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Nov 16, 2013 (4 years and 6 months ago)

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Antennas

Antenna

Electric and Magnetic Fields

Power density

Beamwidth

Pattern Solid Angle

Directivity

Efficiency

Gain

Antennas

Wires passing an alternating current emit, or
,
electromagnetic energy. The shape and size of the
current carrying structure determines how much

Transmitting Antenna:

Any structure designed to
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
a transmitting antenna

Antennas

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
-
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
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:

Antennas

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

Isotropic

antenna
:

electromagnetic waves equally in all directions.

,1
n
iso
P


Antennas

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.

Antennas

A polar plot

A rectangular plot

It is clear in Figure that in some very specific
directions there are zeros, or
nulls
, in the

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.

Beam Width:

Antennas

Pattern Solid Angle

A

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

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

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
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

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
 
R R
R
 
where the antenna resistance consists of
radiation resistance and and a dissipative
resistance.

2
1
2
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.

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

= 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%
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