
ones are tabulated.

Chapter 2: Circuit Concepts and Transmission Lines. The concepts of lumped and dis­tributed systems are established. The focus is placed on the fundamentals and characteristics
of transmission lines. A comparison of various transmission lines and connectors is presented.
The Smith Chart, impedance matching and bandwidth are also addressed in this chapter.

xii Preface

Chapter 3: Field Concepts and Radio Waves. Field concepts, including the plane wave,
intrinsic impedance and polarization, are introduced and followed by a discussion on radio
propagation mechanisms and radio wave propagation characteristics in various media. Some
basic radio propagation models are introduced, and circuit concepts and field concepts are
compared at the end of this chapter.

Chapter 4: Antenna Basics. The essential and important parameters of an antenna (such
as the radiation pattern, gain and input impedance) are addressed from both the circuit point
of view and field point of view. Through this chapter, you will become familiar with antenna
language, understand how antennas work and know what design considerations are.

Chapter 5:Popular Antennas. In this long chapter, some of the most popular antennas (wire­type, aperture-type and array antennas) are examined and analyzed using relevant antenna
theories. The aim is to see why they have become popular, what their major features and
properties are (including advantages and disadvantages) and how they should be designed.

Chapter 6: Computer-Aided Antenna Design and Analysis.Theaimofthis special and unique
chapter is to give a brief review of antenna-modeling methods and software development,
introduce the basic theory behind computer simulation tools and demonstrate how to use
industry standard software to analyze and design antennas. Two software packages (one is
simple and free) are presented with step-by-step illustrations.

Chapter 7: Antenna Manufacturing and Measurements. This is another practical chapter to
address two important issues: how to make an antenna and how to conduct antenna measure­ment, with a focus placed on the measurement. It introduces S-parameters and equipment. A
good overview of the possible measurement systems is provided with an in-depth example.
Some measurement techniques and problems are also presented.

Chapter 8: Special Topics. This final chapter presents some of the latest important develop­ments in antennas. It covers mobile antennas and antenna diversity, RFID antennas, multiband
and broadband antennas, reconfigurable antennas and electrically small antennas. Both the
theory and practical examples are given.

The authors are indebted to the many individuals whoprovidedusefulcomments,suggestions
and assistance to make this book a reality. In particular, we would like to thank Shahzad
Maqbool, Barry Cheeseman and Yang Lu at the University of Liverpool for constructive
feedback and producing figures, Staff at Wiley for their help and critical review of the book,
Lars Foged at SATIMO and Mike Hillbun at Diamond Engineering for their contribution to
Chapter 7 and the individuals and organizations who have provided us with their figures or
allowed us to reproduce their figures.

Yi Huang and Kevin Boyle

Acronyms and Constants

ε

0

μ

0

η

0

AC Alternating current
AF Antenna factor
AM Amplitude modulation
AR Axial ratio
AUT Antenna under test
BER Bit error rate
CAD Computer-aided design
CATR Compact antenna test range
CDF Cumulative distribution function
CEM Computational electromagnetics
CP Circular polarization
CPW Coplanar waveguide
DC Direct current
DCS Digital cellular system
DRA Dielectric resonant antenna
DUT Device under test
EGC Equal gain combining
EIRP Effective isotropic radiated power
EM Electromagnetic
EMC Electromagnetic compatibility
ERP Effective radiated power
FDTD Finite-difference time domain
FEM Finite element method
FNBW First null beamwidth
GPS Global positioning system
GSM Global System for Mobile communications
GTD Geometrical theory of diffraction
HPBW Half-power beamwidth
HW Hansen–Woodyard (condition)
ISI Inter-symbol interference

8.85419 ×10
4π ×10−7H/m

≈ 377 

−12

F/m

xiv Acronyms and Constants

LCP Left-hand circular polarization

Liquid crystal polymer
LPDA Log-periodic dipole antenna
MEMS Micro electromechanical systems
MIMO Multiple-in, multiple-out
MMIC Monolithic microwave integrated circuits
MoM Method of moments
MRC Maximal ratio combining
NEC Numerical electromagnetic code
OATS Open area test site
PCB Printed circuit board
PDF Power density function

