CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
INPUT IMPEDANCE OF UNSYMMETRICAL FOLDED DIPOLES
\l
A comprehensive report submitted in partial
satisfaction of the requirements for the degree of
Master of Science in
Engineering
by
Donald Dean Batson
June 1974
I
I
The comprehensive report of Donald Dean Batson
is approved:
California State University, Northridge
June 1974
ii
'
Acknowledgement
I would like to express my deep gratitude to my
advisor, Prof. E. S. Gillespie, for his invaluable help
and encouragement throughout the course of my graduate
studies at CSUN.
I would also like to thank Michael
Miller for his aid in the construction and impedance
measurement of the antennas.
For the help, patience,
understanding, perserverance and typing, which kept me
at this task, I am eternally grateful to Deanna.
iii
TABLE OF CONTENTS
CHAPTER I
.........
INTRODUCTION . .
CHAPTER II
COMPUTER PROGRAM INVESTIGATION
ABSTRACT ••
2-1
2-2
2-3
2-4
CHAPTER III
3-1
CHAPTER IV
4-1
CHAPTER V
• • • • • • • • •
t
...
viii
1
t
I
I
0
•
•
•
•
•
.
•
•
19
MODELING TECHNIQUES • • • • • • • •
23
Introduction
23
I
Introduction . . . . • •
Pulse Expansion Program .
Piece-Wise Linear Program
Antenna Modeling • • • •
CONCLUSION
•
.
•
•
•
•
•
•
...........
-
MEASURED RESULTS
Introduction
•
.
•
•
4
4
6
12
• • • • • • • • •
47
• • • • • • • • • • •
47
• • • • • • • •
59
• • • • • • • • • • • • • • •
60
• • • • • • • • • • • •
61
APPENDIX B • • • • • • • • • • • • • • • • • • •
71
APPENDIX C •
76
BIBLIOGRAPHY • •
APPENDIX A • • •
APPENDIX D •
•
...•
• • •
•
..................
• • • • • • . . . . . • • • • • • •
iv
89
LIS'r OF FIGURES
Figure
1.1
Unsymmetrical Folded Dipole •
2.1
Geometry of Dipole showing
method of Subsections for
Pulse Expansions . •
o
2.2A
•
•
•
....
•
•
•
•
2.3
10
Input Reactance of Dipole
of Figure 2.1 . . . . • .
11
....
13
......
14
•
Input Resistance of Dipole
with Parasitic Element of
Figure 2.3
o
o
2o4B
..
Linear Dipole with Parasitic
Element • • . . . o o
o
•
o
2o4A
9
Input Resistance of Dipole
of Figure 2.1 .
. •
• • • • • •
o
2.2B
1
•••••
Input Reactance of Dipole
with Parasitic Element of
Figure 2 o 3 . . . . o • o
o
Subsection and Geometry of
Folded Dipole
•
o
o
•
•
•
•
o
•
o
•
•
15
16
2.6
Expansion Function at Sharp
Corner . . • • . . . • . . . .
2.7
A Folded Dipole • •
2.8
A Three Wire Junction .
21
2.9
A Four
21
3.1
Folded
3.2
Folded
3.3
Folded
3.4
Subsectioning of Antennas Group One . • o • • • o • •
•
o
•
•
•
25
Subsectioning of Antennas Group Two • •
• . • .
•
•
•
•
• .
27
3.5
o
..
...
....
.......
Wire Junction . . . .
..
Dipoles - Group I . . . . . .
Dipoles - Group II • . . .
Dipoles - Group III . . . . .
o
v
21
21
24
24
25
Page
Figure
).6
Subsectioning of Antennas Group Three
•
•
..
I
I
I
I
I
• • •
.
29
•
31
• •
32
.
33
• • • •
34
3~7A
Input Resistance - Antenna I
• •
J.7B
Input Reactance - Antenna I
•
J.8A
Input Resistance
J.8B
Input Reactance - Antenna II
J.9A
Input Resistance - Antenna III
•
35
J.9B
Input Reactance - Antenna III • • • •
36
J.10A
Input Resistance
J,10B
Input Reactance
3 .11A
Input Resistance - Antenna V
39
J,11B
Input
40
J.12A
Input
J.12B
Input
J.1JA
Input
J.1JB
Input
J.14A
Input
J,14B
Input Reactance - Antenna VIII
411
Antenna VIII mounted on the
Ground Plane for Impedance
Measurements
•
-
-
.
I
Antenna II • • •
Antenna IV
Antenna IV
•
.
..
• •
37
• •
.
38
•
.• •.
.• ..
Reactance - Antenna V
Resistance - Antenna VI • • • .
Reactance - Antenna VI • • . •
Resistance - Antenna VII • • .
Reactance
Antenna VII
• . •
Resistance - Antenna VIII • . •
~
I
...
.
I
49
Input Impedance - Antenna I
• •
4.4
Input Impedance - Antenna II
.
4.5
Input Impedance
4.6
Input Impedance - Antenna IV
4.7
Input Impedance - Antenna
vi
•
45
48
4.3
v
44
• • •
• • • •
•
43
46
Impedance Measurement Set Up
Antenna III •
42
• • •
4.2
-
41
.
•
51
• • •
52
...
• • .
..•
53
54
55
Figure
~
4.8
Input Impedance - Antenna VI
• • . •
56
4.9
Input Impedance - Antenna VII • • • •
57
4.10
Input Impedance - Antenna VIII
58
vii
. • •
ABSTRACT
INPUT IMPEDANCE OF UNSYMMETRICAL FOLDED DIPOLES
by
Donald Dean Batson
Master of Science in Engineering
June 1974
The input impedance of unsymmetrical folded
dipoles is investigated over the frequency range of 1
to 1.4 GHz.
Eight individual folded dipoles are
presented with both measured and computer calculated
results.
Two computer programs are presented using the
"method of moments" technique, with subsectional basis,
for calculating thin wire antenna problems.
The first
program used pulse current expansion functions and had
incorrect solutions whenever sharp bends or wire junctions were placed into the program.
A second program
uses triangular piece-wise linear current expansion and
is the program which was used for the solution of the
folded dipole problems presented here.
Eight folded dipole antennas were constructed
which correspond to the computer models.
To make the
impedance measurements simple and accurate, one half of
each antenna was constructed for mounting on a ground
plane.
viii
The calculated results on the eight models show
good agreement with the results measured on the image
plane models of the corresponding antennas.
ix
CHAPTER I
1-1
INTRODUCTION
This investigation was undertaken to become
familiar with the Harrington 1 • 2 "method of moments"
technique and,with the aid of the digital computer, to
solve a practical antenna problem.
The input impedance
I .
of a two dimensional unsymmetrical folded dipole was
chosen due to the fact that this structure is not
extensively documented and yet has many practical applications.
The major task of the investigation was to
write or modify an existing computer program that could
be used to calculate the input impedance of the
unsymmetrical folued dipole.
The basic geometry of the
antennasare shown in Figure 1.1.
