APPLICATION
OF
THE
MEASURED
EQUATION
OF
INVARIANCE
TO
RADIATION
AND
SCATTERING
BY
FLAT
SURFACES
Ra ael
POUS:
Ma k P ou y,
and
Kenne l,
I<.
Mei
EECS
Dep .,
Uni .
o
Cali oinia, Be keley
CA
94720
(R. Pouz
is
now wi h he
TSC
Dep .,
Uni .
Poli .
de
Ca alunya,
Apdo.
30.002,
08080
Ba celona,
Spain).
Abs ac
The ewn ly
iii ocliic d
concep o lie Measu ed
Eq
ia ion
o
In a iance
(blE1)
[l,
2.
31
is
used
,o
sol e
.lic-
p oblenl o adia io:] and sca e ing by
la su aces. Becaus on la su aws li elec ic cu en s a e con ined
o
wo
dimensions, a sinil)lc ec o po m .ial o mula ion can be used. The p oblem
o
adia ion and sca ing
by
ec aiigiila s ip dipoles
is
!,ol ed. including he
ans e sal a ia ion
o
he cu m a osq he dipole wid .11.
Also
o
in e es
a lie cu eii s indu ed
(111
an ennas wi h s ep a ia ion: in wid .h, and
wi h
hencl,
and T-junc ioiis.
The
NE1
is
; i1
qua ioii lia caii be used o e mina e
,L
Fini e Di e ence (FD)
iiiesli a bi a ily close o lie objec o in e es . a oiding he use o abso bing bouiid-
a y condi ions, n.hic1i cyiii e lie inesh
o
ex end beyond lie egion o in e es , wi h
aii
inc ease in onipu a ion ime
aid
s o age memo y.
Tlie
iiiexli
us ~l
o
sol e
a
b ip
diliolc-
i.;
21
1e angula. mi sh which es ends only a
ew cells ( ypi dly
!
o
4)
iii
each di w ioii. The poin s
o
his mesh all in o h ee
a ego ics: ex e m
poin x
o
poiu s on lw iiiesh bounda y. poin s on he su ace o
he dipole.
and
iii e ioi poin s no
on
lie
su ace
o
lie dipole. Fo each ca ego y
a
di e en
ype
o
FD
eqiia ion
will
be
used. The uiikiiowii
i:
cliosen
o
be he ec o
po en ial
.i
de inc l as
#.'"''
=
d
x
d
(p
ac o ed in o simpiici y).
I
he la su ace
is pa allel
o
he
, 'y
plaiie. only
.AJ
aiid
d,
need o be conL;ide ed, and hei alues
a
each
mesli
poin
a c
lw
u~ikno~ iis o
lie
FD p oblem. Since wo unknowns a e
associn pd
wi h (wcli ni(4 poin
wo
eqiia ioiis need
o
he
w i en also.
Fo lie
in e io
poin s no
on
liu
me al su ace, lie s ai
da d
FD app oxima ion
o he
wa e
eqiia ioii is
iis4
o
pncli
o
lic,
wo po en ials
,I
=
. .
!)
("2
k, )
.1,1
=
I1
(1)
'This
esea ch
wi s
spon>o l
I,?
I
lac
CaIi1o ni.L
Slaie
hlICRO
p og nm.
xiid
he
indus ial
Spon-
>o
Hughes
.%i e a ('onlpnny
0-7003-1246-5/93/$3.00
0
1993
IEEE
540
Fo he poiii s on he me al su ace, he condi ion ha
ESc
=
-E nc
is used,
yielding he equa ions
This equa. ions caii easily be pu in FD o m. Special ca e needs o be aken a he
edge o he
la
su ace.
Fo
he componen o he ec o po en ial ha is pa allel
o he edge. a sligh a ia. ion
o
he abo e equa ions needs o be used, and o he
no mal componen , we use he ac ha he
no mal
cu en anishes a he edge
(see
[4]
o mo e de ails).
Finally,
o
he
poin s a he mesh bounda ies, we w i e an equa ion o he o m
N-l
.4&l
-
1
(',
'
-A,{,
=
0
l
=
.U,
9
(
4)
j=
I
whe e
Ad"
is he alue o he
ec o
po en ial
a
he bound~y poin and
Ad,
a e he
alues a he
N
--
1
neigl ho ing poin s
(X
may be
4,
5.
01
G
depending
011
whe he
he poin is
011
a
side.
ail
4ge,
o
a
o i c
o
he mesh,. The coe icien s
c,
a e
calcula ed o inake Eqn.
