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Rzvan TAMA
ANTENNA THEORY:
TRADITIONAL VERSUS MODERN
APPROACH
tura
NAUTICA
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Rzvan TAMA
ANTENNA THEORY:
TRADITIONAL VERSUS MODERN
APPROACH
2010
Editura
NAUTICA
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IV
Referent tiinific: Prof. univ. dr. ing. Teodor PETRESCU
(Universitatea POLITEHNICA din Bucureti)
Tehnoredactarea i grafica aparin autorului
Editura NAUTICA, 2010
Editur recunoscut CNCSISStr. Mircea cel Btrn nr.104
900663 Constana, Romnia
tel.: +40-241-66.47.40
fax: +40-241-61.72.60
e-mail: [email protected]
Descrierea CIP a Bibliotecii Naionale a Romniei:
TAMA, RZVANAntenna theory: traditional versus modern approach /Rzvan Tama Constana: Nautica, 2010
Bibliogr.ISBN 978-606-8105-28-4
621.396.67
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V
To the memory of my father.
Foreword
This book is intended to be a comparison of two approaches of the
antenna theory, distinguished in terms of relevant antenna parameters:
the traditional one, based on frequency-domain descriptors, and the
new one, based on energy descriptors.
The later approach was developed in order to better analyze
antenna behavior to pulsed excitations, as new ultra-wide band (UWB)
communications technologies require. Some original contributions of the
author were herein included. This study was supported in part by the
Romanian Ministry of education, research, and innovation National
center for program management (CNMP) under the project SIRADMAR.
The book is mainly addressed to Ph. D students and M. Sc.
students, particularly to those involved in the program Circuits and
Integrated Systems for Communications.
Constana, 2010
The author
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VI
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VII
CONTENTS
1. INTRODUCTION3
1.1. RADIO FREQUENCIES AND RADIO COMMUNICATIONS.ANTENNAS ....31.2. SOURCE, MEDIUM, AND EFFECT: CHARACTERISTIC QUANTITIES .......71.3. MAXWELLS EQUATIONS ................................................................12 1.4. BOUNDARY CONDITIONS ................................................................13 1.5. PROPAGATION OF A UNIFORM, PLANE WAVE IN THE FREE SPACE ....15
2. ANTENNA RADIATION..18
2.1. A SIMPLE INTERPRETATION OF ANTENNA RADIATION .....................182.2. VECTOR AND SCALAR POTENTIALS.................................................20 2.3 RADIATION OF A SMALL CURRENT FILAMENT .................................22
3. ANTENNA PARAMETERS.30
3.1. RADIATION CHARACTERISTIC FUNCTION AND RADIATIONPATTERN........................................................................................................303.2. INTRINSIC AND REALIZED GAIN ......................................................32 3.3. ELECTRICAL INPUT PARAMETERS ...................................................36
4. NOVEL DESCRIPTORS FOR ANTENNAS WITH
PULSED EXCITATION43
4.1. INTRODUCTION TO PULSE OPERATION ............................................43 4.2. ENERGY-BASED DESCRIPTORS ........................................................46
4.2.1. Antenna input mismatch............................................................ 464.2.2. Energy gain............................................................................... 484.2.3. Normalized correlation coefficient ........................................... 50
4.3. APPLICATION: CYLINDRICAL DIPOLE ANTENNAS ............................51 4.4. IMPULSE RESPONSE OF A SHORT DIPOLE .........................................57
4.4.1. Time-domain form of the vector potential.................................574.4.2. Impulse response in transmitting mode .................................... 604.4.3. Impulse response in receiving mode ......................................... 614.4.4. The input capacity of a short dipole..........................................63
5. EXTRACTION OF THE IMPULSE RESPONSE FROM
MEASURED OR SIMULATED DATA...66
5.1. FREQUENCY-DOMAIN VERSUS TIME-DOMAIN APPROACH................66
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VIII
5.2. TIME-DOMAIN EXTRACTION OF THE IMPULSE RESPONSE ................ 675.2.1. Time-domain equation subject to deconvolution...................... 675.2.2. Time-domain deconvolution by the method of moments........... 68
5.3. RESULTS AND VALIDATION ............................................................ 725.3.1. Electrically large antennas....................................................... 72
