Appl. sci. Res.
Section B, Vol. 6
ON THE THEORY OF REFLECTION FROM A WIRE GRID PARALLEL TO AN INTERFACE BETWEEN HOMOGENEOUS MEDIA by JAMES R. WAIT National Bureau of Standards, Boulder, Colorado, U.S.A.
Summary T h e reflection f r o m a wire grid parallel to a p l a n e i n t e r f a c e is considered. T h e r e s p e c t i v e m e d i a are h o m o g e n e o u s a n d e i t h e r or b o t h c a n b e dissipative. T h e grid is c o m p o s e d of t h i n e q u i - s p a c e d wires of finite c o n d u c t i v i t y . T h e p l a n e w a v e s o l u t i o n for a r b i t r a r y i n c i d e n c e is t h e n g e n e r a l i z e d for cylindricalw a v e e x c i t a t i o n . T h e e n e r g y a b s o r b e d f r o m a m a g n e t i c line source b y a grid s i t u a t e d o n t h e surface of a d i s s i p a t i v e h a l f - s p a c e is t r e a t e d m some detail. T h i s l a t t e r p r o b l e m is a t w o - d i m e n s i o n a l a n a l o g y of a v e r t i c a l a n t e n n a w i t h a r a d i a l wire g r o u n d s y s t e m .
w 1. Introduction. There have been m a n y investigations of the electromagnetic properties of thin parallel wires composed of conductive material. The first quantitative study was made by L a m b 1) in 1898 who considered the plane wave incident normally on the grid. He showed that if the diameter 2a of the parallel wires was small, the reflection and transmission could be varied by changing the spacing. In 1914 v o n I g n a t o w s k y 2) made a very exhaustive analysis of the scattering of incident plane waves by single metallic grids including the case where the wire spacing is comparable to the wavelength. His formulae have been reduced, extended and applied by other authors since that time 3-11). A very illuminating treatment has been given by M a c F a r l a n e 5) who indicated that a single grid can be represented by an impedance shunted across an infinite transmission line whose characteristic impedance is proportional to the intrinsic impedance of the surrounding infinite medium. He showed that this shunt impedance was proportional to in (d/2~a) plus a correction factor which is a function of angle of incidence 0 and the spacing d. - -
2 5 9
- -
260
JAMES R. WAIT
I t is the purpose of the present p a p e r to consider the effect of an interface on the equivalent s h u n t impedance of the wire grid. I t can be e x p e c t e d t h a t the evanescent (non-propagating) field of the grid will be modified b y the interface. Since wire grids and meshes are often placed on the surface of the ground to i m p r o v e the efficiency of antennas, it is desirable to consider the effect of the grid on the energy absorption in the lower m e d i u m for a source in the u p p e r medium.
w 2. General theory. T h e grid is illustrated in fig. 1. T h e wires are of circular cross section with radius a and are composed of material having a c o n d u c t i v i t y o, dielectric constant el and per-
