martin on EM
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Lecture-note textbook by R. Gómez Martín, from an Electrodynamics course (Granada, 2006-2007), apparently downloaded for Phil's transmission-line files. The contents cover Maxwell's equations, Poynting's theorem, retarded potentials, multipole radiation, plane waves, reflection and refraction, and TE/TM/TEM modes in waveguides. It ends with rectangular waveguides and attenuation.
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Electromagnetic field theory for physicists and
engineers:Fundamentals and Applications
R. Gómez Martín
2
Contents
0 . 1 P r e f a c i o ................................ i
IE l e c t r o m a g n e t i c field: radiation and propagation 1
1 Electromagnetic field fundamentals 3
1 . 1 I n t r o d u c t i o n .............................. 31.2 Review of Maxwell’s equations ................... 3
1.2.1 Physical meaning of Maxwell’s equations .......... 6
1 . 2 . 2 C o n s t i t u t i v e e q u a t i o n s.................... 81.2.3 Boundary conditions . .................... 1 2
1 . 3 T h e c o n s e r v a t i o n o f e n e r g y .P o y n t i n g ’ s t h e o r e m.......... 1 41.4 Momentum of the electromagnetic fie l d ............... 1 6
1.5 Time-harmonic electromagnetic fie l d s ............... 1 8
1.5.1 Maxwell’s equations for time-harmonic fie l d s ....... 1 9
1 . 5 . 2 C o m p l e x d i e l e c t r i c c o n s t a n t . ................ 2 01.5.3 Boundary conditions for harmonic signals ......... 2 4
1 . 5 . 4 C o m p l e x P o y n t i n g v e c t o r .................. 2 5
1 . 6 O n t h e s o l u t i o n o f M a x w e l l ’ s e q u a t i o n s ............... 2 7
2 Fields created by a source distribution: retarded potentials 29
2 . 1 E l e c t r o m a g n e t i c p o t e n t i a l s ...................... 2 9
2 . 1 . 1 L o r e n z g a u g e ......................... 3 2
2.2 Solution of the inhomogeneous wave equation for potentials . . . 342.3 Electromagnetic fie l d s f r o m a b o u n d e d s o u r c e d i s t r i b u t i o n.... 3 8
2.3.1 Radiation fie l d s........................ 4 3
2.3.2 Fields created by an in fin i t e s i m a l c u r r e n t e l e m e n t ..... 4 5
2 . 3 . 3 F a r - z o n e a p p r o x i m a t i o n s f o r t h e p o t e n t i a l s ........ 5 0
2 . 4 M u l t i p o l e e x p a n s i o n f o r p o t e n t i a l s ................. 5 1
2 . 4 . 1 E l e c t r i c d i p o l a r r a d i a t i o n .................. 5 32 . 4 . 2 M a g n e t i c d i p o l a r r a d i a t i o n ................. 5 52 . 4 . 3 E l e c t r i c q u a d r u p o l e r a d i a t i o n................ 5 7
2.5 Maxwell’s symmetric equations ................... 5 9
2.5.1 Boundary conditions . .................... 6 3
2 . 5 . 2 H a r m o n i c v a r i a t i o n s ..................... 6 4
3
4 CONTENTS
2.5.3 Fields created by an in finitesimal magnetic current element 65
2 . 6 T h e o r e m o f u n i q u e n e s s........................ 6 5
2.6.1 Non-harmonic electromagnetic fie l d ............. 6 6
2.6.2 Time-harmonic fie l d s ..................... 6 7
3 ??Electromagnetic waves 69
3 . 1 W a v e e q u a t i o n ............................ 6 9
3 . 2 H a r m o n i c w a v e s ........................... 7 3
3 . 2 . 1 U n i f o r m p l a n e h a r m o n i c w a v e s ............... 7 43.2.2 Propagation in lossless media ................ 7 6
3.2.3 Propagation in good dielectrics or insulators ........ 7 6
3.2.4 Propagation in good conductors .............. 7 8
3 . 2 . 5 S u r f a c e r e s i s t a n c e ...................... 7 9
3 . 3 G r o u p v e l o c i t y ............................ 8 0
3 . 4 P o l a r i z a t i o n.............................. 8 1
4R e fle c t i o na n dr e f r a c t i o no fp l a n ew a v e s 8 5
4 . 1 N o r m a l i n c i d e n c e . .......................... 8 6
4 . 1 . 1 G e n e r a l c a s e : i n t e r f a c e b e t w e e n t w o l o s s y m e d i a ...... 8 64 . 1 . 2 P e r f e c t / L o s s y d i e l e c t r i c i n t e r f a c e .............. 8 8
4.1.3 Perfect dielectric/Perfect conductor interface ....... 8 9
4 . 1 . 4 S t a n d i n g w a v e s........................ 8 94 . 1 . 5 M e a s u r e s o f i m p e d a n c e s................... 9 1
4 . 2 M u l t i l a y e r s t r u c t u r e s......................... 9 1
4 . 2 . 1 S t a t i o n a r y a n d t r a n s i t o r y r e g i m e s ............. 9 2
4 . 3 O b l i q u e i n c i d e n c e........................... 9 3
4.4 Incident wave with the electric field contained in the plane of
i n c i d e n c e ............................... 9 5
4.5 Wave incident with the electric field perpendicular to the plane
o f i n c i d e n c e .............................. 9 8
5 Electromagnetic wave-guiding structures : Waveguides and trans-
mission lines 101
5 . 1 I n t r o d u c t i o n ..............................1 0 1
5.2 General relations between fie l d c o m p o n e n t s ............1 0 3
5 . 2 . 1 T r a n s v e r s e m a g n e t i c ( T M ) m o d e s ..............1 0 5
5 . 2 . 2 T r a n s v e r s e e l e c t r i c ( T E ) m o d e s...............1 0 6
5 . 2 . 3 T r a n s v e r s e e l e c t r o m a g n e t i c ( T E M ) m o d e s.........1 0 75.2.4 Boundary conditions for TE and TM modes on perfectly
conducting walls .......................1 0 9
5.3 Cuto fff r e q u e n c y ...........................1 1 0
5 . 4 A t t e n u a t i o n i n g u i d i n g s t r u c t u r e s..................1 1 3
5 . 4 . 1 T E a n d T M m o d e s . .....................1 1 3
5 . 4 . 2 T E M m o d e s ..........................1 1 5
CONTENTS 5
6 Some types of waveguides and transmission lines 117
6 . 1 I n t r o d u c t i o n ..............................1 1 76 . 2 R e c t a n g u l a r w a v e g u i d e........................1 1 7
6 . 2 . 1 T M m o d e s i n r e c t a n g u l a r w a v e g u i d e s ...........1 1 8
6 . 2 . 2 T E m o d e s i n r e c t a n g u l a r w a v e g u i d e s............1 2 0
6 . 2 . 3 A t t e n u a t i o n i n r e c t a n g u l a r w a v e g u i d e s...........1 2 3
6 CONTENTS
0.1. PREFACIO i
0.1 Prefacio
Asignatura: Electrodinámica
4oC. Físicas
Curso 2006-2007
(Granada)
ii CONTENTS
Part I
Electromagnetic field:
radiation and propagation
1
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 th e theory of electromagnetic behavior
of matter, explain on a macroscopic scale the properties of the electromagneticfield, the relationships of this field with its sources, and its interaction with
matter. The reader is assumed to be familiar with these equations at least at an
undergraduate level. Next, after reviewin g other fundamental topics such as con-
stitutive parameters and boundary conditions, we apply the energy-conservationlaw to a bounded volume, limited by a surface S, 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 Sthat limits V. This term leads us
to the de finition 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-di fferential
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 fieldKis uniquely determined by its di-
vergence and curl if they are given throughout the entire space and if they approach zero atinfinity at least as 1/r
nwithn> 1. A proof of this theorem is given in Appendix ??
3
4 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
Maxwell’s equations are usually formulated in di fferential form (i.e., as relation-
ships between quantities at the same point in space and at the same instant intime) 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 n dS t o k e s ’( ??)t h e o r e m s .
For stationary media
2, Maxwell’s equations in di fferential and integral forms
are:
Differential form of Maxwell’s equations
∇·D(r,t)=ρ(r,t)(Gauss’ law) (1.1a)
∇·B(r,t)=0 (Gauss’ law for magnetic fields) (1.1b)
∇×E(r,t)=−∂B(r,t)
∂t(Faraday’s law) (1.1c)
∇×H(r,t)=J(r,t)+∂D(r,t)
∂t(Generalized Ampère’s law) (1.1d)
Integral form of Maxwell’s equations
I
SD(r,t)·ds=QT(t)(Gauss’ law) (1.2a)
I
SB(r,t)·ds=0 (Gauss’ law for magnetic fields) (1.2b)
I
ΓE(r,t)·dl=−Z
S∂B(r,t)
∂t·ds(Faraday’s law) (1.2c)
I
ΓH(r,t)·dl=Z
S(J(r,t)+∂D(r,t)
∂t)·ds(Generalized Ampère’s law)
(1.2d)
Maxwell’s equations, involve only macroscopic electromagnetic fields and,
explicitly, only macroscopic densities of free-charge, ρ(r,t),w h i c ha r ef r e et o
move within the medium, giving rise to the free-current densities, J(r,t).T h e
effect of the macroscopic charges and current densities bound to the medium’s
molecules is implicitly included in the auxiliary magnitudes DandHwhich are
related to the electric and magnetic fields,EandBby the so-called constitutive
equations that describe the behavior of the medium (see Subsection 1.2.2). Ingeneral, the quantities in these equations are arbitrary functions of the position(r)a n dt i m e
3(t). The de finitions and units of these quantities are
E=e l e c t r i c field intensity (volts/meter; Vm−1)
2In a stationary medium all quantities are evaluated in a reference frame in which the ob-
server and all the surfaces and volumes are assumed to be at rest. Maxwell’s equations formoving media can be considered in terms of the special theory of relativity, as shown in
chapter ??.
3Throughout the book, in most cases, in order to make the notation more concise, we will
not explicitly indicate the arguments, (r,t),of the magnitudes unless we consider it convenient
to emphasize the dependence on any of the variables.
1.2. REVIEW OF MAXWELL’S EQUATIONS 5
B= magnetic flux density (teslas4or webers/square meter; TorWbm−2)
D= electric flux density (coulombs/square meter; Cm−2)
H= magnetic field intensity (amperes/meter; Am−1)
ρ= free electric charge density (coulombs/ cubic meter; Cm−3)
QT= net free charge, in coulombs ( C), inside any closed surface S
J= free electric current density (amperes/square meter Am−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 sim ilarity 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, ∇×H(r,t)=J(r,t), the displacement current
density term or, in short, the displacement current, ∂D/∂t , as an additional
vector source for the fieldH. T h i st e r mh a st h es a m ed i m e n s i o n sa st h ef r e e
current density but its nature is di fferent 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 atthe constant velocity of light c.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
∇·J=−∂ρ
∂t(1.3)
o r ,i ni n t e g r a lf o r m ,I
J.ds=−dQT
dt(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, f(inN
m
−3), acting on a volume charge density ρmoving at a velocity u(inms−1)
in a region where an electromagnetic field exists,
f=ρ(E+u×B)=ρE+J×B (1.5)
whereJ=ρuis the current density in terms of the mean drift velocity of the
particles5, which is independent of any random velocity due to collisions. The
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, G) is often used instead, 1T=1 04G.
5In general, when there is more than one type of particle the current density its de fined
asJ=S
iρiuiwhereρianduirepresent the volume charge density and drift velocity of the
charges of class i.
6 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
total force Fexerted on a volume of charge is calculated by integrating fin this
volume. For a single particle with charge qthe Lorentz force is
F=q(E+u×B) (1.6)
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 remoteinteraction of charges and currents taking place via the electric and/or magneticfields that they produce. From Eq. (1.6) the work done by an electromagnetic
field acting on a volume charge density ρinside a volume dvduring a time
intervaldtis
dW =f·udtdv =ρ(E+u×B)·udtdv =ρE·udtdv =E·Jdtdv (1.7)
This work is transformed into heat. The corresponding power density P
v(
Wm−3) that the electromagnetic field supplies to the charge distribution is
Pv=dP
dv=dW
dtdv=E·J (1.8)
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 orderivatives of the fields att=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 conditionsand refer to the solution of Maxwell´s equations, with all these conditions, as aboundary-value problem.
Next, we brie fly describe the physical mean ing 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 onthe distance, r, between them, as r
−2. According to Gauss’ law, the divergence
of the vector fieldDis the volume density of free electric charges which are
sources or sinks of the fieldD, i.e. the lines of Dbegin on positive charges
(ρ> 0) and end on negative charges ( ρ< 0). In its integral form, Gauss’
law relates the flux of the vector Dthrough a closed surface S(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 Bfield does
not have scalar sources, i.e., it is divergenceless or solenoidal. This is becauseno free magnetic charges or monopoles have been found in nature (see Section
2.5) which would be the magnetic analogues of electric charges for E. Hence,
there are no sources or sinks where the field lines of Bstart or finish, i.e., the
field lines of Bare closed. In its integral form, this indicates that the flux of
theBfield through any closed surface Sis null.
1.2. REVIEW OF MAXWELL’S EQUATIONS 7
dSr
S V
(a) (b)S dSr
Γ
dSr
S V
(a) (b)S dSr
Γ
Figure 1.1: (a) Closed surface Sbounding a volume V.( b ) O p e n s u r f a c e Sbounded by the
closed loop Γ. The direction of the surface element dSis given by the right-hand rule: the
thumb of the right hand is pointed in the direction of dSand the fingertips give the sense of the
line integral over the contour Γ. ¡¡ ¡Atención: las dSdebe serds
Faraday’s law, (1.1c) or (1.2c), establishes that a time-varying Bfield 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 (RB·ds)
through any surface Sbounded 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 dsaccording 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 oppose s the change in the magnetic flux (Lenz’s law).
Ampère’s generalized law, (1.1d) or (1.2 d), constitutes another connection,
different from Faraday’s law, between EandB. I ts t a t e st h a tt h ev e c t o rs o u r c e s
of the magnetic field may be free currents, J,and/or displacement currents,
∂D/∂t . Thus, the displacement current performs, as a vector source of H,a
similar role to that played by ∂B/∂t as a source of E.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 Sbounded
by a closed loop Γ(Fig. 1.1 ), of both currents: the free current Jand the
displacement current ∂D/∂t .
8 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
1.2.2 Constitutive equations
Maxwell’s equations (1.1) can be written without using the arti ficialfieldsD
andH,a s
∇·E(r,t)=ρall
ε0(r,t) (1.9a)
∇·B(r,t)=0 (1.9b)
∇×E(r,t)=−∂B(r,t)
∂t(1.9c)
∇×B(r,t)=μ0Jall(r,t)+μ0ε0∂E(r,t)
∂t(1.9d)
whereε0=1 0−9/(36π)(farad/meter; Fm−1)a n dμ0=4π10−7(henry/meter;
Hm−1) are two constants called electric per mittivity and magnetic permeability
of free space, respectively. The subscript allindicates that all kinds of charges
(free and bound ) must be individually included in ρandJ.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 extendequations ( ??)a n d( ??) and to obtain Maxwell’s macroscopic equations (1.1),
in which only macroscopic quantities are used and in which only the densi-
ties 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 m icroscopically large-volume elements,
∆v, so that these contain a large number of molecules but at the same time
are macroscopically small enough to rep resent accurate spatial dependence at a
macroscopic scale. As a result of this average, the properties of matter relatedto atomic and molecular charges and currents are described by the macroscopic
parameters, electric permittivity ε, magnetic permeability μ, and electrical con-
ductivity σ. 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 statisticaltheory to describe their collective behav ior. Fortunately, in most of the practi-
cal situations, it is possible to achieve good results using simpli fied microscopic
models. Appendices ??and??present a brief introduction to the microscopic
theory of electric and magnetic media, respectively.
To de fine the electric permittivity and describe the behaviour of the electric
field in the presence of matter, we must introduce a new macroscopic field
quantity, P(Cm
−2), called electric polarization vector, such that
D=ε0E+P (1.10)
1.2. REVIEW OF MAXWELL’S EQUATIONS 9
and de fined as the average dipole moment per unit volume
P= lim
∆v→0PN∆v
n=1pn
∆v(1.11)
whereNis the number of molecules per unit volume and the numerator is
the vector sum of the individual dipolar moments, pn, of atoms and molecules
contained in a macroscopically in finitessimal volume ∆v. For many materials,
called linear isotropic media, Pcan be considered colinear and proportional to
the electric field applied. Thus we have
P=ε0χeE (1.12)
where the dimensionless parameter χe, called the electric susceptibility of the
medium, describes the capability of a dielectric to be polarized. Expression
(1.10) can be written in a more compact form as
D=( 1+χe)ε0E (1.13)
so that
D=ε0εrE=εE (1.14)
where
εr=1+χe (1.15)
and
ε=ε0εr (1.16)
are the relative permittivity and the permittivity of the medium, respectively.
To de fine 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 M(Am−1),s u c h
that
H=B
μ0−M (1.17)
whereMis defined, in a similar way to that of the electric polarization vector,
as the average magnetic dipole moment per unit volume
M=l i m
∆v→0PN∆v
n=1mn
∆v(1.18)
whereNis the number of atomic current elements per unit volume and the
numerator is the vector sum of the individual magnetic moments, mncontained
in a macroscopically in finitessimal volume ∆v.
In general, Mis a function of the history of BorH, which is expressed
by the hysteresis curve. Nevertheless, many magnetic media can be consideredisotropic and linear, such that
M=χ
mH (1.19)
10 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
whereχmis the adimensional magnetic suscep tibility magnitude, being negative
and small for diamagnets, positive and s mall for paramagnets, and positive and
large for ferromagnets. Thus
B=( 1+χm)μ0H=μrμ0H=μH (1.20)
where
μr=( 1+χm) (1.21)
and
μ=μrμ0 (1.22)
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 μrrequires a careful de finition 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 (see
Appendix ??).
In a vacuum, or free space, εr=1 ;μr=1,a n dt h e r e f o r et h e fields vectors
DandE,a sw e l la s BandH, are related by
D =ε0E (1.23a)
B=μ0H (1.23b)
Very often the relation between an electric field and the conduction current
densityJcthat it generates is given, at any point of the conducting material,
by the phenomenological relation, called Ohm’s law
Jc=σE (1.24)
so thatJis linearly related to Etrough the proportionality factor σcalled the conductivity of the
medium. Conductivity is measured in siemens per meter (Sm−1≡Ω−1m−1)o rm h o sp e r
meter (mho m−1). Media in which (1.24) is valid are called ohmic media. A
typical example of ohmic media are metals where (1.24) holds in a wide range
of circumstances. However, in other materials, such as semiconductors, (1.24)it may not be applicable.
For most metals σis a scalar with a magnitude that depends on
the temperature and that, at room temperature, has a very high value of the order of 107mho
m−1.Then very often metals are considered as perfect conductors with an in finite conductivity.
The relations between macroscopic quantities, (1.13), (1.20) and (1.24), are called constitutive
relations. Depending on the characteristics of the constitutive macroscopic parameters ε,μand
σ, which are associated with the microscopic response of atoms and molecules in the medium, this
medium can classi fied as:
Nonhomogeneous or homogeneous: according to whether or not the constitutive parameter of
interest is a function of the position, ε=ε(r),μ=μ(r),o rσ=σ(r).
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., EandD; and/or EandJc; and/or
1.2. REVIEW OF MAXWELL’S EQUATIONS 11
BandH. In anisotropic materials the constitutive parameter of interest is a
tensor (see Chapter ??)
Nonlinear or linear: according to whether or not the constitutive parameters
depend on the magnitude of the applied fields. For instance ε(E),μ(H)orσ(E)
en general función de EyB??
Time-invariant: if the constitutive parameters do not vary with time ε6=
ε(t),μ6=μ(t)orσ6=σ(t)
Dispersive: according to whether or not, for time-harmonic fields, the con-
stitutive parameters depend on the frequency, ε=ε(ω),μ=μ(ω)orσ=σ(ω).
The materials in which these parameters are functions of the frequency arecalled dispersive
6.
Magnetic medium: if μ6=μ0. Otherwise the medium is called nonmagnetic
because its only signi ficant 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 a nd isotropic, time-invariant, nondis-
persive and nonmagnetic. Indeed, this assumption is not very restrictive sincemany electromagnetic phenomena c an be studied using this simpli fication. In
fact, even practical cases of the propagat ion of electromagnetic waves through
nonlinear media (semiconductors, ferrites, nonlinear crystals, etc.) are analysedwith linear models using the so-called small-signal approach. Most of thisbook concerns homogeneous, linear, iso tropic and nonmagnetic media, except
in Chapter ??where anisotropic and magnetic ma terials (ferrites) are consid-
ered.
The effect of the properties of a medium on the macroscopic field can be
emphasized by expressing EandBin Maxwell’s equations (1.1a) and (1.1d) by
(1.10) and (1.17). Thus we have
∇·E=ρ
all
ε0=1
ε0³
ρ−∇·P´
(1.25a)
∇×B=μ0Jall+μ0ε0∂E
∂t=μ0(J+∂P
∂t+∇×M)+ε0μ0∂E
∂t
(1.25b)
6Eqs (1.12), (1.19) and (1.24) are strictly valid only for nondispersive media E ffectively, for
example, because of the dependence of the electric permittivity with frequency we generally have
P(ω)=ε0χe(ω)E(ω). Thus, according to the convolution theorem, for arbitrary time dependence
this expression becomes
P(t)=ε0]t
−∞χe(t−t0)E(t0)dt0
Similarly for magnetization and Ohms’ law we have
M(t)=]t
−∞χm(t−t0)H(t0)dt0
J(t)=]t
−∞σ(t−t0)E(t0)dt0
These expressions indicate that, as for any physical system, the response of the medium to an
applied field is not instantaneous.
12 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
In (1.25a) we have explicitly as scalar sources of Eb o t ht h ef r e ec h a r g e ρ
and the polarization or bounded density of charge, −∇·P. Then in (1.9a) we
have
ρall=ρ−∇·P (1.26)
Similarly, in (1.25b), we have, explicitly as vector sources of B, besides the free current density
J( which includes the conduction current density Jc=σE),the polarization current
∂P/∂t (which results from the motion of the bounded charges in dielectrics), the
displacement current in the vacuum, ε0∂E/∂t and the magnetization current,
∇×M(which takes place when a non-uniformly magnetized medium exists).
Then in (1.9d) we have
Jall(r,t)=J+∂P
∂t+∇×M (1.27)
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 (1.13), (1.20), (1.24), in general the fields
E,B,DandHare discontinuous at points where ε,μandσalso are. Hence
thefield 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 di fferent constitutive
parameters ε,μandσwhere surface density of sources may exist along the
boundary.
The boundary condition for Dcan be calculated using a very thin, small pill-
box that crosses the interface of the two media, as shown in Fig. 1.2. Applying
t h ed i v e r g e n c et h e o r e m7to (1.1a) we have
I
D.ds=Z
Base 1D1.ds+Z
Curved surfaceD.ds+Z
Base 2D2.ds=Z
ρdv (1.28)
whereD1denotes the value of Din medium 1,a n dD2the value in medium
2. Since both bases of the pillbox can be made as small as we like, the total
outward flux ofDover them is (Dn1−Dn2)ds=(D1−D2)·ˆnds,w h e r et h e s e Dn
are the normal components of D,dsis the area of each base, and ˆnis the unit
normal drawn from medium 2to medium 1. At the limit, by taking a shallow
enough pillbox, we can disregard the flux over the curved surface, whereupon
the sources of Dreduce to the density of surface free charge ρson the interface
ˆn·(D1−D2)=ρs (1.29)
7El teorema de la divergencia requiere que las propiedades del medio varíen de forma contínua,
pero puede suponerse una transición rápida pero contínua del medio 1 al 2
1.2. REVIEW OF MAXWELL’S EQUATIONS 13
Medium 1ˆnInfinitesimal loop Pilbox
dh
dl ˆnMedium 2dhMedium 1ˆnInfinitesimal loop Pilbox
dh
dl ˆnMedium 2dh
Figure 1.2: Derivation of boundary conditions at the interface of two media.
Pintar solo la ˆnhacia arriba y las dsu n aH c i aa r r i b ayl ad ea b a j oh a c i aa b a j o
Cuidado pilbox es con dos l
Hence the normal component of Dchanges discontinously across the interface by
an amount equal to the free charge surface density ρson the surface boundary.
Similarly the boundary condition for Bcan be established using the Gauss’
law for magnetic fields (1.1b). Since the magnetic field is solenoidal, it follows
that the normal components of Bare continuous across the interface between
two media
ˆn·(B1−B2)=0 (1.30)
The behavior of the tangential components of Ecan be determined using
ai nfinitesimal rectangular loop at the interface which has sides of lengh dh,
normal to the interface, and sides of lengh dlparallel to it (Fig. 1.2). From
the integral form of the Faraday’s law, (1.2c) and de fining ˆtas the unit tangent
vector parallel to the direction of integration on the upper side of the loop, we
have
(E1·ˆt−E2·ˆt)dl+contributions of sides dh
=−∂B
∂t·ds (1.31)
In the limit, as dh→0,the area ds=dldh bounded by the loop approaches
zero and, since Bisfinite, the flux ofBvanishes. Hence (E1−E2)·ˆt=0
and we conclude that the tangential components of Eare continuous across the
interface between two media. In terms of the normal ˆnto the boundary, this
can be written as
ˆn×(E1−E2)=0 (1.32)
Analogously, using the same in finitesimal rectangular loop, it can be deduced
from the generalized Ampère’s law, (1.2d), that
(H1·ˆt−H2·ˆt)dl+contributions of sides dh
=−Ã
∂D
∂t+J!