5.3.2. Small antennas ......................................................................... 785.3.3. Conclusions .............................................................................. 82
6. TIME-DOMAIN MEASURING TECHNIQUES84
6.1. INTRODUCTION TO TIME-DOMAIN MEASURING............................... 846.2. DIFFERENTIAL TIME-DOMAIN SINGLE-ANTENNA METHOD.............. 856.3. AVERAGING TECHNIQUE FOR ELECTRICALLY LARGE ANTENNAS.... 896.4. EXPERIMENTAL RESULTS AND VALIDATION ................................... 92
7. TIME-DOMAIN PULSE-MATCHED ANTENNASYNTHESIS..100
7.1. INTRODUCTION TO TIME-DOMAIN SYNTHESIS .............................. 1007.2. PULSE-MATCHED SYNTHESIS TECHNIQUE..................................... 1017.3. EXAMPLE OF SYNTHESIS.RESULTS .............................................. 103
REFERENCES.107
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PART I
Traditional approach
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1. Introduction
1.1. Radio frequencies and radio communications. Antennas
A classical communication chain includes at least three elements:
transmitter, communication channel, and receiver (Fig. 1.1).
The transmitter transforms the information from the source user in a
proper form that could be handled by the communication channel. The
reverse transformation is performed by the receiver in order to deliver
the information to the recipient user.
Fig. 1.1. Minimal communication chain
The channel can physically be any propagation medium, e.g. cable,optical fiber, or the free space in the case of radio communications.
The free space is defined as an infinite, isotropic, homogeneous,
and lossless medium, usually vacuum or air.
In the case of radio communications, since propagation is due to
electromagnetic waves, specific interfaces should be inserted between
the transmitter and the channel, and between the channel and the
receiver, respectively.
Transmitter ReceiverChannel
Source
user
Recipient
user
Communication chain
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
4
These two interfaces are called antennas, transmitting antenna and
receiving antenna, respectively (Fig. 1.2.). The transmitting antenna
converts the electric signal from the transmitter into electromagnetic
field that emerges in the propagation medium. The receiving antenna
transforms the energy of the incident electromagnetic field into an
electric signal. There are no antennas specifically designed to only
receive or to only transmit. Antennas are reciprocal, passive devices, so
they can be used both for transmitting and receiving. There are some
products on the market improperly called active antennas or receiving
antennas. Actually, such devices include amplifiers so they are more
than antennas.
Fig. 1.2. Radio communication chain
It should be noted that radio transmission in the free space is
possible only if the spectrum of the informational signal is translated in a
frequency range that allows the propagation of electromagnetic waves.
We call that frequency range radio frequency (RF) range.
Physically, wave means energy transport i.e., propagation from a
source point to a field point. In order to state that an electromagnetic
wave is established, and not an inductive or capacitive coupling, the
distance between those two points should be in the order of the
wavelength,
fc= , (1.1)
ReceiverTransmitter
Transmitting
antenna
Receiving
antenna
Electric signal Electric signalElectromagnetic field
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Introduction
5
where f is the frequency and c is the wave speed (particularly, the
speed of light, c0, in the free space).
It is remembered that the wavelength is the minimal distance
between two points that oscillate in-phase and therefore, the minimal
distance that grants wave behavior.
Nevertheless, antenna dimensions should also be in the order of the
wavelength in order to radiate efficiently.
Hence, the minimum frequency of that range i.e., the maximum
wavelength is determined by feasibility of in terms of physical
dimensions.
A frequency of 3kHz, i.e., =100km, is generally accepted as the
lowest limit of the RF range. Propagation is still possible at such a
frequency although antennas in that order of wavelength are not
feasible and the radiation efficiency is low for practical antenna size.
In practice, the lowest limit is slightly higher, i.e., around 10kHz
(=30km). The 3kHz limit was only set in order to divide the RF rangeinto wavelength decades.
