有限元电磁场计算4

(
)
source
ˆ n
Reading materials for Master’s program
8
Surface integral equations for perfectly conducting scatterers (VI) ---- for thin open target (II)
We wish to treat scattering from an infinitesimally thin open p.e.c. shell, strip or plate. If the surface S collapses to the scatterer surface, the equivalent current densities on either side of the scatterer become superimposed at the location of the thin shell. The equations are unable to distinguish between the two equivalent sources, and we are forced to work woth a single equivalent source that represents the sum of the sources on either side. Since the boundary condition of (1.5) remains valid for infinitesimally thin p.e.c. structures, however, an EFIE of the form Equation (4.5)can be used to treat this type of scattering problem.
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9
Surface integral equations for perfectly conducting scatterers (VII) ---- for thin open target (III)
The MFIE of Equation (4.6) is based on a boundary condition that is NOT valid for extremely thin geometries. Equation (1.6) is actually a special case of the general boundary condition
Region 1 Region 2
µ1 , ε1
ε1 , µ1
(nuHale Waihona Puke Baidul fields)
r r E1 , H1
ˆ n
source
r r ˆ × H1 JS = n
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3
Surface integral equations for perfectly conducting scatterers (I)
p.e.c.
r r E1 , H1
Region 2 (null fields) source
ˆ n
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Equivalent exterior problem associated with perfectly conducting scatterers
r r 2 r inc ∇∇ ⋅ A + k A ˆ × E = −n ˆ× n S jωε 0
(4.5)
Which is an integro-differential equation for the unknown equivalent r surface current density J S . Equation (4.5)holds only for points on the surface S of the scatterer and is one form of the electric field integral equation (EFIE).
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Conclusions
EFIE
Open p.e.c. target and closed p.e.c. target
MFIE
Closed p.e.c. target only
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In deriving the EFIE and MFIE, we imposed only one of the conditions (1.5) and (1.6). Because of this there are scatterers for which the solution of these equations is not unique. The uniqueness issue will be discussed later.
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7
Surface integral equations for perfectly conducting scatterers (V) ---- for thin open target (I)
r H2 r H1
r r r ˆ × H1 − H 2 J =n
r r r ˆ × H1 − H 2 = J S n
(
)
(4.7)
For closed bodies or other situations where the magnetic field vanishes on one side of the surface, Equation (4.7) reduces to (1.6). For infinitesimally thin structures with nonzero fields on both sides of the surface, however, Equation (1.6) is not equivalent to (4.7) and does not actually consititute a boundary condition. Consequently, the MFIE of (4.7) is restricted to closed bodies and cannot be used to describe scattering from an infinitesimally thin p.e.c. structure.
r r inc r ˆ × H = JS − n ˆ ×∇× A n
{
}
S+
(4.6)
Equation (4.6) isralso an integro-differential equation for the unknown surface current J S and is enforced an infinitesimal distance outside the scatterer surface ( S + ). Note that any of the source-field relationship presented before could be employed as alternatives, producing equivalent equations.
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5
Surface integral equations for perfectly conducting scatterers (III) ----MFIE
If the Equation (1.6) is combined with Equation (4.4), we obtain the magnetic field integral equation (MFIE)
(4.3) (4.4)
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Surface integral equations for perfectly conducting scatterers (II) ----EFIE
If the boundary condition of Equation (1.5) is imposed on the surface of the scatterer
Surface integral equations for perfectly conducting scatterers
Part 4
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1
Perfectly conducting scatterers
Region 1
µ1 , ε1
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6
Surface integral equations for perfectly conducting scatterers (IV) ---- how to solve those equations
In principle, either of Equations (4.5) r can be solved to produce r and (4.6) the unknown equivalent source J S . Once J S is determined, the electric and magnetic fields everywhere in space may be found from the sourcefield relationships presented previously, r superimposing the incident field with the scattered fields produced by J S .
It follows that equivalent sources
r r ˆ×H JS = n r KS = 0
can get
(4.1) (4.2)
r inc r inc Assuming that the incident fields E and H are specified, we
r r 2 r inc r r r ∇∇ ⋅ A + k A E (r ) = E (r ) − jωε 0 r r inc r r r H (r ) = H (r ) − ∇ × A