-.A
Figure 1.1
Unsymmetrical Folded Dipole
1
2
The folded dipole antenna has been documented very
extensively by King3 with work on the unsymmetrical
folded dipole being formulated by Harrison.3• 4
Although
extremely accurate, these formulations are limited in
that they apply only to round conductors of a specific
geometry.
There are many applications for the extension of
folded dipole theory.
Using the computer modeling
described here, the input impedance of shunt excited
wings and empennage of aircraft and missiles can be
predicted.
Also the performance of shunt fed vertical
masts and structures used as shipboard antennas are
predictable with the use of these techniques.
The folded dipoles investigated here are thin wire
structures which model the large conducting sections by
a grid of wires.
This is a standard technique used by
Richmond5 and Theile 6 in their modeling of flat plates,
spheres and aircraft structures.
Good agreement between
full scale back-scattered measurements and wire grid
computer simulation models have been verified.
The first technique to model the folded dipole was
to extend the pulse current expansion function developed
by Harrington~
This formulation, as discussed in Chap-
ter II, produced excellent agreement with theoretical
results for straight dipoles or dipoles with parasitic
excitation.
However, when sharp corners or junctions were
attempted, the program did not give the correct answer.
As a result, further investigation and research was
continued until a method was found which would handle
these discontinuities.
The triangular piece-wise linear
current expansion functions, as formulated by Strait and
Chao7, was capable of handling these conditions.
The computer program formulated by Strait and
Kuo 8 was then modified for the special task of solving
for the input impedance of the folded dipoles under
investigation.
Eight different folded dipoles, having different
lengths and shunt element thickness, were calculated and
are presented in Chapter III.
To verify the accuracy of
the program, image plane antenna structures were
constructed and impedance measurements were made across
the same frequency band as the computer models.
These
results are shown and compared in Chapter IV.
Good agreement was achieved between the computer
predictions and the measured antennas.
1
I
CHAPTER II
COMPUTER PROGRAM INVESTIGATION
2-1
INTRODUCTION
Schelkunoff and Friis9 utilized subsectioning of
thin wire conductors to solve for the fields and
radiated power of dipole antennas. Harrington1 in 1966
applied this technique to the general classification of
antennas and wire grid structures.
Harrington's first examples were for thin linear
antennas and used rectangular pulse functions for both
the current expansion function and the ·testing function
("Galerkin's Method").
He used the method of sub-
sectional basis which divides the wire structure into
small linear sections.
Different forms of current
expansion functions were later developed, but the basic
formulations of the "method of moments" and the boundary
conditions have not changed.
For thin wires, the following approximations can
be mader
1)
The currents are assumed to flow only
in the axial direction of each subsection.
2)
The current and charge densities are
approximated by filaments of current
4
5
and charge on the wire axis.
3)
The boundaryconditionsz
..
- X -i
n X -s
E = -n
E
(1)
I .
are applied to the axial component
-
. surface.
of E on each wJ.re
With the use of these approximations, the field
equations which are to be solved by matrix methods
reduce toa
d,
-
"*i
-EI = -jwA
!=~
ff
~
:::
1
T+if£
(J=-
_1_
jw
i1
1
a1
5I
(i) e -jkR d/
R
axis
s
axJ.s
d
dT
s
on
o<-R>
e -jkR d.£
R
[I co]
(2)
(3)
(4)
(5)
When these equations are solved with the selected current
expansion and testing functions, the problem reduces to
setting up the computer program to solve for the elements
of the mutual impedance matrix
[z]
*
[I]
[z]
=
[v J
(6)
is the matrix equation which must be simultaneously solved
to determine the currents on each small subsection of the
6
wire antenna, once the [
The
[zJ
z]
matrix is determined.
)llatrix is an M x M matrix, where M is the
number of current expansion functions.
[v]
generalized voltage matrix of dimension M x 1, and
is the generalized current matrix also of dimension
Mx1.
is obtained by
inverting the matrix
(7)
where
tance
impedance matrix
are basically functions only of
the geometry of the problem and the expansion functions,
while completely independent of any excitations.
To treat the dipole as an antenna, a unit voltage
is applied at the segment (through the
selected as the feed point.
[v J
matrix)
Therefore, the currents
that are calculated by the program automatically
produce the input admittance of the antenna.
2-2
PULSE EXPANSION PROGRAM
Since the pulse expansion formulation is the
simplest and easiest to use, the initial computer
program was based upon this.
7
Harrington1 developed the formula for the elements
of the [
z]
matrix which is given byo
Zmn = jW~
A1 n' ~ J. mY
[ 1£/ (n+,m+l
+ J£ (n- ,m-)
J
Where
JL
and
Rm
=
_1_
A.J-..n
=
(m,n) +
-r<n-,m+l -:;J?<n+,m-l
(8)
J
e -jkRnm
4rr
Rnm
V;'Jm2
+ · (Z-Z m) 2
d_j_
Va2 + z2
For
For
m ~
m
=
n
jj€
'J- (m,n) =
n ~(n,n) =
( 9)
·m
\
n
( 10)
m
=
n
(11)
e -jkRmn
(12)
4 rr R
mn
1
2 rr A ,( n
*log (A l n) - ..il£rr
a
(13)
4n
Utilizing these equations, Ba11 10 wrote a computer
program to solve the special case of linear dipoles in
parallel.
The computer program utilized the equations
just outlined but did not have the capability of X
directed conductors or various combinations of charges
overlapping at junctions that would be required in the
folded dipole case, as shown in Figures 1.1 and 2. 5.
8
His computer program was modified to include junctions and right angle bends, under the assumption that the
basic criteria and boundary conditions, as outlined above,
were held; that is, pulses and currents are· continuous
only over each individual section and that current flows
only in the axial direction of each subsection.
The program, as written, .is presented and discussed
in Appendix A and works very well for thin linear dipole
elements.
However, whenever sharp bends or junctions were
introduced, the program did not give·reasonable answers.
In setting up the program, thirty subsections were used
per wavelength uniformly throughout the structure.
In first testing the program, a thin linear halfwave dipole was used as a test case.
The method of sub-
sectioning and dipole geometry is shown in Figure 2.1.
The plus and minus signs indicate the charge polarity for
that subsection and the dotted portions on the ends
indicate the effective length of the antenna.
The dotted
half-sections are required in the mathematical formulation to satisfy the boundary requirement when single
pulse functions are used for the expansion function.
The plots of the resistance and reactance terms
for the computer output contained in Appendix A are shown
in Figure 2.2.
It should be noted that the well known
impedance for a resonant dipole of 71 + jO.O ohms is met.
9
r
~-·------------------~
...
+I-'
•
I?
1--
/4
.+
.-
-tl--
1'3
. '+
1--
12.
II
.-
+ t--
+
1--
10
•
.-
7.33
1'1--
9
.+
+I-- 7 .
.+
,,
1-
F"EED PT.
= :11.8
X
t--
'
5.0''
+ r---
5'
.
. __j
~+
4
+ 1--
3
.
.
1-
2
.
...
/:i
1=o. '333 ,,
-4- f---
J
f--1
L...,J
_J L.- ct=o.oos"
Figure 2.1
Geometry of Dipole showing
method of Subsection for
Pulse Expansion
10
e-o
&-e
Triangular program
Pulse program
200-
-s
[I)
..c:
0
.......
