Prof. David R. Jackson
ECE Dept.
Spring 2009
Notes 15
ECE 6341
ECE 6341
1
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Arbitrary Line Current and Far-Field Identity - Prof. David Jackson, Study notes of Electrical and Electronics Engineering
Notes from a spring 2009 ece 6341 class on the topic of arbitrary line current and far-field identity. It covers the introduction of fourier transform, the expression of arbitrary line current as a collection of phased line currents, and the derivation of the sommerfeld identity for calculating the far-field of finite sources in cylindrical coordinates.
Typology: Study notes
Pre 2010
1 / 17
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Arbitrary Line Current and Far-Field Identity - Prof. David Jackson and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!
Prof. David R. Jackson
ECE 6341ECE 6341^ Spring 2009 ECE Dept.^ Notes 15
Arbitrary Line CurrentArbitrary Line Current
TMz
:
y
z x
( )I z
( ,^
) ρ A^ zz
Introduce Fourier Transform:
1 ( )^
(^ ) 2
jk z^ z z^
z
I z^
I k^
e^
dk
− −∞ =^
∫
(^ )
(^ )
jk zz
z I k^
I^ z e
dz
+∞
=^ ∫ −∞
Hence, from superposition, the total magnetic vector potential is
(^ )^ (
2
0
0 (^
)
8
jk z^ z
z^
z^
z
A^
I k
H
k^
e^
dk
j
ρ
+∞
−
−∞ =^
Arbitrary Line Current (cont.)^ Arbitrary Line Current (cont.)
ExampleExample
Uniform phased line current
0 0 ( )
jk^ zz
I z
I e
−
0 0 0
(^
)
0 0
0
(^
)
2
(^
) z^
z jk^ z^ z^ z
jk z
z
j k^
k^ z z
z
I k
I^
e^
e^
dz
I^
e^
dz
I^
k^
k
+∞−^ πδ
−∞+∞
− −∞ = = =^
− ∫ ∫
^
y
x
I^ (z)^
= I^0
z
(^ )^
(^12)
jk x^ x
x
x^
e^
dk
δ^
∞ π − −∞ =^
∫
Note:
5
Example (cont.)Example (cont.)
Hence^ (^
(^1) ) 2
2 2
0
(^0) z
k^
k^
k = ρ
− where
y
x
I^ (z)^
= I^0
z 0
(2) 0
0
0 (^
)
4
jk^ z^ z
z
I A^
H^
k^
e
j
ρ
μ
ρ^
−
⎛^
⎞ = ⎜^
⎟ ⎝^
⎠
ExampleExample
Dipole
y
z x
Il ( )
( )
I z
Il^
z δ = ( )z I k
Il=
Example (cont.)Example (cont.)
Hence, A spherical wave is thus expressed as a collection of cylindrical waves.
(^2) ( ) 0
1
(^
)
2
z
jkr
jk z
z
e^
H^
k^
e^
dk
r^
j
ρ ρ
−^
+∞^
−
−∞ =^
∫ “Sommerfeld identity”
Far-Far
-Field IdentityField Identity
This identity is useful for calculating the far-field of finite (3D) sources incylindrical coordinates. Note: we assume that the current decays at
z^ = ±
∞^ fast enough so
that the 3D far field exists.
y
z x
( )I z
r
θ
Far-Far
-Field Identity (cont.)Field Identity (cont.)
From 6340, as
→ ∞r
0 ~^
( ,
)
4
jkre
A^
a r μ^
θ φ
π
−⎛
⎞ ⎜^
⎟ ⎝^
⎠
(^ )
(^
)
'sin^ cos
'sin
sin^
'cos 'cos
0 ( ,^
)^
'^
'^ '
'
(^ ')
'
4
jk x^
y^
z
V
jkr
j kz
a^
J^ r
e^
dx dy dz
e^
I z^
e^
dz
r
θ^ φ
θ^ φ^
θ θ
θ φ
μ π
+^
−^
+∞ −∞
=
⎛^
⎞
=^
⎜^
⎟ ⎝^
⎠ ∫
∫
Hence
0 ~^
(^ cos
)
4
jkr
z
e
A^
I k r μ
θ
π
−⎛
⎞ ⎜^
⎟ ⎝^
⎠
Far-Far
-Field Identity (cont.)Field Identity (cont.)
Hence, comparing these two,
as
(^2) ( )
0
0
0 (^ )
(^
)^
~^
(^ cos
z
jkr
jk z
z^
z
e
I k
H
k^
e^
dk^
I k
j^
r
ρ
−
+∞^
−
−∞
⎛^
⎜^
⎝^
∫^
^
→ ∞r
(^2) ( ) 0 (^ )
(^
)^
(^ cos
z
jkr
jk z
z^
z
e
I k
H
k^
e^
dk^
j^
I k r
−
+∞^
−
−∞
⎛^
⎜^
⎝^
∫^
^
or 14
Far-Far
-Field Identity (cont.)Field Identity (cont.)
Therefore, we have
(^2) ( )
1
(^ )
(^
)^
(^ cos
z
jkr
jk z^
n
z^
n^
z
e
I k
H
k^
e^
dk^
j^
I k r
ρ ρ
θ
−
+∞^
−^
−∞
⎛^
⎜^
⎝^
∫^
^
Note: this is valid for
ρ^ → ∞ ,
r
θ → ∞
≠^
D
Hence, this is valid for 16
Far-Far
-Field Identity (cont.)Field Identity (cont.)
Since the current function is arbitrary, we can write
(^2) ( )
1
(^ )
(^
)^
(^ cos
z
jkr
jk z^
n
z^
n^
z
e
f k
H
k^
e^
dk^
j^
f k r
ρ ρ
θ
−
+∞^
−^
−∞
⎛^
⎜^
⎝^
for^
,^
r
θ → ∞
≠^
D