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c
John
C. Young, July 8, 2010
1
Consider the rectangular waveguide shown below.
Figure 1: Rectangular waveguide geometry.
1
Magnetic Current Green’s Function
1.1
Magnetic Field Due to a Magnetic Current
The magnetic field due to a Magnetic dipole M = l̂δ(r − r′ ) is given.
1.1.1
Mixed Potential Forumulation
The mixed-potential formulation for the magnetic field due to a magnetic current is
H[M; r] = −jωF[M; r] − ∇Ψ[M; r]
(1)
where the electric vector potential is
F[M; r] =
ZZ
S
G F r, r′ · M(r′ )dS ′
(2)
and the magnetic sclar potential is
ZZ
qm (r′ )GF r, r′ dS ′
SZ Z
′
−1
=
∇ · M(r′ ) GF ( r, r′ )dS ′ .
jω
S
Ψ[M; r] =
(3)
(4)
Therefore,
H[M; r] = −jω
ZZ
GF
S
1
r, r · M(r )dS +
∇
jω
′
′
′
ZZ
S
′
∇ · M(r′ ) GΨ ( r, r′ )dS ′
Here, the dyadic Green’s function for the electric vector potential is
G F r, r′ = Fx r, r′ x̂ + Fy r, r′ ŷ + Fz r, r′ ẑ
(5)
(6)
c
John
C. Young, July 8, 2010
2
and
Fx
∞ ∞
nπ mπ nπ mπ ε X X ǫmn
′
sin
r, r =
x sin
x′ cos
y cos
y ′ e−jβmn |z−z |
j2ab
βmn
a
a
b
b
(7)
n=1 m=0
∞
∞ X
X
mπ nπ nπ mπ ǫmn
′
x cos
x′ sin
y sin
y ′ e−jβmn |z−z |
cos
βmn
a
a
b
b
(8)
(9)
n=0 m=0
nπ nπ mπ mπ ǫmn
′
x cos
x′ cos
y cos
y ′ e−jβmn |z−z | .
cos
βmn
a
a
b
b
′
Fy r, r′ =
ε
j2ab
Fz r, r′ =
ε
j2ab
n=0 m=1
∞ X
∞
X
The Green’s function for the magnetic scalar potential is
GΨ r, r′ = Fz r, r′ .
(10)
In the above,
ǫmn = ǫm ǫn ,
(11)
where
ǫn =
nπ 2
−
1 n=0
2 n=
6 0
(12)
and
βmn =
r
k2 −
a
mπ 2
b
, Imagβmn ≤ 0 .
(13)