(!)
150100-
0
§
+'
[I)
50-
•r-i
[I)
(!)
~
.45
·55
·5
.6
ij'A
Figure 2.2A
Input Resistance of Dipole
of Figure 2.1
.65
11
........,
Triangular program
e-e
Pulse program
JOO-
250200-
tr.l
~
150-
..c~
0
Q)
100-
0
§
.p
0
50-
ru
Q)
0::
6-
-so-10
.45
·55
·5
.6
£;~
Figure 2.2B
Input Reactance of Dipole
of Figure 2.1
.65
12
The next test to be given the program was to place
a parasitic element juxtaposed to the driven element.
The antenna system was subsectioned, as shown in Figure
2.3, and input to the computer.
The program ran well
and the input impedance is plotted in Figure 2.4 with
the corresponding point calculated by King indicated at
the value of 1.8 + j20.ohms.
Finally, the initial folded dipole case as subsectioned in Figure 2.5 was input into the program with
the unfortunate result that the answers were totally
erroneous.
After several attempts at re-configuring the
direction of the plus and minus charges, further attempts
at the folded dipole problem with this program were
abandoned.
2-3
PIECE-WISE LINEAR PROGRAM
The "method of moments" technique using triangular
functions in a piece-wise linear current expansion was
found to be capable of accurately handling both the
multi-junction and sharp bends required in this investigation.
The current distribution programmed by Strait?
utilizes subsectional basis, but places a triangular
expansion function across 4 successive subsections
containing 5 points as shown in Figure 2.6.
Although
the amplitude on each section is characterized by an
1J
~
~o.zl
r•
'-+
..--,
---------~
I
15
"30
14-
2.9
1"3
28
12
'27
II
10
£.33 "
13
F£EO PT=-:#8
+
7
'2.2
'
?.1
- 10
19
+
'3
2
/7
-,,______
....
I
~
Figure 2.3
18
__....
..JI _ _ _ _ _ _ _ _ ___,_
Linear Dipole with Parasitic
Element
14
60-
•
50..........
4o-
[/)
s
;:::
0
30-
Q)
0
~
+>
[/)
·r-1
[/)
Q)
10-
~
.45
.5
·55
.6
£1X
Figure 2.4A
Input Resistance of Dipole
with Parasitic Element of
Figure 2.3
.65
1.5
250-
J
225200-
•
175150-125-
--_2
100-
( !)
0
._..
Q)
0
§
7550-
+-'
0
ro
Q)
2.5-
p::
0-25-50-75-10
.45
·.5
·55
.6
L!"'
Figure 2.4B
Input Reactance of Dipole
with Parasitic Element of
Figure 2.3
.65
16
[
,,
-
11
... _
~
Z=Z.5"
Ill
,,
/3
2.0
12.
2.1
II
22
10
23
~
Zlf
F££0 PT=#6
!.5
7
21.
'
27
18
5
Z8
4
2~
3
30
z
3/
I
+
+
""" .,.,
-+-
32
X
Z==-2..5 "
-
~o.z-J
Figure 2 . .5
Sub-Section and Geometry of
Folded Dipole
17
impulse function, the piece-wise linear approximation,
using
'+
segments, has the advantage of meeting all boundary
conditions, both at the end of open wires and by suitable
overlapping, multiple junction problems can also be
handled.
The initial formulation of the piece-wise linear
expansion of this problem, is the same as for the pulse
function and is described in detail by Harrington 1 and
Strait 7 .
The main difference between the two approaches is
in the current expansion formulation.
This is the most
difficult portion of the solution to any of the "method
of moments" problems.
The initial task before inverting
the matrix is to solve for the Zmn values determined by
the geometry of the problem.
for the values of the
[z]
Z ( j 'm) ' ( i 'n)
=
Strait 7 gives the formula
matrix as equal to:
j W)" Z 1 +
(14)
_L Z2
jc.ut:
Where
*tJj i,
1/r (Q j , 2m
- 2
+
2n -2 +
A , Q i , 2n - 2
+
B'~
B
~
( 15 )
1.8
and
=
Z2
*A
*
J.
~= 1
A
j , 2m - 2 + A
~(Qj,
*
n • 2n - 2 + B
l1 ..x..
1,
2m- 2 +A, Qi, 2 n- 2 + B>] (i 6 l
Where
e -jkR
R
P.1,n
ct,£ .
(17)
In evaluating the integral, the amplitude functions
are approximated by four pulses, whose amplitudes are given
by:
c.1, n(1)
c.1, n(2)
=
=
1/).,R_
2
.
1
2n - 1
t1 .L. 1,
. 2n - 1 +..1 L
A£..1r 2n - 1
t1.R..
(4)
,
_.e..
1'
2n
(19)
2n
i , 2n + 1 + .A L i , 2n + n
2n + 1 + A L i , 2n + 2 (20)
c.1,
c.1
+~.c1i.i!
i, 2n - 1 +A
(18)
i' 2n
n
=i ~
L i,
2n + 2
.4 L. i ' 2n + 1 + a
( 21)
l. i'
2n + 2
19
D.
J. '
D.
J.,
n(1) = D.J., n (2) =
n ( 3) = D.J., n(4) =
l1:l·J. '
A:l·J. '
1
(22)
-1
(23)
2n + 2
2n - 1 +41. i' 2n
1!. J.. '
2n + 1 +A
.
The point Q.
is at the center of the nth segment of the
'n
i th wire and R is the distance from the point
Q.J ,m to d.£ .
..
J.
To treat the dipole as an antenna, a unit voltage is
applied at the peak of the triangle which occurs at the
location of the feed point.
For the cases presented here,
this always occurs at the peak of the sixth triangle or
current expansion function.
Therefore, the currents which
are calculated by the program automatically produce the input admittance of the antenna at triangle number 6.
2-4
ANTENNA MODELING
In computer modeling of the antenna configurations
under investigation, it is necessary to select thin wire
approximations for the conductors.
11 he relatively large
areas of conducting material in the antennas were modeled
by grids of wires with multiple sharp bends and junctions.
Figure 2.6 shows a sharp corner placed in a single wire.
The wire is comprised of 9 points and 8 segments.
On this
wire, there are 3 current expansion functions whose
triangular peaks occur at A, B and C.
There are several rules which must be faithfully
20
applied when comnuter modeling the wire grid antennas.
They are:
1)
The number of points comprisinr; any
single wire (no matter how many bends)
must be odd.
2)
When a junction or connection of two
wires is made, the two wires must overlap by two sections as shown in Figure
2.7 for a folded dipole.
The folded
dipole has an odd :number of points, .19·
Although it has ,5 straight sub-wires,
in the program this is considered as
one wire.
Also, note that where the
two ends of the wire ,join, there is an
overlapping of two sections that, in
reality, occupy the same space.
J)
When multiple junctions are encountered,
as in Figures 2.8 and 2.9, the number
of overlapping wires is always one less
than the number of wires entering the
junction.
