California State University, Northridge
SCATTERING OF NON-UNIFORM PLANE WAVES
FROM A TILTED METALLIC STRIP
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
Vaughn Paul Cable
June, 1972
The thesis of Vaughn Paul Cable is approved:
i
I
California State University, Northridge
June, 1972
IL___
_
ii
_J .
ACKNOWLEDGEMENTS
The author wishes to express his
sincerest gratitude to Dr. E. S.
Gillespie, whose patient guidance
and friendship have been an inspiration for further learning.
The
author also wishes to express his
appreciation and his love to his
own wife, Susan, for the contributions she made in the effort to
complete this thesis.
I
l________,________,· - - - - - - -.. _,_j
iii
TABLE OF CONTENTS
Page
Section
ACKNOWLEDGEMENTS •
iii
LIST OF TABLES • •
v
vi
LIST OF FIGURES
LIST OF SYMBOLS
.
.
. viii
ix
ABSTRACT • • • •
I.
II.
III.
IV.
v.
INTRODUCTION
.....
DESCRIPTION OF THE PROBLEM
DEVELOPMENT OF 'rHE GREEN'S FUNCTION
......
....
.
...
THE SURFACE WAVE REFLECTION COEFFICIENT
DERIVATION OF THE INTEGRAL EQUATION
. . ....
...
VII.
RESULTS AND DISCUSSION
........
VIII. DESCRIPTION OF THE EXPERIMENTAL PHASE .
BIBLIOGRAPHY .
..................
APPENDIX.
THE COMPUTER PROGRAM
.........
VI.
DISCUSSION OF THE NUMERICAL SOLUTION
L____.____ ----------·
iv
1
4
9
19
21
24
28
47
53
54
LIST OF TABLES
Page
I •
Strip Scattering Coefficients (k 0 w
II.
Strip Scattering Coefficients (k 0 w
III.
Strip Scattering Coefficients (k 0 w
IV.
=
=
=
9.98)
29
6.29)
30
3.94)
31
Strip Scattering Coefficients (k w = 2.79)
32
0
I
j
l____
v
LIST OF FIGURES
Page.
1.
2.
3.
4.
5.
Surface Wave Guiding System and Location
of Strip
...................
Location and Orientation of Unit Current
Source
...
10
Proper Branch Point and Branch Cut and
contour c
..................
12
Transmission Line Analogy in Transverse
Direction
16
Normalized Input Impedance of Tilted Strip
for k 0 w = 9.98 and k h = 0.5
33
Normalized Input Impedance of Tilted Strip
for k 0 w = 9.98 and k h = 1.0
0
34
Normalized Input Impedance of Tilted Strip
for k w = 9.98 and k h = 1.5
0
0
35
Normalized Input Impedance of Tilted Strip
for k 0 w = 6.29 and k h = 0.5
37
Normalized Input Impedance of Tilted Strip
for k w = 6.29 and k h = 1.0
38
Normalized Input Impedance of Tilted Strip
for k w = 6.29 and k h = 1.5
0
39
Normalized Input Impedance of Tilted Strip
for k ow = 3.94 and k 0 h = 0.5
40
Normalized Input Impedance of Tilted Strip
for k 0 w = 3.94 and k 0 h = 1.0
41
Normalized Input Impedance of Tilted Strip
for k w = 3.94 and k h = 1.5
42
. . . . . . . . . . . . . . .
. . . . .. .
0
6.
7.
8.
0
12.
13.
i
114.
I
.........
. . .. . ....
. .... ....
0
11.
. .. . .....
.........
0
10.
.........
.........
0
9.
5
. . . . . ... .
.........
0
0
.........
I
Normalized Input Impedance of Tilted Strip
for k w = 2.79 and k h = 0.5
431
Normalized Input Impedance of Tilted Strip
for k w = 2.79 and k h = 1.0
44_1
0
0
. .. . . . .
I
I
115.
l
L-·-·---
0
0
--·-·-----·---·-
. . . . . . . . .
·-·---~~~
vi
Normalized Input Impedance of Tilted Strip
for k w = 2.79 and k h = 1.5 • . . . • . •
45
17.
Experimental Setup of Surface Wave Table and
Associated Apparatus . . • • • . . • .
48
18.
Two-Port Model of Strip with Output Terminated
by Sliding Load
• • . . . . • •
• . .
50
Example of Data Points for Determining s
11
(k h = 0.1, k 0 w = 6.29, ~ = -40°)
••..•.
52
16.
0
19.
0
0
__"__j
-----·· - - vii
LIST OF SYMBOLS
i
1+
+
+
+
ta
x' a z
a x' a z
lao
so
Unit vectors in x and z directions
Unit vectors in x and z directions
Transverse attenuation constant of surface wave
Propagation constant of surface wave
jko
Free space wave number
ir
Surface wave reflection coefficient
I
Radian frequency
1:I
'
0
Permittivity of free space
illo
Permeability of free space
iJ
1-l
!
!' •
iK
i
operator
Relative dielectric constant of coating
•t
Thickness of coating
lw
Width of the strip
h
Height of the lower edge of the strip above the
dielectric (also used for transverse wave
number within the dielectric)
d
Height of the midpoint of the strip above the
dielectric
Tilt angle of the strip from the normal to the
dielectric surface
'
L-·-- --· ----·-····--·-- ·······-----·--viii
ABSTRACT
SCATTERING OF NON-UNIFORM PLANE WAVES
FROM A TILTED METALLIC STRIP
by
Vaughn Paul Cable
Master of Science in Engineering
June, 1972
The scattering of a non-uniform plane surface wave by
an arbitrarily tilted thin metallic strip located near a
dielectric coated ground plane is investigated both theoretically and experimentally.
Two of·the scattering coeffi-
cients, s 11 and s 22 , are computed. This entails the
development of a vector Green's function for the problem
and formulation of an integral equation for the determination of the induced current distribution on the strip.
A numerical solution is developed for calculation of
the theoretical values of s 11 and s 22 , and an experiment
is performed for verification of these values. Good agreement was obtained between computed and measured values for
!strip widths up to 1.58 wavelengths, strip midpoint heights
1
up to 1.03 wavelengths, and tilt angles in the range of
I
-40°
<
~
<
I
30°.
·--·--.------·---·-·-----···J
ix
------------------ ·-------·------------
-- -----------------------------------
I.
INTRODUCTION
Although considerable work concerning scatter:.ng of
surface waves has been reported in the literature [1],
little of this has been directed toward the study of scattering from obstacles located near a surface waveguide
system, and in particular, scattering from metallic
obstacles above a planar surface waveguide.
In 1959, Sharma[2] published a paper in which he considered a thin metallic strip attached to the reactive
surface of a corrugated surface waveguide.
His results
were mostly empirical; however, he did use an approximation
for the aperture field distribution above the strip to
obtain an expression for the magnitude of the transmission
coefficient, but not for its angle.
His paper made no men-
tion of any attempt to determine the induced current distribution on the strip, and he treated only the case for
which the plane of the strip was positioned normal to the
direction of propagation.
In 1965, Gillespie and Gustincic[l] published a paper
on scattering by a thin metallic strip located near a surface waveguide.
The strips considered were of arbitrary
width and of arbitrary height above a planar surface waveguide system.
They developed a Green's function for the
problem and used a variational approach to derive a stationary expression for the reflection coefficient of the.:_____ _.
