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CHAPTER FOUR: ELECTROMAGNETIC RADIATION

A system of conductors/material media which is connected to a power source so as to produce a

time varying electromagnetic field in an external region will radiate energy. When this system is

arranged so as to optimize the radiation of energy from some portion of the system while at the

same time minimizing/suppressing radiation from the rest of the system, that portion of the

system is called an antenna.

Antenna Fundamentals

An antenna acts as a transducer for converting a movement of charge on a conductor into

electromagnetic waves propagating in free space (transmitter function) and the reverse process

(receiver function). It is assumed that the antenna is connected to a known power source by

means of a transmission line/wave guide. Reception and transmission antennas have similar

characteristics and therefore the two words will be used synonymously and sometimes the same

antenna is often used for both purposes. The antenna is an integral part of any radio

communication system and thus its design is of paramount importance to a radio engineer.

Vector ( A ) and Scalar ( ) Potentials

The electric and magnetic fields are so closely inter-related that one can never be defined without

the other unlike in electrostatics and magnetostatics. This relationship is shown in Maxwell’s

equations of electromagnetics.

t

BE

(1)

Jt

DH

(2)

D (3)

0 B (4)

Note: In a material media with electrical properties r and r , the constitutive electric and

magnetic field equations are re-written as:

ED r 0 (5a)

HB r 0 (5b)

In electromagnetic waves, the magnetic and electric are related to the vector ( A ) and scalar ( )

potentials. These are in turn also related to their sources which are: current density (J) and charge

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density ( ). Consider the distribution of charge density, tr, which varies with space and

time. The relationship between charge density and current density is manifested in the continuity

equation.

t

trtrJ

,,

(6)

We wish to relate the magnetic and electric fields to their sources, i.e. current density, J and

charge density, .However, equations 1 and 2 are coupled in a complex fashion, with the result

that it is difficult to relate H and E to J and directly.

Taking the curl of 1 and 2 with substitutions of Maxwell’s equations yields;

t

J

t

EE

2

2

(7)

Jt

JJ

2

2

(8)

Using the vector identity: FFF 2 and equation 3 in equation 7 and 8:

t

J

t

EE

2

22

(9)

Jt

HH

2

22 (10)

The LHS of equation 9 and 10 are travelling wave equations.

In order to relate the vector ( A ) and the scalar potentials to the sources J and , it is

necessary to make use of supporting functions, i.e. 0 F and 0 V .

Therefore the vector potential A is defined as:

AB (11)

Thus from equation 1 and 11,

t

AE

(12)

3

0

t

AE (13)

Hence:

t

AE (14)

Equations 2,3 and 14 are used to show how A and are related to their respective sources, J and

. Using equations 2 and 11, we obtain:

Jt

EA

1 (15)

Note: All the vectors have space and time functional relationships, i.e. tr, .

Using 14 and 15 together with the above vector identity gives:

Jtt

AAA

2

221

(16)

Using 3 and 14 also gives:

2

t

A (17)

The partial differential equations 16 and 17 are coupled since each of them contains A and .

Since A is already known, it is also necessary to determine A in order to define A

completely. A vector field is completely specified only if its curl and divergence are defined.

The Lorentz gauge condition (equation 18) defines A completely and is used to decouple A and

.

0

tA

(18)

Substituting 18 in 16 gives:

Jt

AA

2

22 (19)

Substituting 18 in 17 gives:

4

2

22

t (20)

Consider the following cases:

Case 1: is independent of time, then:

2 , such that

r

drr

4

1

Case 2: 0 , then

02

22

t

, such that vtrgvtrftr ,

Where f and g are arbitrary functions.

The solutions of equation 19 and 20 are given as:

r rr

v

rr

trJ

trA

,

4, (21)

r rr

v

rr

tr

tr

,

4

1, (22)

where v

rr

tt

Equation 21 and 22 say that sources which had the configurations J and at a t (previous time

instant) produce a potential A and at a point P at a time t which is later than the time t by an

amount that takes into account the finite velocity of propagation of waves in the medium.

Because of this time delay aspect of solutions, the potentials A and are known as retarded

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potentials and the phenomenon itself is known as retardation. These retarded potentials give rise

to fields only after their sources are brought into existence.

The solutions to 19 and 20 remain unchanged if we change v to –v, i.e.

r rr

v

rr

trJ

trA

,

4, (23)

r rr

v

rr

tr

tr

,

4

1, (24)

This shows that the solutions to 19 and 20 have two parts, i.e. two waves travelling in opposite

directions. The potentials in 23 and 24 are called advanced potentials and they give rise to fields

only before the current and charge distributions are brought into existence. However, in all

physical phenomena, effects should occur after their cause. Consequently Advanced potentials

are outside the scope of this work.