The upper limit of the RF range is given by the absorption of
electromagnetic waves in the free space. As frequency increases,
wavelength becomes shorter and microscopic interaction with the
medium, e.g. molecular losses is more and more evident. Figure 1.3
gives the normalized transmission coefficient of the free space as a
function of frequency.It can be noted that absorption dips are increasingly deep and
numerous as the frequency exceeds 250 GHz. For that reason, the
upper limit of the RF range is generally accepted as 300GHz.
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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Fig. 1.3. Normalized transmission coefficient of the free space
As Table 1.1 shows the RF range is divided into decades orfrequency bands from VLF (Very Low Frequency) to EHF (Extremely
High Frequency).
Table 1.1. Frequency bands
Frequency Wavelength Band
3 30 kHz 100 10 km VLF
30 300 kHz 10 1 km LF0.3 3 MHz 1 0.1 km MF
3 30 MHz 100 10 m HF
30 300 MHz 10 1 m VHF
0.3 3 GHz 1 0.1 m UHF
3 30 GHz 100 10 mm SHF
30 300 GHz 10 1 mm EHF
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Introduction
7
1.2. Source, medium, and effect: characteristic quantities
There are three elements in a propagation problem: the source, the
medium, and the field effect.
Sources are usually included in a finite volume (continuous or not).
A point belonging to the source volume is called source point.
The effect of the sources is observed in the propagation medium at
a certain distance away. A point where the source effect is observed is
called field point.
Source point coordinates are usually referred by prime symbol (Fig.
1.4).
Fig. 1.4. Source point and field point. Notations
Sourcecharacteristic quantities are:
a. the volume current density, J, a vector quantity that is
measured in A/m2
b. the charge density, , a scalar that is measured in C/m3.
r'
P(x,y,z)
P(x,y,z)
Y
X
Z Source volume
Source point:
Field point :
r
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
8
The two above quantities are not independent since a current is a
charge transport. Let V be a volume containing the charge density
and the surface of the volume boundary crossed by the current
density J (Fig. 1.5). Then
=V
sQ d (1.2)
and
= sJ dI (1.3)
whereI is the total current crossing ,Q is the total charge in V, and
sdd = ns .
Fig. 1.5. Current density and charge density
Then
t
QI
= (1.4)
and
n
dsV
J90
Jn =Jn
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Introduction
9
=
+ 0dd
V
st
sJ . (1.5)
Relation (1.5) expresses the charge continuity.
The effectof the sources in a field point is quantified by the following
quantities:
a. the electric field intensity or simply the electric field, E, a vector
with the magnitude measured in V/m
b. the magnetic field intensity or simply the magnetic field, H, a
vector with the magnitude measured in A/m
c. the electric displacement field, D, a vector with the magnitude
measured in FV/m2
or C/m2
d. the magnetic flux field, B, a vector with the magnitude
measured in HA/m2
or T (Tesla).
The mediumis characterized by three quantities, as follows:
a. the conductance, , which is measured in -1m-1 or S/m
b. the electric permittivity, , which is measured in F/m
c. the magnetic permeability, , which is measured in H/m.
The conductance establishes the relationship between the current
density and the electric field as Ohms lawshows
EJ = . (1.6)
If the medium is isotropic, then and do not depend on the
direction. Moreover, if the medium is homogeneous, then and do
not depend on the field point.
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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As stated before, the free space is an example of isotropic and
homogeneous medium. The values of the above quantities for the free
space are:
0=910
36
1
F/m (1.7)
0=410-7
H/m (1.8)
Relative electric permittivity and relative magnetic permeability canbe defined for any generic medium by normalizing andto 0and0,
respectively:
0
=r , (1.9)
0 =r . (1.10)
The electric displacement field and the magnetic flux field are linked
to the electric and magnetic field, respectively through the medium
properties,
D(r)= E(r) (1.11)
B(r)= H(r). (1.12)
A quick inspection of the source, medium, and fieldquantities and of
the corresponding units of measurement reveals the equivalence
between a general propagation problem and a simple circuit problem, as
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Introduction
11
given in Fig. 1.6. That is, a simple circuit problem is a particular case of
the general, propagation problem.
a
b
Fig. 1.6. Comparison: propagation problem (a) versus circuit
problem (b)
The relationship between the source, mediumand fieldquantities is
established by Maxwells equations. Additionally, the continuity law,
Ohms law, and boundary conditions help in solving these equations.