·ds (1.33)
14 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
where, since Disfinite, 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 σof the medium 2,and consequently J
s,i si n finite. This
requires the surface to be a perfect conductor. Thus
ˆn×(H1−H2)=Js (1.34)
the tangential component of His discontinuous by the amount of surface current
densityJs.F o r finite conductivity, the tangential magnetic field is continuous
across the boundary.
A summary of the boundary conditions, given in (1.35), are particularized
in (1.36) for the case when the medium 2is a perfect conductor ( σ2→∞ ).
General boundary conditions
ˆn×(E1−E2)=0 (1.35a)
ˆn×(H1−H2)=Js (1.35b)
ˆn·(D1−D2)=ρs (1.35c)
ˆn·(B1−B2)=0 (1.35d)
Boundary conditions when the medium 2is a perfect conductor ( σ2→∞ )
ˆn×E1=0 (1.36a)
ˆn×H1=Js (1.36b)
ˆn·D1=ρs (1.36c)
ˆn·B1=0 (1.36d)
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 fieldE×Hin a
homogeneous, linear and isotropic finite region Vbounded by a closed surface S.
If we assume that Vcontains power sources (generators) generating currents
J,then, from Maxwell’s equations (1.1c) and (1.1d), we get
∇·(E×H)=H·∇×E−E·∇×H=−H·∂B
∂t−E·∂D
∂t−E·(σE+J)(1.37)
whereJrepresents the source current density distribution which is the primary
origin of the electromagnetic fields8, while the induced conduction current den-
sity is written as Jc=σE(1.24).
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 15
As the medium is assumed to be linear, the derivates with respect to time
can be written as
E·∂D
∂t=εE·∂E
∂t=∂
∂tµ1
2εE2¶
=∂
∂tµ1
2E·D¶
(1.38a)
H·∂B
∂t=μH·∂H
∂t=∂
∂tµ1
2μH2¶
=∂
∂tµ1
2B·H¶
(1.38b)
By introducing the equalities (1.38a) and (1.38b) into (1.37), integrating over
the volume V, applying the divergence theorem, and then rearranging terms,
we have
Z
VJ·Edv =−∂
∂tZ
V1
2(E·D+B·H)dv−Z
VσE2dv−I
S(E×H)·ds(1.39)
To interpret this result we accept that
Uev=1
2D·E (1.40)
and
Umv=1
2B·H (1.41)
represent, as a generalization of their expression for static fields, the instanta-
neous electric energy density, Uev, and magnetic energy density, Umv,s t o r e di n
the respective fields. Thus according to (1.8) the left side of (1.39) represents
the total electromagnetic power supplie d by all the sources within the volume
V. Regarding the right side of (1.39), the first term represents the change rate
of the stored electromagnetic energy within the volume; the second term repre-sents the dissipation rate of electrom agnetic energy within the volume; and the
third term represents the flow of electromagnetic energy per second (power)
through the surface Sthat bounds volume V.D efining Poynting’s vector Pas
P=E×H (W/m
2) (1.42)
we can write I
S(E×H)·ds=I
SP·ds (1.43)
This equation represents the total flow of power passing through the closed sur-
faceSand, consequently, we conclude that P=E×Hrepresents 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 Scould be added to Pwithout a ffecting
the total flow. Nevertheless, this is a natural interpretation that does not con-
tradict any experience. Only when we tr y to particularize (1.39) to steady fields
do we find ambiguous results, because, in static, the location of the electric and
magnetic energy has no physical signi ficance.
16 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
Note that Eq. (1.39) was deduced by assuming a linear medium and that the
losses occur only through conduction currents. Otherwise the equation shouldbe modi fied to include other kinds of losses such as those due to hysteresis or
possible transformations of the electrom agnetic energy into mechanical energy,
etc. When there are no sources within V, (1.39) represents an energy balance
of that flowing through Sversus that stored and dissipated in V.
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 Vbounded by a surface S,i ti s
necessary to include a term that, by means of the Poynting vector P,t a k e s
into account the flow of power through S. We shall now see that when an
electromagnetic field interacts with the charges and currents in V,i ti sa l s o
necessary to consider a momentum as sociated 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 ρandJas
ρ=∇·D (1.44)
J=∇×H−∂D
∂t(1.45)
so that
f=ρE+J×B=³
∇·D´
E−B×(∇×H)+B×∂D
∂t(1.46)
which, taking into account that
B×∂D
∂t=−∂
∂t(D×B)+D×∂B
∂t=
−∂
∂t(D×B)−D×(∇×E) (1.47)
becomes
f=(∇·D)E−B×(∇×H)−∂
∂t(D×B)−D×(∇×E) (1.48)
By adding the term H(∇·B)=0 to this equality to make the final expres-
sions symmetrical, and by reordering, we can write the Lorentz force densityas
f=E(∇·D)−D×(∇×E)+H∇·B−B×(∇×H)−∂
∂t(D×B)(1.49)
1.4. MOMENTUM OF THE ELECTROMAGNETIC FIELD 17
The component αof Lorentz force density can be written, taking into account
the de finition of the Poynting vector P,a s
fα=εo∂
∂β∙
EβEα−1
2δβαE2¸
+μ0∂
∂β∙
HβHα−1
2δβαH2¸
−1
c2∂
∂tPα(1.50)
whereδβαis the Kronecker delta ( δβα=1ifβ=αand zero if β6=α)a n dt h e
indicesα,β =1,2,3correspond to the coordinates x, y, z, respectively, and we
have made use of the Einstein’s summation convention (i.e., the repetition ofan index automatically implies a summa tion over it). To obtain (1.50) we have
made use of the following equalities
E
α∇·D−D×(∇×E)¯¯¯
α=εo∂
∂β∙
EβEα−1
2δβαE2¸
Bα∇·B−B×(∇×H)¯¯¯
α=μ0∂
∂β∙
HβHα−1
2δβαH2¸
D×B¯¯¯
α=Pα
c2(1.51)
Thefirst two summands in (1.50) constitute the αcomponent of the diver-
gence of a tensor quantity, Tem,such that
(∇·Tem)α=∂Tem
βα
∂β(1.52)
whereTemis a symmetric tensor, known as the Maxwell stress tensor, de fined
by
Tem
βα=εo∙
EβEα−1
2δβαE2¸
+μ0∙
HβHα−1
2δβαH2¸
(1.53)
Therefore, from (1.50) and (1.52), we have
f=∇·Tem−1
c2∂P
∂t(1.54)
with
∇·Tem=∙∂
∂x,∂
∂y,∂
∂z¸⎡
⎣Tem
xxTem
xyTem
xz
Tem
yxTem
yyTem
yz
Tem
zxTem
zyTem
zz⎤
⎦ (1.55)
The components of the electromagnetic tensor Tem
βαcan be written as
Tem
βα=Te
βα+Tm
βα=DβEα−1
2δβαEγDγ+BβHα−1
2δβαHγBγ (1.56)
whereTm
βαandTe
βαrepresent, respectively, the electric and magnetic tensors
defined by
Te
βα =DβEα−1
2δβαEγDγ (1.57)
Tm
βα =BβHα−1
2δβαHγBγ (1.58)
18 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
Integrating (1.50) over the volume Vthe total electromagnetic force Fex-
erted on the volume is
F=Z
Vfdv =Z
V(ρE+J×B)dv=Z
Sfsds−1
c2∂
∂tZ
VPdv (1.59)
wherefsis the force per unit of area on S
fs=Tem·ˆn (1.60)
and we have applied the theorem of divergence to the tensor Temi.e.
Z
V∇·Temdv=Z
STem·ds=Z
STem·ˆnd s =Z
Sfsds (1.61)
Thus Z
Sfsds=F+1
c2∂
∂tZ
VPdv (1.62)
Note that the term
1
c2∂
∂tZ
VPdv (1.63)
is not null even in the absence of charges and currents. Since the only elec-
tromagnetic force possible due to the interaction of the fie l dw i t hc h a r g e sa n d
currents is F,the term (1.63) must represent another physical quantity with
t h es a m ed i m e n s i o n sa saf o r c e ,i . e . ,t h er a t eo fm o m e n t u mt r a n s m i t t e db ythe electromagnetic field to the volume V. T h i si se q u i v a l e n tt oa s s o c i a t i n ga
momentum density gwith the electromagnetic field, given by 1/c
2times the
Poynting vector,
g=P
c2(1.64)
which propagates in the same direction as the flow of energy. Thus, Eq. 1.62
represents the formulation for the momentum conservation in the presence ofelectromagnetic fields.
The momentum of an electromagnetic field, which can be determined ex-
perimentally, is inappreciable under n ormal 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 impliesa 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-harmon ic dependence of the sources gives rise
1.5. TIME-HARMONIC ELECTROMAGNETIC FIELDS 19
tofields which, once having reached the steady state, also vary sinusoidally in
time. However, time-harmonic analysis is important not only because manyelectromagnetic systems operate with signals that are practically harmonic, butalso 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 arelinear differential equations, the total fields can be synthesized from its Fourier
components.
Analytically, the time-harmonic var iation is expressed using the complex
exponential notation based on Euler’s formula, where it is understood that thephysical 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(ωt+ϕ),whereωis the angular frequency, is expressed as
E=R e{~Ee
jωt}=1
2(~Eejωt+(~Eejωt)∗)=E0cos(ωt+ϕ) (1.65)
where~Eis the complex phasor,
~E=E0ejϕ(1.66)
of amplitude E0and phase ϕ, 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(r,ω),a n dρ=
ρ(r,ω).In this way, time-dependent (real) quantities, which are represented by
mathematical symbols not in bold, such as E=E(r,t),a n dρ=ρ(r,t),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. Thereal time-dependent quantity associated with a complex phasor is calculated, as
in (1.65), by multiplying it by e
jωtand taking the real part.
1.5.1 Maxwell’s equations for time-harmonic fields
Assuming ejωttime dependence, we can get the phasor form or time-harmonic
form of Maxwell’s equations simply by changing the operator ∂/∂t to the factor
jωin (1.1a)-(1.2d) and eliminating the factor ejωt. 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) (1.67a)
∇·~B =0 (Gauss’ law for magnetic fields) (1.67b)
∇×~E =−jω~B(Faraday’s law) (1.67c)
∇×~H =~J+jω~D(Generalized Ampère’s law) (1.67d)
Integral form of Maxwell’s eq uations for time harmonic fields
20 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
I
S~D·ds=QT(Gauss’ law) (1.68a)
I
S~B·ds=0 (Gauss’ law for magnetic fields) (1.68b)
I
Γ~E·dl=−jωZ
S~B·ds(Faraday’s law) (1.68c)
I
Γ~H·dl=Z
S(~J+jω~D)·ds(Generalized Ampère’s law) (1.68d)
For time-harmonic fields, expressions (1.25a) and (1.25b) become
∇·~E =ρall
ε0=1
ε0³
ρ−∇·~P´
(1.69a)
∇×~B =jωε 0μ0~E+μ0~Jall=jωε 0μ0~E+μ0(~J+jω~P+∇×~M)
(1.69b)
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 PandE(a phase shift between ~Pand~E), and con-
sequently between EandD, and to a loss of electromagnetic energy as heat in
overcoming the damping forces (see Appendix ??).At the macroscopic level this e ffect
is analytically expressed by means of a complex permittivity, εcas
~D=εc~E (1.70)
with
εc=ε0−jε00=ε0εcr (1.71)
whereεcr
εcr=1+χce=ε0
r−jε00
r (1.72)
is the relative complex permittivity and χce=χ0
cer−jχ00
ceris the complex electric susceptibility.
In general both ε0andε00present a strong frequency dependence and they
are closely related to one another by the Kramer-Kronig relations as is shown
in Appendix ??, 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 ~Eand~Jcor between
~Band~Hwhich, at the macroscopic level, is re flected in the corresponding
complex constitutive parameters σc=σ0−jσ00andμc=μ0−jμ00.
For a medium with complex permittivity, the complex phasor form of the
displacement current is
jω~D=jωεc~E=ωε00~E+jωε0~E (1.73a)
1.5. TIME-HARMONIC ELECTROMAGNETIC FIELDS 21
dδ
G
E
deσ=GG
JE'r jωεG G
=J ΕiG
J
dδ
G
E
deσ=GG
JE'r jωεG G
=J ΕiG
J
Figure 1.3: Induced current density in the complex plane.
while the sum, of the displacement and co nduction current, called total induced
current, ~Ji,i s
~Ji=σ~E+jωεc~E=(σ+ωε00)~E+jωε0~E=~Jd+~Jr (1.74)
where ~Jd, called the dissipative current,
~Jd=(σ+ωε00)~E (1.75)
in phase with the electric field, is the real part of the induced current ~Ji(Fig.
1.3) while ~Jr,called the reactive current,
~Jr=jωε0~E (1.76)
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
~Jd=σe~E (1.77)
whereσeis the effective or equivalent conductivity
σe=σ+ωε00(1.78)
which includes the ohmic losses due to σand the damping losses due to ωε00.
Thus the induced current, (1.74), can be written as
~Ji=σe~E+jωε0~E=σec~E (1.79)
whereσecis the complex e ffective conductivity, de fined as
σec=σe+jωε0(1.80)
Thus a medium with conductivity σecand null permittivity is formally equiva-
lent to one with conductivity and permittivity, σandεc, respectively.
22 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
On the other hand, the phase angle δdbetween 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 reactiv e currents) is called the loss tangent
tanδ
d=σe
ωε0(1.81)
and the induced current, (1.79), can be written in terms of the loss tangent as
~Ji=σec~E=jωε0(1−jσe
ωε0)~E=jωε0(1−jtanδd)~E=jωεec~E (1.82)
whereεecis defined as the e ffective complex permittivity
εec=ε0(1−jtanδd)=ε0εer (1.83)
and
εer=( 1−jtanδd)ε0
r (1.84)
denotes the e ffective relative permittivity. Thus, according to (1.79) and (1.82), a medium
can be formally considered alternatively either as a medium of permittivity ε0and effective con-
ductivityσe, or as a dielectric medium of e ffective permittivity εecor as a conducting medium of
effective conductivity σec. In summary, this possibilities are
Permittivity Conductivity
Original medium εc=ε0−jε00σ
Equivalent medium 1ε0σe=σ+ωε00
Equivalent medium 2εec=ε0−j(ε00+σ
ω)0
Equivalent medium 30 σec=σ+ωε00+jωε0
(1.85)
T h el o s st a n g e n ti se q u a lt ot h ei n v e r s eo ft h eq u a l i t yf a c t o r Qof the dielectric which is a
dimensionless quantity de fined as
Q=ωMaximun energy stored per unit volume
Time average power lost per unit volume=ωWv
P0
dv
(1.86)
The average power dissipated per cycle and unit volume, P0
dv, due both to the
Joule effect and to that of dielectric polarization, is given, according to (1.8),
by
P0
dv =1
TZT
0E·Jidt=1
TZT
0E0cosωt·(σeE0cosωt+ωε0E0sinωt)dt
=1
TZT
0σeE2
0cos2ωtdt =1
TZT
0E·Jddt=σeE2
0
2
(1.87)
1.5. TIME-HARMONIC ELECTROMAGNETIC FIELDS 23
whereT=2π/ω is the period of the signal. Note that only the dissipative part
ofJicontributes to the average power. Of this power, the part corresponding
to polarization losses is
1
TZT
0ωε00E2
0cos2ωtdt =ωε00E2
0
2(1.88)
The maximum electric field energy stored per unit of volume is
Wv=1
2ε0E2
0 (1.89)
Thus, dividing (1.89) by (1.87), we have
Q=ωε0
σe=1
tanδd(1.90)
Although both dimensionless quantities, Qand tanδd,c a nb eu s e dt od e fine
the characteristics of a dielectric, we will use the loss tangent throughout thisbook.
Depending on whether the reactive or the dissipative current is predominant
at the operating frequency, a medium is classi fied as a weakly lossy or a strongly
lossy medium respectively. Thus for weakly lossy media, usually called gooddielectrics or insulators, we have, ωε
0>> σe,s ot h a t
tanδd=σe
ωε0<< 1 (1.91)
Or, ifσ=0,
tanδd=ε00
ε0<< 1 (1.92)
Ifσe=0 (i.e. tanδd=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 ε.
If the medium is strongly lossy we have ωε0<< σe,s ot h a t
tanδd=σe
ωε0>> 1 (1.93)
which for good conductors where ε00=0 ;ε0=εsimpli fies to
tanδd=σ
ωε>> 1 (1.94)
being practically ε=ε0.I fσ=∞(i.e. tanδd=∞) the medium is termed a
perfect conductor.
For a homogeneous conducting medium where ε0andσedo not depend on
the position, Gauss’ law (1.1a) and the continuity equation (1.3) can be writenas
∇·E=ρ/ε
0(1.95)
24 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
and
σe∇·E=−∂ρ
∂t(1.96)
respectively. Hence we have
σeρ
ε0+∂ρ
∂t=0 (1.97)
so that the expression for the decay of a charge distribution in a conductor is
given by
ρ=ρ0e−(σe/ε0)t(1.98)
whereρ0i st h ec h a r g ed e n s i t ya tt i m e t=0.The characteristic time
τ=ε0
σe(1.99)
required for the charge at any point to decay to 1/eof its original value is called
the relaxation time.
For most metals τ=1 0−14s, signifying that in good conductors the charge
distribution decays exponentially so quickly that it may be assumed that ρ=0
at any time. In terms of the relaxation time, the loss tangent can be written as
tanδd=σ
εω=(τω)−1(1.100)
Thus the classi fication 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 di fferent
constitutive parameters ε,μandσ, (1.35a)-(1.36d), become
General boundary conditions
ˆn×(~E1−~E2)=0 (1.101a)
ˆn×(~H1−~H2)= ~Js (1.101b)
ˆn·(~D1−~D2)=ρs (1.101c)
ˆn·(~B1−~B2)=0 (1.101d)
Boundary conditions when the medium 2is a perfect conductor ( σ2→∞ )
ˆn×~E1=0 (1.102a)
ˆn×~H1=~Js (1.102b)
ˆn·~D1=ρs. (1.102c)
ˆn·~B1=0 (1.102d)
1.5. TIME-HARMONIC ELECTROMAGNETIC FIELDS 25
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 Poynting vector over a period, i.e. the
time-average power passing through a unit area perpendicular to the direction
ofP. From (1.65) we have
E=R en
~Eejωto
=1
2³
~Eejωt+(~Eejωt)∗´
(1.103a)
H =R en
~Hejωto
=1
2³
~Hejωt+(~Hejωt)∗´
(1.103b)
Thus, the instantaneous Poynting vector (1.42) can be written as
P=E×H=R e{~Eejωt}×Re{~Hejωt}
=1
2Re{~E×~H∗+~E×~He2jωt} (1.104)
where we have made use of the general relation for any two complex vectors A
andB
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
4³
~A×~B∗+³
~A×~B∗´∗´
+1
4³
~A×~B+³
~A×~B´∗´
=1
2Re{~A×~B∗+~A×~B} (1.105)
The time-average value of the instantaneous Poynting vector can be calcu-
lated integrating (1.104) over a period , i.e.,
Pav =1
TZT
0Pdt=1
2TZT
0Re{~E×~H∗+~E×~He2jωt}dt
=1
2Re{~E×~H∗}=1
2Re{Pc} (1.106)
since the time average of ~E×~He2jωtvanishes. The magnitude
Pc=~E×~H∗(1.107)
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
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 la w, (1.67c), and from Ampère’s general
law, (1.67d), in its conjugate complex form, we have
∇×~E =−jωμ~H (1.108a)
∇×~H∗=−jωε~E∗+~J∗+σ~E∗(1.108b)
26 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
where ~J∗represents the complex conjugate of the current supplied by the
sources. Performing a scalar multiplication of Eq. 1.108a by ~H∗and of Eq.
1.108b by ~E, and subtracting the results, we get
∇·³
~E×~H∗´
=~H∗·∇×~E−~E·∇×~H∗
=−jω¡
μH2
0−εE2
0¢
−~E·(~J∗+σ~E∗)(1.109)
where it has been taken into account that ~H·~H∗=H2
0and~E·~E∗=E2
0,
withH0andE0being the amplitude of the two harmonic fields. After dividing
(1.109) by 2we get
∇·µ1
2~E×~H∗¶
=−2jωµ
μH2
0
4−εE2
0
4¶
−σE2
0
2−1
2~J∗·~E (1.110)
The terms μH2
0/4andεE2
0/4represent, respectively, the mean density of the
magnetic and electric energy, while σE2
0/2is the the mean power transformed
into heat9withinV, since the mean value of the square of a sine or cosine
function is 1/2.
By multiplying Equation (1.110) by the volume element dv,i n t e g r a t i n go v e r
an arbitrary volume Vand applying the divergence theorem, we obtain the
complex version of the Poynting theorem
Z
V1
2³
~J∗·~E´
dv =−Z
VσE2
0
2dv−2jωZ
Vµ
μH2
0
4−εE2
0
4¶
dv
−Z
S1
2³
~E×~H∗´
·ds (1.111)
which is the expression corresponding to (1.39) in complex notation and where
thefirst member represents the power supplied by external sources. By sepa-
rating the real and imaginary parts, we obtain the following two equalities
Z
VRe1
2(~J∗·~E)dv =−Z
VσE2
0
2dv−Z
SRe1
2(~E×~H∗)·ds (1.112a)
Z
VIm1
2(~J∗·~E)dv =−2ωZ
Vµ
μH2
0
4−εE2
0
4¶
dv−Z
SIm1
2(~E×~H∗)·ds
(1.112b)
Thefirst member of (1.112a)
Pa=Z
VRe1
2(~J∗·~E)dv (1.113)
represents the active mean power supplied by all the sources within V.O nt h e
right-hand side of (1.112a) the first integral, as commented above, gives the
9Expression (1.112b) can be easily extended to the case of lossy dielectric just substituting σby
the equivalent conductivity σedefin e di n( 1 . 7 8 ) a n d εbyε0defin e di n( 1 . 7 1 ) .
1.6. ON THE SOLUTION OF MAXWELL’S EQUATIONS 27
power transformed into heat within V, while the surface integral represents the
mean flow of power through the surface S.
Regarding to expression (1.112b), the first member
Pr=Z
VImµ1
2~J∗·~E¶
dv (1.114)
is called the reactive power of the sources. On the right-hand side the first
summand is 2ωtimes the di fference 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 S.I f t h e s u r f a c e
integral in (1.112a) is non-zero, the external region is said to be an active chargefor the sources within V. Similarly, if the surface integral of Eq. (1.112b) is non-
zero, the external region is said to be a reactive charge for the sources within V.
In general, both of these surface integr als 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, Maxw ell’s equations are in general not easy
to solve. In fact, even in the most favorable situation of homogeneous, linear
and isotropic media, there are not many problems of interest that can be analyt-
ically solved except for those presenting a high degree of geometrical symmetry.Moreover, the frequency range of scienti fic and technological interest can vary
by many orders of magnitude, expanding from frequency values of zero (or very
low) to roughly 10
14Hertz . The behavior and values of the constitutive para-
meters can change very signi ficantly in this frequency. range. Conductivity, for
example, can vary from 0to107Sm−1. It is even possible to build arti ficial
materials, called metamaterials, which present electromagnetic properties that
are not found in nature. Examples of such as metamaterials are those char-
acterized with both negative permittivity ( ε< 0) and negative permeability
(μ< 0). These media are called DNG (doubl e-negative) metamaterials and,
owing to their unusual electromagnetic properties, they present many potential
technological applications.
Another important factor to study th e interaction of an electromagnetic field
with an object is the electrical size of the body, i.e., the relationship betweenthe 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 complexproblems 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
28 CHAPTER 1. ELECTROMAGNETIC FIELD FUNDAMENTALS
Chapter 2
Fields created by a source
distribution: retardedpotentials
In this chapter, we introduce the scalar electric and magnetic vector potentials
as magnitudes that facilitate the calculation of the fields created by a bounded-
source distribution, paying special attention to the radiation field. Finally, we
extend Maxwell’s equations, in order to make them symmetric, by introducingthe concept of magnetic charges and currents.
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-i nvariant, linear and isotropic medium.
Figure 2.1 represents such a distribution, where, as usual, the coordinates asso-ciated with source points, J=J(r
0,t0),ρ=ρ(r0,t0), are designated by primes,
while those associated with field points or observation points P(r,t)are without
primes. In the following, we will assume the medium surrounding the sourcedistribution to be free space, i.e. μ=μ
0,ε=ε0, although of course all the re-
sulting formulas remain valid for media of constant permittivity and permeabil-
ity, provided that ε0is replaced by εrε0andμbyμrμ0. 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-
tialΦ=Φ(r,t)and the magnetic vector potential A=A(r,t)(Fig. 2.2). Once
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 fieldBis always
29
30CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
'dV 'V'rrRr
rr
OP
(' , ) ; (' , )Jr t r tρrrr
S
ˆrl
θ
'dV 'V'rrRr
rr
OP
(' , ) ; (' , )Jr t r tρrrr
S
ˆrl
θ
Figure 2.1: Time-varying source bounded distribution V0of maximun dimension l.T h e c o o r -
dinates associated with source points of currents and charges J=J(r0,t0),a n dρ=ρ(r0,t0),
respectivelly are designated by primes, while the associate with field points, P(r,t),a r ew i t h o u t
primes.
