Waves and transmission lines

192 Pages·2016·2.07 MB·English

by  Macháč J.

CONTENTS

1. INTRODUCTION .............................................................................................. 6
1.1 Description of an electromagnetic field ………………….….............................. 8
1.2 Wave equation ………………...........................................................…............... 13
1.3 Potentials ………….........................................................................…................. 14
2. ELECTROMAGNETIC WAVES IN FREE SPACE .................................… 17
2.1 Solution of the wave equation ....................................…..................................... 17
2.2 Propagation of a plane electromagnetic wave …………………………………. 18
2.3 Wave polarization ……………………………………………………………… 28
2.4 Cylindrical and spherical waves ……………………………………………….. 32
2.5 Problems ……………………………………………………………………….. 34
3. WAVES ON A PLANE BOUNDARY ………………………………………. 35
3.1 Perpendicular incidence of a plane wave to a plane boundary ………………… 35
3.2 Perpendicular incidence of a plane wave to a layered medium ………………... 42
3.3 Oblique incidence of a plane electromagnetic wave to a plane boundary ……... 45
3.4 Problems ……………………………………………………………………….. 54
4. SOLUTION OF MAXWELL EQUATIONS AT VERY HIGH
FREQUENCIES ………………………………………………………………. 56
5. GUIDED WAVES …………………………………………………………….. 61
6. TEM WAVES ON A TRANSMISSION LINE ………………………………. 65
6.1 Parameters of a TEM wave …………………………………………………….. 65
6.2 Transformation of the impedance along the line ………………………………. 71
6.2.1 An infinitely long line ………………………………………………………….. 71
6.2.2 A line of finite length …………………………………………………………… 72
6.2.3 A line terminated by a short cut and by an open end ……….………………….. 73
6.3 Smith chart ……………………………………………………………………… 76
6.4 Problems ……………………………………………...………………………… 87
7. WAVEGUIDES WITH METALLIC WALLS ……………………………... 88
7.1 Parallel plate waveguide ……………………………………………………….. 88
7.2 Waveguide with a rectangular cross-section …………………………………… 94
7.3 Waveguide with a circular cross-section ………………………………………. 103
7.4 Problems ………………………………………………………………………... 107
8. DIELECTRIC WAVEGUIDES ……………………………………………… 108
8.1 Dielectric layers ………………………………………………………………… 109
8.2 Dielectric cylinders ……………………………………………………………... 114
8.3 Problems ………………………………………………………………………... 115
9. RESONATORS ……………………………………………………………….. 116
9.1 Cavity resonators ……………………………………………………………….. 116
9.2 Problems ………………………………………………………………………... 120
10. RADIATION ………………………………………………………………….. 121
10.1 Elementary electric dipoles …………………………………………………….. 121
10.2 Elementary magnetic dipoles …………………………………………………... 126
10.3 Radiation of sources with dimensions comparable with the wavelength ………. 129
10.4 Antenna parameters …………………………………………………………….. 133
10.5 Antenna arrays ………………………………………………………………….. 134
10.6 Receiving antennas ……………………………………………………………... 137
4

book - 1
10.7 Problems ……………………………………………………………………….. 140
11. WAVE PROPAGATION IN NON-ISOTROPIC MEDIA ………………… 141
11.1 Tensor of permeability of a magnetized ferrite ………………………………… 142
11.2 Longitudinal propagation of a plane electromagnetic wave in a magnetized
ferrite …………………………………………………………………………… 146
11.3 Transversal propagation of a plane electromagnetic wave in a magnetized
ferrite …………………………………………………………………………… 151
11.4 Applications of non-reciprocal devices ………………………………………… 154
11.5 Problems ………………………………………………………………………... 155
12. APPLICATIONS OF ELECTROMAGNETIC FIELDS ………………….. 156
12.1 Introduction to microwave technology …………………………………………. 156
12.2 Antennas ………………………………………………………………………... 162
12.2.1 Wire antennas …………………………………………………………………... 164
12.2.2 Aperture antennas ………………………………………………………………. 165
12.2.3 Broadband antennas …………………………………………………………….. 167
12.2.4 Planar antennas …………………………………………………………………. 168
12.3 Propagation of electromagnetic waves in the atmosphere ……………………… 169
12.4 Optoelectronic ………………………………………………………………….. 173
12.4.1 Optical waveguides …………………………………………………………….. 173
12.4.2 Optical detectors ………………………………………………………………. 176
12.4.3 Optical amplifiers and sources …………………………………………………. 177
12.4.4 Optical modulators and sensors ………………………………………………… 178
13. MATHEMATICAL APPENDIX …………………………………………….. 180
14. BASIC PROBLEMS ………………………………………………………….. 191
15 LIST OF RECOMMENDED LITERATURE ………………………………. 194