Probability density function
PIFA Planar inverted F antenna
PO Physical optics
PTFE Polytetrafluoroethylene
RAM Radio-absorbing material
RCP Right-hand circular polarization
RCS Radar cross-section
RF Radio frequency
RFID Radio frequency identification
RMS Root mean square
SAR Specific absorption rate
SC Selection combining
SI units International system of units (metric system)
SLL Side-lobe level
SNR Signal-to-noise ratio
SWC Switch combining
TE Transverse electric (mode/field)
TEM Transverse electromagnetic (mode/field)
TM Transverse magnetic (mode/field)
TV Television
UHF Ultra-high frequency
UTD Uniform theory of diffraction
UWB Ultra-wide band
VHF Very high frequency
VNA Vector network analyzer
VSWR Voltage standing wave ratio
WLAN Wireless local area network
WiMax Worldwide interoperability of microwave access

1

Introduction

1.1 A Short History of Antennas

Work onantennasstartedmanyyears ago. The firstwell-knownsatisfactoryantennaexperiment
was conducted by the German physicist Heinrich Rudolf Hertz (1857–1894), pictured in
Figure 1.1. The SI (International Standard) frequency unit, the Hertz, is named after him. In
1887 he built a system, as shown in Figure 1.2, to produce and detect radio waves. The original
intention of his experiment was to demonstrate the existence of electromagnetic radiation.

In the transmitter, a variable voltage source was connected to a dipole (a pair of one-meter
wires) with two conducting balls (capacity spheres) at the ends. The gap between the balls
could be adjusted for circuitresonance as well as forthe generation ofsparks. When the voltage
was increased to a certain value, a spark or break-down discharge was produced. The receiver
was asimple loop with two identical conducting balls. The gap between theballs wascarefully
tuned to receive the spark effectively. He placed the apparatus in a darkened box in order to
see the spark clearly. In his experiment, when a spark was generated at the transmitter, he also
observed a spark at the receiver gap at almost the same time. This proved that the information
from location A (the transmitter) was transmitted to location B (the receiver) in a wireless
manner – by electromagnetic waves.

The information in Hertz’s experiment was actually in binary digital form, by tuning the
spark on andoff.Thiscould be considered theveryfirst digital wireless system,which consisted
of two of the best-known antennas: the dipole and the loop. For this reason, the dipole antenna
is also called the Hertz (dipole) antenna.

Whilst Heinrich Hertz conducted his experiments in a laboratory and did not quite know
what radio waves might be used for in practice, Guglielmo Marconi (1874–1937, pictured
in Figure 1.3), an Italian inventor, developed and commercialized wireless technology by
introducing a radiotelegraph system, which served as the foundation for the establishment of
numerous affiliated companies worldwide. His most famous experiment was the transatlantic
transmission from Poldhu, UK to St Johns, Newfoundland in Canada in 1901, employing
untuned systems. He shared the 1909 Nobel Prize for Physics with Karl Ferdinand Braun
‘in recognition of their contributions to the development of wireless telegraphy’. Monopole
antennas (near quarter-wavelength) were widely used in Marconi’s experiments; thus vertical
monopole antennas are also called Marconi antennas.

Antennas: From Theory to Practice Yi Huang and Kevin Boyle



C

2008 John Wiley & Sons, Ltd

2 Antennas: From Theory to Practice

Figure 1.1 Heinrich Rudolf Hertz

During World War II, battles were won by the side that was first to spot enemy aeroplanes,
ships or submarines. To give the Allies an edge, British and American scientists developed
radar technology to ‘see’ targets from hundreds of miles away, even at night. The research
resulted in the rapid development of high-frequency radar antennas, which were no longer just
wire-type antennas. Some aperture-type antennas, such as reflector and horn antennas, were
developed, an example is shown in Figure 1.4.

Variable

Voltage Source

Figure 1.2 1887 experimental set-up of Hertz’s apparatus

Loop

Introduction 3

Figure 1.3 Guglielmo Marconi

Broadband, circularly polarized antennas, as well as many other types, were subsequently
developed for various applications. Since an antenna is an essential device for any radio
broadcasting, communication or radar system, there has always been a requirement for new
and better antennas to suit existing and emerging applications.

More recently, one of the main challenges for antennas has been how to make them broad­band and small enough in size for wireless mobile communications systems. For example,
WiMAX (worldwide interoperability for microwave access) is one of the latest systems aimed
at providing high-speed wireless data communications (>10 Mb/s) over long distances from
point-to-point links tofull mobile cellular-typeaccess over a widefrequencyband. The original
WiMAX standard in IEEE 802.16 specified 10 to 66 GHz as the WiMAX band; IEEE 802.16a

Figure 1.4 World War II radar (Reproduced by permission of CSIRO Australia Telescope National
Facility)

4 Antennas: From Theory to Practice

was updated in 2004 to 802.16-2004 and added 2 to 11 GHz as an additional frequency range.
The frequency bandwidth is extremely wide although the most likely frequency bands to be
used initially will be around 3.5 GHz, 2.3/2.5 GHz and 5 GHz.

