Radiation Pattern

murmerlastUrban and Civil

Nov 16, 2013 (3 years and 1 month ago)

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Antennas


Antenna


Electric and Magnetic Fields


Power density


Radiated Power


Radiation Patterns


Beamwidth


Pattern Solid Angle


Directivity


Efficiency


Gain

Antennas

Wires passing an alternating current emit, or
radiate
,
electromagnetic energy. The shape and size of the
current carrying structure determines how much
energy is radiated as well as the direction of radiation.

Transmitting Antenna:

Any structure designed to
efficiently radiate electromagnetic radiation in a
preferred direction is called a
transmitting antenna
.

We also know that an electromagnetic field will induce
current in a wire. The shape and size of the structure
determines how efficiently the field is converted into current,
or put another way, determines how well the radiation is
captured. The shape and size also determines from which
direction the radiation is preferentially captured.

Receiving Antenna:

Any structure designed to
efficiently receive electromagnetic radiation is called
a transmitting antenna

Antennas


Radiation Power

Let us consider a transmitting antenna (transmitter) is located at the origin of a
spherical coordinate system.

In the far
-
field, the radiated waves resemble plane waves propagating in the
radiation direction and time
-
harmonic fields can be related by the chapter 5
equations.

r
r
1
s o s
s s
o
and


  
 
E a H
H a E


*
1
,,Re
2
s s
r

 
 
 
P E H
The time
-
averaged power density vector of the wave is found by
the Poynting Theorem





,,,,
r P r
 

r
P a




2
,,,,sin
rad
P r d P r r d d
    
 
  
P S
The total power radiated by the antenna is found by integrating over a closed
spherical surface,

Electric and
Magnetic Fields:


Power Density:


Radiated Power:


Antennas


Radiation Patterns

Radiation patterns usually indicate either electric field intensity or power
intensity. Magnetic field intensity has the same radiation pattern as the
electric field intensity, related by

o





max
,,
,
n
P r
P
P



It is customary to divide the field or power component by its
maximum value and to plot a normalized function

Normalized radiation intensity:


Isotropic

antenna
:

The antenna radiates
electromagnetic waves equally in all directions.



,1
n
iso
P


Antennas


Radiation Patterns

A directional antenna radiates and receives
preferentially in some direction.


A polar plot

A rectangular plot

It is customary, then, to take slices of the
pattern and generate two
-
dimensional plots.

The polar plot can also be in terms of decibels.





max
,,
,
n
E r
E
E











,20log,
n n
E dB E
 









,10log,
n n
P dB P
 

It is interesting to note that a normalized electric
field pattern in dB will be identical to the power
pattern in dB.

Radiation Pattern:


Antennas


Radiation Patterns


A polar plot

A rectangular plot

It is clear in Figure that in some very specific
directions there are zeros, or
nulls
, in the
pattern indicating no radiation.


The protuberances between the nulls are
referred to as
lobes
, and the main, or major,
lobe is in the direction of maximum radiation.


There are also
side lobes

and
back lobes
.
These other lobes divert power away from
the main beam and are desired as small as
possible.

One measure of a beam’s directional nature is
the
beamwidth
, also called the half
-
power
beamwidth or 3
-
dB beamwidth.

Radiation Pattern:


Beam Width:


Antennas


Pattern Solid Angle

A
radian

is defined with the aid of Figure a). It is the
angle subtended by an arc along the perimeter of the
circle with length equal to the radius.

A
steradian

may be defined using Figure (b). Here,
one steradian (sr) is subtended by an area r2 at the
surface of a sphere of radius r.

sin.
d d d


A
differential solid angle
,
d

, in sr, is
defined as

2
0 0
sin 4 ( ).
d d sr
 
 
 
 
 
 
For a sphere, the solid angle is found by
integrating

An antenna’s pattern solid angle,



,
p n
P d

  
 
Antenna Pattern Solid Angle:


All of the radiation emitted by the antenna is concentrated in a cone of solid
angle

p

over which the radiation is constant and equal to the antenna’s
maximum radiation value.

Antennas


Directivity







,
,
,
n
n
avg
P
P
D




The
directive gain
,, of an antenna is the ratio of the
normalized power in a particular direction to the
average normalized power, or







max
max
max
,
,
,
n
n
avg
P
D D
P



 
The
directivity, Dmax
, is the maximum directive gain,

max
4
p
D



Directivity:






,
,
4
n
p
n
avg
P d
P
d





 

 
 
Where the normalized power’s average value taken
over the entire spherical solid angle is



max
1
,
n
P


Using



*
2
2
2
*
*
1 1
,,Re Re
2 2
1 1
Re Re
2 2
1
Re sin
2
sin sin
sin sin sin sin

s s
o
o
o s s
j j j j
o o o o o o
r
I
r
I I
r r
I e I e I e I e
r r r r
 
   
   


 

 
 
   

