Electromagnetic ﬁeld theory for physicists and … · Electromagnetic ﬁeld theory for...

Electromagnetic field theory for physicists and

engineers: Fundamentals and Applications

R. Gómez Martín

Contents

1 Electromagnetic field fundamentals 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Review of Maxwell’s equations . . . . . . . . . . . . . . . . . . . 1

1.2.1 Physical meaning of Maxwell’s equations . . . . . . . . . . 4

1.2.2 Constitutive equations . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 10

1.3 The conservation of energy. Poynting’s theorem . . . . . . . . . . 12

1.4 Momentum of the electromagnetic field . . . . . . . . . . . . . . . 14

1.5 Time-harmonic electromagnetic fields . . . . . . . . . . . . . . . 16

1.5.1 Maxwell’s equations for time-harmonic fields . . . . . . . 17

1.5.2 Complex dielectric constant. . . . . . . . . . . . . . . . . 18

1.5.3 Boundary conditions for harmonic signals . . . . . . . . . 22

1.5.4 Complex Poynting vector . . . . . . . . . . . . . . . . . . 23

1.6 On the solution of Maxwell’s equations . . . . . . . . . . . . . . . 25

2 Fields created by a source distribution: retarded potentials 27

2.1 Electromagnetic potentials . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Lorenz gauge . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Solution of the inhomogeneous wave equation for potentials: re-

tarded potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Electromagnetic fields from a bounded source distribution . . . . 36

2.3.1 Radiation fields . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.2 Fields created by an infinitesimal current element . . . . . 44

2.3.3 Far-zone approximations for the potentials . . . . . . . . 48

2.4 Multipole expansion for potentials . . . . . . . . . . . . . . . . . 50

2.4.1 Electric dipolar radiation . . . . . . . . . . . . . . . . . . 51

2.4.2 Magnetic dipolar radiation . . . . . . . . . . . . . . . . . 53

2.4.3 Electric quadrupole radiation . . . . . . . . . . . . . . . . 56

2.5 Fields created by a charge under arbitrary movement: The Lié-

nard Wiechert potentials. . . . . . . . . . . . . . . . . . . . . . . 58

2.5.1 Arbitrarily moving point charge fields . . . . . . . . . . . 60

2.5.2 Angular distribution of the energy radiated by an accel-

erated charge . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.5.3 Larmor’s Formula . . . . . . . . . . . . . . . . . . . . . . 68

3

4 CONTENTS

2.6 Maxwell’s symmetric equations . . . . . . . . . . . . . . . . . . . 69

2.6.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 73

2.6.2 Harmonic variations . . . . . . . . . . . . . . . . . . . . . 73

2.6.3 Fields created by an infinitesimal magnetic current element 74

2.7 Theorem of uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 75

2.7.1 Non-harmonic electromagnetic field . . . . . . . . . . . . . 76

2.7.2 Time-harmonic fields . . . . . . . . . . . . . . . . . . . . . 77

3 Electromagnetic waves 79

3.1 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.2 Harmonic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.2.1 Uniform plane harmonic waves . . . . . . . . . . . . . . . 84

3.2.2 Propagation in lossless media . . . . . . . . . . . . . . . . 86

3.2.3 Propagation in good dielectrics or insulators . . . . . . . . 88

3.2.4 Propagation in good conductors . . . . . . . . . . . . . . 88

3.2.5 Surface resistance . . . . . . . . . . . . . . . . . . . . . . 89

3.3 Group velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.4 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Reflection and refraction of plane waves 95

4.1 Normal incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1.1 General case: interface between two lossy media . . . . . . 95

4.1.2 Perfect/Lossy dielectric interface . . . . . . . . . . . . . . 99

4.1.3 Perfect dielectric/Perfect conductor interface . . . . . . . 99

4.1.4 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1.5 Measures of impedances . . . . . . . . . . . . . . . . . . . 101

4.2 Multilayer structures . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.1 Stationary and transitory regimes . . . . . . . . . . . . . 103

4.3 Oblique incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.3.1 Incident wave with the electric field contained in the plane

of incidence . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3.2 Wave incident with the electric field perpendicular to the

plane of incidence . . . . . . . . . . . . . . . . . . . . . . 109

5 Electromagnetic wave-guiding structures: Waveguides and trans-

mission lines 111

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2 General relations between field components . . . . . . . . . . . . 113

5.2.1 Transverse magnetic (TM) modes . . . . . . . . . . . . . . 115

5.2.2 Transverse electric (TE) modes . . . . . . . . . . . . . . . 116

5.2.3 Transverse electromagnetic (TEM) modes . . . . . . . . . 117

5.2.4 Boundary conditions for TE and TM modes on perfectly

conducting walls . . . . . . . . . . . . . . . . . . . . . . . 119

5.3 Cutoff frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.4 Attenuation in guiding structures . . . . . . . . . . . . . . . . . . 123

5.4.1 TE and TM modes. . . . . . . . . . . . . . . . . . . . . . 123

CONTENTS 5

6 Rectangular waveguide 127

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2 Rectangular waveguide . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.1 TM modes in rectangular waveguides . . . . . . . . . . . 128

6.2.2 TE modes in rectangular waveguides . . . . . . . . . . . . 130

6.2.3 Attenuation in rectangular waveguides . . . . . . . . . . . 133

7 Fundamentals of antenna theory 137

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.2 Linear thin-wire dipole antennas . . . . . . . . . . . . . . . . . . 138

7.3 Qualitative analysis of thin wires antennas. Where and why an-

tennas radiate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.4 Antennas above a perfect ground plane . . . . . . . . . . . . . . . 148

7.5 Aperture antennas . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.6 Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.7 Pattern multiplication principle. Array factor . . . . . . . . . . . 153

7.8 Array factor for uniformly spaced linear arrays . . . . . . . . . . 156

7.8.1 Uniform linear array . . . . . . . . . . . . . . . . . . . . . 156

I Appendixes 163

A Vector algebra and Analysis formulas used in this book 165

6 CONTENTS

CONTENTS i

Prefacio

This book is developed from notes for a semester course about electromag-

netism at the University of Granada (Spain) for physics and engineering students

that are familiar with Maxwell’s equations and vector analysis, at least at an

advanced undergraduate level. The idea is not to replace the many excellent

books that exist in the bibliography about this topic but to help the students

follow my classes. To this end, the book is organized in the way I like to explain

the fundamentals of the radiation and propagation of electromagnetic fields,

that seems to have been successful among the students. This relative success

have encouraged me to make a translation to English with the hope that other

people find it helpful in some way.

On the other hand the book is a work in progress and, although I consider

that can be used as a class text, may contain some small flaws or omissions.

Any feedback from readers is welcome and in particular the opinion of those

who teach or learn radiation and propagation of electromagnetic fields is highly

appreciated.

The bibliography at the end of the book is far from being exhaustive, and it

contains only a list from the many excellent books I have used for learning and

teaching.

Rafael Gómez Martín

Emeritus Professor

Departamento de Electromagnetismo y Física de la Materia

Facultad de Ciencias

Universidad de Granada

Granada (Spain)

ii CONTENTS

Chapter 1

Electromagnetic field

fundamentals

1.1 Introduction

This chapter starts with a brief review of Maxwell’s equations, which are the

fundamental laws that, together with the theory of electromagnetic behavior

of matter, explain on a macroscopic scale the properties of the electromagnetic

field, the relationships of this field with its sources, and its interaction with mat-

ter. The reader is assumed to be familiar with these equations, at least at an

undergraduate level. Next, after reviewing other fundamental topics such as con-

stitutive parameters and boundary conditions, we apply the energy-conservation

law to a bounded volume, limited by a surface , inside of which there exists a

time-variable electromagnetic field. We shall see that when the energy balance

is formulated, there appears a term representing a flow of energy carried by the

electromagnetic field through the surface that limits . This term leads us

to the definition of Poynting’s vector. Similarly, when the law of conservation

of momentum is applied to the same region, we find that the electromagnetic

field also carries a momentum density, which can also be expressed in terms of

Poynting’s vector.

1.2 Review of Maxwell’s equations

The general theory of electromagnetic phenomena is based on Maxwell’s equa-

tions, which constitute a set of four coupled first-order vector partial-differential

equations relating the space and time changes of electric and magnetic fields to

their scalar source densities (divergence) and vector source densities (curl) 1 .

1According to the Helmholtz theorem a vector field is uniquely determined by its di-

vergence and curl if they are given throughout the entire space and if they approach zero at

infinity at least as 1 with 1.

1

2 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS

Maxwell’s equations are usually formulated in differential form (i.e., as relation-

ships between quantities at the same point in space and at the same instant in

time) or in integral form where, at a given instant, the relations of the fields

with their sources are considered over an extensive region of space. The two

formulations are related by the divergence (A.17) and Stokes’ (A.22) theorems.

For stationary media2, Maxwell’s equations in differential and integral forms

are:

Differential form of Maxwell’s equations

∇ · ( ) = ( ) (Gauss’ law) (1.1a)

∇ · ( ) = 0 (Gauss’ law for magnetic fields) (1.1b)

∇× ( ) = −( )

(Faraday’s law) (1.1c)

∇× ( ) = ( ) + ( )

(Generalized Ampère’s law) (1.1d)

Integral form of Maxwell’s equationsI

( ) · = () (Gauss’ law) (1.2a)I

( ) · = 0 (Gauss’ law for magnetic fields) (1.2b)IΓ

( ) · = −Z

( )

· (Faraday’s law) (1.2c)I

Γ

( ) · =

Z

( ( ) + ( )

) · (Generalized Ampère’s law)

(1.2d)

Maxwell’s equations, involve only macroscopic electromagnetic fields and,

explicitly, only macroscopic densities of free-charge, ( ), which are free to

move within the medium, giving rise to the free-current densities, ( ). The

effect of the macroscopic charges and current densities bound to the medium’s

molecules is implicitly included in the auxiliary magnitudes and which are

related to the electric and magnetic fields, and by the so-called constitutive

equations that describe the behavior of the medium (see Subsection 1.2.2). In

general, the quantities in these equations are arbitrary functions of the position

() and time3 (). The definitions and units of these quantities are = electric field intensity (volts/meter; −1)

2 In a stationary medium all quantities are evaluated in a reference frame in which the

observer and all the surfaces and volumes are assumed to be at rest or moving at

being the velocity of light. Maxwell’s equations for moving media can be considered in terms

of the special theory of relativity.3Throughout the book, in most cases, in order to make the notation more concise, we will

not explicitly indicate the arguments, ( ) of the magnitudes unless we consider it convenient

to emphasize the dependence on any of the variables.

1.2. REVIEW OF MAXWELL’S EQUATIONS 3

= magnetic flux density (teslas4 or webers/square meter; or −2) = electric flux density (coulombs/square meter; −2) = magnetic field intensity (amperes/meter; −1) = free electric charge density (coulombs/ cubic meter; −3) = net free charge, in coulombs (), inside any closed surface = free electric current density (amperes/square meter −2).Three of Maxwell’s equations (1.1a), (1.1c), (1.1d), or their alternative inte-

gral formulations (1.2a), (1.2c), (1.2d), are normally known by the names of the

scientists who deduced them. For its similarity with (1.1a), equation (1.1b) is

usually termed the Gauss’ law for magnetic fields, for which the integral formu-

lation is given by (1.2b). These four equations as a whole are associated with

the name of Maxwell because he was responsible for completing them, adding

to Ampère’s original equation, ∇× ( ) = ( ), the displacement current

density term or, in short, the displacement current, , as an additional

vector source for the field . This term has the same dimensions as the free

current density but its nature is different because no free charge movement is

involved. Its inclusion in Maxwell’s equations is fundamental to predict the ex-

istence of electromagnetic waves which can propagate through empty space at

the constant velocity of light The concept of displacement current is also fun-

damental to deduce from (1.1d) the principle of charge conservation by means

of the continuity equation

∇ · = −

(1.3)

or, in integral form, I = −

(1.4)

With his equations, Maxwell validated the concept of "field" previously in-

troduced by Faraday to explain the remote interactions of charges and currents,

and showed not only that the electric and magnetic fields are interrelated but

also that they are in fact two aspects of a single concept, the electromagnetic

field.

The link between electromagnetism and mechanics is given by the empirical

Lorenz force equation, which gives the electromagnetic force density, (in

−3), acting on a volume charge density moving at a velocity (in −1)in a region where an electromagnetic field exists,

= ( + × ) = + × (1.5)

where

= (1.6)

4Given that the tesla is an excessivelly high magnitude to express the values of the magnetic

field usually found in practice, the cgs unit (gauss, ) is often used instead, 1 = 104


is the current density in terms of the mean drift velocity of the particles5 , which

is independent of any random velocity due to collisions. The total force

exerted on a volume of charge is calculated by integrating in this volume. For

a single particle with charge the Lorentz force is

= ( + × ) (1.7)

Maxwell’s equations together with Lorenz’s force constitute the basic mathe-

matical formulation of the physical laws that at a macroscopic level explain and

predict all the electromagnetic phenomena which basically comprise the remote

interaction of charges and currents taking place via the electric and/or magnetic

fields that they produce. From Eq. (1.7) the work done by an electromagnetic

field acting on a volume charge density inside a volume during a time

interval is

= · = ( + × ) · = · = · (1.8)

This work is transformed into heat. The corresponding power density (

−3) that the electromagnetic field supplies to the charge distribution is

=

=

= · (1.9)

This equation is known as the point form of Joule’s law.

In applications, Maxwell’s equations have to be complemented by appropri-

ate initial and boundary conditions. The initial conditions involve values or

derivatives of the fields at = 0, while the boundary conditions involve the

values or derivatives of the fields on the boundary of the spatial region of inter-

est. Usually, we consider the initial conditions as a form of boundary conditions

and refer to the solution of Maxwell´s equations, with all these conditions, as a

boundary-value problem.

Next, we briefly describe the physical meaning of Maxwell’s equations.

1.2.1 Physical meaning of Maxwell’s equations

Gauss’ law, (1.1a) or (1.2a), is a direct mathematical consequence of Coulomb’s

law, which states that the interaction force between electric charges depends on

the distance, , between them, as −2. According to Gauss’ law, the divergenceof the vector field is the volume density of free electric charges which are

sources or sinks of the field , i.e. the lines of begin on positive charges

( 0) and end on negative charges ( 0). In its integral form, Gauss’

law relates the flux of the vector through a closed surface (which can be

imaginary; Fig. 1.1), to the total free charge within that surface.

Gauss’ law for magnetic fields, (1.1b) or (1.2b), states that the field does

not have scalar sources, i.e., it is divergenceless or solenoidal. This is because

5 In general, when there is more than one type of particle the current density its defined

as =

where and represent the volume charge density and drift velocity of the

charges of class


Figure 1.1: a) Closed surface bounding a volume . (b) Open surface bounded

by the closed loop Γ. The direction of the surface element is given by the right-

hand rule: the thumb of the right hand is pointed in the direction of and the

fingertips give the sense of the line integral over the contour Γ

no free magnetic charges or monopoles have been found in nature (see Section

2.6) which would be the magnetic analogues of electric charges for . Hence,

there are no sources or sinks where the field lines of start or finish, i.e., the

field lines of are closed. In its integral form, this indicates that the flux of

the field through any closed surface is null.

Faraday’s law, (1.1c) or (1.2c), establishes that a time-varying field pro-

duces a nonconservative electric field whose field lines are closed. In its integral

form, Faraday’s law states that the time variation of the magnetic flux (R ·)

through any surface bounded by an arbitrary closed loop Γ, (Fig. 1.1), in-

duces an electromotive force given by the integral of the tangential component

of the induced electric field around Γ. The line integration over the contour Γ

must be consistent with the direction of the surface vector according to

the right-hand rule. The minus sign in (1.1c) and (1.2c) represents the feature

by which the induced electric field, when it acts on charges, would produce an

induced current that opposes the change in the magnetic flux (Lenz’s law)

Ampère’s generalized law, (1.1d) or (1.2d), constitutes another connection,

different from Faraday’s law, between and . It states that the vector sources

of the magnetic field may be free currents, and/or displacement currents,

. Thus, the displacement current performs, as a vector source of , a

similar role to that played by as a source of In its integral form the

left-hand side of the generalized Ampere’s law equation represents the integral

of the magnetic field tangential component along an arbitrary closed loop Γ

and the right-hand side is the sum of the flux, through any surface bounded

by a closed loop Γ (Fig. 1.1 ), of both currents: the free current and the

displacement current .


1.2.2 Constitutive equations

Maxwell’s equations (1.1) can be written without using the artificial fields

and , as

∇ · ( ) =0( ) (1.10a)

∇ · ( ) = 0 (1.10b)

∇× ( ) = −( )

(1.10c)

∇× ( ) = 0( ) + 00

( )

(1.10d)

where 0 = 10−9(36) (farad/meter; −1) and 0 = 410

−7 (henry/meter; −1) are two constants called electric permittivity and magnetic permeabilityof free space, respectively. The subscript indicates that all kinds of charges

(free and bound ) must be individually included in and These equations

are, within the limits of classical electromagnetic theory, absolutely general.

Nevertheless, in order to make it possible to study the interaction between an

electromagnetic field and a medium and to take into account the discrete nature

of matter, it is absolutely necessary to develop macroscopic models to extend

equations (1.10a) and (1.10d) and to obtain Maxwell’s macroscopic equations

(1.1), in which only macroscopic quantities are used and in which only the den-

sities of free charges and currents explicitly appear as sources of the fields. To

this end, the atomic and molecular physical properties, which fluctuate greatly

over atomic distances, are averaged over microscopically large-volume elements,

∆, so that these contain a large number of molecules but at the same time

are macroscopically small enough to represent accurate spatial dependence at a

macroscopic scale. As a result of this average, the properties of matter related

to atomic and molecular charges and currents are described by the macroscopic

parameters, electric permittivity , magnetic permeability , and electrical con-

ductivity . These parameters, called constitutive parameters, are in general

smoothed point functions. The derivation of the constitutive parameters of a

medium from its microscopic properties is, in general, an involved process that

may require complex models of molecules as well as quantum and statistical

theory to describe their collective behavior. Luckily, in most of the practical

situations, it is possible to achieve good results using simplified microscopic

models.

To define the electric permittivity and describe the behaviour of the electric

field in the presence of matter, we must introduce a new macroscopic field

quantity, ( −2), called electric polarization vector, such that

= 0 + (1.11)

and defined as the average dipole moment per unit volume

= lim∆→0

P∆=1

∆(1.12)


where is the number of molecules per unit volume and the numerator is

the vector sum of the individual dipolar moments, , of atoms and molecules

contained in a macroscopically infinitessimal volume ∆. For many materials,

called linear isotropic media, can be considered colinear and proportional to

the electric field applied. Thus we have

= 0 (1.13)

where the dimensionless parameter , called the electric susceptibility of the

medium, describes the capability of a dielectric to be polarized. Expression

(1.11) can be written in a more compact form as

= (1 + )0 (1.14)

so that = 0 = (1.15)

where

= 1 + (1.16)

and

= 0 (1.17)

are the relative permittivity and the permittivity of the medium, respectively.

To define the magnetic permeability and describe the behaviour of the mag-

netic field in the presence of magnetic materials, we must introduce another

new macroscopic field quantity, called magnetization vector ( −1), suchthat

=

0− (1.18)

where is defined, in a similar way to that of the electric polarization vector,

as the average magnetic dipole moment per unit volume

= lim∆→0

P∆=1

∆(1.19)

where is the number of atomic current elements per unit volume and the

numerator is the vector sum of the individual magnetic moments, contained

in a macroscopically infinitessimal volume ∆.

In general, is a function of the history of or , which is expressed

by the hysteresis curve. Nevertheless, many magnetic media can be considered

isotropic and linear, such that

= (1.20)

where is the adimensional magnetic susceptibility magnitude, being negative

and small for diamagnets, positive and small for paramagnets, and positive and

large for ferromagnets. Thus

= (1 + )0 = 0

= (1.21)


where

= (1 + ) (1.22)

and

= 0 (1.23)

are the relative magnetic permeability and the permeability of the medium,

respectively, which can reach very high values in magnetic materials such as

iron and nickel.

The concept of requires a careful definition when working with magnetic

materials with strong hysteresis, such as ferromagnetic media. The phenomenon

of hysteresis may also occur in certain dielectric materials called ferroelectric.

In a vacuum, or free space, = 1; = 1, and therefore the fields vectors and , as well as and , are related by

= 0 (1.24a)

= 0 (1.24b)

Very often the relation between an electric field and the conduction current

density that it generates is given, at any point of the conducting material,

by the phenomenological relation, called Ohm’s law

= (1.25)

so that is linearly related to trough the proportionality factor called the

conductivity of the medium. Conductivity is measured in siemens per meter (

−1 ≡ Ω−1−1) or mhos per meter ( −1). Media in which (1.25) is validare called ohmic media. A typical example of ohmic media are metals where

(1.25) holds in a wide range of circumstances. However, in other materials,

such as semiconductors, (1.25) it may not be applicable. For most metals is

a scalar with a magnitude that depends on the temperature and that, at room

temperature, has a very high value of the order of 107 −1Then very oftenmetals are considered as perfect conductors with an infinite conductivity.

The relations between macroscopic quantities, (1.14), (1.21) and (1.25), are

called constitutive relations. Depending on the characteristics of the constitutive

macroscopic parameters , and , which are associated with the microscopic

response of atoms and molecules in the medium, this medium can classified as:

Nonhomogeneous or homogeneous: according to whether or not the consti-

tutive parameter of interest is a function of the position, = (), = (), or

= ().

Anisotropic or isotropic: according to whether or not the response of the

medium depends on the orientation of the field. In isotropic media all the

magnitudes of interest are parallel, i.e., and ; and/or and ; and/or and . In anisotropic materials the constitutive parameter of interest is a

tensor.

Nonlinear or linear: according to whether or not the constitutive parameters

depend on the magnitude of the applied fields. For instance (), () or ().


Time-invariant: if the constitutive parameters do not vary with time 6=(), 6= () or 6= ()

Dispersive: according to whether or not, for time-harmonic fields, the con-

stitutive parameters depend on the frequency, = (), = () or = ().

The materials in which these parameters are functions of the frequency are

called dispersive6.

Magnetic medium: if 6= 0. Otherwise the medium is called nonmagnetic

because its only significant reaction to the electromagnetic field is polarization.

Fortunately, in many cases the medium in which the electromagnetic field ex-

ists can be considered homogeneous, linear and isotropic, time-invariant, nondis-

persive and nonmagnetic. Indeed, this assumption is not very restrictive since

many electromagnetic phenomena can be studied using this simplification. In

fact, even practical cases of the propagation of electromagnetic waves through

nonlinear media (semiconductors, ferrites, nonlinear crystals, etc.) are analysed

with linear models using the so-called small-signal approach. Most of this book

concerns homogeneous, isotropic, linear7 (LIH) and nonmagnetic media.

The effect of the properties of a medium on the macroscopic field can be

emphasized by expressing and in Maxwell’s equations (1.1a) and (1.1d) by

(1.11) and (1.18). Thus we have

∇ · =0

=1

0

³−∇ ·

´(1.26a)

∇× = 0 + 00

= 0(

+

+∇× ) + 00

(1.26b)

In (1.26a) we have explicitly as scalar sources of both the free charge

and the polarization or bounded density of charge, −∇ · . Then in (1.10a) wehave

= −∇ · (1.27)

Similarly, in (1.26b), we have, explicitly as vector sources of , besides the

free current density ( which includes the conduction current density = ),

6Eqs (1.13), (1.20) and (1.25) are strictly valid only for nondispersive media Effectively, for

example, because of the dependence of the electric permittivity with frequency we generally have () = 0()

(). Thus, according to the convolution theorem, for arbitrary time dependence

this expression becomes

() = 0

−∞(− 0)(0)0

Similarly for magnetization and Ohms’ law we have

() =

−∞(− 0) (0)0

() =

−∞(− 0)(0)0

These expressions indicate that, as for any physical system, the response of the medium to an

applied field is not instantaneous.7Materials satisfying Linearity, Isotropy and Homogeneity are known as LIH media or

dielectrics. Most insulators are LIH dielectrics


the polarization current (which results from the motion of the bounded

charges in dielectrics), the displacement current in the vacuum, 0 and

the magnetization current, ∇ × (which takes place when a non-uniformly

magnetized medium exists). Then in (1.10d) we have

( ) = +

+∇× (1.28)

In the following we will assume that there is no magnetization current.

1.2.3 Boundary conditions

As is evident from (1.1a)-(1.1d) and (1.14), (1.21), (1.25), in general the fields and are discontinuous at points where , and also are. Hence

the field vectors will be discontinuous at a boundary between two media with

different constitutive parameters.

The integral form of Maxwell’s equations can be used to determine the

relations, called boundary conditions, of the normal and tangential components

of the fields at the interface between two regions with different constitutive

parameters , and where surface density of sources may exist along the

boundary.

The boundary condition for can be calculated using a very thin, small pill-

box that crosses the interface of the two media, as shown in Fig. 1.2. Applying

the divergence theorem to (1.1a) we have

I =

ZBase 1

1+

ZCurved surface

+

ZBase 2

2 =

Z (1.29)

where 1 denotes the value of in medium 1, and 2 the value in medium

2. Since both bases of the pillbox can be made as small as we like, the total

outward flux of over them is (1−2) = (1− 2)·, where these

are the normal components of , is the area of each base, and is the unit

normal drawn from medium 2 to medium 1. At the limit, by taking a shallow

enough pillbox, we can disregard the flux over the curved surface, whereupon

the sources of reduce to the density of surface free charge on the interface

· (1 − 2) = (1.30)

Hence the normal component of changes discontinously across the interface by

an amount equal to the free charge surface density on the surface boundary.

Similarly the boundary condition for can be established using the Gauss’

law for magnetic fields (1.1b). Since the magnetic field is solenoidal, it follows

that the normal components of are continuous across the interface between

two media

· ( 1 − 2) = 0 (1.31)


Figure 1.2: Derivation of boundary conditions at the interface of two media.

The behavior of the tangential components of can be determined using

a infinitesimal rectangular loop at the interface which has sides of lengh

normal to the interface, and sides of lengh parallel to it (Fig. 1.2). From

the integral form of the Faraday’s law, (1.2c) and defining as the unit tangent

vector parallel to the direction of integration on the upper side of the loop, we

have

(1 · − 2 · ) + contributions of sides

= −

· (1.32)

In the limit, as → 0 the area = bounded by the loop approaches

zero and, since is finite, the flux of vanishes. Hence(1 − 2) · = 0

and we conclude that the tangential components of are continuous across the

interface between two media. In terms of the normal to the boundary, this

can be written as

× ( 1 − 2) = 0 (1.33)

Analogously, using the same infinitesimal rectangular loop, it can be deduced

from the generalized Ampère’s law, (1.2d), that

( 1 · − 2 · ) + contributions of sides

= −Ã

+

!· (1.34)

where, since is finite, its flux vanishes. Nevertheless, the flux of the surface

current can have a non-zero value when the integration loop is reduced to zero,

if the conductivity of the medium 2 and consequently , is infinite. This

requires the surface to be a perfect conductor. Thus

× ( 1 − 2) = (1.35)


the tangential component of is discontinuous by the amount of surface current

density . For finite conductivity, the tangential magnetic field is continuous

across the boundary.

A summary of the boundary conditions, given in (1.36), are particularized

in (1.37) for the case when the medium 2 is a perfect conductor (2 →∞).General boundary conditions

× (1 − 2) = 0 (1.36a)

× ( 1 − 2) = (1.36b)

· (1 − 2) = (1.36c)

· ( 1 − 2) = 0 (1.36d)

Boundary conditions when the medium 2 is a perfect conductor (2 →∞)

× 1 = 0 (1.37a)

× 1 = (1.37b)

· 1 = (1.37c)

· 1 = 0 (1.37d)

1.3 The conservation of energy. Poynting’s the-

orem

Poynting’s theorem represents the electromagnetic energy-conservation law. To

derive the theorem, let us calculate the divergence of the vector field × in a

homogeneous, isotropic and linear finite region bounded by a closed surface .

If we assume that contains power sources (generators) generating currents then, from Maxwell’s equations (1.1c) and (1.1d), we get

∇ · (× ) = ·∇× − ·∇× = − ·

− ·

− · ( + ) (1.38)

where represents the source current density distribution which is the primary

origin of the electromagnetic fields8, while the induced conduction current den-

sity is written as = (1.25).

As the medium is assumed to be linear, the derivates with respect to time

can be written as

·

= ·

=

µ1

22

¶=

µ1

2 ·

¶(1.39a)

·

= ·

=

µ1

22

¶=

µ1

2 ·

¶(1.39b)

8The source current may be maintained by external power sources or generators (this current is

often called driven or impressed current).

1.3. THE CONSERVATION OF ENERGY. POYNTING’S THEOREM 13

By introducing the equalities (1.39a) and (1.39b) into (1.38), integrating over

the volume , applying the divergence theorem, and then rearranging terms,

we haveZ

· = −

Z

1

2( · + · )−

Z

2−I

( × ) · (1.40)

To interpret this result we accept that

=1

2 · (1.41)

and

=1

2 · (1.42)

represent, as a generalization of their expression for static fields, the instanta-

neous electric energy density, , and magnetic energy density, , stored in

the respective fields. Thus according to (1.9) the left side of (1.40) represents

the total electromagnetic power supplied by all the sources within the volume

. Regarding the right side of (1.40), the first term represents the change rate

of the stored electromagnetic energy within the volume; the second term repre-

sents the dissipation rate of electromagnetic energy within the volume; and the

third term represents the flow of electromagnetic energy per second (power)

through the surface that bounds volume . Defining Poynting’s vector P as

P = × (2) (1.43)

we can write I

( × ) · =I

P · (1.44)

This equation represents the total flow of power passing through the closed sur-

face and, consequently, we conclude that P = × represents the power

passing through a unit area perpendicular to the direction of P. This conclu-sion may seem questionable because it could be argued that any vector with an

integral of zero over the closed surface could be added to P without affectingthe total flow. Nevertheless, this is a natural interpretation that does not con-

tradict any experience. Only when we try to particularize (1.40) to steady fields

do we find ambiguous results, because, in static, the location of the electric and

magnetic energy has no physical significance.

Note that Eq. (1.40) was deduced by assuming a linear medium and that the

losses occur only through conduction currents. Otherwise the equation should

be modified to include other kinds of losses such as those due to hysteresis or

possible transformations of the electromagnetic energy into mechanical energy,

etc. When there are no sources within , (1.40) represents an energy balance

of that flowing through versus that stored and dissipated in .


