电磁场数值分析-谢拥军3

(
)
S
2
Located on the surface S and radiating in a homogeneous space having constitutive parameters ε1 and 1 . This property is a fundamental theorem of electromagnetics generally known as Huygens' surface equivalence principle.
Integrating both sides of Equation (3.5) over region 1 and applying the divergence theorem r r (3.6) ∫∫∫ Qdv = ∫∫ Q ndS + ∫∫ Q ndS
Γ1 S S∞
where n is the normal vector on the surface pointing out of region 1, produce r r r r r r r r (3.7) ∫∫ E1 × H 2 E2 × H1 ndS = ∫∫∫ E2 J1 H 2 K1 dv
r r ′ ′ + k 2 E 1 (r ′ ) = j ωε 1 e u r r 4π r r ′
r r r r jk r r ′
(3.12) (3.13)
r r e jk r r ′ H 1 (r ′ ) = ′ × u r r 4π r r ′
Γ2
ε 2 (r ) r 2 (r )
r
r r J 1 , K1
S
Γ1 ε1 , 1
S∞
n
2
Two reg(II)
Throughout region 1, Maxwell's curl equations can be written r r r (3.1) × E1 = jω1H1 K1 r r r × H1 = jωε 1E1 + J 1 (3.2) r × E2 = jω1H 2 (3.3) r r × H 2 = jωε 1E2 (3.4) We can construct
r 1 1 S J S ρe = jω r 1 S KS ρm = jω
Where S is the surface divergence operator.
(3.19) (3.20)
11
�
S
(
)
Γ1
(
)
Equation (3.7) can also be written as
∫∫ [ (
S
r r r r r r r r E1 n × H 2 H1 E2 × n dS = ∫∫∫ E2 J 1 H 2 K1 dv
Γ1
)
(
)]
(
)
(3.8)
4
Huygens' surface equivalence principle (I)
However, we have converted the problem from one requiring the solution of Equations (1.1)-(1.4) to one requiring the solution of Equations (1.13)-(1.16), with equivalent surface currents and charges defined respectively by Equations (3.17)-(3.18) and the continuity equations
r r r r r r r r E1 × H2 E2 × H1 = E2 J1 H2 K1
(
)
(3.5)
Equation (3.5) is a form of the Lorentz reciprocity theorem.
3
Two regions separated by a surface S (III)
7
Extinction theorem (I)
Region 1
ε 2 ( x, y , z ) 2 ( x, y , z )
ε1 , 1
p.e.c.
r r E1 , H1
S
Region 2
r r E2 , H 2
n
source
8
Extinction theorem (II)
r r Consider equivalent source J S and K S located on the mathematical
integration variable in (3.8). For these sources, Equation (3.8) can be written as
r K1 = 0
r r r r r (3.11) u E2 rr = ∫∫ E1 n × H 2 H1 E2 × n dS ′ S r r r Where E1 and H1 are the fields producedrat location r ′ in an r r infinite homogeneous space by sources J 1 and K1 located at r .
surface S and satisfying r r J S = n × H1 r r K S = E1 × n Where n is the outward normal vector.
(3.17) (3.18)
The fields in region 2 are NOT identical to those of the original problem; in fact, null fields are produced throughout region 2 by the combination of the original and equivalent sources. We can remove all the inhomogeneities from region 2 and leave a homogeneous medium with the same constitutive parameters as region 1.
Let us suppose that the sources in region 1 are
r r r δ (r r ′) J1 = u
(3.9) (3.10)
r r r denotes the source point in region 1 and r ′ represents the Where
[ (
)
(
)]
5
Huygens' surface equivalence principle (II)
These fields can be expressed in terms of the first sour-field relationship derived before, to obtain
-3Surface equivalence principle
Xie, Yongjun(谢拥军) Department of Microwave Telecommunication Engineering
1
Two regions separated by a surface S (I)
n
r r J 2 , K2
r r jk r r′ r r jk r r′
r +k r = u u E2 r jωε1
2
∫∫ (
S
r e r e n × H2 r r dS′ u × ∫∫ E2 ×n r r dS′ S 4π r r′ 4π r r′
)
(
)
(3.14) r r Equation 3.14 is a statement that the field produced by J 2 , K 2 at some location outside of region 2 can be expressed in the form of an integration over tangential fields on the surface of region 2. And the field is equivalent to that produced by surface current densities r r (3.15) J S = n × H 2 r r (3.16) K = E × n
9
Extinction theorem (III)
Region 1
ε1 , 1
ε1 , 1
r r E1 , H1
r r K S = E1 × n
Region 2 (null fields)
r r J S = n × H1
n
10
Conclusions
r r In fact, J S and K S are unknowns that remain to be determined.
where k = ω (1ε1 )1 / 2. Note that the derivatives are taken with respect to the primed coordinates.
6
Huygens' surface equivalence principle (III)
The Equation (3.11) can be converted to