1
2
strip in terms of the induced current distribution.
Using
a set of functions to approximate the current distribution
and the Rayleigh-Ritz technique to calculate the reflection
coeffic·ients, they were able to show good agreement with
experimental
data for strip widths up to k 0 w = 5 and strip
.
heights in the range 0.5
~
k0 h
<
5.
Gillespie and Kil-
burg[3] have recently extended this work to include strip
widths up to k 0 w = 15.
They used a numerical approach to
solve an integral equation for the induced current distribution and, in addition to experimental verification of the
calculated reflection coefficients, their results included
calculated impedance properties for the strips and fractions of power reflected, transmitted and radiated.
The
plane of the strips considered by these authors was also
positioned normal to the direction of propagation.
To this author's knowledge, no· one has published any
studies of scattering from obstacles which are distributed
in the direction of propagati?n of the surface wave.
It is
the purpose of this work to consider a special case of this
problem, namely, scattering from a tilted thin metallic
strip located above a dielectric coated ground plane.
It
will require the development of a Green's function and an
appropriate numerical procedure for calculation of current
J
distributions on strips of arbitrary width, height and tilt
1Since the tilted strip is distributed in the direction of
propagation, it can be represented as a two-port network,
and its circuit properties can be expressed in terms of
-·--------·---
--·--·-··---·
3
a 2x2 scattering matrix.
and
The scattering coefficients
s 11
s 22 will be determined both analytically and experi-
mentally.
L____
II.
DESCRIPTION OF THE PROBLEM
The two-dimensional geometry for the problem is shown
in Figure 1.
The dielectric coating is assumed to be loss-
less with relative dielectric constant
thickness t.
K
= eje 0
and uniform
The thickness is adjusted to allow only the
lowest order TM (Transverse Magnetic) surface wave mode to
propagate.
where S
0
This is assured when
(Kk~ - S~) 1 1 2 t
<<
nj2 [4]
is the surface wave propagation constant and k 0 is
the free space wave number.
and the wave is said to be
Under these conditions, S +k0
0
11
loosely bound 11 to the surface.
The surface w.ave incident on the strip is assumed to be
=
launched at z
-oo
and it propagates in the +z direction.
The field components can be written[4] as
=
(1)
=
(2)
( 3)
The relationship between the attenuation constant a
the phase constant S
0
0
and
is expressed by the two-dimensional
separation equation; i.e.,
a2
0
=
S2 0
K2
( 4)
0
!All field quantities for this problem will be uniform in
b
,______ j
direction.
4
X
X
<I>
-
Strip (cr=co)
Inciden·t Surface Wave
Ei,Hi
f
Air (JJ 0 r Eo)
d
Dielectric Coating
llillllooZ
(JJo,~>
Ground.-l?lane.
(cr=co)
I
Figure 1.
Surface Wave Guiding System and Location of Strip
01
6
The metallic strip's location can be described in
either the x-z or the x-z coordinate system shown in Figure
1, and the following transformations apply:
in which
x
=
x cos
~
- z sin
~
+ d
(5)
z
=
x sin
~
+ z cos
~
,
(6)
~
is defined positive for clockwise rotation of th
strip about its midpoint and d is the height of this midpoint in the z
=
0 plane.
The strip width is w and the
length is taken to be infinite in the y dimension.
It is
assumed that the strip is infinitely thin and perfectly
conducting.
The incident wave induces a current distribution on
the strip which only has an x component with a density
Jx(x') per unit length in they direction.
.
.
t~on ~n
Hs.
This distribu.
+s
turn acts as a source for the scattered fleld E ,
A portion of this field is radiated directly away from
the surface wave system.
The remainder is scattered down
toward the air-dielectric interface.
Part of this field is
reintroduced into the surface wave guiding system as forwar
scattered and backward scattered surface waves, the rest is
reflected away as additional radiation field.
In order to
satisfy the boundary condition on the conducting strip, the
scattered electric field Es must cancel the tangential component
where
[
E; of the incident field at all points on the strip,
E~ i~ given
E~ =
·---·---·-·
by
E; cos
~ +
E! sin ~
·-----·
(7)
----------------------
7
A surface wave reflection coefficient for the strip
can be defined in terms of the x components of the incident
and backscattered surface waves; i.e.,
=
where the minus
( 8)
(-) refers to the backscattered surface
wave and r is the reflection coefficient.
If the surface
waveguide is assumed to be infinite in extent in the +z
direction and the strip is tilted to some angle
~
=
~l'
then
the reflection coefficient r can be recognized as just the
scattering coefficient
same strip is tilted to
scattering coefficient
s 11 [4].
~
s 22
=
On the other hand, if the
1 then r is precisely the
-~ ,
for the previous case.
The scattered electric field can be expressed in terms
of the magnetic vector potential A(x,z) as
1
-jwA(x,z) + -.--- [VV • A(x,z)]
J W].l £
=
where
(9)
v is the vector differential operator in the x-z
coordinate system.
The magnetic vector potential is given
by
=
-+
A (x,z)a
X
in which G (x,zlx')
X
X
=
f.·
str1p x
1
G (x,zlx')J (x')~ dx'
X
X
X
(10)
is the Green's function for the problem
and the primed coordinates refer to source points on the
strip and the unprimed coordinates refer to field points.
I
Once the Green's function G
X
!current distribution
lfield
J
ca~ the~.~"_ foun:
is formulated, the induced
can be computed and the scattered
everJ/Where_by use of (9)
and-~=~-· j
8
·---------------·· - - - - - - - - - Once the scattered surface wave to the left of the strip
is known, the reflection coefficient is easily calculated
by use of the following expression:
r
=
I
which is a restatement of (8) .
( 11)
III.
DEVELOPMENT OF THE GREEN'S FUNCTION
The Green's function for this problem is exactly the
magnetic vector potential for a unit point current source
with arbitrary orientation in the presence of the dielectric coated conductor.
Consider the unit localized current
source and associated boundaries shown in Figure 2.
The
source can be located by either x',z' in the x-z coordinate
system or by x• ,0 in the x-z coordinate system.
Its magni-
tude and direction are given by the arbitrarily oriented
-+
unit vector a X which can be expressed in terms of its x and I
i
z components as
=
-+
(ax cos
ct>
-+
+ a 2 sin
cp )
o (x-x ' ) o ( z- z ' )
(12)
where o(x-x')o(z-z') are Dirac delta functions which locate
the source point in the x-z coordinate system.
Each com-
ponent in (12) will give rise to a corresponding vector
component of the Green's function.
The solution for the
I
vertical component G
X
of the Green's function has already
been found by Gillespie and Gustincic[S].
Their result as
i t applies to this geometry can be written as
I Gx
.J.lo~(2)
= - ]4
0
(k r)
0
+
J:__
J1T
I
Iwhere
=
~e
Ic
-j t Ix+x'
1-y I z-z 'I
d) cos
J
jKR. + h tan ht
h tan ht
ct>
(13)
{14)
JKt
r is the position vector of the field point and is given by
Lr_~ ____U_x-x') 2 + (z-z') 2 ] 1 / 2 • The separation equat~~ns _!or
9
I
j
X
:x;
<I>
__(x' ,z')
(x',O)
1
d
z
~~~"o," ~i :s\\")')7 7 ~
z
I
Figure 2. Locati0n and Orientation of Unit Current Source
I
'
L---------------------------------------~
I
._.