1.4 Momentum of the electromagnetic field

As we have seen in the previous section, when we apply the law of conservation

of electromagnetic energy to a finite volume bounded by a surface , it is

necessary to include a term that, by means of the Poynting vector P, takesinto account the flow of power through . We shall now see that when an

electromagnetic field interacts with the charges and currents in , it is also

necessary to consider a momentum associated with the electromagnetic field in

order to guarantee the conservation of momentum. To calculate this momentum,

we will begin by expressing, only in terms of the fields, the Lorentz force density,

(1.5), exerted by the electromagnetic field on the distribution of charges and

current, which we assume to be in free space. For this purpose, let us consider

Maxwell’s equations (1.1a) and (1.1d) to express and as

= ∇ · (1.45)

= ∇× −

(1.46)

so that

= + × =³∇ ·

´ − × (∇× ) + ×

(1.47)

which, taking into account that

×

= −

( × ) + ×

=

−

( × )− × (∇× ) (1.48)

becomes

= (∇ · ) − × (∇× )−

( × )− × (∇× ) (1.49)

By adding the term (∇ · ) = 0 to this equality to make the final expres-sions symmetrical, and by reordering, we can write the Lorentz force density

as

= (∇ · )− × (∇× ) + ∇ · − × (∇× )−

( × ) (1.50)

The component of Lorentz force density can be written, taking into account

the definition of the Poynting vector P, as

=

∙ − 1

2

2

¸+ 0

∙ − 1

2

2

¸− 1

2

P (1.51)

where is the Kronecker delta ( = 1 if = and zero if 6= ) and the

indices = 1 2 3 correspond to the coordinates respectively, and we

1.4. MOMENTUM OF THE ELECTROMAGNETIC FIELD 15

have made use of the Einstein’s summation convention (i.e., the repetition of

an index automatically implies a summation over it). To obtain (1.51) we have

made use of the following equalities

∇ · − × (∇× )¯

=

∙−

1

2

2

¸∇ · − × (∇× )

¯

= 0

∙ − 1

2

2

¸ ×

¯

=

2(1.52)

The first two summands in (1.51) constitute the component of the diver-

gence of a tensor quantity, such that

(∇ · ) =

(1.53)

where is a symmetric tensor, known as the Maxwell stress tensor, defined

by

=

∙−

1

2

2

¸+ 0

∙ − 1

2

2

¸(1.54)

Therefore, from (1.51) and (1.53), we have

= ∇ · − 1

2 P

(1.55)

with

∇ · =

∙

¸⎡⎣

⎤⎦ (1.56)

The components of the electromagnetic tensor can be written as

=

+ = − 1

2 + − 1

2 (1.57)

where and

represent, respectively, the electric and magnetic tensors

defined by

= − 1

2 (1.58)

= − 1

2 (1.59)

Integrating (1.51) over the volume the total electromagnetic force ex-

erted on the volume is

=

Z

=

Z

( + × ) =

Z

− 1

2

Z

P (1.60)


where is the force per unit of area on

= · (1.61)

and we have applied the theorem of divergence to the tensor i.e.Z

∇ · =

Z

· =Z

· =

Z

(1.62)

Thus Z

= +1

2

Z

P (1.63)

Note that the term1

2

Z

P (1.64)

is not null even in the absence of charges and currents. Since the only elec-

tromagnetic force possible due to the interaction of the field with charges and

currents is the term (1.64) must represent another physical quantity with

the same dimensions as a force, i.e., the rate of momentum transmitted by

the electromagnetic field to the volume . This is equivalent to associating a

momentum density with the electromagnetic field, given by 12 times the

Poynting vector,

=P2

(1.65)

which propagates in the same direction as the flow of energy. Thus, Eq. 1.63

represents the formulation for the momentum conservation in the presence of

electromagnetic fields.

The momentum of an electromagnetic field, which can be determined ex-

perimentally, is inappreciable under normal conditions and its value is often

below the limits of the measurement error. However, in the domain of atomic

phenomena, the momentum of an electromagnetic field can be comparable to

that of particles, and plays a crucial role in all the processes of interaction with

matter. The transfer of momentum to a system of charges and currents implies

a reduction in the field momentum, and the loss of momentum by the system,

for example by radiation, leaves to an increase in the momentum of the field.

1.5 Time-harmonic electromagnetic fields

A particular case of great interest is one in which the sources vary sinusoidally

in time. In linear media the time-harmonic dependence of the sources gives rise

to fields which, once having reached the steady state, also vary sinusoidally in

time. However, time-harmonic analysis is important not only because many

electromagnetic systems operate with signals that are practically harmonic, but

also because arbitrary periodic time functions can be expanded into Fourier

series of harmonic sinusoidal components while transient nonperiodic functions

can be expressed as Fourier integrals. Thus, since the Maxwell’s equations are

1.5. TIME-HARMONIC ELECTROMAGNETIC FIELDS 17

linear differential equations, the total fields can be synthesized from its Fourier

components.

Analytically, the time-harmonic variation is expressed using the complex

exponential notation based on Euler’s formula, where it is understood that the

physical fields are obtained by taking the real part, whereas their imaginary

part is discarded. For example, an electric field with time-harmonic dependence

given by cos(+ ) where is the angular frequency, is expressed as

= Re~E = 1

2(~E + (~E)∗) = 0 cos(+ ) (1.66)

where ~E is the complex phasor,

~E = 0 (1.67)

of amplitude 0 and phase , which will in general be a function of the angular

frequency and coordinates. The asterisk ∗ indicates the complex conjugate,and Re represents the real part of what is in curly brackets.Throughout the book, we will represent both complex phasor magnitudes

(either scalar or vector) by symbols in bold, e.g. ~E = ~E( ), and ρ =

ρ( ) In this way, time-dependent (real) quantities, which are represented by

mathematical symbols not in bold, such as = ( ), and = ( ) can be

distinguished from complex phasors which do not depend on time. In general,

as indicated, these complex phasors may depend on the angular frequency. The

real time-dependent quantity associated with a complex phasor is calculated, as

in (1.66), by multiplying it by and taking the real part.

1.5.1 Maxwell’s equations for time-harmonic fields

Assuming time dependence, we can get the phasor form or time-harmonic

form of Maxwell’s equations simply by changing the operator to the factor

in (1.1a)-(1.2d) and eliminating the factor . Maxwell’s equations in

differential and integral forms for time-harmonic fields are given below.

Differential form of Maxwell’s equations for time-harmonic fields

∇ · ~D = ρ (Gauss’ law) (1.68a)

∇ · ~B = 0 (Gauss’ law for magnetic fields) (1.68b)

∇× ~E = − ~B (Faraday’s law) (1.68c)

∇× ~H = ~J + ~D (Generalized Ampère’s law) (1.68d)

Integral form of Maxwell’s equations for time harmonic fields


I

~D · = Q (Gauss’ law) (1.69a)I

~B · = 0 (Gauss’ law for magnetic fields) (1.69b)IΓ

~E · = −Z

~B · (Faraday’s law) (1.69c)IΓ

~H · =

Z

( ~J + ~D) · (Generalized Ampère’s law) (1.69d)

For time-harmonic fields, expressions (1.26a) and (1.26b) become

∇ · ~E =ρ0

=1

0

³ρ−∇ · ~P

´(1.70a)

∇× ~B = 00~E+ 0

~J = 00~E + 0(

~J + ~P +∇× ~M)

(1.70b)

1.5.2 Complex dielectric constant.

Over certain frequency ranges, due to the atomic and molecular processes in-

volved in the macroscopic response of a medium to an electromagnetic field,

there appear relatively strong damping forces that give rise to a delay between

the polarization vector and (a phase shift between ~P and ~E), and con-

sequently between and , and to a loss of electromagnetic energy as heat in

overcoming the damping forces . At the macroscopic level this effect is analyti-

cally expressed by means of a complex permittivity, as

~D = ~E (1.71)

with

= 0 − 00 = 0 (1.72)

where = 1 + = 0 − 00 (1.73)

is the relative complex permittivity and = 0−00 is the complex electricsusceptibility. In general both 0 and 00 present a strong frequency dependenceand they are closely related to one another by the Kramer-Kronig relations

where the dependence with the frequency of the dielectric constant is studied.

Similar processes occur in magnetic and conducting media, and, within a

given frequency range, there may be a phase shift between ~E and ~J or between~B and ~H which, at the macroscopic level, is reflected in the corresponding

complex constitutive parameters = 0 − 00 and = 0 − 00.For a medium with complex permittivity, the complex phasor form of the

displacement current is

~D = ~E = 00 ~E + 0 ~E (1.74a)


ddE

d es= J E

'r jwe

=J Ei

J

ddE

d es= J E

'r jwe

=J Ei

J

Figure 1.3: Induced current density in the complex plane.

while the sum, of the displacement and conduction current, called total induced

current, ~J , is

~J = ~E + ~E = ( + 00)~E + 0 ~E = ~J + ~J (1.75)

where ~J, called the dissipative current,

~J = ( + 00)~E (1.76)

in phase with the electric field, is the real part of the induced current ~J (Fig.

1.3) while ~J called the reactive current,

~J = 0 ~E (1.77)

is the imaginary part of the induced current which is in phase quadrature with

the electric field. The dissipative current can be expressed in a more compact

form as~J = ~E (1.78)

where is the effective or equivalent conductivity

= + 00 (1.79)

which includes the ohmic losses due to and the damping losses due to 00.Thus the induced current, (1.75), can be written as

~J = ~E + 0 ~E = ~E (1.80)

where is the complex effective conductivity, defined as

= + 0 (1.81)

Thus a medium with conductivity and null permittivity is formally equiva-

lent to one with conductivity and permittivity, and , respectively.


On the other hand, the phase angle between the induced and reactive

currents, (Fig. 1.3), is called the loss or dissipative angle, and its tangent (i.e.,

the ratio of the dissipative and reactive currents) is called the loss tangent

tan =

0(1.82)

and the induced current, (1.80), can be written in terms of the loss tangent as

~J = ~E = 0(1−

0)~E = 0 (1− tan ) ~E = ~E (1.83)

where is defined as the effective complex permittivity

= 0 (1− tan ) = 0 (1.84)

and

= (1− tan )0 (1.85)

denotes the effective relative permittivity. Thus, according to (1.80) and (1.83),

a medium can be formally considered alternatively either as a medium of per-

mittivity 0 and effective conductivity , or as a dielectric medium of effective

permittivity or as a conducting medium of effective conductivity . In

summary, this possibilities are

Permittivity Conductivity

Original medium = 0 − 00

Equivalent medium 1 0 = + 00

Equivalent medium 2 = 0 − (00 + ) 0

Equivalent medium 3 0 = + 00 + 0

(1.86)

The loss tangent is equal to the inverse of the quality factor of the dielectric

which is a dimensionless quantity defined as

= Maximun energy stored per unit volume

Time average power lost per unit volume=

0(1.87)

The average power dissipated per cycle and unit volume, 0, due both to theJoule effect and to that of dielectric polarization, is given, according to (1.9),

by

0 =1

Z

0

· = 1

Z

0

0 cos · ( 0 cos+ 0 0 sin)

=1

Z

0

20 cos

2 =1

Z

0

· = 20

2

(1.88)


where = 2 is the period of the signal. Note that only the dissipative part

of contributes to the average power. Of this power, the part corresponding

to polarization losses is

1

Z

0

0020 cos2 =

00202

(1.89)

The maximum electric field energy stored per unit of volume is

=1

2020 (1.90)

Thus, dividing (1.90) by (1.88), we have

=0

=

1

tan (1.91)

Although both dimensionless quantities, and tan , can be used to define

the characteristics of a dielectric, we will use the loss tangent throughout this

book.

Depending on whether the reactive or the dissipative current is predominant

at the operating frequency, a medium is classified as a weakly lossy or a strongly

lossy medium respectively. Thus for weakly lossy media, usually called good

dielectrics or insulators, we have, 0 , so that

tan =

0 1 (1.92)

Or, if = 0,

tan =00

0 1 (1.93)

If = 0 (i.e. tan = 0), the medium is termed a perfect or ideal dielectric,

in which case the reactive current coincides with the displacement current, and

the dielectric is characterized by a real permittivity .

If the medium is strongly lossy we have 0 , so that

tan =

0 1 (1.94)

which for good conductors where 00 = 0; 0 = simplifies to

tan =

1 (1.95)

being practically = 0. If = ∞ (i.e. tan = ∞) the medium is termed a

perfect conductor.

For a homogeneous conducting medium where 0 and do not depend on

the position, Gauss’ law (1.1a) and the continuity equation (1.3) can be writen

as

∇ · = 0 (1.96)


and

∇ · = −

(1.97)

respectively. Hence we have

0+

= 0 (1.98)

so that the expression for the decay of a charge distribution in a conductor is

given by

= 0−(0) (1.99)

where 0 is the charge density at time = 0 The characteristic time

=0

(1.100)

required for the charge at any point to decay to 1 of its original value is called

the relaxation time.

For most metals = 10−14, signifying that in good conductors the chargedistribution decays exponentially so quickly that it may be assumed that = 0

at any time. In terms of the relaxation time, the loss tangent can be written as

tan =

= ()−1 (1.101)

Thus the classification of a medium as a good or poor conductor depends on

whether the relaxation time is short or long compared with the period of the

signal.

1.5.3 Boundary conditions for harmonic signals

For harmonic signals the boundary conditions of the normal and tangential

components of the fields at the interface between two regions with different

constitutive parameters , and , (1.36a)-(1.37d), become

General boundary conditions

× (~E1 − ~E2) = 0 (1.102a)

× ( ~H1 − ~H2) = ~J (1.102b)

· ( ~D1 − ~D2) = ρ (1.102c)

· ( ~B1 − ~B2) = 0 (1.102d)

Boundary conditions when the medium 2 is a perfect conductor (2 →∞)

× ~E1 = 0 (1.103a)

× ~H1 = ~J (1.103b)

· ~D1 = ρ (1.103c)

· ~B1 = 0 (1.103d)


1.5.4 Complex Poynting vector

In formulating the conservation-energy equation for time-harmonic fields, it is

convenient to find, first, the time-average Poynting9 vector over a period, i.e. the

time-average power passing through a unit area perpendicular to the direction

of P. From (1.66) we have

= Ren~E

o=1

2

³~E + (~E)∗

´(1.104a)

= Ren~H

o=1

2

³~H + ( ~H)∗

´(1.104b)

Thus, the instantaneous Poynting vector (1.43) can be written as

P = × = Re~E ×Re ~H=

1

2Re~E × ~H

∗+ ~E × ~H2 (1.105)

where we have made use of the general relation for any two complex vectors

and

Re ~A ×Re ~B =1

2( ~A+ ~A

∗)× 1

2( ~B + ~B

∗)

=1

4( ~A× ~B

∗+ ~A

∗ × ~B) +1

4( ~A× ~B + ~A

∗ × ~B∗)

=1

4

³~A× ~B

∗+³~A× ~B

∗´∗´+1

4

³~A× ~B +

³~A× ~B

´∗´=

1

2Re ~A× ~B

∗+ ~A× ~B (1.106)

The time-average value of the instantaneous Poynting vector can be calcu-

lated integrating (1.105) over a period , i.e.,

P =1

Z

0

P = 1

2

Z

0

Re~E × ~H∗+ ~E × ~H2

=1

2Re~E × ~H

∗ = 1

2ReP (1.107)

since the time average of ~E × ~H2 vanishes. The magnitude

P = ~E × ~H∗

(1.108)

is termed the complex Poynting vector. Thus the time-average of the Poynting

vector is equal to one-half the real part of the complex Poynting vector

9 In most applications we are interested in the Poynting vector averaged over time rather

than in its instantaneous value.


For a more complete view of the meaning of the complex Poynting vector,

let us again formulate Poynting’s theorem particularized for sources with time-

harmonic dependence. From Faraday’s law, (1.68c), and from Ampère’s general

law, (1.68d), in its conjugate complex form, we have

∇× ~E = − ~H (1.109a)

∇× ~H∗= −~E∗ + ~J

∗+ ~E

∗(1.109b)

where ~J∗represents the complex conjugate of the current supplied by the

sources. Performing a scalar multiplication of Eq. 1.109a by ~H∗and of Eq.

1.109b by ~E, and subtracting the results, we get

∇ ·³~E × ~H

∗´= ~H

∗ ·∇× ~E − ~E ·∇× ~H∗

= − ¡20 − 2

0

¢− ~E · ( ~J ∗ + ~E∗) (1.110)

where it has been taken into account that ~H · ~H∗ = 20 and

~E · ~E∗ = 20

with 0 and 0 being the amplitude of the two harmonic fields. After dividing

(1.110) by 2 we get

∇ ·µ1

2~E × ~H

∗¶= −2

µ20

4−

204

¶− 20

2− 12~J∗ · ~E (1.111)

The terms 204 and

204 represent, respectively, the mean density of the

magnetic and electric energy, while 202 is the the mean power transformed

into heat10 within , since the mean value of the square of a sine or cosine

function is 12

By multiplying Equation (1.111) by the volume element , integrating over

an arbitrary volume and applying the divergence theorem, we obtain the

complex version of the Poynting theoremZ

1

2

³~J∗ · ~E

´ = −

Z

202

− 2Z

µ20

4−

204

¶

−Z

1

2

³~E × ~H

∗´ · (1.112)

which is the expression corresponding to (1.40) in complex notation and where

the first member represents the power supplied by external sources. By sepa-

rating the real and imaginary parts, we obtain the following two equalitiesZ

Re1

2( ~J∗ · ~E) = −

Z

20

2 −

Z

Re1

2(~E × ~H

∗) · (1.113a)Z

Im1

2( ~J∗ · ~E) = −2

Z

µ20

4−

20

4

¶ −

Z

Im1

2(~E × ~H

∗) ·

(1.113b)

10Expression (1.113b) can be easily extended to the case of lossy dielectric just substituting by

the equivalent conductivity defined in (1.79)and by 0 defined in (1.72).

1.6. ON THE SOLUTION OF MAXWELL’S EQUATIONS 25

The first member of (1.113a)

=

Z

Re1

2( ~J∗ · ~E) (1.114)

represents the active mean power supplied by all the sources within . On the

right-hand side of (1.113a) the first integral, as commented above, gives the

power transformed into heat within , while the surface integral represents the

mean flow of power through the surface .

Regarding to expression (1.113b), the first member

=

Z

Im

µ1

2~J∗ · ~E

¶ (1.115)

is called the reactive power of the sources. On the right-hand side the first

summand is 2 times the difference of the average energies stored in the

electric and magnetic fields, while the second represents the flow of reactive

power that is exchanged with the external medium through . If the surface

integral in (1.113a) is non-zero, the external region is said to be an active charge

for the sources within . Similarly, if the surface integral of Eq. (1.113b) is non-

zero, the external region is said to be a reactive charge for the sources within .

In general, both of these surface integrals are non-zero and the external region

becomes both an active and a reactive charge for the sources.

1.6 On the solution of Maxwell’s equations

Despite their apparent simplicity, Maxwell’s equations are in general not easy

to solve. In fact, even in the most favorable situation of homogeneous, linear and

isotropic media, called LIH media, there are not many problems of interest that

can be analytically solved except for those presenting a high degree of geomet-

rical symmetry. Moreover, the frequency range of scientific and technological

interest can vary by many orders of magnitude, expanding from frequency val-

ues of zero (or very low) to roughly 1014 . The behavior and values of the

constitutive parameters can change very significantly in this frequency. range.

Conductivity, for example, can vary from 0 to 107 −1. It is even possible tobuild artificial materials, called metamaterials, which present electromagnetic

properties that are not found in nature. Examples of such as metamaterials

are those characterized with both negative permittivity ( 0) and negative

permeability ( 0). These media are called DNG (double-negative) metama-

terials and, owing to their unusual electromagnetic properties11, they present

many potential technological applications.

Another important factor to study the interaction of an electromagnetic field

with an object is the electrical size of the body, i.e., the relationship between

11Metamaterials may be described roughly as man-made materials with a structure on the

scale of nanometers which gives them unusual electromagnetic properties not to be found in

nature. It should be emphasized that these artificial materials not violate any known principle

of physics.


the wavelength and the body size, which can also vary by several orders of mag-

nitude. All these circumstances make it in general necessary to use analytical,

semi-analytical or numerical methods appropriate to each situation. In partic-

ular, numerical methods are fundamental for simulating and solving real com-

plex problems that do not admit analytical solutions. Today numerical methods

make up the so-called computational electromagnetics, which together, with ex-

perimental and theoretical or analytical electromagnetics, constitute the three

pillars supporting research in Electromagnetics. Of course, both the develop-

ment of analitycal, numerical or experimental tools, as well as the interpretation

of the results, require theoretical knowledge of electromagnetic phenomena

Chapter 2

Fields created by a source

distribution: retarded

potentials

In this chapter we introduce the scalar electric and magnetic vector potentials

as magnitudes to facilitate the calculation of the fields created by a time-varying

source bounded distribution. Of these fields we will pay special attention to the

radiation fields. Next we will find the fields created by a charged particle under

arbitrary movement. Then we extend Maxwell’s equations, in order to make

them symmetric, by introducing the concept of magnetic charges and currents

and finally we study the conditions for which we can state that there exists a

unique electromagnetic field that satisfies, simultaneously, Maxwell’s equations

and the given boundary conditions.

2.1 Electromagnetic potentials

A basic problem in electromagnetism is that of finding the fields created for a

time-varying source distribution of finite size, which we assume to be in a non-

magnetic, lossless, homogeneous, time-invariant, linear and isotropic medium.

Figure 2.1 represents such a distribution, where, as usual, the coordinates asso-

ciated with source points, = (0 0), = (0 0), are designated by primes,while those associated with field points or observation points ( ) are without

primes. In the following, we will assume the medium surrounding the source

distribution to be free space, i.e. = 0 = 0, although of course all the re-

sulting formulas remain valid for media of constant permittivity and permeabil-

ity, provided that 0 is replaced by 0 and by 0. While the expressions

for the fields can be derived directly from their sources, the task can often be

facilitated by calculating first two auxiliary functions, the scalar electric poten-

tial Φ = Φ( ) and the magnetic vector potential = ( ) (Fig. 2.2). Once

27

28CHAPTER 2. FIELDS CREATEDBYA SOURCEDISTRIBUTION: RETARDED POTENTIAL

Figure 2.1: Time-varying source bounded distribution 0 of maximun dimen-sion . The coordinates associated with source points of currents and charges = (0 0), and = (0 0), respectivelly are designated by primes, whilethe associate with field points, ( ), are without primes.

the potentials are obtained, it is a simple matter to calculate the fields from

them. In this section, we formulate the general expressions for these potentials.

Since, according to (1.1b), the divergence of the magnetic field is always

zero, we can express it as the curl of an electromagnetic vector potential as

= ∇× (2.1)

Inserting this expression into (1.1c) we get

∇×µ +

¶= 0 (2.2)

Since any vector with a zero curl can be expressed as the gradient of a scalar

function Φ, called the scalar potential, we can write

+

= −∇Φ (2.3)

or

= −∇Φ−

(2.4)

where is the nonconservative part of the electric field with a non-vanishing

curl. When the vector potential is independent of time, expression (2.4)

reduces to the familiar () = −∇Φ().

2.1. ELECTROMAGNETIC POTENTIALS 29

( ', ), ( ', )r t J r t

;E B

, A

( ', ), ( ', )r t J r t

;E B

, A

Figure 2.2: While the expressions for the fields can be derived directly from

their sources, the task can often be facilitated by calculating first two auxiliary

functions, the scalar electric potential Φ = Φ( ) and the magnetic vector

potential = ( )

According to the relations (2.1) and, (2.4) the fields and are completely

determined by the vector and scalar potentials and Φ However, the fields

do not uniquely determine the potentials. For instance, it is clear that the

transformation = 0 +∇Ψ (2.5)

where Ψ = Ψ ( ) is any arbitrary, single-valued, continuously differentiable,

scalar function of position and time that vanishes at infinity, leaves unchanged

= ∇× = ∇× 0 +∇×∇Ψ = ∇× 0 (2.6)

Inserting (2.5) into (2.4), it follows that

= −∇µΦ+

Ψ

¶− 0

(2.7)

so that the value of , obtained from 0, also remains unchanged provided thatΦ is replaced by the scalar potential

Φ0 = Φ+Ψ

(2.8)

Thus different sets of potentials and Φ give rise to the same set of fields1

and . The joint transformation (2.5) and (2.8) leaves the electromagnetic field

invariant. The different forms of choosing the potentials and Φ leaving the

fields unchanged are called gauge transformations, and the function Ψ is called

the gauge function. The degree of freedom provided by the gauge transforma-

tions facilitates the calculation of the potentials and hence of the fields because,

1The liberty to select the value of is understandable taking into account that by (2.1) the

magnetic field fixes only ∇× . However, Helmholtz’s theorem posits that, to determine the

(spatial) behavior of completely, ∇ (which is still undetermined) must also be specified.

Thus, we can choose it in any way we consider suitable for facilitating the calculation of the



once the potentials are known, the fields are easily derived by differentiation

from (2.1) and (2.4). An example of gauge transformation is the Lorenz gauge,

also called the Lorenz condition.

2.1.1 Lorenz gauge

Inserting (2.1) and (2.4) into the generalized Ampère’s law, (1.1d), and Gauss’

law, (1.1a), using (A.16d) and rearranging terms, we get two, coupled, second-

order partial-differential equations

002

2−∇2 = 0

−∇∙∇ · + 00

Φ

¸(2.9a)

002Φ

2−∇2Φ =

0+

∙∇ · + 00

Φ

¸(2.9b)

These equations could be considerably simplified if we could force (without

changing the fields) the potentials to satisfy the auxiliary relation

∇ · + 00Φ= 0 (2.10)

called the Lorenz gauge (or Lorenz condition)2. Fortunately, as we will show

below, we can always take advantage of the freedom in choosing the potentials

so that they fulfil the Lorenz condition and consequently simplify Eqs (2.9) to

the inhomogeneous Helmholtz wave equations

002

2−∇2 = 0

(2.11a)

002Φ

2−∇2Φ =

0(2.11b)

The advantage of having applied the Lorenz condition is that the equations

(2.11) for the potentials are uncoupled and each one depends on only one type

of source. This makes it easier to calculate the potentials than the fields (see

Section 3.1).

It remains to be shown that it is always possible to force the potentials to

satisfy the Lorenz condition (2.10). To this end, let us consider two potentials,0 and Φ0, which fulfil Equations (2.5) and (2.8) and check whether it is

possible to select them so that they satisfy equations (2.11). By inserting (2.5)

2A very interesting property of the Lorenz condition is that it is covariant, i.e., if it holds

in one particular inertial frame then it automatically holds in all other inertial frames.

2.1. ELECTROMAGNETIC POTENTIALS 31

and (2.8) into (2.9) and by rearranging, we get

∇2 0 − 002 0

2= −0 +∇

µ∇ · 0 +∇2Ψ+ 00

Φ0

− 00

2Ψ

2

¶(2.12a)

∇2Φ0 − 002Φ0

2= −

0−

µ∇ · 0 +∇2Ψ+ 00

Φ0

− 00

2Ψ

2

¶(2.12b)

and, given that the scalar functionΨ is arbitrary, we can choose it as the solution

to the differential equation

∇2Ψ− 002Ψ

2= −∇ · 0 − 00

Φ0

(2.13)

Thus (2.12a) and (2.12b) become (2.11a) and (2.11b), respectively, meaning

that 0 and Φ0 fulfil the Lorenz condition.Expressions (2.11a) and (2.11b) are the inhomogeneous wave equations for

the potentials, and their solutions, which are provided in the next section, rep-

resent waves propagating at the velocity = 1√00 ' 3× 108 of light in

free space. They take the form

∇2 − 1

22

2= ¤ = −0 (2.14a)

∇2Φ− 1

22Φ

2= ¤Φ = −

0(2.14b)

where the symbol ¤ represents the D’Alembertian operator defined by

¤ ≡ ∇2 − 1

22

2(2.15)

The Lorenz gauge (2.10) for harmonic fields simplifies to

∇ · ~A+

2Φ = 0 (2.16)

such that

Φ =2∇ · ~A

(2.17)

while (2.14a) and (2.14b) simplify to

∇2 ~A+ 2

2~A = −0 ~J (2.18a)

∇2Φ+ 2

2Φ = − ρ

0(2.18b)

In addition to Lorenz’s gauge, other gauge conditions may sometimes be

useful. For instance, in quantum field theory, where the potentials are used to


describe the interaction of the charges with the electromagnetic field instead of

being used to calculate the fields, it is useful to use Coulomb’s gauge, in which

∇ · = 0. By taking the divergence of (2.4) and the curl of (2.1), and taking

into account the generalized Ampère’s law and Gauss’ law , we can easily see

that with Coulomb’s gauge the expressions for the potential wave equations are

∇2Φ = −

0(2.19)

∇2 − 1

22

2− 1

2∇Φ

= −0 (2.20)

As can be seen from (2.19), in Coulomb’s gauge the scalar potential is deter-

mined by the instantaneous value of the charge distribution, using an equation

similar to Poisson’s expression in electrostatics. The vector potential, however,

is considerably more difficult to calculate. According (2.19), a time change in

implies an instantaneous change in Φ. This fact denotes the non-physical nature

of Φ since real physical magnitudes can change only after a delay determined

by the propagation time between the perturbation and the measurement point.

In this book we will use only the Lorenz condition, but it should be made

clear that the and fields calculated from the potentials with the Coulomb

or Lorenz gauges must be identical.

The complete solutions of the inhomogeneous wave equations for the poten-

tials (2.14) are linear combinations of the particular solutions and of the general

solutions for the corresponding homogeneous wave equations. The next section

is devoted to finding these particular solutions, which express the potentials in

terms of integrals over the source distributions and .

2.2 Solution of the inhomogeneous wave equa-

tion for potentials: retarded potentials.