(' , ) ,(' , )rtJ rtρrrr;EBrr
,AΦr(' , ) ,(' , )rtJ rtρrrr;EBrr
,AΦr
Figure 2.2: Quitar o poner argumentos pero uni ficar
2.1. ELECTROMAGNETIC POTENTIALS 31
zero, we can express it as the curl of an electromagnetic vector potential Aas
B=∇×A (2.1)
Inserting this expression into (1.1c) we get
∇×µ
E+∂
∂tA¶
=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
E+∂
∂tA=−∇Φ (2.3)
or
E=−∇Φ−∂A
∂t(2.4)
where∂A/∂t is the nonconservative part of the electric field with a non-vanishing
curl. When the vector potential Ais independent of time, expression (2.4)
reduces to the familiar E(r)=−∇Φ(r).
According to the relations (2.1) and, (2.4) the fieldsBandEare completely
determined by the vector and scalar potentials AandΦ.However, the fields
do not uniquely determine the potentials. For instance, it is clear that thetransformation
A=A
0+∇Ψ (2.5)
whereΨ=Ψ(r,t)is any arbitrary, single-valued, continuously di fferentiable,
scalar function of position and time that vanishes at in finity, leaves Bunchanged
B=∇×A=∇×A0+∇×∇Ψ=∇×A0(2.6)
Inserting (2.5) into (2.4), it follows that
E=−∇µ
Φ+∂Ψ
∂t¶
−∂A0
∂t(2.7)
so that the value of E, obtained from A0, also remains unchanged provided that
Φis replaced by the scalar potential
Φ0=Φ+∂Ψ
∂t(2.8)
Thus different sets of potentials AandΦgive rise to the same set of fields1B
andE. The joint transformation (2.5) and (2.8) leaves the electromagnetic field
1The liberty to select the value of Ais understandable taking into account that by (2.1) the
magnetic fieldfixes only ∇×A. However, Helmholtz’s theorem posits that, to determine the
(spatial) behavior of Acompletely, ∇.A(which is still undetermined) must also be speci fied.
Thus, we can choose it in any way we consider suitable for facilitating the calculation of theelectromagnetic fields.
32CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
invariant. The di fferent forms of choosing the potentials AandΦleaving the
fields unchanged are called gauge transformations, and the function Ψis called
the gauge function. The degree of freed om provided by the gauge transforma-
tions facilitates the calculation of the potentials and hence of the fields because,
once the potentials are known, the fields are easily derived by di fferentiation
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 gener alized Ampère’s law, (1.1d), and Gauss’
law, (1.1a), using ( ??) and rearranging terms, we get two, coupled, second-order
partial-di fferential equations
μ0ε0∂2A
∂t2−∇2A=μ0J−∇∙
∇·A+μ0ε0∂Φ
∂t¸
(2.9a)
μ0ε0∂2Φ
∂t2−∇2Φ=ρ
ε0+∂
∂t∙
∇·A+μ0ε0∂Φ
∂t¸
(2.9b)
These equations could be considerably simpli fied if we could force (without
changing the fields) the potentials to satisfy the auxiliary relation
∇·A+μ0ε0∂Φ
∂t=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 ful fil the Lorenz condition and consequently simplify Eqs (2.9) to
the inhomogeneous Helmholtz wave equations
μ0ε0∂2A
∂t2−∇2A=μ0J (2.11a)
μ0ε0∂2Φ
∂t2−∇2Φ=ρ
ε0(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 ??).
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,
A0andΦ0,w h i c h f u l filE q u a t i o n s( ??)a n d( ??) and check whether it is possible
2A very interesting property of the Lorenz condition is that, as shown in ( ??), 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 33
to select them so that they satisfy equations (2.11). By inserting (2.5) and (2.8)
into (2.9) and by rearranging, we get
∇2A0−μ0ε0∂2A0
∂t2=−μ0J+∇µ
∇·A0+∇2Ψ+μ0ε0∂Φ0
∂t−μ0ε0∂2Ψ
∂t2¶
(2.12a)
∇2Φ0−μ0ε0∂2Φ0
∂t2=−ρ
ε0−∂
∂tµ
∇·A0+∇2Ψ+μ0ε0∂Φ0
∂t−μ0ε0∂2Ψ
∂t2¶
(2.12b)
and, given that the scalar function Ψis arbitrary, we can choose it as the solution
to the di fferential equation
∇2Ψ−μ0ε0∂2Ψ
∂t2=−∇·A0−μ0ε0∂Φ0
∂t(2.13)
Thus (2.12a) and (2.12b) become (2.11a) and (2.11b), respectively, meaning
thatA0andΦ0fulfil 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 c=1/√
μ0ε0'3×108m/s of light in
free space. They take the form
∇2A−1
c2∂2A
∂t2=¤A=−μ0J (2.14a)
∇2Φ−1
c2∂2Φ
∂t2=¤Φ=−ρ
ε0(2.14b)
where the symbol ¤represents the D’Alembertian operator de fined by
¤≡∇2−1
c2∂2
∂t2(2.15)
The Lorenz gauge (2.10) for harmonic fields simpli fies to
∇·~A+jω
c2Φ=0 (2.16)
such that
Φ=jc2∇·~A
ω(2.17)
while (2.14a) and (2.14b) simplify to
∇2~A+ω2
c2~A =−μ0~J (2.18a)
∇2Φ+ω2
c2Φ =−ρ
ε0(2.18b)
34CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
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
∇·A=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 seethat with Coulomb’s gauge the expressions for the potential wave equations are
∇
2Φ=−ρ
ε0(2.19)
∇2A−1
c2∂2A
∂t2−1
c2∇∂Φ
∂t=−μ0J (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 equationsimilar to Poisson’s expression in electrostatics. The vector potential, however,
is considerably more di fficult to calculate. According (2.19), a time change in ρ
implies an instantaneous change in Φ. This fact denotes the non-physical nature
ofΦsince 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 EandHfields 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 homoge neous wave equations. The next section
is devoted to finding these particular solutions, which express the potentials in
terms of integrals over the source distributions Jandρ.
2.2 Solution of the inhomogeneous wave equa-
tion for 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 isot ropic medium of conductivity zero that
we assume to be free space (Fig. 2.1). From (2.14) we see that the scalar po-tentialΦas well as each of the three components A
i,(i=1,2,3), of the vector
potential Asatisfy inhomogeneous scalar wave equations with the general form
¤Ψ(r,t)=∇2Ψ(r,t)−1
c2∂2Ψ(r,t)
∂t2=−g(r,t) (2.21)
where the operator ¤acts on the coordinates r,tof the field point, while the
sources coordinates are r0,t0.
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. SOLUTION OF THE INHOMOGENEO US WAVE EQUATION FOR POTENTIALS 35
g(r,t)as constructed from a sum of weighted space-time Dirac delta function
sources, i.e.,
g(r,t)=Zt
t0=−∞Z
V0g(r0,t0)δ(r−r0)δ(t−t0)dv0dt0(2.22)
whereV0is a volume containing all the sources. Thus, (2.21) can be solved in
two steps, using Green’s method in the time domain, as follows.
a) The first step is to calculate the response, G(r,r0,t,t0),generated by the
space-time Dirac δ−function source, δ(r−r0)δ(t−t0),located at position r0and
applied at time t0which obeys the inhomogeneous wave equation
¤G(r,r0,t,t0)=∇2G(r,r0,t,t0)−1
c2∂2G(r,r0,t,t0)
∂t2=−δ(r−r0)δ(t−t0)
(2.23)
and satis fies the boundary conditions of the problem. The function G(r,r0,t,t0)
is called Green’s free-space function, which, because of the homogeneity of the
space, must be a spherical wave centred at position r0at timet0. This function
depends on the relative distance, R=|r−r0|, between the point source and
the observation or field point and on the time di fferenceτ=t−t0.T h u s
G(r,r0,t,t0)=G(R,τ)and (2.23) can be written, using spherical coordinates,
as
¤G(R,τ)=1
R∂2(RG)
∂R2−1
c2∂2G
∂τ2=−δ(R)δ(τ) (2.24)
where, ( ??),
∇2G=1
R∂2(RG)
∂R2(2.25a)
∂2G
∂τ2=∂2G
∂t2(2.25b)
b) The second step is to findΨ(r,t)from Green’s function. Owing to the
linearity of the problem and, from (2.22), if the solution of (2.23) is G, then the
solution of (2.21) is3
Ψ=Zt
t0=−∞Z
V0g(r0,t0)G(R,τ)dv0dt0. (2.26)
Because Gfulfils the boundary conditions, so too does Ψ(r,t).
Tofind the Green’s function let us consider first a general point R6=0such
that equation (2.24) simpli fies to
¤G(R,τ)=1
R∂2(RG)
∂R2−1
c2∂2G
∂τ2=0. (2.27)
Multiplying this equation by Rand de finingG0=RGwe have the homogeneous
wave equation
∂2G0
∂R2−1
c2∂2G0
∂τ2=0 (2.28)
3Note that Eq. (2.27) represents the spatial and temporal convolution of g(r,t)andG(r,t)
36CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
The general solution of the above expression, as can be veri fied by direct sub-
stitution, is
G0(R,τ)=f(τ−R/c)+h(τ+R/c) (2.29)
wheref(τ−R/c)andh(τ+R/c)are two arbitrary functions of their respective
arguments and they represent waves propagating along Rin the positive and
negative directions, respectively. Therefore
G(R,τ)=f(τ−R/c)
R+h(τ+R/c)
R(2.30)
The potential that results from s ubstituting Green’s function h(τ+R/c)/Rin
(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 clearlynot consistent with our ideas about causality, according to which the potentialat(t,r)can depend only on sources at earlier times. Thus, in (2.30) we must
consider only the retarded f(τ−R/c)/Rsolution as physically meaningful.
To determine f(τ−R/c)/R, we integrate the di fferential equation (2.23) in
a very small volume around the singular point R=0. Thus, taking into account
that forR→0the function Gbehaves as f(τ)/R,w eh a v e
Z
V0µ
∇2G(R,τ)−1
c2∂2G(R,τ)
∂τ2¶
R→0dv0(2.31)
=Z
V0µ
∇2µf(τ)
R¶
−1
c2∂2
∂τ2µf(τ)
R¶¶
dv0(2.32)
=−Z
V0δ(r−r0)δ(τ)dv0=−δ(τ) (2.33)
or, since ∇2(1/R)=−4πδ(R)anddv0=4πR2dR,
−Z
V04πf(τ)δ(R)dv0+4π
c2Z
V0R∂2f(τ)
∂τ2dR=−δ(τ) (2.34)
AsR→0, the second integral can be eliminated and therefore
f(τ)=δ(τ)
4π(2.35)
As the function fdepends on τ−R/c andf(τ)=f(τ−R/c)|R=0,w eh a v e
f(τ−R/c)=δ(τ−R/c)
4π(2.36)
and the solution of (2.24) is given by
G(R,τ)=δ(τ−R/c)
4πR=δ(t−t0−R/c)
4πR(2.37)
2.2. SOLUTION OF THE INHOMOGENEO US WAVE EQUATION FOR POTENTIALS 37
This is Green’s time-dependent retarded function, which takes into account the
time needed for the electromagnetic perturbation to reach the observation pointfrom the point source. Substituting this function in (2.26), we have
Ψ(r,t)=Z
t
t0=−∞Z
V0g(r0,t0)δ(τ−R/c)
4πRdv0dt0. (2.38)
and, integrating in t0,w e finally find that, under the assumption of causality,
the solution of the inhomogeneous wave equation for potentials is given by
Ψ(r,t)=1
4πZ
V0g(r0,t−R/c)
Rdv0=1
4πZ
V0[g]
Rdv0. (2.39)
where
[g]=g(r0,t−R
c)=g(r0,t0) (2.40)
is the value of the source densities evaluated at the retarded times t0=t−R/c,
w h i c hi ng e n e r a la r ed i fferent for each source point, R/c 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(r,t)=1
4πε0Z
V0[ρ]
Rdv0(2.41a)
A(r,t)=μ0
4πZ
V0[J]
Rdv0(2.41b)
where the bracket symbol []indicates that the enclosed magnitude must be
evaluated at the retarded time t0=t−R/c.T h a ti s
[ρ]=ρ(r0,t0)=ρ(r0,t−R/c) (2.42)
[J]=J(r0,t0)=J(r0,t−R/c) (2.43)
are the charge and current densities, respectively, evaluated in the retarded
timest0.
Expressions (2.41a) and (2.41b), which are called retarded potentials, in-
dicate that the potentials created by a distribution at the field point Pare
determined, at a given time t,by the values of the the charge and current den-
sities at the source points evaluated at previous times t0,w h i c hg e n e r a l l yd i ffer
for each source poin. It is easy to check that these potentials, together with the
continuity equation, verify Lorenz’s condition (2.10).
It should be noted that (2.39) is a particular solution of (2.21), to which a
complementary solution of the homogeneous wave equation ¤Ψ(r,t)=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)
38CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
is (2.39). These conditions, establishi ng the uniqueness of (2.39), can be found
in Appendix ??.
For sources with time-harmonic dependence
ρ(r0,t)=R e©
ρ(r0)ejωt)ª
(2.44)
J(r0,t)=R e {~J(r0)ejωt} (2.45)
the expressions of the retarded potentials ΦandAsimplify to
A(r,t)=μo
4πRe½Z
V01
R~J(r0)ejω(t−R
c)dv0¾
=R e{~A(r)ejωt}(2.46a)
Φ(r,t)=1
4πεoRe½Z
V01
Rρ(r0)ejω(t−R
c)dv0¾
=R e©
Φ(r)ejωtª
(2.46b)
where
~A(r)=μo
4πZ
V01
R~J(r0)e−jkRdv0(2.47a)
Φ(r)=1
4πεoZ
V01
Rρ(r0)e−jkRdv0(2.47b)
wherek=ω/c =2π/λ is the wavenumber in the unbounded medium and λis
the wavelength in the medium. For harmonic signals the time delay R/c, when
multiplied by ω, becomes a phase shift given by kR.
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
fieldfirst and for the electric field afterwards4.
Magnetic field
Starting from the equation
B=∇×A=μo
4πZ
V0∇×[J]
Rdv0(2.48)
and transforming the integrand by the vector analysis formulas ( ??)a n d( ??)
of Appendix ??,w i t hΨ=1/RandA=J, we can directly find the magnetic
field equation
B(r,t)=μo
4πZ
V0⎛
⎝[J]×R
R3+1
ch
∂J
∂ti
×R
R2⎞
⎠dv0(2.49)
4An alternative way of obtaining the electromagnetic fields is indicated in Section ??.
2.3. ELECTROMAGNETIC FIELDS FROM A BOUNDED SOURCE DISTRIBUTION 39
where [J]is the retarded current density at the source point r0andh
∂J/∂ti
=
∂[J]/∂t0=∂[J]/∂tis its time derivative at the instant t0=t−R/c.
Expression (2.49) can be written as the sum of the two components B=
Bbs+Brad,w h i c ha r ed e fined below.
The Biot-Savart term, Bbs:
Bbs=μo
4πZ
V0[J]×R
R3dv0(2.50)
which is formally analogous to the Biot-Savart expression of magnetostatics,
although here with the sources evaluated at the retarded times. As this termdecreases with 1/R
2, its contribution is appreciable only at short distances.
The radiation term, Brad:
Brad=μo
4πcZ
V0h
∂J
∂ti
×R
R2dv0(2.51)
which depends on 1/R, 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 (2.49) simpli fies to the Biot-Savart
expression of magnetostatics
Bbs=μo
4πZ
V0J×R
R3dv0(2.52)
Electric field
From (2.4) and (2.41) we see that
E=−1
4πε0Z
V0∇[ρ]
Rdv0−μ0
4πZ
V0∂
∂t[J]
Rdv0(2.53)
Taking into account that ∂/∂t0=∂/∂t and that ∇Ψ(R)=(dΨ/dR)∇Rwe
have
∇[ρ]
R=[ρ]∇1
R+1
R∇[ρ]=[ρ]Ã
−R
R3!
+R
R2∂[ρ]
∂R(2.54a)
∂[ρ]
∂R=∂[ρ]
∂t0dt0
dR=∙∂ρ
∂t¸µ
−1
c¶
(2.54b)
∇[ρ]
R=[ρ]Ã
−R
R3!
−R
cR2∙∂ρ
∂t¸
(2.54c)
40CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
which, when substituted in (2.53), and taking into account the continuity equa-
tion∇·J=−∂ρ/∂t ,g i v e s
E(r,t)=1
4πε0Z
V0⎛
⎝[ρ]R
R3−1
c2h
∂J
∂ti
R−R
R2ch
∇0·Ji⎞
⎠dv0(2.55)
To get the exact form of the radiation term, which depends on the distance as
1/R, we need to transform the integrand of this expression by developing ∇0·[J]
as5
[∇0·J]=∇0·[J]−R·[∂J
∂t]
cR(2.56)
thus we can rewrite the third term on the right-hand side of (2.55) as
−Z
V0R
R2ch
∇0·Ji
dv0=−Z
V0R
R2c∇0·[J]dv0+Z
V0³h
∂J
∂ti
·R´
R
c2R3dv0(2.57)
The calculation of the first term on the right-hand side can be facilitated by
calculating just one component, for example the xcomponent
−Z
V0Rx
R2c∇0·[J]dv0=Z
V0[J]
c·∇0Rx
R2dv0−Z
V0∇0·µRx
R2c[J]¶
dv0
=Z
V0[J]
c·∇0Rx
R2dv0
=Z
V0[J]
R2c·∇0Rxdv0+Z
V0Rx[J]
c·∇01
R2dv0
=Z
V0⎛
⎝−[Jx]
cR2+2³
[J]·R´
Rx
cR4⎞
⎠dv0(2.58)
w h e r ew eh a v eu s e d( ??), applied the divergence theorem, and integrated over
an external surface that encloses the sources in which [J]=0 . Therefore,
generalizing to three dimensions and inserting the result in (2.55), we get
4πεoE=Z
V0[ρ]R
R3dv0+Z
V0⎛
⎝2³
[J]·R´
R−[J]³
R·R´
cR4⎞
⎠dv0+
+1
c2Z
V0³h
∂J
∂ti
×R´
×R
R3dv0(2.59)
5∇0·[J]=(∇0·J)t0+∂[J]
∂t0·∇0t0=(∇0·J)t0−∂[J]
∂t0·∇t0=(∇0·J)t0+R
cR·∂[J]
∂t0
Thus
(∇0·J)t0=[∇0·J]=∇0·[J]−R·∂[J]
∂t0
cR
2.3. ELECTROMAGNETIC FIELDS FROM A BOUNDED SOURCE DISTRIBUTION 41
which can be expressed as the sum of the three components E=Ec+Ei+Erad,
which are de fined below.
Coulomb’s term, Ec,
Ec=1
4πεoZ
V0[ρ]R
R3dv0(2.60)
This term is similar to the static Coulom b’s expression except concerning the
time delay.
Induction term, Ei,
Ei=1
4πεoZ
V0⎛
⎝2³
[J]·R´
R
cR4−[J]
cR2⎞
⎠dv0(2.61)
Because of their dependence on 1/R2, the contribution to the field of the terms
(2.60) and (2.61) decrease quickly with distance.
Radiation term, Erad,
Erad =1
4πεoc2Z
V0³h
∂J
∂ti
×R´
×R
R3dv0=μ0
4πZ
V0³h
∂J
∂ti
×R´
×R
R3dv0
(2.62)
This term, which depends on 1/R, is the electric field component that predom-
inates for long distances. Together with (2.51), this component is of interest in
radiation phenomena (see next subsection) .
At the static limit, expression (2.59) simpli fies to Coulomb’s expression of
electrostatics
E=1
4πεoZ
V0ρR
R3dv0(2.63)
Alternatively, the electric field can be expressed only in terms of the current
density, by using the continuity equation. In fact, from (2.56) we have
[ρ]=−Zt
−∞[∇0·J]dt0=−Zt
−∞Ã
∇0·[J]−R·[∂J
∂t]
cR!
dt0(2.64)
Inserting (2.64) into (2.59) and operating in a similar way to (2.58), we obtain
another alternative expression for the electric field created by a bounded source
distribution
E=1
4πεoZ
V0Zt
−∞⎛
⎝3³
[J]·R´
R
R5−[J]
R3⎞
⎠dt0dv0
+1
4πεoZ
V0⎛
⎝3³
[J]·R´
R
cR4−[J]
cR2⎞
⎠dv0
+1
4πεo1
c2Z
V0³h
∂J
∂ti
×R´
×R
R3dv0(2.65)
42CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
Fields created by a time-harm onic source distribution
For time-harmonic dependence of the sources, the field expressions (2.4) and
(2.1) simplify to
~B =∇×~A (2.66a)
~E =−∇Φ−jω~A (2.66b)
and equation (2.49) for the magnetic field becomes
~B=μo
4πZ
V0(~J×R)µ1
R3+jk
R2¶
e−jkRdv0
while the di fferent expressions for the electric field, (2.55), (2.59) and (2.65)
become, respectively,
~E =1
4πεoZ
V0ρe−jkRR
R3dv0+
jk
4πεoZ
V0Ã
ρR
R−~J(r0)
c!
e−jkR
Rdv0(2.68a)
~E =1
4πεoZ
V0ρR
R3e−jkRdv0+1
4πεoZ
V0⎛
⎝2³
~J·R´
R
cR4−~J
cR2⎞
⎠e−jkRdv0+
+jk
4πεocZ
V0³
~J×R´
×R
R3e−jkRdv0(2.68b)
~E =j
4πωεoZ
V0⎛
⎝~J
R3−3³
~J·R´
R
R5⎞
⎠e−jkRdv0+
1
4πεoZ
V0⎛
⎝3³
~J·R´
R
cR4−~J
cR2⎞
⎠e−jkRdv0+
+jk
4πεocZ
V0³
~J×R´
×R
R3e−jkRdv0(2.68c)
and the radiation fields (2.51) and (2.62) become
~B =jωμo
4πcZ
V0~J×R
R2e−jkRdv0(2.69a)
~E =jωμo
4πZ
V0³
~J×R´
×R
R3e−jkRdv0(2.69b)
2.3. ELECTROMAGNETIC FIELDS FROM A BOUNDED SOURCE DISTRIBUTION 43
2.3.1 Radiation fields
Examining the total fields (2.49) and (2.65) generated by a bounded distribution
of sources with arbitrary time dependence, we find that in general the near-zone
terms, which depend on 1/Rn(n> 1), are negligible compared to the radiation
terms, (2.51) and (2.62), which depend on 1/R, when the condition
R> >c¯¯¯[J]¯¯¯
¯¯¯d[J]/dt¯¯¯(2.70)
is fulfilled for any of the in finitesimal volume elements into which the source can
be subdivided. For time-harmonic fields, this condition becomes
R> >λ (2.71)
Hence, the radiation term predomin ates 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 atlower 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 r> >r
0,
or equivalently r> >l ,w h e r elis the largest dimension of the source distribu-
tion, then it is possible to make some gen eral approximations in the expressions
(2.51) and (2.62) which greatly simplify the calculations. To con firm this, let
us write Rin Fig. 2.1 as
R=|r−r0|=¡
r2−2r·r0+r02¢1/2(2.72)
Since the reference origin is close to or within the source distribution, we can
calculate the radiation fields at distances r> >r0by expanding the binomial
(2.72) as a series in powers of the small parameter r0/rand take only the linear
terms of the expansion
R=rµ
1−2r·r0
r2+r02
r2¶1/2
=r−r·r0
r+...'r−r0·ˆr=r−r0cosθ
(2.73)
whereθis the angle between ˆrandr0. This approximation is equivalent to
considering that, far away from the sources, randRbecome parallel.
Thus, as r0/r << 1, in the expressions (2.51) and (2.62), we can make the
approximation
R'r (2.74)
in the denominator. This is equivalent to ignoring, in the modulus of the con-
tribution of each source point to the total field, the di fference in the distance
44CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
travelled by the signal. Thus (2.70) becomes
r> >c¯¯¯[J]¯¯¯
¯¯¯d[J]/dt¯¯¯(2.75)
and (2.71) becomes
r> >λ (2.76)
In the retarded time, t0=t−R/c, the approximation (2.74) is not valid
because the sources can be very sensitive to small changes in the delay timeR/c. Thus, for the delay time, at distances r> >r
0we need to keep at least
the two linear terms of the expansion (2.73). Therefore
t0=t−R
c=t−r
c+r0·ˆr
c=t0
0+r0cosθ
c(2.77)
wheret0
0=t−r/c.