5

book - 1
1. INTRODUCTION

This textbook is aimed at students of the Faculty of Electrical Engineering, Czech
Technical University taking a course in Waves and Transmission Lines. The textbook builds
on basic knowledge of time varying electromagnetic fields gained from courses in physics
and electromagnetic field theory. The textbook introduces all the basic knowledge that an
electrical engineer specializing in radio engineering and telecommunications should have and
that is necessary for further courses such as microwave engineering, antennas and
propagation, optical communications, etc. Sequential mastering of wave theory contributes to
the final objective of university studies, which is to enable graduates to do creative work.
The course in Waves and Transmission Lines studies the theory and applications of
classical electrodynamics. It is based on Maxwell’s equations. This course provides a basis
for understanding the behavior of all high frequency electric circuits and transmission lines,
starting from those applied to transmitting and receiving electric energy, processing signals,
microwave circuits, optical fibers, and antennas. The main applications lie in wireless
communications, radio engineering and optical systems.
The text follows the classical approach to macroscopic electrodynamics. All quantities
are assumed to be averaged over the material, which by its nature has a microscopic structure
consisting of atoms. This confines the description of electromagnetic effects using
macroscopic theory on the high frequency side, as the wavelength must be much longer than
the dimensions of the atoms and molecules. This boundary lies in the range of ultraviolet
light. Nevertheless, the spectrum of frequencies in which electromagnetic effects can be
treated using this macroscopic theory is really huge – over 17 orders. And this whole
spectrum really is used in a variety of different applications. Modern communication systems
use electromagnetic waves with ever shorter wavelengths. The spectrum of electromagnetic
waves is shown in Fig. 1.1.
First, we review the basic relations from electromagnetic field theory and introduce
potentials describing a time varying electromagnetic field. The concept of a plane
electromagnetic wave is carefully reviewed. In addition, a cylindrical wave together with a
spherical wave are briefly introduced. The behaviour of a plane electromagnetic wave on the
boundary between two different materials is studied in detail. Here we will start treating the
incidence of a wave perpendicular to the plane boundary and to a layered medium. Oblique
incidence is studied in general, and then special effects such as total transmission and total
reflection are treated. Specific aspects of solving Maxwell equations at very high frequencies
are discussed separately.
Transmission lines are designed to transmit guided waves. After introducing the
general properties of guided waves the TEM wave is treated, and the transformation of
impedances along a line is described. The Smith chart, a very effective graphical tool for
analysis and design of high frequency circuits, is described and its basic applications are
explained through particular problems. Waves propagating along waveguides with metallic
walls of rectangular and circular cross-sections are studied. Dielectric waveguides are treated
separately. They form the basis of optical fibers. Cavity resonators, unlike low frequency L-C
resonant circuits, are able to resonate on an infinite row of resonant frequencies. Several kinds
of such resonators are analyzed in the text.
Attention is paid to problems of radiation of electromagnetic waves. This covers
antenna theory. Elementary sources of an electromagnetic field, such as an electric dipole and
a magnetic dipole, are studied and compared. Then radiation from sources with dimensions
comparable with the wavelength is described. The basic antenna parameters are defined. The
basic idea of antenna arrays is built up. Finally, receiving antennas are dealt with, and the
effective antenna length and effective antenna surface are derived.
6

book - 1
Non-isotropic materials are introduced, and the tensor of permeability of a magnetized
ferrite is calculated. The propagation of a plane electromagnetic wave in this ferrite material
homogeneously filling an unbounded space is studied, in particular when the wave propagates
both in the direction parallel with the magnetizing field and in the direction perpendicular to
the magnetizing field. Some devices using non-isotropic materials are mentioned.
frequency wavelength classification applications
(Hz) (m)
1018
10-9 ultraviolet radiation
nm
360
violet
460
blue
argon laser 490 nm
1015 green 560
visible light yellow
10-6 He-Ne laser 630 nm
red 660
infrared radiation 760
1012 quasi optical waves
10-3
m
millimeter waves i radar, space investigation
10-2 cr
o
centimeter waves w radar, satellite commun.
a
v
109 e
decimeter waves s radar, TV, navigation
100
very short waves TV, FM radio, services
10
short waves radio, services
102
106
medium waves AM radio
103
long waves
104
103
phone, audio
106
The spectrum of electromagnetic waves.
Fig. 1.1
The applications of electromagnetic fields in particular branches of electrical
engineering are briefly introduced. An introduction to microwave technology is given. Here
scattering parameters are introduced. The paragraph on antennas represents a continuation of
Chapter 10, introducing basic types of antennas. Particular mechanisms of wave propagation
in the atmosphere are explained. The transmission formula and radar equations are derived.
The basics of optoelectronics are presented. This involves a characterization of optical
waveguides, detection and optical detectors, optical amplifiers, and the sources of optical
7