~ 0
x
E~
Fig. 1. The wire grid )arallel to a plane interface between two homogeneousmedia. m e a b i l i t y #1. The wires are c o n t a i n e d in the plane x = h and are spaced a distance d between centers. I t is assumed t h a t a ~ d. T h e half space, defined b y x > 0, surrounding the wires has a dielectric constant e and a p e r m e a b i l i t y #. The half space x < - - h has a dielectric constant e' and a p e r m e a b i l i t y /~. The electric field of the incident wave with the phase reference at the origin is t a k e n for a time factor e *~t to be
E~=
(E x, ~ 0, E~) = (E0 sin 0, 0, E0 cos 0) exp [ik(x cos 0 -- z sin 0)]
(1)
where k = 2~/wavelength, where E0 is the a m p l i t u d e of the incident wave and where 0 is the angle of incidence. The currents on
REFLECTION
FROM A WIRE
261
GRID
the wires t h e n h a v e the f o r m I e -*kz sin 0 where I is the u n k n o w n c u r r e n t at z = 0. T h e field E w of the c u r r e n t s on the wires can be derived f r o m an electric H e r t z v e c t o r w i t h only a z c o m p o n e n t I I ~ since the induced c u r r e n t s are essentially in the z direction. Therefore
E w = k2H~' iz + g r a d ~HW &
(2)
where iz is a unit v e c t o r in the z d i r e c t i o n . ' T h e H e r t z v e c t o r for the c u r r e n t s on a wire grid is easily o b t a i n e d b y a d d i n g t h e contrib u t i o n s f r o m each of the wires, so t h a t
k2H~ -
#o)I +oo 4 ~e-ikzslnOH(02) [ k c o s O ~ / ( n d - - y ) 2 + n=
(x--h)21
(3)
--oo
where H~02) is the H a n k e l function of order zero of the second kind. T h e axial electric field of the c u r r e n t s on the wire grid is t h e n given b y E~ = k~H~' cos 20. F o r the p r e s e n t a p p l i c a t i o n it is desirable to t r a n s f o r m the H a n k e l function series to a simpler form. This is effected b y using a t r a n s f o r m a t i o n f o r m u l a given p r e v i o u s l y 11) so t h a t
/U=
i#cole-ikz s i n 0 4xk~
+oo
ei2'~mY/a m=-oo
exp [-
2~ Ix -
hi V m 2 - - (d cos
O/~)2/d~
V m 2 - - (d COS 0/k)2
(4)
To satisfy b o u n d a r y conditions at the interface, x ---- 0, it is now necessary to i n t r o d u c e s e c o n d a r y or s c a t t e r e d fields. F o r e x a m p l e in the absence of the grid the field could be derived f r o m the following H e r t z v e c t o r s :
ilk+n;-
Eoz
e-ikz sin 0 (eilCx cos 0 @ R o e -~kx cos 0)
(5a)
(k cos 0)2
for x > 0 a n d
EozTo (k' cos 0')2
e-~kz sin O eik'x cos O"
(5b)
262
J A M E S R. W A I T
for x < 0 w h e r e k ' = N k , or N = (e'/e)i = ()./).'), s i n 0 ' = N s i n 0 , R0=
T0--
cos O' -- N cos 0
1=
and (5c)
cos 0' + N cos 0
T o a c c o u n t for t h e p r e s e n c e of t h e grid a n d its r e a c t i o n on t h e i n t e r f a c e a t x = 0, it is n e c e s s a r y to c o n s i d e r t h e a d d i t i o n a l H e r t z vectors
i/z~o I e - i k z 4~k2
Hzw + H w' =
sin 8
+co
Z m
e x p V-- 2~r Ix - - hi ~ / m 2 - - (d cos
~
e i2~'mv/a.