Therefore, from (2.77), the retarded time has two components. One, r/c,i s
the time needed for the electromagnetic field to reach the field point from the
origin of the coordinates. The other, r0·ˆr/c, represents the time necessary for
the propagation of the electromagnetic pe rturbation within the geometric limits
of the source distribution. This term, given that the largest dimension of thesource distribution is l, (Fig. 2.1), has a magnitude of
r
0·ˆr/c∼l/c << r/c (2.78)
Hence, using the approximations (2.74) and (2.77) the integrands of the
radiation fields (2.51) and (2.62) simplify to
Brad =μo
4πcrZ
V0∂J(r0,t0
0+r0·ˆr
c)
∂t׈rdv0(2.79a)
Erad =1
4πεoc2rZ
V0Ã
∂J(r0,t0
0+r0·ˆr
c)
∂t׈r!
׈rdv0(2.79b)
or, for time-harmonic dependence,
~Erad =jωμo
4πrZ
V0³
~J׈r´
׈re−jkRdv0(2.80a)
~Brad =jωμo
4πcrZ
V0~J׈re−jkRdv0=jkηo
4πcrZ
V0~J׈re−jkRdv0(2.80b)
A comparison of Eqs. (2.79a) and (2.79b), shows that the radiation fields
are perpendicular to each other and to the direction of propagation. They arerelated by
E=η
0H׈r (2.81)
2.3. ELECTROMAGNETIC FIELDS FROM A BOUNDED SOURCE DISTRIBUTION 45
where the ratio η0is defined as
η0=E
H=(μo/εo)1
2= 120πΩ (2.82)
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 Erad×Hradis given
by
Zt
−∞Praddt=Zt
−∞(Erad×Hrad)dt (2.83)
and the total flow of power passing through the closed surface Ssituated in the
far-field zone is
Zt
−∞Z
SPrad·dsdt=Zt
−∞Z
S(Erad×Hrad)·dsdt (2.84)
In summary, the assumptions involved in using (2.79) and (2.80) to calculate
the radiation fields created by a bounded source distribution in the far- field zone
are:
a)r> > (c|[J]|/|d[J]/dt|)or, equivalently, r> >λ for any wavelength of
the radiation spectrum which allows us to neglect 1/r2terms.
b)r> >l ,w h e r elis the largest dimension of the source distribution which
allows us to make the approximations (2.74) and (2.77).
2.3.2 Fields created by an in finitesimal current element
The simplest case of a bounded source distribution is that of an in finitesimal
current element i(t),w h i c hi sa s s u m e dt ob eo r i e n t e do nt h e zaxis (Fig. 2.3)
and to have arbitrary time dependence. This current is mathematically de fined,
in terms of the Dirac delta function, as
J(r,t)=i(t)δ(x0)δ(y0)ˆz−∆z
2<z0<∆z
2(2.85)
Thefields of this current element can be easily calculated by substituting (2.85)
in (2.49) and (2.65). Thus, we have
46CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
z
/2zΔθ
()it
/2z−Δy
xEθHϕ
ˆrz
/2zΔθ
()it
/2z−Δy
xEθHϕ
ˆr
Figure 2.3: Infinitesimal current element solo campos de radiacion ¡ ¡¡falta la rdel
radio vector del punto campo
Fields created by an in finitesimal current eleme nt with arbitrary-time
dependence:
H(r,t)=∆z
4πµ1
crd[i]
dt+[i]
r2¶
(ˆz׈r)=
∆z
4πµ1
crd[i]
dt+[i]
r2¶
sinθˆϕ (2.86a)
E(r,t)=∆z
4πεoµ1
r3Zt
−∞[i]dt+[i]
cr2¶
(3 (ˆz·ˆr)ˆr−ˆz)+
∆z
4πεo1
c2rd[i]
dt(ˆr×(ˆr׈z)) =∆z
4πεoµ1
r3Zt
−∞[i]dt+[i]
cr2¶
(2 cosθˆr+s i nθˆθ)+
∆z
4πεo1
c2rd[i]
dtsinθˆθ (2.86b)
where [i]=i(t−r/c).
For time-harmonic dependence of the current element, i=R e©
Iejωtª
,e q u a -
tions (2.86a) and (2.86b) simplify to
2.3. ELECTROMAGNETIC FIELDS FROM A BOUNDED SOURCE DISTRIBUTION 47
Figure 2.4: Radiation field separates from the source and propagates to in finity
Dibujar el dipolo. Note that there is not radiation in the direction in which thecurrent element is pointing.
pp 259 del panofsky : -Como puede verse en la figura
las lineas de campo de raciación representa una familia de lazos, atravezados
por las líneas de campo magnético, que se propagan hacia el in finito ( i.e. waves
see chapter tal)
Fields created by an in finitesimal current element with time-harmonic
dependence:
~H(r)=I∆z
4πjkµ
1+1
jkr¶e−jkr
rsinθˆϕ (2.87a)
~E(r)=I∆z
4πjkη0Ã
1+1
jkr−1
(kr)2!
e−jkr
rsinθˆθ+
I∆z
2πjkη0Ã
1
jkr−1
(kr)2!
e−jkr
rcosθˆr (2.87b)
These expressions can be also derived di rectly from the vector potential (2.41b),
which in this case simpli fies to
A=ˆzμ0
4πZ∆z
2
−∆z
2[i]
rdz0'ˆz[i]μ0
4πr∆z (2.88)
Thus, the magnetic field is given by6
48CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
H =1
μ0∇×A=1
μ0∇×(Aˆz)=1
μ0(∇A׈z+A(∇׈z)) =1
μ0∇A׈z
(2.89)
where we have applied the vector identity ( ??)a n dt a k e ni n t oa c c o u n tt h a tt h e
curl of a constant vector is zero. Hence , using spherical coordinates, we get
H =1
μ0∇A׈z=∆z
4π∂
∂rµ[i]
r¶
ˆr׈z
=∆z
4πµ
−1
crd[i]
dt−[i]
r2¶
ˆr׈z=∆z
4πµ1
crd[i]
dt+[i]
r2¶
sinθˆϕ
(2.90)
which of course coincides with (2.86a).
The electric field (2.86b) can be calculated from (2.90), taking into account
that from (1.1d), in source-free regions, we have7
E(r,t)=1
ε0Zt
−∞∇×H(r,t)dt (2.93)
From the relation between the charge and current, i(t)=dq(t)/dt,w eh a v e
i4z=dq
dt4z=dp
dt= ˙p (2.94)
wherep=q4zis the dipole moment of a time-varying electric dipole8,t h e
so-called Hertzian dipole, formed by two point charges with values of +q(t)
6For a given vector fieldA,t h e field lines are de fined by the condition that, at any point,
t h el i n ee l e m e n t dland the field are parallel i.e. dl×A=0.F o r t h e field created by a current
element, from Eqs (2.87a), the magnetic field has only ˆϕcomponent and consequently their field
lines are closed around the Zaxis. The radiation electric field has ˆθand ˆrcomponents, although
the radiation electric field has only ˆθcomponent which becomes null in the region θ→0.T h e ni n
this region predominates the Er=E·ˆrcomponent and consequently the electric field lines close(
see Fig 2.4) as would be expected from Maxwell’s equations since, outside the sources, there onlyexist curl sources.
7Note that once calculed H=1/μ0∇×Awe can obtain Eusing (1.1d) or (1.67d) and taking
into account that, in source-free regions, we have
E=1
ε0]
∇×Hd t =c]
∇×(∇×A)dt (2.91)
or
E=1
jε0ω∇×H=1
jk∇×(∇×A) (2.92)
for arbitrary or harmonic time dependence respectively. Thus we do not need necessarily to
calculate Φto obtain the fields.
8T h et i m ev a r y i n ge l e c t r i cd i p o l ei sd e fined as two time varying charges of opposite magnitude
±q(t)separated by a constant distance ∆zmuch less than the field point r. The dipole moment
p(t)is given by the magnitude of the charge times the distance ∆zbetween them and the de fined
direction is toward the positive charge i.e. p(t)=q(t)∆z.. Alternatively it would be possible to
model the oscillating dipole as two constant point charges of opposite sign separated by oscillat-
ing distance ∆z(t).However, the fields created for such accelerated charges need from the theory
d e v e l o p e di nC h a p t e r ??.
2.3. ELECTROMAGNETIC FIELDS FROM A BOUNDED SOURCE DISTRIBUTION 49
and−q(t)and the dot indicates di fferentiation with respect to time. Thus the
time-varying current element is equivalent to
i(t)=1
4zdp
dt(2.95)
or, for time-harmonic dependence,
I=jωp
4z(2.96)
Introducing (2.95) into (2.86a) and (2.86b), and (2.96) into (2.87a) and (2.87b),
we get the field created by an in finitesimal current element (hertzian dipole) in
terms of its dipole moment as:
Fields created by a Hertzian dipole with arbitrary-time dependence:
H =1
4πrµ[˙p]
r+[¨p]
c¶
sinθˆϕ (2.97a)
E=1
4πrε0µ[p]
r2+[˙p]
rc+[¨p]
c2¶
sinθˆθ+
1
2πrεoµ[p]
r2+[˙p]
rc¶
cosθˆr (2.97b)
Fields created by a Hertzian dipole with time-harmonic dependence:
~H =jωp
4πµ1
r+jk¶e−jkr
rsinθˆϕ (2.98a)
~E =p
4πε0µ1
r2+jk
r−k2¶e−jkr
rsinθˆθ+
p
2πεoµ1
r2+jk
r¶e−jkr
rcosθˆr (2.98b)
The radiation fields created by an in finitesimal current element can be ex-
pressed, from (2.86a) to (2.87b), in terms of its current amplitude or of its
equivalent dipolar moment.
Radiation fields created by an in finitesimal current element:
For arbitrary-time dependence
Hrad =∆z
4π1
crd[i]
dtsinθˆϕ (2.99a)
Erad =∆z
4πεo1
c2rd[i]
dtsinθˆθ (2.99b)
50CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
For time-harmonic dependence
~Hrad =I∆z
4πjke−jkr
rsinθˆϕ (2.100a)
~Erad =I∆z
4πjkη0e−jkr
rsinθˆθ (2.100b)
and from (2.97a) to (2.98b), we have the radiation fields in terms of its equivalent
Hertzian dipole:
Radiation fields created by an electric dipole:
For arbitrary-time dependence
Hrad =1
4πr[¨p]
csinθˆϕ (2.101a)
Erad =1
4πrε0[¨p]
c2sinθˆθ (2.101b)
For time-harmonic dependence
~Hrad =−ωpk
4πe−jkr
rsinθˆϕ (2.102a)
~Erad =−pk2
4πε0e−jkr
rsinθˆθ (2.102b)
More details about this elemental radiators and how they can be physical
approximated are given in subsubsection ??of chapter ??.
2.3.3 Far-zone approximations for the potentials
The general expressions (2.49) and (2.59) 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 in finitesimal 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, (2.49) and (2.59), with those
for the potentials (2.41a) and (2.41b). Because of the vector product in the
integrand of (2.51) and (2.62), this argument continues being true even when
we are interested only in the radiation fields. In the far zone, we can make the
approximations (2.74) and (2.77) for the potentials. Hence, the integrands ofthe retarded potentials (2.41) simplify to
Φ(r,t)=1
4πε0Z
V0ρ(r0,t0)
Rdv0'1
4πε0rZ
V0ρ(r0,t0
0+r0·ˆr
c)dv0
(2.103a)
A(r,t)=μ0
4πZ
V0J(r0,t0)
Rdv0'μ0
4πrZ
V0J(r0,t0
0+r0·ˆr
c)dv0(2.103b)
2.4. MULTIPOLE EXPANSION FOR POTENTIALS 51
The magnetic field can now be calculated from (2.1), using ( ??), as
B=∇×A=μ0
4πZ
V0∇×J(r0,t0
0+r0·ˆr
c)
rdv0
=μ0
4πZ
V0∇×J(r0,t0
0+r0·ˆr
c)
rdv0−μ0
4πZ
V0J(r0,t0
0+r0·ˆr
c)×∇1
rdv0
(2.104)
where, if we are interested only in the radiation field, the second term can be
ignored since it depends on 1/r2, and therefore
H=∇×A
μ0=1
4πZ
V0∇×J(r0,t0
0+r0·ˆr
c)
rdv0(2.105)
Furthermore, from ( ??), we have ∇×J(Ψ)=∇Ψ×dJ/dΨwithΨ=t0
0+r0·ˆr/c.
Thus, it follows that
∇×J(r0,t0
0+r0·ˆr
c)=−∇r
c×∂J(r0,t0
0+r0·ˆr
c)
∂t
=−ˆr
c×∂J(r0,t0
0+r0·ˆr
c)
∂t(2.106)
and therefore
H=−1
μ0cˆr×∂A
∂t(2.107)
which, as would be expected, leads to (2.79a). If the time variations of the
sources are harmonic the expressions (2.103a) , (2.103b) and (2.107) become
Φ =1
4πε0re−jkrZ
V0ρ(r0)ejk·r0dv0(2.108a)
~A =μ0
4πre−jkrZ
V0~J(r0)ejk·r0dv0(2.108b)
~H =−jω
μ0cˆr×~A (2.108c)
The radiation electric field can be calculated from (2.107) or (2.108c) simply
using (2.81).
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 (2.103b) or (2.108b) in order to calculate the vector potential. However,
if we assume that the charge distribution does not change appreciably over time
52CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
l/c, we may expand the integrands of (2.103) in a Taylor series about t0
0in terms
of the parameter r0·ˆr/c. For example, for the vector potential, we have
J(r0,t0
0+r0·ˆr
c)=J(r0,t0
0)+∂J(r0,t0)
∂t0¯¯¯¯¯
t0=t0
0r0·ˆr
c+... (2.109)
where we have omitted higher-order terms in r0·ˆr/c. Thus after inserting (2.109)
in (2.103b), we can write Aas the power-series expansion
A'A1+A2+...=
μ0
4πrZ
V0J(r0,t0
0)dv0+μ0
4πcrZ
V0∂J(r0,t0)
∂t0¯¯¯¯¯
t0=t0
0r0·ˆrdv0+...
(2.110)
Therefore the first two terms of the expansion (2.110), A1andA2,a r eg i v e nb y
A1=μ0
4πrZ
V0J(r0,t0
0)dv0(2.111a)
A2=μ0
4πcrZ
V0∂J(r0,t0)
∂t0¯¯¯¯¯
t0=t0
0r0·ˆrdv0
=μ0
4πcrZ
V0∂
∂t0J(r0,t0)r0·ˆrdv0¯¯¯¯
t0=t0
0(2.111b)
If the time dependence of the sources is sinusoidal the condition that the source
distribution does not change appreciably over time l/cis equivalent to assuming
thatl/c << T (whereTis the period of the signal) or equivalently l/λ << 1,
i.e., that the dimension of wavelength is much greater than that of the source
distribution
λ> > l (2.112)
In this case, we can perform the series expansion
ejk·r0=ejkˆr·r0≈1+jkˆr·r0−1
2k2(ˆr·r0)2+... (2.113)
which, after substituting in (2.108b), leads to
~A=~A1+~A2+... (2.114)
where
~A1=μ0
4πe−jkr
rZ
V0~J(r0)dv0(2.115a)
~A2=jkμ0
4πe−jkr
rZ
V0~J(r0)ˆr·r0dv0(2.115b)
2.4. MULTIPOLE EXPANSION FOR POTENTIALS 53
which are the Fourier transforms of (2.111a) and (2.111b), respectively.
Of course, there are analogous expressions for the terms of Φ
Φ=Φ1+Φ2+...
=1
4πε0rZ
V0ρ(r0,t0
0)dv0+1
4πε0crZ
V0∂ρ(r0,t0)
∂t0¯¯¯¯
t0=t0
0r0·ˆrdv0+...
(2.116)
where
Φ1=1
4πε0rZ
V0ρ(r0,t0
0)dv0(2.117a)
Φ2=1
4πε0crZ
V0∂ρ(r0,t0)
∂t0¯¯¯¯
t0=t0
0r0·ˆrdv0(2.117b)
Note that, since the contribution of each point source to the integral in
(2.117a) is evaluated at the same time t0
0, this integral represents the total
charge 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 Φ1created by that charge depends on r−2and consequently it does
not contribute to the radiation.
The expansion (2.110) 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 (ormagnetic) 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, (2.111a) and (2.111b), of (2.110).
2.4.1 Electric dipolar radiation
The evaluation of the term (2.111a) of the power-series expansion of Acan be
facilitated by calculating just one component ofR
V0J(r0,t0
0)dv0, for example the
xcomponent
Z
V0Jx(r0,t0
0)dv0=Z
V0J(r0,t0
0)·ˆxdv0=Z
V0J(r0,t0
0)·∇0x0dv0
=Z
V0∇0·³
x0J(r0,t0
0)´
dv0−Z
V0x0∇0·J(r0,t0
0)dv0
=−Z
V0x0∇0·J(r0,t0
0)dv0(2.118)
since Z
V0∇0·³
x0J(r0,t0
0)´
dv0=0 (2.119)
54CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
as can be seen by applying the divergence theorem and by integrating over
an external surface, where J(r0,t0
0)=0 , that encloses the sources. Therefore,
generalizing to three dimensions we have
Z
V0J(r0,t0
0)dv0=−Z
V0r0∇0·J(r0,t0
0)dv0(2.120)
and using the equation of continuity
∇0·J(r0,t0
0)=−∂ρ(r0,t0
0)
∂t(2.121)
we get
Z
V0J(r0,t0
0)dv0=Z
V0r0∂ρ(r0,t0
0)
∂tdv0(2.122)
which, when substituted in (2.111a), gives
A1=μ0
4πrZ
V0r0∂ρ(r0,t0
0)
∂tdv0=μ0
4πr∂
∂tZ
V0r0ρ(r0,t0
0)dv0(2.123)
The integralR
V0r0ρ(r0,t0
0)dv0is by de finition the electric dipole moment, [p],
evaluated at the retarded time t0
0, of the time-varying source distribution, i.e.,
[p]=Z
V0r0ρ(r0,t0
0)dv0=Z
V0r0ρ(r0,t−r
c)dv0(2.124)
Thus we have
A1=μ0
4πr∂[p]
∂t=μ0·
[p]
4πr(2.125)
The magnetic radiation field, from (2.107), is given by
Hrad=−ˆr×[··
p]
4πrc=[··p]s i nθ
4πrcˆϕ (2.126)
w h e r ew eh a v ea s s u m e d t h ed i r e c t i o no f pparallel to the polar zaxis. This
expression, as might be expected, coinc ides with the radiation term, (2.101a),
of (2.97a). From (2.126), the electric radiation field, given by (2.101b), can
be obtained using (2.81). Of course the corresponding expressions for time-harmonic fields are given by (2.102a) and (2.102b). Therefore, in a preliminary
approximation, the original source distribution can be replaced by an electric
dipole located at the origin of coordinates.
2.4. MULTIPOLE EXPANSION FOR POTENTIALS 55
2.4.2 Magnetic dipolar radiation
The analysis of the term (2.111b), can be facilitated by expressing the integrand
as follows
J(r0,t0
0)(ˆr·r0)=1
2³
J(r0,t0
0)(r0·ˆr)−r0³
J(r0,t0
0)·ˆr´´
+1
2³
J(r0,t0
0)(r0·ˆr)+r0³
J(r0,t0
0)·ˆr´´
=1
2ˆr׳
J(r0,t0
0)×r0´
+1
2³
J(r0,t0
0)(ˆr·r0)+r0³
J(r0,t0
0)·ˆr´´
(2.127)
Then, substituting in A2,w eg e t
A2=A2m+A2q (2.128)
where
A2m=μ0
8πcrZ
V0∂
∂tˆr׳
J(r0,t0
0)×r0´
dv0(2.129)
and
A2q=μ0
8πcr∂
∂tZ
V0³
J(r0,t0
0)(ˆr·r0)+r0³
J(r0,t0
0)·ˆr´´
dv0(2.130)
The integral (2.129) can be written as
A2m=μ0
4πcr∂[m]
∂t׈r (2.131)
where
[m]=Z
V0r0×J(r0,t0
0)
2dv0(2.132)
is by de finition the magnetic dipolar moment about O, evaluated at the retarded
timet0
0, of the source distribution. Thus, under the assumption that m=mˆz,
the magnetic radiation field given by (2.107) is
Hrad=1
4πc2rˆr×(ˆr×[··
m]) =1
4πr[··m]
c2sinθˆθ (2.133)
From (2.81) the electric radiation field is given by
Erad=−μ0
4πr[··m]
csinθˆϕ (2.134)
For time-harmonic dependence, we have
~Hrad=−k2msinθ
4πre−jkrˆθ (2.135)
56CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
and
~Erad=k2msinθ
4πrη0e−jkrˆϕ (2.136)
where
~m=Z
V0r0×~J(r0)
2dv0(2.137)
These expressions are similar to (2.101a)-(2.102b), 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 andmagnetic dipoles.
In the particular case of a current loop of radius a,F i g . ??,f o rw h i c ht h e
currentidoes not change appreciably over time a/c(or equivalently a< <λ for
any frequency involved), (2.132), becomes
m=iZ
Γr0×dl
2=iS (2.138)
whereΓis the countour of loop and Sis the vector area of the surface subtended
by the contour Γ.I nt h i se x p r e s s i o n , Jdv has been changed to idl.T h es u r f a c e
vectorSis 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 (2.133)-(2.136), can be written, for arbitrary time dependence,
as
Hrad =S
4πr[··
i]
c2sinθˆθ (2.139a)
Erad =−μ0S
4πr[··
i]
csinθˆϕ (2.139b)
whereiis evaluated at t0
0. These equations, for time-harmonic dependence
become
~Hrad =−k2ISsinθ
4πre−jkrˆθ (2.140a)
Erad =k2ISsinθ
4πrη0e−jkrˆϕ (2.140b)
It should be mentioned that the magnetic moment is important only when
there exists no radiation of the electric moment of the system. Otherwise theone due to the magnetic moment may be ignored. E ffectively, comparing Eqs.
(2.101b) and (2.134), and using E
pandEmto indicate the amplitudes of the
electric radiation fields from an electric and a magnetic dipole, respectively, we
have
Eprad
Emrad=c¨p
¨m(2.141)
2.4. MULTIPOLE EXPANSION FOR POTENTIALS 57
RG
Iθz
y
xarG
OmGP
RG
Iθz
y
xarG
OmGRG
Iθz
y
xarG
OmGP
Figure 2.5: Ponerm=iS,p o n e rien vez de Iye lc o n t o u r Γ.A circular loop of
current in the x-y plane y dibujar campos como en el elemento de corriente. Hacer el dibujo igualque el del electrico
or for time-harmonic variation with both dipoles oscillating at the same fre-
quency,
Eprad
Emrad=cp0
m0(2.142)
Since from (2.137) we have
m0=Z
V0r0×J0
2dv0=1
2Z
V0ρ0r0×udv0(2.143a)
p0=Z
V0ρ0r0dv0(2.143b)
and consequently
m0∼up0 (2.144)
whereuis the velocity of motion of the charges. Thus from (2.142) we have, for
u< <c ,
Eprad>> Emrad (2.145)
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, A2q,o fA2in (2.130), is associated with the electric quadru-
pole radiation, but to see this we must transform it further. To this end let us
58CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
consider the xcomponent of the first summand of the integralR
V0³
J(r0,t0
0)ˆr·r0´
dv0
i.e.
Z
V0ˆr·r0³
J(r0,t0
0)·ˆx0´
dv0=Z
V0ˆr·r0³
J(r0,t0
0)·∇0x0´
dv0
=Z
V0∇0·³
x0(ˆr·r0)J(r0,t0
0)´
dv0−Z
V0x0∇0·³
(ˆr·r0)J(r0,t0
0)´
dv0
(2.146)
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. Thus
Z
V0ˆr·r0³
J(r0,t0
0)·ˆx0´
dv0=−Z
V0x0∇0·³
(ˆr·r0)J(r0,t0
0)´
dv0
=−Z
V0x0∇0(ˆr·r0)·J(r0,t0
0)dv0
−Z
V0x0(ˆr·r0)∇0·J(r0,t0
0)dv0
(2.147)
but
∇0(ˆr·r0)=ˆr (2.148a)
∇0·J(r0,t0
0)=−∂ρ(r0,t0
0)
∂t(2.148b)
therefore
Z
V0Jx(r0,t0
0)ˆr·r0dv0
=−Z
V0x0³
J(r0,t0
0)·ˆr´
dv0+Z
V0x0(ˆr·r0)∂ρ(r0,t0
0)
∂tdv0(2.149)
Generalizing to three dimensions
Z
V0J(r0,t0
0)(ˆr·r0)dv0=−Z
V0r0³
J(r0,t0
0)·ˆr´
dv0+Z
V0r0(ˆr·r0)·∂ρ(r0,t0
0)
∂tdv0
(2.150)
and therefore, substituting in (2.130), we have
A2q=μ0
8πcr∂2
∂t2Z
V0r0(ˆr·r0)ρ(r0,t0
0)dv0(2.151)
The magnetic radiation field, given by (2.107), is
H2qrad=−1
8πc2rˆr×∂3
∂t3Z
V0r0(ˆr·r0)ρ(r0,t0
0)dv0(2.152)
2.5. MAXWELL’S SYMMETRIC EQUATIONS 59
The above expression can be written in a more useful form by adding the
term ˆrr02ρ(r0,t0
0)to the integrand
H2qrad=−1
24πc2rˆr×∂3
∂t3Z
V0¡
3r0(ˆr·r0)−ˆrr02¢
ρ(r0,t0
0)dv0(2.153)
Note that, since ˆr׈rr02=0, the added term do no a ffect 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 Q, called electric quadrupole-
moment tensor of the source distribution, and the vector ˆr
Z
V0¡
3r0(ˆr·r0)−ˆrr02¢
ρ(r0,t0
0)dv0=[Q]ˆr (2.154)
The elements of [Q]are
[Qαβ]=Z
V0¡
3x0
αx0β−r02δαβ¢
ρ(r0,t0
0)dv0(2.155)
and [Q]ˆris a vector with components
X
α[Qαβ]ˆrβ (2.156)
Therefore the radiation magnetic field from a varying electric quadrupole is
given by
H2qrad=−1
24πc2rˆr×∂3[Q]ˆr
∂t3=−1
24πc2rˆr×[...
