Page 365 - Electromagnetics

Symmetric field decomposition. Field symmetries may be applied to arbitrary
source distributions through a symmetry decomposition of the sources and fields. Con-
i
i
sider the general impressed source distributions (J , J ). The source set
m
ie 1 i i
J (x, y, z) = J (x, y, z) + J (x, y, −z) ,
x
x
x
2
ie 1 i i
J (x, y, z) = J (x, y, z) + J (x, y, −z) ,
y
y
y
2
ie 1 i i
J (x, y, z) = J (x, y, z) − J (x, y, −z) ,
z z z
2
ie 1 i i
J (x, y, z) = J (x, y, z) − J (x, y, −z) ,
mx mx mx
2
ie 1 i i
J (x, y, z) = J (x, y, z) − J (x, y, −z) ,
my
my
my
2
ie 1 i i
J (x, y, z) = J (x, y, z) + J (x, y, −z) ,
mz
mz
mz
2
is clearly of even symmetric type while the source set
io 1 i i
J (x, y, z) = J (x, y, z) − J (x, y, −z) ,
x
x
x
2
io 1 i i
J (x, y, z) = J (x, y, z) − J (x, y, −z) ,
y y y
2
io 1 i i
J (x, y, z) = J (x, y, z) + J (x, y, −z) ,
z z z
2
io 1 i i
J (x, y, z) = J (x, y, z) + J (x, y, −z) ,
mx mx mx
2
io 1 i i
J (x, y, z) = J (x, y, z) + J (x, y, −z) ,
my
my
my
2
io 1 i i
J (x, y, z) = J (x, y, z) − J (x, y, −z) ,
mz mz mz
2
ie
i
io
i
ie
io
is of the odd symmetric type. Since J = J + J and J = J + J , we can decompose
m
m
m
any source into constituents having, respectively, even and odd symmetry with respect
to a plane. The source with even symmetry produces an even field set, while the source
with odd symmetry produces an odd field set. The total field is the sum of the fields
from each field set.
Planar symmetry for frequency-domain fields. The symmetry conditions intro-
duced above for the time-domain fields also hold for the frequency-domain fields. Because
both the conductivity and permittivity must be even functions, we combine their effects
and require the complex permittivity to be even. Otherwise the field symmetries and
source decompositions are identical.
Example of symmetry decomposition: line source between conducting planes.
˜
Consider a z-directed electric line source I 0 located at y = h, x = 0 between conducting
planes at y =±d, d > h. The material between the plates has permeability ˜µ(ω) and
c
complex permittivity ˜ (ω). We decompose the source into one of even symmetric type
˜
with line sources I 0 /2 located at y =±h, and one of odd symmetric type with a line
© 2001 by CRC Press LLC