Notes 21 Spring 2005
Antenna Principles
Consider
a
differential
current
filament
(antenna)
of
length
d
carrying
a
current
.
Note that this current is
uniform
along the length, i.e., it is
position independent a temporary simplification.
( )
(
)
ω
t
cos
I
t
i
o
=
φ
d
θ
l
d
p
v
E
°
=
φ
90
θ
d
P
φ
l
d
θ
Wire filament
H
i
x
o
r
θ
(
)
θ
sin
r
o
°
=
φ
0
d
z
y
Earlier in the semester we determined that a magnetic field exists near a wire carrying a
steady current.
It follows then that a timevarying
field exists if a timevarying
current is present in the wire.
And through Faraday’s law of induction we know that the
timevarying
field in turn produces a timevarying
field and the timevarying
E

field produces a timevarying
field via Amperes law, etc., etc., and an EM wave will
propagate away from the wire at velocity
.
H
H
E
H
p
v
It can be shown that in spherical coordinates these fields in phasor form are represented
by
⎟
⎠
⎞
⎜
⎝
⎛
λ
π
−
∗
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
λ
π
∗
θ
π
φ
=
φ
r
2
j
exp
r
1
r
2
j
sin
4
d
I
ˆ
H
2
o
r
⎟
⎠
⎞
⎜
⎝
⎛
λ
π
−
∗
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
π
λ
−
∗
θ
π
η
=
r
2
j
exp
r
2
j
r
1
cos
2
d
I
rˆ
E
3
2
o
r
r
⎟
⎠
⎞
⎜
⎝
⎛
λ
π
−
∗
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
π
λ
−
+
λ
π
∗
θ
π
η
θ
=
θ
r
2
j
exp
r
2
j
r
1
r
2
j
sin
4
d
I
ˆ
E
3
2
o
r
Together, these are
known as the
near
fields.
1
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Notes 21 Spring 2005
It can be seen in these equations that as the distance (r) from the filament increases the
terms involving
become negligible and the resulting equations are called the
far field equations or the
radiation fields
.
Traditionally there is a rule of thumb where
this transition occurs at
:
3
2
r
and
r
λ
>
10
r
⎟
⎠
⎞
⎜
⎝
⎛
λ
π
−
∗
θ
λ
φ
=
φ
r
2
j
exp
sin
r
2
d
I
j
ˆ
H
o
r
0
E
r
=
r
Together, these are
known as the far fields
or
radiation fields.
⎟
⎠
⎞
⎜
⎝
⎛
λ
π
−
∗
θ
λ
η
θ
=
θ
r
2
j
exp
sin
r
2
d
I
j
ˆ
E
o
r
These far field equations show that
φ
θ
η
=
H
E
, which is precisely the relationship
between the fields that we established for the uniform plane wave!
The time domain
expressions are
(
)
and
r
t
sin
sin
r
2
d
I
ˆ
H
2
o
λ
π
φ
−
ω
⎟
⎠
⎞
⎜
⎝
⎛
θ
λ
φ
−
=
r
(
)
r
t
sin
sin
r
2
d
I
ˆ
E
2
o
λ
π
θ
−
ω
⎟
⎠
⎞
⎜
⎝
⎛
θ
λ
η
θ
−
=
r
We note that
represents the equatorial or xy plane.
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 Fall '10
 Czarnecki
 Fundamental physics concepts, Coaxial cable, plane wave, Transmission line, Dipole antenna

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