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Antennas & Propagation
Antennas
&
Propagation
Mischa Dohler
King’s College London
Centre for Telecommunications Research
Lecture II, 1. Oct. 2001
Antennas & Propagation
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Lecture II, 1. Oct. 2001
Antennas & Propagation
Overview of Lecture II
-
Review of Lecture I
-
Philosophy of Antennas
- Analytical
Tools
-
Wave Equation
-
Hertzian Dipole
Lecture II, 1. Oct. 2001
Antennas & Propagation
Review
Lecture II, 1. Oct. 2001
Antennas & Propagation
Fourier Transform
S ( f )   s (t )  e
 j 2ft
dt
“The steeper the signal in time, the more
high frequency components are
required to build such signal.”
The FT does tell us which frequencies are
used, but NOT when.
Lecture II, 1. Oct. 2001
Antennas & Propagation
Vector Analysis
Gradient (grad φ):
Characterises the changes of a scalar field.
Divergence (div E):
Characterises, “how much a field diverges.”
Rotation (rot H):
Characterises, “how much a field rotates.”
Nabla Vector
 




x y z
x
y
z



x
y
z
x
y
z
D x D y D z
D 


x
y
z
  grad 
  D  div D
  H  ...
  H  rot H
Lecture II, 1. Oct. 2001
Antennas & Propagation
Maxwell’s Equations
div D  
div B  0
D   0  E
B  0   H
B
rot E  
t
They seem
coupled.
D
rot H  J 
t
rot and div merely characterise the
change in location, yet not in time!
Lecture II, 1. Oct. 2001
Antennas & Propagation
Maxwell’s Equations
B
rot E  
t
D
rot H  J 
t
THE KEY TO ANY OPERATING ANTENNA
Suppose:
1. There does exist an electric medium, which
provides a current I and thus a current density J.
2. This causes location varying magnetic field H
3. This causes location varying magnetic flux B, but no
time varying magnetic flux. Thus no rot E, thus no
time varying electric flux. Thus no wave!
Lecture II, 1. Oct. 2001
Antennas & Propagation
Maxwell’s Equations
B
rot E  
t
Suppose:
D
rot H  J 
t
1. There is a time varying current density J.
2. This causes location and time varying magnetic
field H
3. This causes location and time varying magnetic
flux B.
4. This causes location and time varying electric field E.
5. This causes location and time varying electric flux D.
6. This causes location and time varying magnetic
field H, even if without current density J.
Lecture II, 1. Oct. 2001
Antennas & Propagation
Philosophy of
Antennas
Lecture II, 1. Oct. 2001
Antennas & Propagation
The Transmitting Antenna
H
rot E    0
t
E
rot H  J  0
t
THE TAKE-OFF RUNWAY FOR A TRANSMITTING ANTENNA
Thus, we only need a medium, which is capable of
carrying a time-variant current.
We will call this medium: Antenna.
Outside the Antenna the electromagnetic field can
propagate on its own without the source J, since
both fields are coupled through the formulas!
Lecture II, 1. Oct. 2001
Antennas & Propagation
The Receiving Antenna
H
rot E    0
t
E
rot H  J  0
t
THE LANDING RUNWAY FOR A RECEIVING ANTENNA
Thus, we only need a medium, which has free
electrons to generate a current out of a timevarying electromagnetic field.
In any case, there is always a time-variant current density necessary:

J0
t

I 0
t
2
Q0
t 2
Lecture II, 1. Oct. 2001
Antenna Philosophy
Antennas & Propagation
Blackboard!
Accelerated Charges
1. Time-varying electric current
2. Discontinuities in the wire
- harmonic current
- bent wire
- modulated information
- sharp edges, etc
Decoupled, thus propagating waves
Efficiency?
Lecture II, 1. Oct. 2001
Antennas & Propagation
Antenna Philosophy
An Antenna is
an efficient way of converting a guided wave
into a radiating wave or vice versa.
rod, wave guide, micro strip, transmission line
free space traveling wave
Lecture II, 1. Oct. 2001
Antennas & Propagation
Antenna Philosophy
Transmission Line
Current Distribution
V
Mutual Cancellation
(Half-wave) Dipole
V
Radiation
Lecture II, 1. Oct. 2001
Antennas & Propagation
Dipole
r … (radial) distance
-
Coordinate system
θ … Elevation
z
φ … Azimuth
θ
Tr. Line
r
y
Load
φ
x
-
Electric and Magnetic Field Vector
H
r
E
The “longer” the vectors E & H at point r, the more energy is
available at that point.
BUT! We are also interested in the changes from location to location.
Lecture II, 1. Oct. 2001
Antennas & Propagation
Dipole Radiation Pattern
Radiation Pattern is defined as …
“… the variation of the magnitude of the electric or magnetic field
as a function of direction (at a distance far from the antenna).”
very short dipole
half wave
one wave length
1.5 wave length
Lecture II, 1. Oct. 2001
Antennas & Propagation
Analytical Tools
Lecture II, 1. Oct. 2001
Vector Relationships
Antennas & Propagation
Blackboard!
rot rot H  0
vector
rot H
vector
div rot H  0
vector
div D
rot grad    0
scalar
scalar
grad 
vector
div grad    0
Lecture II, 1. Oct. 2001
Vector Relationships
Antennas & Propagation
Blackboard!
a) div rot H      H  0
b) rot grad         0
c) div grad       
2
2
2
 2  2  2
x
y
z
 2   ... Laplace operator
d) 2 A  A 