Q]ˆr (2.157)
or, for, time-harmonic dependence,
~H2qrad=jck3
24πrej(ωt−kr)ˆr×Qˆr (2.158)
The radiation electric field can be calculated as usual by (2.81). 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 Hertziandipole (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 Maxwell’s symmetric equations
It can be observed from (1.1a)-(1.1d) that Maxwell’s equations present a certainsymmetry that, except in free space an dw i t hn os o u r c et e r m s ,i sn o tc o m p l e t e
because of the absence of magnetic charges and currents. Indeed, despite many
experimental attempts, no free magnet ic charges or monopoles have been found
60CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
in nature nor, therefore, would magnetic currents be created9. Nevertheless,
from a purely theoretical standpoint, nothing prevents us from assuming theexistence of magnetic monopoles; theref ore, to complete Maxwell’s equations
we must add the necessary magnetic source terms in order to achieve com-
plete symmetry 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 in-
troducing, on their right-hand side, hy pothetical magnetic current densities J
m
(Vm−2) and magnetic charge densities ρm(Wb/m3), respectively, as additional
source terms. With these new quantities included, we can rewrite Maxwell’sequations 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
∇·D =ρ (2.159a)
∇·B=ρ
m (2.159b)
∇×E=−Jm−μ0∂H
∂t(2.159c)
∇×H =J+ε0∂E
∂t(2.159d)
Integral form of Maxwell’s symmetric equations
I
SD·ds=QT (2.160a)
I
SB·ds=Qm (2.160b)
I
ΓE·dl=−Z
SJm·ds−∂
∂tZ
SB·ds (2.160c)
I
ΓH·dl=Z
SJ·ds+∂
∂tZ
SD·ds (2.160d)
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 (2.159c) and using (2.159b)
∇·∇×E=−∇·Jm−∂∇·B
∂t=0 (2.161)
9It 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
gwould explain the quantization
of the electric charge e. We refer to the magnetically charged particles as magnetic monopoles or
simply monopoles.
2.5. MAXWELL’S SYMMETRIC EQUATIONS 61
we get the equation of continuity
∇·Jm=−∂ρm
∂t(2.162)
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, E,D,HandB, into the sum of two components
D =De+Dm=ε0³
Ee+Em´
=ε0E (2.163a)
B=Be+Bm=μ0³
He+Hm´
=μ0H (2.163b)
where the quantities with the esubscript depend only on the “true” electric
sourcesρandJwhile the quantities with the msubscript depend only on the
“hypothetical” magnetic sources ρmandJm. In this way, we divide Maxwell’s
equations into two groups corresponding to the fie l dc o m p o n e n t sa s s o c i a t e dw i t h
the electrical and magnetic sources, respectively; that is
∇·De=ρ (2.164a)
∇·Be=0 (2.164b)
∇×Ee=−μ0∂He
∂t(2.164c)
∇×He=J+ε0∂Ee
∂t(2.164d)
∇·Dm=0 (2.165a)
∇·Bm=ρm (2.165b)
∇×Em=−Jm−μ0∂Hm
∂t(2.165c)
∇×Hm=ε0∂Em
∂t(2.165d)
Note that the sum of each expression (2 .164), added to its equivalent (2.165),
gives (2.159) and that the set (2.164) coincides with the conventional Maxwell’sequations (2.159), and that Eqs. (2.164) 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 ΦandA. Thus, from (2.4) and (2.1), we have
B
e=∇×A (2.166)
Ee=−∇Φ−∂A
∂t(2.167)
62CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
where
∇·A+μ0ε0∂Φ
∂t=0 (2.168)
and where AandΦfulfil the wave equations (2.14a) and (2.14b)
∇2A−μ0ε0∂2A
∂t2=−μ0J (2.169)
∇2Φ−μ0ε0∂2Φ
∂t2=−ρ
ε0(2.170)
the solutions to which are the retarded potentials (2.41a) and (2.41b)
Φ=1
4πε0Z
V0[ρ]
Rdv0(2.171)
A=μ0
4πZ
V0[J]
Rdv0(2.172)
Thefields created by the magnetic sources ρmandJmcan be deduced by ob-
serving that equations (2.164) are transformed into (2.165) and vice versa with
the simultaneous replacement of the following quantities, called duals
Eedual of Hm
Hedual of −Em
ε0dual of μ0
μ0dual of ε0
ρdual of ρm
Jdual of Jm(2.173)
ThefieldsEeandHeassociated with the electric sources can be calculated
from the magnetic vector potential Aand the electric scalar potential Φby
means of (2.171) and (2.172). To calculate the fieldsHmandEmwe can use the
same formalism de fining two new potentials, termed ”electric vector potential”
Fand ”magnetic scalar potential” ψ, such that
A dual of F
Φ dual of ψ(2.174)
Hence
ψ=1
4πμ0Z
V0[ρm]
Rdv0(2.175)
F=ε0
4πZ
V0[Jm]
Rdv0(2.176)
which are the dual expressions of (2.171) and (2.172).
2.5. MAXWELL’S SYMMETRIC EQUATIONS 63
By substituting the magnitudes in the first column of (2.173 and 2.174) for
their duals in the equations from (2.166) to (2.172) we get
Dm=εoEm=−∇×F (2.177a)
Hm=−∇ψ−∂F
∂t(2.177b)
∇·F+μ0ε0∂ψ
∂t=0 (2.177c)
in which ψandFsatisfy wave equations that are analogous to (2.169) and
(2.170):
∇2F−μ0ε0∂2F
∂t2=−ε0Jm (2.178)
∇2ψ−μ0ε0∂2ψ
∂t2=−ρm
μ0(2.179)
Thus, by the superposition principle, if both current densities JandJmexist
simultaneously in a region of free space, the total fieldEproduced at any point
is the sum of EeandEmgiven by (2.167) and (2.177a). Hence
E=Ee+Em=−∇Φ−∂A
∂t−1
ε0∇×F=
1
c2∇Z
∇·Adt−∂A
∂t−1
ε0∇×F (2.180)
where Lorenz gauge Eq. (2.10) has been used to express Ein terms of Aand
F.
The total fieldHis determined analogously from (2.166) and (2.177b)
H=He+Hm=−∇ψ−∂F
∂t+1
μ0∇×A (2.181)
In practice, it is not necessary to use the latter expression, because once E
has been calculated using (2.180), by substituting the result in (2.159c), with
Jm=0we obtain H.
2.5.1 Boundary conditions
It is easy to show, ejercicio, that the boundary conditions corresponding to Maxwell’s
symmetric equations are a logical extension of (1.35); that is,
ˆn·³
D1−D2´
=ρs (2.182a)
ˆn·³
B1−B2´
=ρsm (2.182b)
ˆn׳
E1−E2´
=−Jsm (2.182c)
ˆn׳
H1−H2´
=Js (2.182d)
64CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
in which ˆnis the normal unit vector that goes from region 2to region 1.E q u a -
tions (2.182b) and (2.182c) show the additional e ffects of the imaginary sur-
face magnetic charges and currents, ρsmandJsm, at the interface. According
to (2.182c) and (2.182d), the tangential components of the fields on a real or
imaginary surface Scan be written in terms of surface distributions of electric
currents
ˆn×H¯¯¯
S=Js (2.183)
and magnetic ones
−ˆn×E¯¯¯
S=Jsm (2.184)
2.5.2 Harmonic variations
For harmonic variations, the symmetric equations (2.159) simplify to
∇·~D =ρ (2.185a)
∇·~B =ρm (2.185b)
∇×~E =−~Jm−jμ0ω~H (2.185c)
∇×~H =~J+jε0ω~E (2.185d)
and the wave equations for the magnetic scalar potential ψ,the electric vector
potential F, and the Lorenz relations are
∇2ψ+ω2μ0ε0ψ=−ρm
μ0(2.186a)
∇2~F+ω2μ0ε0~F =−ε0~Jm (2.186b)
ψ=j∇·~F
ωε0μ0(2.186c)
with the solutions to (2.186a) and (2.186b) being
ψ=1
4πμ0Z
V0ρme−jkR
Rdv0(2.187)
~F =ε0
4πZ
V0~Jme−jkR
Rdv0(2.188)
The total field~Eproduced at any point is the sum of ~Eeand~Em,a n di s
given by
~E=~Ee+~Em=−jc2
ω∇³
∇·~A´
−jω~A−1
ε0∇×~F (2.189)
while for the total field~Hwe have
~H=−jc2
ω∇³
∇·~F´
−jω~F+1
μ0∇×~A. (2.190)
whereAis given by (2.47a).
2.6. THEOREM OF UNIQUENESS 65
2.5.3 Fields created by an in finitesimal magnetic current
element
From (2.86a) and (2.86b), using the dual equations (2.173), we deduce that the
fields generated by an in finitesimal magnetic current element,
Jm(r,t)=im(t)δ(x0)δ(y0)ˆz−∆z
2<z0<∆z
2(2.191)
are given by, (Fig. 2.6),
E=−∆z
4πµ1
crd[im]
dt+[im]
r2¶
sinθˆϕ (2.192a)
H =∆z
4πμoµ1
r3Zt
−∞[im]dt+[im]
cr2¶
(2 cosθˆr+s i nθˆθ)+∆z
4πμo1
c2rd[im]
dtsinθˆθ
(2.192b)
or, for time-harmonic variation
~E =−∆zIm
4πjkµ
1+1
jkr¶e−jkr
rsinθˆϕ (2.193a)
~H =Im∆z
4πjωε 0µ
1+1
jkr−1
k2r2¶e−jkr
rsinθˆθ+Im∆z
2πjωε 0µ
−1
k2r2+1
jkr¶e−jkr
rcosθˆr
(2.193b)
Comparing the radiation terms of th ese equations to (2.139a)-(2.140b), we find
that
im∆z=μ0Sdi
dt(2.194)
or for time-harmonic dependence.
Im∆z=jωμ0IS (2.195)
2.6 Theorem of uniqueness
Whenever we have to resolve a di fferential equation, it is desirable to know
the conditions that must be ful filled in order to state that a unique solution is
possible. In our context, this means to s eek the conditions for which we can state
that there exists a single electromagnetic field that satis fies, simultaneously,
Maxwell’s equations and the given boundary conditions.
Next, we establish these conditions fo r non-harmonic and time-harmonic
electromagnetic fields.
66CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
z
/2zΔθ
()mit
/2z−Δy
xEϕ−
Hθ
ˆrz
/2zΔθ
()mit
/2z−Δy
xEϕ−
Hθ
ˆr
Figure 2.6: solo los campos de radiacion se representan”’
2.6.1 Non-harmonic electromagnetic field
A non-harmonic electromagnetic field that varies in a linear region Vbounded
by a surface Sis uniquely determined from an initial time, t=t0,i ft h ef o l l o w i n g
are known:
i) The values of the sources at each point and at each time for every t>t 0
within the region.
ii) The values of the electromagnetic field (EandH) at each point of Vat
the initial time t=t0.
iii) The tangential components of the electric fieldEor of the magnetic field
Hon the entire the surface Sfor allt>t 0, or, alternatively, the tangential
components of the electric fieldEin any part of Sand of the magnetic field
Hin the remaining part of S, for allt>t 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 de fined the three above conditions within a volume
V, there exist two di fferent electromagnetic fields, (E
1andH1)a n d(E2and
H2), respectively, which are solutions to the problem. Given the linearity of
Maxwell’s equations, any linear combinati on of these two solutions must in itself
be a solution. In particular, the di fference between the two aforementioned
fields, i.e. the field de fined by ( E0=E1−E2andH0=H1−H2), must also
be a solution to the problem. Given that, from the hypothesis, the sources arethe same for the fields (E
1andH1)a n d(E2andH2), the field (E0,H0)i s
source-free in V.Thus, if we apply the Poynting theorem (1.39) to ( E0,H0), we
2.6. THEOREM OF UNIQUENESS 67
get
0=∂
∂tZ
V1
2(E0·D0+B0·H0)dv+Z
VσE02dv+I
S(E0×H0)·ds(2.196)
It is straightforward to show that if the tangential components of the electric
fieldEand/or of the magnetic fieldHare uniquely determined on surface S,
thefinal term in (2.196) is null. By integrating this expression with respect to
the time from t0totand, taking into account that the initial values for t=t0
are de fined for all V,w efind that
0=Z
V1
2(E0·D0+B0·H0)dv+Zt
t0µZ
VσE02dv¶
dt (2.197)
As both of the terms on the second member in (2.197) are positive, this
equality can be ful fil l e do n l yw h e nb o t h E0andH0are null (i.e. when E1=E2
andH1=H2) ,w h i c hi sw h a tw es e to u tt op r o v e .
2.6.2 Time-harmonic fields
In the case of harmonic variations, the uniqueness theorem states that a field
in a lossy ( σ6=0)10region is uniquely determin ed by the sources within the
region together with the tangent ial components of the electric fieldEor of
the magnetic fieldHonS, or, alternatively, the tangential components of the
electric fieldEin any part of Sand of the magnetic fieldHin the remaining
part ofS.
Proof By a reasoning similar to that used for the above case, but using the
expression (1.111), we get
0=Z
VσE02
0
2dv+2jωZ
VµμH02
0
4−εE02
0
4¶
dv (2.198)
By making the real and the imaginary pa rts equal to zero, we see that these
two equalities imply that H0
0andE0
0are both equal to zero only if σ6=0.T h i s
is why we started from the premise that the medium occupying the volume hasa conductivity that may be arbitrarily small but which is non-zero at all points.Thefield in a lossless region can be considered the limit to the lossy case when
such losses tend to zero.
10The reason why we need the extra condition of the space to be lossy for time-harmonic
s i g n a l si st h a t ,b yd e finition, a pure harmonic signal has an in finite duration.
68CHAPTER 2. FIELDS CREATED BY A SOURCE DISTRIBUTION: RETARDED POTENTIA L
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 far away from the sources. 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 su fficiently large
distances from the sources, in a su fficiently 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 radiocommunications the radiatedfield at su fficient distance from the antenna can be considered to be a plane
wave.
Moreover, it is possible to demostrate that, in general, an electromagnetic field can puede
descomponerse como suma lineal de ondas planas ( see Appendix ??) In this Chapter we consider
this kind of waves in a linear homogeneous isotropic medium libre de fuentes. Then incidencianormal y oblicua. Ondas esféricas , desarrollo en ondas planas?
Harmonic..Electromagnetic waves are not limited in wavelength and in fact cover the spec-
trum from gamma rays (wavelengths of ¿¿¿¿ ¿¿ 10-12 cm???????) through X-rays, visible light,
microwaves, and radio waves, to long waves (hundreds of kilometers long).
3.1 Wave equation
For time-varying electromagnetic fields it is possible to combine Maxwell’s equa-
tions to eliminate one of the fields,HorE, to obtain two uncoupled second-order
differential equations, one in Eand the other in H, known as wave equations.
To formulate these wave equations, let us consider a non-magnetic ( μ=μ0),
homogeneous, linear and isotropic re gion where, in general, source terms Jand
ρmay exist. Taking the curl of (1.1c) and using the vector relation ( ??)w e
1Wave-front is de fined as a surface that, at any time t, is orthogonal to the propagation
vector ˆnat all the points on the surface.
69
70 CHAPTER 3. ??ELECTROMAGNETIC WAVES
have
∇×∇×E=∇(∇·E)−∇2E=−∂∇×B
∂t
=−μ0∂
∂t(Jc+J+∂D
∂t)⇒
∇2E=∇(∇·E)+μ0∂
∂t(Jc+J+∂D
∂t)
=1
ε∇ρ+μ0σ∂E
∂t+μ0∂J
∂t+μ0ε∂2E
∂t2
(3.1)
whereJandJcare the source and induced conduction density of the currents,
respectively. Thus, rearranging terms, we get
∇2E−μ0σ∂E
∂t−μ0ε∂2E
∂t2=∇ρ
ε+μ0∂J
∂t(3.2)
which is known as the inhomogeneous vector-wave equation for the electric field.
A similar equation can be written for the magnetic fieldHby taking the
curl of (1.1d),
∇2H−μ0σ∂H
∂t−μ0ε∂2H
∂t2=−∇×J (3.3)
For a lossless media (3.2) and (3.3) reduce to
∇2E−μ0ε∂2E
∂t2=1
ε∇ρ+μ0∂J
∂t(3.4a)
∇2H−μ0ε∂2H
∂t2=−∇×J (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.
E(r,t)=−1
4πεZ
V0∇[ρ]+1
c2h
∂J
∂ti
Rdv0(3.5a)
H(r,t)=1
4πZ
V0∇×h
Ji
Rdv0(3.5b)
from which, by means of straightforward operations, we can obtain the expres-
sions (2.49) and (2.55) for the fields created by a bounded distribution of finite
densities of charges and currents with arbitrary space and time dependence.
In source-free regions ( J=0 ;ρ=0, except the charge and current densi-
ties induced by the presence of the fields, which are expressed in terms of the
3.1. WAVE EQUATION 71
constitutive parameters) the equations (3.2) and (3.3) simplify to
∇2E−μ0ε∂2E
∂t2−μ0σ∂E
∂t=0 (3.6a)
∇2H−μ0ε∂2H
∂t2−μ0σ∂H
∂t=0 (3.6b)
which are the homogeneous wave equations that determine the propagation of
thefieldsEandHin a sourceless homogeneous, linear and isotropic medium.
The solutions to these wave equations mu st be compatible with Maxwell’s equa-
tions and the coe fficients of the solutions must be derived from the boundary
conditions.
Uniform plane waves are de fined 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 ξfrom the origin to the plane (fig.6.1) .
Therefore, if ˆn=ξ/ξis the unit vector that is normal to the plane, the del
operator ∇becomes ∇=∂/∂ξ ˆnand Maxwell’s equations simplify to
ˆn·∂D
∂ξ=0 (3.7a)
ˆn·∂B
∂ξ=0 (3.7b)
ˆn×∂E
∂ξ=−∂B
∂t(3.7c)
ˆn×∂H
∂ξ=σE+∂D
∂t(3.7d)
and the wave equations become
∂2E
∂ξ2−μ0ε∂2E
∂t2−μ0σ∂E
∂t=0 (3.8a)
∂2H
∂ξ2−μ0ε∂2H
∂t2−μ0σ∂H
∂t=0 (3.8b)
These equations, which describe the propagation of plane waves in a homoge-
neous conducting medium, are called the “telegrapher’s equations”. For nondis-sipative media, for example the free space, these equations simplify to
∂
2E
∂ξ2−μ0ε0∂2E
∂t2=∂2E
∂ξ2−1
c2∂2E
∂t2=0 (3.9a)
∂2H
∂ξ2−μ0ε0∂2H
∂t2=∂2H
∂ξ2−1
c2∂2H
∂t2=0 (3.9b)
72 CHAPTER 3. ??ELECTROMAGNETIC WAVES
ct(, 0 )fz (,)fztf
ξv
ct(, 0 )fz (,)fztf
ξv
Figure 3.1: The wave tal, in a lossless medium, propagates at velocity talto
the right without changing shape
ˆnz
xrG
yξ
Oˆnz
xrG
yξ
O
Figure 3.2: poner plane wave front
3.2. HARMONIC WAVES 73
3.2 Harmonic waves
For time-harmonic fields, when the medium presents a conductivity σand, at
the operating frequency, acomplex dielectric constant, εc=ε0−jε00,(1.71), the
wave equation (3.6a) can be written as a time-independent wave equation
∇2~E−jωμ0σ~E+μ0ω2εc~E =∇2~E−jωμ0σe~E+μ0ω2ε0~E
=∇2~E+ω2μ0ε0(1−jtanδd)~E
=(∇2+ω2μ0εec)~E=0 (3.10)
whereσe=σ+ωε00,tanδd=σe/ωε0,a n dεec=ε0(1−jtanδd),a r et h e effective
conductivity, the loss tangent and the e ffective complex permittivity de fined
in (1.78), (1.81), and (1.83) respectively.
A c c o r d i n gt oS u b s e c t i o n( ??), depending on the characteristics of the medium,
the values of the term jtanδdin Eq. (3.10) may range from << 1(zero for a
perfect dielectric or lossless medium) to >> 1(infinite for a perfect conductor).
In a highly conductive medium tanδd>> 1and 1−jtanδd'−jtanδd,a n d
thus Eq. (3.10) becomes the so-called time-independent di ffusion equation for
the electric field~E
∇2~E−jωμ0σ~E=0 (3.11)
which is of the same type as the one that determines the propagation of heat by
conduction or by di ffusion. As commented in Subsection ( ??), for most metals
the relaxation time τis10−14s, which is a low value compared with the period
for all frequencies lower than the optical ones. Thus, since tanδd=(τω)−1,t h e
diffusion equation is adequate for me tals at all these frequencies.
Equation (3.10) can be written more concisely as
∇2~E−γ2~E=0 (3.12)
whereγis in general a complex quantity ca lled the complex propagation con-
stant, which, from (3.10) and (3.12), is given by
−γ2=ω2μ0(εc−jσ
ω)
=ω2μ0(ε0−j(ε00+σ
ω))
=ω2μ0ε0(1−jtanδd)=k2(1−jtanδd)=ω2μ0εec
(3.13)
where
k=ωp
μoε0 (3.14)
is the wavenumber corresponding to an unbounded lossless medium with a real
dielectric constant ε0.
Analogously, for the magnetic field, we have
∇2~H−γ2~H=0 (3.15)
74 CHAPTER 3. ??ELECTROMAGNETIC WAVES
3.2.1 Uniform plane harmonic waves
For uniform plane waves, we have ∇2=∂2/∂2ξand Eqs. (3.12) and (3.15)
simplify to
∂2~E
∂ξ2−γ2~E =0 (3.16a)
∂2~H
∂ξ2−γ2~H =0 (3.16b)
The complex propagation constant γis usually written as2
γ=jk(1−jtanδd)1/2=α+jβ (3.17)
where the imaginary part, β, is termed the phase constant, whereas the real
part,α, is called the attenuation constant of the wave. Thus, from, (3.13) and
(3.17), we can easily calculate the explicit expressions for βandα
β=ωµμ0ε0
2¶1
2h
(1 + tan2δd)1/2+1i1/2
=ωp
μoε0
√
2Ãr
1+³σe
ωε0´2
+1!1/2
=k√
2Ãr
1+³σe
ωε0´2
+1!1/2
(3.18a)
α=ωµμ0ε0
2¶1
2h
(1 + tan2δd)1/2−1i1/2
=ωp
μoε0
√
2Ãr
1+³σe
ωε0´2
−1!1/2
=k√
2Ãr
1+³σe
ωε0´2
−1!1/2
(3.18b)
The dimensions of αandβarem−1and they are referred to as neper and
radian, respectively, to indicate their attenuative and phase meanings in waveexpressions. For lossless media we have σ
e=0,α=0and the phase constant
becomes γ=jβ=jk.
2Recordemos que los valores del factor de atenuación, tal como se han calculado, vienen
expresados en nepers/metro y que multiplicados por 80868 se convierten en dB/m .
3.2. HARMONIC WAVES 75
Equations (3.16) have solutions of the form ~Eeγξand~Heγξso that the
instantaneous values for the fields are given by wave equations
E=R e {~Ee(jωt−γξ)}=R e{~Ee(jωt−γ·r)}=R e{~Ee−αξej(ωt−βξ)}
(3.19a)
H =R e {~He(jωt−γξ)}=R e{~He(jωt−γ·r)}=R e{~He−αξej(ωt−βξ)}
(3.19b)
where the so-called complex propagation vector γ=γˆn(with module γand
direction of the unit vector ˆnnormal to the wave-front planes) has been in-
troduced and ris the position of any point on the wave-front plane so that
ˆn·r=ξ.
Equations (3.19) represent waves traveling at a speed given by the phase
velocityvp
vp=ω
β(3.20)
w h i c hi ng e n e r a l ,a s βis given by (3.18a), depends on the frequency (dispersive
media).
The penetration factor δis defined as
δ=1
α(3.21)
This is the distance at which, due to the attenuation α,thefield module de-
creases from an initial given value to 1/eof this value.