1
a
leas
squa e
e o
i
o
he
hl
po en ials (which we
call
measu ing
iinc ion.~)
whe e
.J;,
J;
a c
wo se s o linea ly independen cu en i, which we call
me ons.
Fo
a
ec angula pla e o leng h
I
and wid h
u .
hese
me ons
a e
])La
713
(my
-
1)ay
Jyly(.T,
y)
=
4111
___
15
m,
5
M,
1s
my
5
My
(6)
1
5
my
5
~4~
(7)
1
c 5
W
(
7zz
-
.
sin
Jym=n'qX,y)
=
CO5
1
5
ma
5
M,
I
W
Figu e
1
shows
he longi udid cu en on a s ip dipole o leng h and wid h
I
=
18.5~
=
1.42X.
when ed o -cen e
a
1/4,
and he inpu admi ance
o
a
cen e -
ed hin s ip dipole. Figu e
2
shows he cu en s in he wo di ec ions when he
wis ed dipole is ed a he cen e . Bo h he longi udinal
and
ans e sal a ia ions
o
he cu en s a e ob ained. as
well
as
a
clea pic u e
o
how he cu en s beha e
lieax he
90
deg ee bend. Figu e
3
shows he monos a ic ada c oss sec ion
o
a
squa e pla e
o
side
n,
coinpa cd
o
he ineasu emen s epo ed in
[GI.
Re e ences
[l]
K.
I<. Mei.
R.
Pous,
Z.
Chen,
Y.
W. Liu, and
M.
D
P ou y, "The Measu ed
Equa ion o In a iance: a new concep in ield compu a ion,"
IEEE
IPmns.
An-
ennas
P opaga .,
submi ed o publica ion.
541
mA/1,’
V
10
8
1G
14
12
10
S
F
4
2
0
Re .
[3]
)-
0 0.2 0.4 0.G
0.8
1
1.2
1.4
1.6
1.8
[/A
Figu e
1.
Cu en on
an
o - en e ed s lip dipole an enna wi h
1
=
18.5~
=
1.42X,
ed a
1/4
( op) niid inpu admi ance o
a
en e - ed s ip dipole an enna wi h
I
=
18.5~
e sus
equen ).
compa ed wi h he
MOM
esul p esen ed in
[5]
(bo om).
[2]
R.
Pous,
‘The Measu ed Equa. ion
o
In a iance:
a
new concep in ield compu-
a ion,” Ph.D. diss a ion, Uni . o Cali o nia a Be kdey, 1992.
(31 I<.
K.
Mei,
R.
Pous.
h.1.
D.
P ou y, and
Y.
U’.
Liu, “Fu he insigh in o he Mea-
su ed Equa ion o In a iance,”
IEEE An ennas and
P paga .
In l.
Sumposium,
Ann A bo , Michigan, 1993.
[4]
M.
D.
P ou y,
R.
Pous,
and
I<.
K.
hlei, “Applica ion
o
he Measu ed Equa ion
o In a iance o ansmission
lilies
and
discon inui ies.”
IEEE
An ennas
and
P opaga .
h l.
Synqi~siil m,
Ann
h bo ,
Michigan, 1993
151
R.
F.
Ha sii g on,
Field
Conipi n ion
by
Momen Me hods,
Robe
E.
K iege
Publishiiig Company. hlalaba , Flo ida,
1952.
[GI
S.
M.
Rao. D.
R.
Wil oii.
and
A.
W.
Glisson, “Elec omagne ic sca e ing
by
su aces o a. bi a y shape,”
IEEE
Z’ uiis.
An ennas
and
P opaga .,
ol. 30, pp.
409-418,
May 1982.
542
Figu e
2:
Cu eli 3
on
a
wis ed
dipole.
o
X
=
In
( e ical w eii
on
he le ,
and
lio izoii al cu en
on
li
igli
J.
10
1
0.1
a
jxz
0.01
0.001
0.0001 0 0.2
O.G
0,5
1
1.2
l/X
Figu e
3:
I Iono ;i ic
ad;iI
C~OS~
scsc ioii
s.
equen y
o
:I
squa e pla e
o
side
a,
o
no mal iiic.idcn e.
c.ompa ed
o
lw measu emen s epo ,d
in
[GI
( he cdcula ioii
is
o
a
ze o- liicli~iehs
pla e.
and
he
iiieaw enieii s a e
o
a
pla e
o
liickiiess
0
00012iX).
543