0
11
the regions above and below the air-dielectric interface
are given by
and
R,2
=
y
h2
=
y
2
2
+ k2
0
+
(15)
k2
(16)
K 0
where y is the complex propagation constant in the z direction.
The first term in (13) represents the direct radia-
tion field from the source point in the absence of the
grounded dielectric.
The second term accounts for the
"retrapped" surface waves and the indirect radiation
reflected up from the surface.
have shown that, for z
<
0 {z
Gillespie and Gustincic[S]
<
z' for this geometry), the
contour of integration C can be warped into the contour cb
around the branch cut shown in Figure 3.
of the original path c assures that G
X
Note the position
represents bounded
outward propagating or evanescent waves at infinity; i.e.,
Re { .t }
>
Im{.t}
<
0
(1 7 )
0
(18)
where R. is taken to be the negative branch of (y 2 + k 2 ) 1 1 2 .
0
Integration around Cb accounts for the indirect radiation
field, and integration around the surface wave pole at
y
=
-js
0
gives the surface wave field.
desired for the region z
1
I
>
z', then the contour Cis closed
in the right half y plane and y
surface wave pole.
If the fields are
=
+j(3
0
becomes the proper
The details for this case are given in
Collin[6].
l_______________ ----
_____j
'
12
Im
c
Proper Branch
y Plane
Cut~
-----·----
_____ .,--- --,
cb
1
')
I
-jk (
\
~1--I
o, ..... ../
,.-j 6 {
--Proper Branch Point
.......
·+--surface Wave Pole
0 ,_ ,/
Figure 3.
-·-----·--------·-
Proper Branch Point and Branch
cut and Contour C
-----·--
13
= jS 0
Surface wave poles show up at both y
y
=
-ja
0
and
when the denominator of (14) equals zero; i.e.,
when
jK~
=
(19)
h tan ht
This equation, along with (15) and (16), gives the discrete
modes that are possible for the geometry in terms of h and
For this problem, only the lowest order TM mode propa-
2.
gates and the values for 2 and h are 2
h =
y
=
2
2 1 2
[k (K-l) - a J 1 .
0
-ja
-ja
and
0
The integration around the pole at
0
0
=
(z < z') is introduced when contour Cis warped
into the left half y plane.
This is a simple pole and
Cauchy's residue theorem can be easily applied to evaluate
the integral.
Finally, the backscattered surface wave con-
tribution of Gx can be written[S] as
=
. )lORXO
J 2B
e
-a 0 (x+x')+jS 0 ]z-z']
cos
( 20)
~
0
where R
is related to the residue of the integrand in
xo
(13) evaluated at y
=
-ja
0
and is given by
-2 a
=
k
K.
2
0
[k
The horizontal component G
z
0
(K -1) t
2
0
(K
2
+ a0
(K
2
-l)t]
°
( 21)
-1)
of the Green's function
must satisfy the inhomogeneous scalar wave equation,
=
-Jl
0
o(x-x')o(z-z')
sin~
,
(22)
in the region above the dielectric and, within the dielectric, the homogeneous scalar wave equation,
14
=
(23)
0 •
Boundary conditions for this problem are that the tangen-
E and H fields be continuous across the air-dielectric
boundary (x = 0), and that the tangential E field be zero
tial
at the dielectric-ground plane boundary (x = -t) .
This
leads to the following boundary conditions on Gz:
1.
2
1
(-a-~< az2
aGZ
2
+ Kk 0 )Gz continuous at x
=
continuous at x
= Q.
0.
2•
ax
3.
(_a-+
2
az
4.
Gz must represent bounded outward propagating
or evanescent waves at infinity.
2
k2)G Z
=
K 0
0 at x
= -t.
Solutions for Gz will be obtained under the bilateral
Laplace transformation given by
Gz(x,y)
= f
00
Gz(x,z)eyzdz
(24)
-oo
which removes the z dependency.
The transformed wave equa-
tions are written as
=
for x
>
L
0
o(x-x')e Yz'
sin~-
(25)
O, and
d 2G
l
-~
K_k_~_)_G_·
----------dx---2_z_+__'_Y_2_+__
__=_ _
o_______
(26)
15
for -t < x < 0.
The previously stated boundary conditions
remain invariant under the transformation and can be written as follows:
1
1.
K
(y
2
2
+ Kk )G
0
Z
continuous at x = 0.
2.
dGZ
dx continuous at x
3.
Gz
4.
G must represent bounded outward propagating
z
or evanescent waves at infinity.
=
0 at x
=
=
0.
-t.
Two additional boundary conditions are imposed on Gz
at the source point x'.
These conditions are obtained by
integration of (25) across the source point and the results
are given as follows:
dG
Z
dx discontinuous by a factor of
5.
-~
0
e
Yz
I
sin
~
atx=x'.
6.
Gz continuous at x
= x'.
Solutions for G in terms of arbitrary constants can
z
be written immediately with the use of the transmission
line analogy shown in Figure 4:
=
Gz
2
Gz3
~n_wh-ich
=
Region I
2
( 27)
c fej~(x-x') + Rze-j~(x+x')] , Region II
( 2 8)
=
(29]!I
Region III
the transverse wave numbers
~
and h are given in
-------------~-----
16
I
x
=
x' -
rv
lI
X
Region II
=0 -
1r
X
Region I
Region III
= -t
Figure 4.
Transmission Line Analogy
in Transverse Direction
L.____;____
17
(15) and (16).
The arbitrary constants can be determined
by the imposition of the boundary conditions.
Note that
boundary condition 3 is already satisfied by (29).
Gz
Solutions for
in Regions I and II can be combined
into one expression subject to the condition x
11
-
=
GZ12
where
Rz
=
~-jJI.Ix-xll
o e
2j
Jl.
>
0; i.e.,
Re-jJI.Ix+xiiJ
z
yz 1 •
+
Jl.
e
Sln ~
(30)
jKJI. + h tan ht
jKJI. - h tan ht ·
(31)
Equation (30) is the desired solution above the dielectric
and is the region of interest since it contains the strip.
The z dependency is re-established by the use of the inversion transformation and the resultant horizontal component
of the Green's function above the dielectric can be written
as
11
=
. o
-J--
4
+ ~
J7f
[
1
J
7f
J
c
J e -j Jl.l x-x 1-y I z-z
I
~
c
R e -j Jl.l x+x I
z
Jl.
I
Jl.
1-y I z-z
I
I
I
dy
]
dy
sin ~-
(32)
In this form, the first term can be recognized to be
the Hankel function of the second kind of order zero.
Equation (32) is, in fact, identical to (13) with the
exception of the factor Rz' which is equal to -Rx of (14).
The same procedure for evaluation of the integral over contour C can be applied, and the final result for the surface
l:ave-:ontribution of Gz can be written as
18
=
JJoRxo
-j ·
e
2
e
-a.o (x+x') +j So] z-z' I .
s1n
cj>
(33)
0
where Rxo is given by
(21) •
The contributions of the two components of the Green's
function can be combined to.give
=
-j
llo~(2)
4Lo
+ cos
2
<1>
-
]'IT
(k r)
o
sin
2
cj>
f
Rxe -jR.Ix+x' 1-YI.z...;z• I
J
dy
R,
c
(34)
as the Green's function for the total scattered field; also
=
(35)
is the Green's function for the backscattered surface wave.
