电磁场数值计算

(2.8) (2.9)
6
General description of a scattering problem (IV)
(4) Radiation conditions in a two-dimensional problem (the Sommerfeld radiation conditions)
of the magnetic .according to
(2.12)
jωε 0
r H
s
=
∇×
r A+
∇∇
⋅
r F
+
k
2
r F
(2.13)
jωµ0
And the vector potential satisfy
∇2
r A
+
k
2
r A
=
−
r J
∇2
r F
+
k
2
r F
Байду номын сангаас
=
−Kr
(2.14） （2.15）
8
7
Source-field relationships in homogeneous space
---- a first form (I)
How to solve equations (2.5) and (2.6)?
TvEehrces t=ocrl∇aps∇osti⋅ecAnratl+iaaklp2pAArrro−aa∇cnhd×itsFhretoeelexcptrreicssvethcetofrieplodtseinntitaelrmFrs
those in the right.
___r________________________r_______
Er
Hr Jr
K
_
Hr E
r
_KJr
ρe
ρm
ρm
_ ρe
εµr
µ
εr
Ar
_Fr
__F________________________A_________
14
Conclusions
•General description of a scattering problem •Source-field relationships in homogeneous space ---- four forms •Principle of Duality
(2.31)
( ) G mk
=
1
jωµ 0
∇∇ + k 2 I G
(2.32)
13
Duality relationships
Principle of Duality: the equations describing electromagnetic fields
remain valid if all the quantities in the left column are replaced by
∫∫∫ Ar(rr) =
( ) r
J
rr′
e− jk rr−rr′
4π rr′
drr′
(2.19)
9
Source-field relationships in homogeneous space ---- a first form (III)
The two-dimensional Green’s function is
The description of a scattering problem
Part 2
1
Quick summary of last class (I)
ε0
µ0
r E
inc
kˆ
r H
inc
ε 0ε r
µ0µr
r Er
H
s
s
2
Quick summary of last class (II)
How to describe the electromagnetic problems? Maxwell’s equations and boundary conditions
( ) G
=
1 4j
H
(2
0
)
k
ρr
(2.20)
To summarize, the above procedure is the integration-followed-bydifferentiation procedure, which is NOT well suited for numerical implementation.
rr
A = J *G
r F
=
r K
*G
(2.16) (2.17)
where the scalar function G is the well-known three-dimensional
Green’s function
G
=
e − jk rr
4π rr
(2.18)
And the asterisk (*)denote three-dimensional convolution, that is,
general solutions ( 2.16) and (2.17)and vector Helmholtz equations
appearing in (2.5) and (2.6). Formally, we can write the solutions to
Equations (2.5) and (2.6) directly as
lim ∂Ezs
ρ→∞ ∂ρ
= − jkEzs
lim
∂H
s z
ρ→∞ ∂ρ
=
−
r jkH
s z
(2.10) (2.11)
For the TM and TE polarizations, respectively, where ρ is
the radial variable in cylindrical coordinates.
Source-field relationships in homogeneous space ---- a first form (II)
The solution to equations (2.14)and (2.15) satisfying the radiation
condition can be concisely written in the form
field, symbolically denoted
( ) G ej =
1 j ωε 0
∇∇
+ k2I G
( ) G ek = −∇ × I G
(2.29) (2.30)
and and are the dyadic Green’s functions for the magnetic
field,
( ) G mj = ∇ × I G
15
r E
s
r H
s
= =
−
jωµr0
r A
∇× A−
j−ω∇εΦ0Fre
− −
∇
×
r F
∇Φ m
(2.21) (2.22)
where Φe and Φm are scalar potential functions and given by
Φe
=
ρe ε0
*G
(2.23)
Φm
=
ρm µ0
*G
(2.24)
10
Source-field relationships in homogeneous space ---- a second form (I)
As an alternative to the pure vector potential source-field relationship, a
mixed-potential formulism can be developed
r E
s
r H
s
= =
− ∇
jωµ0
r J
×
r J
−
r + ∇∇ ⋅ J
jωε 0
jωε
0
r K
+
−
∇
×
r K
∇∇
⋅
r K
jωµ0
*G *G
(2.25) (2.26)
A differentiation followed by an integration.
12
Source-field relationships in homogeneous space
(3) Radiation conditions in a three-dimensional problem
lim
rˆ
×
∇
×
r E
s
=
r jkE
s
r→∞
lim
rˆ
×
∇
×
r H
s
=
r jkH
s
r→∞
where r is the conventional spherical coordinate variable.
(2.5) (2.6)
5
General description of a scattering problem (III)
Some additional explanations:
( ) (1) The vector Laplacian:
∇2
r E
=
∇
∇
⋅
r E
−
∇
×
∇
×
r E
(2.7)
(2)our interests is the case of an excitation produced by some source in the far zone. Often, we will consider the incident field to be a uniform plane wave.
Scattered
fields
r E
s
r H
s
4
General description of a scattering problem (II)
The original fields in the presence of the scatterer can be written as:
r E
=
r E
where the asterisk again denotes multidimensional convolution.
11
Source-field relationships in homogeneous space ---- a third form (I)
Using an analogy between Equations (2.14) and (2.15) and their
Volumetric equivalence principle The inhomogenerous material can be replaced by equivalent induced sources radiating in free space
3
General description of a scattering problem (I)
---- a fourth form (I)
A simple interchange of integration and differentiation results in
r E
s
=
r J * Gej
+
r K
* Gek
r H
s
=
r J * Gmj
+
r K
* Gmk
(2.27) (2.28)
Where and are the dyadic Green’s functions for the electric
Splitting the fields into two parts
One associated with the primary source located somewhere outside the scatterer
Incident fields
r E
inc
r H
inc
Another associated with the equivalent induced source
inc
+
r E
s
(2.1)
r H
=
r H
inc
+
r H
s
(2.2)
where
∇2
r E
inc
+
k
2
r E
inc
=
0
(2.3)
∇2
r H
inc
+
k
2
r H
inc
=
0
(2.4)
∇2 ∇2
r E
s
r H
s
+
k
2
r E
s
+
k
2
r H
= s=
j−ω∇µ0×JrJr−+∇jj∇ωωεε⋅ 0J0rKr+−∇∇×j∇ωKrµ⋅ K0r