--oo
0/2)2/d]
+
X/m 2 - - (d cos 0/).) 2
Rm e x p E-- 2z~(x + h) V/m e - - (d cos O/).)2/d]
+
(6a)
V/m 2 - - (d cos 0/).) 2 for x > 0 a n d i/z~o I e -ikz sin0 4x(k') 2
+oo
e i2~rmy/a T m
e x p E2z~x%/m 2 - (dcos O'/).')2/dJ
,~=-oo
V ~
2z~h 9e x p I - - - - d - %/m2 -
7 cos 2 0 (d cos 0I).)2 |_J cos 9' 0'
-
(d cos
o/).)2
(6c)
for x < 0. B y i n v o k i n g t h e c o n d i t i o n t h a t t h e t a n g e n t i a l electric a n d m a g n e t i c fields are c o n t i n u o u s a t x = 0, t h e coefficients c a n be readily found, whence
Rm = Tm
--
1
~--
cos2 0' V ' m 2 -
(d cos 0/4) 2 - - cos2 0 V ' m 2 - - (d cos 0'/2') 2
(7)
cos 2 0' %/m 2 - - (d cos 0/).) 2 + cos 2 0 ~ / m 2 - - (d cos 0'/2') 2 I t n o w r e m a i n s to solve for t h e u n k n o w n c u r r e n t . T h i s is a c c o m p l i s h e d b y i m p o s i n g t h e c o n d i t i o n t h a t t h e axial electric field a t t h e wires is e q u a l to I e x p ( - - ikz sin O)Z, w h e r e Zi is t h e i n t e r n a l i m p e d a n c e of t h e wires. Since a ~ d, it c a n b e a s s u m e d t h a t t h e field is u n i f o r m a r o u n d t h e wires, a n d h e n c e Zi c a n be c a l c u l a t e d b y k n o w n m e t h o d s 11) a n d is g i v e n b y z,
-
~llo(Ta)
- -
- -
2~all(Ta)
(8)
REFLECTION
FROM A WIRE
263
GRID
where [i~tlO)/(ffl -~- iO)el)l89 ~ = [i#lO)(al -1- i~el)J 89
71 =
I0 and 11 are modified Bessel functions of the first and second t y p e of order zero. F o r metallic wires, the displacement currents are negligible since, even at microwave frequencies, (oel ~ al. In addition, the f r e q u e n c y is usually sufficiently high t h a t I~al >~ 1 and hence - - k 2al / \ 2 ~ a - - / "
(9)
With the restriction a ~. d, it t h e n follows t h a t
-- IZt
= k2 cos 2 0 [ H v
+ H~ + H~ +
Hy']~=h+ ~ = y=0
i#wI
e-27ralml/d
cos 2 0 ! +oo
E'
47t
|tm=-~o +
Im[
Ro e-2~(2h+a)tml/d
+
exp [-- 2~a ~/m 2 -- (d cos O/~)2/d~ +
E'
~/m2 -- (d cos 0/~)2
~=-oo +
+
Im]
Rm exp
[-- 2~(2h +
a)A/m 2 -- d cos O/,~)2/d]
V~m2 -- (d cos 0/~) 2 -
-
e-2~ralml/d+ ROe2~r2(h+a[m l a[m /l [
@ e-i2"a c~
+ R~ cos e-i2~r(a+2h~ 0
[
+ Eo cos 0(e/2'~(h+a) cos oA + R o e -i2"(h+a) cos o/A
(10)
where the prime over the s u m m a t i o n indicates t h a t the t e r m for m ~ 0 is to be omitted. In the above R 0 = [Rmja=o which is not to be confused with Ro ~ [Rm]m=o. Utilizing the i d e n t i t y la) e_CZm
9 m=l
--
ln(1 -- e - ~ ) ~ - l n ~
for e ~
1,
(11)
(1--e-2"(2h+a)/a)+A 1 +
dZ,(12)}
m
the determining e q u a t i o n for I can be t r a n s f o r m e d to I = - - Eo cos 0 d(e 12"h cos 0A + Ro e -2~h cos 0A) :
/ ~ cos 0 0/A) :/---2-(1 + Ro e-~,.h cos + + -i#~od - - - cos 2 0 I In - -d- - 2a 2~a
Roln
264
JAMES R. W A I T
where
1 Jr- Rm exp [-- 4~h ~/m 2 -- (d cos 0/2)2/d] m=l
Vm2
-
(d cos 0/2)2
1 -k Ro exp [-- 4zlhm/d] --
(13) m
In the preceding equation a has been considered small compared to both d and 2. A can be regarded as a correction factor which becomes negligible for d ~ 2.