Source
J [A/m2]
Field
E [V/m]
H [A/m]
Medium
[F/m]
[H/m]
[-1m-1]
Generator
Ig [A],
Vg[V]
Load
V[V]
I[A]
Circuit
C [F]
L [H]
R[]
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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1.3. Maxwells equations
The study of radio waves propagation is placed into the context of
the electrodynamics since source and field quantities are time-domain
variable.
We may assume time harmonic variation for source and field
quantities [1], [2]. It is the case of most classical, narrowband
applications.
Maxwells equations actually describe a set of known physical
phenomena that are more obvious when integral expressions are used.
However, in the study of radio wave propagation differential expressionsare preferred instead since it is easier to use them. The differential form
is derived from the integral form by applying Stokess theorem and
Gauss Ostrogradsky (or divergence) theorem, respectively in order to
achieve the same order of integration in both left-hand and right-hand
members.
The four Maxwells equations in differential form are
HE jrot = , (1.13)
also called Faradays law,
EJH jrot += (1.14)
also called Ampres generalized law,
div =E (1.15)
also called Gausss law for the electric field, and
0div =H (1.16)
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Introduction
13
also called Gausss law for the magnetic field.
By applying Gauss Ostrogradsky theorem in (1.5) a differential
form of the continuity law can be derived,
div J+j= 0 (1.17)
1.4. Boundary conditions
The above differential form of Maxwells equations only stands for
homogeneous an isotropic media. In a standard propagation problem as
stated in 1.2, although the propagation medium was supposed to be
homogeneous, the presence of source boundaries or other obstructing
objects induce step-like variations of the medium characteristic
quantities. Hence, the differential operators in Maxwells equations
should be considered as distributional operators since they are
supposed to act on distributions [3]. As a result, each equation can be
splitted into a functional part, similar to the form given 1.2, and a
purely distributional part called boundary condition.
We consider two media divided by a boundary, (Fig. 1.7). Let 1,
1, 2, 2
be the media characteristic quantities and E1, H1, D1, B1,
E2, H2, D2, B2the fields on the two sides of the boundary.
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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Fig. 1.7. Boundary between two media
The four corresponding boundary conditions are
nE1nE2 = 0, (1.18)
i.e., the tangential electric field is the same on both sides of the
boundary,
nH1nH2 = Js, (1.19)
with Jsthe surface current density on the boundary, measured in [A/m],
nD1nD2 = s, (1.20)
with sthe surface charge density on the boundary, measured in [C/m2],
nB1nB2 = 0, (1.21)
i.e., the normal magnetic field is the same on both sides of the
boundary.
n
Medium 1
1, 1
Medium 2
2, 2E1,
H1,
D1,
B1
E2,
H2,
D2,
B2
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Introduction
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1.5. Propagation of a uniform, plane wave in the free space
The simplest application of Maxwells equations is the case of a
uniform, plane wave propagating in the free space. In that case, those
equations yield analytical solutions.
Planewave means that the wavefront, i.e., the locus of the equiphase
points, is a plane. Uniformmeans that there is no variation of the electric
or magnetic field in that plane.
Let OZbe the direction of propagation (Fig. 1.8).
Fig. 1.8. Plane wave
Since the wave is plane then 0=
=
yx. By using (1.13) and
(1.14) it can easily be demonstrated that a plane wave is a transversal
electromagnetic (TEM) wave i.e., the electric and magnetic fields are
perpendicular each to other and are both included in the (XOY) plane.
Without loss of generality we can accommodate the choice of OXand
Y
X
ZO
Wavefront
Ey
Hx
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
16
OY axes so that the electric field is along OY and the magnetic field
along OX.