A

x

A

y

Az  z
x
y
x 2
y 2
z 2
2
2
Equivalent value
for vector
2
e) rot rot H       H  
 grad div H   2H
   H   2H
Lecture II, 1. Oct. 2001
Wave Equation
Antennas & Propagation
Blackboard!
(1)
div D  
(2)
D   0  E
B  0   H
(3)
B
rot E  
t
    H  0
(4)
div B  0
(5)
D
rot H  J 
t
     0
    H    H  2H
Lecture II, 1. Oct. 2001
Wave Equation
Antennas & Propagation
Blackboard!
The magnetic vector potential A
The electric scalar potential Φ
is defined such that
is defined such that
A
   E 
t
  A  B(r , t )
They are normalised through the
Lorentz condition:

A  
0
t
1  2A
 A  2 2     J (r , t )
c t
2
2
1


 (r , t )
2
  2 2 
c t

Lecture II, 1. Oct. 2001
Wave Equation
Antennas & Propagation
Blackboard!
Time-dependent inhomogeneous wave equation
2
1


2
   2 2   F (r , t )
c t
Find : A, 
B(r, t )    A
A
E   
t
1
E     H  dt

Lecture II, 1. Oct. 2001
Wave Equation
Antennas & Propagation
Blackboard!
Harmonic excitation:
 r' , t   Re r'   e jt 
Ar' , t   ReAr'   e jt 
H  A/
E    H / j
Helmholtz Equations
2A(r)  k 2A(r)    J(r)
2(r)  k 2(r)    (r) / 
k


2
c

... wave number
Lecture II, 1. Oct. 2001
Antennas & Propagation
Wave Equation
Advantageous procedure to solve radiation problems.
1. Represent signal to be transmitted through current density J.
2. Resolve J into its harmonics.
3. Find the harmonic magnetic vector potential A.
4. To find the magnetic field H, solve :
H  A/ 
5. To find the electric field E, solve :
E    H / j
6. To find the overall field of the signal, apply inverse FT.
Lecture II, 1. Oct. 2001
Solution of Wave Equation
Antennas & Propagation
Blackboard!
Time-dependent inhomogeneous wave equation
r … (radial) distance
θ … Elevation
z
Point Charge Q(t)
at (0,0,0)
φ … Azimuth
θ
r
y
φ
x
2
1


Q (t )   (0)
2
  2 2 
c t

 2  2  2

 2  2
2
x
y
z
2
2 
  
1   2  
  
r

f

f


 1

2
2

r r  r 
  
  
Lecture II, 1. Oct. 2001
Solution of Wave Equation
Antennas & Propagation
Blackboard!
1   2   1  2
Q(t )   (0)
r
 2 2  
2
r r  r  c t

Outside the source charge.
1   2   1  2
r
 2 2 0
2
r r  r  c t
Substitution: R=r·Φ
2R 1 2R
 2 2 0
2
r
c t
Solution: wave

R( r , t )  f t  r
c

Lecture II, 1. Oct. 2001
Solution of Wave Equation
Antennas & Propagation
Blackboard!