From (3.7) the following equalities may be deduced
γ·~E =0 (3.22a)
γ·~H =0 (3.22b)
γ×~E =jμ0ω~H (3.22c)
γ×~H =−jεecω~E (3.22d)
From these equations, we see that ~E,~Hand ˆnare perpendicular to one
another and that they form a right-handed system in the order ~E,~H,ˆn.F o r t h i s
reason these waves are often referred to a s transverse electromagnetic (TEM)
waves. The magnitudes of ~E,~Hare related by
H=E
ηc=γE
jωμ0(3.23)
where the quantity ηc, known as the complex characteristic impedance of the
medium, is given, taking into a ccount (3.13) and (3.17), by
ηc=E
H=jωμ0
γ=µμ0
εec¶1/2
=ωμ0
α2+β2(β+jα)=|ηc|ejθ(3.24)
76 CHAPTER 3. ??ELECTROMAGNETIC WAVES
Thus, its module and phase is given by
|ηc|=¡μ0
ε0¢1/2
[1 + (σe
ωε0)2]1/4(3.25a)
θ=t a n−1α
β=1
2tan−1σe
ε0ω=δd
2(3.25b)
Therefore, in general there is a phase shift θbetween ~Eand~H.
3.2.2 Propagation in lossless media
By particularizing the above expressions for a lossless medium where, ε0=ε=
εrε0,ε00=0,a n dσ=0,w et h u sh a v e tanδd=0;γ=jk;a n dγ=k=kˆn,a n d
consequently equations (3.12) and (3.15) simplify to
∇2~H+k2~H =0 (3.26a)
∇2~E+k2~E =0 (3.26b)
and the complex characteristic impedance of the medium, (3.25), simpli fies to
ηc=η=³μ0
ε´1/2
=µμ0
ε0εr¶1/2
=η0
ε1/2
r=120π
ε1/2
r
θ=0 (3.27)
so that the impedance is re al and constant. In particular, when the medium is
free space, ηsimpli fies to the impedance of free space
η=η0=µμ0
ε0¶1/2
= 120π (3.28)
Consequently, in unbounded lossless media, there is no phase shift between ~E
and~Hand the attenuation is null ( α=0). Thusγ=jkandδ=∞and Eqs
(3.22) simplify to
k·~E =0 (3.29a)
k·~H =0 (3.29b)
k×~E =μ0ω~H (3.29c)
k×~H =−jεω~E (3.29d)
3.2.3 Propagation in good dielectrics or insulators
In a good dielectric (see Subsection ??) the reactive current predominates on
the dissipative current and according to (1.91), tanδd=σe/ωε0<< 1.I n t h i s
3.2. HARMONIC WAVES 77
ξ
ξ
Figure 3.3: Cuidado¡¡¡ estan normalizada a η0possitiveξtraveling fields of a uniform plane in
dissipative medium
ξ
ξ
Figure 3.4: Uniform plane wave propagating in the +ξdirection in a lossless medium
78 CHAPTER 3. ??ELECTROMAGNETIC WAVES
case, we can develop the complex propagation constant (3.17) to get
γ=jk(1−jtanδd)1/2=jω(μ0ε0)1/2µ
1−jtanδd
2+tan2δd
8+..¶
'
jω(μ0ε0)1/2µ
1−jtanδd
2¶
(3.30)
and therefore
α'ω(μ0ε0)1/2tanδd
2=σe
2³μ0
ε0´1/2
(3.31a)
β'k=ω(μ0ε0)1/2(3.31b)
Thus the propagation velocity can be approximated by
ω
β'1
(μ0ε0)1/2(3.32)
From (3.31a) it can be seen that αis small and therefore so is the wave atten-
uation. Moreover, since σe/ωε0<< 1, the intrinsic impedance of the medium
(3.25) is usually simpli fied to
ηc'η=³μ0
ε0´1/2
(3.33a)
θ=0 (3.33b)
3.2.4 Propagation in good conductors
For a good conductor (see Subsection ??) the dissipative current predominates
on the reactive current and according to (1.94), tanδd=σ/ωε >> 1.I n t h i s
case, from (1.93) and (3.13) we have
γ=jk(1−jtanδd)1/2'jk(−jtanδd)1/2=jk³σ
2εω(1−j)( 1−j)´1/2
=( 1 + j)³μ0σω
2´1/2
(3.34)
and consequently from (3.17),
α=β=³μ0σω
2´1/2
(3.35)
Thus the electric field from (3.19a), simpli fies to
E=R e{~Ee−ξ/δej(ωt−ξ/δ)} (3.36)
whereδ
δ=1
α=µ2
μ0ωσ¶1/2
(3.37)
3.2. HARMONIC WAVES 79
is the penetration factor (3.21) particularized by a good conductor. Thus, for
good conductors, the penetration factor δhas a very low value which decreases
as the frequency increases. Thus the fields are con fined within a very short
distance from the surface of the conductor. For a perfect conductor, σ→∞
andδ=0. Furthermore, the dielectric constant and the complex impedance
are reduced to
εec=εµ
1−jσ
ωε¶
'−jσ
ω(3.38)
and, respectively
ηc=µμ0
εec¶1/2
=³
−jμ0ω
σ´1/2
=( 1+j)³μ0ω
2σ´1/2
=( 1+j)ωμ0δ
2
(3.39)
Thus the phase shift between EandHis45o.
3.2.5 Surface resistance
Let us consider an area element perpendicular to the direction of propagation ξ.
Since the wave amplitudes of EandHdecrease exponentially according to the
factore−αξ, the complex Poynting vector (1.107), and consequently the mean
power per unit of area, (1.106), attenua tes along the direction of propagation
by the factor e−2αξ. Therefore
Pav=1
2Re{~E×~H∗}=Pav(0)e−2αξ(3.40)
wherePav(0)is the mean power per unit area at ξ=0.T h u s t h e t o t a l p o w e r
per unit area transmitted by the wave to the medium along the distance ξ=l
is given by
dP
ds=Pav(0)−Pav(l)=Pav(0)(1−e−2αl) (3.41)
This can be also calculated, according to (1.87), as
dP
ds=σe
2ÃZl
0(E2
0e−2αξ)dξ!
=σeE2
0
4α(1−e−2αl) (3.42)
This expression for l=∞,o rf o rad i s t a n c e lsuch that the magnitude of the
fields becomes negligible, simpli fies to
dP
ds=σeE2
0
4α=1
2Re{η−1
c}E2
0=1
2Re{ηc}H2
0= (3.43)
since
Re{η−1
c}=σe
2α(3.44)
80 CHAPTER 3. ??ELECTROMAGNETIC WAVES
For a good conductor, expression (3.43) simpli fies, from (3.39), to
dP
ds=H2
0
2³μ0ω
2σ´1/2
=1
2RsH2
0 (3.45)
whereRsis the so-called surface resistance
Rs=³μ0ω
2σ´1
2=1
σδ(3.46)
andδis the penetration factor given by (3.37).
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 (phasevelocity) of a harmonic wave depends on its frequency. In practice, the ideal
situation of a pure harmon ic wave which extends to in finity both backward
and forward in time never arises and, moreover, such a wave could not carryinformation. What in fact happens is that a transmitter emits a given signalf(ξ,t)for a finite period of time that, according to Fourier’s theorem, can be
expanded into a continuous spectrum of amplitudes A
ωsuch that
f(ξ,t)=Z∞
−∞Aωej(ωt−βξ)dω (3.47)
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 di fferent velocity and, as a consequence, the signal will deform as
it propagates. When, as commonly occurs in practice, the spectrum of thesignal is narrow
3and 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 whichthe envelope or energy of the wave group propagates in the medium is calledgroup velocity. To calculate this, let us consider a wave group centered on a
frequency ω
0such that Aω'0except for ω=ω0±4ω/2(Fig. 7.4). Under
these conditions, Eq. (3.47) simpli fies to
f(ξ,t)=Z
4ωAωej(ωt−βξ)dω (3.48)
extended to the values of ωin which Aω6=0.G i v e n t h a t β=β(ω),i tc a nb e
developed into a Taylor series around the frequency ω0
β(ω)=β(ω0)+∂β
∂ω¯¯¯¯
ω0(ω−ω0)+∂2β
∂ω2¯¯¯¯
ω0(ω−ω0)2
2(3.49)
3Note that a concentration of the field in space does not imply a concentration in the frequency
spectrum, but just the opposite, in accordance with the scale change property of the Fourier transform,
which indicates that an inverse relation exists between the duration of a signal and its bandwidth.
3.4. POLARIZATION 81
If the dispersive medium is such that the dependence of the phase velocity vp
on the frequency is so slowly that we can consider (as a good approximation)
that there exists a linear relation between βandω, then (3.49) simpli fies to
β(ω)=β0+∂β
∂ω¯¯¯¯
ω0(ω−ω0) (3.50)
whereβ0=β(ω0).
By substituting (3.50) in (3.48) we get
f(ξ,t)=ej
∂β
∂ω|ω0ω0ξ−β0ξZ
4ωAωejω
t−∂β
∂ω|ω0ξ
dω (3.51)
which, taking into account (3.48), can be written as a function of f(0,t)in the
following way
f(ξ,t)=fÃ
0,t−∂β
∂ω¯¯¯¯
ω0ξ!
ej
∂β
∂ω|ω0ω0ξ−β0ξ
(3.52)
This means that, at a point ξ, the signal has the same amplitude as at the origin
after a time t=∂β/∂ω |ω0ξand a phase shift given by ∂β/∂ω |ω0ω0ξ−β0ξ.
Consequently, the velocity at which the signal, and thus its associated energy,
propagates is
vg=dξ
dt=dω
dβ¯¯¯¯
ω0
=d
dβ(vpβ)¯¯¯¯
ω0=vp+βdvp
dβ¯¯¯¯
ω0
=vp−λdvp
dλ¯¯¯¯
ω0=1
dβ/dω¯¯¯¯
ω0(3.53)
If the phase velocity varies slowly with the frequency, then a pulse may
travel through a dispersive medium a certain distance without a signi ficant
change. If this condition is not satis fied and the medium is very dispersive
the shape of signal changes rapidly and the concept of group velocity is notlonger valid. The sign of dv
p/dω determines whether vgi sg r e a t e ro rl e s st h a n
vp.I f t h e p h a s e v e l o c i t y vpincreases with the frequency, it is termed normal
dispersion. On the contrary, when vpdecreases with the frequency, it is termed
anomalous dispersion. In an ideal dielectric where vp6=vp(β),s ot h a ta l lt h e
wavelengths propagate at the same velocity vp=vg, the signal propagates
without deformation.
3.4 Polarization
A st h ew a v ee q u a t i o ni sal i n e a rd i fferential equation, it ful fils the superposition
principle and any sum of solutions is also a solution of the di fferential equation.
82 CHAPTER 3. ??ELECTROMAGNETIC WAVES
In particular, let us consider the sum of two plane waves propagating in direction
z(one with the electric field lying along the xaxis and the other along the
ya x i s )a ti d e n t i c a lf r e q u e n c ies but, in general, with di fferent amplitudes ( a
andb) and phases ( δ1andδ2), 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 xdirection and the other in the ydirection. However,
in an electromagnetic wave the direction of the electric field generally changes
and traces out an ellipse as the wave propagates4. To see this, let us consider
the total time-varying electric field, which is sum of the two linearly polarized
waves, given by
~E(z,t)=(aejδ1ˆx+bejδ2ˆy)ej(ωt−kz)(3.54)
Let us determine the time evolution in a plane z=cteof the electric field
vector resulting from the composition of these two plane waves. We will assumea homogeneous, isotropic, lossless medium (although the e ffects of losses as an
exponential factor common to all the field components do not in fluence the
polarization).
At the plane z=0, for example, we have
E
x=acos(ωt+δ1) (3.55a)
Ey=bcos(ωt+δ2) (3.55b)
Ez=0 (3.55c)
Using the trigonometric identity for the sum of two angles ,s o l v i n gf o r cosωt
and sinωtin terms of aandb,d efiningδ=δ1−δ2as the relative phase
difference between the two components and after some simpli fications based on
simply trigonometric identities, we find
E2
x
a2+E2
y
b2−2ExEy
abcosδ=s i n2δ (3.56)
which is the equation of an ellipse with its major axis tilted depending on the
value ofδ. This means that at a plane z=cte, as the time goes on, the electric
fielddelineates 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 angular velocity of the vector→
Et=Ex
ˆx+Eyˆyis given by
·ϕ=dϕ
dt=d
dt(tan−1Ey
Ex)=Ex·
Ey−Ey·
Ex
|Et|2(3.57)
whereϕis ..........
The sense of rotation together with the direction of propagation de fine 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.
4In a unpolarized wave, the vector Eis subject to random changes of amplitud and phase
3.4. POLARIZATION 83
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 a,b,δ , and the polarization
ellipse may degenerate into a centred ellipse, a circle or a straight line.
Whena6=bandδ=mπ/ 2,w i t hm=±1,±3,±5,..the polarization ellipse
(3.56) becomes a centred ellipse with the major and minor axis oriented alongthex,ydirections, i.e.
E
2
x
a2+E2
y
b2=1 (3.58)
Ifa=b,t h e n
E2
x+E2
y=a2(3.59)
which is the equation of a circumference.
Whenδ=±mπ,w i t hmbeing an integer, the equation (3.56) becomes
∙Ex
a±Ey
b¸2
=0 (3.60)
which represents the equation of a straight line
Ey=∓b
aEx (3.61)
intersecting the origin. The wave is then linearly polarized and the components
ofEare
Ex=acos(ωt−kz) (3.62a)
Ey=bcos(ωt−kz±mπ) (3.62b)
T h ea n g l eo ft h es l o p ew i t ht h e xaxis is
tanϕ=t a nEy
Ex=(−1)mb
a(3.63)
84 CHAPTER 3. ??ELECTROMAGNETIC WAVES
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 (assumedto be plane and inde finite) separating two linear , nonmagnetic, homogeneous
and isotropic dielectrics having di fferent electromagnetic characteristics. The
change in the constitutive parameter s ,a st h ew a v ep a s s e sf r o mo n em e d i u mt o
the other, is assumed to take place in an electrically very narrow region witha thickness much less than λ. In general, when a wave propagating through
a medium strikes the interface (incident wave), part of its energy is re flected
and propagates through the same medium (re flected wave), while another part
is transmitted to the second medium (transmitted, or refracted wave). Thecharacteristics of re fle c t e da n dt r a n s m i t t e dw a v e sc a nb ec a l c u l a t e df r o mt h o s eo f
t h ei n c i d e n tw a v eb yf o r c i n gt h et o t a l fie l do nt h ei n t e r f a c et of u l fil 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,
1.
and then the more general case of oblique incidence. This study has extensive
applications in optics where the interfac e of many optical devices, such as lenses
andfiber-optic transmission lines, has a radius of curvature much larger than
the wavelength of the incident wave. Thus the interface can be considered quiteaccurately as a plane interface. In the f ollowing, with no loss of generality, we
will assume the interface to be parallel to the xyplane.
1La incidencia normal tiene muchas analogías con la líneas de transmisión que se estudiarán en
el capítulo Tal
85
86CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
ˆn
01 1Medium 1
;;μεσ
02 2Medium 2
;;μεσ
i
xEr
i
yHrˆiPr
xEr
t
xEr
t
yHrr
yHr
ˆtP
ˆrPˆn
01 1Medium 1
;;μεσ
02 2Medium 2
;;μεσ
i
xEr
i
yHrˆiPr
xEr
t
xEr
t
yHrr
yHr
ˆtP
ˆrP
Figure 4.1: Poner los vectores de pynting PEl subindice de campo electrico
incidente ponerlo mejor
4.1 Normal incidence.
4.1.1 General case: interface between two lossy media
normally incident from a lossy medi a, characterized by the parameters μ0,ε1=
ε0
1−jε00
1,σ1to the surface of another one with di fferent constitutive parameters
μ0,ε2=ε0
2−jε00
2,σ2
Considering two semi-inde finite lossy media that are separated by the plane
z=0,s e e figure 4.1 , let us assume that a harmonic plane wave propagates
through the first medium in the positive sense of the zaxis with the electric field
parallel to the xaxis. The wave impinges with normal incidence on this plane.
Due to the discontinuity of the constitutive parameters, μ=μ0,εci=ε0
i−jε00
i,
andσiwhere subindex i(i=1,2) refers to medium 1or2,p a r to ft h ew a v e
is propagated through medium 2and part is re flected back through medium
1. Therefore, the total field in medium 1(wherez< 0) and medium 2(where
z> 0)i sg i v e nb y
Medium 1
Ex1=Ei
x1e−α1ze−jβ1z+Er
x1eα1zejβ1z=Ei
x1e−γ1z+Er
x1eγ1z
(4.1a)
Hy1=Ei
x1
ηc1e−α1ze−jβ1z−Er
x1
ηc1eα1zejβ1z=Ei
x1
ηc1e−γ1z−Er
x1
ηc1eγ1z
(4.1b)
Medium 2
Ex2=Et
x2e−α2ze−jβ2z=Et
x2e−γ2z(4.1c)
Hy2=Et
x2
ηc2e−α2ze−jβ2z=Et
x2
ηc2e−γ2z(4.1d)
4.1. NORMAL INCIDENCE. 87
The ¿¿¿superindices?? i,r,a n dtindicate the incident wave (medium 1), the re flected 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 re flected wave propagates in the −ˆzdirection. In these expressions,
ηci=p
μ0/εecirepresents the impedance (3.24) of medium iwhileγiis the
complex propagation factor (3.17),
γi=αi+jβi (4.2)
whereαiandβiare the attenuation and propagation constants (3.18a)and
(3.18b), respectively
βi=ωp
μoε0
i√
2∙q
1+(σe/ωε0
i)+1¸1/2
(4.3)
αi=ωp
μoε0
i√
2∙q
1+(σe/ωε0
i)−1¸1/2
(4.4)
The time dependence of the fields is achieved by adding the factor ejωtto (4.1).
For each instant of time, by imposing the boundary conditions in the plane z=0 (2.182b)
onto the tangential components of EandH,
E1t=E2t (4.5a)
H1t=H2t (4.5b)
we obtain
ωi=ωt=ωr=ω (4.6a)
Er
x1=ΓLEi
x1 (4.6b)
Et
x2=TLEi
x1=( 1+ΓL)Ei
x1 (4.6c)
Hr
y1=−ΓLHi
y1 (4.6d)
Ht
y2=ηc1
ηc2TLHi
y1 (4.6e)
whereΓLis the re flection coe fficient in the plane z=0 defined by
ΓL=ηc2−ηc1
ηc2+ηc1=|ΓL|ejΦL(4.7)
andTLis the transmission coe fficient in the same plane, de fined by
TL=2ηc2
ηc2+ηc1(4.8a)
TL=1 +ΓL=|TL|ejΨL(4.8b)
If there is an impedance adaptation ( ηc2=ηc1) then there is no re flected
wave, and so all the incident energy is absorbed by the second medium.
88CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
From (4.1a) and (4.6b), the total electric field in the first medium can be
expressed as
Ex1=Ei
x1e−α1ze−jβ1z(1 +ΓLe2α1ze2jβ1z) (4.9)
=Ei
x1e−α1ze−jβ1z(1 +Γ(z)) =Ei
x1e−γ1z(1 +Γ(z)) (4.10)
whereΓ(z),d efined as
Γ(z)=ΓLe2α1ze2jβ1z(4.11)
is the re flection coe fficient in the plane z=z. Similarly, for the magnetic field,
we have
Hy1=Ei
x1
ηc1e−γ1z(1−Γ(z)) (4.12)
The impedance associated with the total field at a coordinate point zin the
first medium is de fined as
ηinp(z)=Ex1
Hy1¯¯¯¯
z=ηc11+Γ(z)
1−Γ(z)=ηc1ηc2−ηc1tanh(γ1z)
ηc1−ηc2tanh(γ1z)(4.13)
The impedance ηinp(z)is continuous through the interface, because the tan-
gential components Ex1andHy1are similarly continuous, while the re flection
coefficientΓis discontinuous.
4.1.2 Perfect/Lossy dielectric interface
In the particular case in which the first dielectric is perfect, i.e. lossless ( σ1=0
andε00
1=0,ε1=ε0
1,γ1=jk1), the characteristic impedances reduce to
η1=rμ0
ε1(4.14)
and the coe fficient of re flection (4.7) at the interface ( z=0) becomes
ΓL=1−q
εec2
ε1
1+q
εec2
ε1(4.15)
Sinceσ1=0andε00
1=0,it follows that α1=0and therefore
Ex1=Ei
x1e−jk1z(1 +ΓLe2jk1z)=Ei
x1e−jk1z(1 +Γ(z)) (4.16)
and
Hy1=Ei
x1
η1e−jk1z(1−Γ(z)) (4.17)
where
Γ(z)=ΓLe2jk1z(4.18)
The input impedance (4.13) simpli fies to
ηinp(z)=η1ηc2−η1tan(k1z)
η1−ηc2tan(k1z)(4.19)
4.1. NORMAL INCIDENCE. 89
4.1.3 Perfect dielectric/Perfect conductor interface
Another particular case arises when the second medium is a perfect conductor
(η2=0) and therefore TL=0andΓL=−1. Then the fields in the first medium
are
Ex1=Ei
x1e−jk1z¡
1−e2jk1z¢
=Ei
x1¡
e−jk1z−ejk1z¢
=−2jEi
x1sin (k1z) (4.20a)
Hy1=Ei
x1
η1¡
e−jk1z+ejk1z¢
=2Ei
x1
η1cos (k1z) (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.
a) Lossless case
For the first medium, the expression of the total electric field is,
Ex1=Ei
x1e−jk1z+Er
x1ejk1z=( 1+ΓL)Ei
x1e−jk1z+ΓLEi
x1(ejk1z−e−jk1z)
=TLEi
x1e−jk1z+|ΓL|Ei
x1(ej(ΦL+k1z)−ej(ΦL−k1z))
=|TL|Ei
x1ej(ΨL−k1z)+2|ΓL|Ei
x1sin(k1z)ej(ΦL+π/2)(4.21)
By including the time dependence, and assuming an initial phase ϕ=0,w e
obtain the following expression for the total field
Ex1(z,t)=|TL|Ei
0x1cos(ωt−k1z+ΨL)−2|ΓL|Ei
0x1sin(k1z)s i n (ωt+ΦL)
(4.22)
where the first summand of the second member corresponds to a wave that is
propagating, while the secon d 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 coe fficient of transmission, while that of
the standing wave depends on the coe fficient of re flection. The envelope of the
equation (4.22) is termed the diagram of the standing wave. If the coe fficient
of transmission Tis 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
E0x1=Ei
0x1|1+Γ(z)| (4.23a)
H0y1=1
η1Ei
0x1|1−Γ(z)| (4.23b)
90CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
The maximum values of E0x1(the minima of H0y1)a r eg i v e nb y
E0x1(z)max=Ei
0x1+Er
0x1 (4.24)
at the coordinate points
zmax=−ΦL+2nπ
2k1n=0,1,... (4.25)
and the minimum values of E0x(the maxima of H0y), assuming Ei
0x1>Er
0x1,
are given by
E0x(z)min=Ei
0x1−Er
0x1 (4.26)
at the points
zmin=−ΦL+( 2n+1 )π
2k1;n=0,1,... (4.27)
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
SWR =E0x1(z)max
E0x1(z)min=Ei
0x1+Er
0x1
Ei
0x1−Er
0x1=1+|Γ(z)|
1−|Γ(z)|=1+|ΓL|
1−|ΓL|(4.28)
Its value ranges from 1(no re flected wave) to in finity (pure standing wave), i.e.
1≤SWR≤∞ (4.29)
b) Lossy case
In this case, the expression of the total electric field in the first medium is
Ex1(z)= Ei
x1e−α1ze−jβ1z+Er
x1eα1zejβ1z=Ei
x1(e−γ1z+ΓLeγ1z)
=TLEi
x1e−γ1z+ΓLEi
x1(eγ1z−e−γ1z) (4.30)
and, by including the time dependence, we get
Ex1(z,t)= |TL|Ei
0x1e−α1zcos(ωt−β1z+ΨL)+
2|ΓL|Ei
0x1sinh(γ1z)c o s (ωt+ΦL) (4.31)
In these media, it makes no sense to de fine theSWR parameter because the
maxima and minima are not constant.
4.2. MULTILAYER STRUCTURES 91
4.1.5 Measures of impedances
Assuming that the first medium is lossless, from (4.27) the firstfield minimum
occurs at φL+2k1zmin=π. Consequently, we can determine the phase angle φL,
assuming that k1is known and that zminis determined experimentally (by using
an appropriate device to detect the firstfield minimum). If k1is not known, it
can be calculated from the distance between two consecutive minima.
The value of |ΓL|can be found from the ratio between the maximum and
minimum field values E0x1m a x/E0x1m i n =SWR =( 1+ |ΓL|)/(1−|ΓL|).
Note that if the incident wave has an amplitude of one, then |ΓL|is identical
toEr
0and thus we need to measure only this amplitude. Thus η2is 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 astructure in which there are more than two media separated by parallel planes.
To simplify the analysis, we consider th e case of three lossless dielectrics, as
shown in Fig. tal. The generalization to more media, including the possibilityof losses, is straightforward. Clearly, for a wave that is propagating to the rightin medium 2, the problem is analogous to the two-layer cases discussed above.