In Figure 2.8 there are 3 wires forming a junction,
with 2 overlaps and 5 straight sub-wires.
In Figure 2.9
there are 4 wires with 3 overlaps forming the junction
and 7 straight sub-wires.
21
\'''"
'
X
B
II
<
)'
A\
/4
,,
\
I
F'igure 2. 6
11
3
I
Expansion Function
Figure 2.7
at sharp cornel7
A Folded Dipole
, I, I
I,'
IJ/
4-,24-
I, 2.
• •
IJ"f.
•
I
•
•
•
3 ,,.
313
. • .
z,a
1,7
4,~
"3, 19
17
2)10
•
3)23
''3J/ij
z,s•
~2.(.
I
L. ~
1-,to
~IZ
~)1
Figure 2.8
A Three Wire Junction
Figure 2.9
A Four Wire
Junction
22
Strait? in his computer program does not input the
number of straight sub-wires.
In tailoring the pror,ram
to the folded dipole application, the input format has
been changed to accept sections of straight wires and
has the program select the points on the wire (after given
the number of subsections per wire).
This is explained
in Appendix A.
To check out the computer program, the dipole
described in Section 2-2 was, again, programmed using
the triangular piece-wise linear formulation and placed
in the computer.
Figure 2.2 shows the computed
resistance and reactance terms and are plotted along with
values computed from the pulse function and also with the
Hallen value of 71 +
,jO.O
ohms for a resonant dipole.
These results indicated that the program was accurate
and might be used for the required folded dipole
solutions.
CHAPTER III
MODELING TECHNIQUES
3-1
INTRODUCTION
The computer program which utilizes the triangular
piece-wise linear current expansion requires that the
antenna under investigation be accurately modeled by subsectional basis for input into the program.
The eight antennas under investigation were broken
into small individual segments for input into the
computer program.
The eight structures and their dimen-
sions are shown in Figures 3.1 through 3.4.
Antennas I through III as shown in Figure 3.1 are
comprised of a single wire . 061+ in diameter and are fed
at the center of the antenna.
The same size wire is used
for the shunt element and the end wires.
The second set of dipoles shown in Figure 3.2
comprise antennas IV through VI and are characterized by
the shunt element which is
0.5 inches wide.
To synthesize
the larger thickness of the shunt element, it is broken
into a grid comprising four squares.
The heights are
then varied in the three antennas.
The numbers VII and VIII antennas shown in Figure
3.3 have correspondingly larger width shunt elements and
are modeled by increasing the number of grids in the X
dimension.
It was intended at the beginning of the
23
o. 5" -...,
-...,
r-
-
7·5"
10.0"
--
-
I
II
III
Figure 3.1
Folded Dipoles d
1
Group One
= d2
1. 0"
,r
0.5"
1.0"
0.5"
1. 0" ~
0.5"
~
~r
-l
~
-
-
5.0"
t--
-
~ ~r
7·5"
1--
-
-
10.0"
~ 1--
1--
IV
v
VI
Figure 3.2
Folded Dipoles d 2
Group Two
= 7.81
d
1
-
25
~ 2.5'1
I
0.5 .. ~
r'""-
o. 5"
n
5
0"
·
I
I
5.0"
5· 0"
II
VIII
VII
Figure 3.3
. .,J~x'~'"",,
/
,
\/)
{
v
,1
1\
I\
v
I
<
0.0635
III
0.0127
o.o
o.o
o.o
)11/J
)1114
I
(
= 31.23 d1
II
0.09525
I
Antenna a
2.S'
<."\
Folded Dipoles d2
Group rrhree
and ct 2 = 70.31 d1
I
v
I'
3L.___.X
0
II
><
-0.0635
.
-0.09525
-0.0127
0
II
><
Note:
Figure 3.4
All dimensions in meters.
Subsectioning of Antennas - Group One
-
-
- ....,.
26
program to also vary the height of antennas VII and VIII
to correspond to the previous groups of antennas, however,
the computer cost involved was becoming excessive for
the amount of information supplied and were, therefore,
omitted.
The subsectioning for the individual antennas were
grouped in three divisions as shown in Figures 3.4, 3.5,
and 3.6.
The first group of antennas, i.e., I through III
are characterized by the subsections of Figure 3. L~.
This
antenna is modeled as 1 wire with 47 points, made up of
5 straight sub-wires.
Where sub-wire 1 goes from point 1
to point 3, sub-wire 2 goes from point 3 to point 23, subwire 3 goes from point 23 to 25 and so forth.
The different dimensions to cover the varying
heights or widths of the antenna in the program are made
simply by changing the size of each segment in the
computer program.
The second classification of antennas are described
in Figure J.2.
The details of subsectioning for
antennas IV through VI are shown in Figure 3. 5.
'I'he
antenna model is composed of 138 points, 12 wires and 32
straight sub-wires.
Wire 3, for instance, is composed of 3 straight
sub-wires and comprises points 35 through 49.
Wire 4 is
composed of J straight sub-wires beginning at point 50,
..,..,...
-
--
"JJIIf
27
X
Dimensions
r---
-ao
<\I<S'l
0 0
0
0
I t
-C\i
\!)-
-
N
oO
0
0
~
~
(.\J
0
0
II
Z Dimensions
Antenna No!
IV
0. 0"5J 0./2 7
-0.0508
.07(.. 2 ,/O/(.
-o.o445
,0,(,7
-0.0381
.0$72 .07(,2
-o.OZ5"4
-0.0/~1
.0381 .0~08
.028'- .0381
.:...__ 0.0 12 7
.0191
Notes
~
-o.o
o.o
.0881
.02~4-
/
FEED PT. /
-
VJ
O·OC.35"
-
/,
V
< ~13
''
'
o.o
-
-.00'4 -.00~$' -.0127
-.o127 -.OI'JI -.02~4-
-
-.DZ.54 -,038/ -.0508
-
-.0318- .047' -.0,3$'
-
-.044
-.0'' 7 -.088~
All dimensions in meters.
Figure
3.5
Subsectioning of Antennas - Group Two
28
ending at 60.
The overlapping at the junctions is required to
provide the correct boundary conditions for the current.
Antenna IV has a minimum of 40 divisions per wavelength
and the four large sections are more than adequate to
simulate a solid metal sheet.
However, the 7 1/2 inch
.and 10 inch dipole impedance would be more accurate if
the analytical model had maintained the subsections at
40 divisions per wavelength.
The final subsectioning for antennas VII and VIII as
shown in Figure ).6, is comprised of eight major squares
to simulate the solid metal of the shunt section.
It is
comprised of 223 points, 21 wires and 54 straight subwires.
The size of the antennas are changed by altering
the dimensions of the segments and point locations as
indicated by the scales accompanying the drawing.
In all cases, the wire diameters have remained
constant at .032 inches.
This wire diameter is chosen
smaller than that used in the impedance models described
in Chapter IV.
Because the length of each segment must
be greater than five times the radius of the wire?
described, a smaller wire diameter would allow finer subsections to be taken in the computer model.
All of the computer models described are assumed to
29
Dimensions
X
.....
N
N
0
()
-
VI If?
