1. An antenna acts as a transducer that converts electromagnetic waves between free space and a transmission line. It is an integral part of any radio communication system. 2. Maxwell's equations relate the electric and magnetic fields through vector and scalar potentials. For time-varying electromagnetic waves, the vector potential A and scalar potential φ are defined in terms of the current and charge densities using retarded potentials. 3. Radiation from an infinitesimal current element called a Hertzian dipole is analyzed. The vector potential at a point due to a constant current element of length dl is derived in terms of the current moment dlI0.

1. 1
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
B
E


 (1)
J
t
D
H 


 (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

2. 2
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
tr
trJ



,
,

(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
E
E





  2
2
(7)
J
t
J
J 


 2
2
 (8)
Using the vector identity:   FFF 2
 and equation 3 in equation 7 and 8:
t
J
t
E
E











 


 2
2
2
(9)
J
t
H
H 


 2
2
2
 (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
A
E


 (12)

3. 3
0










t
A
E (13)
Hence:




t
A
E (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:
  J
t
E
A 


 

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:
   J
tt
A
AA 













 2
2
21
(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



t
A

 (18)
Substituting 18 in 16 gives:
J
t
A
A  


 2
2
2
(19)
Substituting 18 in 17 gives:

4. 4


 


 2
2
2
t
(20)
Consider the following cases:
Case 1:  is independent of time, then:


 2
, such that 
r
dr
r



4
1
Case 2: 0 , then
02
2
2




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

5. 5
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
erJ
rA
4
(25)
    



 


 

r
rrj
rr
er
r
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.

6. 6
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
erJ
rA
rrj
z
z
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 1a and also the length of
the element is infinitensimally small, then it is proper to omit r in equation 27. Thus:
   
 








r
rJe
rA 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:
  dlIrJ
r
z 0
 (29)
Hence
z
y
x
I

7. 7
    rj
z e
r
dlI
rA 

 

4
0
(30)
dlI0 is called the moment of current (current moment).
Expressing the vector potential at P in spherical coordinates we obtain:
      


 
cos
4
cos 0 rj
zr e
r
dlI
rArA 
 (31a)
   
 



 
 sin
4
sin 0 rj
z e
r
dlI
rArA 
 (31b)
  0rA (31c)
Thus:
     




aae
r
dlI
rA r
rj
sincos
4
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.:
   rA
j
r 


1
(33)
Hence,
   






 2
0 1
4
cos
rr
j
j
edlI
r
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.
   rA
r
rH 
1
(35)
Thus:
   





a
rr
jedlI
rH
rj








2
0 1
sin
4
(36)
Also:

8. 8
   rH
j
rE 

1
(37)
Thus:
     











a
rjrr
jedlI
a
rjr
edlI
rE
rj
r
rj














32
0
32
0 11
sin
4
11
cos
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
edlI
jrH
rj
sin
4
0

 (39)
   





a
r
edlI
jrE
rj
sin
4
0

 (40)
The inverse 2
r 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 2
r term represents the electric field intensity of an
electric dipole. The inverse 3
r 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 2
r terms are equal in magnitude
when:
rv
1


(41)
Therefore:
62





v
r (42)

9. 9
Far-zone and Near-zone Fields
The far zone and near zone are defined respectively by the inequalities 1r and 1r . The
terms of the fields whose amplitudes vary as the inverse of r are called the far zone fields, i.e.
  

a
r
eH
rH
rj
 0
(43)
  

 a
r
eE
rE
rj
 0
(44)
Where
  


sin
4
0
0
dlI
jEHo  (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 rj
e 
is expanded into a power series in r and since 1r ;
    

 sin
4 2
0
r
dlI
rH  (47)
    

 cos
2 3
0
r
dlI
rEr  (48)
    

 sin
4 3
0
r
dlI
rE  (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.

10. 10
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
1
mWHEPPav  (51)
Since the far zone fields of the current element are perpendicular to each other, then;
  2
0 sin
322
1







r
dlI
HEPav


(52)
Hence,
 
rav a
r
dlI
P
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
0
312











 dlI
dlIPrad (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. 11
22
..
3
2








 dlI
P 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








 dl
Rrad (59)
In free space, :asgivenissoand120 radR 







2
2
80


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

12. 12
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:

13. 13
100% 




 

C
LH
f
ff
BW (63)
Where fH and fL are determined by  2VSWR511  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
2
g
Aeff  (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.