w 3. The distant field. Although the complete solution of the problem has not been obtained, it is very desirable to focus attention on the distant scattered field. For example, if J x - hi >~ 2, it is evident that only the terms for m = 0 are significant if d/2 and d/2' < 1. The higher values of m correspond to evanescent waves which are highly damped in the positive and negative x directions. For larger values of d, additional undamped waves can be scattered from the grid. The discussion will be limited here to the smaller grid spacings satisfying the above inequality. The distant magnetic field, which has only a y component, is given by E
H y = H o e x p [i(2~/,~)(xcosO--zsinO)]-- HoRo
I cos 0
2d
cos 01.~@
+ R0 e-~2~h cos 0A)~ exp [i(2~/2)(-- xcos0 -- z sin 0)~
(14)
for large positive x, and
My = 1( go -k -/-c~o-s O e_i27rh cos OA) .exp [i(2;z/2')(x cos 0' + y sin 0')][
ToN
cos 0
cos O'
(15)
for large negative x, where
R o = T o - - 1 : ( K ' - - K)/(K' + K) with K = ~ cos O, K' = r]' cos 0', ~ = (t,/s) ~, ~' = (#/s') 89 The current I can now be written r =
--Eo cos 0 d {exp [i(2~/2)h cos 0J +R0 exp [--i(2~/2)h cos 0)} (16) (~1cos 0/2)(1 + Ro exp [-- i(4z~/2)h cos 0]) + Zg
265
REFLECTION FROM A WIRE GRID
where
it*mdcos 2 0
Zg
[-In
L
2~ COS2 0 '
--
d 2:m
COS 2 0
In (1
cos 20' + cos 2 0
-
-
e-2~rl2h+a)/d) +
A~ @ dZ,
(17)
where A=E
~
m=l
+
1
[ I+
V/~b 2 - - (d cos 0/2) 2
COS20'/%/m
2
--
(d cos 0'/2') 2 --
cos 20/%/m 2 --
COS2 0 ' / ~ / m 2 - - ( d COS 0'/2') 2 -~- COS2 0 / % / 7
exp 4~h~/m2_(dcosO/2)2~
(d cos 0/).) 2
- - (d COS 0/2) 2 "
1 (l_~C~176
d
--m-
)}
cos 2 0 ' @ c o s 2 0 e - 4 1 r h m / d
"
(18)
The equivalent circuit which m a y be taken as the analogue of the wire grid is shown in fig. 2. The space to the right of the interface, x > 0, is represented b y a transmission line of characteristic impedance K and propagation constant F. The line constants for the space to the left, x > 0, are K' and F'. At x = h the line is shunted b y an impedance Z~. The voltage V across the line can now be identified with the electric field Ez and the magnetic field Hv, respectively. The propagation constants are given b y / ' = i(2~/2) cos 0 and At normal become
incidence,
F'= i(2~/).') cos 0'.
the constants
of the equivalent
circuit
K = ~ 7, K ' = ~ ' , Z~ --
it~c~ (ln 2@a + A) + dZ~, 2:~
(19)
where now A =x m=i
§
Vm2
-
(d/2p
~v/m2--(d/).)2--Vm2--(d/).')
1+ 2
~ / m 2 - - (d/).) 2@ V1yt2__ (d/).,) 2
exp
[--4~h~/m--(d/).)2/d]~ - 1__. _1
m
(20)
266
JAMES R. W A I T
When the grid is located in the interface and returning to the case of oblique incidence, it follows that
Zg
--
i#oJd 2~
cos 20
( 2 cos 2 0' ln~+A \cos ~-6 + c~s2 0 ' 2~a
)
+dZi
(21)
where now A
2 cos 2 0' 2~
m = 1 cos2 0' v / m 2 - (d cos 0/a) 2 + cos2 0 ~ / m 2 _ (d cos 0'/~)
2 cos 20' --
(cos 2 0' + cos~
O)m
(22)
The special case of equation (22) for normal incidence which of course is the same as equation (20) with h ---- 0 is in agreement with a formula quoted to me b y G. D. M o n t e a t h of the B.B.C.* Then Z o is given b y equation (19) and oo 2 A =Z m = x V m -- (d/2) 2 + V m -- (d/2')2
1
(23)
m
On previous occasions 12) it has been assumed that the equivalent shunt impedance Zg for a wire grid is only dependent on the properties of the media in which it is immersed. In other words
Zg -- i#~~ c~ 2~
0 (ln
d ) 2z~---~+ A + dZi
(24)
where co
1
1
A ----X= =1 ~/m2 -
(25) (d cos 0/a) 2
m
It can be seen from equation (17) that this is only justified if h >~ d. In the case of large angles of incidence this condition becomes more stringent.
w 4. Generalization to a arbitrary incidence. In the preceding analysis it has been assumed that the electric vector is polarized in the plane of incidence (i.e., E u ---- 0). It is of interest to consider the case were the incidence is arbitrary such that the electric field of the incident wave is given b y E~ = A exp [ik(x cos ~ cos 0 + y sin r cos 0 -- z sin 0)l * Personal communicatoin.