(\1
0
VII 0
lf'O"\
"'0
'-J"'
<:::>
o· 6
0
1:'-\1"\
N...,
~
0
0
I
oo
0
I I
I
-d
(\J
1(1
N
0
r-
00
1.n
CS'\
0
(j
fl
Q
0
0 6
0
if'~
d)
C'J
""
It\
-..s
0
0
0
0
I
I
Z Dimensions
J,ts: .. <"l
i/J
• • : :
4,4¥
~~~·
r.
t,.:.
'5,1S"
f'
t.,,., •
• •
., lt,f>i. ·p
,
9'U
I
1''.61 .
/I
• •
~~~~;·t,
,,B,
~=~"I "·'''".
r. 1 7o
.
(~
'0,"'~
10) 10(,
"' ,,.,
4,rt.
.!
,.~·
.'8,188
b
·~3
· tsp
,.~
~,,,
~/'IS'
,~,;,-.a..-: 12,"4
l ll
~.z.o,zo'
i,3.,1'"
-
0.04-4-5
-
0.0'381
-
0. 02.54
0.01" I
-0.012
-
7
0,00'4
-0.0
-·00,~
--,Ot2.7
-·0154,a~207
..,.".. ,
•t'··
p
14}'~•
--o.os-os
1(,,,,
.]
,, 4-¥
• • •
-.03( 8
-.04-45
2.11 ZZ.'
t,._l~2 lt.ll
1,3
Note a
All dimensions in meters.
Figure ).6
Subsectioning of Antennas - Group Three
JO
be fed by a unit voltage applied at point 13 at the peak
of triangle number 6, which corresponds to the Z=O,
X=O on the coordinate system and corresponds to the image
plane models discussed in Chapter IV.
The computer models, as shown, were each in turn
programmed and placed in an IBM 370 digital computer.
The output of the program, as detailed in Appendix D,
prints out the input impedance and admittance at
triangle peak number 6 as well as the current at each of
the remaining triangle peaks.
The typical computer
output for antenna VII is recorded in Appendix D.
The input impedance for each antenna is plotted in
Figures 3.7 through 3.14 in terms of their resistance
and reactance components.
As can be seen from the plots,
the impedances are well-behaved and correspond to the
predicted impedance values.
J1
1200
1100
1000
900
800
.....--.
{/)
.a
700
(1)
600
0
C.J
§
+'
{/)
·~
{/)
500
(1)
0::
400
JOO
200
1.0
1.1
1.2
1.J
Frequency GHz
Figure J.7A
Input Resistance - Antenna I
1.4
'I
300200100-
-E.
0-
Ul
0
.._,
(l)
C)
§
-200-
+'
C)
ro
(l)
p::
-300-400-500
1.0
1.1
1.2
1.3
Frequency GHz
Figure 3.7B
Input Reactance - Antenna I
1.4
33
900800........
UJ
700s
..c::
0
._,
Q)
()
600-
§
-P
[/)
•rl
500-
[/)
Q)
~
4003002001000
1.0
1.2
1.3
Frequency GHz
Figure J.8A
Input Resistance - Antenna II
1.4
J4
400
JOO
200
100
0
-too-··
.........
rn
-200-
.20
-JOO(])
(.)
§
+'
-400-
(.)
ro
(])
0:::
-500-600-·
-900--10001---~------~---------~---------~------~---
1.0
1.2
1.J
Frequency GHz
Figure J.8B
Input Reactance - Antenna II
1.4
35
1100
....-..
rn
.20
Q)
700-
0
§
+'
rn
rn
600-
•r-1
Q)
~
500400300-.
2001000
1.0
1.1
1.2
1.3
1.4
Frequency GHz
Figure 3.9A
Input Resistance - Antenna III
-----
~
-
36
120011001000900800700-
6oo.-..
fl.l
.500-
.E
0
.......,
400-
Q)
(.)
§ JOO-
.p
(.)
ctl
Q)
0::
2001000 -·
-100-200-
-400--~------~--------~------~~------~---
1.0
1.1
1.2
1.3
1.4
Frequency GHz
Figure J.9B
Input Reactance - Antenna III
37
.........
Ul
~0
700
Q)
()
600
..._,
§
.p
....
Ul
500 .
Ul
Q)
~
400
JOO
200
100
0
1.0
Figure ).lOA
1.1
1.2
Frequency GHz
1. 4
Input Resistance - Antenna IV
J8
300
200
100
0
-100 .
........
tJ)
.@
0
.._.,
Q)
()
§
-200
-300
..p
()
ro
Q)
-400
ll::
-500
-600
-700
-800
-900
-1000
1.0
1.1
1.2
1.3
Frequency GHz
Figure ).lOB
Input Reactance - Antenna IV
1.4
39
600--;
.E
0
-(1)
500-
.
400-
0
g
+'
rn
......
rn
~
200-
100-
1.0
Figure J.llA
1.1
1.2
Frequency GHz
1.J
Input Resistance - Antenna V
1.4
40
0-
-too-200-.300.........
til
.a
-400-
0
........
(I)
0
-500-
~
+>
0
ro
-6oo-
(I)
tl::
-700-
-Boo-90
1.0
1.1
1.2
1..3
Frequency GHz
Figure J.llB
Input Reactance - Antenna V
1.4
I
l
;I
600500........
Ul
~0
400-
( ])
300-
-0
@
..p
Ul
•r-1
Ul
200-
(])
~
100-
1.0
Figure 3.12A
1.1
1.2
Frequency GHz
1.3
1.4
Input Resistance - Antenna VI
L~2
1000-
500.........
fJ)
400
s
..t:!
0
...._,
JOO-
Q)
()
§ 200-
+'
()
ro
Q)
0:::
10
0
-100-.
-JOO
-4001~L-------~------~------~------~~1.0
1.1
1.2
1.4
1.3
Frequency GHz
Figure J.l2B
Input Reactance - Antenna VI
43
1000
900
800
700
,-...
Cll
~0
Q)
0
6oo500-
§
.p
Cll
•r-1
Cll
400-
Q)
p::
300200100-.
0
1.0
Figure J,lJA
1.1
1.2
Frequency GHz
1.J
1.4
Input Resistance - Antenna VII
2001000-100-200-300
-400-
-.2
Ul
-500-
0
........
-600-
Q)
C)
~
..p
C)
-700-
cd
Q)
ll::
-800-900-.
-1000-1100-1200
1.0
1.1
1.2
1.4
Frequency GHz
Figure J.lJB
Input Reactance - Antenna VII
45
{/)
.E0
1.0
1.1
1.2
1.4
Frequency GHz
Figure J.l4A
Input Resistance - Antenna VIII
46
0-100-200-300.........
[/)
-400-
.E
0
..._.,
-500-
Q)
()
@
+'
()
cd
-600-
Q)
0:::
-700-800-
-100~~------~---------L--------~------~---
1.0
Figure 3.14B
1.1
1.2
Frequency GHz
1.3
1.4
Input Reactance - Antenna VIII
CHAPTER IV
MEASURED RESULTS
4-1
INTRODUCTION
To verify the calculated impedance data presented
in Chapter III, image plane antennas similar to the
computer models were constructed.