R( r , t )  f t  r
c

 r, t  
R(r, t )  r  r, t 

f tr
c

r
f t 
 r, t  
r
r0
Neglecting the time derivative!
Q (t )   (0)
 f (t ) 
  


 r 
2
2
Poisson’s equation for the electrostatic potential.
Lecture II, 1. Oct. 2001
Antennas & Propagation
Solution of Wave Equation
 
2
Q (t )   (0)

The Solution for Poisson’s equation is the Coulomb potential:
Q (t )
 r, t  
4  r
Q (t )
f t  
4
2
1


Q (t )   (0)
2
  2 2 
c t

f t 
 r, t  
r
 r, t  

f tr

c
r
Q (t  r )
c
 r, t  
4  r
Lecture II, 1. Oct. 2001
Antennas & Propagation
Solution of Wave Equation
Q (t  r )
c
 r, t  
4  r
d r, t  
1
 dQ (t  r )
c
4  r
z
|r-r’|
d r, t  
|r|
dQ
y
dQ    dV '
|r’|
x
Volume Charge
Q(r’,t) in V’
 r, t   
V'
1
 dQ t  | r  r' | c 
4 | r  r' |
d r, t  
 r' , t  | r  r' | c 
 dV'
4 | r  r' |
 r' , t  | r  r' | c 
 dV'
4 | r  r' |
Lecture II, 1. Oct. 2001
Antennas & Propagation
Retarded Potentials
 r, t  
 r' , t  | r  r' | c 
1
4 
V'

A r, t  
4

V'
| r  r' |
 dV'
J r' , t  | r  r' | c 
 dV'
| r  r' |
 r' , t   Re r'   e jt 
J r' , t   ReJ r'   e jt 
 r  
1
4 
V'

A r  
4

V'
 r' e  jk|r r'|
| r  r' |
 dV'
J r' e  jk|r r'|
 dV'
| r  r' |
Lecture II, 1. Oct. 2001
Antennas & Propagation
Wave Equation
Advantageous procedure to solve radiation problems.
1. Represent signal to be transmitted through current density J.
2. Resolve J into its harmonics.
solved
3. Find the harmonic magnetic vector potential A.
4. To find the magnetic field H, solve :
H  A/ 
5. To find the electric field E, solve :
E    H / j
6. To find the overall field of the signal, apply inverse FT.
Lecture II, 1. Oct. 2001
Antennas & Propagation
Hertzian Dipole
Lecture II, 1. Oct. 2001
Hertzian Dipole
Antennas & Propagation
Blackboard!
Impact of current along infinitesimal small wire.
r … (radial) distance
θ … Elevation
z
Current along z of
φ … Azimuth
a wire length ΔL
θ
r
y
ΔL
φ
x

A r  
4

V'
J r' e  jk|r r'|
 dV'
| r  r' |
-
current constant?
-
coordinate system?
-
distance r = r’?
Lecture II, 1. Oct. 2001
Hertzian Dipole
Antennas & Propagation
Blackboard!

A r  
4
| r  r' || r | r

V'
J  J z
J r' e  jk|r r'|
 dV'
| r  r' |
I   J  dA'
dV'  dA'dz'
A'

  jkr 
  J z dA  dz'
A r  
e



4r
L A


A r  
I  L  e  jkr z
4r
Lecture II, 1. Oct. 2001
Hertzian Dipole
Antennas & Propagation
Blackboard!

A r  
I  L  e  jkr z
4r
 
A θ 
A  sin      r 

 
 
 1 A r 1  rA  


  θ 
r r 
 r sin  
1   rA θ  A r 


φ
r  r
 
1
B  H    A r  
sin 
Ar  Az cos
Aφ  0
A  Az sin 
Lecture II, 1. Oct. 2001
Antennas & Propagation
Hertzian Dipole
2

 1  
1
1
2
 jkr
Hφ  
I  L  k  sin   e 
 
 
4
 jkr  jkr  
E    H / j
2
3

 1  

2
 jkr  1 
Er  
I  L  k  cos  e 
  
 
2
 jkr   jkr  
2
3


1  1   1  
2
 jkr
E  
I  L  k  sin   e 
 
  
 
4
 jkr  jkr   jkr  


k

 
Intrinsic impedance (120π  377ohm for free space)
Lecture II, 1. Oct. 2001
Antennas & Propagation
Hertzian Dipole
Near Field Approximation
Far Field Approximation
k  r  1
k  r  1
  I  L  cos
Er 
j  2kr3
Er  0
  I  L  sin 
E 
j  4kr3
E    H φ
I  L  sin 
Hφ 
4r 2
I  L  k  sin   jkr
Hφ  j
e
4r
E & H are in quadrature phase, thus
E & H are in phase, thus they carry
merely energy storage
energy!
Lecture II, 1. Oct. 2001