Equations (1.13) and (1.14) become
x
yHj
z
E0=
(1.22)
and
yx Ej
zH 0= . (1.23)
By substituting Ey in (1.22) and Hx in (1.23) one can obtain two
formally identical differential equations,
0002
2
2
=+
x
x
Hz
H
, (1.24)
and
0002
2
2
=+
y
yE
z
E . (1.25)
The solutions of the two above equations are:
)exp( 00 zjkEEy = (1.26)
and
)exp( 00 zjkHHx = (1.27)
with k0the phase constant in the free space,
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Introduction
17
00
0
0
2
===
ck . (1.28)
By substituting (1.26) in (1.22) one can obtain that
yx Ek
H0
0
= . (1.29)
That is, the ratio between Ey andHx is a medium constant, 0, called
free space wave impedance
=== 37712000
0
x
y
H
E. (1.30)
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2. Antenna radiation
2.1. A simple interpretation of antenna radiation
Let us consider an open ended transmission line (Fig. 2.1).
Depending on the length of the line it can be assimilated either to a
distributed capacitor or to a distributed inductance. Consequently, if a
time-harmonic voltage source is connected to the other end of the line,
opposite and equal currents are established on the two wires. Since the
electric field is mostly confined between the wires and the two opposite
currents almost annihilate the resulting magnetic fields an open line
would not put electromagnetic energy into the surrounding space.
As Fig. 2.1 shows, by pulling out the two wires till they become
collinear, an open electric field structure, called dipole, is achieved. Yet,
currents flow on the two wires of that degenerated line and the magnetic
fields created by the two currents have now the same sense.
Fig. 2.1 Transition from transmission line to dipole
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Antenna radiation
19
The magnetic field lines are circular, around each dipole arm (Fig.
2.2). Since the source is time-variant, the magnetic field is too. As Eq.
(1.13) shows, an electric field in a perpendicular plane (since
EE rot ) is generated. As there is no charge in the free space, (1.15)
becomes div E = 0, that is, the electric field lines are closed.
Next, as Eq. (1.14) shows, the time-variant electric field generates a
magnetic field in a perpendicular plane (since HH rot ). It should be
noted that J=0 in (1.14) since the free space is a dielectric. As (1.16)
states since 0div =H the magnetic field lines are closed.So on the electromagnetic energy propagates in the free space.
Fig. 2.2. Wave propagation
Source
H
H
E E
I
I
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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2.2. Vector and scalar potentials
Vector and scalar potentials were mainly defined as calculus
quantities for antenna analysis, although some publications give them a
physical sense.
Gausss law for the electric field in the free space can be written as
div B = 0 (2.1)
or, by using the alternative nabla symbol notation,
(2.2)
As nabla acts as a vector, the above dot product type relation stands for
any vector orthogonal to , of type
AAB rot== . (2.3)
We shall call Avector potential.
By applying (2.3) in (1.13) one can find that
0)( =+ AE j (2.4)
That is, AE j+ should be a vector collinear to , of type
==+ gradAE j , (2.5)
0=B
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Antenna radiation
21
where is called scalar potential. It should be noted that for static fields
the time derivative of the vector potential i.e., Aj , vanishes, so the
classical definition of the potential is found. That explains the negative
sign before the right-hand member.
By using (2.3) and (2.5) in (1.14) then
)()( 000 AJA jj += (2.6)
The double cross product in the left-hand member can be expanded
as follows:
AAA2)()( = , (2.7)
so
AAJA 0022
000 )( ++=+ j . (2.8)
Since A and were independently defined, a relation can be
enforced between the two quantities, in order to achieve a simpler form
of the above differential equation:
000 =+ jA . (2.9)
The above relation is called Lorentz gauge and it leads to the
following differential, inhomogeneous equation, also called the
Helmholtz equation for the vector potential:
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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JAA 02
0
2 =+ k . (2.10)
2.3 Radiation of a small current filament
The current filament is a wire radiator (i.e., zero thickness),
infinitesimal long crossed by a constant, axial current. The study of the
radiation emerging from a current filament is essential since any current
distribution can be divided into current filaments and the overall
radiation can then be evaluated by adding up the contributions of all
individual filaments.