Let us now calculate the expression of the potentials created by an arbitrary

bounded source distribution (charges and currents) in an unbounded homoge-

neous, time-invariant, linear and isotropic medium of conductivity zero that

we assume to be free space (Fig. 2.1). From (2.14) we see that the scalar po-

tential Φ as well as each of the three components , ( = 1 2 3), of the vector

potential satisfy inhomogeneous scalar wave equations with the general form

¤Ψ( ) = ∇2Ψ( )− 1

22Ψ( )

2= − ( ) (2.21)

where the operator ¤ acts on the coordinates of the field point, while the

sources coordinates are 0 0.To facilitate the solution of this equation, we can use, owing to the linearity

of the problem, the superposition principle and consider a source distribution

2.2. SOLUTIONOF THE INHOMOGENEOUSWAVEEQUATION FOR POTENTIALS: RETARDED POTEN

( ) as constructed from a sum of weighted space-time Dirac delta function

sources, i.e.,

( ) =

Z

0=−∞

Z 0

( 0 0) ( − 0)(− 0)00 (2.22)

where 0 is a volume containing all the sources. Thus, (2.21) can be solved intwo steps, using Green’s method in the time domain, as follows.

a) The first step is to calculate the response, ( 0 0) generated by thespace-time Dirac −function source, (− 0)(−0) located at position 0 andapplied at time 0 which obeys the inhomogeneous wave equation

¤( 0 0) = ∇2( 0 0)− 1

22( 0 0)

2= −( − 0)(− 0)

(2.23)

and satisfies the boundary conditions of the problem. The function ( 0 0)is called Green’s free-space function, which, because of the homogeneity of the

space, must be a spherical wave centred at position 0 at time 0. This functiondepends on the relative distance, = | − 0|, between the point source andthe observation or field point and on the time difference = − 0. Thus

( 0 0) = ( ) and (2.23) can be written, using spherical coordinates,

as

¤( ) = 1

2 ()

2− 1

22

2= −()() (2.24)

where, (A.13d),

∇2 =1

2 ()

2(2.25a)

2

2=

2

2(2.25b)

b) The second step is to find Ψ( ) from Green’s function. Owing to the

linearity of the problem and, from (2.22), if the solution of (2.23) is , then the

solution of (2.21) is 3

Ψ =

Z

0=−∞

Z 0

( 0 0)( )00 (2.26)

Because fulfils the boundary conditions, so too does Ψ( ).

To find the Green’s function let us consider first a general point 6= 0 suchthat equation (2.24) simplifies to

¤( ) = 1

2 ()

2− 1

22

2= 0 (2.27)

Multiplying this equation by and defining 0 = we have the homogeneous

wave equation20

2− 1

220

2= 0 (2.28)

3Note that Eq. (2.27) represents the spatial and temporal convolution of ( ) and ( )


The general solution of the above expression, as can be verified by direct sub-

stitution, is

0( ) = ( −) + ( +) (2.29)

where ( −) and ( +) are two arbitrary functions of their respective

arguments and they represent waves propagating along in the positive and

negative directions, respectively. Therefore

( ) =( −)

+

( +)

(2.30)

The potential that results from substituting Green’s function ( + ) in

(2.26) is termed the advanced potential and is a function of the value of the

sources at the future observation instant. This advanced potential is clearly

not consistent with our ideas about causality, according to which the potential

at ( ) can depend only on sources at earlier times. Thus, in (2.30) we must

consider only the retarded ( −) solution as physically meaningful.

To determine ( −), we integrate the differential equation (2.23) in

a very small volume around the singular point = 0. Thus, taking into account

that for → 0 the function behaves as (), we haveZ 0

µ∇2( )− 1

22( )

2

¶→0

0 (2.31)

=

Z 0

µ∇2µ()

¶− 1

22

2

µ()

¶¶0 (2.32)

= −Z 0

( − 0)()0 = −() (2.33)

or, since ∇2 (1) = −4() and 0 = 42,

−Z 04()()0 +

4

2

Z 0

2()

2 = −() (2.34)

As → 0, the second integral can be eliminated and therefore

() =()

4(2.35)

As the function depends on − and () = ( −)|=0, we have

( −) =( −)

4(2.36)

and the solution of (2.24) is given by

( ) =( −)

4=

(− 0 −)

4(2.37)

2.2. SOLUTIONOF THE INHOMOGENEOUSWAVEEQUATION FOR POTENTIALS: RETARDED POTEN

This is Green’s time-dependent retarded function, which takes into account the

time needed for the electromagnetic perturbation to reach the observation point

from the point source. Substituting this function in (2.26), we have

Ψ( ) =

Z

0=−∞

Z 0

( 0 0)( −)

400 (2.38)

and, integrating in 0, we finally find that, under the assumption of causality,the solution of the inhomogeneous wave equation for potentials is given by

Ψ( ) =1

4

Z 0

( 0 −)

0 =

1

4

Z 0

[]

0 (2.39)

where

[] = ( 0 −

) = ( 0 0) (2.40)

is the value of the source densities evaluated at the retarded times 0 = −,

which in general are different for each source point, being the delay time

due to the finite propagation velocity of the electromagnetic perturbations. In

the following the physical magnitudes evaluated in retarded times are shown in

brackets.

By analogy with (2.39) the solutions to the inhomogeneous equations for the

potentials are

Φ ( ) =1

40

Z 0

[]

0 (2.41a)

( ) =04

Z 0

[ ]

0 (2.41b)

where the bracket symbol [ ] indicates that the enclosed magnitude must be

evaluated at the retarded time 0 = −. That is

[] = ( 0 0) = ( 0 −) (2.42a)

[ ] = ( 0 0) = ( 0 −) (2.42b)

are the charge and current densities, respectively, evaluated in the retarded

times 0.Expressions (2.41a) and (2.41b), which are called retarded potentials, in-

dicate that the potentials created by a distribution at the field point are

determined, at a given time by the values of the the charge and current den-

sities at the source points evaluated at previous times 0, which generally differfor each source point (i.e. the retardation 0 = − forces us to evaluate

[] at different times for different parts of the charge distribution). It is easy

to check that these potentials, together with the continuity equation, verify

Lorenz’s condition (2.10)4.

4 It should be noted that (2.39) is a particular solution of (2.21), to which a com-


For sources with time-harmonic dependence

( 0 ) = Re©ρ(0))

ª(2.43)

( 0 ) = Re ~J( 0) (2.44)

the expressions of the retarded potentials Φ and simplify to

( ) =4Re

½Z 0

1

~J( 0)(−

)0

¾= Re ~A() (2.45a)

Φ( ) =1

4Re

½Z 0

1

ρ( 0)(−

)0

¾= Re

©Φ()

ª(2.45b)

where

~A() =4

Z 0

1

~J( 0)−0 (2.46a)

Φ() =1

4

Z 0

1

ρ( 0)−0 (2.46b)

where = = 2 is the wavenumber in the unbounded medium and is

the wavelength in the medium. For harmonic signals the time delay when

multiplied by , becomes a phase shift given by .

2.3 Electromagnetic fields from a bounded source

distribution

The fields created by a bounded source distribution (charges and currents in

free space) of arbitrary time dependence can be determined by inserting (2.41a)

and (2.41b) into (2.1) and (2.4). Next, we find the expression for the magnetic

field first and for the electric field afterwards.

plementary solution of the homogeneous wave equation ¤Ψ( ) = 0 can be added in

order to arrive at other possible solutions of (2.39). Thus, other conditions must be

imposed to ensure that the only possible solution of (2.21) is (2.39). The general so-

lution is given, according Kirchhoff ’s integral theorem by: Ψ( ) = 14

0

[]

0 +

14

01

Ψ( 00)0 −Ψ( 0 0)

01+ 1

Ψ( 00)0

0

0 where the volume integral is a

particular solution and represents the contribution of all the sources inside . The sur-

face integral is the complementary solution of (2.21) and represents the potential in of all

the sources outside . If Ψ and its first derivatives are known all over , Ψ is completely

determined at all interior points. However as the field propagates with a finite velocity; con-

sequently , if all sources are located within a finite distance of some fixed point of reference

and if they have been established within some finite period in the past, one may allow the

surface to recede beyond the first wave front. It lies then entirely within a region unreached

by the disturbance at time and within which Ψ and its derivatives are zero and therefore

the complementary solution is null.

2.3. ELECTROMAGNETIC FIELDS FROMABOUNDED SOURCEDISTRIBUTION37

Magnetic field

Starting from the equation

= ∇× =4

Z 0∇× [

]

0 (2.47)

and transforming the integrand by the vector analysis formula (A.16g) with

Ψ = 1 and = , we can directly find the magnetic field equation

( ) =4

Z 0

⎛⎝ [ ]×

3+1

h

i×

2

⎞⎠ 0 (2.48)

where [ ] is the retarded current density at the source point 0 andh

i=

[ ]0 = [ ] is its time derivative at the instant 0 = −.

Expression (2.48) can be written as the sum of the two components = + , which are defined below.

The Biot-Savart term, :

=4

Z 0

[ ]×

30 (2.49)

which is formally analogous to the Biot-Savart expression of magnetostatics,

although here with the sources evaluated at the retarded times. As this term

decreases with 12, its contribution is appreciable only at short distances.

The radiation term,

=4

Z 0

h

i×

20 (2.50)

which depends on 1, and consequently its contribution to the magnetic field

predominates at long distances from the sources.

At the static limit, when the sources do not change with time (i.e., for a

stationary current distribution) equation (2.48) simplifies to the Biot-Savart

expression of magnetostatics

=4

Z 0

×

30 (2.51)

Electric field

From (2.4) and (2.41) we see that

= − 1

40

Z 0∇ []

0 − 0

4

Z 0

[ ]

0 (2.52)


Taking into account that 0 = and that ∇Ψ () = (Ψ)∇ we

have

∇ []

= []∇ 1+1

∇ [] = []

Ã−

3

!+

2 []

(2.53a)

[]

=

[]

00

=

∙

¸µ−1

¶(2.53b)

∇ []

= []

Ã−

3

!−

2

∙

¸(2.53c)

which, when substituted in (2.52), and taking into account the continuity equa-

tion ∇ · = −, gives

( ) =1

40

Z 0

⎛⎝ [] 3− 1

2

h

i−

2

h∇0 ·

i⎞⎠ 0 (2.54)

To get the exact form of the radiation term, which depends on the distance as

1, we need to transform the integrand of this expression by developing ∇0 · [ ]as5

[∇0 · ] = ∇0 · [ ]−·[

]

(2.55)

thus we can rewrite the third term on the right-hand side of (2.54) as

−Z 0

2

h∇0 ·

i0 = −

Z 0

2∇0 · [ ]0 +

Z 0

³h

i· ´

230 (2.56)

The calculation of the first term on the right-hand side can be facilitated by

calculating just one component, for example the component

−Z 0

2∇0 · [ ]0 =

Z 0

[ ]

·∇0

20 −

Z 0∇0 ·

µ

2[ ]

¶0

=

Z 0

[ ]

·∇0

20

=

Z 0

[ ]

2·∇0

0 +Z 0

[ ]

·∇0 1

20

=

Z 0

⎛⎝− []2

+2³[ ] ·

´

4

⎞⎠ 0 (2.57)

5∇0 · [ ] = (∇0 · )0 + []

0 ·∇00 = (∇0 · )0 −[]

0 ·∇0 = (∇0 · )0 +· []0

Thus

(∇0 · )0 = [∇0 · ] = ∇0 · [ ]−· []

0


where we have used (A.16f), applied the divergence theorem, and integrated

over an external surface that encloses the sources in which [ ] = 0. Therefore,

generalizing to three dimensions and inserting the result in (2.54), we get

4 =

Z 0

[]

30 +

Z 0

⎛⎝2³[ ] ·

´− [ ]

³ ·

´4

⎞⎠ 0 +

+1

2

Z 0

³h

i×

´×

30 (2.58)

which can be expressed as the sum of the three components = + + ,

which are defined below.

Coulomb’s term, ,

=1

4

Z 0

[]

30 (2.59)

This term is similar to the static Coulomb’s expression except concerning the

time delay.

Induction term, ,

=1

4

Z 0

⎛⎝2³[ ] ·

´

4− [ ]

2

⎞⎠ 0 (2.60)

Because of their dependence on 12, the contribution to the field of the terms

(2.59) and (2.60) decrease quickly with distance.

Radiation term, ,

=1

42

Z 0

³h

i×

´×

30 =

04

Z 0

³h

i×

´×

30

(2.61)

This term, which depends on 1, is the electric field component that predom-

inates for long distances. Together with (2.50), this component is of interest in

radiation phenomena (see next subsection) .

At the static limit, expression (2.58) simplifies to Coulomb’s expression of

electrostatics

=1

4

Z 0

30 (2.62)

Alternatively, the electric field can be expressed only in terms of the current

density, by using the continuity equation. In fact, from (2.55) we have

[] = −Z

−∞[∇0 · ]0 = −

Z

−∞

Ã∇0 · [ ]−

·[ ]

!0 (2.63)


Inserting (2.63) into (2.58) and operating in a similar way to (2.57), we obtain

another alternative expression for the electric field created by a bounded source

distribution

=1

4

Z 0

Z

−∞

⎛⎝3³[ ] ·

´

5− [

]

3

⎞⎠ 00

+1

4

Z 0

⎛⎝3³[ ] ·

´

4− [ ]

2

⎞⎠ 0

+1

4

1

2

Z 0

³h

i×

´×

30 (2.64)

In summary the radiation electric and magnetic fields are given by

=04

Z 0

³h

i×

´×

30 (2.65)

=4

Z 0

h

i×

20 (2.66)

which from (1.6) can be written as

=04

Z 0

³h()

i×

´×

30 (2.67)

=4

Z 0

h()

i×

20 (2.68)

and, assuming that the charge density does not depend on time, we have

=04

Z 0

³[]×

´×

30 (2.69a)

=4

Z 0

[]×

20 (2.69b)

so that radiation fields exist only when the charge density is accelerated.

Fields created by a time-harmonic source distribution

For time-harmonic dependence of the sources, the field expressions (2.4) and

(2.1) simplify to


~B = ∇× ~A (2.70a)

~E = −∇Φ− ~A (2.70b)

and equation (2.48) for the magnetic field becomes

~B =4

Z 0( ~J × )

µ1

3+

2

¶−0

while the different expressions for the electric field, (2.54), (2.58) and (2.64)

become, respectively,

~E =1

4

Z 0

ρ −

30 +

4

Z 0

Ãρ

−~J( 0)

!−

0 (2.72a)

~E =1

4

Z 0

ρ

3−0 +

1

4

Z 0

⎛⎝2³~J ·

´

4−

~J

2

⎞⎠ −0 +

+

4

Z 0

³~J ×

´×

3−0 (2.72b)

~E =

4

Z 0

⎛⎝ ~J

3−3³~J ·

´

5

⎞⎠ −0 +

1

4

Z 0

⎛⎝3³~J ·

´

4−

~J

2

⎞⎠ −0 +

+

4

Z 0

³~J ×

´×

3−0 (2.72c)

and the radiation fields (2.50) and (2.61) become

~B =4

Z 0

~J ×

2−0 (2.73a)

~E =4

Z 0

³~J ×

´×

3−0 (2.73b)

2.3.1 Radiation fields

Examining the total fields (2.48) and (2.64) generated by a bounded distribution

of sources with arbitrary time dependence, we find that in general the near-zone


terms, which depend on 1 ( 1), are negligible compared to the radiation

terms, (2.50) and (2.61), which depend on 1, when the condition

¯[ ]¯

¯[ ]

¯ (2.74)

is fulfilled for any of the infinitesimal volume elements into which the source can

be subdivided. For time-harmonic fields, this condition becomes

(2.75)

Hence, the radiation term predominates when distances from the sources

are great compared to any wave-length involved. The zone where the radiation

fields predominate can be called by several names: far zone, wave zone and

Fraunhofer zone. Note that the far zone is farther away from the sources at

lower time dependence (i.e., at lower frequencies) and there is no far zone at the

static limit.

Let us select the reference origin close to or within the source distribution,

(Fig. 2.1). If the field point is far away from any source point such that 0,or equivalently , where is the largest dimension of the source distribu-

tion, then it is possible to make some general approximations in the expressions

(2.50) and (2.61) which greatly simplify the calculations. To confirm this, let

us write in Fig. (2.1) as

= | − 0| = ¡2 − 2 · 0 + 02¢12

(2.76)

Since the reference origin is close to or within the source distribution, we can

calculate the radiation fields at distances 0 by expanding the binomial(2.76) as a series in powers of the small parameter 0 and take only the linearterms of the expansion

=

µ1− 2 ·

0

2+

02

2

¶12= − · 0

+ ' − 0 · = − 0 cos

(2.77)

where is the angle between and 0. This approximation is equivalent toconsidering that, far away from the sources, and become parallel.

Thus, as 0 1, in the expressions (2.50) and (2.61), we can make the

approximation

' (2.78)

in the denominator. This is equivalent to ignoring, in the modulus of the con-

tribution of each source point to the total field, the difference in the distance

travelled by the signal. Thus (2.74) becomes

¯[ ]¯

¯[ ]

¯ (2.79)


and (2.75) becomes

(2.80)

In the retarded time, 0 = − , the approximation (2.78) is not valid

because the sources can be very sensitive to small changes in the delay time

. Thus, for the delay time, at distances 0 we need to keep at leastthe two linear terms of the expansion (2.77). Therefore

0 = −

= −

+

0 ·

= 00 + 0 cos

(2.81)

where 00 = − .

Therefore, from (2.81), the retarded time has two components. One, , is

the time needed for the electromagnetic field to reach the field point from the

origin of the coordinates. The other, 0 · , represents the time necessary forthe propagation of the electromagnetic perturbation within the geometric limits

of the source distribution. This term, given that the largest dimension of the

source distribution is , (Fig. 2.1), has a magnitude of

0 · ∼ (2.82)

Hence, using the approximations (2.78) and (2.81) the integrands of the

radiation fields (2.50) and (2.61) simplify to

=4

Z 0

( 0 00 + 0·)

× 0 (2.83a)

=1

42

Z 0

Ã ( 0 00 +

0·)

×

!× 0 (2.83b)

or, for time-harmonic dependence,

~E =4

Z 0

³~J ×

´× −0 (2.84a)

~B =4

Z 0~J × −0 =

4

Z 0~J × −0 (2.84b)

A comparison of Eqs. (2.83a) and (2.83b), shows that the radiation fields

are perpendicular to each other and to the direction of propagation. They are

related by = 0

× (2.85)

where the ratio 0 is defined as

0 =

= ()

12 = 120 Ω (2.86)

and is called the intrinsic impedance of free space.

According to Poynting’s theorem the total radiated energy passing through

the unit area perpendicular to the direction of the vector × is given

by


Z

−∞P =

Z

−∞( × ) (2.87)

and the total flow of power passing through the closed surface situated in the

far-field zone isZ

−∞

Z

P · =Z

−∞

Z

( × ) · (2.88)

In summary, the assumptions involved in using (2.83) and (2.84) to calculate

the radiation fields created by a bounded source distribution in the far-field zone

are:

a) ( |[ ]| |[ ]|) or, equivalently, for any wavelength of

the radiation spectrum which allows us to neglect 12 terms.

b) , where is the largest dimension of the source distribution which

allows us to make the approximations (2.78) and (2.81).

2.3.2 Fields created by an infinitesimal current element

The simplest case of a bounded source distribution is that of an infinitesimal

current element (), which is assumed to be oriented on the axis (Fig. 2.3)

and to have arbitrary time dependence. This current is mathematically defined,

in terms of the Dirac delta function, as

( ) = ()(0)(0) − ∆2

0 ∆

2(2.89)

The fields of this current element can be easily calculated by substituting (2.89)

in (2.48) and (2.64). Thus, we have

Fields created by an infinitesimal current element with arbitrary-time

dependence:

( ) =∆

4

µ1

[]

+[]

2

¶( × ) =

∆

4

µ1

[]

+[]

2

¶sin (2.90a)

( ) =∆

4

µ1

3

Z

−∞[] +

[]

2

¶(3 ( · ) − ) +

∆

4

1

2

[]

( × ( × )) =

∆

4

µ1

3

Z

−∞[] +

[]

2

¶(2 cos + sin ) +

∆

4

1

2

[]

sin (2.90b)

where [] = (− ).

For time-harmonic dependence of the current element, = Re©I

ª, equa-

tions (2.90a) and (2.90b) simplify to


Figure 2.3: Radiation fields from an infinitesimal current element. Transverse

field components, and , are orthogonal to each other.

Fields created by an infinitesimal current element with time-harmonic

dependence:

~H() =I∆

4

µ1 +

1

¶−

sin (2.91a)

~E() =I∆

40

Ã1 +

1

− 1

()2

!−

sin +

I∆

20

Ã1

− 1

()2

!−

cos (2.91b)

These expressions can be also derived directly from the vector potential (2.41b),

which in this case simplifies to

= 04

Z ∆2

−∆2

[]

0 '

[]04

∆ (2.92)

Thus, the magnetic field is given by

=1

0∇× =

1

0∇× () = 1

0(∇× +(∇× )) =

1

0∇×

(2.93)

where we have applied the vector identity (A.16g) and taken into account that

the curl of a constant vector is zero. Hence, using spherical coordinates, we get


Figure 2.4: Fields created by an infinitesimal current element with time-

harmonic dependence. Radiation field separates from the source and propa-

gates to infinity. Note that there is not radiation in the -direction in which the

current element is pointing.

=1

0∇× =

∆

4

µ[]

¶ ×

=∆

4

µ− 1

[]

− []

2

¶ × =

∆

4

µ1

[]

+[]

2

¶sin

(2.94)

which of course coincides with (2.90a).

The electric field (2.90b) can be calculated from (2.94), taking into account

that from (1.1d), in source-free regions, we have6

( ) =1

0

Z

−∞∇× ( ) (2.97)

6 Note that once calculed = 10∇× we can obtain using (1.1d) or (1.68d) and taking

into account that, in source-free regions, we have

=1

0

∇× =

∇× (∇× ) (2.95)

or

=1

0∇× =

1

∇× (∇× ) (2.96)

for arbitrary or harmonic time dependence respectively. Thus we do not need necessarily to

calculate Φ to obtain the fields.


From the relation between the charge and current, () = (), we have

4 =

4 =

= (2.98)

where = 4 is the dipole moment of a time-varying electric dipole7, the

so-called Hertzian dipole, formed by two point charges with values of +()

and −() and the dot indicates differentiation with respect to time. Thus thetime-varying current element is equivalent to

() =1

4

(2.99)

or, for time-harmonic dependence,

I =p

4(2.100)

Introducing (2.99) into (2.90a) and (2.90b), and (2.100) into (2.91a) and (2.91b),

we get the field created by an infinitesimal current element (hertzian dipole) in

terms of its dipole moment as:

Fields created by a Hertzian dipole with arbitrary-time dependence:

=1

4

µ[]

+[]

¶sin (2.101a)

=1

40

µ[]

2+[]

+[]

2

¶sin +

1

2

µ[]

2+[]

¶cos (2.101b)

Fields created by a Hertzian dipole with time-harmonic dependence:

~H =p

4

µ1

+

¶−

sin (2.102a)

~E =p

40

µ1

2+

− 2

¶−

sin +

p

2

µ1

2+

¶−

cos (2.102b)

The radiation fields created by an infinitesimal current element can be ex-

pressed, from (2.90a) to (2.91b), in terms of its current amplitude or of its

equivalent dipolar moment.

7The time varying electric dipole is defined as two time varying charges of opposite magnitude

±() separated by a constant distance ∆ much less than the field point . The dipole moment

() is given by the magnitude of the charge times the distance ∆ between them and the defined

direction is toward the positive charge i.e. () = ()∆.. Alternatively it would be possible to

model the oscillating dipole as two constant point charges of opposite sign separated by oscillating

distance ∆() However, this model would need the fields created for such accelerated charges.


Radiation fields created by an infinitesimal current element:

For arbitrary-time dependence

=∆

4

1

[]

sin (2.103a)

=∆

4

1

2

[]

sin (2.103b)

For time-harmonic dependence

~H =I∆

4

−

sin (2.104a)

~E =I∆

40

−

sin (2.104b)

and from (2.101a) to (2.102b), we have the radiation fields in terms of its equiv-

alent Hertzian dipole:

Radiation fields created by an electric dipole:

For arbitrary-time dependence

=1

4

[]

sin (2.105a)

=1

40

[]

2sin (2.105b)

For time-harmonic dependence

~H = −p4

−

sin (2.106a)

~E =−p240

−

sin (2.106b)

2.3.3 Far-zone approximations for the potentials

The general expressions (2.48) and (2.58) for the fields due to an arbitrary

source distribution of finite size are of theoretical and sometimes of practical

interest. However, except for the case of the infinitesimal current element, it

is much easier to calculate the fields created by a given source distribution via

the potentials, as indicated in Fig. 2.2. This can be seen simply by comparing

the complexity of the expressions for these fields, (2.48) and (2.58), with those

for the potentials (2.41a) and (2.41b). Because of the vector product in the

integrand of (2.50) and (2.61), this argument continues being true even when

we are interested only in the radiation fields. In the far zone, we can make the

approximations (2.78) and (2.81) for the potentials. Hence, the integrands of


the retarded potentials (2.41) simplify to

Φ ( ) =1

40

Z 0

( 0 0)

0 ' 1

40

Z 0

( 0 00 + 0 · )0

(2.107a)

( ) =04

Z 0

( 0 0)

0 ' 04

Z 0

( 0 00 + 0 · )0 (2.107b)

The magnetic field can now be calculated from (2.1), using (A.16g), as

= ∇× =04

Z 0∇×

( 0 00 + 0·)

0

=04

Z 0

∇× ( 0 00 + 0·)

0 − 0

4

Z 0

( 0 00 + 0 · )×∇1

0

(2.108)

where, if we are interested only in the radiation field, the second term can be

ignored since it depends on 12, and therefore

=∇×

0=1

4

Z 0

∇× ( 0 00 + 0·)

0 (2.109)

Furthermore we have ∇× (Ψ) = ∇Ψ × Ψ with Ψ = 00 + 0 · . Thus,it follows that

∇× ( 0 00 + 0 · ) = −∇

× ( 0 00 +

0·)

= − × ( 0 00 +

0·)

(2.110)

and therefore

= − 1

0 ×

(2.111)

which, as would be expected, leads to (2.83a). If the time variations of the

sources are harmonic the expressions (2.107a) , (2.107b) and (2.111) become

Φ =1

40−

Z 0ρ( 0)

· 00 (2.112a)

~A =04

−Z 0~J( 0)

· 00 (2.112b)

~H = −

0 × ~A (2.112c)

The radiation electric field can be calculated from (2.111) or (2.112c) simply

using (2.85).


2.4 Multipole expansion for potentials

In many cases, such as the study of most antennas, in order to calculate the

radiation fields, we cannot make any approximation concerning the potentials

other than those assumed above. For example, we need to carry out the integra-

tion in (2.107b) or (2.112b) in order to calculate the vector potential. However,

if we assume that the charge distribution does not change appreciably over time

, we may expand the integrands of (2.107) in a Taylor series about 00 in termsof the parameter 0 · . For example, for the vector potential, we have

( 0 00 + 0 · ) = ( 0 00) +

( 0 0)0

¯¯0=00

0 ·

+ (2.113)

where we have omitted higher-order terms in 0·. Thus after inserting (2.113)in (2.107b), we can write as the power-series expansion

' 1 + 2 + =

04

Z 0

( 0 00)0 +

04

Z 0

( 0 0)0

¯¯0=00

0 · 0 +

(2.114)

Therefore the first two terms of the expansion (2.114), 1 and 2, are given by

1 =04

Z 0

( 0 00)0 (2.115a)

2 =04

Z 0

( 0 0)0

¯¯0=00

0 · 0

=04

Z 0

0( 0 0) 0 · 0

¯0=00

(2.115b)

If the time dependence of the sources is sinusoidal the condition that the source

distribution does not change appreciably over time is equivalent to assuming

that (where is the period of the signal) or equivalently 1,

i.e., that the dimension of wavelength is much greater than that of the source

distribution

(2.116)

In this case, we can perform the series expansion

· 0 = ·

0≈ 1 + · 0 − 1

22( · 0)2 + (2.117)

which, after substituting in (2.112b), leads to

~A = ~A1 + ~A2 + (2.118)

2.4. MULTIPOLE EXPANSION FOR POTENTIALS 51

where

~A1 =04

−

Z 0~J(0)0 (2.119a)

~A2 = 04

−

Z 0~J(0) · 00 (2.119b)

which are the Fourier transforms of (2.115a) and (2.115b), respectively.

Of course, there are analogous expressions for the terms of Φ

Φ = Φ1 +Φ2 +

=1

40

Z 0

( 0 00)0 +

1

40

Z 0

( 0 0)0

¯0=00

0 · 0 +

(2.120)

where

Φ1 =1

40

Z 0

( 0 00)0 (2.121a)

Φ2 =1

40

Z 0

( 0 0)0

¯0=00

0 · 0 (2.121b)

Note that, since the contribution of each point source to the integral in

(2.121a) is evaluated at the same time 00, this integral represents the totalcharge of the source distribution. Thus, if the net charge of the distribution is

zero, we have Φ1 = 0. If the net charge is not zero, the constant, the electrostatic

potential Φ1 created by that charge depends on −2 and consequently it doesnot contribute to the radiation.

The expansion (2.114) allows us to decompose the electromagnetic field

created by a time-varying source distribution of finite dimension in terms of

elementary time-varying source distributions, called electric and magnetic mul-

tipoles, located at the origin. This is similar to the well-known multipolar ex-

pansion of the electrostatics (or magnetostatics) to decompose the field created

by a stationary source distribution of charge (or current) in terms of electric (or

magnetic) multipoles. However, now the original distribution is time-varying

and produces both electric and magnetic fields. Thus, as result of the expan-

sion, we will obtain both, electric and magnetic multipoles. To verify this, we

next analyze the first two terms, (2.115a) and (2.115b), of (2.114).

2.4.1 Electric dipolar radiation

The evaluation of the term (2.115a) of the power-series expansion of can be

facilitated by calculating just one component ofR 0 (

0 00)0, for example the


componentZ 0

(0 00)

0 =

Z 0

( 0 00) · 0 =Z 0

( 0 00) ·∇000

=

Z 0∇0 ·

³0 ( 0 00)

´0 −

Z 0

0∇0 · ( 0 00)0

= −Z 0

0∇0 · ( 0 00)0 (2.122)

since Z 0∇0 ·

³0 ( 0 00)

´0 = 0 (2.123)

as can be seen by applying the divergence theorem and by integrating over

an external surface, where ( 0 00) = 0, that encloses the sources. Therefore,

generalizing to three dimensions we haveZ 0

( 0 00)0 = −

Z 0

0∇0 · ( 0 00)0 (2.124)

and using the equation of continuity

∇0 · ( 0 00) = −( 0 00)

(2.125)

we get Z 0

( 0 00)0 =

Z 0

0( 0 00)

0 (2.126)

which, when substituted in (2.115a), gives

1 =04

Z 0

0( 0 00)

0 =

04

Z 0

0( 0 00)0 (2.127)

The integralR 0

0( 0 00)0 is by definition the electric dipole moment, [],

evaluated at the retarded time 00, of the time-varying source distribution, i.e.,

[] =

Z 0

0( 0 00)0 =

Z 0

0( 0 −

)0 (2.128)

Thus we have

1 =04

[]

=

0

·[]

4(2.129)

The magnetic radiation field, from (2.111), is given by

= − × [··]

4=[··] sin

4 (2.130)

where we have assumed the direction of parallel to the polar axis. This

expression, as might be expected, coincides with the radiation term, (2.105a),


of (2.101a). From (2.130), the electric radiation field, given by (2.105b), can

be obtained using (2.85). Of course the corresponding expressions for time-

harmonic fields are given by (2.106a) and (2.106b). Therefore, in a preliminary

approximation, the original source distribution can be replaced by an electric

dipole located at the origin of coordinates.