These are the desired Green's functions for the problem.
IV.
THE SURFACE WAVE REFLECTION COEFFICIENT
The magnetic vector potential for the backscattered
surface wave is obtained by the substitution of (35) into
(10);
i.e.,
Asw
J.l R
. o xo(
2
S
cos
¢ 2 0
=
a;
J
e
-ru
~o
X+J'a
~o
Z
. 2
¢)
s~n
J
e
-a X 1 -J'
o
0
~o
Z1
Jx(x')dX'
strip x'
(36)
s -
The backscattered electric field components (Ex}
sw
and
(E~>;w are found by the insertion of (36} into {9) and
taking the indicated derivatives with respect to the x-z
coordinates.
This necessitates the use of the transforma-
s The (Ex)sw component is
tions given by {5) and (6).
resolved in the following way:
s (Ex) sw
=
( Es)X SW COS ¢ -
(Es)sin o/~
Z SW
(37)
•
Now that the backscattered surface wave is known, the
reflection coefficients can be computed with the use of
(11);
i.e.,
R
r
=
e
-a 0 d
xo
2we: S
0
(cos
2
¢ - sin
2
¢)~cos ¢ -
Q sin ¢)
0
w/2
. f
(38)
-w/2
where
---------·----------------
19
20
and
p
=
(-a. 0 cos
cj>
+ jSo sin cJ>f + k20
Q
=
(-a. 0 cos
cj>
+ jSo sin
cJ>) ( a.o
(39)
sin
cj>
+ jSo cos
cj>).
(40)
The only unknown in ( 38) is
J
X
.
Once this is determined,
then numerical values for r can be obtained.
I
L_________________._
-----·-------
'·"
--
V.
DERIVATION OF THE INTEGRAL EQUATION
On the surface of the conducting strip, the total tangential electric field is zero; i.e.,
=
( 41)
0
where Ei is given by (7). The application of this condition
X
to (9) results in the following inhomogeneous equation:
2
(-a- + k2)A
ax
2
0
X
=
.
-Jw]l
0
e: Ei
0
( 42)
X
A complete solution to this equation, together with (10),
yields an integral equation for which J
X
is the only
unknown.
The complete solution to (42) is the sum of a particular solution and a general solution.
A particular solution
of (42) is of the following form:
A
xP
=
CEi •
x
( 43)
The constant C can be determined by recalling that the inci
dent field satsifies the homogeneous equation given by
=
where k
X
(44)
0
is the wave number in the x direction; i.e.,
(45)
A comparison of (44) and (42) reveals that
L.
21
22
=
c
(46)
and since the general solution to the homogeneous equation
for this geometry is a linear combination of sines and
cosines, the complete solution for
j
A
X
=
W)l
0
£
0
k2- k2
0
Ei +
X
A
A
X
is given by
sin k ox +
cos k 0 x
B
( 4 7)
X
A and B are arbitrary constants to be determined.
The results of Gustincic's previous work[S] indicate
that the following approximation for the Green's function
may be used, provided that the strip heights are given by
G
X
• R(-jk )H( 2 ){k
0
0
0
j.Kl/ 2 + tan
=
where
[(x+x')
K
2
+
( z- z I ) 2 ] 1 I 2 }]
I
( 48)
l/2k t
0
(49)
jKl/ 2 - tan
The desired form for the integral equation is given by
=
l
k2
Ei + A' sin k x + B'
X
0
COS
k
0
X
(50)
X
where A' and B' are new constants to be determined by the
23
-
x
= w/2.
~---~---- ---------~----------
~------
This equation will be solved numerically for J X
and the result will be used to calculate
----------------------
r.
VI.
DISCUSSION OF THE NUMERICAL SOLUTION
In order to numerically solve the integral equation
given by (SO), the integral on the left side is replaced
by a finite summation of N terms; i.e., the strip is
o=
divided into N segments each of width
w/N.
This is
equivalent to numerical integration by use of Simpson's
rule since the end points are zero (edge condition), and
the result is a set of N equations in N unknowns.
The jth
term for this set of equations can be written as
o
-4W€0
where
G ..
J1
N
L
i=l
=
G .. J.
~--1 --~ E. +A'S. + B'C.
=
J1 1
k2
k2
0
X
J
J
J
(51)
H( 2 ){k [(x.-x.) 2 + (z.-z.) 2 ] 1 1 2 }
0
0
J 1
J 1
• H( 2 ) {k ((x .+x.) 2 + (z .-z.) 2 ] 1 1 2 } ,
0
J
0
1
J
1
(52)
(53)
and
E.
J
=
E (x.,O)
X
J
s.J
=
sink x.
0 J
c.
=
J
(54)
,
(55)
(56)
The i subscripts refer to source points and the j subscript
refer to field points on the strip.
The arbitrary constant
A' and B' are determined analytically by imposition of the
t~~ge condition in (51);.~ .e_:_:~-=- JN--=-~~-~his yields two
24
25
equations in two unknowns, A' and B'.
These equations can
be written as
N-1
0
4w~
0
.I
1=2
1
=
Gl.J.
1 1
0
N-1
4we
0
I
i=2
k2
0
I
(57)
X
1
=
GN.J.
1 1
El + A'Sl + B'C 1
k2 - k2
-
k
2 EN + A'SN + B'CN '
(58)
X
and their simultaneous solution yields the following:
A'
~
=
N-1
1:!. - 4 W£
0
·
I
1= 2
(c-N
__ Gl.1 -ClGN.)
1 J 1· (59)
B'
and
=
~
N-1
-1:!. ----4
0
W£
0
I
•
1=
2
(SNGl.-SlGN.)J.
1
1 1 (60)
( 61)
where
One difficulty remains before the numerical solution
for J
X
can be completed.
function when i
point coincide.
=
j
1
A singularity occurs in the G ..
]1
i.e., when the source point and field
'l'he same procedure used by Gillespie and
Kilburg[3] will be used here.
The small argument form for
the Hankel function is integrated analytically over the
I
interval in question and the average value over this interival used as the final result.
The small argument form for
Hankel function for this case can be expressed as
H~ 2 ) (k 0 lx-x'
L-·----·
I>
-+
x-+x'
·-----···--
1- j
~
yk
ln
°
I x-x' I
2
I
~J
26
where y = .5772 (Euler's Constant) and ]x-x'] is the distance between the field point and the source point.
The
average value of this expression over the interval is written as follows:
81
=
f
x'+o/2[
2 ln
·
1 - j
X 1 - o/2
1T
yk 0
]x-x']J
dx'
2
( 63)
=
!
where
o = w/N.
1
;rr, 2
+ j
oyk
J
ln ~
(64)
,
This expression is used to evaluate the
=
first tenn in (52) whenever i
j.
The numerical solution to (51) can be expressed in
rna trix form as
[Ji_]
=
4 •
{k2~k2 [Ej]
0
+ A' [Sj] + B' [Cj ]} • {Gjifl
X
( 65)
where the column matrix J!~ has been scaled by a factor of
o/we 0 and kx is given in (45).
applied to J
X
The same scale factor is
in (38) and the final expression for numerica
evaluation of r is given by
r
Rxo
= '2i3
e
-aod
(cos
2
~
- sin
2
~)
(P cos
~
- Q sin
~)
0
N-1
.I
( 66)
~=2
where Rxo' P and Q are obtained from (21),
~
~he
appropriate value for a
0
(39) and (40).
can be calculated from the
approximate formula[6] given by
-------••
----.-.-w--··--·••••--•-•,~·--·------·_..
