1
1
Chapter 3 : Vector potential and
radiation integrals
• Vector potentials
• Far-field radiation
• Duality theorem
2
Vector Potentials
M
J
r
r
,
ξ
)
,
,
(
z
y
x
)
'
,
'
,
'
(
z
y
x
R
r
'
r
r
r
r
H
E,
)
,
(
fields
magnetic
and
electric
find
)
,
(
currents
magnetic
and
electric
given
:
H
E
Problem
M
J
r
r
)
,
(
potentials
vector
magnetic
and
electric
evaluate
First,
F
A
r
r
3
Vector differential equations
Excitation:
voltage/current source
or incident field
Sources J,M
Vector potential
Radiated fields
integration
differentiation
Analytic or
numerical
techniques
Assuming Lorentz gauge:
M
F
k
F
J
A
k
A
r
r
r
r
r
r
ε
µ
−
=
+
∇
−
=
+
∇
2
2
2
2
Inhomogeneous Helmholtz equations
4
Solution for unbounded
homogeneous regions
'
)
'
(
4
)
'
(
4
)
(
'
)
'
(
4
)
'
(
4
)
(
dv
R
e
r
M
r
F
d
r
F
dv
R
e
r
J
r
A
d
r
A
jkR
V
V
V
V
jkR
V
V
−
−
∫∫∫
∫∫∫
∫∫∫
∫∫∫
=
=
=
=
r
r
r
r
r
r
r
r
r
r
r
r
π
ε
π
ε
π
µ
π
µ
'
)
'
(
4
)
'
(
4
)
(
'
)
'
(
4
)
'
(
4
)
(
ds
R
e
r
M
r
F
d
r
F
ds
R
e
r
J
r
A
d
r
A
jkR
S
S
s
S
jkR
S
S
−
−
∫∫
∫∫
∫∫
∫∫
=
=
=
=
r
r
r
r
r
r
r
r
r
r
r
r
π
ε
π
ε
π
µ
π
µ
2
/
1
2
2
)
cos
'
2
'
(
|
|
;
ˆ
;
'
ξ
rr
r
r
R
R
R
R
R
r
r
R
−
+
=
=
=
−
=
r
r
r
r
r
For volume currents:
]
[V/m
],
[A/m
2
2
V
V
M
J
r
r
]
[V/m
],
[A/m
S
S
M
J
r
r
For surface currents:
5
Solution for unbounded
homogeneous regions (2)
'
)
'
(
4
)
'
(
4
)
(
'
)
'
(
4
)
'
(
4
)
(
dl
R
e
r
M
r
F
d
r
F
dl
R
e
r
J
r
A
d
r
A
jkR
L
l
L
L
jkR
l
L
−
−
∫
∫
∫
∫
=
=
=
=
r
r
r
r
r
r
r
r
r
r
r
r
π
ε
π
ε
π
µ
π
µ
)
2
(
)
(
1
)
(
)
1
(
1
)
(
)
(
F
j
F
j
A
r
F
A
j
A
j
r
r
r
r
r
r
r
r
r
⋅
∇
∇
−
−
×
∇
=
×
∇
−
⋅
∇
∇
−
−
=
ωµε
ω
µ
ε
ωµε
ω
H
E
For line currents:
]
[V
],
[A
M
J l
r
r
Once the vector potentials are found, the fields can be obtained
from
6
Solution for unbounded
homogeneous regions (3)
From (1) and (2), it can be shown that
)
4
(
1
1
)]
'
(
ˆ
[
4
)
(
)
(
1
)]
'
(
ˆ
(
ˆ
[
2
)
3
(
)
(
1
1
)]
'
(
ˆ
ˆ
[
4
)
(
2
2
R
e
kR
r
J
R
jk
r
R
e
kR
kR
j
r
J
R
R
j
R
e
kR
kR
j
r
J
R
R
j
r
jkR
jkR
jkR
−
−
−
∫
∫
∫
−
×
−
=