2.4.2 Magnetic dipolar radiation

The analysis of the term (2.115b), can be facilitated by expressing the integrand

as follows

( 0 00)( · 0) =1

2

³( 0 00)(

0 · )− 0³( 0 00) ·

´´+1

2

³( 0 00)(

0 · ) + 0³( 0 00) ·

´´=

1

2 ×

³( 0 00)× 0

´+1

2

³( 0 00)( · 0) + 0

³( 0 00) ·

´´(2.131)

Then, substituting in 2, we get

2 = 2 + 2 (2.132)

where

2 =08

Z 0

×

³( 0 00)× 0

´0 (2.133)

and

2 =08

Z 0

³( 0 00)( · 0) + 0

³( 0 00) ·

´´0 (2.134)

The integral (2.133) can be written as

2 =04

[]

× (2.135)

where

[] =

Z 0

0 × ( 0 00)2

0 (2.136)

is by definition the magnetic dipolar moment about , evaluated at the retarded

time 00, of the source distribution. Thus, under the assumption that = ,

the magnetic radiation field given by (2.111) is

=1

42 × ( × [

··]) =

1

4

[··]

2sin (2.137)

From (2.85) the electric radiation field is given by

= − 04

[··]

sin (2.138)


For time-harmonic dependence, we have

~H =−2m sin

4− (2.139)

and

~E =2m sin

40− (2.140)

where

~m =

Z 0

0 × ~J(0)2

0 (2.141)

These expressions are similar to (2.105a)-(2.106b), which were obtained for

the electric field of the radiation of the electric dipole. In fact, as we will see

in the next section, there exists a duality in the analysis of the electric and

magnetic dipoles.

In the particular case of a current loop of radius , Fig.(2.5), for which the

current does not change appreciably over time (or equivalently for

any frequency involved), (2.136), becomes

=

ZΓ

0 ×

2= (2.142)

where Γ is the countour of loop and is the vector area of the surface subtended

by the contour Γ. In this expression, has been changed to . The surface

vector is directed normal to the loop according to the right-hand rule for

the direction of the current in the loop. Thus, for the circular current loop, the

radiation fields (2.137)-(2.140), can be written, for arbitrary time dependence,

as

=

4

[··]

2sin (2.143a)

= −04

[··]

sin (2.143b)

where is evaluated at 00. These equations, for time-harmonic dependencebecome

~H =−2I sin

4− (2.144a)

E =2I sin

40− (2.144b)

It should be mentioned that the magnetic moment is important only when

there exists no radiation of the electric moment of the system. Otherwise the

one due to the magnetic moment may be ignored. Effectively, comparing Eqs.

(2.105b) and (2.138), and using and to indicate the amplitudes of the


R

Iq

z

y

x

a

r

O

m

P

R

Iq

z

y

x

a

r

O

m

R

Iq

z

y

x

a

r

O

m

P

Figure 2.5: A circular loop of current in the x-y plane ( = ).

electric radiation fields from an electric and a magnetic dipole, respectively, we

have

=

(2.145)

or for time-harmonic variation with both dipoles oscillating at the same fre-

quency,

=0

0

(2.146)

Since from (2.141) we have

0 =

Z 0

0 × 0

20 =

1

2

Z 0

00 × 0 (2.147a)

0 =

Z 0

000 (2.147b)

and consequently

0 ∼ 0 (2.148)

where is the velocity of motion of the charges. Thus from (2.146) we have, for

,

(2.149)

i.e., the magnetic dipolar radiation may be ignored in comparison with the

electric dipolar radiation.


2.4.3 Electric quadrupole radiation

The second term, 2, of 2 in (2.134), is associated with the electric quadru-

pole radiation, but to see this we must transform it further. To this end let us

consider the component of the first summand of the integralR 0

³( 0 00) · 0

´0

i.e. Z 0

· 0³( 0 00) · 0

´0 =

Z 0

· 0³( 0 00) ·∇00

´0

=

Z 0∇0 ·

³0 ( · 0) ( 0 00)

´0 −

Z 0

0∇0 ·³( · 0) ( 0 00)

´0

(2.150)

where the first integral is null, as can be seen using the divergence theorem to

convert the volume integral in a surface integral with the surface of integration

outside of the source distribution. ThusZ 0

· 0³( 0 00) · 0

´0 = −

Z 0

0∇0 ·³( · 0) ( 0 00)

´0

= −Z 0

0∇0 ( · 0) · ( 0 00)0

−Z 0

0 ( · 0)∇0 · ( 0 00)0

(2.151)

but

∇0 ( · 0) = (2.152a)

∇0 · ( 0 00) = −(0 00)

(2.152b)

therefore Z 0

(0 00) · 00

= −Z 0

0³( 0 00) ·

´0 +

Z 0

0 ( · 0) (0 00)

0 (2.153)

Generalizing to three dimensionsZ 0

( 0 00) ( · 0) 0 = −Z 0

0³( 0 00) ·

´0+

Z 0

0 ( · 0)·(0 00)

0

(2.154)

and therefore, substituting in (2.134), we have

2 =08

2

2

Z 0

0 ( · 0) ( 0 00)0 (2.155)


The magnetic radiation field, given by (2.111), is

2 = −1

82 × 3

3

Z 0

0 ( · 0) ( 0 00)0 (2.156)

The above expression can be written in a more useful form by adding the

term 02( 0 00) to the integrand

2 = −1

242 × 3

3

Z 0

¡30 ( · 0)− 02

¢( 0 00)

0 (2.157)

Note that, since × 02 = 0, the added term do no affect to the value of the

integral. The advantage of including this term is that, now, the integrand can

be written as the product of a second rank tensor , called electric quadrupole-

moment tensor of the source distribution, and the vector Z 0

¡30 ( · 0)− 02

¢( 0 00)

0 = [] (2.158)

The elements of [] are

[] =

Z 0

¡30

0 − 02

¢( 0 00)

0 (2.159)

and [] is a vector with componentsX

[ ] (2.160)

Therefore the radiation magnetic field from a varying electric quadrupole is

given by

2 = −1

242 × 3[]

3= − 1

242 × [

...] (2.161)

or, for, time-harmonic dependence,

~H2 =3

24(−) ×Q (2.162)

The radiation electric field can be calculated as usual by (2.85). It can be

shown that quadrupole radiation fields are of the same order as the magnetic

dipole moment and thus much less than that corresponding to the Hertzian

dipole (Ejercicio)..

Of course, if we continued analyzing other terms in the expansion tal, we

would find other multipole moments, such as magnetic quadrupole radiation,

electric octupole radiation, etc. However, for this, other more complex mathe-

matical methods provide the results more systematically.


2.5 Fields created by a charge under arbitrary

movement: The Liénard Wiechert poten-

tials.

The fields created by a single point charge under arbitrary movement along a

specified trajectory (0) can be obtained from the retarded potentials (2.41)

expressing the retarded charge [] and current densities [ ], see (2.42), in terms

of the delta functions as

[] = (0(0)− 0(0)) (2.163a)

[ ] = (0)((0(0)− 0(0)) (2.163b)

where is the three-dimensional Dirac delta function, 0 = 0(0) is the positionvector describing the trajectory (0) of the point charge , 0 is the fixed

position vector of the field or observer’s point , see Fig. (2.6), [0] = 0(0) isthe retarded position of at the retarded time 0 = − (0), is the timeat the field point and (0) = [] = |0 − [0]| = |0 − 0(0)| is the separationdistance of point charge at its retarded position to the observer’s point ,

besides

[] = (0) =(0(0))

(2.164)

is the velocity of the charge at the retarded time 0. Substituting (2.163) into(2.41) we get

Φ (0 ) =

40

Z 0

( 0(0)− 0(0))|0 − 0(0)|

0 (2.165a)

(0 ) =0

4

Z 0

(0)(0(0)− 0(0))

|0 − 0(0)|0 (2.165b)

The evaluation of these integrals is not simple because, as the field time is

fixed, both terms in the argument of delta function varies (integration over 0

means that 0 also varies). To facilitate the evaluation we will rewrite them by

inserting a second delta function (0 − 0) that replaces 0 with 0

Φ (0 ) =

40

Z Z 0

(0 − 0)|0 − 0(0)|(

0(0)− 0(0))0 (2.166a)

(0 ) =0

4

Z Z 0

(0 − 0)|0 − 0(0)|(

0)( 0(0)− 0(0))0(2.166b)

and after integrating over the spatial delta distribution first, we obtain

Φ (0 ) =

40

Z(0 − 0)|0 − 0(0)|

0 =

40

Z(0 − 0)(0)

(2.167a)

(0 ) =0

4

Z(0)

(0 − 0)(0)

0 (2.167b)

2.5. FIELDS CREATED BYACHARGEUNDERARBITRARYMOVEMENT: THE LIÉNARDWIECHERT P

Figure 2.6: A point charge in arbitrary motion describing the trajectory (0)defined by the radius vector 0(0).

To evaluate this integral we need the identity

((0)) =

X

(0 − 0)¯|(0)0|

¯ (2.168a)

where each is a zero of . Applying this to the present case and taking into

account that there is only one zero at retarded time 08we have

(0 − 0) =(0 − 0)

0 (

0 − 0)0=0=

(0 − 0)0 (

0 − (− 1|0 − 0(0)|))0=0

=

=(0 − 0)

1− 0 (− 1

|(0 − 0(0))(0 − 0(0))|12)0=0

=1

1− (0)(0)(2.169)

where = . Then after substituting (2.169) into (2.167a) and following

exactly the same procedure for the calculation of , we obtain

Φ( ) =

4

1

[]= (2.170a)

( ) =

4

[]

[]=[]

2Φ( ) (2.170b)

where [] = (0), given by

80(0) is the only position of the charge on (0) where (0) = − 0


(0) =

Ã[]− [

] · []

!= []

µ1− [] · []

¶; [] = [][] (2.171)

is a function of both, the field point and the source point. The expressions

(2.170a) and (2.170b), valid for any charge velocity, are the so-called Liénard-

Wiechert potentials. It should be noted that the velocity[] = (0) of thecharged particle and the radiovector [] = (0) from the position of the particleto the point at which the potential is evaluated must be taken not at time but

at 0 = − .

2.5.1 Arbitrarily moving point charge fields

The fields can be calculated from the potentials in the usual manner9,(2.4) and

(2.1),

= −∇Φ−

= ∇×

where Φ and are given by (2.170a), (2.170b). Thus, assuming that we know

the position of the charge as a function of the time, we can calculate the electric

and magnetic fields observed at position 0 and time due to an arbitrarily

moving charge for which the retarded position and time are 0(0) and 0,respectively (Figure 2.6). To calculate the fields, we must first transform

and ∇, using the chain rule for derivatives and (A.16a)

= −∇Φ−

= −

4∇µ1

¶− 0

4

µ

¶=

42∇− 0

4

µ

−

¶(2.172)

= ∇× =

4∇×

=

4

∙∇×

+∇

µ1

¶×

¸(2.173)

In these expressions the components of the gradient operator ∇ are the

partial derivatives at constant observation time but not at constant retarded

time 0. However, the trajectory, velocity and acceleration of the charge aregiven with respect to the retarded time 0. Thus we must transform (2.172) and(2.173) into expressions involving the operators 0 and ∇0 = ∇|0=. Tothis end, we use the following transformation for the derivatives

=

0

0; 0 = −(0) (2.174)

9 In the following since we know that the variables in (2.170) must be evaluated at retarded

times, 0 = − (0), in order to make the notation more concise, we will not explicitlyindicate it unless we consider it convenient.


where0

=

(−)

= 1− 1

0

0

(2.175)

and thus0

=1

1 + 10

(2.176)

The quantity 0 can be found taking into account that

0= ·

0= − · (2.177)

therefore

0= −−

·

= − · (2.178)

where = (Fig. 2.6). Inserting (2.178) in (2.176) we have

0

=1

1− · =

(2.179)

and therefore

=

0

0=

0(2.180)

Furthemore since = (− 0), we have

∇0 = ∇µ−

¶= −∇

= −1

µ∇0+

0∇0

¶(2.181)

hence

∇0 = −

(− · ) = −

(2.182)

and, in general

∇ = ∇0 −

0(2.183)

Equations (2.180) and (2.183) are the basic transformation rules from the coor-

dinates of the field point to those of the retarded field point. These rules allow

us to compute the derivatives of the Liénard-Wierchert potentials to obtain

=

4

∙

23

0−

22+

1

2∇0− 1

3

0

¸(2.184a)

=

4

∙1

µ×

23

0− ×

22+

∇0 × −

2∇0×

¶¸(2.184b)


where = 0 is the charge acceleration at 0. Expressions (2.184a) and

(2.184b) can be simplified, observing that

0=

0

µ− ·

¶=

0−

0· −

·

= − ·

+2

−

·

(2.185)

and

∇0 =

− 1

∇0³ ·

´=

−

(2.186)

To obtain (2.186) we must take into account that, when deriving while keeping

0 constant, the vector remains constant. After combining the terms contain-ing to form a triple vector product, and performing similar calculations for, we get10

=

43

∙µ−

¶µ1− 2

2

¶+1

2×

µµ−

¶×

¶¸(2.187a)

=

4

∙∇×

¸=

4

∙∇×

+∇

µ1

¶×

¸(2.187b)

It should be noted that in these expressions, and must be evaluated

at retarded time 0.Moreover, as = (0), we have

(∇× ) =

−

=

0

0−

0

0

= − [×∇0] (2.188)

and taking into account (2.182)

∇× = − [×∇0] = −×

(2.189)

Finally, we obtain

=

4

"−×

2− 1

2(∇× )

#(2.190)

which, using (2.187a), can be written as

10The electromagnetic field due to an arbitrarily moving point charge can be also calcu-

lated directly from (2.48)and (2.54) using a similar procedure,to the followed here, however

the more elegant way is to find first the covariant form of the potentials in an instantaneous

proper frame, i.e., a frame in which the particle is at rest at a given instant of time and then

generalize the result for any other inertial frame using the Lorentz-transform of the special

relativity.


=1

"

×

#=1

h×

i(2.191)

Thus the magnetic field is always perpendicular to and to the vector .

The electric field (2.187a) created by the charge can be divided into two

parts, = + where

=

4

∙1

3

µ−

¶µ1− 2

2

¶¸(2.192)

is the induction or near field, which depends solely on the velocity of the charge

and varies spatially as −2.The other component is the radiation field

=

42

∙1

3

½×

µµ−

¶×

¶¾¸(2.193)

which depends on the velocity and on the acceleration and decrease as −1andpredominates at large distances from the charge and so at such distances ' . However at short distances the induction fields predominates on the

radiation fields. Needless to say that according (2.191), we have that the

magnetic radiation field from the particle is always perpendicular to its electric

field

( ) =1

([]× ( )) (2.194)

From (2.193) the radiation field is always perpendicular to and (transver-

sal field). Due to the inverse factor −2, large acceleration values are needed toproduce significant radiation. .

Equation (2.187) may be written as

( ) =

403[(− )

22+

×³(− )×

´2

] (2.195)

( ) =1

([]× ( )) (2.196)

and the radiation fieds as

( ) =

403[×

³(− )×

´2

] (2.197a)

( ) =1

([]× ( )) (2.197b)

where


= [1− · ]] (2.198a)

= (2.198b)

= (1− 2)−12 (2.198c)

When = 0 and = 0 then = = 0 so that the radiation fields are

nulls and (2.195) reduces to the electrostatic Coulomb’s field (40). Note

that (2.197a) shows that the radiated far field is proportional to the charge ac-

celeration. This formula can be particularized for the two significant cases of

synchrotron radiation, where the charge velocity and acceleration are perpen-

dicular (point charge in circular motion), and bremsstrahlung11 radiation, where

the acceleration is parallel to the velocity. In this case (2.197a) simplifies to

( ) =

403[× (× )

2] (2.199)

( ) =1

(× ( )) (2.200)

2.5.2 Angular distribution of the energy radiated by an

accelerated charge

The rate of energy loss for an accelerated charge per unit of solid angle Ω is

(0)Ω

=

0Ω=

Ω

0=

()

Ω

(2.201)

where we have taken into account that, according to (2.179), 0 = .

Using (2.194) and (2.193) to calculate the Poynting vector and remembering

the definition of differential of solid angle (Ω = · 2), we get the generalexpression for the power radiated per unit of solid angle of the particle,

(0)Ω

= P() · = 2

16231

4

³×

h³−

´×

i´2(1− · )5

(2.202)

Although this expression is in general complex, it is easy to see that there

is no radiation in the direction in which × ( − ) is annulled. In the

case of velocities close to the speed of light, the radiation intensity (2.202) is

high, due to the high exponential power of (1 − · ) in the denominator,within the narrow angular interval in which (1 − · ) is small. Thus, the11Bremsstrahlung is a German term that means “braking radiation” and is the radiation

produced by the acceleration or especially the deceleration of a charged particle


Figure 2.7: 2D radiation pattern for q and several values of .

Figure 2.8: 3D radiation pattern for q and several values of .


Figure 2.9: 2D radiation pattern for ⊥ and several values of .

Figure 2.10: 3D radiation pattern for ⊥ and several values of .


particle radiates mainly in the direction of its movement. Let us now consider

two particular cases:

1) Velocity and acceleration are parallel. It is straightforward to see that

(0)Ω

=2

1623

"2 sin2

(1− cos )5

#(2.203)

where is the angle formed by (or ) and . The 2D and 3D radiation

patterns12 for this case are shown in Figures (2.7) and (2.8) respectively.

The power radiated is dependent on the magnitude squared of the accelera-

tion. However, as the angle does not depend on the acceleration. It depends on

the speed. The faster the particle moves the more is the radiation concentrated

in the forward direction.

For ¿ the distribution acquires the functional form of the dipolar radia-

tion sin2 , but for relativistic velocities, the maxima are strongly forward-tilted.

The large exponent of the denominator means that for ultrarelativistic particles

( 1) and for values of cos ≈ 1 the denominator is very small. Conse-

quently, almost all the radiation is concentrated within the region of small

but there is no radiation for = 0.

The total radiated power can be determined from the expression 2.203 inte-

grating over every direction, thus obtaining

(0) =226

63(2.204)

The factor 6 means that the radiated energy increases enormously as the

charge (or particle) velocity approaches the speed of the light.

An example of applying (2.203) is the calculation of the radiation, called

“braking radiation” or “bremsstrahlung”, which occurs during the deceleration

of charged particles passing through a material medium, as for example when

a high-speed electron strikes a metal target. For an exact calculation, we need

to know the dependence of deceleration on time; however, for an approximate

calculation, we can assume that is constant while velocity decreases from the

value 0 to zero. Thus (2.203) becomes

=22

1623

Z 0

0

sin2 0Ω

(1− cos )5

(2.205)

Ω= − 2 sin2 2

64202 cos

"1

(1− 0 cos )4− 1#

(2.206)

This equation is usually used to estimate the efficiency of a low-voltage -ray

tube.

12Radiation pattern refers to the angular dependence of the strength of the radiation field.


2) Acceleration is perpendicular to the velocity. By developing the numerator

of (2.202), we get

(0)Ω

=22

16231

(1− cos )3

"1− sin2 cos2

2 (1− cos )2

#(2.207)

where is again the angle formed by and , while is the azimuthal angle

of the vector with respect to the plane containing and . As in the case of

bremsstrahlung, the velocity causes the radiation pattern to tilt in the forward

direction, i.e. towards the direction of velocity.

The radiation diagram presents a maximum that is mainly in the direction

of velocity such that, each time the velocity of the particle is towards a static

observer, radiation pulses would be seen .

Figure (2.9) shows the 2D radiation patterns in the plane of the orbit ( = 0)

for some values of = . Figure (2.10) shows the 3D pattern.

The total power can be calculated by integrating the expression (2.207).

(0) =224

63(2.208)

A comparison of (2.204) and (2.208) shows that for the same magnitude of

force applied, the power radiated for linear acceleration is 2 times greater than

that of transversal acceleration. Therefore, for relativistic particles accelerating

in an arbitrary direction, the effect of acceleration in the direction of the move-

ment predominates (in terms of energy loss by radiation) over the effect of the

instantaneous component of circular movement. Note that, since the angular

distribution depends on squared, the radiation does not change whether the

particle is under acceleration or deceleration and, in both cases, the radiation

is focused on the velocity-forward direction.

2.5.3 Larmor’s Formula

If the velocity of an accelerated charge is small compared to the speed of

light then → 0 and from (2.197a), we have

( ) =

40[× (× )

2] (2.209)

The energy flux is given by the (instantaneous) Poynting vector

P = ( )× ( ) =2

162023|× (× )|2

(2.210)

and the rate of energy per unir solid angle is

Ω= P2 = 2

16203||2 sin (2.211)

2.6. MAXWELL’S SYMMETRIC EQUATIONS 69

where is the angle between and .

From the vector products, we see that the radiation is polarized in the plane

of , , perpendicular to . Finally, the integral over angles yields 83, so that

=2

603||2 (2.212)

This is the Larmor formula for the power radiated from a nonrelativistic

accelerated point charge.

2.6 Maxwell’s symmetric equations

It can be observed from (1.1a)-(1.1d) that Maxwell’s equations present a certain

symmetry that, except in free space and with no source terms, is not complete

because of the absence of magnetic charges and currents. Indeed, despite many

experimental attempts, no free magnetic charges or monopoles have been found

in nature nor, therefore, would magnetic currents be created13. Nevertheless,

from a purely theoretical standpoint, nothing prevents us from assuming the ex-

istence of magnetic monopoles; therefore, to complete Maxwell’s equations we

must add the necessary magnetic source terms in order to achieve complete sym-

metry between electric and magnetic quantities. To this end, we can reformulate

Faraday’s law (1.1c) and Gauss’ law for magnetic fields (1.1b) by introducing,

on their right-hand side, hypothetical magnetic current densities ( −2)and magnetic charge densities (Wb/m3), respectively, as additional source

terms. With these new quantities included, we can rewrite Maxwell’s equations

for the case that both, electric as well as magnetic sources, exist in free space,

in the following completely symmetric manner:

Differential form of Maxwell’s symmetric equations

∇ · = (2.213a)

∇ · = (2.213b)

∇× = − − 0

(2.213c)

∇× = + 0

(2.213d)

13 It should be emphasized that, although there is no experimental evidence for the existence

of magnetic charges, such existence does not violate any known principle of physics. In fact,

from a purely theoretical viewpoint, Dirac showed [P.A.M. Dirac, Proc Roy. Soc.Lond. A133,

60 (1931)] that the existence of magnetic monopoles with magnetic charge would explain

the quantization of the electric charge . We refer to the magnetically charged particles as

magnetic monopoles or simply monopoles.


Integral form of Maxwell’s symmetric equationsI

· = (2.214a)I

· = (2.214b)IΓ

· = −Z

· −

Z

· (2.214c)IΓ

· =

Z

· +

Z

· (2.214d)

It should be emphasized that the symmetrization of Maxwell’s equations

is a powerful mathematical tool which greatly facilitates the solution of many

practical problems such as the radiation and scattering from aperture antennas

or permeable bodies.

Taking the divergence of (2.213c) and using (2.213b)

∇ ·∇× = −∇ · − ∇ ·

= 0 (2.215)

we get the equation of continuity

∇ · = −

(2.216)

which expresses the conservation of magnetic monopoles and has the same form

as that for the electric charges (1.3).

In linear media, we can apply the superposition principle and split each one

of the field quantities, and , into the sum of two components

= + = 0

³ +

´= 0 (2.217a)

= + = 0

³ +

´= 0

(2.217b)

where the quantities with the subscript depend only on the “true” electric

sources and while the quantities with the subscript depend only on the

“hypothetical” magnetic sources and . In this way, we divide Maxwell’s

equations into two groups corresponding to the field components associated with

the electrical and magnetic sources, respectively; that is

∇ · = (2.218a)

∇ · = 0 (2.218b)

∇× = −0

(2.218c)

∇× = + 0

(2.218d)


∇ · = 0 (2.219a)

∇ · = (2.219b)

∇× = − − 0

(2.219c)

∇× = 0

(2.219d)

Note that the sum of each expression (2.218), added to its equivalent (2.219),

gives (2.213) and that the set (2.218) coincides with the conventional Maxwell’s

equations (2.213), and that Eqs. (2.218) are formally identical to Eqs. (1.1a)-

(1.1d) and therefore can be solved as in the previous sections by means of the

scalar and vector potentials Φ and . Thus, from (2.4) and (2.1), we have

= ∇× (2.220)

= −∇Φ−

(2.221)

where

∇ · + 00Φ

= 0 (2.222)

and where and Φ fulfil the wave equations (2.14a) and (2.14b)

∇2 − 002

2= −0 (2.223)

∇2Φ− 002Φ

2= −

0(2.224)

the solutions to which are the retarded potentials (2.41a) and (2.41b)

Φ =1

40

Z 0

[]

0 (2.225)

=04

Z 0

[ ]

0 (2.226)

The fields created by the magnetic sources and can be deduced by ob-

serving that equations (2.218) are transformed into (2.219) and vice versa with

the simultaneous replacement of the following quantities, called duals

dual of

dual of −

0 dual of 00 dual of 0 dual of dual of

(2.227)


The fields and associated with the electric sources can be calculated

from the magnetic vector potential and the electric scalar potential Φ by

means of (2.225) and (2.226). To calculate the fields and we can use the

same formalism defining two new potentials, termed ”electric vector potential” and ”magnetic scalar potential” , such that

dual of

Φ dual of (2.228)

Hence

=1

40

Z 0

[]

0 (2.229)

=0

4

Z 0

[ ]

0 (2.230)

which are the dual expressions of (2.225) and (2.226).

By substituting the magnitudes in the first column of (2.227 and 2.228) for

their duals in the equations from (2.220) to (2.226) we get

= = −∇× (2.231a)

= −∇ −

(2.231b)

∇ · + 00

= 0 (2.231c)

in which and satisfy wave equations that are analogous to (2.223) and

(2.224):

∇2 − 002

2= −0 (2.232)

∇2 − 002

2= −

0(2.233)

Thus, by the superposition principle, if both current densities and exist

simultaneously in a region of free space, the total field produced at any point

is the sum of and given by (2.221) and (2.231a). Hence

= + = −∇Φ−

− 1

0∇× =

1

2∇Z∇ · −

− 1

0∇× (2.234)

where Lorenz gauge Eq. (2.10) has been used to express in terms of and .


The total field is determined analogously from (2.220) and (2.231b)

= + = −∇ −

+1

0∇× (2.235)

In practice, it is not necessary to use the latter expression, because once

has been calculated using (2.234), by substituting the result in (2.213c), with = 0 we obtain .

2.6.1 Boundary conditions

It is easy to show, ejercicio, that the boundary conditions corresponding to Maxwell’s

symmetric equations are a logical extension of (1.36); that is,

·³1 − 2

´= (2.236a)

·³1 − 2

´= (2.236b)

×³1 − 2

´= − (2.236c)

×³1 − 2

´= (2.236d)

in which is the normal unit vector that goes from region 2 to region 1. Equa-

tions (2.236b) and (2.236c) show the additional effects of the imaginary sur-

face magnetic charges and currents, and , at the interface. According

to (2.236c) and (2.236d), the tangential components of the fields on a real or

imaginary surface can be written in terms of surface distributions of electric

currents

× ¯= (2.237)

and magnetic ones

− × ¯= (2.238)

2.6.2 Harmonic variations

For harmonic variations, the symmetric equations (2.213) simplify to

∇ · ~D = ρ (2.239a)

∇ · ~B = ρ (2.239b)

∇× ~E = − ~J − 0~H (2.239c)

∇× ~H = ~J + 0 ~E (2.239d)


and the wave equations for the magnetic scalar potential the electric vector

potential , and the Lorenz relations are

∇2ψ + 200ψ = −ρ0

(2.240a)

∇2 ~F + 200~F = −0 ~J (2.240b)

ψ =∇ · ~F00

(2.240c)

with the solutions to (2.240a) and (2.240b) being

ψ =1

40

Z 0

ρ−

0 (2.241)

~F =0

4

Z 0

~J−

0 (2.242)

The total field ~E produced at any point is the sum of ~E and ~E, and is

given by

~E = ~E + ~E = − 2

∇³∇ · ~A

´− ~A− 1

0∇× ~F (2.243)

while for the total field ~H we have

~H = − 2

∇³∇ · ~F

´− ~F +

1

0∇× ~A (2.244)

where is given by (2.46a).

2.6.3 Fields created by an infinitesimal magnetic current

element

From (2.90a) and (2.90b), using the dual equations (2.227), we deduce that the

fields generated by an infinitesimal magnetic current element,

( ) = ()(0)(0) − ∆

2 0

∆

2(2.245)

are given by

= −∆4

µ1

[]

+[]

2

¶sin (2.246a)

=∆

4

µ1

3

Z

−∞[] +

[]

2

¶(2 cos + sin ) +

∆

4

1

2

[]

sin

(2.246b)

2.7. THEOREM OF UNIQUENESS 75

z

/ 2z

( )mi t/ 2z

y

x

E

H

r

z

/ 2z

( )mi t/ 2z

y

x

E

H

r

Figure 2.11: Radiation fields created by an infinitesimal magnetic current ele-

ment

or, for time-harmonic variation

~E = −∆I4

µ1 +

1

¶−

sin (2.247a)

~H =I∆

40

µ1 +

1

− 1

22

¶−

sin +

I∆

20

µ− 1

22+

1

¶−

cos

(2.247b)

Comparing the radiation terms of these equations to (2.143a)-(2.144b), we find

that

∆ = 0

(2.248)

or for time-harmonic dependence.

I∆ = 0I (2.249)

Fig. 2.11) shows the radiation fields created by an infinitesimal magnetic current

element.

2.7 Theorem of uniqueness

Whenever we have to resolve a differential equation, it is desirable to know

the conditions that must be fulfilled in order to state that a unique solution is

possible. In our context, this means to seek the conditions for which we can state

that there exists a single electromagnetic field that satisfies, simultaneously,

Maxwell’s equations and the given boundary conditions.