book - 1
radiation, namely lasers, and finally modulators of an optical beam and a short review of
optical fiber sensors.
The book has a mathematical appendix summarizing the necessary knowledge of
mathematics. Basic problems in the form of questions are summarized at the end of the text.
They help the students in preparing for their examinations. A list of suggested literature is
given.
The textbook treats time varying electromagnetic fields. Harmonic dependence on
time is assumed throughout most of the text. Such fields are described using symbolic
complex quantities called phasors, and time dependence in the form ejωt is assumed. These
phasors are not marked by special symbols. When we need to emphasize an instantaneous
value, we will mark it by showing dependence on time, e.g., E(t). Vectors will be marked by
bold characters, e.g., E.

1.1 Description of an electromagnetic field
The sources of an electromagnetic field are electric charge Q [C] and electric current I
[A], which is nothing else than the flow of charge. Charge is often distributed continuously in
a space, on a surface, or along a curve. It is convenient in this case to define the corresponding
charge densities. The charge volume density is defined as the charge amount stored in a unit
volume

∆q
ρ = lim [C/m3] . (1.1)
∆V→0∆V

The charge surface density is defined by analogy

∆q
σ = lim [C/m2] . (1.2)
∆S→0∆S

The charge linear density is defined as the charge stored along a line or a curve of unit
length

∆q
τ= lim [C/m] , (1.3)
∆l→0 ∆l

Electric current is created by a moving charge. This is defined as the passing charge
per time interval ∆t

∆q
I = lim [A=C/s] . (1.4)
∆t→0 ∆t

It is useful to define current densities, which are vector quantities, as it is necessary to define
the current flow direction. Current density is defined as

∆I
J = lim i [A/m2] , (1.5)
∆S→0∆S 0

where i is a unit vector describing the current flow direction. It is sometimes useful to use the
0
abstraction of a surface current passing along a surface, see Fig. 1.2. The linear density of
8

book - 1
this current is defined as

∆I
K = lim i [A/m] . (1.6)
∆x→0∆x 0

The surface current and its density are an
abstraction used to simplify the mathematical I
i
description of the current passing a conductor at a 0
K
high frequency. Due to the skin effect, current
S
flows through only a very thin layer under the ∆x
material surface. In fact the surface current
defined by (1.6) represents the finite current
passing a cross-section of zero value. This Fig. 1.2
requires infinite material conductivity. Another
widely used abstraction is a current filament representing a conductor of negligible cross-
section (e.g., a line) carrying a finite current.
The total electric current crossing a closed surface is related to the charge accumulated
inside the volume surrounded by this surface by the continuity equation in an integral form

∫∫J⋅dS+ jωQ =0 , (1.7)
S

or in a differential form

divJ+ jωρ=0 . (1.8)

where Q is the total charge accumulated in volume V with boundary S. The current density is
related by Ohm’s law in the differential form to an electric field

J=σE , (1. 9)

where σ is conductivity in S/m. It should be noted that in spite of the movement of the free
electrons, a conductor passed by an electric current stays electrically neutral, as the charge of
the electrons is compensated by the positive charge of the charged atomic lattice.
The electric field is described by the vector of electric field intensity E, the unit of
which is V/m. The magnetic field is described by the vector of magnetic field intensity H, the
unit of which is A/m. These vectors are related to the induction vectors by material relations

B = µH , (1.10)

D=εE , (1.11)

where ε=εε and µ= µµ are permittivity and permeability, respectively, and
0 r 0 r
ε =10−9/36πF/m and µ = 4π10−7H/m are the permittivity and permeability of a vacuum,
0 0
respectively. Vectors E and H are related by the set of Maxwell’s equations. Their
differential form reads

( )
rotH = σ+ jωεE+J , (1.12)
S

9

book - 1
rotE = −jωµH , (1.13)

( )
div εE = ρ , (1.14)
0

( )
div µH =0 , (1.15)

where J is a current supplied by an external independent source, ρ is the volume density of a
S 0
free charge supplied by an external independent source. The differential form of Maxwell’s
equations is valid only at those points where the field quantities
are continuous and are continuously differentiated functions of n
position. They are not valid, for example, on a boundary between 1 ε1 µ1 σ1
two different materials where the material parameters change
ε µ σ
step-wise, Fig. 1.3. For this reason we have to append 2 2 2 2
corresponding boundary conditions to these equations. We
suppose that the boundary between two materials contains a free Fig. 1.3

electric charge with density σ, and electric current K passing
0
along this boundary. The boundary conditions in vector form can be expressed