______
_.._,,._,~•·~-""'
i
27
=
(6 7)
which holds for t small.
The program is written in standard FORTRAN IV language
1
and a version of that program appears in the Appendix.
I
!
I
L----·---
I
____ _j
VII.
RESULTS AND DISCUSSION
Computed and measured values of s 11 and s 22 are tabulated in Tables I-IV. In general, they show that best
agreement between computed and measured values occurs at
the smaller tilt angles <1~1 ~ 20°) and at the lower strip
heights (k 0 h
~
1.0).
It is believed that, for the larger
tilt and greater height cases, the experimental error is
the principal cause of the discrepancy.
A probable cause
is that part of the radiation field couples into the VSWR
detector probe (see Figure 17, Section VIII, for the relative positions of the probe and the strip).
This contam-
ination of the surface wave field in the vicinity of the
probe leads to errors in the VSWR measurement.
It is convenient to plot the data on Smith charts.
When
s 11 is located on such a chart, its coordinates auto-
matically give the normalized input impedance rotated to
the z
=0
reference plane for the strip located above a
matched surface wave system.
of
Computed and measured values
s 11 are shown plotted on Smith charts
in Figures 5-16.
In each case, the tilt angle is varied over the range
-40°
~ ~ ~
30° in 10° increments.
Figures 5, 6 and 7 are for the widest strip, k 0 w =
n
98
They show that, for positive tilt angles, the imped]
ce.is predominantly capacitive.
___ _____________
When tilted negatively,
e ___strip
impedance becomes predominantly inductive.
,_.
,"
28
For
29
r------------------------------------------------------~
TABLE I
STRIP SCATTERING COEFFICIENTS (k w
0
k h*
0
0.5
1.0
1.5
I
*NOTE:
1
l
Tilt
Angle
=
9.98)
s22
sll
cp
Computed
Measured
computed
Measured
oo
.769/191°
.84/193°
.769/191°
.84[193°
10°
.460[238°
.54[252°
.616/139°
.56/134°
20°
.154/-48°
.25/-46°
.308/64°
.34/79°
30°
.050~
.10~
.127/-22°
.34~
40°
.013/146°
.037[260°
.5 L-68°
oo
.651/189°
.68/184°
.651/189°
.68/184°
10°
.385/237°
.47/250°
.515/138°
.46/124°
20°
.129/-48°
.20[-43°
.255/64°
.28/70°
30°
.042~
.07~
.113/-13°
.26/-29°
40 °
.010/147°
.029/-87°
.50/240°
oo
.537/188°
.56/192°
.537/188°
.56/192°
10 °
.318[237°
.38[244°
.423/137°
.40/137°
20°
.107/-48°
.18[-57°
.213~
.23/70°
30°
.034~
.06~
.108/-11°
.18L-44°
40°
.008/148°
.028/-79°
.35/218°
k 0 h refers to initial height of bottom edge of
strip above dielectric surface at cp
- - - --------------
=
0° tilt.
I
·-·--·--·-·'
30
TABLE II
STRIP SCATTERING COEFFICIENTS (k w = 6.29)
0
k h
0
0.5
1.0
1.5
Tilt
Angle
sll
s22
Computed
Measured
computed
Measured
oo
.• 672/189 °
.69/197°
.672/189°
.69Ll97°
10 °
.436/208°
.48/221°
.637/165°
.64Ll59°
20°
.124/238°
.17L258°
.420/128°
.49/134°
30 °
.064L.::£:.
.13/16°
.227/81°
.32/107°
40°
.034/21°
.062/50°
.17/91°
oo
.565/188°
.58/191°
.565/188°
.58/191°
10°
.369/208°
.38L224°
.520/163°
.52Ll58°
20°
.111/239°
.16L255°
.318/126°
.41/129°
30°
.050~
.1lill..:
.154L.l.2..:
.35ill.:
40°
.027m_:
.046/48°
.29~
oo
.468/188°
.47/190°
.468/188°
.47/190°
10°
.310/208°
.34/221°
.422/163°
.41/162°
20°
.096/237°
.14/257°
.253/130°
.31/127°
30°
.037L=.2.:
• 09L_::L
.116~
.32~
40°
.022~
.035~
• 34L.2..Q..:
cj>
II
I
l__._ . . -"·---.. ··---··---·-----------·--·-·--·
31
TABLE III
STRIP SCATTERING COEFFICIENTS {k 0 w
k h
0
0.5
1.0
1.5
Tilt
Angle
=
3.94)
s22
sll
~
Computed
Measured
Computed
Measured
oo
.732/197°
.78/197°
.732/197°
.78/197°
10°
.612/197°
.66/202°
.677/192°
.74/188°
20°
.384/195°
.50/206°
.484/183°
.58/182°
30°
.171/189°
.32/206°
.247/172°
.36,{173°
40°
.038,{179°
.058/162°
.14/160°
oo
.615,{193°
.64,{198°
.615,{193°
.64/198°
10°
.504/196°
.54/202°
.571/188°
.62/190°
20°
.312/195°
.42/208°
.401/180°
.52/184°
30°
.138/190°
.24/203°
.200/169°
.38/175°
40°
.031/181°
.047/160°
.22/174°
oo
.497,{192°
.50,{198°
.497,{192°
.50,{198°
10°
.409,{195°
.42,{204°
.461,{186°
.49,{192°
20°
.256,{196°
.31,{205°
.320/178°
.42/174°
30°
.115,{192°
.19/200°
.157/169°
.32/164°
40°
.026,{182°
.038,{160°
.23,{157°
32
TABLE IV
STRIP SCATTERING COEFFICIENTS (k 0 w = 2.79)
k h
0
0.5
1.0
1.5
Tilt
Angle
8
sll
22
cj>
Computed
Measured
Computed.
Measured
00
.605/223°
.67/227°
.605/223°
.67/227°
10°
.554/224°
.62/235°
.555/220°
.63/218°
20°
.408/223°
.54/240°
.418/217°
.52/213°
30°
.216/221°
.38/241°
.229/212°
.36/200°
40°
.053/218°
.057/207°
.19/186°
oo
.570/221°
.62/230°
.570/221°
.62/230°
10°
.507/221°
.56/237°
.519/218°
.60/226°
20°
.356/219°
.49/246°
.376/213°
.49L214°
30°
.180/217°
.36L252°
.196/207°
.36L213°
40°
.043/216°
.047/204°
.19/211°
oo
.485/216°
.53/210°
.485/216°
.53/210°
10°
.424/217°
.50/222°
.437/213°
.50/205°
20°
.288/217°
.44/236°
.306L209°
.46L2ooo
30°
.143L216°
.35L245°
.155L205°
.32L196°
40°
.035/216°
.038/203°
.20L192°
I
I
!.__···-.. -··---·---····· ·-·--·------·--------------·---·-.. ------..--·-·-·--··-----·-·--·
33
Computed
=
=
~i'O
a~·o
Figure 5.