Next, we establish these conditions for non-harmonic and time-harmonic



2.7.1 Non-harmonic electromagnetic field

A non-harmonic electromagnetic field that varies in a linear region bounded

by a surface is uniquely determined from an initial time, = 0, if the following

are known:

i) The values of the sources at each point and at each time for every 0within the region.

ii) The values of the electromagnetic field ( and ) at each point of at

the initial time = 0.

iii) The tangential components of the electric field or of the magnetic field on the entire the surface for all 0, or, alternatively, the tangential

components of the electric field in any part of and of the magnetic field in the remaining part of , for all 0.

Proof

This theorem can be proven by a reduction to absurdity— that is, by showing

that to assume the opposite of what is postulated would lead to a contradiction.

Let us assume that having defined the three above conditions within a volume

, there exist two different electromagnetic fields, (1 and 1) and ( 2 and2), respectively, which are solutions to the problem. Given the linearity of

Maxwell’s equations, any linear combination of these two solutions must in itself

be a solution. In particular, the difference between the two aforementioned

fields, i.e. the field defined by ( 0 = 1 − 2 and 0 = 1 − 2), must also

be a solution to the problem. Given that, from the hypothesis, the sources are

the same for the fields ( 1 and 1) and ( 2 and 2), the field (0 0) issource-free in Thus, if we apply the Poynting theorem (1.40) to ( 0 0), weget

0 =

Z

1

2(0 · 0 + 0 · 0) +

Z

02 +I

(0 × 0) · (2.250)

It is straightforward to show that if the tangential components of the electric

field and/or of the magnetic field are uniquely determined on surface ,

the final term in (2.250) is null. By integrating this expression with respect to

the time from 0 to and, taking into account that the initial values for = 0are defined for all , we find that

0 =

Z

1

2( 0 · 0 + 0 · 0) +

Z

0

µZ

02¶ (2.251)

As both of the terms on the second member in (2.251) are positive, this

equality can be fulfilled only when both 0 and 0 are null (i.e. when 1 = 2

and 1 = 2), which is what we set out to prove.

2.7. THEOREM OF UNIQUENESS 77

2.7.2 Time-harmonic fields

In the case of harmonic variations, the uniqueness theorem states that a field

in a lossy ( 6= 0) 14 region is uniquely determined by the sources within the

region together with the tangential components of the electric field or of

the magnetic field on , or, alternatively, the tangential components of the

electric field in any part of and of the magnetic field in the remaining

part of .

Proof By a reasoning similar to that used for the above case, but using the

expression (1.112), we get

0 =

Z

0202

+ 2

Z

µ 02

0

4− 020

4

¶ (2.252)

By making the real and the imaginary parts equal to zero, we see that these

two equalities imply that 00 and 00 are both equal to zero only if 6= 0. This

is why we started from the premise that the medium occupying the volume has

a conductivity that may be arbitrarily small but which is non-zero at all points.

The field in a lossless region can be considered the limit to the lossy case when

such losses tend to zero.

14The reason why we need the extra condition of the space to be lossy for time-harmonic

signals is that, by definition, a pure harmonic signal has an infinite duration.

Chapter 3

Electromagnetic waves

In chapter 2 the fields created by a bounded time-variyng source distribution

were calculated and in particular we found that the radiation field propagates

energy as a wave far away from the sources. Given that energy propagates as a

wave, we will first obtain the electromagnetic wave equation from the Maxwell

equations and then, of all the possible solutions for the wave equation, we will

examine primarily the properties of their plane-wave solutions, i.e., waves for

which the wave-front are planes1. Plane waves constitute a good approximation

to actual waves in many situations because at sufficiently large distances from

the sources, in a sufficiently small region, any wave front can be treated as

a plane wave. For example, a great deal of optics is founded on the plane-

wave approximation and, similarly, in radiocommunication the radiated field at

sufficient distance from the antenna can be considered to be a plane wave. In

this Chapter we consider this kind of waves in a linear homogeneous isotropic

medium where there are no sources.

3.1 Wave equation

For time-varying electromagnetic fields it is possible to combine Maxwell’s equa-

tions to eliminate one of the fields, or , to obtain two uncoupled second-order

differential equations, one in and the other in , known as wave equations.

To formulate these wave equations, let us consider a non-magnetic ( = 0),

homogeneous, linear and isotropic region where, in general, source terms and

may exist. Taking the curl of (1.1c) and using the vector relation (A.16d) we

1Wave-front is defined as a surface that, at any time , is orthogonal to the propagation

vector at all the points on the surface.

79

80 CHAPTER 3. ELECTROMAGNETIC WAVES

have

∇×∇× = ∇(∇ · )−∇2 = −∇×

= −0

( + +

)⇒

∇2 = ∇(∇ · ) + 0

( + +

)

=1

∇+ 0

+ 0

+ 0

2

2

(3.1)

where and are the source and induced conduction density of the currents,

respectively. Thus, rearranging terms, we get

∇2 − 0

− 0

2

2=∇+ 0

(3.2)

which is known as the inhomogeneous vector-wave equation for the electric field.

A similar equation can be written for the magnetic field by taking the

curl of (1.1d),

∇2 − 0

− 0

2

2= −∇× (3.3)

For a lossless media (3.2) and (3.3) reduce to2

∇2 − 02

2=

1

∇+ 0

(3.4a)

∇2 − 02

2= −∇× (3.4b)

These Eqs are analogous to the inhomogeneous wave equation for the vector

potential (2.14a), and consequently their solutions take the form of the retarded

vector potential given by Eq. (2.41b), i.e.

( ) = − 1

4

Z 0

∇ [] + 12

h

i

0 (3.5a)

( ) =1

4

Z 0

∇×hi

0 (3.5b)

from which, by means of straightforward operations, we can obtain the expres-

sions (2.48) and (2.54) for the fields created by a bounded distribution of finite

densities of charges and currents with arbitrary space and time dependence.

2These equation are not decoupled and their solutions are of the form (3.5) which are

much more complex than (2.41).

3.1. WAVE EQUATION 81

In source-free regions ( = 0; = 0), except the charge and current densi-

ties induced by the presence of the fields, which are expressed in terms of the

constitutive parameters) the equations (3.2) and (3.3) simplify to

∇2 − 02

2− 0

= 0 (3.6a)

∇2 − 02

2− 0

= 0 (3.6b)

which are the homogeneous wave equations that determine the propagation of

the fields and in a sourceless homogeneous, linear and isotropic medium.

The solutions to these wave equations must be compatible with Maxwell’s equa-

tions and the coefficients of the solutions must be derived from the boundary

conditions.

Uniform plane waves are defined as waves with a field amplitude that, at

any instant, is the same at all points of the wave-front plane. Thus, the field

amplitude depends only on the distance from the origin to the plane (Fig.

3.1). Therefore, if = is the unit vector that is normal to the plane, the

operator ∇ becomes ∇ = and Maxwell’s equations simplify to

·

= 0 (3.7a)

·

= 0 (3.7b)

×

= −

(3.7c)

×

= +

(3.7d)

and the wave equations become

2

2− 0

2

2− 0

= 0 (3.8a)

2

2− 0

2

2− 0

= 0 (3.8b)

These equations, which describe the propagation of plane waves in a homoge-

neous conducting medium, are sometimes called the “telegrapher’s equations”.

For nondissipative media, for example the free space, these equations simplify

to

2

2− 00

2

2=

2

2− 1

22

2= 0 (3.9a)

2

2− 00

2

2=

2

2− 1

22

2= 0 (3.9b)


nz

x

r

y

x

O

nz

x

r

y

x

O

Figure 3.1: Plane wave front.

Figure 3.2: Wave in a lossless medium travelling to the right without changing

shape. The velocity of propagation in unbounded free space is the speed of light.

The most general solution of these one dimensional wave equations are of the

form which are plane waves travelling to the right or left (Fig. (3.2))

( ± ) (3.10a)

3.2 Harmonic waves

For time-harmonic fields, when the medium presents a conductivity and, at

the operating frequency, a complex dielectric constant, = 0− 00,(1.72), thewave equation (3.6a) can be written as a time-independent wave equation

3.2. HARMONIC WAVES 83

∇2 ~E − 0~E + 0

2 ~E = ∇2 ~E − 0~E + 0

20 ~E

= ∇2 ~E + 200 (1− tan ) ~E

= (∇2 + 20)~E = 0 (3.11)

where = +00, tan = 0, and = 0(1− tan ), are the effective

conductivity, the loss tangent and the effective complex permittivity defined

in (1.79), (1.82), and (1.84) respectively.

According to Subsection (1.5), depending on the characteristics of the medium,

the values of the term tan in Eq. (3.11) may range from 1 (zero for a

perfect dielectric or lossless medium) to 1 (infinite for a perfect conductor).

In a highly conductive medium tan 1 and 1 − tan ' − tan , andthus Eq. (3.11) becomes the so-called time-independent diffusion equation for

the electric field ~E

∇2 ~E − 0~E = 0 (3.12)

which is of the same type as the one that determines the propagation of heat by

conduction or by diffusion. As commented in Subsection (1.5), for most metals

the relaxation time is 10−14, which is a low value compared with the periodfor all frequencies lower than the optical ones. Thus, since tan = ()

−1, thediffusion equation is adequate for metals at all these frequencies.

Equation (3.11) can be written more concisely as

∇2 ~E − 2 ~E = 0 (3.13)

where is in general a complex quantity called the complex propagation con-

stant, which, from (3.11) and (3.13), is given by

−2 = 20( −

)

= 20(0 − (

00+

))

= 200 (1− tan ) = 2 (1− tan ) = 20

(3.14)

where

= p

0 (3.15)

is the wavenumber corresponding to an unbounded lossless medium with a real

dielectric constant 0.Analogously, for the magnetic field, we have

∇2 ~H − 2 ~H = 0 (3.16)


3.2.1 Uniform plane harmonic waves

For uniform plane waves, we have ∇2 = 22 and Eqs. (3.13) and (3.16)

simplify to

2 ~E

2− 2 ~E = 0 (3.17a)

2 ~H

2− 2 ~H = 0 (3.17b)

The complex propagation constant is usually written as

= (1− tan )12

= + (3.18)

where the imaginary part, , is termed the phase constant, whereas the real

part, , is called the attenuation constant of the wave3. Thus, from, (3.14) and

(3.18), we can easily calculate the explicit expressions for and

=

µ0

0

2

¶ 12 h(1 + tan2 )

12 + 1i12

=p

0√2

Ãr1 +

³

0

´2+ 1

!12

=√2

Ãr1 +

³

0

´2+ 1

!12(3.19a)

=

µ0

0

2

¶ 12 h(1 + tan2 )

12 − 1i12

=p

0√2

Ãr1 +

³

0

´2− 1!12

=√2

Ãr1 +

³

0

´2− 1!12

(3.19b)

The dimensions of and are −1 and they are referred to as neper andradian, respectively, to indicate their attenuative and phase meanings in wave

expressions. For lossless media we have = 0, = 0 and the phase constant

becomes = = .

3The term attenuation constant, also called attenuation parameter or attenuation coeffi-

cient is measured in nepers/metro (Np/m). The neper and dB are related by the following

relationships 1Np=20 log dB= 8 686 dB.


Equations (3.17) have solutions of the form ~E and ~H so that the

instantaneous values for the fields are given by wave equations

= Re~E(−) = Re~E(−·) = Re~E−(−)(3.20a)

= Re ~H(−) = Re ~H(−·) = Re ~H−(−)(3.20b)

where the so-called complex propagation vector = (with module and

direction of the unit vector normal to the wave-front planes) has been in-

troduced and is the position of any point on the wave-front plane so that

· = .

Equations (3.20) represent waves traveling at a speed given by the phase

velocity4

=

(3.21)

which in general, as is given by (3.19a), depends on the frequency (dispersive

media).

The penetration factor is defined as

=1

(3.22)

This is the distance at which, due to the attenuation the field module de-

creases from an initial given value to 1 of this value.

From (3.7) the following equalities may be deduced

· ~E = 0 (3.23a)

· ~H = 0 (3.23b)

× ~E = 0~H (3.23c)

× ~H = − ~E (3.23d)

From these equations, we see that ~E, ~H and are perpendicular to one

another and that they form a right-handed system in the order ~E, ~H . For this

reason these waves are often referred to as transverse electromagnetic (TEM)

waves. The magnitudes of ~E, ~H are related by

H =E

=

E

0(3.24)

where the quantity , known as the complex characteristic impedance of the

medium, is given, taking into account (3.14) and (3.18), by

4The phase velocity is the speed with which a position of constant phase moves in a

traveling wave.


=E

H=

0

=

µ0

¶12=

0

2 + 2( + ) =| | (3.25)

Thus, its module and phase is given by

|| =

¡00¢12

[1 + ( 0 )

2]14(3.26a)

= tan−1

=1

2tan−1

0=

2(3.26b)

Therefore, in general there is a phase shift between ~E and ~H (see Fig (3.3)).

3.2.2 Propagation in lossless media

By particularizing the above expressions for a lossless medium where, 0 = =

0, 00 = 0, and = 0, we thus have tan = 0; = ; and = = , and

consequently equations (3.13) and (3.16) simplify to

∇2 ~H + 2 ~H = 0 (3.27a)

∇2 ~E + 2 ~E = 0 (3.27b)

and the complex characteristic impedance of the medium, (3.26), simplifies to

= =³0

´12=

µ00

¶12=

0

12

=120

12

= 0 (3.28)

so that the impedance is real and constant. In particular, when the medium is

free space, simplifies to the impedance of free space

= 0 =

µ00

¶12= 120 (3.29)

Consequently, in unbounded lossless media, there is no phase shift between ~E

and ~H and the attenuation is null ( = 0). Thus = and = ∞ and Eqs

(3.23) simplify to (see Fig (3.4))

· ~E = 0 (3.30a)

· ~H = 0 (3.30b)

× ~E = 0~H (3.30c)

× ~H = − ~E (3.30d)


xx

Figure 3.3: Uniform plane wave propagating in the + direction in a dissipative

medium.

xx

Figure 3.4: Uniform plane wave propagating in the + direction in a lossless

medium.


3.2.3 Propagation in good dielectrics or insulators

In a good dielectric the reactive current predominates on the dissipative current

and according to (1.92), tan = 0 1. In this case, we can develop the

complex propagation constant (3.18) to get

= (1− tan )12

= (00)12

µ1− tan

2+tan2

8+

¶'

(00)12

µ1− tan

2

¶(3.31)

and therefore

' (00)12 tan 2

=

2

³00

´12(3.32a)

' = (00)12 (3.32b)

Thus the propagation velocity can be approximated by

' 1

(00)12

(3.33)

From (3.32a) it can be seen that is small and therefore so is the wave atten-

uation. Moreover, since 0 1, the intrinsic impedance of the medium

(3.26) is usually simplified to

' =³00

´12(3.34a)

= 0 (3.34b)

3.2.4 Propagation in good conductors

For a good conductor (see Subsection 1.5) the dissipative current predominates

on the reactive current and according to (1.95), tan = 1. In this

case, from (1.94) and (3.14) we have

= (1− tan )12 ' (− tan )12 =

³

2(1− ) (1− )

´12= (1 + )

³02

´12(3.35)

and consequently from (3.18),

= =³0

2

´12(3.36)

Thus the electric field from (3.20a), simplifies to

= Re~E−(−) (3.37)


where

=1

=

µ2

0

¶12(3.38)

is the penetration factor (3.22) particularized by a good conductor. Thus, for

good conductors, the penetration factor has a very low value which decreases

as the frequency increases. Thus the fields are confined within a very short

distance from the surface of the conductor. For a perfect conductor, → ∞and = 0. Furthermore, the dielectric constant and the complex impedance

are reduced to

=

µ1−

¶' −

(3.39)

and, respectively

=

µ0

¶12=³− 0

´12= (1 + )

³02

´12= (1 + )

0

2

(3.40)

Thus the phase shift between and is 45.

3.2.5 Surface resistance

Let us consider an area element perpendicular to the direction of propagation .

Since the wave amplitudes of and decrease exponentially according to the

factor −, the complex Poynting vector (1.108), and consequently the meanpower per unit of area, (1.107), attenuates along the direction of propagation

by the factor −2. Therefore

P = 1

2Re~E × ~H

∗ = P(0)−2 (3.41)

where P(0) is the mean power per unit area at = 0. Thus the total powerper unit area transmitted by the wave to the medium along the distance =

is given by

= P(0)− P() = P(0)(1− −2) (3.42)

This can be also calculated, according to (1.88), as

=

2

ÃZ

0

(20−2)

!=

20

4(1− −2) (3.43)

This expression for =∞, or for a distance such that the magnitude of thefields becomes negligible, simplifies to

=

20

4=1

2Re−1 20 =

1

2Re2

0 = (3.44)


since

Re−1 =

2(3.45)

For a good conductor, expression (3.44) simplifies, from (3.40), to

=

20

2

³02

´12=1

2

20 (3.46)

where is the so-called surface resistance

=³02

´ 12

=1

(3.47)

and is the penetration factor given by (3.38).

3.3 Group velocity

So far, we have considered the ideal case of a plane harmonic wave, i.e. one

in which the wave number and the frequency are fixed. When this type of

wave propagates through a dispersive medium, the propagation velocity (phase

velocity) of a harmonic wave depends on its frequency. In practice, the ideal

situation of a pure harmonic wave which extends to infinity both backward

and forward in time never arises and, moreover, such a wave could not carry

information. What in fact happens is that a transmitter emits a given signal

( ) for a finite period of time that, according to Fourier’s theorem, can be

expanded into a continuous spectrum of amplitudes such that

( ) =

Z ∞−∞

(−) (3.48)

When the signal propagates through a dispersive medium, i.e. a medium

where the phase velocity depends on the frequency, each spectral component

travels at a different velocity and, as a consequence, the signal will deform as

it propagates. When, as commonly occurs in practice, the spectrum of the

signal is narrow and the transmission medium is only slightly dispersive, then a

single velocity, termed the group velocity, may be assigned to the signal which

is usually known as a wave group or wave package. The velocity with which

the envelope or energy of the wave group propagates in the medium is called

group velocity. To calculate this, let us consider a wave group centered on a

frequency 0 such that ' 0 except for = 0 ±42 (Fig. 7.4). Under

these conditions, Eq. (3.48) simplifies to

( ) =

Z4

A(−) (3.49)

extended to the values of in which A 6= 0. Given that = (), it can be

developed into a Taylor series around the frequency 0

() = (0) +

¯0

( − 0) +2

2

¯0

( − 0)2

2(3.50)

3.4. POLARIZATION 91

If the dispersive medium is such that the dependence of the phase velocity on the frequency is so slowly that we can consider (as a good approximation)

that there exists a linear relation between and , then (3.50) simplifies to

() = 0 +

¯0

( − 0) (3.51)

where 0 = (0).

By substituting (3.51) in (3.49) we get

( ) =

|00−0

Z4

−

|0 (3.52)

which, taking into account (3.49), can be written as a function of (0 ) in the

following way

( ) =

Ã0 −

¯0

!

|00−0

(3.53)

This means that, at a point , the signal has the same amplitude as at the origin

after a time = |0 and a phase shift given by |0 0 − 0.

Consequently, the velocity at which the signal, and thus its associated energy,

propagates is

=

=

¯0

=

()

¯0

= +

¯0

= −

¯0

=1

¯0

(3.54)

If the phase velocity varies slowly with the frequency, then a pulse may

travel through a dispersive medium a certain distance without a significant

change. If this condition is not satisfied and the medium is very dispersive

the shape of signal changes rapidly and the concept of group velocity is not

longer valid. The sign of determines whether is greater or less than

. If the phase velocity increases with the frequency, it is termed normal

dispersion. On the contrary, when decreases with the frequency, it is termed

anomalous dispersion. In an ideal dielectric where 6= (), so that all the

wavelengths propagate at the same velocity = , the signal propagates

without deformation.

3.4 Polarization

As the wave equation is a linear differential equation, it fulfils the superposition

principle and any sum of solutions is also a solution of the differential equation.


In particular, let us consider the sum of two plane waves propagating in direction

(one with the electric field lying along the axis and the other along the

axis) at identical frequencies but, in general, with different amplitudes (

and ) and phases (1 and 2), respectively. Each of these waves, because the

direction of their electric field does not change with time, is said to be linearly

polarized, one in the direction and the other in the direction. However,

in an electromagnetic wave the direction of the electric field generally changes

and traces out an ellipse as the wave propagates5. To see this, let us consider

the total time-varying electric field, which is sum of the two linearly polarized

waves, given by

~E( ) = (1 + 2 )(−) (3.55)

Let us determine the time evolution in a plane = of the electric field

vector resulting from the composition of these two plane waves. We will assume

a homogeneous, isotropic, lossless medium (although the effects of losses as an

exponential factor common to all the field components do not influence the

polarization).

At the plane = 0, for example, we have

= cos(+ 1) (3.56a)

= cos(+ 2) (3.56b)

= 0 (3.56c)

Using the trigonometric identity for the sum of two angles, solving for cos

and sin in terms of and , defining = 1 − 2 as the relative phase

difference between the two components and after some simplifications based on

simply trigonometric identities, we find

22+

2

2− 2

cos = sin2 (3.57)

which is the equation of an ellipse with its major axis tilted depending on the

value of . This means that at a plane = , as the time goes on, the electric

field delineates an ellipse or, equivalently, that the electric field delineates an

elliptical helix in the direction of propagation. The resulting polarization is

referred to as elliptical polarization.

The sense of rotation together with the direction of propagation define left-

handed polarized versus right-handed polarized waves, according to the right-

hand rule: the thumb of the right hand is pointed in the direction of propagation.

Thus, if the fingertips are curling in the direction of the rotation of the electric

field, the wave is right-handed polarized, and in the contrary case the wave is

left polarized.

Particular cases occur depending on the values of , and the polarization

ellipse may degenerate into a centred ellipse, a circle or a straight line.

5 In a unpolarized wave, the vector is subject to random changes of amplitud and phase

3.4. POLARIZATION 93

Figure 3.5: Linearly, circularly and elliptically right hand polarized electromagnetic

waves.

When 6= and = 2, with = ±1±3±5 the polarization ellipse(3.57) becomes a centred ellipse with the major and minor axis oriented along

the directions, i.e.

22+

2

2= 1 (3.58)

If = , then

2 +2 = 2 (3.59)

which is the equation of a circumference.

When = ±, with being an integer, the equation (3.57) becomes∙

±

¸2= 0 (3.60)

which represents the equation of a straight line

= ∓

(3.61)

intersecting the origin. The wave is then linearly polarized and the components

of are

= cos(− ) (3.62a)

= cos(− ±) (3.62b)

The angle of the slope with the axis is

tan = tan

= (−1)

(3.63)

Chapter 4

Reflection and refraction of

plane waves

In the previous chapter, we studied the characteristics of harmonic plane waves,

and now consider what happens when such waves reach the interface (assumed

to be plane and indefinite) separating two linear, nonmagnetic, homogeneous

and isotropic dielectrics having different electromagnetic characteristics. The

change in the constitutive parameters, as the wave passes from one medium to

the other, is assumed to take place in an electrically very narrow region with

a thickness much less than . In general, when a wave propagating through

a medium strikes the interface (incident wave), part of its energy is reflected

and propagates through the same medium (reflected wave), while another part

is transmitted to the second medium (transmitted, or refracted wave). The

characteristics of reflected and transmitted waves can be calculated from those of

the incident wave by forcing the total field on the interface to fulfil the boundary

conditions. We will consider first the simplest case of normal incidence, i.e.

when the interface is perpendicular to the propagation direction of the wave

and then the more general case of oblique incidence. This study has extensive

applications in optics where the interface of many optical devices, such as lenses

and fiber-optic transmission lines, has a radius of curvature much larger than

the wavelength of the incident wave. Thus the interface can be considered quite

accurately as a plane interface. In the following, with no loss of generality, we

will assume the interface to be parallel to the plane.

4.1 Normal incidence.

4.1.1 General case: interface between two lossy media

Considering two semi-indefinite lossy media that are separated by the plane

= 0, see figure (4.1), let us assume that a harmonic plane wave propagates

through the first medium in the positive sense of the axis with the electric field

95

96 CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES

Figure 4.1: Normal incidence: Fields distribution of the incident, reflected and

transmitted waves

parallel to the axis. The wave impinges with normal incidence on this plane.

Due to the discontinuity of the constitutive parameters, = 0 = 0 − 00 ,and where subindex ( = 1 2) refers to medium 1 or 2, part of the wave

is propagated through medium 2 and part is reflected back through medium

1. Therefore, the total field in medium 1 (where 0) and medium 2 (where

0) is given by, see figure (4.2),

Medium 1

E1 = E1−1−1 +E

111 = E

1−1 +E

11

(4.1a)

H1 =E1

1−1−1 − E

1

111 =

E1

1−1 − E

1

11

(4.1b)

Medium 2

E2 = E2−2−2 = E

2−2 (4.1c)

H2 =E2

2−2−2 =

E2

2−2 (4.1d)

The superscripts , , and indicate the incident wave (medium 1), the

reflected wave (medium 1) and the transmitted wave (medium 2), respectively.

The minus sign for the reflected wave of the magnetic field is associated with

the fact that the Poynting vector of the reflected wave propagates in the −direction. In these expressions, =

p0 represents the impedance (3.25)

of medium while is the complex propagation factor (3.18),

4.1. NORMAL INCIDENCE. 97

Figure 4.2: Electric fields of the incident, reflected and transmitted waves in the

general case of normal incidence between two semi-indefinite lossy media that

are separated by the plane = 0.

= + (4.2)

where and are the attenuation and propagation constants (3.19a)and

(3.19b), respectively

=p

0√

2

∙q1 + (

0) + 1

¸12(4.3)

=p

0√

2

∙q1 + (

0)− 1

¸12(4.4)

The time dependence of the fields is achieved by adding the factor to (4.1).

For each instant of time, by imposing the boundary conditions in the plane

= 0 (2.236b) onto the tangential components of and ,

1 = 2 (4.5a)

1 = 2 (4.5b)

98 CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES

we obtain

= = = (4.6a)

E1 = Γ E

1 (4.6b)

E2 = E

1 = (1 + Γ)E

1 (4.6c)

H1 = −ΓH

1 (4.6d)

H2 =

12

H1 (4.6e)

where Γ is the reflection coefficient in the plane = 0 defined by

Γ =2 − 12 + 1

= |Γ| Φ (4.7)

and is the transmission coefficient in the same plane, defined by

=22

2 + 1(4.8a)

= 1 + Γ = || Ψ (4.8b)

If there is an impedance adaptation (2 = 1) then there is no reflected

wave, and so all the incident energy is absorbed by the second medium.

From (4.1a) and (4.6b), the total electric field in the first medium can be

expressed as

E1 = E1−1−1(1 + Γ2121) (4.9)

= E1−1−1(1 + Γ()) = E

1−1(1 + Γ()) (4.10)

where Γ(), defined as

Γ() = Γ2121 (4.11)

is the reflection coefficient in the plane = . Similarly, for the magnetic field,

we have

H1 =E1

1−1(1− Γ()) (4.12)

The impedance associated with the total field at a coordinate point in the

first medium is defined as

() =E1

H1

¯

= 11 + Γ()

1− Γ() = 12 − 1 tanh(1)

1 − 2 tanh(1)(4.13)

The impedance () is continuous through the interface, because the tan-

gential components 1 and 1 are similarly continuous, while the reflection

coefficient Γ is discontinuous.

4.1. NORMAL INCIDENCE. 99

4.1.2 Perfect/Lossy dielectric interface

In the particular case in which the first dielectric is perfect, i.e. lossless (1 = 0

and 001 = 0 1 = 01 1 = 1), the characteristic impedances reduce to

1 =

r01

(4.14)

and the coefficient of reflection (4.7) at the interface ( = 0) becomes

Γ =1−

q21

1 +q

21

(4.15)

Since 1 = 0 and 001 = 0 it follows that 1 = 0 and therefore

E1 = E1−1(1 + Γ21) = E

1−1(1 + Γ()) (4.16)

and

H1 =E1

1−1(1− Γ()) (4.17)

where

Γ() = Γ21 (4.18)

The input impedance (4.13) simplifies to

() = 12 − 1 tan(1)

1 − 2 tan(1)(4.19)

4.1.3 Perfect dielectric/Perfect conductor interface

Another particular case arises when the second medium is a perfect conductor

(2 = 0) and therefore = 0 and Γ = −1. Then the fields in the first mediumare

E1 = E1−1 ¡1− 21

¢= E

1

¡−1 − 1

¢= −2E

1 sin (1)(4.20a)

H1 =E1

1

¡−1 + 1

¢= 2

E1

1cos (1) (4.20b)

4.1.4 Standing waves

It is well known that two waves with the same frequency that are propagating in

opposite directions interfere and form what are termed standing (or stationary)

waves. To examine this concept, let us first consider the case in which the

first medium is lossless, and then analyse the case in which the first medium is

dissipative.

100CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES

a) Lossless case

For the first medium, the expression of the total electric field is,

E1 = E1−1 +E

11 = (1 + Γ)E

1−1 + ΓE

1(

1 − −1)

= E1−1 + |Γ|E

1((Φ+1) − (Φ−1))

= ||E1

(Ψ−1) + 2 |Γ|E1 sin(1)

(Φ+2) (4.21)

By including the time dependence, and assuming an initial phase = 0, we

obtain the following expression for the total field

1( ) = ||01 cos(− 1 +Ψ)− 2 |Γ|

01 sin(1) sin(+Φ)

(4.22)

where the first summand of the second member corresponds to a wave that is

propagating, while the second summand represents a standing wave, i.e., one in

which the mean energy transported by the wave is null. The amplitude of the

propagating wave is determined by the coefficient of transmission, while that of

the standing wave depends on the coefficient of reflection. The envelope of the

equation (4.22) is termed the diagram of the standing wave. If the coefficient

of transmission is null (which occurs when the second medium is a perfect

conductor) the wave of the first medium becomes a pure standing wave.

From (4.16) and (4.17) the magnitudes of the fields are

01 = 01 |1 + Γ()| (4.23a)

01 =1

101 |1− Γ()| (4.23b)

The maximum values of 01 (the minima of 01) are given by

01()max = 01 +

01 (4.24)

at the coordinate points

max = −Φ + 221

= 0 1 (4.25)

and the minimum values of 0 (the maxima of 0), assuming 01

01,

are given by

0()min = 01 −

01 (4.26)

at the points

min = −Φ + (2+ 1)21

; = 0 1 (4.27)

4.2. MULTILAYER STRUCTURES 101

Ratio of the standing wave

The relation between the maximum and minimum values of the diagram of the

standing wave is called the ratio of the standing wave, and is described by

=01()max

01()min=

01 +

01

01 −

01

=1 + |Γ()|1− |Γ()| =

1 + |Γ|1− |Γ| (4.28)

Its value ranges from 1 (no reflected wave) to infinity (pure standing wave), i.e.