For time harmonic variation of current and charge density, the expressions for the retarded

potentials are given as:

r

rrj

rr

erJrA

4 (25)

r

rrj

rr

err

4 (26)

where

2

v is the wave number.

Once the retarded vector and scalar potentials are obtained, then the magnetic and electric fields

at a point away from the sources J and can also be obtained using equations 11 and 14.

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Radiation from a Current Element

Characteristics of a current element

It should be of negligible thickness and if its length is dl , then dl

The current in the element should vary harmonically with time and have a constant

amplitude along the length of the element.

A constant current element of infinitesimal length cannot be realized practically but it is however

important to study s radiation characteristics as a foundation to understand how antennas work.

This infinitensimal length current element is called a Hertzian dipole. The dipole we consider

is a cylindrical tube of length dl and we wish to find the vector potential at a point ,,rP .

Assuming the current density on the cylindrical tube to be only in the z-direction, then the

resultant vector potential at P is also in the z-direction and is given by the equation 25 i.e

rr

erJrA

rrj

zz

4 (27)

The position vector to any point on the cylindrical tube is denoted as r . If the radius a , of the

tube is very small in comparison to the wave length , such that 1a and also the length of

the element is infinitensimally small, then it is proper to omit r in equation 27. Thus:

r

rJerA z

rj

4 (28)

The current density integrated over the cross section of the cylindrical tube gives the total current

0I which we assumed to be constant along the length of the current element. Thus:

dlIrJr

z 0

(29)

Hence

z

y

x I

7

rj

z er

dlIrA

4

0 (30)

dlI0 is called the moment of current (current moment).

Expressing the vector potential at P in spherical coordinates we obtain:

cos

4cos 0 rj

zr er

dlIrArA (31a)

sin4

sin 0 rj

z er

dlIrArA (31b)

0rA (31c)

Thus:

aae

r

dlIrA r

rj sincos4

0 (32)

The scalar potential can now be easily obtained from the vector potential by considering the

harmonic time variation equation of the Lorentz gauge condition, i.e.:

rAj

r

1

(33)

Hence,

2

0 1

4

cos

rr

j

j

edlIr

rj

(34)

Magnetic and Electric fields from a Current Element

The magnetic and electric fields due to the current element can now be easily obtained from the

vector and scalar potentials obtained above, i.e.

rAr

rH 1

(35)

Thus:

a

rr

jedlIrH

rj

2

0 1sin

4 (36)

Also:

8

rHj

rE

1 (37)

Thus:

a

rjrr

jedlIa

rjr

edlIrE

rj

r

rj

32

0

32

0 11sin

4

11cos

2 (38)

Where

is the intrinsic impedance of the media between the source and observation

points.

It is recognized that the magnetic and electric field components involve inverse terms of 32 ,, rrr .

Since the antenna’s primary function is to radiate energy to distant points, it is possible and valid

most of the time to ignore the components that do not contribute to energy radiation. In this case,

we can neglect the higher order terms for large distances, and thus call these fields (that are

reverse functions of r) radiation fields. These are given as:

a

r

edlIjrH

rj

sin4

0

(39)

a

r

edlIjrE

rj

sin4

0

(40)

The inverse 2r term is called the induction field and this term dominates at short distances (i.e.

r ). Essentially it is the field you find near the source (current element). At r , the

radiation field dominates. It is possible to determine the induction field from the Biot-Savart law.

In the case of the electric field, the inverse 2r term represents the electric field intensity of an

electric dipole. The inverse 3r term is called the electrostatic field term. Since we are dealing

with antennas, both the induction and electrostatic field terms will be dropped.

On close examination of rH and rE , the inverse r and 2r terms are equal in magnitude

when:

rv

1

(41)

Therefore:

62

vr (42)

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Far-zone and Near-zone Fields

The far zone and near zone are defined respectively by the inequalities 1r and 1r . The

terms of the fields whose amplitudes vary as the inverse of r are called the far zone fields, i.e.

ar

eHrH

rj

0 (43)

ar

eErE

rj

0 (44)

Where

sin

4

00

dlIjEHo (45)

It’s observed that the ratio of the magnitude of the far-zone electric field to the magnitude of the

far-zone magnetic field is equal to the intrinsic impedance, of the medium i.e.

rH

rE

For a lossless medium, the intrinsic impedance is real. The electric and magnetic vectors are thus

both in time and space phase. The electric field, magnetic field and the direction of propagation

form a triad of mutually perpendicular right handed system of vectors; thus in the far-zone, the

fields due to a current element constitute a plane transverse electromagnetic wave (TEM mode).

In the near-zone, the exponential rje is expanded into a power series in r and since 1r ;

sin4 2

0

r

dlIrH (47)

cos2 3

0

r

dlIrEr (48)

sin4 3

0

r

dlIrE (49)

Therefore the fields in the near-zone are equivalent to the field obtained due to a current element

by application of the laws of magnetostatics.