(26)
267
REFLECTION FROM A WIRE GRID
where A is the vector m a g n i t u d e of the field. Under the assumption t h a t the radius of the wires is small compared with the wavelength, only the z component of the electric field will excite currents in the grid. In the present instance there will be a difference of phase of the incident field at adjacent wires of kd sin r cos 0 radians together with a phase change of k sin 0 radians per unit length along each wire. The currents on the wires m a y t h e n be represented b y the expression I e ikna sinr
cos
0 e-fkz sin
(n ---- O, + 1, •
0
2 ...)
where I is the current on the reference wire (n----O) at x----h, y=0, z=0. The subsequent analysis for this problem is very similar to the special case (r = 0) treated above. The algebra is, however, v e r y cumbersome so further details will be o m i t t e d and the final result will be quoted directly in terms of the parameters of the equivalent circuit which has the same form as fig. 2. The voltage on the line is now to be identified with Ez component and the current is to be identified with the H v component. * The characteristic impedances and propagation constants are given b y K = ~ cos O/cos 4, K ' ---- ~' cos O'/cos 4', Y = i(22~/2) cos 0 cos r where ~'/~ = 2'/2 ~- (e/e') 8 9
O' =
sin
F ' = i(2z~/2') cos 0' cos $',
l/N,
(I/N) sin 0,
sin r ~-
sin r cos 0 ~ / N 2 -- sin 2 0
The equivalent shunt impedance is now given b y
zg--i#~~176 + 2 Iln2+a ~ COS 2 0 {-
_
--
cos 20'
7
_ cos 2 0 + r cos 2 0' In (1 --
e-2~(2h+a)/d) -~ AJ + dZi
(27)
where ~, m=l
1+ R m exp
[--4ah~/(m+d
cos 0 sin 4/2) 2 - (d cos
~ / ( m + d cos 0 sin r
0/2)2/d]
2 - (d cos 0/2) 2
* The E a a n d Ey c o m p o n e n t s of the field are u n a f f e c t e d b y the grid b u t , of course, t h e y are modified b y the dielectric interface.
268 +
JAMESR. WAIT
1+
R-m
e x p [ - - 4~h ~ / ( m - - d cos 0 sin r ~ / ( m - - d cos 0 sin r
_ (d cos
0/,1)2/d]
_ (d cos 0/~) 2
2 I1-- COS20--COS20' m
cos 2 0 + cos 2 0'
)]
e-4~hmm
. (28)
T h e e q u a t i o n for Rm is
I Rm =
COS20' ~ / ( m + d cos 0' sin r
_ (d cos 0'/X') 2 - -
_
] •
cos 2 0 ~ / ( m + d cos 0 sin r
_ (d cos 0/,1) ~
COS20' X
~ / ( m + d cos 0' sin r
_ (d cos 0'/,1') 2 + cos 2 0
+
~ / ( m + d cos 0 sin r
]-1
(29)
_ (d cos 0/~) 2
a n d t h e c o r r e s p o n d i n g e q u a t i o n for R - m is o b t a i n e d b y r e p l a c i n g m w i t h - - m. A n i m p o r t a n t special case of (27) is w h e n t h e electric v e c t o r is a l w a y s p a r a l l e l to t h e wires (i.e., E v = E x = 0) so t h a t 0 = 0' = 0 and then
zg
if~~ ( ln d ) 2:r 2:r~ + A + dZ,
(30)
where
A = 89 ~
1 -4- Rm e x p
~=1
[ - - 4~h 1 / ( m + d sin r
~ / ( m + d sin r
__
(d/2)e/d] +
_
(d/,1)2
1 + R_mexp[--4~hV'(m--dsinr + where
~ / ( m - - d sin r
~v/(m+d sin r Rm= V~(m+d sin r
2 - - (d/A) 2
(d/,1)2_
r
sin
r
Zg -- iff~ c~ 0 [ln
(31)
--
~v/(m-+dsin
A n o t h e r l i m i t i n g case is w h e n
2~
2]
h >~ d,
so t h a t
d
]
2:~ + A + dZ~
(d/,1,)2 2"
(32)
(33)
REFLECTION
FROM A WIRE
269
GRID
and oo
1
+
A ~,~1 X/(m -- d cos 0 sin r
2 -- (d cos 0/4) 2 1
+
V'(m + d cos 0 sin r
2
2 -- (d cos 0/,~)2
(34)
m
In this case the equivalent s h u n t impedance only depends on the properties of the media in which the grid is immersed. I t is in a g r e e m e n t with a formula derived previously 15).