1 ≤ ≤ ∞ (4.29)

b) Lossy case

In this case, the expression of the total electric field in the first medium is

E1() = E1−1−1 +E

111 = E

1(−1 + Γ1)

= E1−1 + ΓE

1(

1 − −1) (4.30)

and, by including the time dependence, we get

1( ) = ||01

−1 cos(− 1 +Ψ) +

2 |Γ|01 sinh(1) cos(+Φ) (4.31)

In these media, it makes no sense to define the parameter because the

maxima and minima are not constant.

4.1.5 Measures of impedances

Assuming that the first medium is lossless, from (4.27) the first field minimum

occurs at +21min = . Consequently, we can determine the phase angle ,

assuming that 1 is known and that min is determined experimentally (by using

an appropriate device to detect the first field minimum). If 1is not known, it

can be calculated from the distance between two consecutive minima.

The value of |Γ| can be found from the ratio between the maximum and

minimum field values 01max01min = = (1 + |Γ|)(1− |Γ|).Note that if the incident wave has an amplitude of one, then |Γ| is identical

to 0 and thus we need to measure only this amplitude. Thus 2 is determined

from this information and from expression (4.7).

4.2 Multilayer structures

Let us now consider the normal incidence of an electromagnetic wave on a

structure in which there are more than two media separated by parallel planes.

To simplify the analysis, we consider the case of three lossless dielectrics, as

shown in Fig.(4.3). The generalization to more media, including the possibility


Figure 4.3: Structure formed of three perfect dielectrics.

of losses, is straightforward. Clearly, for a wave that is propagating to the right

in medium 2, the problem is analogous to the two-layer cases discussed above.

Therefore, the coefficient of reflection in the = 0 plane is

Γ23 =3 − 23 + 2

= Γ( = 0) (4.32)

where subindex 23 refers to the surface that separates medium 2 from medium

3. Particularizing (4.13) for = − we have the load impedance

= ( = −) = 21 + Γ23

−2

1− Γ23−2 (4.33)

Taking into account that () is continuous at an interface, the coefficient

of reflection (4.7) at = −, becomes

Γ = Γ( = −) = − 1 + 1

by introducing (4.33) into this equation and then operating, we get

Γ =Γ12 + Γ23

−2

1 + Γ12Γ23−2(4.34)

where

Γ = −

+ (4.35)

Thus, for an electromagnetic wave with an amplitude of one, impinging

normally from the first medium onto the structure, the amplitude of the reflected

wave is given by (4.34)

4.2. MULTILAYER STRUCTURES 103

Quarter-wave layer

For a quarter-wave layer, = 4 (−2 = −1), equation (4.34) becomes

Γ =Γ12 − Γ231− Γ12Γ23 (4.36)

Thus, to transmit the incident energy completely (adaptation of impedance),

the coefficient of reflection must be null, and so

Γ12 − Γ23 = 0 (4.37)

Taking into account equation (4.35) we have

2 =√13 (4.38)

as a condition for impedance adaptation to exist.

Half-wave layer

For a half-wave layer, i.e. = 2 (−2 = 1), expression (4.34) is reduced to

Γ =Γ12 + Γ23

1 + Γ12Γ23(4.39)

For impedance adaptation to exist, the following must be fulfilled

Γ12 + Γ23 = 0 (4.40)

By replacing the coefficients by the values given in (4.35), we have

(3 = 1) (4.41)

and thus Γ = 0 irrespectively of 2. Thus any material with a thickness of 2

is adapted so long as the impedances of media 1 and 3 are the same.

4.2.1 Stationary and transitory regimes

The above analyses are valid for monochromatic waves in a stationary regime. It

should be noted that such a regime is the limit of a transitory process involving

multiple reflected and transmitted waves within media 1 and 2. To illustrate

this limit process, let us consider the normal incidence of a wave that impinges

upon a structure formed of three perfect dielectrics, as shown in Fig. (4.3). From

the process of multiple reflections and transmissions, we find that in medium 1

a reflected field is given by

E1 =

01(Γ12 + 12Γ2321−22 + 12Γ

223Γ2121

−42

+12Γ323Γ

22121

−62 + ) (4.42)


Observing the second member, we can see that the summands following

the first one constitute a geometric progression of common ratio Γ21Γ23−22.

Thus the coefficient of reflection can be written as

Γ = Γ12 +12Γ2321

−22

1− Γ23Γ21−22 (4.43)

which, taking into account the equalities

Γ21 = −Γ12 (4.44)

12 = 1 + Γ12 (4.45)

21 = 1− Γ12 (4.46)

is reduced to equation (4.34).

4.3 Oblique incidence

As a more general case than the normal incidence, let us now consider the oblique

incidence of a plane wave on a plane interface separating two media. In medium

1 there is generally an incident and a reflected wave, while the transmitted (also

called refracted) wave is in medium 2. To study the oblique incidence, we will

use the geometry shown in Fig.(4.4), where the waves have been represented,

as usual, by arrows (called rays1) in the direction of propagation. These rays

are perpendicular to the equiphase planes (wavefronts). The oblique incident

has extensive applications in optics where the interface of many optical devices,

such as lenses and fiber optic waveguides, has a radius of curvature much larger

than the wavelength of the incident wave. Thus the interface can be considered

very approximately as a plane interface.

In principle, we make no assumption that the three rays are coplanar, al-

though they are shown as such in Fig.(4.4). The plane of incidence is defined

by vector = and by the axis. Let us assume that is in the plane

= 0 and forms an angle with the axis. In the general case of two lossy

media, the electric fields of the incident, reflected, and refracted waves can be

written, respectively, as

= Re0(−·) (4.47a)

= Re0(−·) (4.47b)

= Re0(−·) (4.47c)

In = 0, the tangential component of the electric field must be continuous,

and thus we have +

=

(4.48)

1The technique of ray optics and its generalizations provide approximate solutions to

Maxwell’s equations that are valid when the wavelength is small as compared to significant

dimensions of the objects involved. Ray theory does not describe phenomena such as in-

terference and diffraction, which require wave theory since are related to the phase of the

wave.

4.3. OBLIQUE INCIDENCE 105

Figure 4.4: Oblique incidence.

so that

Re 0

(−·)+Re 0

(−·) = Re 0

(−·) (4.49)

A similar relation must be fulfilled between the components of the fields with

respect to the axis. These conditions can be satisfied only if

= = = (4.50)

and

· =

· = · (4.51)

Since lies on the = 0 plane, from (4.51) it follows that = = 0,

signifying that the reflected and refracted waves are coplanar with the incident

wave. Thus we have

· =

1

[ sin + cos ] (4.52a)

· =

1

[ sin + cos ] (4.52b)

· =

2

[ sin + cos ] (4.52c)

where is the complex index of refraction of the medium2, such that

=

(4.53)

2Usually is assumed that the refractive index of a material is positive due to the absence

of naturally occurring materials with negative index . However presently it is possible to

develop artificial materials (called metamaterials, see Section (1.6)) with a structure such that

it exhibits properties not usually found in natural materials, especially a negative refractive


By substituting (4.52) in (4.49), and by making the coefficients of equal,

we get

= = (4.54a)

1 sin = 2 sin (4.54b)

These equations, together with the coplanarity of the rays, constitute Snell’s

laws.

For lossless media Eq. (4.54b) simplifies to

1 sin = 2 sin (4.55)

where3

=

= ()

12 (4.56)

or, for nonmagnetic media,

= ()12 (4.57)

Next, we study the relations between the amplitudes of the incident, trans-

mitted and reflected waves by making use of the boundary conditions at the

interface between the two media. For this we will assume lossless media al-

though the generalization to lossy media is straightforward4. Let us analyze

the problem in two stages: first, where the electric field oscillates in the

incidence plane, and, secondly, where it oscillates perpendicularly to the same

plane. Any other case can be considered a superposition of these two situations.

4.3.1 Incident wave with the electric field contained in the

plane of incidence

From the continuity of the tangential components of and (Eqs. (4.5a) and

(4.5b), we obtain (Fig. (4.5)),

Ek cos +E

k cos −E

k cos = 0 (4.58a)

1

1(E

k −Ek)−

1

2Ek = 0 (4.58b)

index. Metamaterials are promising because enable the extreme miniaturization of existing

optical devices. and to design entirely new devices that can be used in communication, optics,

radar and imaging applications.

Snell’s law is still satisfied when one of the materials has a negative refractive index, but

the direction of the light ray is ‘mirror-imaged’ about the normal to the surface.

3The index of refraction is a measure of how much slower thye wave travels in a dielectric

medium that in free space.

4 If the medium is lossy, we must replace by ( )


Figure 4.5: Incident wave with the electric field contained in the plane of inci-

dence

where the subindex k indicates that the physical magnitude in question lies inthe incidence plane and

~E

k = 0k−· (4.59a)

~E

k = 0k−· (4.59b)

From (4.58), we find that

Γk =Ek

Ek=

2 cos − 1 cos

2 cos + 1 cos (4.60a)

k =Ek

Ek=

22 cos

2 cos + 1 cos (4.60b)

Where Γk and k are the coefficients of reflection and transmission, respectively.If medium 2 is a perfect conductor from (3.40) we have 2 = 0 and then Γk = −1.For lossless non-magnetic materials (1 = 2 = 0) such that

12 =21=

2

1=

1

2

=sin

sin (4.61)

with 12 being the ratio of the indices of refraction of medium 1 and medium

2, expressions (4.60a) and (4.60b) are reduced to


Ek

Ek

=tan ( − )

tan ( + )(4.62a)

Ek

Ek

=2 sin cos

sin ( + ) cos ( − )(4.62b)

The total electric field ~E1

k in medium 1 is given by

~E1

k = ~E

k + ~E

k (4.63)

where

~E

k = Ek cos −E

k sin (4.64a)

~E

k = Ek cos +E

k sin (4.64b)

Thus we have

~E1

k = cos ³0k−· +

0k−·

´+ sin

³0k−· −

0k−·

´

(4.65)

That is,

E1k =

0k cos ³−

· + Γk−·

´(4.66a)

E1k =

0k sin ³Γk−· − −

·´

(4.66b)

Taking into account that

· = − cos + sin (4.67a)

· = cos + sin (4.67b)

and by substituting these equations in (4.66), we find that

E1k =

0k−( sin + cos )

³1 + Γk

2 cos ´cos (4.68a)

E1k = −

0k−( sin + cos )

³1− Γk2

cos ´sin (4.68b)

When the time factor is introduced, the term (− sin ) represents

a wave that is propagating in the direction of the axis, while the term³−

cos + Γk cos

´ (4.69)


From (4.68) we have the superposition of two waves that are propagating with

respect to the axis, but in opposite directions. In other words, a stationary

wave overlies a traveling one such that the energy that is transported in direction

, from medium 1 to medium 2, is transported by the traveling wave. For the

case of a perfect conductor, Γk = −1 and there exits only a standing wave alongthe axis.

The total magnetic field ~H1

⊥ in medium 1 is

~H1

⊥ = 0⊥−· +

0⊥−· =

1

1

³0k−· −

0k−·

´ =

0⊥−( sin + cos )

³1− Γk2

cos ´

(4.70)

where the symbol ⊥ indicates that the magnitude in question is perpendicularto the incidence plane

In the case of a perfect conductor, it is straightforward to show that there

is no energy flow towards , but there is towards , as the mean time value of

the Poynting vector towards is zero.

4.3.2 Wave incident with the electric field perpendicular

to the plane of incidence

As above, the continuity equations are used for the tangential components of

and and thus (Fig.(4.6)):

~E

⊥ + ~E

⊥ = ~E

⊥ (4.71a)

~E

⊥ − ~E

⊥1

cos =~E

⊥2

cos (4.71b)

where the subindex ⊥ indicates that the physical magnitude in question corre-sponds to the case in which the electric field of the incident wave is perpendicular

to the plane of incidence.

By solving the two Eqs. (4.71), we get

Γ⊥ =E⊥

E⊥=

2 cos − 1 cos

2 cos + 1 cos (4.72a)

⊥ =E⊥

E⊥=

22 cos

2 cos + 1 cos (4.72b)

where the parameters Γ⊥ = E⊥E

⊥and ⊥ = E

⊥E⊥ are the coefficients of

reflection and transmission, respectively.

For lossless non-magnetic materials, Eqs. (4.72) are transformed into


Figure 4.6: Wave incident with the electric field perpendicular to the plane of

incidence

E⊥

E⊥

=sin ( − )

sin ( + )(4.73a)

E⊥

E⊥

=2 sin cos

sin ( + )(4.73b)

By operating in a similar way to that described for the previous case of k

we arrive at the following for the total electric and magnetic fields in a lossy

medium 1

~E1

⊥ = 0⊥− sin

³−

cos + Γ⊥ cos

´ (4.74a)

~H1

k =0⊥1

cos ³−

· − Γ⊥−·´

−0⊥1

sin ³−

· + Γ⊥−·

´ (4.74b)

As in the case of ~Ek, the field is formed by a travelling wave towards andas a travelling wave, superposed over a standing one, towards . The formu-

las (4.62) and (4.73) are known as Fresnel’s formulas, which give the relations

between the amplitudes and phase of the incident, reflected, and tranmitted

waves.

Chapter 5

Electromagnetic

wave-guiding structures:

Waveguides and

transmission lines

5.1 Introduction

There are many engineering applications in which it is necessary to use devices

to confine the propagation of the electromagnetic waves in order to transmit

electromagnetic energy from one point to another with a minimum of interfer-

ence, radiation, and heat losses. Although such transmission systems can take

many different forms, a common characteristic is that they are uniform. That

is, their cross-sectional geometry and constitutive parameters do not change in

the direction of the wave propagation for wavelengths numerous enough to

make border effects negligible. In general, any device used to transmit con-

fined electromagnetic waves can be considered a waveguide; however, when the

transmission device contains two or more separate conductors the term "trans-

mission line" is generally used instead of "waveguide". Figure (5.1) shows the

cross-sectional shape of some guiding transmission systems: two-wire trans-

mission lines; coaxial transmission lines formed by two concentric conductors

separated by a dielectric; two hollow (or dielectric-filled) metal tubes of rectan-

gular and circular cross section (i.e. a rectangular and a circular waveguide);

two planar transmission lines (the stripline and microstrip); and two dielectric

(without conducting parts) waveguides: the circular dielectric waveguide (or

homogeneous dielectric rod) and the optical fiber.

In hollow conducting pipes waves propagate within the tube, whereas in

transmission lines formed by two or more conductors, the waves propagate

in the dielectric medium between the conductors. In homogeneous dielectric

111

112CHAPTER 5. ELECTROMAGNETICWAVE-GUIDING STRUCTURES:WAVEGUIDES AND

waveguides the field decays exponentially away from the dielectric in the trans-

verse plane, and consequently the electromagnetic waves are confined mainly

within the dielectric medium. Optical fibers, used mostly at optical wavelengths,

consist of a cylindrical core surrounded by a cladding and are usually circular

in cross section. The light is essentially confined to the core (which has a larger

refractive index than the cladding) by total internal reflection as it propagates

along the fiber and the wave is confined without need of any conducting walls.

The choice of a specific transmission system depends on the application

and should take into account aspects such as frequency range, losses, power-

transmission capacity, and production costs. For example, the two-wire trans-

mission lines, which are usually covered by polyethylene, are relatively inexpen-

sive to manufacture, but radiation losses (mainly at discontinuities and bends)

make them inefficient for transferring electromagnetic energy farther than the

lower range of microwaves. Coaxial lines and hollow metal pipe waveguides are

more efficient than two-wire lines for transferring electromagnetic energy be-

cause the fields are completely confined by the conductors. For the transmission

of large amounts of power at high frequencies, waveguides are the most appro-

priate means. In a coaxial cable, significant wave attenuation occurs at high

frequencies because of the large current densities carried by the central conduc-

tor, which has a relatively small surface area. On the other hand, waveguides are

intrinsically dispersive and consequently incapable of transmitting large band-

width signals without distortion. However, coaxial lines can guide signals of

much higher bandwidths than waveguides can.

As shown in the next chapter, the dimension of the cross section of a

waveguide is related to the wavelength of the guided wave. Thus, for very low-

frequency waveguides the cross section would be too large and thus impractical

for frequencies lower than 1 GHz. On the other hand, at optical frequencies

the size of a metal waveguide must be too small (in the range of the ) and,

moreover, at these frequencies the study of the interaction of the electromagnetic

field with the metal walls requires of quantum mechanical theory.

As a result of the development in solid-state microwave and millimeter tech-

nology, planar transmission lines are used instead of waveguides in many ap-

plications because these lines are inexpensive, compact, and simple to match

solid-state devices using printed-circuit technology. Planar lines allow different

configurations, usually including a dielectric substrate material with a ground

plane and one or more conducting strips on the upper surface. The most com-

monly used of these are striplines and microstrips.

The field configurations that can be supported for any guiding structure

must satisfy Maxwell’s equations and the corresponding boundary conditions.

The different field distributions that satisfy this requirement are termed modes.

Although the electromagnetic field distribution in ideal guiding transmission

systems (composed of perfect conductors separated by a lossless dielectric).can

be expressed as a superposition of plane waves, the study of the propagation is

greatly simplified when we seek other kinds of solutions called transverse mag-

netic (TM) modes, transverse electric (TE) modes, or transverse electromagnetic

(TEM) modes. These terms indicate that, in the direction of propagation, the

5.2. GENERAL RELATIONS BETWEEN FIELD COMPONENTS 113

Conductor

Dielectric

Dielectric 2

Dielectric 1

Conductors

Dielectric

Conductor

Dielectric

(1)

Conductors

(2) (3) (4)

(7) (8)

Dielectric

(6)

ConductorsDielectric

(5)

DielectricConductors

Conductor

Dielectric

Conductor

Dielectric

Dielectric 2

Dielectric 1

Dielectric 2

Dielectric 1

ConductorsConductors

Dielectric

Conductor

Dielectric

Conductor

Dielectric

(1)

Conductors

(2) (3)

Conductors

(2) (3) (4)

(7) (8)

Dielectric

(6)


(6)


(5)


(5)


Figure 5.1: Examples of waveguides and transmission systems: (1) Two wire

transmission line (2) Coaxial transmission line (3) Rectangular waveguide (4)

Circular waveguide (5) Stripline (6) Microstrip (7) Circular dielectric waveguide

(8) Optical fiver cable.

TM modes have no magnetic field component, the TE modes have no electric

field component, and the TEM modes have neither electric nor magnetic field

components. In practice, these modes form a complete set of orthogonal func-

tions and, hence, any propagating electromagnetic field in the guiding structure

can be expressed as a linear combination of these modes. As discussed below,

there are two important properties that distinguish TEM from TE and TM

modes:

1) TM and TE modes have a cutoff frequency below which they cannot

propagate, which depends on the cross-sectional dimension of the guiding struc-

ture.

2) TEM modes cannot exist within a waveguide formed a single perfect

conducting pipe while transmission lines can in general support TE, TM and

TEM modes.

In this chapter, we present some general aspects of the propagation of time-

harmonic electromagnetic waves in guiding systems formed by perfect con-

ductors and only one homogeneous lossless dielectric in which the guided field

propagates. Nevertheless, the results can serve as a basis for structures in which

the cross-section contains more than one dielectric medium. The effect of lossy

media is analyzed in the final section.

5.2 General relations between field components

Let us assume that a time-harmonic wave propagates along the -axis, in the

+ direction, in a lossless guiding transmission system. Thus the dependence

on and time is given by the factor (−)and the fields are of the general


form

Re

(0

(−)

0(−)

)= Re

½~E

~H

¾(5.75)

where ~E = 0−and ~H = 0

− and is the wavenumber of the

guided wave. Because the geometry and constitutive parameters do not change

along the -axis, 0 and 0 are functions only of the transverse coordinates.

To determine ~E and ~H, we will first show that it is possible to express

their transverse components, ~E and ~H, in terms of their -components, ~E

and ~H. For this, we divide the three dimensional Laplacian operator ∇2 inthe homogeneous Helmholtz wave equations (3.27) into two parts1. One part,

22, acts only on the axial coordinate, , and the other, ∇2 , on the transverseones only2, i.e.

∇2 = 2

2+∇2 (5.76)

Since ≡ −, the wave equations can be written as

¡∇2 + 2¢½ ~E

~H

¾=¡∇2 + 2

¢½ ~E~H

¾= 0 (5.77)

where

2 = 2 − 2 (5.78)

and = ()12 is the wavenumber for the wave propagating in an unbounded

medium of the matter which fills the transmission system. By particularizing

(5.77) for the field component, we have

¡∇2 + 2¢½E

H

¾= 0 (5.79)

This equation, when solved together with the boundary conditions of a given

structure, has solutions for an infinite but discrete number, , of characteristic

values (eigenvalues) , i.e.¡∇2 + 2¢½E

H

¾= 0 (5.80)

where

2 = 2 − 2 (5.81)

with E or H being the corresponding functions characteristic (eigen-

functions) which satisfy the equations (5.80) and the corresponding boundary

conditions, which are determined by the geometry of the system.

1The partial homogeneous Helmholtz wave equations is usually solved by separation of

variables method, in cartesian, spherical or cylindrical coordinates (see for example Section

(6.2))2For example in Cartesian coordinates we have ∇ = ∇ +

where ∇ =

+

so

that ∇2 = 2

2+ 2

2.


Now we are going to demonstrate that, once equation (5.79) has been solved,

we can obtain ~E or ~H from E and H. From Maxwell’s equations (1.68c)

and (1.68d), in a sourceless region, we have

∇× ~E = − ~H (5.82a)

∇× ~H = ~E (5.82b)

The transverse components of these equations can be written as

(∇× ~E) = ∇ × ~E +∇ × ~E = − ~H (5.83a)

(∇× ~H) = ∇ × ~H +∇ × ~H = ~E (5.83b)

Thus, as ~E and ~H are assumed to be known, we have a system of two equa-

tions and two unknowns, ~E and ~H, the solutions to which are

~H =2

³∇ × ~E − ∇H

´(5.84a)

~E = − 2

³∇ × ~H + ∇E

´(5.84b)

According to (5.84), once the components of the fields are known, the trans-

verse components can also be calculated. Moreover, in ideal guiding structures,

we can express any field propagating in the homogeneous guiding transmis-

sion structure as a linear superposition of TE, TM and TEM waves or modes.

Clearly, it is not possible to find specific expressions for the field distribution of

any of these modes without previously knowing the geometry and characteris-

tics of the transmission system. However, as shown below, we can study some

of their general characteristics.

5.2.1 Transverse magnetic (TM) modes

Let us first consider TM modes so that = 0 in (5.84). Thus we have

~E = − 2∇E = ∇

12

= ∇Φ (5.85a)

~H =2∇ × ~E = −

2 ×∇E =

× ~E =

1

× ~E

(5.85b)

where we have defined the scalar potential for the TM modes, Φ , as

Φ = 12

(5.86)

To obtain (5.85b), we have used the equality ∇× ~E = −×∇E and defined

the frequency-dependent quantity , as

==

(5.87)


where = ()12 is the intrinsic impedance of the dielectric that fills the trans-

mission system. The quantity , which has the dimensions of impedance, is

called the wave impedance for the TM modes. From (5.85b), we can see that~E, ~H, and form a right-handed system when the wave propagates in the

-positive direction.

Thus, from (5.85a), in TM modes, ~E can be written as the gradient of a

scalar function Φ . This result could have been obtained by simple reasoning

from Faraday’s law (5.82a), taking into account that, since ~H has only trans-

verse components the same is true for ∇× ~E. Therefore, from Stokes’ theorem,

we have Z

(∇× ~E) · =IΓ

~E · = 0 (5.88)

where is a transverse surface normal to the axis. But cannot contribute

to the line integral because the integration path Γ lies on the transverse plane.

Therefore IΓ

~E · =IΓ

~E · = 0 (5.89)

which implies that is conservative and, therefore, can be written as the

gradient of a scalar function Φ .

5.2.2 Transverse electric (TE) modes

For TE modes, from equations (5.84), with E = 0, we have

~H = − 2∇H = ∇

12

= ∇Φ (5.90a)

~E = − 2∇ × ~H =

2

×∇H = − × ~H = − × ~H

(5.90b)

where

Φ =12

(5.91)

is the scalar potential for TE waves and

==

(5.92)

is the wave impedance for the TE mode. From (5.90b) we can see that ~E, ~H,

and form a right-handed system when the wave propagates in the -positive

direction.

The fact that, according to (5.90a), ~H can be expressed as the gradient of

the scalar function Φ can be explained by Ampere’s law, (5.82b), following a

reasoning similar to that used in the case of TE modes.


5.2.3 Transverse electromagnetic (TEM) modes

For TEM modes, since E = 0 and H = 0, substituting these values in (5.84),

we can get no null or trivial solutions only if = 0. Consequently, from (5.78),

for TEM modes, we have

2 = 2 = 2 (5.93)

This means that a TEM mode in a transmission system has the same propa-

gation constant as a uniform plane wave traveling in the unbounded dielectric

between the conductors. Since = 0 and ~E = ~E and ~H = ~H, (5.77) reduces

to

∇2 ~E = ∇2 ~E = 0 (5.94a)

∇2 ~H = ∇2 ~H = 0 (5.94b)

Thus the distribution of the electric and magnetic fields on a transverse

plane satisfies the same bidimensional Laplace’s equation as for the static fields.

This means that, for TEM modes, on a transverse plane, is conservative, and

derivable from a scalar function Φ by means of the gradient function, i.e.

~E = −∇Φ (5.95)

Hence, the electric field distribution in the cross-sectional plane has the same

spatial dependence as the electrostatic field created by static charges located on

the conductors of the transmission system. Consequently, a TEM mode cannot

exist within a waveguide formed by a single perfect conducting tube of any

cross section since no electrostatic field can exist within a sourcesless region

completely enclosed by a conductor. When two or more separated conductors

exist, as for example in coaxial, two-wire or stripline transmission lines, TEM

waves can be propagated along the dielectric separating the conductors.

It is straightforward from (5.83b) that

~E = − × ~H = − × ~H (5.96)

where

= =³

´ 12

(5.97)

is the wave impedance for the TEM mode, which coincides with the character-

istic impedance of the dielectric that fills the transmission system.

Now we will demonstrate that, for TEM modes, Maxwell’s equations can be

used to derive a pair of coupled differential equations which enable us to study

the propagation of these modes in transmission lines as voltage and current

waves (instead of electromagnetic waves), using elemental circuit theory.

From (5.95), according to the fundamental property of the gradient, in

the transverse plane the line integral of the electric field is path independent

and consequently voltage and potential difference Φ2 − Φ1 will be the same.


l

I x

y

1C

2C

I

ll

I x

y

1C

2C

I

I x

y

1C

2C

I

Figure 5.2: Two conductor transmission line.

Then for TEM waves (using, without loss of generality, Cartesian coordinates)

we have

= Φ(2)−Φ(1) = −Z

= −Z

+ (5.98)

whereΦ(2) and Φ(1) are the values of the scalar function Φ at the the conductors

1 and 2 and where is any line that joins the equipotential transverse sections of

these conductors (Fig.5.2). Deriving with respect to and taking into account

Faraday’s law, (1.1c), particularized for the source-free region, outside the

conductors, we get

= −

Z

+

= −

Z

−+ (5.99)

Note thatR− + is the magnetic flux through the area swept,

along a unit of length in the direction , by the line joining the conducting

surfaces. This flux can be expressed by using the magnetostatic definition of

coefficient of self-inductance per unit of length, as the product . Therefore

we have

= −

(5.100)

On the other hand, from Ampere’s law, (1.1d), for the source-free dielectric

region, we have

=

IΓ

=

IΓ

+ (5.101)

where Γ is a closed path around one of the wires (see Fig. 5.2). Deriving with

respect to , we have

=

I

+

= −

IΓ

− (5.102)


Figure 5.3: Lumped-element equivalent circuit of a lossless line composed of

two perfect conductors separated by a lossless dielectric. A finite length of

transmission line can be viewed as a cascade of these elements.

where, in a similar way as above, − + represents the flow of vector per unit length in the direction . Using the magnetostatic definition of

capacitance per unit length, this flux can be expressed as the product .

Thus we have

= −

(5.103)

Note that, from (5.98) and (5.101), and must have the same dependence

as and , respectively. Thus, and are also traveling waves.

In summary, according (5.100) and (5.103) we have

= −

(5.104a)

= −

(5.104b)

which are the coupled differential equations that voltage and current satisfy at

any cross section of an ideal line composed of perfect conductors separated by a

lossless dielectric (see Fig. 5.3 ). Equations (5.104) are called ideal "transmission

line equations".

5.2.4 Boundary conditions for TE and TM modes on per-

fectly conducting walls

For a guiding transmission system with perfectly conducting walls, the general

boundary conditions on the walls require that the tangential component ~E of


~E and the normal component H of ~H be null, i.e.

~E = × ~E = 0 (5.105a)

H = · ~H = 0 (5.105b)

where is the unit vector normal to the conducting walls. However, for TM

and TE modes, as shown below, these conditions can be simplified and reduced

to equivalent ones which are expressed only in terms of the component of the

fields. For example for TM modes, the requirement that

E = 0 (5.106)

on the perfectly conducting guide walls suffices to ensure that Eqs. (5.105) are

fulfilled. From (5.85a) and the gradient properties, we can see that ~E is normal

to the lines where E = and, therefore, to the boundary of the conductor,

since it represents a line with E = 0. Given that ~E and ~H are perpendicular

to each other, the magnetic field is tangential to the conductor and thus E = 0

is equivalent to Eqs. (5.105).

For TE modes, the necessary and sufficient condition to ensure that Eqs.

(5.105) are fulfilled is that the normal derivative of be null on the perfect

conducting parts of the guiding structure. That is

H

= ∇H · = (∇ +∇)H · = 0 (5.107)

where we have divided ∇ into its transverse and axial components. Taking intoaccount (5.90a), we see that

∇H · = ~H · = 0 (5.108a)

which means that ~H is tangencial to the conductor and therefore, due to the

perpendicularity of the fields, we have

× ~E = 0 (5.109)

In summary, the necessary and sufficient boundary conditions on the perfect

conducting walls of the propagation system are

Boundary conditions on the perfect conducting walls

For TM modes

E = 0(5.110a)

For TE modes

= 0

(5.110b)

5.3. CUTOFF FREQUENCY 121

5.3 Cutoff frequency

From (5.75) and (5.78) we see that, for propagation to exist, must be real,

and consequently,

2 2 (5.111)

For this reason, , defined as

= =2

(5.112)

is called the cutoff wavenumber, and is the cutoff wavelength. Thus, from

Eq. (5.78), we have

2 = 2 − 2 (5.113)

and, consequently1

2=1

2− 1

2(5.114)

where is the wavelength of a plane wave in the unbounded lossless dielectric

medium filling the waveguide, and is that of the wave in the guide. Thus

we have

=2

; =

2

; =

2

(5.115)

The cutoff frequency is defined3 as

=

2=

2√=

2

(5.116)

where = is the phase velocity in the unbounded medium filling the

waveguide. Thus, from (5.78), the wavenumber can be expressed in terms of

, as

=

s1−

µ

¶2(5.117)

and the corresponding wavelength in the guide is

=2

=

r1−

³

´2 (5.118)

which is greater than . According to (5.117) the wavenumber is imaginary for

modes with frequencies below the cutoff frequency , i.e. (or ).