L£'0
Normalized Input Impedance of Tilted
Strip for k w = 9.98 and k h = o.s
0
0
----------~-----
9.98
0.5
34
~------------
---------·-
= 9.98
= 1.0
os-
Figure 6.
ii'O
ti'O
st·o
~~·o
Normalized Input Impedance of Tilted
Strip for k 0 w = 9.98 and k 0 h = 1.0
J
~----------------~----------------------------------------
.-
35
---Computed
Figure 7.
IL..-----
=
0.12
0.13
0.38
0.37
9.98
= 1.5
Normalized Input Impedance of Tilted
Strip for k 0 w = 9.98 and k 0 h = 1.5
.-
36
the larger k h values, the locus of the impedance points
0
tends to close up slightly (see Figure 7), but remains in
the same general regions of the chart.
Note also that the
locus is nearly symmetric about the real axis.
Figures 8, 9 and 10 are for the case k 0 w = 6.29.
In
general, they differ only slightly from the curves for the
previous case, k 0 w
=
9.98.
For this case, however, the
loci are slightly smaller in area (more closed) and exhibit
less symmetry about the real axis.
They do have approxi-
mately the same normalized impedance characteristics over
the same range of angles and tend to also close up slightly
at the greater k 0 h values.
Figures 11, 12 and 13 are for the case k 0 w
= 3.94
and,
for all three heights, the loci have collapsed quite notice
ably onto themselves, although not completely.
The imped-
ance properties are now almost predominantly capacitive
except for the large negative angles
(~
= -30°,
-40°).
Figures 14, 15 and 16 are for the narrowest strip,
k0 w
= 2.79.
They show that the loci of impedance points
are now almost completely collapsed onto themselves and,
hence, for the corresponding posi·tive and negative tilt
angles
(~
= +10°,
±20°, +30°), the strip reflection proper-
ties are nearly independent of the direction of tilt and
only dependent on its magnitude; i.e., s 11 ~ s 22 . Also
note tha.t the strip impedance properties are capacitive at
l
all angles shown and that the phase of
tively constant.
s 11 remains rela-
37
Figure 8.
0.12
0.13
o.38
0.37
Normalize~;:Inp~~:
Impedance of Tilted
Strip fork w = 6.29 and k 0 h = 0.5
----------------------~----------~~----------------~----------
=
=
6.29
0.5
J
38
= 6.29
=
21'0
Figure 9.
No:rmalize~~~put
Impedance of Tilted
Strip for k 0 w = 6.29 and k 0 h
= 1.0
1.0
J
..___.__,.._.._-~-
39
0.12
0.13
o.38
0.37
= 6.29
= 1.5
oeZI"O
l
Figure 10.
£1"0
Normalized·K:~~~:·I~pedance
=
Strip for k w
0
of T i l t e d j
= 1.5
6.29 and k 0 h
·-------------------------------------~-------------
40
Figure 11.
0.12
0.13
0.38
0.37
Normalized Input Impedance of Tilted
Strip fork W·= 3~94 and k h = 0.5
=
=
. - - = - - - 0 -~
3.94
0.5
J
41
Figure 12.
Normalized Input Impedance of Tilted
Strip for k w = 3.94 and k 0 h = 1.0
0
42
= 3.94
= 1.5
l__
Figure 13.
Normalized Input Impedance of Tilted
Strip for k w = 3.94 and k h = 1.5
43
---Computed
l
0.13
0.12
. 8£'0
Figure 14.
Lt 0
Normalized Input Impedance of Tilted
Strip for k w = 2.79 and k 0 h = 0.5
0
=
=
2.79
0.5
44
0.12
0.13
o.3B
0.37
11Cf0
Figure 15.
I
I
I
1Ltl0 I
=
2.79
=
1.0
J
Normalized Input Impedance of Tilted
~----------------~s_t_r_1_·p~f_o_r__k_0_w__=
__2__._79__a_n_d__k_0_h__=
___l_.__
o ---.--
45
= 2.79
= 1.5
.
l
Figure 16.
~•·o
ti'O
8''0
LIO
Normalized Input Impedance of Tilted
Strip fork 0 w = 2.79 and k 0 h = 1.5
·-·----,-------
46
The comparison between the theoretical and experimenta
data given here has lent a considerable degree of credibility to the analytical model chosen and to the numerical
approach taken in this analysis.
The last step in the
complete characterization of the strip as a two-port is to
find an expression for the transmission coefficient s 12 •
However, this is beyond the scope of the study presented
here.
I
_j
VIII.
DESCRIPTION OF THE EXPERIMENTAL PHASE
A block diagram of the experimental setup is shown in
Figure 17.
It is located in the Microwave Laboratory of
California State University, Northridge, and it is the same
apparatus which was used by Gillespie and Gustincic£1,5]
and Kilburg [ 3] •
A frequency of 9.375 GHz was used for this experiment.
At this frequency, the width of the surface wave table is
approximately 18 wavelengths and the length is approximately 54 wavelengths.
The dielectric coating was 0.06
inches thick and had a relative dielectric constant
K
= 2. 55
The transverse attenuation constant used for the numerical
solution was computed from (67); a 0
=
0.908 nepers/inch.
The tunable detector probe shown in Figure 17 was
mounted in a sliding carriage which allowed approximately
two wavelengths of longitudinal probe movement along the
centerline of the table.
This was the slotted line section
of the surface wave system, and the relative position of
the probe was read on a dial indicator (to the nearest 0.00
inches) •
'l'he measurement technique used required the use
of a sliding load.
For this experiment, it consisted of a
block of x-band microwave absorber mounted to a positioning
apparatus.
Its position was also read on a dial indicator.
The metallic strips used were of two types, goldplated steel and beryllium copper, and both had a thickness
-·---·-------
47
.~
SW Table
-22~o
"'I .
I
""Normal to
-20ft.-"-~
-12~o-l
~----.1_1
o
~
r---
Slotted
ine
r
.
I
Tunable Sliding
1
Detector Mount~
T
d
Sliding
Load
(Microwave
Absorber)
~Tilted
f
7rstrip
f
T
::I
1
VSWR
~ndicator
Xl3
Klystron
& P.S.
~Alclad
NOTE:
Aluminum
Table width - 18
where ft. is the
ft.
I
l
0
0
20
Coupler
Oscilloscope
...,,..
...........
t--tor
Display
free space wavelength
Figure 17. Experimental Setup of Surface Wave
Table and Associated Apparatus
.co.
(X)
49
of 0.010 inches.
They were positioned above the dielectric
surface by nonmetallic supports which were attached to the
sides of the table.
The strips were held in tension to
minimize sag when tilted.
made at
~
=
0° tilt.
The initial height settings were
The strips were then rotated about
their midpoints to the various tilt angles used (-40° to
+30°).
For angles greater than these, the experimental
errors were thought to be significant and therefore, angles
outside this range were not used.
In all cases, k h refers
0
to the height of the lower edge of the strip above the
dielectric surface for the untilted case
initial heights were used (k h
0
strip widths were used (k 0 w
=
=
(~
=
0°).
Three
0.50, 1.00, 1.50) and four
2.79, 3.94, 6.29, 9.98).
The method of measureme_nt of the strip scattering
coefficients
s 11
and
s 22
was derived from the slotted line
impedance measurement technique given in the literature[?].
For this case, the strip was viewed as a two-port network
with its output terminated by the sliding load as shown in
Figure 18.