The large flat sec-
tions of the shunt portion of the dipoles were
constructed as flat sheets of .032 brass.
The eight
models were constructed and attached to the 36 inch by 72
inch ground plane as shown in Figure
L~.
1.
The equipment and measuring set up is detailed in
Figure 4.2.
Figures 4.3 through 4.10 show the measured
impedance characteristics of the eight antennas.
Also
on the same plot are the calculated impedance values for
the antennas.
The values calculated by the computer were
for balanced antenna structures, while the measured
antennas were mounted upon an image plane.
Therefore,
the impedance values for the measured antennas were
multiplied by two before being plotted.
The general shape of the calculated and measured
curves are very similar.
The difference between the
calculated and measured impedance lies primarily with
the accuracy with which the very high impedances can be
measured.
The impedances measured on the antennas vary from
47
L~8
I. . ,____
72
fD
Figure 4.1
Antenna VIII mounted on the
Ground Plane for Impedance
Measurements
49
(Source)
Unit Osc
1218B &
Pwr Supp
G.R. 1263-C
2000 !VJHz
Low Fass
Filter
I
10db
Pad
G.R.
Admittance
Constant
1602-P4
Meter
Impedance
1-Term Std,-1602B
Adjustable
Line (20cm,
874-IJK20L
/
v
RG-9
Coax
Cable
50"
(14")
L..-
Adj.
I
I
l~
I
Stub~~~----~
Mixer
8 74 MRL
t-----t
I-F
Amp.
1216-A
~--~
I
I
(Local)
I
Unit
__ j
Osc.
G.R.1218A
Figure
~.
2
'
Referenc e
Plane
Impedance Measurement Set Up
50
the calculated value not only from dimensional
inaccuracies, but from losses in the system associated
with poor junctions, cable attenuation and instrumentation
errors.
The first time the impedance of the antennas
was measured, there was significant discrepancy between
the measured and calculated values that an investigation
of the measuring set up was undertaken.
'rhe four feet of
RG 9 cable used to measure the antenna and the 20 em.
line stretcher used for short calibration were removed
and a 14 inch RG 9 cable was substituted in their place.
The measurements were retaken on six of the
antennas and the results replotted.
As can be seen from
the variation of plotted data, the second set of
measurements agree more closely to those of the calculated
prediction.
The effect of cable attenuation can be seen
quite well in cases 2 and 5 as well as the total spread
of measurement variation between the two successive
impedance measurements.
51
0--0 Computer calculated value
0····0 Measured with 14 inch cable
®--~
Figure
Measured with 50 inch cable
4.3
Input Impedance - Antenna I
52
Z0
~
==
50 ohms
Computer calculated value
0····0 Measured with 14 inch cable
®---® Measured with 50 inch cable
Figure 4.4
Input Impedance - Antenna II
53
Z0
= 50
ohms
0-0 Computer calculated value
®---® Measured with 50 inch cable
Figure
4.5
Input Impedance - Antenna III
.
54
'
l
I
(
I
\•
!
Z
0
= 50 ohms
0-0 Computer calculated value
0···-G Measured with 14 inch cable
®---e
Measured with 50 inch cable
Figure 4.6
Input Impedance - Antenna IV
55
Z
0
==
50
ohms
0--0 Computer calculated value
0····0 Measured with 14 inch cable
®---® Measured with
Figure
4.7
50
inch cable
Input Impedance - Antenna V
56
Z0
~
= 50 ohms
Computer calculated value
®---® Measured with 50 inch cable
Figure 4.8
Input Impedance - Antenna VI
57
Z0
= 50
ohms
0--0 Computer calculated value
0···-0 Measured with 14 inch cable
Figure
4.9
Input Impedance- Antenna·VII
58
Z0
= 50
ohms
0--0 Computer calculated value
0···-0 Measured with
Figure 4.10
tL~
inch cable
Input Impedance - Antenna VIII
-CHAPr.I.'ER V
5-1
CONCLUSION
The input impedance of eight unsymmetrical folded
dipoles have been measured and computer calculated using
the "method of moments" formulation and the IBM 370
digital computer.
Eight image plane antennas were constructed and
their input admittance measured over the frequency
range of 1 to 1.4 GHz.
The calculated values agree quite
closely with the measured input impedance.
Two computer programs· which ar:e presented in the
Appendix are both useful in the solution of thin wire
antenna models.
·The program which uses triangular
piece-wise linear expansion functions is more useful
since it can handle sharp bends and multiple wire
junctions.
The pulse expansion program is very accurate
for linear dipoles and arrays of dipoles with and without
parasitic elements.
59
60
BIBLIOGRAPHY
1.
Harrington, R. F. "Matrix Methods for Field
Problems," Proceedings IEEE, Vol. 55, No.2,
pp. 136-149, February-r9b~
2.
Harrington, Roger F. Field Computation by Moment
Methods, New York: The Macmillan Company, 1968.
3.
King, R. W. P. The Theory of Linear Antennas,
Cambridge, Mass.: Harvard University Press, 1956,
PP• 286-312.
4.
Harrison, C. J. Jr. "Folded Antennas," Doctoral
Dissertation, Harvard University, 1954. Cruft
Laboratory Technical Report No. 193.
5.
Richmond, J. H. "A Wire-Grid Model for Scattering
by Conducting Bodies,~ Electro Science Laboratory,
Ohio State University, Tech. Report 2097-5.
6.
Mi ttra, R. Computer 'J.1echniq ues for Electromagnet ics,
New York: Pergamon Press.
7.
Strait, Bradley J. and Chao, Hu H. "Computer
Programs for Radiation and Scattering by Arbitrary
Configuration of Bent Wires," Scientific Report
No. 7, on Contract F19628-68-C0180, AFCRL-70-0)74;
September 1970.
8.
Strait B. J. and Kuo, D. "Improved Program for
Analysis of Radiation and Scattering by Configuration of Arbitrarily Bent Thin Wires," Scientific
Report No. 15 on Contract F19628-68-C-0180,
January 1972.
9.
Schelkunoff, S.A. and Friis, H.T. Antenna Theory
and Practice, New York: John Wiley & Sons, 1952.
10.
Ball, J.J. "Input Impedance of Closely Spaced
Dipole Antennas," Masters Thesis, San Fernando Valley
State College, Northridge, California, December 1971.
'
APPENDIX A
PULSE EXPANSION PROGRAM
61
62
The computer program presented in this Appendix is
applicable only for unconnected dipoles in parallel.
The
program will run with inter-connected dipoles, however, the
answers are incorrect.
63
PULSE EXPANSION PROGRAM
IN'rRODUCTION
The computer program described here is for the
calculation of Zmn as defined in equation 8 and the
subsequent solution of the current In as defined in
equation 7.
A unit voltage is applied over the segment
defined by N which automatically produces the admittance
at the feed point of the antenna just defined.
By
taking the reciprocal of the real and imaginary portions
of the current (input admittance) the input impedance
is calculated.