The UWB (ultra-wide band) wireless system is another example of recent broadband radio
communication systems. The allocated frequency band is from 3.1 to 10.6 GHz. The beauty of
the UWB system is that the spectrum, which is normally very expensive, can be used free of
charge but the power spectrum density is limited to −41.3 dBm/MHz. Thus, it is only suitable
for short-distance applications. The antenna design for these systems faces many challenging
issues.

The role of antennas is becoming increasingly important. In some systems, the antenna is
now no longer just a simple transmitting/receiving device, but a device which isintegrated with
other parts of the system to achieve better performance. For example, the MIMO (multiple-in,
multiple-out) antenna system has recently been introduced as an effective means to combat
multipath effects in the radio propagation channel and increase the channel capacity, where
several coordinated antennas are required.

Things have been changing quickly in the wireless world. But one thing has never changed
since the very first antenna was made: the antenna is a practical engineering subject. It will
remain an engineering subject. Once an antenna is designed and made, it must be tested. How
well it works is not just determined by the antenna itself, it also depends on the other parts of
the system and the environment. The standalone antenna performance can be very different
from that of an installed antenna. For example, when a mobile phone antenna is designed, we
must take the case, other parts of the phone and even our hands into account to ensure that it
will work well in the real world. The antenna is an essential device of a radio system, but not
an isolated device! This makes it an interesting and challenging subject.

1.2 Radio Systems and Antennas

A radio system is generally considered to be an electronic system which employs radio waves,
a type of electromagnetic wave up to GHz frequencies. An antenna, as an essential part of a
radio system, is defined as a device which can radiate and receive electromagnetic energy in
an efficient and desired manner. It is normally made of metal, but other materials may also be
used. For example, ceramic materials have been employed to makedielectricresonatorantennas
(DRAs). There are many things in our lives, such as power leads, that can radiate and receive
electromagnetic energy but they cannot be viewed as antennas because the electromagnetic
energy is not transmitted or received in an efficient and desired manner, and because they are
not a part of a radio system.

Since radio systems possess some unique and attractive advantages over wired systems,
numerous radio systems have been developed. TV, radar and mobile radio communication
systems are just some examples. The advantages include:

r

mobility: this is essential for mobile communications;

r

good coverage: the radiation from an antenna can cover a very large area, which is good for
TV and radio broadcasting and mobile communications;

r

low pathloss: this is frequency dependent. Since the loss of a transmission line is an expo­nential function of the distance (the loss in dB =distance ×per unit loss in dB) and the loss

Introduction 5

of a radio wave is proportional to the distance squared (the loss in dB = 20 log10(distance)),
the pathloss of radio waves can be much smaller than that of a cable link. For example,
assume that the loss is 10 dB for both a transmission line and a radio wave over 100 m; if the
distance is now increased to 1000 m, the loss for the transmission line becomes 10 × 10 =
100 dB but the loss for the radio link is just 10 + 20 = 30 dB! This makes the radio link
extremely attractive for long-distance communication. It should be pointed out that optical
fibers are also employed for long-distance communications since they are of very low loss
and ultra-wide bandwidth.

Figure 1.5 illustrates a typical radio communication system. The source information is
normally modulatedand amplified in the transmitter and then passed on to thetransmit antenna
via a transmission line, which has a typical characteristic impedance (explained in the next
chapter) of 50 ohms. The antenna radiates the information in the form of an electromagnetic
wave in an efficient and desired manner to the destination, where the information is picked up
by the receive antenna and passed on to the receiver via another transmission line. The signal
is demodulated and the original message is then recovered at the receiver.

Thus, the antenna is actually a transformer that transforms electrical signals (voltages and
currents from a transmission line) into electromagnetic waves (electric and magnetic fields),
or vice versa. For example, a satellite dish antenna receives the radio wave from a satellite and
transforms it into electrical signalswhich areoutput to a cable tobe further processed. Our eyes
may be viewed as another example of antennas. In this case, the wave is not a radio wave but
an optical wave, another form of electromagnetic wave which has much higher frequencies.