These modes, called evanescent modes, are attenuated and cannot propagate

along the guide. Thus, waveguides behave as high-pass filters for the TE and TM

modes since they cannot transmit any of these modes for which the wavelengths,

3For a guiding transmission system with more than one dielectric the cutoff frequency can

be defined in a different manner than (5.116).


in the unbounded medium filling the waveguide, exceed the value of the cutoff

wavelength.

In terms of the cutoff frequency, the expressions of the wave impedances for

the TM and TE modes (5.87) and (5.92) for and become, respectively

=

s1−

µ

¶2(5.119a)

=r

1−³

´2 (5.119b)

From (5.119a) and (5.119b), we can see that and and they

become imaginary below the cutoff frequency. Thus, for , the waveguide

behaves, in this respect, as a reactive impedance.

From (5.78), we obtain the dispersion relation (i.e. the relation between

phase constant and the frequency )

=¡2 + 2

2

¢12(5.120)

The plot of the phase constant as a function of the frequency (dispersion

diagram) is shown in Fig. (5.4). The transversal broken line corresponds to

= 0, i.e. to an unbounded lossless, nondispersive medium in which the wave

propagates at the phase velocity regardless of its frequency. The solid-line

curve represents Eq. (5.120) and shows that the waveguide is very dispersive

close to the cutoff frequency . For frequencies such that their

wavelengths are much smaller than the transversal waveguide dimensions, the

walls do not affect the propagation and the velocity tends to .

The group velocity, , within the guide is given by

=

=

s1−

µ

¶2(5.121)

which is smaller than the phase velocity in the unbounded medium. The

phase velocity within the waveguide, , is given by

=

= =

r1−

³

´2 (5.122)

which is always higher than that in the unbounded medium and is frequency

dependent. Hence single conductor waveguides are dispersive transmission sys-

tems. Note that

· = 2 (5.123)

For TEM modes, from (5.93), we have = which is real and independent

of the frequency. Thus, all frequencies propagate along a lossless transmission

line at the same phase velocity as that of the unbounded homogeneous di-

electric filling the waveguide and there is no cutoff frequency.

5.4. ATTENUATION IN GUIDING STRUCTURES 123

Figure 5.4: Dispersion relation =¡2 + 2

2

¢12between phase constant

and the frequency .

5.4 Attenuation in guiding structures

For a propagating mode an attenuation constant , owing to energy dissipation

within the waveguide, can arise from losses in the non-perfect conducting

walls () and in the non-perfect dielectric filling the waveguide (). Thus,

the attenuation constant consists of two parts = + . Dielectric losses

are generally negligible when waveguides are filled with air, which has a lower

dielectric loss than do conventional dielectrics.

First, we analyze the losses for TE and TM due to a non-perfect dielectric

and afterward the ones due to non-perfect walls. In any case, as generally occurs

in practice, these losses are assumed to be very small.

5.4.1 TE and TM modes.

Dielectric Losses

When the dielectric filling the waveguide is lossy the attenuation can be easily

taken into account if in the expressions obtained for ideal dielectrics the real

propagation constants and are replaced by − and −, respectively,where = + and = + are the complex propagation constants

in the unbounded dielectric filling the waveguide and in the waveguide, respec-

tively. Then, from equations (3.18), (5.78) and (5.112), we have

2(1− tan ) = −2 = −2 + 2 (5.124)


Using the above expressions for and and neglecting the term 2, because

the attenuation constant is very small, we find

2 = 2 − 2 (5.125a)

=2

2tan =

2 + 2

2tan (5.125b)

Thus, the attenuation factor is proportional to the loss tangent, tan , of

the dielectric filling the waveguide. On the other hand, (5.125a) coincides

with Eq. (5.113) for waveguides with ideal dielectric, and consequently the

phase constant (and thus the wavelength) remains practically the same as those

for a lossless waveguide. The dependence of the attenuation factor on the

frequency (assuming a range of frequencies in which the permittivity of the

dielectric remain unchanged) can be deduced by substituting the expressions of

tan and , given by (1.92) and (5.117), respectively, in (5.125b). Thus we

get

=

2(1−()2)12(5.126)

where is the intrinsic impedance of the dielectric given in (3.34a) and is

its effective or equivalent conductivity (1.79). From (5.126) we can see that becomes very high at frequencies close to the cutoff value, then decreases to a

minimum value, and afterwards increases with the frequency, becoming almost

proportional to it.

Wall losses

When the conductivity is finite the tangential magnetic field induces currents

which are not restricted to the surface and, according to Ohm’s law, are asso-

ciated with a tangential electric field (i.e. = = × ) in the walls. The

vector product of the fields and at the surface of the walls represents a

flux of power directed towards the inner of the wall. This power coincides with

the dissipation in the conductor caused by the Joule effect and is subtracted

from the mode that propagates along the waveguide. As a consequence, the

amplitude of the electric and magnetic fields of the mode are attenuated accord-

ing to −, where is the attenuation constant due to wall losses. We candetermine the value of for a given propagating mode by taking into account

that the time-average power transmitted through the cross-section of the

guiding transmission system is

=

Z

P · = 1

2

Z

Re(~E × ~H∗ ) · (5.127)

Since, due to the losses, the amplitude of the field wave varies according to

−, then, will vary according to −2. Moreover, the law of energy

conservation requires that the rate of the decrease of with distance along

the transmission system equals the time-average power loss on the surface of the

walls per unit length, 0, in the direction of propagation. Therefore, we have

5.4. ATTENUATION IN GUIDING STRUCTURES 125

Figure 5.5: In the lossy transmission line model, the series resistance and di-

electric conductance are introduced into the equivalent circuit model:.

0 = −

= 2 (5.128)

and thus

= 02

(5.129)

If ~H is the magnetic field existing near the walls, the time-average power dissi-

pated per unit of length in the walls is given, according to (3.46), by

0 =1

2

ZΓ

20 =

1

2

ZΓ

~H · ~H∗ (5.130)

where is the surface resistance given by (3.47) and Γ is the cross-sectional

contour of the non-perfect conducting walls. Thus the coefficient of attenuation

of the -th TE or TM mode is found to be

=Γ· ∗

4P·

=Γ· ∗

2Re(×

∗ )·

(5.131)

This equation will be applied in next chapter to the calculation losses for TE and

TM modes in waveguides. In strict terms, the modes we have found assuming

perfect conducting walls are no longer valid since non-perfect conducting walls

represent a change in the boundary conditions because in this case the tangential

component of the electric field is not null. However, if the losses are small, we can

make an approximate analysis (known in Mathematical Physics as "first order

perturbation method") by assuming that the field configurations or modes in

the waveguide coincide with those found for ideal-wall waveguides.

To include losses in the analysis of the propagation of transverse electromag-

netic (TEM) modes in transmission lines the element of the line of lengh

in Fig. (5.3) is modelled by four parameters, resistance, capacity, conductance

and inductance per unit of length, as shown in Fig. (5.5) where the resistance

represents the resistance due to the finite conductivity of the conductors, and

the shunt conductance is due to dielectric loss in the material between the


conductors4. Thus, using Kirchhoff’s voltage and current laws of the elemental

circuit theory, for a line composed of two perfect conductors with finite con-

ductivity separated by a lossy dielectric , the increment in potential difference

− is equal to +() and the increment in current − is equal to

+ () and thus we have the general transmission-line equations5

= −

µ +

¶(5.132a)

= −

µ +

¶(5.132b)

4The amount of inductance, capacitance and resistance depends on the length ofthe line, the shape and size of the conducting wires , the spacing between the wiresand the dielectric ( air or insulating medium) between the wires

5Additional Information regarding transmission lines can be found in numerous textbooks

on microwaves.

Chapter 6

Rectangular waveguide

6.1 Introduction

In the previous chapter we examined some general properties of the propaga-

tion modes that may exist in an ideal guiding transmission system which has

no sources and is constituted by perfect conductors and one ideal homogeneous

dielectric. Specific expressions for such modes can be determined only when

the particular geometry of the guide is given. In this chapter we will first an-

alyze in some detail the homogeneously filled rectangular and circular metallic

waveguides. After this, as a simple example of non homogeneous guiding struc-

ture in which the electromagnetic field propagates in more than one dielectric,

we will study the dielectric slab waveguide. Then, we will give some basic ideas

on propagation in strip and microstrip lines. Finally, we will consider cavity res-

onators which are basically constituted by a dielectric region totally enclosed

by conducting walls. This region, when excited by an electromagnetic field,

presents resonance with a very high-quality factor . In particular, we will

study the common simple cases of rectangular and circular cavity resonators.

6.2 Rectangular waveguide

Figure (6.1) shows a rectangular waveguide of sides and with , and

homogeneously filled with a perfect dielectric. Following the theory developed

in the previous chapter, in order to calculate the TE and TM modes that can

propagate in this waveguide, we start by solving the wave equation for the lon-

gitudinal components of the field with the corresponding boundary conditions

determined by the geometry of the system. The transverse components are then

calculated from these longitudinal ones.

With axis chosen as shown in the figure, the expressions for the fields in

127

128 CHAPTER 6. RECTANGULAR WAVEGUIDE

a

b

y

z

xa

b

y

z

x

Figure 6.1: Rectangular waveguide of width and height Rellenar en negro.

(5.75), take the form

~E = 0( )− (6.133a)

~H = 0( )− (6.133b)

Next, we are going to find the expression for these fields, first for TM modes

and afterwards for the TE modes.

6.2.1 TM modes in rectangular waveguides

For the TM modes, the differential equation (5.79) for

E = 0( )− (6.134)

can be solved by using the standard method of separation of variables in rec-

tangular Cartesian coordinates. For this, we assume, for 0, solutions in the

form of the product

0( ) = () () (6.135)

in which () and () are, respectively, functions only of and .

By substituting (6.135) in (5.79) and dividing by 0, we get

1

2

2+1

2

2+ 2 = 0 (6.136)

As each summand depends on a different variable, it should be verified that

1

2

2= −2 (6.137a)

1

2

2= −2 (6.137b)

2 = 2 = 2 + 2 (6.137c)

6.2. RECTANGULAR WAVEGUIDE 129

where we have substituted, according to (5.112), by the cutoff wavenumber

and where and are the separation constants to be determined from the

boundary condition (5.110a) at the guide walls. This boundary condition for

the geometry of Figure 6.1 implies

0 = 0 at

⎧⎪⎪⎨⎪⎪⎩ =

½0

=

½0

(6.138)

The solution of the Eqs. (6.137a) and (6.137b) are, respectively,

= 1 sin+ 2 cos (6.139a)

= 3 sin + 4 cos (6.139b)

where the coefficients are arbitrary constants to be determined from bound-

ary conditions. Therefore the general solution (6.135) for 0 takes the form

0 = (1 sin+ 2 cos) (3 sin + 4 cos) (6.140)

From the boundary conditions (6.138), we find that 2 = 4 = 0 and

=

(6.141a)

=

(6.141b)

and thus, from (6.137c),

2 =¡

¢2+¡

¢2(6.142)

where and are integers. The different solutions achieved by giving values to

and are termed TM modes and each set of values of and indicates

a specific mode. Thus, from (6.134), (6.140) and (6.141), for TM modes, we

have

E = − sin

sin

(6.143)

where the product of the constants 1 and 3 has been replaced by a new

constant .

Once we know the longitudinal component E, we can calculate the trans-

verse components ~E by means of (5.85a) and then, by using (5.85b), which

implies that

=E

H

= −E

H

(6.144)

we can obtain ~H. As a result, we get the following general expressions for the


components of the TM modes in a rectangular waveguide

TM modes in rectangular waveguides

(E)TM=

− sin sin

(~E)TM=³−

2

− cos

sin

´−³

2

− sin

cos

´

( ~H)TM=³

−1 2

− sin

cos

´−³

−1

2

− cos

sin

´

(6.145)

6.2.2 TE modes in rectangular waveguides

To analyze the TE modes, we can follow a procedure similar to that used for

the TM modes but now solving for H and imposing the boundary condition

(5.110b), H = 0, on the guide walls. This, for the geometry of Figure

6.1, implies that

H

= 0 at

½ = 0

=

H

= 0 at

½ = 0

= (6.146)

Then, using (5.90a) and (5.90b), and after steps analogous to those followed for

TM modes, we get

TE modes in rectangular waveguides

(H)TE=

− cos cos

( ~H)TE=³

2

− sin

cos

´+³

2

− cos

sin

´

(~E)TE=³

2

− cos

sin

´−³

2

− sin

cos

´

(6.147)

Note that, for both, TM and TE modes, the subindexes and ,

indicate the number of half-wave variations of the field in the and directions,

respectively. For a TM mode with or equal to zero, from (6.143), we

have (E)TE00 = 0 and consequently, from Eqs (6.145), (~E)TE00 = 0 and

( ~H)TE00 = 0. Hence, there is no TM mode in which or is equal to zero.


This was to be expected because a TM wave with = 0 would degenerate to

become a TEM wave which, as we saw in Subsection 5.2.3, cannot propagate

within a waveguide.

For TE modes, it is easy to see from (6.147) that either or may be

equal to zero but not both at the same time, since in this case the expression

of ()TEin (6.147) reduces to

(H)TE00 = 00− (6.148)

while ( )TE00 = 0 and = ()TE00 = 0 such that only ()TE00 exists.

This field does not fulfil Maxwell’s equations, since a time-varying field

should generate an electric field . Therefore the TE00 mode cannot exist.

Cutoff frequencies in a rectangular waveguide

From (5.116), (6.137c), and (6.141), we see that the cutoff frequency for either

a TE or a TM mode is given by

() =

2

∙³

´2+³

´2¸ 12(6.149)

where is the phase propagation velocity of the wave in the unbounded

medium filling the waveguide. The wavelength and wavenumber in the waveguide

are given, respectively, by

() =2h¡

¢2+¡

¢2i 12 (6.150a)

() =

∙2 −

³

´2−³

´2¸ 12(6.150b)

From (6.149) we see that the cutoff frequency of the modes depends on

the dimensions of the cross-section of the waveguide. Values of the cutoff

wavelengths and frequencies for several modes are

()TE10 = 2; ()TE10 =

2(6.151a)

()TE01 = 2; ()TE01 =

2(6.151b)

()TE20 = ; ()TE20 =

(6.151c)

()TE11 = ()TM11=

2

(2 + 2)12

; ()TE11 = ()TM11=

¡2 + 2

¢ 12

2

(6.151d)

Note that if = the cutoff frequencies of TE10 and TE01 and the two modes

are equal except for a rotation of 2. Figure (6.2) shows the ratio of the cutoff

frequency of several modes to that of the TE10 mode as a function of .


Figure 6.2: Rectangular waveguide: ratio of the cutoff frequency of several

modes to that of the TE10 mode as a function of .

The dominant TE10 mode

In practice, we usually wish to have only the mode which has the lowest cut-

off frequency, called fundamental or dominant mode, propagating through the

guide. Thus, in the case of a rectangular waveguide, if , such that

()TE10 ()TE01 , the waveguide is usually designed so that only the TE10mode can be propagated. The cutoff frequency of the dominant TE10 mode is

selected by means of the dimension . The ratio of the cutoff frequency of each

mode to that of the TE10 mode as a function of is plotted in Figure (6.2).

We see that the separation of the cutoff frequencies for different modes is larger

for higher values of the ratio of the and dimensions Note that if ' 2, thenthe cutoff frequencies of the modes TE01 and TE20 are nearly the same and in

the frequency range 2 2 only the TE10 mode can be propagated.

Moreover, if 2, then ()TE20 ()TE01 . As we will see in the next Section,

losses due to non-perfectly conducting walls increase as decreases. Thus, to

have the greatest frequency range in which only the TE10 mode can propagate

and, at the same time, to have the smallest losses possible, we usually choose

the dimensions of the guide such that ' 2. Under this condition, only TE10modes will propagate in the frequency range ()TE10 2()TE10 . For the

dominant TE10 mode the general expressions (6.147) simplify to those given in

(6.152) where the constant 10 has been replaced by 0.


Rectangular TE10 mode

=

=2

=

q2 − ¡

¢2H = 0

− cos

H = 0

− sin

H = 0

E = 0

E = 0

E = −0

− sin

(6.152)

6.2.3 Attenuation in rectangular waveguides

Losses due to a non-perfect dielectric filling the waveguide and to non-perfect

conducting walls can be calculated using the expressions (5.126) and (5.131),

respectively. For a given mode, to obtain the attenuation due to dielectric

losses, we simply need to use, in the formula (5.126), the value of the cutoff

frequency of the mode, given by (6.149), and the values of the constitutive

parameters of the dielectric at the work frequency. However, to find the the

attenuation constant due to wall losses for any TE or TM mode, though

not complicated, is quite laborious. Here, to illustrate the procedure, we will

consider the particular case of the dominant TE10 mode

Attenuation of the TE10 mode For the TE10 mode, the integrals of the

formula (5.131) can be calculated from the general expressions for the field

components (6.152). Thus, for the denominator, we have

TE10 =

Z

(P)TE10 · = −1

2

Z

0

Z

0

(EH∗)TE10 =µ

0

2

¶2 (6.153)

Regarding the integral of the numerator in (5.131), because in the dominant

mode TE10 in a rectangular waveguide the magnetic field has only and


components, this integral takes the formZΓ

~H · ~H∗ = 2(Z

0

³|H|2 + |H|2

´+

Z

0

³|H|2 + |H|2

´

)(6.154)

Using the expressions of and given in (6.152) and by operating, we obtain

ZΓ

~H · ~H∗ = 220

"

2

Ã1 +

2

2

!+

#= 22

0

"

2

µ

¶2+

#(6.155)

where the last expression is obtained from (5.117). By substituting (6.153) and

(6.155) in (5.131) and after operating, we finally obtain the following expression

for the attenuation factor ()TE10

()TE10 =

1+ 2

( )

2

1−( )

2= 1

µ

(1−( )2)

¶ 12∙1 + 2

³

´2¸Np/m

(6.156)

Following a similar analysis,we can show that the general expresions for the

attenuation constant due to wall losses for any TE mode is

()TE=

2r

1−³

´2(µ

1 +

¶µ

¶2+

Ã0

2−µ

¶2!

©( )2 + 2

ª¡

¢22 + 2

)(6.157)

where

0 =n1 =02 6=0 (6.158)

While for TM mode is

TM=

2

r1−

³

´2 ( )32 + 2

2¡

¢2+ 2

(6.159)

These expressions show the dependence of the attenuation on the frequency.

Reedactar: Computed values of for a few TE and TM modes are given

in Fig. (6.3). In practice, surfaces imperfections, the value of may be greater

than the theoretical values. This effect can be reduced using well polished walls.


5 10 20 50 100 200

0.01

0.02

0.05

0.1

0.2

0.5

f(GHz)

c(n

p/m

)

TM11

TE10

TE20

TE11

Figure 6.3: Attenuation characteristics of rectangular waveguides modes. They

tend to infinite when is close to the cutoff frequency and decrease toward an

optimum frequency (minimum value of ) an then increase almost linearly

with .

Chapter 7

Fundamentals of antenna

theory

7.1 Introduction

Antennas are devices specifically designed for the efficient radiation (or recep-

tion) of electromagnetic energy to (or from) the surrounding medium, which we

will assume to be free space. Depending on the expected characteristics of the

antenna, such as frequency of work, electrical size, power, bandwidth, polariza-

tion, or optimization of the radiated power in a specific direction, antennas may

have very different configurations. For example, an antenna may be a piece of

conducting wire with a specific dimension and shape, an aperture or slot in a

guiding system, or an arrangement (called array antenna or simply array) of

individual antennas working together as a unit. A detailed study of antennas

is beyond the scope of this book, and therefore in this chapter and the follow-

ing one we will consider only some basic aspects. In antenna problems, we are

interested mainly in radiation or far-zone fields1.

From Chapter 2 we know that, given the current distribution of an antenna,

we can calculate the electromagnetic fields that it generates. The problem is

that, even for simple antenna structures, it is difficult to determine the exact

current distribution on the antenna under a given excitation, this requiring

numerical methods to be accurately calculated. Nevertheless, for some kinds

of antennas, such as the conducting thin-wire antennas studied in the following

Section, it is possible to make a physically reasonable analytical estimate for the

current distribution and from it to derive analytical expressions for the radiation

fields.

Antenna theory is developed largely under the assumption that the antenna

is excited by time-harmonic current waves. This is not only because most anten-

1 It can be shown that any antenna presents similar radiation characteristics regardless

of whether it is transmitting or receiving. This property is called reciprocity. However, it is

normally easier to study antennas when they are transmitting.

137

138 CHAPTER 7. FUNDAMENTALS OF ANTENNA THEORY

nas are designed to work in this situation but also because analysis for arbitrary

time dependence of the current is usually complex. Nevertheless, for straight

thin-wire antennas, there is a simple way of obtaining the analytical expression

of the radiated field produced by an arbitrary current signal traveling along the

wire. This analysis may help us to reach a better understanding of the radia-

tion mechanisms in antennas. Section (7.3) is devoted to this subject. The last

section is devoted to the study of arrays of antennas.

7.2 Linear thin-wire dipole antennas

One very simple but practical antenna is that formed by a straight perfect-

conducting thin wire of length 2 with a small feeding gap at the center

(Fig. 7.1), and an electrically very small radius (usually ≤ 10). For

these kinds of antennas, called linear dipole antennas, it is possible to estimate

the current distribution along the wires assuming the following simplifications

to achieve, at least as a first approximation, very good results:

1. Since the wire radius is electrically very small, it is considered to be zero

(i.e. infinitely thin wire) and the current is assumed to be null at the ends

of the antenna.

2. According to the principle of charge conservation, in order not to have

charge accumulation at the feeding source, the current entering one termi-

nal must at any instant of time be equal to the current exiting the other

terminal.

3. As the wire is considered to be a perfect electric conductor the current

attenuation due to ohmic and radiation losses is ignored. Thus, the cur-

rent travels, without deformation, along the wire at the phase velocity

corresponding to the surrounding exterior medium.

4. The traveling current wave is reflected backwards at the ends of the wire

with no attenuation. Thus the current distribution along the antenna

becomes a standing wave formed by two traveling waves of the same am-

plitude propagating in opposite directions.

5. The effects of the presence of the ground and any object near the antenna

(including the antenna’s own structure) are ignored and the current dis-

tribution in the antenna is assumed to be isolated in space.

If we apply these rules to an antenna of total length 2 and placed along the

-axis, Fig. 7.1, which is fed at its center by a time-harmonic source (usually by

a transmission line such as a coaxial line), we arrive at the following expressions

for the current distribution along the antenna

(0) = sin ( − 0)(0)(0) 0 0 (0) = sin ( + 0)(0)(0) − 0 0

(7.1)

7.2. LINEAR THIN-WIRE DIPOLE ANTENNAS 139

z

'z l

y

x

'z l

' 0z

( ')I z

R

(Field point)P

r

r

z

'z l

y

x

'z l

' 0z

( ')I z

R

(Field point)P

r

r

Figure 7.1: Center-fed dipole antena.

where denotes the maximun current amplitude occurring along the antenna.

These expressions specify the spatial dependence of the current at each

source position 0 in the upper and lower arms of the antenna. The contri-bution of an infinitesimal current element (0)0 to the radiation field of thethin-wire antenna at the field point , is given, from (2.104b), by

E = E = 0(

0)0

4− sin (7.2)

The total radiation field can be calculated by integrating (7.2), in order to add

together the contributions of all the current elements, over the total length of

the antenna, i.e.

E =

Z

−E =

0

4sin

Z

−(0)

−

0 (7.3)

If we assume that the field point is in the far zone ( ) and use the

approximations (2.77) and (2.78) for the radiation fields (i.e. we replace by

− 0 cos , with 0 = 0 in the exponential, and by in the denominator),

then (7.3) simplifies to

E = 0 sin

4−

Z

−(0)

0 cos 0 (7.4)


Substituting (7.1) into the integral of (7.4) and integrating we get

E = 0 sin

4−

µZ 0

−sin ( + 0)

0 cos 0

+

Z

0

sin ( − 0)0 cos 0

!

=0

2−

µcos( cos )− cos

sin

¶=

60

− () = 0H (7.5)

where we have used the value 0 = 120 for the intrinsic impedance of free

space and introduced the radiation antenna factor () defined as

() =cos( cos )− cos

sin (7.6)

The last equality in (7.5) is attained by using the general relationship (2.85)

between the radiation fields and . From expression (7.5) we can see the

following characteristics of the far-zone radiation fields:

a) The radiation fields are linearly polarized and proportional to the excitation-

current amplitude .

b) The radiation fields and are independent of the azimuthal angle

, i.e. they present axial symmetry about the axis.

The dependence on of the radiation field is given by the antenna factor

(), which gives, at a constant radius , the three-dimensional graphic repre-

sentation (called radiation pattern) of the amplitude (normalized to a maximum

value of unity) of the radiation field. In general the antenna factor is associ-

ated with the interference of the fields generated by each source element of the

antenna, i.e. their constructive or destructive sum due to their phase difference.

From (7.5), the time-average value of the Poynting vector, (1.107), is given

byP = 12ReP = 12Re × ∗ = (7.7)

that depends on 2(), which therefore, represents the radiation power pattern.

Figure (7.2) shows typical three-dimensional and -plane radiation patterns for

four different lengths of straight thin-wire dipole antennas. It is worth noting,

that dipole antennas do not radiate in the direction given by = 0.

Antennas such as straight-wire dipoles for which the radiation pattern de-

pends only on (i.e. presents axial symmetry), are called omnidirectional.

However, in general, the radiation pattern of antennas depends on both co-

ordinates, and , and it is usually plotted either in polar or rectangular

coordinates. The radiation pattern normally presents lobes or beams, (i.e.

zones of radiation field bounded by zones of relatively weak or null radiation

intensity) with a main lobe in the direction of maximum radiation and other

series of minor lobes called sidelobes. A lobe radiating in the counter direction


to the desired radiation direction is called back lobe. Regions for which the

radiation is very weak are called nulls.

The polarization of an antenna is defined as the polarization of the wave

radiated by the antenna. Thus, from (7.5), the straight-wire dipoles are linearly

polarized antennas. For linearly polarized antennas the radiation pattern is

normally represented in two main orthogonal planes: the electric field or -

plane and the magnetic field or -plane. The -plane is the one containing the

electric field vector and the direction of maximum radiation while the -plane

is the one containing the magnetic field vector and the direction of maximum

radiation.

The total time-average radiated power or, in short, the total radiated power

, can be calculated from (7.7) by integrating P over a sphere of radious enclosing the antenna and using the expressions of the fields given in (7.5).

Thus we have

=

Z

P ·

=1

2

02

422

Z

0

Z 2

0

[ ()]22 sin =

02

4

Z

0

[ ()]2sin

(7.8)

The radiation resistance of an antenna is the value of an hypothetical

resistance that, when the current in the resistance is equal to the peak amplitude

of the current on the antenna, would dissipate the same amount of power

that is radiated by the antenna, i.e.

=1

2

2 (7.9)

Evidently, a desirable property of an antenna is to have high radiation resis-

tance.

In general, the input current at the terminals of an antenna differs from

. Thus, alternativally, we can write radiated power of the antenna as

=1

2

2 (7.10)

so that

=

22

(7.11)

where represents the radiation input resistance of the antenna. The two

radiation resistances and coincide only if = , as for example for

thin-wire antennas of dimension 2 = 2, being an odd integer, for instance

when = 1 3, in Fig. (7.2). When is even (e.g. when = 2 4 in Fig. (7.2) the

input current is theoretically zero, inplying infinite input resistance. In practice

the input current is very small but not null and is large but finite.

The input resistance represents only the real part of the input impedance,

= + , which normally is a complex magnitude that characterizes the


22

l

l

l

2 l

l

l

32

2l

l

l

2 l

l

l

22

l

l

l

22

l

l

l

l

l

2 l

l

l

2 l

l

l

l

l

32

2l

l

l

32

2l

l

l

l

l

l

l

l

l

l

l

l

l

2

2 0.5l

2l

2 1.5l

2 2l

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

22

l

l

l

2 l

l

l

32

2l

l

l

2 l

l

l

22

l

l

l

22

l

l

l

l

l

2 l

l

l

2 l

l

l

l

l

32

2l

l

l

32

2l

l

l

l

l

l

l

l

l

l

l

l

l

2

22

l

l

l

2 l

l

l

32

2l

l

l

2 l

l

l

22

l

l

l

22

l

l

l

l

l

2 l

l

l

2 l

l

l

l

l

32

2l

l

l

32

2l

l

l

l

l

l

l

l

l

l

l

l

l

2

22

l

l

l

2 l

l

l

32

2l

l

l

2 l

l

l

22

l

l

l

22

l

l

l

l

l

2 l

l

l

2 l

l

l

l

l

32

2l

l

l

32

2l

l

l

l

l

l

l

l

l

l

l

l

l

2

2 0.5l

2l

2 1.5l

2 2l

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

0.2 0.4 0.6 0.8 1

30

210

60

240

90270

120

300

150

330

180

0

Figure 7.2: Current distribution and E-plane radiation patterns for center-fed linear

dipole antennas of different length.


Figure 7.3: Dependence of the radiation resistance with the antenna length expressed

in terms of the wavelength.

antenna as a circuit element. The imaginary component of , i.e. the antenna

reactance , is associated with energy stored in the electric and magnetic near

fields, as explained in Subsection (1.5.4).

From (7.8) and (7.10) the radiation resistance of a straight dipole antenna

is given by

=02

Z

0

[ ()]2sin = 60

Z

0

[ ()]2sin (7.12)

The integral in (7.12) may be expressed in terms of the special mathematical

functions sine, (), and cosine, (), integral functions defined as

() =

Z

0

sin

(7.13a)

() = −Z ∞

cos

(7.13b)

Thus we get

= 60

µ + ln(2)− (2) +

1

2sin(2) [(4)− 2(2)]

¶+

+60

µ1

2cos(2) [ + ln() + (4)− 2(2)]

¶(7.14)

where = 005772 is the Euler-Mascheroni constant. The dependence of the ra-diation resistance with the antenna length expressed in terms of the wavelength,

is given in Figure (7.3).