The program·is capable of handling
arrays of parallel antennas up to 100 segments without
alteration in the program.
If larger numbers of segments
are required to describe the antenna elements, the
,
i
~
dimension statements of line 1 of the program must be
increased.
The program is written in Fortran IV, G Level, for
operation on the IBM 370 digital computer.
To save time inputting wire segments into the
program, a special DO loop is included which will subdivide straight J.;inear wires into a specified number of
increments.
This is DO loop 16.
For a straight wire
antenna, the only input required is the start of the
X coordinate and Z coordinate, the end coordinate in
X and Z of the line and the number of increments into
.
64
which the straight wire is to be divided.
COMPUTER PROGRAM
INPUT DATA
Card Item
Descri12tion
Column
Format
1-10
8F10.5
in inches.
11-20
8F10.5
1-10
7I10
(feed point).
11-20
7I10
21-.30
7I10
1-10
F10.5
11-20
F10.5
21-.30
F10.5
31-40
F10.5
41-50
I10
1
FREQ
Start frequency in GHz.
1
DIST
Length of driven element
2
NW
Number of straight wires.
2
N
Number of segment at which
admittance will be given
2
MT
Total number of segments.
.3
xs
X-Coordinate of starting
point of first straight
wire.
.3
XE
X-Coordinate of end point of
first straight wire.
.3
zs
Z-Coordinate of starting
point of first straight
wire.
.3
ZE
Z-Coordinate of end point of
first straight wire.
.3
NINC
Integer number of segments
that the wire is divided
into.
65
Repeat card 3 until NW is satisfied.
The frequency range is varied in DO loop 30 by
changing the number of increments (M1 :::: 1, 15) and the
incremental frequency from .025 GHz to the desired
quantity in the line after 135 CONTINUE.
'l1 he
wire
radius is modified by changing the value in R after
Statement 40 from .008 to the desired radius.
The segment
at which the feed point of the antenna occurs is given by
N.
DESCRIPTION OF THE PROGRAM
The computer program is logically organized in
three major sections.
The first part of the program, after
the necessary data input, is comprised of subsectioning
all of the straight wires which comprise the antenna and
takes place in DO loop 16.
The program prints out the
number of wire segments, the mid-point coordinates of
each segment, its length and the coordinates of the
negative and positive charges on the segment, i.e., the
end points.
DO loop 100 are the calculations for the
Zmn matrix co-efficients and extends from approximately
Statement 36 to Statement 100.
Statement 300 to the
end of the program (Statement 1}5) is the solution of
the simultaneous equations by the Gauss Reduction Method
to solve for the currents in each subsection.
The program prints out the frequency in GHz, the
66
length of the driven antenna element in wavelengths, the
real and imaginary admittance at the feed point, the real
and imaginary impedance and the resultant magnitude of
the impedance.
A copy of the computer input and output is
presented in the examples given in Appendix B.
D fll. f N5! 0111 XP I 1 0 0 I • XM I 1 00 I 'XN I I 00 I oZP C1 00 I o ZM ClOU I oZN C100 I oVI( C1 00 I o
ltZI ClOOolOOI
10~ l L 11'10 ltl I 10'l o100
Pl•3ol4!~9?653~~
TPI•i'o•PI
10
12
13
RfAOI~tiOI
FQ:~oOIST
REAOI~t12l
~~tNt~T
FC'lP~'AlCM'lOoSl
r('P.~•ATI7ll0l
FQk~ATI4FlO,StliOI
)o1•0
DO 16 I •1 t'lt.'
Rf.AOI~•lll
~qiTEI&olJI
xs.xr,z~,ZEo~INC
X~o1Etl~oZEol
FNI•N:'IC
IF
0! e
IZ~.~T.ZEI ~0
• ~. • ( i. ( - Z S. l I C. ~ l
~,I'J
TO ;> 0
]7
~;
70
(,..'IT I'I'J(
••
1~
TO 17
J
~•llS-ZEI/F~I
.
l~:;.c,T,~EI
GQ
TO 22
~X·,~•IXt-JSt/r~l
GJ
21
24
T C (t.
C••.~•IXS-Y[l/~~1
CC.'IT 1-'•vt
1~ J•lt'll'l':
::J'l
~L'.'+
1
~rLLI"l•~o•C~X+CZI
FJ•i•J-1
r: cl<,r,T,l"l ::;::- Tl'l ?6
1'·'1 ,. ) :l!; 5. ~ ..... .) z
z ;..1 ( ... 1 & J ·~ ' v ) + ) :.
z· ~ c ~.. 1 s .~ .... ' t' , - :> z
G::l TO 211
2~
..
zut~J·l~-~J·~l
z:•('-'1•.'"1''1-':ll
Z'liVJ•i."I'-'I+Jl.
2ft
c:-.'I;T 1'1: rr
1r
IX~ 0 0ToXfl
,0 TO 31
X 'J ( ' ' ) • X S + .- .J • ) )(
Xr" ( Y.): XV P·!J +')X
X'il
v I•X" I '1.)-')X
<><l TO 34
31
XMI~I•X5-FJ•OX
34
X;:> I ~~ l • X''· ( )'. l - ~ )(
X'II f,l l =X II, I V l +()X
CO~H PI;JE
15
crJ·~r
r·•, v
<iRITt.l~oiOI
16
FMolNCMl
,z~~IMI tl~IMitXNIMioXI·:IMitXPIMloOELLIMl
!tt.....'E
CC'HI~IVE
W'liTE16t401
40
F01MATI/13Xt4MFREOt5Xo6HL/WAVEt4Xo4HCONOt6Xt5HSUSPTo5Xo~HY
1 t;H;::HAL, SX • !>liZ I '-'AGo SX o411Z·'1AG//I
R•oO:l8
DO 36 L•lo'1T
1110 L l•o. oo
36
CO:iTikvE
VKINl•-1•
MAG•5Xo
68
co 1 0 \11 • 1 ' 1 ~
C1•0,.,32l4 1 FIIrO
0:1 1 no l•l•'H
DO 10~ J•lo'n
RrO • C.O" T I l"l• ~~+I
H~ I I l -1. ''I J l l • • 2 +I X'o1 ( I l-X 'o1 I J l l • •11
FIPP • 50~ 1 ( l"l• f'+ ( lP ( I l-lO I J ll .. 7.+ t"X P ( I l-XP ( J I l ";>I
RP-Hl • 5(JRT ( '" r• +I VH I I -l'H J l l* •2+ I¥ N ( ll-XN I J II
I
R•;PoO,':;QT (rl•fi+(/.Nt 11-lPI Jll•*2•1XNI I l-XP(JII*'11
RPN • ~:;.r1 l I R• P +I l PI I I -/I'll J I l • o 2 +I XPI I l -X "'I J l l• • 2 I
C7•DlLLI II•GtLLIJJ•0·0~54•789~.116•FREO
XX '• • I J[ LL I I 1/ 2 • • ~UflT I I LJl LL I I II 2, I • • 2+R •R 11/R
C3•17.?/IFREu•Oa0254l
IF I lal~eJI GO To 400
IF IROOaLTa,~ll GO TO 200
••?