Now it is clear that the antenna is actually a transformer of voltage/current to electric/
magnetic fields, it can also be considered a bridge to link the radio wave and transmission line.
An antennasystem isdefined asthe combinationof theantenna andits feedline. Asan antenna
is usually connected to a transmission line, how to best make this connection is a subject of
interest, since the signal from the feed line should be radiated into the space in an efficient
and desired way. Transmission lines and radio waves are, in fact, two different subjects in
engineering. To understand antenna theory, one has to understand transmission lines and radio
waves, which will be discussed in detail in Chapters 2 and 3 respectively.

In some applications where space is very limited (such as hand-portables and aircraft), it is
desirable to integrate the antenna and its feed line. In other applications (such as the reception
of TV broadcasting), the antenna is far away from the receiver and a long transmission line
has to be used.

Unlike other devices in a radio system (such as filters and amplifiers), the antenna is a very
special device; itdeals with electrical signals (voltages and currents) as well aselectromagnetic
waves (electric fields and magnetic fields), making antenna design an interesting and difficult

Transmission

Transmitter

Line

Antenna

Electromagnetic

wave

Figure 1.5 A typical radio system

Antenna

Receiver

6 Antennas: From Theory to Practice

subject. For different applications, the requirements on theantenna may be verydifferent, even
for the same frequency band.

In conclusion, the subject of antennas is about how to design a suitable device which will be
well matched with its feed line and radiate/receive the radio waves in an efficient and desired
manner.

1.3 Necessary Mathematics

To understand antenna theory thoroughly requires a considerable amount of mathematics.
However, the intention of this book is to provide the reader with a solid foundation in antenna
theory andapply the theory to practical antenna design. Here weare justgoing tointroduce and
review the essential and important mathematics required for this book. More in-depth study
materials can be obtained from other references [1, 2].

1.3.1 Complex Numbers

In mathematics, a complex number, Z, consists of real and imaginary parts, that is

Z = R + jX (1.1)

where R is called the real part of the complex number Z , i.e. Re(Z), and X is defined as the
imaginary part of Z , i.e. Im(Z). Both R and X are real numbers and j (not the traditional
notation i in mathematics to avoid confusion with a changing current in electrical engineering)
is the imaginary unit and is defined by

√

j =

−1 (1.2)

Thus

2

j

=−1 (1.3)

Geometrically, a complex number can be presented in a two-dimensional plane where the
imaginary part is found on the vertical axis whilst the real part is presented by the horizontal
axis, as shown in Figure 1.6.

In this model, multiplication by −1 corresponds to a rotation of 180 degrees about the
origin. Multiplication by j corresponds toa 90-degree rotation anti-clockwise,and the equation

2

j

=−1 is interpreted as saying that if we apply two 90-degree rotations about the origin, the
net resultis asingle 180-degree rotation. Notethat a 90-degree rotation clockwise also satisfies
this interpretation.

Another representation of a complex number Z uses the amplitude and phase form:

Z = Ae

jϕ

(1.4)

Introduction 7

jX

Z (R, X)

A

ϕ

R

Figure 1.6 The complex plane

where A is the amplitude and ϕ is the phase of the complex number Z; these are also shown
in Figure 1.6. The two different representations are linked by the following equations:

Z = R + jX = Ae

√

A =

R2+ X2,ϕ= tan−1(X/R)

jϕ

;

(1.5)

R = A cos ϕ, X = A sin ϕ

1.3.2 Vectors and Vector Operation

A scalar is a one-dimensional quantity which has magnitude only, whereas a complex number
is a two-dimensional quantity. A vector can be viewed as a three-dimensional (3D) quantity,
and a special one – it has both a magnitude and a direction. For example, force and velocity
are vectors (whereas speed is a scalar). A position in space is a 3D quantity, but it does not
have a direction, thus it is not a vector. Figure 1.7 is an illustration of vector A in Cartesian

z

A

z

A

A

x

x

Figure 1.7 Vector A in Cartesian coordinates

A

y

y

8 Antennas: From Theory to Practice

coordinates. It has three orthogonal components (Ax, Ay, Az) along the x, y and z directions,
respectively. To distinguish vectors from scalars, the letter representing the vector is printed
in bold, for example A or a, and a unit vector is printed in bold with a hat over the letter, for
exampleˆx orˆn.