Therefore, the coe fficient of re flection in the z=0plane is
Γ
23=η3−η2
η3+η2=Γ(z=0 ) (4.32)
where subindex 23refers to the surface that separates medium 2from medium
3. Particularizing (4.13) for z=−lwe have the load impedance ηL
ηL=ηinp(z=−l)=η21+Γ23e−2jkl
1−Γ23e−2jkl(4.33)
Taking into account that ηinp(z)is continuous at an interface, the coe fficient
of reflection (4.7) at z=−l, becomes
ΓL=Γ(z=−l)=ηL−η1
ηL+η1
by introducing (4.33) into this equation and then operating, we get
ΓL=Γ12+Γ23e−2jkl
1+Γ12Γ23e−2jkl(4.34)
where
Γij=ηj−ηi
ηj+ηi(4.35)
Thus, for an electromagnetic wave wi th an amplitude of one, impinging
normally from the first medium onto the structure, the amplitude of the re flected
wave is given by (4.34)
92CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
Quarter-wave layer
For a quarter-wave layer, l=λ/4(e−2jkl=−1), equation (4.34) becomes
Γ=Γ12−Γ23
1−Γ12Γ23(4.36)
Thus, to transmit the incident energy completely (adaptation of impedance),
the coefficient of re flection must be null, and so
Γ12−Γ23=0 (4.37)
Taking into account equation (4.35) we have
η2=√η1η3 (4.38)
as a condition for impedance adaptation to exist.
Half-wave layer
For a half-wave layer, i.e.l=λ/2(e−2jkl=1 ), expression (4.34) is reduced to
Γ=Γ12+Γ23
1+Γ12Γ23(4.39)
For impedance adaptation to exist, the following must be ful filled
Γ12+Γ23=0 (4.40)
By replacing the coe fficients by the values given in (4.35), we have
(η3=η1) (4.41)
and thus ΓL=0irrespectively of η2. Thus any material with a thickness of λ/2
is adapted so long as the impedances of media 1and 3are 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 re flected and transmitted waves within media 1and 2.T o i l l u s t r a t e
this limit process, let us consider the normal incidence of a wave that impingesupon a structure formed of three perfect dielectrics, a ss h o w ni nF i g .t a l . From
the process of multiple re flections and transmissions, we find that in medium 1
ar eflected field is given by
E
r
x1=Ei
0x1(Γ12+T12Γ23T21e−2jk2d+T12Γ2
23Γ21T21e−4jk2d
+T12Γ3
23Γ221T21e−6jk2d+...) (4.42)
4.3. OBLIQUE INCIDENCE 93
iθrθ
tθiθrθ
tθ
Figure 4.2: Poner sistemas de ejes
Observing the second member, we can see that the summands following
thefirst one constitute a geometric progression of common ratio Γ21Γ23e−2jk2d.
Thus the coe fficient of re flection can be written as
ΓL=Γ12+T12Γ23T21e−2jk2d
1−Γ23Γ21e−2jk2d(4.43)
which, taking into account the equalities
Γ21=−Γ12 (4.44)
T12=1+Γ12 (4.45)
T21=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. Ingeneral, in medium 1there exists an incident and a re flected 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. ??, where the waves have been
represented, as usual, by arrows (called rays) in the direction of propagation.These rays are perpendicular to the equiphase planes (wavefronts). The oblique
incident has extensive applications in op tics 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 canbe 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 Tal. The plane of incidence is de fined
94CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
by vector γiand by the zaxis. Let us assume that γiis in the plane y=0
and forms an angle θiwith the zaxis. In the general case of two lossy media,
the electric fields of the incident, re flected, and refracted waves can be written,
respectively, as
Ei=R e {Ei
0e(jωit−γi·r)} (4.47a)
Er=R e {Er
0e(jωrt−γr·r)} (4.47b)
Et=R e {Et
0e(jωtt−γt·r)} (4.47c)
Inz=0, the tangential component of the electric field must be continuous,
and thus we have
Ei
x+Er
x=Et
x (4.48)
so that
Re{Ei
0xe(jωit−γi·r)}+R e{Er
0xe(jωrt−γr·r)}=R e{Et
0xe(jωtt−γt·r)} (4.49)
A similar relation must be ful filled between the components of the fields with
respect to the yaxis. These conditions can be satis fied only if
ωi=ωt=ωr=ω (4.50)
and
γi·r=γr·r=γt·r (4.51)
Sinceγilies on the y=0 plane, from (4.51) it follows that γr
y=γt
y=0,
signifying that the re flected and refracted waves are coplanar with the incident
wave. Thus we have
γi·r=Nc1ω
c[xsinθi+zcosθi] (4.52a)
γr·r=Nc1ω
c[xsinθr+zcosθr] (4.52b)
γt·r=Nc2ω
c[xsinθt+zcosθt] (4.52c)
whereNcis the complex index of refraction of the medium, such that
γ=Ncω
c(4.53)
By substituting (4.52) in (4.49), and by making the coe fficients of xequal,
we get
θi=θr=θ (4.54a)
Nc1sinθ=Nc2sinθt (4.54b)
These equations, together with the copla narity of the rays, constitute Snell’s
laws.
4.4. INCIDENT WAVE WITH THE ELECTRIC FIELD CONTAINED IN THE PLANE OF INCIDENCE 95
For lossless media Eq. (4.54b) simpli fies to
N1sinθ=N2sinθt (4.55)
where
Ni=c
vpi=(μriεri)1
2 (4.56)
or, for nonmagnetic media,
Ni=(εri)1
2 (4.57)
Next we study the relations between the amplitudes of the incident, trans-
mitted and re flected 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 straightforward
2. Let us analyze
the problem in two stages, firstly where the electric field~Eioscillates in the
incidence plane, and then where it oscilla tes perpendicularly to the same plane.
Any other case can be considered a superposition of these two situations.
4.4 Incident wave with the electric field con-
tained in the plane of incidence
From the continuity of the tangential components of EandH(Eqs. (4.5a) and
(4.5b), we obtain (Fig. 9.2)
Ei
kcosθ+Er
kcosθ−Et
kcosθt=0 (4.58a)
1
η1(Ei
k−Er
k)−1
η2Et
k=0 (4.58b)
where the subindex kindicates that the physical magnitude in question lies in
the incidence plane and
~Ei
k=Ei
0ke−jki·r(4.59a)
~Er
k=Er
0ke−jkr·r(4.59b)
From (4.58) we find that
Γk=Er
k
Eik=η2cosθt−η1cosθ
η2cosθt+η1cosθ(4.60a)
τk=Et
k
Eik=2η2cosθ
η2cosθt+η1cosθ(4.60b)
WhereΓkandτkare the coe fficients of re flection and transmisison, respectively.
If medium 2is a perfect conductor ( η2=0)t h e nΓk=−1.
96CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
iEr
rEr
tEr
tHrrHriHr
θθ
tθinˆrnˆ
tnˆiEr
rEr
tEr
tHrrHriHr
θθ
tθinˆrnˆ
tnˆ
Figure 4.3: cuidado con superindices y subindices...
¿¿¿ For lossless non-magnetic materials ?? ( μ1=μ2=μ0) such that cuidado notacion de vel
fase.
N12=η2
η1=v2
v1=N1
N2=sinθt
sinθ(4.61)
withN12 being the ratio of the indices of refraction of medium 1 and medium 2, expressions
(4.60a) and (4.60b) are reduced to
Er
k
Eik=tan (θt−θ)
tan (θt+θ)(4.62a)
Et
k
Eik=2s i nθtcosθ
sin (θt+θ)c o s(θt−θ)(4.62b)
The total electric field~E1
kin medium 1is given by
~E1
k=~Ei
k+~Er
k (4.63)
where
~Ei
k=Ei
kcosθˆx−Ei
ksinθˆz (4.64a)
~Er
k=Er
kcosθˆx+Er
ksinθˆz (4.64b)
Thus we have
~E1
k=c o sθ³
Ei
0ke−jki·r+Er
0ke−jkr·r´
ˆx+s i nθ³
Er
0ke−jkr·r−Ei
0ke−jki·r´
ˆz
(4.65)
2If the medium is lossy, we must replace { η,jk }b y(ηc,γ)
4.4. INCIDENT WAVE WITH THE ELECTRIC FIELD CONTAINED IN THE PLANE OF INCIDENCE 97
that is,
E1
kx=Ei
0kcosθ³
e−jki·r+Γke−jkr·r´
(4.66a)
E1
kz=Ei
0ksinθ³
Γke−jkr·r−e−jki·r´
(4.66b)
Taking into account that
kr·r=−krzcosθ+krxsinθ (4.67a)
ki·r=kizcosθ+kixsinθ (4.67b)
and by substituting these equations in (4.66), we find that
E1
kx=Ei
0ke−jki(xsinθ+zcosθ)³
1+Γke2jkizcosθ´
cosθ (4.68a)
E1
kz=−Ei
0ke−jki(xsinθ+zcosθ)³
1−Γke2jkizcosθ´
sinθ (4.68b)
When the time factor ejωtis introduced, the term e(jωt−jkixsinθ)represents
a wave that is propagating in the direction of the xaxis, while the term
³
e−jkizcosθ+Γkejkizcosθ´
ejωt(4.69)
¿¿¿¿ ¿¿¿¿ ¿¿ From (4.68a) or the corresponding one from (4.68b) gives us the
superposition of two waves that are propagating with respect to the zaxis, but
in opposite directions. In other words, a stationary wave overlies a traveling
one such that the energy that is transported in direction z,f r o mm e d i u m 1to
medium 2, is transported by the traveling wave.????. For the case of a perfect
conductor, Γk=−1and there exits only a standing wave along the zaxis.
The total magnetic field~H1
⊥in medium 1is
~H1
⊥=Hi
0⊥e−jki·r+Hr
0⊥e−jkr·r=1
η1³
Ei
0ke−jki·r−Er
0ke−jkr·r´
ˆy=
Hi
0⊥e−jki(xsinθ+zcosθ)³
1−Γke2jkizcosθ´
(4.70)
where the symbol ⊥indicates that the magnitude in question is perpendicular
to the incidence plane
In the case of a perfect conductor, it is straightforward to show that there is
no energy flow towards z, but there there is towards x, as the mean time value
of Poynting’s vector towards zis zero.
98CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
iErrEr
tErtHrrHr
iHr
θθ
inˆrnˆ
tnˆtθiErrEr
tErtHrrHr
iHr
θθ
inˆrnˆ
tnˆtθ
Figure 4.4: cuidado el re flejado tiene mal el sentido del campo electrico
4.5 Wave incident with the electric field perpen-
dicular to the plane of incidence
As above, the continuity equations are used for the tangential components of E
andHand thus (Fig.9.3):
~Ei
⊥+~Er
⊥=~Et
⊥ (4.71)
~Ei
⊥−~Er
⊥
η1cosθ=~Et
⊥
η2cosθt (4.72)
where the subindex ⊥indicates that the physical magnitude in question corre-
s p o n d st ot h ec a s ei nw h i c ht h ee l e c t r i c field of the incident wave is perpendicular
to the plane of incidence.
By resolving the two Eqs. (4.70) and (4.71), we get
Γ⊥=Er
⊥
Ei⊥=η2cosθ−η1cosθt
η2cosθ+η1cosθt(4.73a)
τ⊥=Et
⊥
Ei⊥=2η2cosθ
η2cosθ+η1cosθt(4.73b)
where the parameters Γ⊥=Er
⊥/Ei
⊥andτ⊥=Et
⊥/Ei
⊥are the coe fficients of
reflection and transmission, respectively.
For lossless non-magnetic materials, Eqs. (4.73) are transformed into
Er
⊥
Ei
⊥=sin (θt−θ)
sin (θt+θ)(4.74a)
Et
⊥
Ei⊥=2s i nθtcosθ
sin (θt+θ)(4.74b)
4.5. WAVE INCIDENT WITH THE ELECTRIC FIELD PERPENDICULAR TO THE PLANE OF INCIDENC E
By operating in a similar way to that described for the case of Ei
k,we arrive
at the following for the total electric and magnetic fields in a lossy medium 1
~E1
⊥=Ei
0⊥e−jkixsinθ³
e−jkizcosθ+Γ⊥ejkizcosθ´
ˆy (4.75a)
~H1
k=Ei
0⊥
η1cosθ³
e−jki·r−Γ⊥e−jkr·r´
ˆx
−Ei
0⊥
η1sinθ³
e−jki·r+Γ⊥e−jkr·r´
ˆz (4.75b)
As in the case of ~Ek,t h e field behaves as a travelling wave towards xand
as a travelling wave overlying a standing one towards z. The formulas (4.62)
and (4.74) are known as Fresnel’s formulas, which give the relations between
the amplitudes and phase of the incident, re flected, and tranmitted waves.
100CHAPTER 4. REFLECTION AND REFRACTION OF PLANE WAVES
Chapter 5
Electromagnetic
wave-guiding structures :
Waveguides andtransmission lines
5.1 Introduction
There are many engineering applications in which it is necessary to use devices
to con fine the propagation of the electrom agnetic waves in order to transmit
electromagnetic energy from one point to another with a minimum of interfer-
ence, radiation, and heat losses. Althou gh 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 zfor wavelengths numerous enough to
make border e ffects 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 thecross-sectional shape of some guiding t ransmission 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
101
102CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
waveguides the field decays exponentially away from the dielectric in the trans-
verse plane, and consequently the electromagnetic waves are con fined 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 con fined to the core (which has a larger
refractive index than the cladding) by total internal re flection as it propagates
along the fib e ra n dt h ew a v ei sc o n fined without need of any conducting walls.
T h ec h o i c eo fas p e c i fic 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 ine fficient for transferring electromagnetic energy farther than the
lower range of microwaves. Coaxial lines and hollow metal pipe waveguides aremore efficient than two-wire lines for transferring electromagnetic energy be-
cause the fields are completely con fined 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, signi ficant 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 ofmuch 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 impracticalfor frequencies lower than 1GHz. On the other hand, at optical frequencies
the size of a metal waveguide must be too small (in the range of the μm) 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 inexp e n s i v e ,c o m p a c t ,a n ds i m p l et om a t c h
solid-state devices using printed-circuit technology. Planar lines allow di fferent
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, which are brie fly described
in Section ??.
The field con figurations 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 se parated by a lossless dielectric).can
be expressed as a superposition of plane waves, the study of the propagation is
greatly simpli fied when we seek other kinds of solutions called transverse mag-
netic (TM) modes, transverse electric (TE) modes, or transverse electromagnetic
5.2. GENERAL RELATIONS BETWEEN FIELD COMPONENTS 103
Conductor
Dielectric
Dielectric 2
Dielectric 1Conductors
DielectricConductor
Dielectric(1)Conductors
(2) (3) (4)
(7) (8)Dielectric
(6)ConductorsDielectric
(5)Dielectric
ConductorsConductor
DielectricConductor
Dielectric
Dielectric 2
Dielectric 1Dielectric 2
Dielectric 1ConductorsConductors
DielectricConductor
DielectricConductor
Dielectric(1)Conductors
(2) (3)Conductors
(2) (3) (4)
(7) (8)Dielectric
(6)ConductorsDielectric
(6)ConductorsDielectric
(5)Dielectric
Conductors
(5)Dielectric
Conductors
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.
(TEM) modes. These terms indicate tha t, in the direction of propagation, the
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 TMmodes:
1) TM and TE modes have a cuto fffrequency below which they cannot
propagate, which depends on the cross-sect ional 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 andTEM 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 lossl ess 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 e ffect of lossy
media is analyzed in the final section. The study of some speci fic geometries is
left for the next chapter.
5.2 General relations between field components
Let us assume that a time-harmonic wave propagates along the z-axis, in the
+zdirection, in a lossless guiding transmission system. Thus the dependence
104CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
onzand time tis given by the factor ej(ωt−βgz)and the fields are of the general
form
Re(
E0ej(ωt−βgz)
H0ej(ωt−βgz))
=R e½~Eejωt
~Hejωt¾
(5.76)
where ~E=E0e−jβgzand~H=H0e−jβgzandβgis the wavenumber of the
guided wave. Because the geometry and constitutive parameters do not change
along the z-axis,E0andH0are functions only of the transverse coordinates.
To determine ~Eand~H,w ew i l l first show that it is possible to express
their transverse components, ~Etand~Ht, in terms of their z-components, ~Ez
and~Hz. For this, we divide the three dimensional Laplacian operator ∇2in the
homogeneous Helmholtz wave equations (3.26) into two parts. One part, ∂2/∂z2,
acts only on the axial coordinate, z, and the other, ∇2
t, on the transverse ones
only1, i.e.
∇2=∂2
∂z2+∇2
t (5.77)
Since∂/∂z≡−jβg, the wave equations can be written as
¡
∇2+k2¢½~E
~H¾
=¡
∇2
t+h2¢½~E
~H¾
=0 (5.78)
where
h2=k2−β2
g (5.79)
andk=ω(με)1
2is the wavenumber for the wave propagating in an unbounded
medium of the matter which fills the transmission system. By particularizing
(5.78) for the zfield component, we have
¡
∇2
t+h2¢½Ez
Hz¾
=0 (5.80)
This equation, when solved together with the boundary conditions of a given
structure, has solutions for an in finite but discrete number, m, of characteristic
values (eigenvalues) hm, i.e.
¡
∇2
t+h2
m¢½Ezm
Hzm¾
=0 (5.81)
where
h2
m=k2−β2
gm (5.82)
withEzm,orHzmbeing the corresponding functions characteristic ( eigen-
functions) which satisfy the equations (5.81) and the corresponding boundary
conditions, which are determined by the geometry of the system.
1For example in Cartesian coordinates we have ∇=∇t+ˆz∂
∂zwhere∇t=∂
∂xˆx+∂
∂yˆyso
that∇2
t=∂2
∂x2+∂2
∂y2.
5.2. GENERAL RELATIONS BETWEEN FIELD COMPONENTS 105
Now we are going to demonstrate that, once equation (5.80) has been solved,
we can obtain ~Etor~HtfromEzandHz. From Maxwell’s equations (1.67c)
and (1.67d), in a sourceless region, we have
∇×~E =−jωμ~H (5.83a)
∇×~H =jωε~E (5.83b)
The transverse components of these equations can be written as
(∇×~E)t=∇t×~Ez+∇z×~Et=−jωμ~Ht (5.84a)
(∇×~H)t=∇t×~Hz+∇z×~Ht=jωε~Et (5.84b)
Thus, as ~Ezand~Hzare assumed to be known, we have a system of two equa-
tions and two unknowns, ~Etand~Ht, the solutions to which are
~Ht=j
h2³
ωε∇t×~Ez−βg∇tHz´
(5.85a)
~Et=−j
h2³
ωμ∇t×~Hz+βg∇tEz´
(5.85b)
According to (5.85), once the zcomponents of the fields are known, the trans-
verse components can also be calculated. Moreover, in ideal guiding ¿¿struc-
tures?? , we can express any field propagating in the homogeneous guiding
transmission structure as a linear superposition of TE, TM and TEM waves ormodes. Clearly, it is not possible to find speci fice x p r e s s i o n sf o rt h e field dis-
tribution of any of these modes without previously knowing the geometry and
characteristics of the transmission sy stem. 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 Hz=0in (5.85). Thus we have
~Et=−jβg
h2∇tEz=∇t1
h2∂Ez
∂z=∇tΦTM (5.86a)
~Ht=jωε
h2∇t×~Ez=−jωε
h2ˆz×∇tEz=ωε
βgˆz×~Et=1
ZTMˆz×~Et
(5.86b)
w h e r ew eh a v ed e fined the scalar potential for the TM modes, ΦTM,a s
ΦTM=1
h2∂Ez
∂z(5.87)
To obtain (5.86b), we have used the equality ∇t×~Ez=−ˆz×∇tEzand de fined
the frequency-dependent quantity ZTM,a s
ZTM=βg
ωε=ηβg
k(5.88)
106CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
whereη=(μ/ε)1
2is the intrinsic impedance of the dielectric that fills the trans-
mission system. The quantity ZTM, which has the dimensions of impedance, is
called the wave impedance for the TM modes. From (5.86b), we can see that
~Et,~Ht,a n d ˆzform a right-handed system when the wave propagates in the
z-positive direction.
Thus, from (5.86a), in TM modes, ~Etc a nb ew r i t t e na st h eg r a d i e n to fa
scalar function ΦTM. This result could have been obtained by simple reasoning
from Faraday’s law (5.83a), taking into account that, since ~Hhas only trans-
verse components the same is true for ∇×~E. Therefore, from Stokes’ theorem,
we have
Z
S(∇×~E)·ˆzds=I
Γ~E·dl=0 (5.89)
whereSis a transverse surface normal to the zaxis. But Ezcannot contribute
to the line integral because the integration path Γlies on the transverse plane.
Therefore
I
Γ~E·dl=I
Γ~Et·dl=0 (5.90)
which implies that Etis conservative and, therefore, can be written as the
gradient of a scalar function ΦTM.
5.2.2 Transverse electric (TE) modes
For TE modes, from equations (5.85), with Ez=0,w eh a v e
~Ht=−jβg
h2∇tHz=∇t1
h2∂Hz
∂z=∇tΦTE (5.91a)
~Et=−jωμ
h2∇t×~Hz=jωμ
h2ˆz×∇tHz=−ωμ
βgˆz×~Ht=−ZTEˆz×~Ht
(5.91b)
where
ΦTE=1
h2∂Hz
∂z(5.92)
is the scalar potential for TE waves and
ZTE=ωμ
βg=ηk
βg(5.93)
is the wave impedance for the TE mode. From (5.91b) we can see that ~Et,~Ht,
and ˆzform a right-handed system when the wave propagates in the z-positive
direction.
The fact that, according to (5.91a), ~Htcan be expressed as the gradient of
the scalar function ΦTEcan be explained by Ampere’s law, (5.83b), following a
reasoning similar to that used in the case of TE modes.
5.2. GENERAL RELATIONS BETWEEN FIELD COMPONENTS 107
5.2.3 Transverse electromagnetic (TEM) modes
For TEM modes, since Ez=0andHz=0, substituting these values in (5.85),
we can get no null or trivial solutions only if h=0. Consequently, from (5.79),
for TEM modes, we have
β2
g=k2=ω2με (5.94)
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 dielectricbetween the conductors. Since h=0and~E=~E
tand~H=~Ht, (5.78) reduces
to
∇2
t~E =∇2
t~Et=0 (5.95a)
∇2
t~H =∇2
t~Ht=0 (5.95b)
Thus the distribution of the electric and magnetic fields on a transverse
plane satis fies the same bidimensional Laplace’s equation as for the static fields.
This means that, for TEM modes, on a transverse plane, Eis conservative, and
derivable from a scalar function Φby means of the gradient function, i.e.
~E=−∇Φ (5.96)
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 sy stem. 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 conductorsexist, as for example in coaxial, two-wi re or stripline transmission lines, TEM
waves can be propagated along the dielectric separating the conductors.
It is straightforward from (5.84b) that
~E
t=−ZTEM ˆz×~Ht=−ηˆz×~Ht (5.97)
whereZTEM
ZTEM =η=³μ
ε´1
2(5.98)
is the wave impedance for the TEM mode, which coincides with the character-
istic impedance ηof the dielectric that fills the transmission system. cuidado en lo de lineas: usar
o no negritas..?- Now we will demonstrate that, for TEM modes, Maxwell’s equations can be
used to derive a pair of coupled di fferential equations which enable us to study
the propagation of these modes in trans mission lines as voltage and current
waves (instead of electromagnetic waves), using elemental circuit theory.
From (5.96), 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 Vand potential di fferenceΦ2−Φ1will be the same.
108CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
l
Ixy
1CΓ
2CIll
Ixy
1CΓ
2CI
Ixy
1CΓΓ
2CI
Figure 5.2: two conductor transmission linecambiar ejes¡ ¡ Comprobar si los ejes
y el texto coincide. Los conductores se ponen en negro.entero no¿ decir que
c1 son los conductores y que los sentidos de la corrinte in idreccion opuesta en
cada conductor son indicadas. Poner origen coincidiendo con el conductor¡ ¡verdibujos de Salva. en todo ccaso la flecha delde acuerdo con libro de siempre
es al revés
Then for TEM waves (using, without loss of generality, Cartesian coordinates)
we have
V=Φ(2)−Φ(1) =−Z
lEt.dl=−Z
lExdx+Eydy (5.99)
whereΦ(2)andΦ(1)are the values of the scalar function Φat the the con-
ductors 1and 2and where lis any line that joins the equipotential transverse
sections of these conductors ( fig.5.2). Deriving with respect to zand taking into
account Faraday’s law, (1.1c), particula rized for the source-free region, outside
the conductors, we get
∂V
∂z=−Z
l∂Ex
∂zdx+∂Ey
∂zdy=−∂
∂tZ
l−Bydx+Bxdy (5.100)
Note thatR
l−Bydx+Bxdyis the magnetic flux through the area swept,
along a unit of length in the direction z, by the line ljoining the conducting
surfaces. This flux can be expressed by using the magnetostatic de finition of
coefficientLof self-inductance per unit of length, as the product LI. Therefore
we have∂V
∂z=−L∂I
∂t(5.101)
On the other hand, from Ampere’s law, (1.1d), for the source-free dielectric
region, we have
I=I
ΓHt.dl=I
ΓHxdx+Hydy (5.102)
5.2. GENERAL RELATIONS BETWEEN FIELD COMPONENTS 109
R L
CGR L
CG
Figure 5.3:
whereΓis a closed path around one of the wires (see Fig. 5.2). Deriving with
respect to z,w eh a v e
∂I
∂z=I∂Hx
∂zdx+∂Hy
∂zdy=−∂
∂tI
ΓDxdy−Dydx (5.103)
where, in a similar way as above, −Dydy+Dxdxrepresents the flow of vector
Dper unit length in the direction z. Using the magnetostatic de finition of
capacitance Cper unit length, this flux can be expressed as the product CV.