Since the source is punctiform it is convenient to use spherical
coordinates (Fig. 2.3). Table 2.1 gives the Lam coefficients that are
used for calculating differential operators such as div, grad, rot,
2 (i.e., Laplacean) in spherical coordinates.
Fig. 2.3. Spherical coordinates
Z
X
Y
P(r, , )
rO
a ra
a
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Table 2.1. Lam coefficients for spherical coordinates
Coordinate Lam coefficient
r h1 = 1
h2 = r
h3 = rsin
As Fig. 2.4 shows the surface element is spherical coordinates ca
be written as
ds = r2
sin d d (2.11)
and the volume element
dv = r2 sin drd d. (2.12)
Z
X
Y
d
d
O
rsin d
rsin
r
drrd
ds
dv
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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Fig. 2.4. Surface element and volume element in spherical coordinates
When calculating 2 in spherical coordinates in Eq. (2.10) the
derivatives with respect to and are zero due to the particular
symmetry. As the current flows along the OZ axis the Helmholtz
equation becomes
zzz JAkr
Ar
rr0
2
0
2
2
1=+
(2.13)
The first step in solving the above equation is to find the solutions of
its homogeneous form. By denoting
zAr= (2.14)
the homogeneous form of equation (2.13) can be written as
02
02
2
=+
k
r. (2.15)
The solutions of (2.15) are
)exp( 0rjkC = , (2.16)
that is,
r
rjkCAz
)exp( 0= (2.17)
where Cis a constant.
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Antenna radiation
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By enforcing the boundary conditions the final solution can be found
as:
r
rjkzIAz
)exp('d 00
= (2.18)
or
zr
rjkzI aA )exp('d 00 = . (2.19)
Since spherical coordinates are used the unit vector za of the OZ
axis should be projected as Fig. 2.5 shows,
sincos aaa = rz . (2.20)
Fig. 2.5. Projections of za on spherical coordinates
ra
a
cos ra
sina
O
Z
za
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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The magnetic field can be found by using (2.3) so we have to
calculate
AhAhAh
r
hhh
hhh
r
r
321
321
321
1rot
=
aaa
A (2.21)
By using (2.19), (2.20) and the Lam coefficients as given in Table
2.1 one can show that
aaAH )exp(
1
4
sin'drot
100
0
Hrjkr
jkr
zI=
+== . (2.22)
By applying Ampres generalized law for the free space (i.e., with
J=0) the electric field can be then found as
=
==
)sin()sin(sin
1
sin00
sin
sin
1rot
1
2
0
2
00
Hrr
rHrrj
Hr
r
rr
rjj
r
r
aa
aaa
HE
(2.23)
By substitutingH from (2.22) and by computing the derivatives the
electric field can be written as
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Antenna radiation
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aa
a
aE
)exp(1sin4
'd
)exp(1
cos2
'd
032
0
2
0
0
0
032
0
0
0
EE
rjkrr
jkr
kk
zIj
rjkrr
jk
k
zIj
rr
r
+=
++
+=
(2.24)
The field components emerging from a current filament are shown
in Fig. 2.6.
Fig. 2.6. Components of the field emerging from a current filament
If
2
1
0
=>>
k
r i.e., 6.1>r then the terms proportional to2
1
r
and3
1
rin (2.22) and (2.24) can be neglected. We shall call far-field
zonethe locus of the field points with 6.1>r .
The far-field components are
Z
X
Y
P(r, , )
r
O
aH
rrEa
aE
I(z)
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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aaE )exp(
4
sin'd 0
00 Erjkr
zIkj== (2.25)
and
aaH )exp(
4
sin'd0
0 Hrjkr
zIjk== . (2.26)
There are only two far-field field components,EandH (Fig. 2.7).