   
 

 
   
 
 
   
 
   
 
 
 
 
       
 
 
       
 
       
 
 

P E H
a a
a a a a
a a


2
2
r
2
1
sin
2
o
o
I
r

 
 

 
 
a
8.1: In free space, suppose a wave propagating radially away from an antenna
at the origin has

Example

sin
s
s
I
r



H a
where the driving current phasor

j
s o
I I e


Find (1)
E
s



r r r
sin sin sin
s s o s
s o s o o
I I I
r r r
  

  
  
      
  
a a a
E a H a a
Find (2)
P
(
r,

,

)



2
2
2
,,
1
sin
2
o
o
r
I
P
r

 

Magnitude:







2
2
2
2
2
2
2 3
0 0
2
2
3
2
0 0
2
2
4
3
,,,,sin
sin


1
sin
2
1
sin
2
1
sin
2
1 4
2
2 3
rad
rad
rad
rad
rad
o
o
o o
o
o
o
o
P
P r d P r r d d
P r d d
P d d
P d
I
r
I
I
d
r
I
r

 
    
  
 


 
 
  
 

 



 
 
 
 
 
 
  
 
  
 
 
  
 
 
 
 
 
 
 
  


 
 
P S
2
o o
I

Find (3)
P
rad

Find (4) P
n
(
r,

,

) Normalized Power Pattern





max
,,
,
n
P r
P
P





2
2
2
,,
1
sin
2
o
o
r
I
P
r

 

2
max
2
1
2
o
o
I
P
r




2
,
sin
n
P



We make use of the formula

3
3

cos
sin cos
3
d


 
  

3
3
0
0
3 3

cos
sin cos
3
cos cos 0
cos cos0
3 3
1 1 2 4
1 1 2
3 3 3 3
d




 


 
  
 
 
 
   
     
 
   
   
 
 
   
       
   
 
   
 

Find (5) Beam Width



2
,
sin
n
P



2
1
sin
2
HP


1
sin
2
HP

 
1
sin
2
HP

 
,1
45
HP


,2
135
HP


and



135 45 90
Beamwidth BW
  
,1
45
HP


,2
135
HP


90
BW

0.5
n
P

0.5
n
P

z
(6) Pattern Solid Angle Ω
p
(
Integrate over the entire sphere!
)



2 2
2 3 3
0 0 0 0

4 8
sin sin sin sin 2
3 3
P
d d d
d d d
  
  

     
  
 
     
  
 
 
  
    


,
p n
P d

  
 
(7) directivity
D
max

max
4 4 2
1.5
8
3
3
P
D
 

   

(8) Half
-
power Pattern Solid Angle Ω
p,HP
(
Integrate over the beamwidth!
)



2 135 135 2
2 3 3
,
0 0
45 45

5 5 2
sin sin sin sin 2
3
3 2
P HP
d d d
d d d
 
  

     
 
 
 
     
 
 
 
 
 
 
 
    


,
,
p HP n
P d

  
 








135
3 3
135
3
3
45
45

cos 135 cos 45
cos
sin cos cos 135 cos 45
3 3 3
1 1 1 1 2 2 10 5
2 6 2 2 6 2 2 6 2 6 2 3 2
d


 
 
   
 
 
   
        
 
   
 
 
   
 
 
   
        
 
   
   
 

Power radiated through the beam width

,
5 2
5 2
3
0.88 (or) 88%
8
8
3
P HP
BW
P
P



   

BW
z
= 88%
BW
P
Antennas


Efficiency

Power is fed to an antenna through a T
-
Line and
the antenna appears as a complex impedance

Efficiency

.
ant ant ant
Z R jX
 
ant rad dis
R R
R
 
where the antenna resistance consists of
radiation resistance and and a dissipative
resistance.

2
1
2
rad o rad
P I R

2
1
2
diss o diss
P I R

The power dissipated by ohmic losses is

The power radiated by the antenna is

An antenna efficiency e can be defined as the ratio of the radiated power
to the total power fed to the antenna.

rad rad
rad diss rad diss
P R
e
P P R R
 
 
For the antenna is driven by phasor current

j
o s
I I e


Antennas


Gain

Gain





,,
G eD
 

The power gain, G, of an antenna is very much like its directive gain, but
also takes into account efficiency

The maximum power gain

max max
G eD

The maximum power gain is often expressed in dB.





max max
10
10log
G G
dB

Example

D8.3: Suppose an antenna has D = 4, R
rad

= 40


and R
diss

= 10

. Find
antenna efficiency and maximum power gain. (Ans: e = 0.80, G
max

= 3.2).

40
10 40
0.8 (or) 80%
rad
rad diss
R
e
R R

 
 
Antenna efficiency

Maximum power gain





max max
4 0.8 3.2
G eD

 






max max
10 10
10log 10log
3.2 5.05 dB
G G
dB

 
Maximum power gain in dB