Thus we have
∂I
∂z=−C∂V
∂t(5.104)
Note that, from (5.99) and (5.102), VandImust have the same zdependence
asEandH, respectively. Thus, VandIare also traveling waves.
In summary, according (5.101) and (5.104) we have
∂Φ
∂z=−L∂I
∂t(5.105a)
∂I
∂z=−C∂V
∂t(5.105b)
which are the coupled di fferential equations that voltage and current satisfy at
anyzcross section of an ideal line composed of perfect conductors separated
by a lossless dielectric. Equations (5.105) are called ideal "transmission lineequations". The use of these equations to study the propagation of TEMwaves in transmission lines is considered in Chapter ??.
5.2.4 Boundary conditions for TE and TM modes on per-
fectly conducting walls
For a guiding transmission system with p erfectly conducting walls, the general
boundary conditions on the walls require that the tangential component ~ETof
~Eand the normal component Hnof~Hbe null, i.e.
~ET=ˆn×~E=0 (5.106a)
Hn=ˆn·~H=0 (5.106b)
110CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
where ˆnis the unit vector normal to the conducting walls. However, for TM
and TE modes, as shown below, these conditions can be simpli fied and reduced
to equivalent ones which are expressed only in terms of the zcomponent of the
fields. For example for TM modes, the requirement that
Ez=0 (5.107)
on the perfectly conducting guide walls su ffices to ensure that Eqs. (5.106) are
fulfilled. From (5.86a) and the gradient properties, we can see that ~Etis normal
to the lines where Ez=cteand, therefore, to the boundary of the conductor,
since it represents a line with Ez=0.G i v e n t h a t ~Etand~Htare perpendicular
to each other, the magnetic field is tangential to the conductor and thus Ez=0
is equivalent to Eqs. (5.106).
For TE modes, the necessary and su fficient condition to ensure that Eqs.
(5.106) are ful filled is that the normal derivative of Hzbe null on the perfect
conducting parts of the guiding structure. That is
∂Hz
∂n=∇Hz·ˆn=(∇t+∇z)Hz·ˆn=0 (5.108)
w h e r ew eh a v ed i v i d e d ∇into its transverse and axial components. Taking into
account (5.91a), we see that
∇tHz·ˆn=~Ht·ˆn=0 (5.109a)
which means that ~Htis tangencial to the conductor and therefore, due to the
perpendicularity of the fields, we have
ˆn×~Et=0 (5.110)
In summary, the necessary and su fficient boundary conditions on the perfect
conducting walls of the propagation system are
Boundary conditions on the perfect conducting walls
For TM modes
Ez=0(5.111a)
For TE modes
∂Hz
∂n=0(5.111b)
5.3 Cuto fffrequency
From (5.76) and (5.79) we see that, for propagation to exist, βgmust be real,
and consequently,
k2>h2(5.112)
5.3. CUTOFF FREQUENCY 111
For this reason, βc,d efined as
βc=h=2π
λc(5.113)
is called the cuto ffwavenumber, and λcis the cuto ffwavelength. Thus, from
Eq. (5.79), we have
β2
g=k2−β2
c (5.114)
and, consequently
1
λ2
g=1
λ2−1
λ2
c(5.115)
whereλis the wavelength of a plane wave in the unbounded lossless dielectric
medium filling the ( ¡ ¡better guiding structure¡¡ ¡ ) waveguide, and λgis that of
the wave in the guide. Thus we have
k=2π
λ;βc=2π
λc;βg=2π
λg(5.116)
The cuto fffrequency fcis defined2as
fc=ωc
2π=βc
2π√με=vpβc
2π(5.117)
wherevp=ω/k is the phase velocity in the unbounded medium filling the (
¡¡better guiding structure¡¡¡ ) waveguide. Thus, from (5.79), the wavenumber βg
can be expressed in terms of fc,a s
βg=ks
1−µfc
f¶2
(5.118)
and the corresponding wavelength λgin the ( ¡¡better guiding structure¡ ¡¡ )
guide is
λg=2π
βg=λr
1−³
fc
f´2(5.119)
which is greater than λ. According to (5.118) the wavenumber is imaginary for
modes with frequencies below the cuto fffrequency fc, i.e.f<fc(orλ>λc).
These modes, called evanescent modes, are attenuated and cannot propagate along the guide. Thus,
( ¡¡better guiding structure¡¡¡) waveguides behave as high-pass fil t e r sf o rt h eT Ea n dT Mm o d e ss i n c e
they cannot transmit any of these modes for which the wavelengths, in the unbounded medium filling
the ( ¡¡better guiding structure¡¡¡) waveguide, exceed the value of the cuto ffwavelength.
2For a guiding transmission system with more than one dielectric the cuto fffrequency can be
defin e di nad i fferent manner than (5.117). See for example Section ??.
112CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
In terms of the cuto fffrequency, the expressions of the wave impedances for
the TM and TE modes (5.88) and (5.93) for ZTMandZTEbecome, respectively
ZTM =ηs
1−µfc
f¶2
(5.120a)
ZTE =ηr
1−³
fc
f´2(5.120b)
From (5.120a) and (5.120b), we can see that ZTM<η andZTE>ηand they
become imaginary below the cuto fffrequency. Thus, for f<fc,t h ew a v e g u i d e
behaves, in this respect, as a reactive impedance.
From (5.79), we obtain the dispersion relation
ω=¡
ω2
c+v2
pβ2
g¢1/2(5.121)
which is analogous to that obtained in ( ??) for the transverse electromagnetic
waves in a nonmagnetized plasma. The plot of the phase constant as a func-tion of the frequency ω(dispersion diagram) is shown in Figure 5.2.3. The
transversal broken line corresponds to ω
c=0, i.e. to an unbounded lossless,
nondispersive medium in which the wave propagates at the phase velocity vp
regardless of its frequency. The solid-line curve represents Eq. (5.121) and
shows that the waveguide is very dispersive close to the cuto fffrequency ωc.For
frequencies ω> >ω csuch that their wavelengths are much smaller than the
transversal ( ¡¡better guiding structure¡ ¡¡ ) waveguide dimensions, the walls do
not affect the propagation and the velocity tends to vp.
JV: texto Dispersion diagram. Es la figura de plasmas. ver tb pp 444 del Jonk
The group velocity, vgg, within the guide is given by
vgg=dω
dβg=vps
1−µfc
f¶2
(5.122)
5.4. ATTENUATION IN GUIDING STRUCTURES 113
which is smaller than the phase velocity vpin the unbounded medium. The
phase velocity within the waveguide( ¡ ¡better guiding structure¡¡ ¡ ),vpg,i s
given by
vpg=ω
βg=fλg=vpr
1−³
fc
f´2(5.123)
which is always higher than that in the unbounded medium and is frequency
dependent. Hence single conductor ( ¡¡better guiding structure¡¡¡ ) waveguides
are dispersive transmission systems. Note that
vpg·vgg=v2
p (5.124)
For TEM modes, from (5.94), we have βg=kwhich is real and independent
of the frequency. Thus, all frequencies propagate along a lossless transmission
line at the same phase velocity vpas that of the unbounded homogeneous di-
electric filling the waveguide and there is no cuto fffrequency .
5.4 Attenuation in guiding structures
For a propagating mode an attenuation constant α, owing to energy dissipation within the
waveguide, can arise from losses in the non-perfect conducting walls ( αc)a n d
in the non-perfect dielectric filling the waveguide ( αd). Thus, the attenuation
constant αconsists of two parts α=αd+αc.Dielectric losses are generally
negligible when ( ¡¡better guiding structure¡ ¡¡ )w a v e g u i d e sa r e filled with air,
which has a lower dielectric loss than do conventional dielectrics.
hay que decir TE y TM y TEM..
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 occursin 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 realpropagation constants kandβ
gare replaced by −jγand−jγg, respectively,
whereγ=α+jβandγg=αd+jβgare the complex propagation constants
in the unbounded dielectric filling the waveguide and in the waveguide, respec-
tively. Then, from equations (3.17), (5.79) and (5.113), we have
k2(1−jtanδd)=−γ2=−γ2
g+β2
c (5.125)
114CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
Using the above expressions for γandγgand neglecting the term α2
d, because the
attenuation constant αdis very small, we find
β2
c=k2−β2
g (5.126a)
αd=k2
2βgtanδd=βg2+β2
c
2βgtanδd (5.126b)
Thus, the attenuation factor is proportional to the loss tangent, tanδd, of the dielectric fill-
ing the waveguide . On the other hand, (5.126a) coincides with Eq. (5.114) for
waveguides with ideal dielectric, and c onsequently the phase constant (and thus
the wavelength) remains practically the same as those for a lossless waveguide.
The dependence of the attenuation factor αdon the frequency (assuming a range
of frequencies in which the permittivity of the dielectric remain unchanged) can
be deduced by substituting the expressions of tanδdandβg, given by (1.91)
and (5.118), respectively, in (5.126b). Thus we get
αd=σeη
2(1−(fc/f)2)1/2 (5.127)
whereηis the intrinsic impedance of the dielectric given in (3.33a) and σeis its
effective or equivalent conductivity (1.78). From (5.127) we can see that αdbecomes very
high at frequencies close to the cuto ffvalue, then decreases to a minimum value,
and afterwards increases with the frequ ency, becoming almost proportional to
it.
Wall losses
When the conductivity is finite the tangential magnetic field induces currents
which are not restricted to the surfac e and, according to Ohm’s law, are asso-
ciated with a tangential electric field (i.e. J=σE=ˆn×H) in the walls. The
vector product of the fieldsEandHat 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 e ffect and is subtracted
from the mode that propagates along the waveguide. As a consequence, theamplitude of the electric and magnetic fields of the mode are attenuated accord-
ing toe
−αcz,w h e r eαcis the attenuation constant due to wall losses. We can
d e t e r m i n et h ev a l u eo f αcfor a given propagating mode by taking into account
that the time-average power Pavtransmitted through the cross-section Sof the
guiding transmission system is
Pav=Z
SPav·ds=1
2Z
SRe(~Et×~H∗
t)·ds (5.128)
Since, due to the losses, the amplitude of the field wave varies according to
e−αcz, then,Pavwill vary according to e−2αcz. Moreover, the law of energy
conservation requires that the rate of the decrease of Pavwith distance along
the transmission system equals the time-average power loss on the surface of the
walls per unit length, P0
d, in the direction of propagation. Therefore, we have
5.4. ATTENUATION IN GUIDING STRUCTURES 115
P0
d=−dPav
dz=2αcPav (5.129)
and thus
αc=P0
d
2Pav(5.130)
If~His the ¿¿magnetic?? field existing near the walls, the time-average power
dissipated per unit of length in the walls is given, according to (3.45), by
P0
d=1
2RsZ
ΓH2
0dl=1
2RsZ
Γ~H·~H∗dl (5.131)
whereRsis the surface resistance given by (3.46) and Γis the cross-sectional
contour of the non-perfect conducting walls. Thus the coe fficient of attenuation
of then-th TE or TM mode is found to be
αc=RsU
ΓH·H∗dl
4U
SPav·ds=RsU
ΓH·H∗dl
2U
SRe(Et×H∗
t)·ds(5.132)
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 con figurations or modes in the waveguide coincide
with those found for ideal-wall ( ¡¡better guiding structure¡ ¡¡ ) waveguides.
5.4.2 TEM modes
The coupled di fferential equations (5.105) for ideal transmission lines can be easily extended to lines
with a non perfect of dielectric (constitutive parameters ε,μ,σ ) separating the perfect conductors.
Inthis case, at any zcross-section of the line, an additional current increment
∆Ileaves ..
assuming that the dielectric has a conductivity σsuch that
∆I=gV (5.133)
in which gdeno tes the conductance .....
pp 487 del Jonk: the series and shunt low-parameters randg...
∂Φ
∂z=−L∂I
∂t(5.134a)
∂I
∂z=−C∂V
∂t−gV (5.134b)
116CHAPTER 5. ELECTROMAGNETIC WAVE-GUIDING STRUCTURES: WAVEGUIDES AN D
Chapter 6
Some types of waveguides
and transmission lines
6.1 Introduction
In the previous chapter we examined some general properties of the propagation modes that may
exist in an ideal guiding transmission system which has no sources and is constituted by perfectconductors and one ideal homogeneous dielectric. Speci fic expressions for such modes can be deter-
mined only when the particular geometry of the guide is given. In this chapter we will first analyze
in some detail the homogeneously filled rectangular and circular metallic waveguides. After this,
as a simple example of non homogeneous guiding structure in which the electromagnetic field prop-
agates 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
Q. 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 aandb,witha>b , 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 longitudinal components zof the field with the corresponding boundary conditions
determined by the geometry of the system. The transverse components are then calculated fromthese longitudinal ones.
With axis chosen as shown in the figure, the expressions for the fields in (5.76), take the form
~E =E0(x,y)e−jβgz(6.135a)
~H =H0(x,y)e−jβgz(6.135b)
117
118CHAPTER 6. SOME TYPES OF WAVEGUIDES AND TRANSMISSION LINES
aby
zx
aby
zx
Figure 6.1: Rectangular waveguide of width aand height bRellenar en negro.
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 di fferential equation (5.80) for
Ez=E0z(x,y)e−jβgz(6.136)
c a nb es o l v e db yu s i n g the standard method of separation of variables in rectangular
Cartesian coordinates. For this, we assume, for E0z, solutions in the form of the
product
E0z(x,y)=X(x)Y(y) (6.137)
in which X(x)andY(y)are, respectively, functions only of xandy.
By substituting (6.137) in (5.80) and dividing by E0z,w eg e t
1
Xd2X
dx2+1
Yd2Y
dy2+h2=0 (6.138)
As each summand depends on a di fferent variable, it should be veri fied that
1
Xd2X
dx2=−h2
x (6.139a)
1
Yd2Y
dy2=−h2
y (6.139b)
h2=β2
c=h2
x+h2
y (6.139c)
where we have substituted, according to (5.113), hby the cuto ffwavenumber
βcand where hxandhyare the separation constants to be determined from
the boundary condition (5.111a) at the guide walls. This boundary condition
6.2. RECTANGULAR WAVEGUIDE 119
for the geometry of Figure 6.1 implies
E0z=0 at⎧
⎪⎪⎨
⎪⎪⎩x=½0
a
y=½0
b(6.140)
The solution of the Eqs. (6.139a) and (6.139b) are, respectively,
X =C1sinhxx+C2coshxx (6.141a)
Y=C3sinhyy+C4coshyy (6.141b)
where the Cicoefficients are arbitrary constants to be determined from bound-
ary conditions. Therefore the general solution (6.137) for E0ztakes the form
E0z=(C1sinhxx+C2coshxx)(C3sinhyy+C4coshyy) (6.142)
From the boundary conditions (6.140), we find thatC2=C4=0and
hx=πm
a(6.143a)
hy=πn
b(6.143b)
and thus, from (6.139c),
β2
c=¡πm
a¢2+¡πn
b¢2(6.144)
wheremandnare integers. The di fferent solutions achieved by giving values to mandnare
termed TM mn modes and each set of values of mandnindicates a speci ficm o d e . T h u s ,f r o m
(6.136), (6.142) and (6.143), for TM mn modes, we have
Ez=Amne−jβgzsinπm
axsinπn
by (6.145)
where the product of the constants C1andC3has been replaced by a new
constant Amn.
Once we know the longitudinal component Ez, we can calculate the trans-
verse components ~Etby means of (5.86a) and then, by using (5.86b), which
implies that
ZTM=Ex
Hy=−Ey
Hx(6.146)
we can obtain ~Ht. As a result, we get the following general expressions for the
120CHAPTER 6. SOME TYPES OF WAVEGUIDES AND TRANSMISSION LINES
components of the TM modes in a rectangular waveguide
TMmnmodes in rectangular waveguides
(Ez)TMmn=Amne−jβgzsinπm
axsinπn
by
(~Et)TMmn=³
−jAmnβg
β2
cπm
ae−jβgzcosπm
axsinπn
by´
ˆx−³
jAmnβg
β2
cπn
be−jβgzsinπm
axcosπn
by´
ˆy
(~Ht)TMmn=³
jAmnη−1k
β2
cπn
be−jβgzsinπm
axcosπn
by´
ˆx−³
jAmnη−1k
β2
cπm
ae−jβgzcosπm
axsinπn
by´
ˆy(6.147)
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 Hzand imposing the boundary condition
(5.111b), ∂Hz/∂n =0, on the guide walls. This, for the geometry of Figure
6.1, implies that
∂Hz
∂x=0 at½x=0
x=a
∂Hz
∂y=0 at½y=0
y=b(6.148)
Then, using (5.91a) and (5.91b), and after steps analogous to those followed for
TM modes, we get
TEmnmodes in rectangular waveguides
(Hz)TEmn=Bmne−jβgzcosπm
axcosπn
by
(~Ht)TEmn=³
jBmnβg
β2
cπm
ae−jβgzsinπm
axcosπn
by´
ˆx+³
jBmnβg
β2
cπn
be−jβgzcosπm
axsinπn
by´
ˆy
(~Et)TEmn=³
jBmnηk
β2
cπn
be−jβgzcosπm
axsinπn
by´
ˆx−³
jBmnηk
β2
cπm
ae−jβgzsinπm
axcosπn
by´
ˆy(6.149)
Note that, for both, TM mn and TEmn modes, the subindexes mandn,indicate the num-
ber of half-wave variations of the field in the xandydirections, respectively. dedidir si subindexes or
subindices .F o r a T M mnmode with mornequal to zero, from (6.145), we have (Ez)TE00=0
and consequently, from Eqs (6.147), (~Et)TE00=0 and (~Ht)TE00=0. Hence,
t h e r ei sn oT Mm o d ei nw h i c h mornis equal to zero. This was to be expected
6.2. RECTANGULAR WAVEGUIDE 121
because a TM wave with Ez=0 would degenerate to become a TEM wave
which, as we saw in Subsection 5.2.3, cannot propagate within a waveguide.
For TE mnmodes, it is easy to see from (6.149) that either mornmay be
equal to zero but not both at the same time, since in this case the expression
of(Hz)TEmnin (6.149) reduces to
(Hz)TE00=B00e−jβgz(6.150)
while (Ht)TE00=0 andE=(Et)TE00=0,such that only (Hz)TE00exists.
This field does not ful fil Maxwell’s equations, since a time-varying fieldH
should generate an electric fieldE. Therefore the TE 00mode cannot exist.
Cutofffrequencies in a rectangular waveguide
From (5.117), (6.139c), and (6.143), we see that the cuto fffrequency for either
aT Emnor a TM mnmode is given by
(fc)mn=vp
2∙³m
a´2
+³n
b´2¸1
2
(6.151)
wherevpis 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
(λc)mn =2
h¡m
a¢2+¡n
b¢2i1
2(6.152a)
(βg)mn =∙
k2−³mπ
a´2
−³nπ
b´2¸1
2
(6.152b)
From (6.151) we see that the cuto fffrequency of the modes depends on
the dimensions of the cross-section of the waveguide. Values of the cuto ff
wavelengths and frequencies for several modes are
(λc)TE10=2a;(fc)TE10=vp
2a(6.153a)
(λc)TE01=2b;(fc)TE01=vp
2b(6.153b)
(λc)TE20=a;(fc)TE20=vp
a(6.153c)
(λc)TE11=(λc)TM 11=2ab
(a2+b2)1
2;(fc)TE11=(fc)TM 11=vp¡
a2+b2¢1
2
2ab
(6.153d)
Note that if a=bthe cutofffrequencies of TE 10and TE 01and the two modes
are equal except for a rotation of π/2.
122CHAPTER 6. SOME TYPES OF WAVEGUIDES AND TRANSMISSION LINES
1 1.5 2 2.5 3 3.5 40123456789
TE10TE20TE01m,n
1,0
a/b TE12,TM12
TE22
TE02 TE21,TM21
TE11,TM11
Figure 6.2: Rectangular waveguide: ratio of the cuto fffrequency of several modes to that of the
TE10mode as a function of a/b . mas grande las letras de las coordenadas
The dominant TE 10mode
In practice, we usually wish to have only the mode which has the lowest cuto ff
frequency (¿¿called fundamental or dominant mode??) propagating through
the guide. Thus, in the case of a rectangular waveguide, if a>b , such that
(fc)TE10<(fc)TE01, the waveguide is usually designed so that only the TE 10
mode can be propagated. The cuto fffrequency of the dominant TE 10mode is
selected by means of the dimension a.T h e r a t i o o f t h e c u t o fffrequency of each
mode to that of the TE 10mode as a function of a/bis plotted in Figure 6.2. We
see that the separation of the cuto fffrequencies for di fferent modes is larger for
higher values of the ratio of the aandbdimensions Note that if a'2b,t h e n
the cutofffrequencies of the modes TE 01and TE 20are nearly the same and in
the frequency range v/2a<f<v / 2bonly the TE 10mode can be propagated.
Moreover, if a> 2b,t h e n (fc)TE20<(fc)TE01. As we will see in the next Section,
losses due to non-perfectly conducting walls increase as bdecreases. Thus, to
have the greatest frequency range in which only the TE 10mode can propagate
and, at the same time, to have the smallest losses possible, we usually choosethe dimensions of the guide such that a'2b. Under this condition, only TE
10
modes will propagate in the frequency range (fc)TE10<f< 2(fc)TE10.F o rt h e
dominant TE 10mode the general expressions (6.149) simplify to those given in
(6.154) where the constant B10has been replaced by H0.
6.2. RECTANGULAR WAVEGUIDE 123
Rectangular TE10mode
βc=π
a
fc=vp
2a
βg=q
k2−¡π
a¢2
Hz=H0e−jβgzcosπ
ax
Hx=jH0βga
πe−jβgzsinπ
ax
Hy=0
Ez=0
Ex=0
Ey=−jH0ωμa
πe−jβgzsinπ
ax(6.154)
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 (5.127) and (5.132),
respectively. For a given mode, to obtain the attenuation due to dielectric
losses, we simply need to use, in the f ormula (5.127), the value of the cuto ff
frequency of the mode, given by (6.151), and the values of the constitutiveparameters of the dielectric at the work frequency. However, to find the the
attenuation constant α
cdue to wall losses for any TE or TM mode, though
not complicated, is quite laborious. He re, to illustrate the procedure, we will
consider the particular case of the dominant TE 10mode
Attenuation of the TE 10mode For the TE 10mode, the integrals of the
formula (5.132) can be calculated from the general expressions for the field
components (6.154). Thus, for the denominator, we have
PTE10=Z
S(Pav)TE10·ds=−1
2Zb
0Za
0(EyH∗
x)TE10dxdy =
µaH0
2π¶2
ωμabβg (6.155)
Regarding the integral of the numerator in (5.132), because in the dominant
mode TE 10in a rectangular waveguide the magnetic fieldHhas only Hxand
124CHAPTER 6. SOME TYPES OF WAVEGUIDES AND TRANSMISSION LINES
Hzcomponents, this integral takes the form
Z
Γ~H·~H∗dl=2(Za
0³
|Hx|2+|Hz|2´
dx+Zb
0³
|Hx|2+|Hz|2´
dy)
(6.156)
Using the expressions of HxandHzgiven in (6.154) and by operating, we obtain
Z
Γ~H·~H∗dl=2H2
0"
a
2Ã
1+β2
g
β2c!
+b#
=2H2
0"
a
2µf
fc¶2
+b#
(6.157)
where the last expression is obtained from (5.118). By substituting (6.155) and
(6.157) in (5.132) and after operating, we finally obtain the following expression
for the attenuation factor ( αc)TE10
(αc)TE10=Rs
1+2b
a(fc
f)2
ηbt
1−(fc
f)2=1
ηbµ
μπf
σ(1−(fc
f)2)¶1
2∙
1+2b
a³
fc
f´2¸
Np/m
(6.158)
Following a similar analysis,we can show that the general expresions for the
attenuation constant αcdue to wall losses for any TE mnmode is
(αc)TEmn =2Rs
bηr
1−³
fcmn
f´2(µ
1+b
a¶µfcmn
f¶2
+
Ã
δ0n
2−µfcmn
f¶2!
b
a©
(b
a)m2+n2ª
¡b
a¢2m2+n2)
(6.159)
where
δ0n=n
1q=0
2q6=0(6.160)
While for TM mnmode is
αcTMmn=2Rs
bηr
1−³
fcmn
f´2(b
a)3m2+n2
m2¡b
a¢2+n2(6.161)
These expressions show the dependence of the attenuation on the frequency. Reedactar: ¿ ¡¡Com-
puted values of αcfor a few TE mn and TM mn modes are given in Figure 6.3In practice, surfaces
imperfections, the value of αcmay be greater than the theoretical values. This e ffect can be reduced
using well polished walls.
6.2. RECTANGULAR WAVEGUIDE 125
5 10 20 50 100 2000.010.020.050.10.20.5
f(GHz)
α
c(np/m
)
TM11
TE10TE20TE11
Figure 6.3: Atenuacion en guias rectangulares: a commom characteristic It
tends to in finite when fis close to the cuto fffrequency, decreases toward an
optimum frequency (minimum value of αc) an then increases almost linearly
withf