( )
n⋅ εE −εE =σ ,
1 1 2 2 0

( )
n× E −E =0 ,
1 2

( )
n⋅ µH −µH =0 , (1.16)
1 1 2 2

( )
n× H −H =K ,
1 2

( )
n⋅ J −J =0 ,
1 2

n is the unit vector normal to the boundary. A scalar form using the normal and tangential
components of vectors is

εE −εE =σ , (1.17)
1 n1 2 n2

E = E , (1.18)
t1 t2

µH = µH , (1.19)
1 n1 2 n2

H −H = K , (1.20)
t1 t2

J = J . (1.21)
n1 n2

Specially on the surface of an ideal conductor with conductivity σ→∞, and since the
electric and magnetic fields are zero inside this conductor, we have

εE =σ , E =0, H =0, H = K . (1.22)
1 n1 0 t1 n1 t1

The first Maxwell equation (1.12) has three terms on its right hand side. Term jωE
10

book - 1
represents the displacement current, term σE represents the conducting current which causes
conducting losses in the material, see (1.9), and J is the current supplied by an internal
S
source. Equation (1.12) can be simplified by introducing a complex permittivity

 σ  ( )
rotH = jωεεE+σE+J = jωε ε − j E+J = jωεε 1− jtgδ E+J =
0 r S 0 r ωε  S 0 r e S
 0 
jωεεE+J
0 c S

where complex permittivity ε is defined
c

σ
( )
ε =ε − j =ε 1− jtgδ =ε'−jε'' , (1.23)
c r ωε r e
0

and term

σ
tgδ = , (1.24)
e ωεε
0 r

is called the loss factor, or thea loss tangent, and δ is the loss angle. This loss factor is
e
frequently used to define the losses in a material in spite of the fact that it is frequency
dependent. The reason is that this loss factor, similarly as the imaginary part of permittivity
ε’’, contains in practice not only the conducting losses, but also polarization and another kinds
of losses. Similarly we can introduce complex permeability µ representing all kinds of
c
magnetic losses

( )
µ = µ 1− jtgδ = µ'−jµ'' . (1.25)
c r m

In material relations (1.9) – (1.11) conductivity σ , permittivity ε and permeability µ
represent the electric and magnetic properties of a material. In a linear material these
parameters do not depend on field quantities, while in a nonlinear material they depend on E
or H. These parameters can depend on space coordinates, which is the case of a non-
homogeneous material. In a homogeneous material σ, ε, and µ do not depend on
coordinates. An isotropic material has parameters that are constant in all directions, ε, µ, σ
are scalar quantities. Non-isotropic materials possess different behaviour in different
directions. Their permittivity, permeability, or conductivity are tensor quantities that can be
expressed by matrices. An example is provided by magnetized ferrite or magnetized plasma.
E.g., equation (1.11) for magnetized plasma can be rewritten into

D=εE , (1.26)

which gives three particular scalar equations

D =ε E +ε E +ε E ,
x xx x xy y xz z

D =ε E +ε E +ε E , (1.27)
y yx x yy y yz z

D =ε E +ε E +ε E .
z zx x zy y zz z
11

book - 1
Vector D has a different direction from vector E. Permittivity is then not a scalar quantity, but
a tensor quantity

ε ε ε 
xx xy xz
 
ε= ε ε ε . (1.28)
 yx yy yz
ε ε ε 
 zx zy zz

Similarly we can express the permeability and conductivity of an anisotropic material.
The density of power transmitted by an electromagnetic wave is described by the
complex Poynting vector. This vector is defined as

1
S = E×H* =S + jQ , (1.29)
av
2

where S is the average value of the transmitted power, which represents the density of active
av
power

[ ]
1
S = ReE×H* . (1.30)
av
2

Q is the density of reactive power

[ ]
1
Q = ImE×H* . (1.31)
2

Poynting’s theorem represents the balance of power in an electromagnetic system in volume
V. It can be read by dividing power into active and reactive power

P = P +P , (1.32)
S J R

( )
Q = 2ωW −W +Q , (1.33)
S mav eav R

where P and Q are the average values of the active and reactive power supplied by an
S S
external source. The active power supplied by an external source is

{ }
1
P = ∫∫∫Re E⋅J* dV , (1.34)
S S
2
V

J is the current supplied by this source. The active power is partly lost in materials, power P ,
S J
and partly radiated outside of our volume, power P . These quantities are
R

1
P = ∫∫∫σE 2dV , (1.35)
J
2
V

{( )}
P =∫∫S ⋅dS = ∫∫Re E×H* ⋅dS , (1.36)
R av
S S

12

book - 1