Under these conditions, the input reflection
coefficient for the network can be written[4] as
=
( 68)
where rL is the reflection coefficient of the sliding load.
r.1n was measured using the minimum shift method.
This was
repeated for various positions of the sliding load, i.e.,
1
rL was changed in angle only.
L___
For sliding load increments
·-------------··----
------------------------------------1
I
I"
sll
sl2
sl2
s22
r
J.n. -
Figure 18. Two-Port Model of Strip With Output
Terminated by Sliding Load
!
I
L_________·------------------------------------------------------------------~------·
l11
0
51
over a range of 1/2 guide wavelength, a plot of rin on a
Smith chart resulted in a circle of data points.
example of this is shown in Figure 19.
An
Note that a line
drawn between the center of the data points and the center
s 11
of the Smith chart represents the phasor
r plane.
s 22
I
When the strip is tilted in the opposite directiot
to the negative angle,
,and
in the complex!
-~,
the relative positions of
in (68) are interchanged, and
s 22
s 11
·
results from a
!plot of rin on the Smith chart.
I
I
I
I
I
I
I
Ii
Ii
I
'
j
L_ _______________ _
----·----·- ---------·-------- --·--···--·· -----· -----------· ··-
---- - ----·---·
-
·-·---·--·---- ----····--
·-------··
--------- ·-------- j
52
Figure 19.
Example of Data Points for Determining
s 11 (k 0 h = 0.1, k 0 w = 6.29, ~ = -40°)
1.••• --~~~-~·----·----· -~~-·--~·-··---·-·--·--":'--·-··· · - - · · -·" · · · · - - - - - - -
·--------------------
BIBLIOGRAPHY
Gillespie, E. s., and J. J. Gustincic.
"The Scattering of a TM Surface Wave by a Perfectly
Conducting Strip," IEEE Transactions on Microwave Theory and Technique, MTT-13 (September
1965) 1 630-40
o
Sharma, K. P.
"An Investigation of the Excitation
of Radiation by Surface Waves," Proceedings of
the IEEE, 106B (March 1959), 116-122.
i [ 3]
l
!
Kilburg, F. J. "Impedance and Scattering Properties
of a Metallic Strip Above a Dielectric Coated i
Conducting Plane." Unpublished Master's Thesisf
California State University, Northridge, 1972. 1
I
Collin, R. E. Foundations for Microwave Engineeringl
1
New York: McGraw-Hlll Book Company, 1966.
: [ 5]
[6]
i [7]
Gillespie, E. S., and J. J. Gustincic. "The Scatter-i
ing of a Plane Surface Wave by a Perfectly
'
Conducting Strip," University of California,
Los Angeles, Report No. 64-56, December 1964.
Collin, R. E. Field Theory of Guided Waves.
York: McGraw-Hill Book Company, 1960.
New
Terman, F. E., and J. M. Pettit. Electronic Measurements. New York: McGraw-Hill Book Company~
1952.
53
[
--------------_------------- ------ -----------------------------------------------
--1
I
I
I
APPENDIX
THE COMPUTER PROGRAM
I
. . . . . .. I
54
55
C
COMPUTE SURFACE WPVE REFLECTIO~ COEFFICIE~T
COMPLEX J,px,Ax,~,L,SUM,GAMMA,PSI<70,70>,psx,psz,
+JI<7~),EC70>,S<7~l,CC70>,COLC49~U>
C
REAL LAM,KAP,NN,K,KW,KH,
SET CONSTANTS
IEND=~J
J=<e .c:J, 1 .vJ>
FED=9·375E'.1
KAP=2·55
T=·lf)6
LAM=2·998El0/CFR0*2·54>
t' I =3 • 1 Ll} 59
K=2·*Pl1LAM
ALF=K**2*<KAP-1·>*T/KAP
BTA=SORTCALF**2+K**2>
EC=·5772
ANG=SGRT<KAP>*K*T
RX=<J•SQRT<KAP>+SI~CANGl/COSCANG>>ICJ*SOPTCKAP>-
+SIN<ANG>/COS<ANG>>
RX0=-2·*ALF/Clo+CALF•<ALF*KAP+K**2*<KAP-1•>*T+ALF**2
+•<KAP**2-l·>*Tl/CKAP•<K**2*<KAP-1·>-ALF**2>>>>
C READ IN STRIP WIDTH, NUMBER OF SEGMENTS AND END DATA
600 READ 9,W,NN,IEND
9 FORMATC2F5·2,T72,Il>
C CHECK FOR END OF DATA
IFCIEND>1000,5000,1000
500Ql N=NN
ILINE=l00
H=0·0
C SET STRIP HEIGHT
DO 900 Il=l,3
H =H + • 1
PHID=-50·
C SET STRIP TILT ANGLE
DO 90vJ I2=1,9
PHID=PHID+ 10·
PH I= PH I D*PI/ 18 !fJ •
C COMPUTEK HEIGHT OF MIDPOINT
D=H+l;J/2.
C COMPUTE WIDTH OF STRIP SEGMENT
DEL= 1..J/NN
C COMPUTE DIMENSIONLESS QUANTITIES
KitJ=K*W
KH=K-*H
C
C
C
'··
.-
COMPUTE MIDPOINT OF FIRST SEGMENT
X 1 =-t.J/2 .+DEL/2 •
BEGIN COMPUTATION OF GREEN'S FUNCTION MATRIX
DO 5(1 M= 1, N
COMPUTE FIELD POINT
X=X1*COS<PHI>+D
Z=X1*SIN<PHI>
'··
........
56
C
C
C
C
C
C
C
C
C
C
C
C
X2=-1J,'/2 .+DEt/2.