w 5. Line source excitation. To reduce the power absorbed in the g r o u n d from an antenna, a wire mesh is often placed on or just below the surface of the ground. F o r example, in b r o a d c a s t antennas, this s y s t e m takes the place of radial wires e m a n a t i n g from the base of the vertical mast. I t is helpful to consider a two-dimensional analogue of this problem in order to f u r t h e r justify some of the a p p r o x i m a t i o n techniques previously e m p l o y e d in the threedimensional c o u n t e r p a r t la). Again the grid is assumed to consist of thin parallel wires and is located in the interface between the two half-spaces (i.e., h = O) as indicated in fig. 1. A line source is now located at x = x0 carrying a magnetic c u r r e n t V (in volts) from y = -- co to y = + co. The p r i m a r y field of this line source has only a y c o m p o n e n t and is given b y eco V
HPv = ~ - - H ( s 2) {k[(x -- x0) 2 + z2]89
(35)
This can be r e w r i t t e n in integral form as follows: +oo
is(oV f U-1 e-UlX-Xol e isz ds H~-4~
(36)
--oo
where u = ~/s 2 -- k 2 with the c o n t o u r being i n d e n t e d u p w a r d b y a small semi-circle at s = k and d o w n w a r d at s = k.* The integration here can be regarded as a superposition over all plane waves whose angle of incidence O, measured from the z axis, is related to s b y s = k sin 0. * A l t e r n a t i v e l y one m a y r e m o v e t h e m d e n t a t m n s of t h e c o n t o u r if k is c o n s i d e r e d to have a vanishing small negative i m a g i n a r y part.
270
J A M E S R. W A I T
It is clear that the plane wave spectrum must include both real and complex angles of incidence. Utilizing the results of the plane wave solution for the case of the H vector parallel to the wires, it readily follows that the resultant field is given b y +00
HY -- ie~o 4:zV f u-1 [e-uix-x~ + R(s) e-U(z+xo)] d sz ds
(37)
--co
for x > 0 where
Z(s) K(s) + Z(s) ' K(~)
R(s) --
Z(s) =
Zg(s) - -
--
(38)
ff~o ~oo ~ 1 -- (s/k) 2 , K(s) = --~2- u = k
(39)
K'(s)Zo(s) K'(s) = - K'(s) + Zg(s) ' k'
(40)
iffoJd 2~z
(1--
~/1 -- (s/k') 2,
[- 2(1-(s/k')2) , )2 In a
(s/k)2), - .... L2_(s/k)2_(s/k
1 (41)
+A(s)
and oo
A(s)
---- X 2 I1 - - ( s / k ' ) 2 ]
{[1 - - (~/k')23 V m 2
- - (s2 - - k 2 ) ( d / 2 ~ ) ~
+
m=l
-~- [1 - -
(s/k) 2] V m 2 -~- (s 2 - - k '2) 2E1 -
[2-
(s/k')2)
(d/2~)~}-1
(s/k')2j -
(~/k)2j m
.
-
(42)
When the field is observed at some large distance from the grid (say x >~ 2), the integral in (37) can be evaluated b y the principle of stationary phase. The saddle point is at s = k sin 0 where = arctan z / ( x - Xo). Since the function R(s) is slowly varying compared to the exponential factor, it can be taken outside the integral and replaced b y R ( k sin 0). The far field is then given b y eoJV 2i V---e Hv~--- 2 ~ - ~ ~kr
-~kr [1 + R(k sin ~) e-i2Xo ~ cos 0q
where r = ~/(x -- x0) 2 + z 2.