Fig. 2.7. Far-field components produced by a current filament
It should be noted that a current filament produces a spherical, TEM
wavesince HE and 0
=H
E. As in the electric circuits theory when
U=R I, the proportionality betweenE andHthrough a real constant
i.e., 0 shows that only those two field components carry real powerthat
is, radiated power. The rest of the terms in (2.22) and (2.24) are
associated with the reactive powerconfined in the near-field zone of the
antenna.
Z
X
Y
P(r, , )
r
O
aH
aE
I(z)
*
2
1HES =
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Antenna radiation
29
The sense of propagation is given by the sense of the complex
Poynting vector (Fig.2.7)
*
2
1
HES =P . (2.27)
The real part of the magnitude of the complex Poynting vector gives
the radiated power density, measured in [W/m2]. In our case, the
magnitude of SP is real (i.e., E and H are both radiated field
components), that is
222
0
2
022sin)'d(
32
1zIk
rSP = . (2.28)
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30
3. Antenna parameters
3.1. Radiation characteristic function and radiation pattern
The radiation characteristic functionof an antenna, F(, ), shows
how the magnitude of the far-field is distributed in the space at a given
distance, compared to the maximum field value. Since E is proportionalto H in the far-field zone there is a unique radiation characteristic
function for both field components.
As an example, the radiation characteristic function for a current
filament is
sin),( =F . (3.1)
The geometrical representation of F(, ) is called radiation
pattern. There are mainly two types of radiation patterns:
a. Three-dimensional radiation patterns, when both and are
variable; a 3D surface is thus obtained.
b. Two-dimensional radiation patterns, when either or is
fixed; plane curves are therefore obtained as vertical or
horizontal cuts of the 3D radiation pattern. For a linear
antenna along the OZ axis as in Fig. 2.7, the radiation
pattern in a vertical plane, F(, =fixed), is called E-plane
radiation pattern(Fig. 3.1) since the electric field is contained
in that plane. Correspondingly, the radiation pattern in a
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Antenna parameters
31
horizontal plane, F(=fixed, ), is called H-plane radiation
pattern.
Fig. 3.1. E-plane and H-plane
The radiation patterns for the current filament are given in Fig. 3.2.
a.
Fig. 3.2. Radiation patterns for a current filament: a 3D,
b 2D, E-plane, c 2D, H-plane
y
x
c.
zLobe
Extinction
b.
H-plane
E-plane
Z
Y
X
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
32
As relation (3.1) shows, for an antenna with cylindrical symmetry as
the current filament is, there is no field variation with respect to i.e., in
the H-plane. In that case the 3D radiation pattern is a torus (Fig. 3.2a).
On a radiation pattern one can note maxima and minima (Fig.
3.2.b). A maximum between two minima is called lobe. The main lobe
refers to the absolute maximum on the pattern diagram.
The -3dB beamwidthis the angular width of a lobe at 3dB below its
maximum. If not specified otherwise, the -3dB beamwidth refers to the
main lobe. As an example, the -3dB beamwidth for the current filament
is 90(Fig. 3.3).
Fig. 3.3. -3dB beamwidth
3.2. Intrinsic and realized gain
An isotropic radiatoris a fictious radiator that radiates the same
amount of power density in all directions.
The directivityis a function of angle coordinates that shows how
an antenna concentrates the radiated power compared to the
isotropic radiator:
z
dB3
Emax
2/maxE
2/maxE
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Antenna parameters
33
)4/(
),)(d/(d
anglesolidunitperradiatedpowerAverage
),(anglesolidunitperradiatedPower),(
r
r
P
P
D
=
=
(3.2)
Let (0, 0) be the direction of the mail lobe. The figure of merit
defined as
Gi=D(0, 0) (3.3)
is called intrinsic gain or simply gain. It solely takes into account the
radiation properties of the antenna.
The antenna input mismatch can be included in order to achieve a
complete figure of merit, that is, the realized gain
)4/(
),)(d/(d
inputantennaatPower
),(anglesolidunitperradiatedPower
00ii
t
r
t
r
r
GGP
P
P
P
G
==
=
=
(3.4)
with the antenna efficiency.