DO 40 I=l,N
COMPUTE SOURCE POINT
XI=X2*COS<PHI>+D
ZI =X2*SI N< PH I>
CHECK FOR COINCIDENCE Or FIELD POINT AND SOURCE POINT
Ir <I-M> 19,20,19
COMPUTE DIRECT RADIATION TEAM ARGUMENT
19 ARG=K*SORT<<X-XI>**2+<Z-ZI>**2>
COMPUTE DIRECT RADIATION TERM USING HANKEL FUNCTION
SUBROUTINE WHICH IN TERN USES STANDARD CANNED BESSEL
FUNCTION SUBROUTINES BESJ AND BESY
CALL HKL<ARG,A>
GO TO 30
COMPUTE AVERAGE HANKEL FUNCTION OVER SEGMENT INTERVAL
WHEN SOURCE AND FIELD POINTS COINCIDE
20 A=l·-2~*J/PI*CALOG<EC*DEL*K14·>-1•>
COMPUTE INDIRECT RADIATION TER~ ARGUMENT
30 ARG=K*SORT<<X+XI>**2+<Z-ZI>**2>
COMPUTE INDIRECT RADIATION. TERM USING HANKEL FUNCTION
CALL HKL <ARG,L>
COMPUTE X COMPONENT Or GREEN'S FUNCTION
PSX=-.25*CA+RX*L>*COS<PHI)
COMPUTE Z COMPONENT Or GREEN'S FUNCTION
PSZ=-.25*<A-RX*L>*SINCPHI~
C
COMPUTE VECTOR GREEN'S FUNCTION AND STORE IN MATRIX CNXN>
PSICM,I>=PSX*COS<PHI>+PSZ*SIN<PHI>
C SET NEXT SOURCE POINT
40 X2=X2+DEL
C COMPUTE INCIDENT FIELD AND STORE IN COLUMN MATRIX <N>
ECM>=EXPC-ALr*X>*<COSCBTA*Z>-J*SIN<BTA*Z>>*<COSCPHI>
++J*ALr/BTA*SIN<PHI>9
C COMPUTE GENERAL SOLUTION TERMS FOR INTEGRAL EQUATION
C AND STORE IN COLUMN MATRIX <N>
S<M>=SINO<*Xl>
C<M>=COSCK*Xl>
C SET NEXt FIELD POINT
50 Xl=Xl+DEL
C COMPUTE CONSTANTS FOR INTEGRAL EQUATION
AX=<<ALr*COS<PHI>+J*BTA*SINCPHI>>**2+K**2>
DTA=t.I<C<l>*S<N>-C<N>*S<t>>
C ELIMINATt 2 ROWS AND 2 COLUMNS rBBM GREEN'S FUNCTION
C MATRIX AND COMBINE CONSTANTS INTO SET Or tOUATIONS· ALSO
C ELIMINATE FIRST AND LAST ROW FROM COLUMN MATRICES·
NL=N-1
DO 9trj M=2, NL
DO 80 I:.2,NL
80 PSICM,I>=PSICM,J>+DTA*S<M>*<C<N>*PSI<l,I>-C<l>*PSICN, >
+>-DTA*C<M>*<S<N>*PSI<I,I>-S<l>*PSI(N,I>>
SCM>=DTA/C-AX>*S<M>*<E<l>~CCN>-E<N>*C<l>>
C<M>=DTA/AX*C<M>*<E<I>*S<N>-E<N>*S<l>>
90 ECM>=ECM)/C-AX>
57
C
C
C
C
C
C
C
C
RESTORE ARRAYS
NL=N-2
DO 100 1'1= 1, NL
MM=!'i+ 1
DO 95 I:.: 1, NL
11=1+1
95 PSICM,I>=PSICMM,II>
S cfvD =S ct·1Ml
c cr-1> =c cr-Jt-1 >
1 0 0 E Ci•D =E CMM l
III=NL**2
CONVERT SQUARE PSI MATRIX TO COLUMN COL MATRIX
CALL ARRAY C2,NL,NL,N,N,COL,PSI>
ADD ECM>+SCM>+CCM> AND STORE IN ECMl
CALL GMADD CE,s,c,NL,ll
INVERT COL MATRIX AND RESTORE IN COL
CALL MINV CCOL,NL,s,c>
COMPUTE CURRENT DISTRIBUTION BY MULTIPLYING COL BY E<M>
AND STORE RESULT IN JICI>
CALL GMPRD CCOL,E,JI,NL,NL,1>
COMPUTE REFLECTION COEFFICIENT GAMMA BY NUMERICAL INTEGR TION
SU\'"1=<0·0,0·0>
Xt=-W/2.+D2L/2.+DEL
DO 127 I=t,NL
B=EXPC-ALF*Xl*COSCPHI>>*<COSCBTA*Xl*SINCPHI>>
F=EXPC~ALF*X1*COSCPHI>>*<-SINCBTA*X1*SINCPHI>>
C
C
C
C
SUM=SUM+CMPLXCB,F>*JI<I>
127 X1=X1+DEL
GAMMA=RXO*SUM*EXPC-ALF*D>*<COS<PHI>**2-SINCPHI>**2>
+/C2·*BTA>*<<<J*BTA*SINCPHI>-ALF*COSCPH1>>**2+K**2>
+*COS<PHI>-<J*BTA*SIN<PHI>-ALF*COSCPHI>>*CALF*SINCPHI
++J*BTA*COS<PHI>>*SINCPHI>>
CONVERT GAMMA TO POLAR FORM
GAMMAR=REALCGAMMA>
GAMMAI=AIMAG<GAMMA)
COMPUTE ANGLE OF GAMMA BY MODIFIED ARCTAN SUBROUTINE
WHICH IS GOOD FOR ALL 4 QUADRANTS
CALL ATANG CGAMMAR,GAMMAI,ANl
cr·•J=Cf~BS CGAfvjlvJA >
Ai\J=AN* 180 • /PI
CHECK TO SEE IF OUTPUT SHOULD BEGIN ATN TOP OF NEXT PAG
IF <ILINE> 700,750,750
7510 PRINT 10
10 FORIVJAT< 1H1,T5, ~~VIDTH 1 ,Tl5, 1 HEIGHT',T26, 'TILT',T37,
+ KltJ', TLJ8, '1-\H', !63, GAivJriJ?\ 1T6, (IN) ', T 16~ (IN) T25,
+ i CDf:GS). ,!35; CHAOS) ,TL!6,. CRI\DS>. ~T59, MAG. ,!69,
1
1
1
I
I
1
1
1
,
I
+'/\NG'/).
!LINE=0
0 PRINT 165~w,H,PHID,KW,KH,GM,AN
5 FORMATCT5,F5:2,T15,F5·2,T25,F6·2,T35~F6·2~T46,F6·2,
+T57,F7·3,T67,F7·2>
-----------------------------------------·
58
900 ILINE=ILINE+l
GO TO 6vJO
1 v.l(tJ(.J
C
ST0i 1
END
MODIFIED ARCTAN SUBROUTINE
SUBROUTINE ATANG(X,Y,AN>
AN=ATANCY/X)
IF" CX/Y) 2,2,LI
2 IF" <Y) 5f21,5!1J,6
6 AN=AN+3 ·1 416
GO TO 50
4 IF" CY> 10, HJ,50
1 liJ AN=AN+3 •l Ltl6
50 F~ETUnN
END
C
HANKEL FUNCTION SUBROUTINE
SUBROUTINE HKL CARG,Al
COI'-1PLEX A
t<EAL*8 J, Y
YY=SNGL<-YC0,ARG>>
BB=SNGLCJC0,ARG>>
A=CI•1PLX CBB, YY l
RETURN
END
NOTE: J AND Y ARE POLYNOMIAL APPROXIMATION SUBF"UNCTIONS
FOR THE BESSEL FUNCTIONS OF THE FIRST AND SECOND
KIND RESPECTIVELY-THEY ARE FASTER THAN THE IBM
SSP PROGRAMS BESJ AND BESY MENTIONED PREVIOUSLY·
SUBROUTINES ARRAY, GMADD, MINV, AND GMPRD ARE
STANDARD IBM SSB CANNED PROGRAMS MODIFIED TO
HANDLE COMPLEX QUANTITIES·
;.;·_li\1 TJivll.i: Fo,·l CtY·1P!IT't.\TIOi\J OF O:\J[ D1"!T1\ POINT(),'-/
317(} ',·! . ~S Al'f'}:? 1JXTI'vl.Cl,'l"f·~l.Y 1 L1 iVJt:'l;)TF:S•
'·(!J:~ 'f'T:·,ii;:
U N J l.l''1 ;~ f:, 0 - 9 1 F 0 H 1 ? p1\ T ~~ 1·> 0 f N 'l' .'.1 1-.1 1\ S fl P ~:ll ~ 0 >< I 1'·'l /'\ T 1-:: L
7 (·") J. 1\JI. JT f.~ S /.J 5 S !•= C 0 1\J D :; •
CDC
·----------··----