(43/
REFLECTION
FROM A WIRE
GRID
271
It is of interest to consider the energy absorption, from a magnetic line source at x = x0, in the lower half-space (x < 0) of dielectric constant e' which can have a finite negative i m a g i n a r y part. * The power flow into the space x > x0 is denoted b y S+ and t h a t for x < Xo is S-. F r o m P o y n t i n g ' s theorem it then follows t h a t +00
S ~ = ( f P~dz)x=xo•
(44)
--oo
with d being a vanishing small positive q u a n t i t y and Px ---- 89Re E~H~ where the asterik denotes a complex conjugate. The tangential fields are given b y +00
H u - - iecoV 4~ f u-1 [e~(X-Xo)U + R(s) e-(X+Xo,u] e iSz ds
and
(48)
-00 OHy
1 ~'Z
--
9
~eoJ
--
Ox +oo
I T e T(x-x~
- - R(s)e-(x+xo) , _ ~ e Isz ds
(46)
where the upper signs are t a k e n for x > Xo and the lower signs for 0 < x < x0. The expression for the power flow is then a threefold infinite integral +00
S• = 89Re with
(ffff-
/(s)g(s')e ~(s-d)z ds ds' dz
(47)
-~ [(s) = T e -uS - - R(s)e -2uh, g(s) - -
iew
u* [e-U*8 + R(s*) e-2U*h].
Noting t h a t the unit impulse function S(~) at cr = 0 can be expressed in the Stieltjes sense, as +00 1
S(~) - - - r e 'a~ d2 2~ d
(48)
--oo
* T h e c o m p l e x d i e l e c t r i c c o n s t a n t e' in t h i s case is r e p l a c e d b y e' - - i a / w w h e r e n o w # is r e a l a n d a is t h e c o n d u c t i v i t y . * The asterisk denotes a complex conjugate.
272
JAMES
R.
WAIT
it then follows that +co
S~: = Re 16:~
(49)
/(s)g(s')S(s -- s')dsds'. --oo
Now utilizing the sifting property of the impulse function, the integration with respect to s' can be carried out to yield 2700
S~=Re
16~
(50a)
/(s)g(s)ds --co
+co
V2e~o f 1 16~ ~-[T
--= Im
e -u*8 :~ R*(s)e-U*2Xo -- R(s)e-U2Xo~ d•. (505)
--oo
The preceding expression specifies the power flowing into the half-spaces above or below the line source. The total power which must be supplied b y the line source is given b y +00 s_
-
s+ -
,o v2e Imfu-i
(51a)
Ee-u~ + R(s)e-U2Xo I ds,
--oo +0o
--u
S---S+=er~176
(52b) --oo
-
-
8
if
700
[1 + Jo(2kxo)~ -- - - Im
}
JR(s) -- 11 e-eux0 u -1 ds . (53b)
7~
--oo
The integral term in the preceding equation can be regarded as a correction term which accounts for the imperfectly reflecting properties of the grid in the interface at x = 0. The first term is the power radiated from the line source when it is located a height x0 over a perfectly conducting flat surface. The additional power A S which the line source must be supplied to account for the losses in the lower half-space is therefore given b y +00
A S = -- Im -e~V2 8:~ f ~R(s) -- I I e-2UXo u -1 ds. --co
(54)
REFLECTION
FROM A WIRE
273
GRID
In the preceding equation -- 2K'Zg(s)
R(~)- 1=
(55)
KZ~' + Zg(s)(Z~' + K)
where
#~
K-
k
K'--
-
Vi
(~/k)~ = -
ik~
~-
iea~
(56)
,
@1 - - (s/k') 2
/~o
(57)
k'
and Zg(s) is given by equation (41). Since R ( s ) is slowly varying compared to the exponential factor in the integrand of equation (54), it can be replaced, subject to Zg ~,1, by R(s)-
2eco
1--
,/'Zg , r]' 4-Zg
(58)
with '1 - \ e ' /
'
Za--
2z
2erda 4-A
(59)
and
a=lZ
1
oo
~ / m 2 -- (d/).) 2 4 -
/=1
2
.]