The magnitude of vector A is given by



2

2

|A|

= A =

A

+ A

x

y

+ A

2

z

(1.6)

Now let us consider two vectors A and B:

ˆ

ˆ

A = A

B = B

x + A

x

ˆ

x + B

x

y

y

y + A

ˆ

y + B

ˆ

z

z

ˆ

z

z

The addition and subtraction of vectors can be expressed as

A + B = (A
A − B = (A

+ Bx)ˆx + (Ay+ By)ˆy + (Az+ Bz)ˆz

x

− Bx)ˆx + (Ay− By)ˆy + (Az− Bz)ˆz

x

(1.7)

Obviously, the addition obeys the commutative law, that is A + B = B + A.

Figure 1.8 shows what the addition and subtraction mean geometrically. A vector may
be multiplied or divided by a scalar. The magnitude changes but its direction remains the
same. However, the multiplication of two vectors is complicated. There are two types of
multiplication: the dot product and the cross product.

The dot product of two vectors is defined as

ArB =|A||B|cos θ = A

+ AyBy+ AzB

xBx

z

(1.8)

where θ is the angle between vector A and vector B and cos θ is also called the direction
cosine. The dotrbetween A and B indicates the dot product, which results in a scalar; thus, it
is also called a scalar product. If the angle θ is zero, A and B are in parallel – thedot product is

A–B

A+B

B

B

A

Figure 1.8 Vector addition and subtraction

A

Introduction 9

C

Right-Hand

Rule

B

A

Figure 1.9 The cross product of vectors A and B

maximized – whereas for an angle of 90 degrees, i.e. when A and B are orthogonal, the dot
product is zero.

It is worth noting that the dot product obeys the commutative law, that is, ArB = BrA.
The cross product of two vectors is defined as

A × B =ˆn|A||B|sin θ = C

=ˆx(A

− AzBy) +ˆy( AzBx− AxBz) +ˆz( AxBy− AyBx)

yBz

(1.9)

whereˆn is a unit vector normal to the plane containing A and B. The cross × between A
and B indicates the cross product, which results in a vector C; thus, it is also called a vector
product. The vector C is orthogonal to both A and B, and the direction of C follows a so-called
right-hand rule, as shown in Figure 1.9. If the angle θ is zero or 180 degrees, that is, A and B
are in parallel, the cross product is zero; whereas for an angle of 90 degrees, i.e. A and B are
orthogonal, the cross product of these two vectors reaches a maximum.Unlike the dot product,
the cross product does not obey the commutative law.

The cross product may be expressed in determinant form as follows, which is the same as

Equation (1.9) but may be easier for some people to memorize:

A × B =




ˆ

xˆy




A





AyA

x

BxByB




ˆ

z





z




z

(1.10)

Another important thing about vectors is that any vector can be decomposed into three
orthogonal components (such as x, y and z components) in 3D or two orthogonal components
in a 2D plane.

Example 1.1: Vector operation. Given vectors A = 10

ˆ

x + 5ˆy + 1ˆz and B = 2ˆy, find:

A + B; A − B; A • B; and A × B

10 Antennas: From Theory to Practice

x

Solution:

A + B = 10ˆx + (5 +2)ˆy + 1ˆz = 10ˆx + 7ˆy + 1ˆz;
A − B = 10ˆx + (5 −2)ˆy + 1ˆz = 10ˆx + 3ˆy + 1ˆz;
A • B = 0 +(5 ×2) + 0 = 10;
A × B = 10 ×2ˆz + 1 × 2ˆx = 20ˆz + 2ˆx

1.3.3 Coordinates

In addition to the well-known Cartesian coordinates, spherical coordinates (r, θ,φ), as shown
in Figure 1.10,will also be used frequentlythroughout this book.These two coordinatesystems
have the following relations:

x = r sin θ cosφ
y = r sin θ sinφ
z = r cos θ

and

(1.11)



r =

θ = cos

φ = tan

x2+ y2+ z

−1



x2+ y2+ z

y

−1

;0≤ φ ≤ 2π

x

z

θ

r

φ

2

z

;0≤ θ ≤ π (1.12)

2

P

y

Figure 1.10 Cartesian and spherical coordinates

Introduction 11

The dot products of unit vectors in these two coordinate systems are:

r

ˆ

ˆ

x

r = sin θ cosφ;ˆy

r

ˆ

ˆ

x

θ = cos θ cos φ;ˆy

r

ˆ

ˆ

x

φ =−sin φ;ˆy

r

ˆ

r = sin θ sinφ;ˆz

r

ˆ

θ = cos θ sin φ;ˆz

r

ˆ

φ = cosφ;ˆz

r

ˆ

φ = 0

r

ˆ

r = cos θ

r

ˆ

θ =−sin θ

(1.13)