The directivity ( ) of a given antenna is a measure of how the antenna

concentrates the radiated power in a given direction compared with an ideal


(non physically realizable) isotropic point radiating source which would radiate

the same total power uniformly in all directions producing an isotropic power

radiation pattern. Thus the directivity is determined by the dimensionless quan-

tity

( ) =Ω

(4)=

( )

(7.15)

where ( ) called radiation intensity, is the radiated power per unit solid

angle or steradian, i.e.

( ) =

Ω(7.16)

and is the radiation intensity of the ideal isotropic radiator, i.e.

=

4(7.17)

The maximum value of the directivity (in the same direction in which the radi-

ation intensity is maximum) is

=max

=4max

(7.18)

Although the directivity is in general a function of the angles and , usually

only its maximum value (called in short simply directivity) is given

From (7.8) and (7.15), the directivity of a dipole antenna is given by

() =12

02

42[ ()]

2

02

162

R 0[ ()]

2sin

=2 [ ()]

2R 0[ ()]

2sin

(7.19)

The directivity as defined in (7.15) is based on radiated power . Because

the power loss, , in the antenna itself as well as nearby lossy structures

including the ground, is less than the total input power . Thus we have

= + (7.20)

and the gain of the antenna is defined as

( ) =4( )

(7.21)

and, consequently, the maximum gain is

= 4 (7.22)

The ratio of the gain to the directivity of an antenna is the radiation efficiency,

=

=

(7.23)

normally the efficiency of very well constructed antennas is very close to 100%.

KK

7.3. QUALITATIVE ANALYSIS OF THINWIRES ANTENNAS.WHEREANDWHYANTENNAS RADIATE

Half-wave dipole antenna

One particular case of practical interest is the half-wave dipole corresponding to

= 4 so that the total length equals 2. For this case the radiation electric

field, (7.5), becomes

= 0 =0

2−

cos(2cos )

sin =

60

−

cos(2cos )

sin (7.24)

Thus the time-average Poynting vector is given by

P =1

2Re~E × ~H

∗ =

02

822

∙cos(

2cos )

sin

¸2 =

1522

∙cos(

2cos )

sin

¸2

(7.25)

and the total power radiated is

=

Z

P · = 02

4

Z

0

£cos2(

2cos )

¤sin

= 302

Z

0

£cos2(

2cos )

¤sin

(7.26)

The integral in (7.26) can be evaluated numerically to give ' 36052.Thus, from (7.9) and (7.10), = = 73Ω. The input impedance of this

antenna also has a positive (i.e. inductive) reactance component , which can

be calculated to be = 425Ω. Thus the total input impedance for the

half-wave dipole antenna is = 73 + 425 Ω and is of this order for dipoles

of electrically very small but finite radius. In practice the reactance can be

made zero, = 0, by using a length slightly shorter than 2. Hence we can

easily match this antenna to a standard coaxial line for which the characteristic

impedance is approximately 75 ohms. The maximun directivity of the half-

wavelengh antennas occur in the direction normal to the dipole, = 2, and

is found to be

' 164 (7.27)

7.3 Qualitative analysis of thin wires antennas.

Where and why antennas radiate.

To understand qualitatively why and where an antenna radiates, let us consider

the same antenna of Fig.(7.1) but excited by a single short electromagnetic

pulse. As was established in Section (7.2), the current distribution of a di-

pole antenna excited by a harmonic source is a standing wave formed by two

harmonic traveling current waves: the one generated at the feed source and

the ones reflected at the ends. However, if the antenna is excited by a tran-

sient current signal or short pulse the resulting transient response consists of


Figure 7.4: Left: Pulses of charge and current propagating along the antenna

at a given time before they reach the ends of the antenna. They are generated

at the feed source and propagate towards the ends of the wire where they will

be reflected. Upper Right: One wavefront is formed at the feed point when the

antenna is excited (wavefront 1) and, each time the current pulses reach the ends

of the antenna and are reflected new wavefront are created (wavefronts 2 and

3). Lower Right: The process of generating radiation wavefronts will continue

taking place each time that the reflected pulses at the ends of the antenna reach

again the opposite ends. In a real antenna the pulses attenuate by ohmic losses

in the conductors and by the radiation itself.

7.3. QUALITATIVE ANALYSIS OF THINWIRES ANTENNAS.WHEREANDWHYANTENNAS RADIATE

Figure 7.5: Time evolution of the current pulses along a thin wire antenna.

two current pulses as shown in (Fig.(7.4)) each having the same shape as the

exciting transient current. Taking into account that the direction of flow of

electrons is opposite to the direction of flow of current, these two current pulses,

both propagating in different senses, correspond to two identical charge pulses

(one positive and the other negative) propagating in the same senses than the

current pulses (see Fig.(7.4)). Thus assuming that the losses in the antenna

are negligible (ohmic loss in the conductors, loss due to imperfect ground, etc.)

the current distribution along the antenna are two traveling transient charge

waves propagating, with no attenuation and phase velocity corresponding to

surrounding free space, towards the ends of the antenna2. From expression

(2.69), each time that the charges that form the propagating pulses are accel-

erated must take place the generation of a wavefront of radiated field. In the

our antenna this acceleration occurs once at the feed point (when the excita-

tion source is applied and the charge pulses are formed) and each time that the

pulses, after propagating along the antenna without radiating, reach the ends

of the antennas where they stops propagating and are reflected changing their

propagation direction. As results the total radiated field must be formed by

the sum of different wavefronts: the one generated at the feed point when the

antenna is excited and the ones formed each time that the current pulses reach

the ends of the antenna and are reflected. Figure (7.5) shows the time evolution

of current pulses along the antenna.

2 It should be note that the total electrical charge of the antenna is null at every moment


7.4 Antennas above a perfect ground plane

In practice, antennas very often work in the presence of a conducting plane, and

the total electromagnetic field is due to the currents in the antenna plus the cur-

rents induced on the plane, which is usually assumed, as a good approximation,

to be infinite and perfectly conducting and referred to as perfect ground plane.

The performance of an antenna over a perfect ground plane can be studied us-

ing the method of images in which the original linear inhomogeneous problem

is replaced by a linear homogeneous equivalent one, where the conducting plane

is removed and its effects accounted for by an image of the antenna located

below the surface. The equivalent problem, formed for the original and image

sources in a homogeneous medium (with the same constitutive parameters as

the original upper half-plane medium) must satisfy Maxwell’s equations and

present the same boundary conditions as the original problem in the region of

interest (i.e. the tangential component of the electric field has to vanish at

any point along the plane where the original perfect ground interface was situ-

ated). Then the uniqueness theorem (see Section 2.7) ensures that the original

and equivalent problems have the same solution above the perfect ground plane,

being the equivalent problem much easier to solve.

In general, the original problem contains "real" electric current ( ) and

”hypothetical” magnetic current ( ) sources(see Section 2.6). Figure (7.6(a))

shows one element of both kinds of currents above a perfect ground plane. The

equivalent problem is shown in Figure (7.6(b)). The image electric current

density has been built equal in magnitude to but 180 out of phase.

The image magnetic current density has been built such that it is just

equal in magnitude to . . Thus, according (2.243) and taking into account

the properties of the vector differential operator, ∇, we have

× ~E = × (~E + ~E) = × (− 2∇(∇ · ~A)− ~A− 1

∇× ~F ) = 0

(7.28)

where ~E = (~E + ~E) is the total field due to the real (~E) and magnetic

(~E) currents and is the unit vector normal to the original perfect ground

interface plane. With these source distributions, the tangential electric field

at any point of the original perfect ground interface plane is zero. Then the

original and the equivalent problem presents the same tangential components

for the electric field, and consequently both generate the same electromagnetic

fields in the region above the ground plane.

As an example of an antenna over a perfect ground plane, let us consider the

monopole antenna, which is composed of a thin straight wire fed by a voltage

source between the wire and a conducting ground plane. From image theory, the

monopole antenna is equivalent to a dipole antenna in a homogeneous region.

The equivalent dipole is twice the length of the monopole and is driven with

twice the antenna source voltage. These equivalent antennas generate the same

fields in the region above the ground plane. For example, as shown in Figure

7.4. ANTENNAS ABOVE A PERFECT GROUND PLANE 149

(a) (b)

ie ed d=

ed md

im md d=

Original problem

J

Equivalent problem

mJ

mJ

imJ

J

iJ

Interface free space-perfect ground plane Perfect ground plane removedned md n

(a) (b)

ie ed d=

ed md

im md d=

Original problem

J

Equivalent problem

mJ

mJ

imJ

J

iJ

Interface free space-perfect ground plane Perfect ground plane removedned md n

Figure 7.6: (a) Actual or original physical problem (b) Equivalent image prob-

lem

Figure 7.7: (a) A vertical quarter-wave monopole over conducting ground. (b) Equiv-

alent half-wave dipole. Dibujar tierra como en la de imágenes

(7.7), a quarter-wave monopole is equivalent to a half-wave dipole and the field

created above the ground plane by the monopole coincides with the one created

by a 2 dipole given by (7.24). To calculate the radiated power by a monopole

antenna, the integral in Eq. (7.8) extends from 0 to 2 i.e.

=0

2

4

Z 2

0

[ ()]2sin (7.29)

and consequently the radiation resistance is one half that of the complete dipole

of length 2. Thus the radiation resistance of the 4 monopole is =

3605Ω.


7.5 Aperture antennas

Aperture antennas are those in which the radiated field is produced by an aper-

ture or slot in some kind of guiding structure3. As was indicated above, for this

kind of antenna, the current distributions are often unknown and difficult to

determine or approximate and it is much easier and efficient to calculate (using

the equivalence principle) the radiated field of the antenna from the fields , over all the points of the aperture. From these fields, we can calculate the

Huygens sources of the radiated fields, i.e. the equivalent surface electric, ,

and magnetic, , currents

= × (7.30a)

= −× (7.30b)

where is the unit vector normal to the surface. Although the exact aperture

fields, and , are in general not available, in some aperture antennas

it is possible to make a physically reasonable estimate for them based on the

assumption that the aperture is electrically large (i.e. the aperture dimensions

are much larger than those of a wavelength) and thus the border effects can

be neglected. Thus, from (7.30), it is possible to derive the analytical expres-

sions for the radiation fields as in the following simple example of a uniform

rectangular aperture.

Let us consider a uniform plane wave propagating perpendicularly to a

rectangular aperture (see Fig. 7.8) which is assumed to be cut into a thin,

flat, infinite perfect ground plane . The fields over on the aperture area are

considered constant and equal to the incident fields. That is

= || ≤

2; || ≤

2(7.31)

= =

0 =

0 || ≤

2; || ≤

2(7.32)

while the fields elsewhere over the infinite ground plane containing the aperture

are zero. From (7.30) the equivalent surface currents on the aperture are

= × = −

0 (7.33a)

= −× = − (7.33b)

Thus, from (2.46a) and (2.242), the electric, , and magnetic, , vector

potentials in terms of the aperture fields (taking into account that the volume

3Although in most cases the aperture antenna coincides with a physical aperture in some

kind of structure, this is not always the situation as happens, for example, in the case of

microstrip antenna

7.5. APERTURE ANTENNAS 151

( )cz

5a b ( )b( )a

R

r

ab

z

E

H

y

x

rsmJ

sJ

n

( )cz

5a b ( )b( )a

R

r

ab

z

E

H

y

x

rsmJ

sJ

( )a

R

r

ab

z

E

H

y

x

rsmJ

sJ

n

Figure 7.8: Rectangular aperture antenna and radiation diagram of a rectangu-

lar aperture antenna for = = 5.

integrations reduce to surface integrations over the aperture), become

~A( ) = −

40

Z 0=2

0=− 2

Z 0= 2

0=− 2

−(−sin (0 cos+0 sin))00

= −

40−

sin

sin

(7.34a)

~F ( ) = − 0

4−

sin

sin

(7.34b)

where , , and are the field point coordinates and we have introduced the

variables and

=

2sin cos (7.35a)

=

2sin sin (7.35b)

To obtain the expressions (7.34) we have used spherical coordinates for the far

field approximation (2.77), so that

' − · 0 = − sin (0 cos+ 0 sin) (7.36)

The radiation electric field can be found by introducing (7.34) into (2.243)

and retaining only the terms decreasing as 1. Thus we have

~E( ) =

2−

∙(1 + cos )

2

sin

sin

(cos − sin)

¸(7.37)


The components of the radiation magnetic field are given, from (2.85), by

H =E

0(7.38a)

H = −E

0(7.38b)

The bracketed factor in (7.37) represents the normalized field magnitude or

antenna factor, i.e.

() =( )

max=(1 + cos )

2( ) (7.39)

where max = 2 and

( ) =sin

sin

(7.40)

is a quantity that depends on the dimensions of the aperture. According to

(7.39) the radiation pattern is determined mainly by ( ) since the factor

(1+ cos 2) (called the Huygens factor) has the effect of only slightly reducing

the side-lobe amplitudes. The radiation pattern simplifies at the two principal

planes, the = 0 (-plane) and = 0 (-plane), corresponding to = 0 and

= 2, where we have, respectively,

( 0) =sin(

2sin )

2sin

(-plane) (7.41a)

(

2) =

sin(2sin )

2sin

(-plane) (7.41b)

Fig. (7.8) shows the three-dimensional field pattern of Eq.(7.39) for aperture

dimensions = = 5. The radiation pattern consists of a narrow mainlobe in

the direction = 0 and several sidelobes. The halfpower beamwidth ( ) of

the principal beam is defined as the angular separation, 20, of the points where

the main beam of the power pattern equals one-half the maximum value or,

equivalently, 1√2 of the maximum value of the field amplitude.For example,

in the -plane ( = 0) the half-power points can be calculated by setting

(0 0) =1√2= 0707 (7.42)

from which, since (sin) = 0 707 at 0 = 1 39, we have

0sin 0 = 139 (7.43)

Thus the expression of the beamwith, given by 20, is

20 = 2 sin−1µ044

0

¶rad (7.44)

7.6. ANTENNA ARRAYS 153

If the aperture width is sufficiently large compared to 0 then this expression

can be approximated by

20 ' 0880rad ' 500

deg (7.45)

This result shows that the beamwidth in the principal plane of a uniformly

excited aperture is inversely proportional to the aperture width measured in

that principal plane.

Una forma análoga tiene el estudio y análisis del diagrama de radiación de

para = 2 .

Este tipo de abertura uniformemente iluminada se usa como referencia de

comparación con otras iluminaciones en las que la dependencia funcional de la

distribución de campo en la abertura es diferente.

Uniform antennas leads to the most efficient use of the aperture antenna.

7.6 Antenna Arrays

For some applications, single-element antennas are unable to fulfil certain re-

quired radiation characteristics such as sidelobe levels and directivity. Thus

the solution may be to use several of these single antennas or radiating elements

forming an array and working together as an unit. In practice the arrays are

usually formed by identical and identically oriented radiating elements which

may be single-element antennas of any class, such as thin-wire or aperture an-

tennas. In this chapter, we will deduce some basic general characteristics of

the arrays assuming that mutual coupling effects are ignored. Mutual effects

depend on the type of antennas, their relative orientation (for example, mutual

impedance for wire antennas placed end to end is not as strong as when they

are placed parallel) and in general decreases with increasing separation between

elements. In particular, we will show that the radiation field can be expressed

as the product of two magnitudes, one being the field created by one of the ele-

ments in the array, and the other being the array factor, which is a magnitude

characterizing the array and which depends exclusively on the excitations and

on the relative positions of the radiating elements in the array. The process

of separating the field into these two parts is known as pattern multiplication

principle. or factorization,

7.7 Pattern multiplication principle. Array fac-

tor

Let us consider an antenna array of elements, as shown in (Fig. 7.9). Let

us take one point of a given antenna, called the reference radiating element

= 1, as the origin of a coordinate system. Thus the position of other

radiating element (1 ≤ ≤ ) of the array can be specified by means of

a vector from a given point in the reference antenna to the analogous point


o

Nr

'Nr

'nr

nA

1A NA

NR

r

'1r

nr

nR

ro

Nr

'Nr

'nr

nA

1A NA

NR

r

'1r

nr

nR

r

Figure 7.9: Antenna array of radiating elements ≈ − 0 · .

in the antenna . Thus, if 01 is the position vector of a point on the referenceradiating element, and 0 is the vector that determines the analogous point onthe element , we have

0 = + 01 (7.46)

On the other hand, as happens frequently in practice, we will assume that

current distribution ~J of the radiating element is the same as in the refer-

ence element 1, i.e. ~J1 = ~J , except for a constant complex coefficient I4,

I = (7.47)

Thus ~J is given by~J = I ~J1 = ~J1

(7.48)

where without loss of generality we can assume that 1 = 1 and 1 = 0 as will

be done from now on.

The radiation electric field for the element in the radiation zone, is given

by

~E =04

−Z 0

³~J(

0)×

´× ·

00 (7.49)

4 (0) = 1(

01)

() =04

−

=1

0(

0)×

× ·

00 =

04

−

=1

0(

0)×

× ·(+

01)0 =

04

−

=1

0 1(

01)×

× ·

01·0 =

04

−(

=1

·0) 01(

01)×

× ·

0101

7.7. PATTERN MULTIPLICATION PRINCIPLE. ARRAY FACTOR 155

o

Nr

'Nr

'nr

nA

1A NA

NR

r

'1r

nr

nR

ro

Nr

'Nr

'nr

nA

1A NA

NR

r

'1r

nr

nR

r

Figure 7.10: Antenna array of isotropic radiating elements

Adding together the contributions of the elements and using (7.48) the

total radiation pattern () of the array is given by

~E() = ~E1()AF = 0~H × (7.50)

This expression is the mathematical formulation of the principle of pattern

multiplication, which states that the radiation pattern of an array of identical

radiating elements is the product of the dimensionless complex magnitude ,

called array factor, given by

AF =

X=1

I 0 =

X=1

( 0+) (7.51)

and the radiation pattern of the reference radiating element i.e.

~E = ~E1 =04

−Z 0

³~J1(

01)×

´× ·

0101 (7.52)

For example, for an array of half-wave dipole antennas, 1 is given by (7.24).

The array factor does not depend on the type of radiating element used to

built the array and represents the radiation pattern of an array of ideal isotropic

point sources. The array factor depends on number of radiating elements and

their complex excitation coefficients (i.e. amplitude and phase of the cur-

rent excitation of the individual elements) and on their geometric configuration

(linear, circular, rectangular, etc.) and relative position . These parameters

can be used as degrees of freedom to synthesize an antenna array with a given

radiation pattern. The study of antenna arrays is based on obtaining the array


y

x

2R NR

d z1Ao

2A

d

nR3R

3A nA NA

1R ry

x

2R NR

d z1Ao

2A

d

nR3R

3A nA NA

1R r

Figure 7.11: Linear array of radiating elements placed along an axis with

a uniform spacing .

factor for a given distribution of isotropic radiating elements (analysis) or on

finding the distribution of isotropic radiating elements for which its array factor

coincides with a given one (synthesis). Regarding the single antennas or radiat-

ing elements forming the array, the designer have to make an adequate choice

according to the technical requirements demanded.

7.8 Array factor for uniformly spaced linear ar-

rays

The discussion in this text will be limited to arrays that meet these conditions....

7.8.1 Uniform linear array

One very common and simple array is the linear array, which is formed by

placing the radiating elements uniformly spaced5 along a line. For example, let

us consider a linear array of radiating elements placed along an axis with a

uniform spacing (Fig. (7.11)) so that the element locations are = (− 1),1 = and 0 = ( − 1) . Thus the expression of the array factor (7.51)

becomes

AF =

X=1

I(−1) cos =

X=1

(−1) cos + (7.53)

5When the elements arrays are of unequally spaced, there exist an additional degree of

freedom that can sometimes be an advantage. Such arrays have been used to obtain greater

gains and smaller secondary lobes than is possible with equally-spaced arrays with the same

number of elements. In arrays with irregularly-spaced elements, it is easier to maintain a

constant excitation amplitude of the elements. Thus, this type of array can be useful in many

applications.

7.8. ARRAY FACTOR FOR UNIFORMLY SPACED LINEAR ARRAYS 157

To simplify the analysis of this array, we will consider the simple practical case

in which the magnitude of the complex coefficient of each radiating element

is identical to that of the reference element 1. This magnitude can be assumed,

without loss of generality, to be the unity (i.e. = 1 = 1). Thus (7.47)

becomes

I = (7.54)

If, moreover, there is a linearly progressive phase from element to element,

i.e.

= (− 1) (7.55)

where is the phase shift of any element with respect to the nearest previous

one, the array is referred to as a uniform array. Thus the array factor (7.53)

simplifies to

AF =

X=1

(−1)( cos +) =X=1

(−1)Ψ (7.56)

where the phase function Ψ

Ψ = cos + (7.57)

is a function of the element spacing, phase shift, frequency, and azimuthal angle

Expression (7.56) is a geometric progression of terms and common ratio

Ψ and therefore can be summed up in a closed form. Thus (7.56) reduces to

AF = (−1)Ψ2sin(Ψ

2)

sin(Ψ2)

(7.58)

The phase factor (−1)Ψ2 is not usually of interest and can be neglectedunless the field has to be added to the one created from another radiating

antenna. Moreover, if the position of the array is shifted so that the center

of the array is located at the origin, (Fig.(7.12)) the phase factor disappears.

Therefore, the array factor is normally written as

=sin(Ψ

2)

sin(Ψ2)

(7.59)

From (7.59) it is evident that the function sin(Ψ2) sin(Ψ2) determines

the radiation properties of the array. The maximun value of this function is

and occurs when

Ψ = cos 0 + = 0 (7.60)

Thus the angle of maximum radiation is given by

0 = arccos³−

´(7.61)


5A1A

r

z

y

x

5A1A

r

z

y

x

Figure 7.12: Array shifted so that the center of the array is located at the origin.

Figure 7.13: Plot of the normalized array factor for = 2 3 8


Moreover, the function (7.59) presents nulls for

Ψ2 = ± = 1 2 3 (7.62)

where is an integer and, between each two consecutive nulls, there is a max-

imum for which the magnitude decreases as increases. By dividing by

its maximum value , we obtain the normalized array factor (see Figure

7.13)

=sin(Ψ

2)

sin(Ψ2)

(7.63)

which is symmetric with respect to Ψ = 180.

Note that from (7.61), for a given distance , we can control the angle 0of the main beam of the array by changing the current phase angle of the

excitation current of the radiating elements while the amplitude of the currents

remains unchanged.

A particular case arises when the input is uniform and identical for all the

elements, i.e. when = 0 in (7.54). Thus, from (7.61), we have 0 = ±2;that is, the direction of maximum radiation is perpendicular to the line of the

array and is independent of the distante . This type of array is known as a

broadside array.

Another particular case is that in which the direction of maximum radiation

coincides with the axis of the array ( i.e. 0 = 0 or 0 = or cos 0 = ±1).Such arrays are termed endfire arrays. By imposing these conditions on (7.60)

the phase angle is given by

= ± = ±2 (7.64)

In the particular cases of = 4 or = 2, we have = ± 2 and = ± ,

respectively.

The radiation patterns in the plane = 0 corresponding to an array of 8

elements separated by a distance = 4, of the broadside and endfire types,

are shown in Figure (7.14). It is apparent from the figures that the directivity is

less for the endfire than for the broadside configuration. In the following, we will

examine more closely the directional characteristics of these two configurations.

Directional properties of uniform arrays

The parameter beamwidth between first nulls (BWFN) is defined as the angle

2 formed by the directions of the two nulls adjacent to the main lobe, i.e.

1 = 0 − ; 2 = 0 + ; = 2 (7.65)

where 0 is the angle of the main lobe and 1 and 2 are, respectively, the

directions corresponding to its two adjacent nulls. Thus, from (7.57) and (7.62),

we have

+ cos 2 = −2

(7.66a)

+ cos 1 =2

(7.66b)


0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

( )a

( )d

( )c

( )b

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

( )a

( )d

( )c

( )b

Figure 7.14: Radiation patterns in the plane = 0 corresponding to an array of

8 elements separated by a distance = 4, of the broadside (a) and (b) and

endfire types (c) and (d).


From (7.60), (7.65) and (7.66a), we get

cos 2 − cos 0 = cos (0 + )− cos 0 = − 2

(7.67)

Assuming 1, we can make the third-order series approximation of the

sine function

cos (0 + ) ' cos 0 − sin 0 − 2

2cos (7.68)

which, when introduced into (7.67), leads to

2 cos 0 + 2 sin 0 − 2

= 0 (7.69)

From this equation, we can determine the width of the main lobe for a

broadside array

0 =

2; BWFN = 2 =

2

(7.70)

and for an endfire array

0 = 0; BWFN = 2 = 2

µ2

¶ 12

(7.71)

Therefore, as the relation ( = ) increases, the beam narrows, and the

farther the main lobe is from the direction 2, the more the beam widens, and

consequently the less directivity it has. Generally, the endfire arrays have wider

main lobes than do broadside arrays, while the number of secondary lobes in

enfire types is greater than for broadside ones. This implies that a substantial

part of the energy radiated would be lost in undesired directions. Thus, normally

in a design, it is necessary to find a compromise between these two parameters:

width of the main lobe and level of the secondary ones.

Part I

Appendixes

163

Appendix A

Vector algebra and Analysis

formulas used in this book

In the following may be any well-behaved vectors and ΦΨ any well-

behaved scalars

Vector identities

× ( × ) = ( · ) − ( · ) (A.1)

( × )× = ( · ) − ( · ) (A.2)

Vector differential operators in rectangular, cylindrical and spher-

ical coordinate systems Rectangular coordinates

∇Ψ = Ψ

+

Ψ

+

Ψ

(A.3)

∇ · =

+

+

(A.4)

∇× =

µ

−

¶+

µ

−

¶+

µ

−

¶(A.5)

∇2Ψ =2Ψ

2+

2Ψ

2+

2Ψ

2(A.6)

∇2 = ∇2 + ∇2 + ∇2 (A.7)

165

166APPENDIX A. VECTORALGEBRAANDANALYSIS FORMULAS USED IN THIS BOOK

Cylindrical coordinates

∇Ψ = Ψ

+

1

Ψ

+

Ψ

(A.8)

∇ · =1

() +

1

+

(A.9)

∇× =

µ1

−

¶+

µ

−

¶+

1

µ ()

−

¶(A.10)

∇2Ψ =1

µΨ

¶+1

22Ψ

2+

2Ψ

2(A.11)

∇2 = ∇³∇ ·

´−∇×∇× (A.12)

Spherical coordinates

∇Ψ = Ψ

+

1

Ψ

+

sin

Ψ

(A.13a)

∇ · =1

2

¡2

¢+

1

sin

( sin ) +

1

sin

(A.13b)

∇× =

sin

µ

( sin )−

¶+

µ1

sin

−

()

¶+

µ

()−

¶(A.13c)

∇2Ψ =1

2

µ2Ψ

¶+

1

2 sin

µsin

Ψ

¶+

1

2 sin2

2Ψ

2

=1

2 (Ψ)

2+

1

2 sin

µsin

Ψ

¶+

1

2 sin2

2Ψ

2(A.13d)

∇2 = ∇³∇ ·

´−∇×∇× (A.13e)

∇× (Ψ) = ∇Ψ×

Ψ(A.14)

∇(Ψ ) = (∇Ψ) −Ψ(∇ ) (A.15)

167

∇× (∇Ψ) = 0 (A.16a)

∇ · (∇× ) = 0 (A.16b)

∇ · (∇Ψ) = ∇2Ψ (A.16c)

∇× (∇× ) = ∇(∇ · )−∇2 (A.16d)

∇ (ΦΨ) = Φ∇Ψ+Ψ∇Φ (A.16e)

∇ · (Ψ ) = Ψ∇ · + · (∇Ψ) (A.16f)

∇× (Ψ ) = Ψ∇× − ×∇Ψ (A.16g)

∇ · ( × ) = ·∇× − ·∇× (A.16h)

∇( · ) = ×∇× + ( ·∇) + ×∇× + ( ·∇) (A.16i)

∇× ( × ) = ∇ · − ( ·∇) − ∇ · + ( ·∇) (A.16j)

∇ · (Ψ) = ∇Ψ ·

Ψ(A.16k)

∇ (Ψ) = ∇Ψ

Ψ(A.16l)

∇× (Ψ) = ∇Ψ×

Ψ(A.16m)

∇(Ψ ) = (∇Ψ) −Ψ(∇ ) (A.16n)

Integral relations

In the following is a three-dimensional volume with volume element and

is the closed surface bounding with surface element = where is

the outer unit normal vector (see Fig. (1.1))Z

∇ · =

Z

Divergence theorem (A.17)Z

∇Ψ =

Z

Ψ (A.18)Z

∇× =

Z

× (A.19)Z

¡Φ∇2Ψ+∇Ψ ·∇Φ¢ = Z

Φ ·∇Ψ Green’s firts identity (A.20)Z

¡Φ∇2Ψ−Ψ∇2Φ¢ = Z

(Φ∇Ψ−Ψ∇Φ) · Green’s second identity

(A.21)

168APPENDIX A. VECTORALGEBRAANDANALYSIS FORMULAS USED IN THIS BOOK

In the following is an open surface and Γ is the contour bounding it, with

line element . The normal to is defined by the right-hand rule in relation

to the sense of the line integral around Γ (see Fig. (1.1))Z

³∇×

´· =

IΓ

· Stokes’ theorem (A.22)Z

×∇Ψ =

IΓ

Ψ (A.23)

Bibliography

[1] Umran S. Inan, Aziz Inan; Electromagnetic Waves; Prentice Hall, 1999.

[2] David J. Griffiths; Introduction to Electrodynamic; Pearson, 1998.

[3] David K. Cheng; Field and Wave Electromagnetics, Addison-Wesley, 1989.

[4] Carlo G. Someda; Electromagnetic Waves, Chapman & Hall, 1998.

[5] Foundations for Microwave Engineering; Robert E. Collin; McGraw-Hill,

1992.

[6] David M Pozar; Microwave Engineering; Wiley, 2012

[7] Constantine A. Balanis; Antenna Theory: Analysis and Design; Wiley-

Interscience, 2005.

[8] Stutzman, C Thiele; Antenna Theory and Design; Wiley, 2012.

[9] Matthew N.O. Sadiku; Numerical Techniques in Electromagnetics; CRC

Press, 2000.

[10] John David Jackson;Classical Electrodynamics; Wiley, 1998.

[11] Adams Stratton Julius; Electromagnetic Theory, Mcgraw Hill, 1941.

[12] Wolfgang K H Panofsky and Melba Phillips; Classical Electricity and Mag-

netism; Dover, 2005.

169

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