(60)
~ / m 2 -- (tiN~').) 2
with N = X/e'/e. Zg is now the normal surface impedance of the grid when it is situated in the interface between media whose intrinsic impedances are ~ and 9'. The integral for the energy absorption is then given by the approximate form +oo
AS-
(e~~
Re
rl' + Zg
- - - d2s . u
(61)
--oo
It is interesting to observe that this result is equivalent to the following -boo
AS=
Rel
~' + Z g --oo
where H ~ is the tangential magnetic field of the line source over the surface x = 0, assuming it to be a perfect conductor. To A p p l . sci. Res. B 6
274
JAMES R. WAIT
demonstrate this equivalence H~~ is expressed in integral form as +00
Hvco -- iaoV --~ f e-UXoe +r ds
(63)
--oo
and substituted into (62), forming a triple integral which can be readily reduced to (61) by making further use of the sifting property of the impulse function. The factor ~'Zg/(~' + Zg) can be interpreted as the normal surface impedance at the interface x = 0, being composed on the surface impedance of the ground in parallel with that of the grid. Equation (62) can be derived directly by an
I
Zg
K', r"
K,F
9.-- h q Fig. 2. T h e e q u i v a l e n t circuit c o n s i s t i n g of t w o semi-infinite t r a n s m i s s i o n lines w i t h a s h u n t e l e m e n t across one.
application of the compensation theorem if one knows, a priori, the appropriate value to use for the composite surface impedance at the interface 13). To express AS in terms of tabulated integrals it is desirable to start with the well-known result co
u -1 e -us ds = =- [H ) (k~)]
(64)
--co
and integrate both sides with respect to e from ~ to oo to give oo
+co
f u 2 e -u~ ds = T J | H 0(~) (k~) d~. --oo
(65)
a
Therefore as-
4
Im
'Zg f 2)
~+Zg
where the integral is tabulated by W a t s o n
15).
(66)
R E F L E C T I O N FROM A W I R E GRID
275
w 6. Conclusion. A complete analysis has been given for the response of a wire grid in, or parallel to, an interface between homogeneous media. The results are valid for a plane wave with arbitrary polarization and angle of incidence. It is seen that the case for normal incidence or parallel polarization leads to considerably simpler formulae. However, subject to the smallness of the wire diameters, the equivalent circuit of the grid is always a pure shunt element and the respective media are homogeneous transmission lines. The energy absorbed from a magnetic line source in the lower (dissipative) medium is shown to be appreciably modified by the presence of the wire grid in, or near, the interface. Some justification is given for the approximation techniques employed previously for the energy computations for monopole antennas with radial wire ground systems. Received 4th S e p t e m b e r , 1956.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
L a m b , H., Proc. L o n d o n Math. Soc. 2 9 (1898) 523. I g n a t o w s k y , W. v o n , Ann. Phys. 44 1914) 369. G a n s , R., Ann. Phys. 61 (1920) 447. W e s s e l , W., Hochfrequenztechnik 54 (1939) 62. M a c F a r l a n e , G. G., J. I n s t n Eleetr. Engrs 93 no, 10 (IIIA) (1946) 7. H o n e r j a g e r , R., Ann. Phys. 4 (1948) 25. L e w i s , E. A. and J. C a s e y , J. Appl. Phys. 23 (1952) 605. G r o v e s , W. E., J. Appl. Phys. 24 (1953) 845. T r e n t i n i , G. y o n , Z. angew. Phys. 5 (1953) 221. W a i t , J. R., Can. J. Phys. 3 ~ (1954) 571. W a i t , J. R., Appl. Sci. Res. B 4 (1954) 393. J o n e s , E. M. T. and S. B. C o h n , J. Appl. Phys. 26 (1955) 452. W a i t , J. R. a n d W . A. P o p e , Wireless E n g r 3 2 (1955) 131. W h e e l on, A. D., J. Appl. Phys. 2 5 (1954) 113. See also L. W. B. J o l l e y , S u m m a t i o n of Series, C h a p m a n and Hall Ltd, Londort 1925. 15) W a t s o n , G. N., Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press 194. N.B. In 11) the form cos 0 cos ~oZt d/~lo should be replaced by cos --1 0 cos ~o Z~ d/y?,, m eq. (25).