Thus, we can express a quantity in one coordinate system using the known parameters in the
other coordinate system. For example, if A

r

A

x

ˆ

= A

x = Arsin θ cosφ + Aθcos θ cosφ − Aφsin φ

, Aθ, Aφare known, we can find

r

1.4 Basics of Electromagnetics

Now let us use basic mathematics to deal with antennas or, more precisely, electromagnetic
(EM) problems in this section.

EM waves cover the whole spectrum; radio waves and optical waves are just two examples
of EM waves. We can see light but we cannot see radio waves. The whole spectrum is divided
into many frequency bands. Some radio frequency bands are listed in Table 1.1.

Although the whole spectrum is infinite, the useful spectrum is limited and some frequency
bands, such as the UHF, are already very congested. Normally, significant license fees have to
be paid to use the spectrum, although there are some license-free bands: the most well-known
ones are the industrial, science and medical (ISM) bands. The 433 MHz and 2.45 GHz are just
two examples. Cable operators do not need to pay the spectrum license fees, but they have to
pay other fees for things such as digging out the roads to bury the cables.

The wave velocity, v, is linked to the frequency, f , and wavelength, λ, by this simple
equation:

λ

v =

f

It is well known that the speed of light (an EM wave) is about 3 ×10

8

m/s in free space. The

(1.14)

higher the frequency,the shorter the wavelength. An illustration of how the frequency is linked

Table 1.1 EM spectrum and applications

Frequency Band Wavelength Applications

3–30 kHz VLF 100–10 km Navigation, sonar, fax
30–300 kHz LF 10–1 km Navigation

0.3–3 MHz MF 1–0.1 km AM broadcasting
3–30 MHz HF 100–10 m Tel, fax, CB, ship communications
30–300 MHz VHF 10–1 m TV, FM broadcasting

0.3–3 GHz UHF 1–0.1 m TV, mobile, radar
3–30 GHz SHF 100–10 mm Radar, satellite, mobile, microwave links
30–300 GHz EHF 10–1 mm Radar, wireless communications

0.3–3 THz THz 1–0.1 mm THz imaging

12 Antennas: From Theory to Practice

8

10

6

10

4

10

2

10

0

10

-2

10

Wavelength (m)

-4

10

-6

10

-8

10

-10

10

0 102 104 106 108 1010 1012 1014 1016

10

Optical

Communications

(1.7μm−0.8μm)

Conventional RF

1Hz−300MHz/1GHz

Microwave

300MHz−30GHz

(1m−1cm)

Millimeter Wave

30GHz−300GHz

Light

(0.76μm−0.4μm)

Frequency (Hz)

Figure 1.11 Frequency vs wavelength

to the wavelength is given in Figure 1.11, where both the frequency and wavelength are plotted
on a logarithmic scale. The advantage of doing this is that we can see clearly how the function
is changed, even over a very large scale.

Logarithmic scales are widely used in RF (radio frequency) engineering and the antennas
community since the signals we are dealing with change significantly (over 1000 times in
many cases) in terms of the magnitude. The signal power is normally expressed in dB and is
defined as

P(dBW) = 10log

10

P(W)

; P(dBm) = 10 log

1W

10

P(W)

1mW

(1.15)

Thus, 100 watts is 20 dBW, just expressed as 20 dB in most cases. 1 W is 0 dB or 30 dBm
and 0.5 W is −3 dB or 27 dBm. Based on this definition, we can also express other parameters
in dB. For example, since the power is linked to voltage V by P = V



2

R (so P ∝ V

2

), the

voltage can be converted to dBV by

V (dBV) = 20log



10

V (V )

1V



(1.16)

Thus, 3 kVolts is 70 dBV and 0.5 Volts is –6 dBV (not −3 dBV) or 54 dBmV.