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Power Radiated by a Current Element & Radiation Resistance

The power flow per unit area (power density) at the point P due to a current element will be

given by Poynting’s vector at that point.

2/ mWHEP (50)

The time-averaged power density is then obtained from;

2/2

1

2

1mWHEPPav (51)

Since the far zone fields of the current element are perpendicular to each other, then;

2

0 sin

322

1

r

dlIHEPav

(52)

Hence,

rav ar

dlIP

2

0 sin

32

(53)

Then the total power radiated, )(WPav by the current element is obtained by carrying out the

integration of the time-averaged power density over the closed spherical shell surrounding the

element, i.e.

v

ravavrad addrPdsPP sin2 (54)

Making the necessary substitutions, the total power is given as;

2

02

0312

dlIdlIPrad (55)

radP is observed to be a real quantity which shows that the far zone fields of the current element

give rise to transport of time-averaged power only. 0I in (55) is the peak value (amplitude) of the

current, which can be expressed in terms of the r.m.s (root mean square)value of current, smrI .. ,

i.e.

smrII ..0 2 (56)

Thus:

11

22

..

3

2

dlIP smr

rad (57)

The coefficient of 2

.. smrI in (57) has the dimensions of resistance and is called the radiation

resistance, Rrad of the current element (antenna), i.e.

2

.. smrradrad IRP (58)

Where:

2

3

2

dlRrad

(59)

In free space, :asgiven is so and 120 radR

2

280

dl

Rrad (60)

Antenna properties

There are several properties/characteristics that determine the operation of all antennas in any

wireless communication network. The following are some of these properties:

Antenna Power Gain, g

Antenna gain, g is the measure of the antenna’s ability to radiate the power that has been input

into its terminals into the media surrounding it (i.e. free space). It is defined as a ratio of the

radiated power density at a given point, P distant r from the test antenna, to the radiated power

density at the same point due to an isotropic antenna, both antennas having the same input

power.

G (61)

Where and represent the radiated power densities at a distance r from the test and isotropic

antennas respectively.

Antenna Directive Gain, gd

Antenna directive gain, gd is the measure of the antenna’s ability to concentrate the radiated

power/energy in a particular direction ,r . It is defined as the power density at a point P in a

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given direction, distant r from the test antenna, to the power density at the same point due to an

isotropic antenna radiating the same power.

The maximum value of the directive gain of an antenna is commonly referred to as the antenna

directivity, D.

Antenna gain and directive gain seem quite similar but they slightly differ and are related to each

other through equation (62) where k is the efficiency factor. Antenna gain is usually less than

directive gain because of the losses that occur within the antenna.

dkgg (62)

Radiation pattern

The radiation pattern from an antenna is a three dimensional plot of the radiated power density

from an antenna (at a given distance r) as the directional parameters ( and in spherical

coordinates) are varied. The radiation pattern will always give an indication of the direction in

which the maximum power is radiated from an antenna.

The radiation patterns of an antenna can either be field or power patterns and their shapes vary

with the different antenna types. The figure below shows the power pattern for the Hertzian

dipole antenna (current element).

Polarization

This is the orientation of the far zone electric field vector within the radiated electromagnetic

wave from an antenna. It describes the locus of the tip of the electric field vector. If this locus is

a straight line constantly parallel to a constant direction, then the polarization is linear. Circular

or elliptical polarization are obtained when the loci are either circular or elliptical respectively.

Depending on the antenna design, different antenna polarisations can be achieved with each

having its merits and demerits. However, it’s important to note that in any wireless system

design, the transmitting and receiving antennas should always have the same polarization.

Antenna bandwidth, BW

This is the range of frequencies (centred about the resonant/design frequency, fc) that can be used

by antennas to radiate electromagnetic waves. At resonant frequencies, the antenna has zero

input reactance and will radiate/deliver maximum power due to the fact that the matching has

been achieved at this frequency. Depending on the application, some antennas are designed with

narrowband (i.e. narrowband antennas)while others may have wider bandwidth (i.e. broad band

antennas). The percentage band width of an antenna can be obtained using equation 63 below:

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100%

C

LH

f

ffBW (63)

Where fH and fL are determined by 2VSWR 511 dbS th which accounts for approximately

88.9% of the power being radiated (transmitted/received) by the antenna.

Effective Area, Aeff

This is the area of an antenna on to which the power density of a radiated electromagnetic

wave is incident. Equation 64 gives the relationship between the effective area of an antenna effA ,

and the antenna’s gain, G.

4

2gAeff (64)

where is the operating centre wavelength of the antenna.

The product of this area with the power density , gives the power received Pr, by an antenna

from a passing wave. The effective area therefore measures the antenna’s ability to extract

electromagnetic energy from an incident/radiated electromagnetic wave.