Transcript Antennas

Antennas
• Hertzian Dipole
– Current Density
– Vector Magnetic Potential
– Electric and Magnetic Fields
– Antenna Characteristics
Hertzian Dipole
Step 1: Current Density
Let us consider a short line of current placed along
the z-axis.
i(t)  Io cos t   
j
Where the phasor I o  I s e
The stored charge at the ends resembles an electric
dipole, and the short line of oscillating current is then
referred to as a Hertzian Dipole.
The current density at the origin seen by the
observation point is
J ds 
Is
S
e  j  Rdo a z
A differential volume of this current element is
dvd  Sdz
J ds dvd  I s e
 j  Rdo
dza z
Hertzian Dipole
Step 2: Vector Magnetic Potential
The vector magnetic potential equation is

A os  o
4
l 2

l 2
I s dza z e
 j  Rdo
Rdo
A key assumption for the Hertzian dipole is
that it is very short so
Rdo  r
o I s l e  j  r
A os 
az
4
r
The unit vector az can be converted to its
equivalent direction in spherical coordinates using
the transformation equations in Appendix B.
az  cos  ar  sin  a
o I s l e  j  r
A os 
 cos  ar  sin  a 
4
r
This is the retarded vector magnetic potential at the observation point resulting
from the Hertzian dipole element oriented in the +az direction at the origin.
Hertzian Dipole
Step 3: Electric and Magnetic Fields
The magnetic field is given by
Bos    A os
H os =
B os
o

1
o
  A 
os
I s l e j  r 
1
Hos 
 j    sin  a
4 r 
r
It is useful to group  and r together
H os 
I s l  2 e j r  j
4
The electric field is given by

 
 sin  a
2

r

r

  
Eos  oar  Hos .
1
Eos  jo
In the far-field, we can neglect the
second term.
Far-field condition:
H os  j
I s l  e j r
4
r
r

2
sin  a
1
1
r
  r 2
I sl  e j r
4
r
sin  a .
Hertzian Dipole
Step 4: Antenna Parameters
Power Density:
P  r, ,   
1
2
Re  E os  H *os 
 o  2 I o2 l 2  2
P  r,   
sin  a r
2 2 
 32 r 
Maximum Power Density: Pmax
o  2 I o2 l 2

32 2 r 2
Antenna Pattern Solid Angle:
 p    sin 2  d     sin 2  sin d d
p 
8
3
Directivity:
Dmax 
4
p
 1.5
Hertzian Dipole
Step 4: Antenna Parameters
Total Radiated Power and Radiation Resistance :
The total power radiated by a Hertzian dipole can be calculated by
Prad  r 2 Pmax  p
 o  2 I o2 l 2
Prad  r 
2 2
 32 r
2
The power radiated by
the antenna is

2  l 
2


40

  Io
 P


2
Prad  I o2 Rrad
Circuit Analysis
Field Analysis
Rrad
l 
 80  

2
2
Hertzian Dipole - Example
Example
Electric Field:
Power density:
Maximum Power density:
Normalized Power density
Example
Antenna Pattern Solid Angle:
 p    Pn  ,   sin  d d d     sin 3  cos 2 d d
p 
  sin  d   cos  d 
3
Radiated Power:
Radiated Resistance:
2