All the above figures are usually expressed in dBi i.e., decibels with
respect to the isotropic radiator; for instance,
Gi, dBi = 20 log10Gi. (3.5)
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
34
As an example, for a current filament
222
0
2
022sin)'d(
32
1
d
dzIk
rs
PS rP == . (3.6)
Since the solid angle is defined on a sphere as (Fig. 3.4)
= s/r2 (3.7)
Fig. 3.4. Definition of the solid angle
then
222
0
2
02sin)'d(
32
1
d
dzIk
Pr =
. (3.8)
The average power radiated per unit of solid angle
r
sO
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Antenna parameters
35
= dsin)'d(321 222
0
2
02
zIkPr . (3.9)
By using (2.11) and (3.7) in (3.9)
ddsinsin)'d(32
1 20
2
0
22
0
2
02 = zIkPr . (3.10)
The integral can be calculated by substituting u=sin :
3
8d)1(2ddsinsin 2
1
1
2
0
2
0
== uu . (3.11)
so
22
0
2
0 )'d(12
1zIkPr
= . (3.12)
Finally, from (3.2), (3.8), and (3.12) it comes out that
2sin5.1),( =D . (3.13)
The maxima of the directivity is achieved for 0 = /2, which gives
Gi=1.5 or Gi,dBi=1.76 dBi.
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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3.3. Electrical input parameters
We shall consider an antenna in transmitting mode. The antenna
radiates power so real electrical power should be absorbed at the
antenna input form the source (transmitter). In order to accept real
power, the input impedance should have non-zero real part, Ra (Fig.
3.5), even if the antenna is considered as lossless. That resistance is
therefore called radiation resistance.
Fig. 3.5. Antenna input impedance
Hence, the radiation resistance can be found from the equality
between the radiated power and the power absorbed form the source.
For a current filament, relation (3.12) gives
22
0
2
0
2)'d(
12
1
2
1zIkIRa
= , (3.14)
so
Ra
Xa
Za
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Antenna parameters
37
]['d
80
2
2
=
zRa . (3.15)
The above formula also applies to short antennas i.e., antennas of atotal length shorter than /20. In that case, the current distribution is of
quasi-linear shape and can therefore be assimilated to a constant
distribution of magnitude equal to half input current.
As an example, a short dipole of length /20 would have Ra=
1.973. In practice, transmission lines used to feed antennas have
higher characteristic impedance, typically 50 . Consequently, an
impedance matching network should be inserted between the antenna
and the transmission line. The radiation resistance is typically in the
order of the series loss resistance of the impedance matching network
so using short antennas might be impractical. Longer antennas exhibit
more appropriate values forRa.
The input reactance, Xa, mainly describes the behavior of the
antenna as a transmission line (Fig. 2.1). Let ZCa be the characteristic
impedance of the antenna. Consequently,
lkZX Caa 0cot . (3.16)
The characteristic impedance of a thin, cylindrical dipole antenna
can be calculated by assimilating it to a degenerated conical antenna
[1]:
= 1ln0
a
LZCa
, (3.17)
withL=2l the total length of the dipole and a the radius of the wire.
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ANTENNA THEORY: TRADITIONAL VERSUS MODERN APPROACH
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LetZCbe the characteristic impedance of the transmission line used
to feed the antenna (Fig. 3.6). Then the input reflection coefficientis
Ca
Ca
ZZ
ZZ
+
= . (3.18)
Note that 1 and 0= when the input is matched i.e.,
Ca
ZZ = .
Fig. 3.6. Antenna feed circuit
In practice, it is convenient to express the input mismatch by a real
figure called voltage standing-wave ratio(VSWR):
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Antenna parameters
+=
1
1VSWR . (3.19)
Obviously, 1VSWR with 1=VSWR when the input is matched.
The input impedance of an antenna can vary dramatically with the
frequency so the VSWR does. That is, the matching is strictly achieved
only at a certain number of frequencies in a given band. For the rest of
the frequencies, a VSWR less than 3 might be accepted in practice
assuming a loss of input power of maximum 30%.