r tamas antenna theory part i

7/30/2019 R Tamas Antenna Theory Part I

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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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6

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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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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10

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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12

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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14

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

15

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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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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20

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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22

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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Antenna radiation

23

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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24

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

25

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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26

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

27

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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28

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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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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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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36

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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38

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%.

r tamas antenna theory part i

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