1.4.1 The Electric Field

The electric field (in V/m) is defined as the force (in Newtons) per unit charge (in Coulombs).
From this definition and Coulomb’s law, the electric field, E, created by a single point

Introduction 13

charge Q at a distance r is

F

E =

Q

where

F is the electric force given by Coulomb’s law (F =
ˆ

ris a unit vector along the r direction, which is also the direction of the electric field E;

=

Q

ˆ

r (V /m) (1.17)

2

4πεr

Q1Q

2

ˆ

r);

2

4πεr

ε is the electricpermittivity (it is also calledthe dielectric constant,but is normallya function

of frequency and not really a constant, thus permittivity is preferred in this book) of the
material. Its SI unit is Farads/m. In free space, it is a constant:

ε

= 8.85419 × 10

0

−12

F/m (1.18)

The product of the permittivity and the electric field is called the electric flux density, D,

which is a measure of how much electric flux passes through a unit area, i.e.

where ε

D = εE = ε

= ε/ε0is called the relative permittivity or relative dielectric constant. The relative

r

E(C/m2) (1.19)

rε0

permittivities of some common materials are listed in Table 1.2. Note that they are functions
of frequency and temperature. Normally, the higher the frequency, the smaller the permittivity
in the radio frequency band. It should also be pointed out that almost all conductors have a
relative permittivity of one.

The electric flux density is also called the electric displacement, hence the symbol D.Itis
also a vector. In an isotropic material (properties independent of direction), D and E are in
the same direction and ε is a scalar quantity. In an anisotropic material, D and E may be in
different directions if ε is a tensor.

If the permittivity isa complex number, it meansthatthe material hassome loss. The complex
permittivity can be written as



− j ε



(1.20)

ε = ε

The ratio of the imaginary part to the real part is called the loss tangent, that is



tan δ =

ε



ε

(1.21)

It has no unit and is also a function of frequency and temperature.

The electric field E is related to thecurrent density J (in A/m

2

), another importantparameter,

by Ohm’s law. The relationship between them at a point can be expressed as

J = σ E (1.22)

where σ is the conductivity, which is the reciprocal of resistivity. It is a measure of a material’s
ability to conduct an electrical current and is expressed in Siemens per meter (S/m). Table 1.3

14 Antennas: From Theory to Practice

Table 1.2 Relative permittivity of some common materials at 100 MHz

Material Relative permittivity Material Relative permittivity

ABS (plastic) 2.4–3.8 Polypropylene 2.2
Air 1 Polyvinylchloride (PVC) 3
Alumina 9.8 Porcelain 5.1–5.9
Aluminum silicate 5.3–5.5 PTFE-teflon 2.1
Balsa wood 1.37 @ 1 MHz PTFE-ceramic 10.2

1.22 @ 3 GHz
Concrete ∼8 PTFE-glass 2.1–2.55
Copper 1 RT/Duroid 5870 2.33
Diamond 5.5–10 RT/Duroid 6006 6.15@3GHz
Epoxy (FR4) 4.4 Rubber 3.0–4.0
Epoxy glass PCB 5.2 Sapphire 9.4
Ethyl alcohol (absolute) 24.5 @ 1 MHz Sea water 80

6.5 @ 3 GHz
FR-4(G-10)
–low resin 4.9 Silicon 11.7–12.9
–high resin 4.2
GaAs 13.0 Soil ∼10
Glass ∼4 Soil (dry sandy) 2.59@1MHz

2.55 @ 3 GHz

Gold 1 Water (32

(68
(212

◦

F) 88.0

◦

F) 80.4

◦

F) 55.3

Ice (pure distilled water) 4.15 @ 1 MHz Wood ∼2

3.2 @ 3 GHz

Table 1.3 Conductivities of some common materials at room temperature

Material Conductivity (S/m) Material Conductivity (S/m)

Silver 6.3 ×10
Copper 5.8 × 10
Gold 4.1 ×10
Aluminum 3.5 ×10
Tungsten 1.8 ×10
Zinc 1.7 × 10
Brass 1 ×10
Phosphor bronze 1 ×10
Tin 9 × 10
Lead 5 ×10
Silicon steel 2 ×10
Stainless steel 1 × 10
Mercury 1 ×10
Cast iron ≈ 10

7

7

7

7

7

7

7

7

6

6

6

6

6

6

Graphite ≈ 10
Carbon ≈ 10
Silicon ≈ 10
Ferrite ≈ 10
Sea water ≈ 5
Germanium ≈ 2
Wet soil ≈ 1
Animal blood 0.7
Animal body 0.3
Fresh water ≈ 10
Dry soil ≈ 10
Distilled water ≈ 10
Glass ≈ 10
Air 0

5

4

3

2

−2

−3

−4

−12