~1•SINIC1'~001'(2tPCO
IF IRr>r•,LT .,1'111 GO TO A~O
IF ll"l~~.LT.,~1l GO TO 800
IF IP~~.Llaal'lll 00 TO 720
I~
(RP~.LT,,~ll GO TO 740
A4•1-SI~IC1•1PPl/R~Pl+ISIN1Cl•NNPI/RNPI
A5•A4-151~1Cl*RNNI/RN~I+I51~1Cl•RPNJ/RPNI
GO TO 700
820
A~•ISIN1Cl
1
RNPI/RNP)+ISIN!Cl'RPNI/RPNJ-!SINICJ•RNNI/HNNI
A~•.:.3•Cl
GO TO 700
800 A~•I-SI~ICl•RPPI/PPPI+ISINICJ•RNPI/RNPI+ISINI~l'RPNI/HPNI
&5•t3•Cl
Gl'l TO 700
710 ·~·1-Si~ICl•RPPl/RPPI+I~I~ICJ•HNPI/R~PI-ISJNICl*RNNI/RNNI
A~<A) .. Cl
c.n To 1r,o
740 A)•I-SINICl•RPPJ/FIPPI+ISINICl•RPNI/RPNI-ISINICl•RNNI/RNNI
A5•A3+Cl
'
700 A2•A5 1 C~
Zl I oJJ•IIll .. A.?J/12.5664
(7•12./Dh.Lf lJI•t.LO<:IXX41
El•C0Sifl 1 K00J•C7/ROO
IF IQPPaLTo.~~OI GO TO 860
IF IQ~~·L'··~laJ Go·ro e4o
tr (~PN,LT,,~lOI G0 TO 47n
IF IR.'I"aLT.,'>l(ll GO.TO 440
(4•1-COSICl•RPPI/RPPI~ICOS!Cl•R~PJ/qNPJ
ES•E4~I(OSICl•RPNI/RPNl-ICOSICl•RNNJ/P.NNI
c.o To •so
860 E3•1C051Cl*RNPI/RNP!+ICOSICl*RPNI/RPNI-ICUSICl•RNNI/RNNI
ES•E3•E7
c,o T0 4 '!) 0
840 E3•1-COS!Cl+RPPI/RPP!+!COSICl•RNPI/RNP1+((0SICl*RPNI/RPNI
ES•f"I+E7
GO T0 4!>0
420 E;ai~COSICJ•RPPI/RPPI+ICOSICl•RNPI/P.NPJ-ICOSICl*RNNI/RNNI
E~•E3•E?
440
(,C) TO 450
E~•I-COSICl*RPPI/RPPI+ICOSICl•RPNI/RPNI-ICOSI(J*RNN)/RNNI
E5•E3+ET
4!10
f2~<E!>*C3
ZllloJI•IEl+E21/l2o5664
GO TO 500
f
r
1
I
l
ZOO
400
XXI• llJC:U.I l I/-;, +'..Of<T I ltJfLL I l I /Z, I ••2•fl•R I liP
(4•1Cl•6,'(l, 3•FRE0•·0?'j4J•Dllll I' I(!ELI IJI+Cl•C,/6,:?632
l I I • •I' L ( 4- c )/ 1 ;>.~I. 4. I I 5 !Ill c l .,, •., ' /~PI' , • I ~ "'~ (( !•RN"' II RNN
( '•. ( I;> t, (,.(,IF f) l Q •• 0 7 "· • c tL l. I I ' • ,, L CG I "X} ' '
XX7 • I l>fl L I l I I 4 , J • ~.c; fl T I fCl ~ L L I I I I 1 , I • • 7 + f'l• f!l I
Cb• I C3 I I 'i. 141 ~ • ort L I I J J J t AI OG I XX I I
XX 3 • 11> • ll • A l () r:.r • X I J I
C7 • I-(,;> 11, 1• F Q E C • ,I);>~'· • [,£ l L I I J • l XX~ •X X3 I • I C 1 . . 2 II
ZllloJI•C5•Cbt(7-C3•C051C!•RPPJ/16.28l2•RPP)
(,0 T0 5 00
XAJ•IDfLLI I 1/;>,+'.C.,TfiCELLI 11/2, I**~•P.•RIJ/R
C4•1Cl•676,J•FPEO•a02~41•D£LLI11••2-ICI•C3/6,28,,1
l I I ' I I • ('1. • ~ I .•; I ( I • R14 '' I • Cl I I 6 • 2 H3? • 'W" I
0•1 !25(.• 6•F"R(O•a0?~4•NLL I I I•ALOGlXXl I I
XX;> • I 0( l L I l 114 1 I • SCR T I I Dll L I l I 17, I • • ?+R •R I
Cf. • I - CV I ~, I •• 1 ~ • ::>( LL I l I I I • II L0 G I XX1 I
)(T)o(Do~•Aln~IXY.111
C7•1-6;>~.J•FRt0•,0254•DELllli•IXX2+XX,J•IC1••2JI
lllloii•(~+C6+C7•C3•CO~ICl•RN~I/I6e2832•~'~NPI
5 00 COl<!' iiiUE
100 cr~.rtr.vE
t-t.'\1 o \! T
00 30C) J•l·~~
llJo14~+1
Z!l
300
J•-VI(IJJ
Jol4'l•l I •Oo
CO~T
tr.•JE
NL • f.;.\•1
DO 1 0 ~ K •l • '< ~
U•liO::oKI
V•lll K' 0:: I
A"(r•U•u·~··v
I~
117
IA~GI
116•117•116
L•O:
118 COlli r.'<UE
L•L +I
IF ll -NN I 119 • 119 fi 2 0
119 U• ll L..• r. I
AVG•U•U•V•V
IF IA''GI 1?1tll8•i2l
120
122
kRITrl~o12ZI
roR~AIIl7H
GO T 0
121 DOllS
JNCON~ISTANCYI
13 5
J~l•NL
A~•liO:oJI
AC•lllo<oJI
ZlO:oJI•ll..tJI
lJIO::oJI=ZIIL•JI
ZlloJI•AR
ZIIL•JI:A(
12~
CCNTJNllf.
116
DO 126 LL•l,NL..
X•ZI'<•LLI
Y•ZJ.IK1LLI
ZIKoLLI•IX*U•Y*VI/A~G
ZIIKoLLJ•IY•U-V•XI/A~G
1.26 Cr>NT II·HIE
00 127 l•loNN
I' 11-KI. 128•127•128
II
70
1211 U•ZIIo(l
V•ll I I ,I( I
CO 1 7 9 .JN •lt NL
X •ZI I( t J~ll
Y•lf!KtJ~I
llloJ'li•LIItJ~r-III.•U-Y•Ifl
111 I•J~I•lllltJ~I-tX•V•Y•UI
121
1<'7 cr•p I'<..:E
10!1 c:;·,r 1:1Jt.
I I • :1
;,.r~LC.A~f~~)
.h~C'JliZIIIo'l"<•li . . ;>+Z
C,; • D I ::. 1 • ,- 1J t' 'jl 1 I • H0 2 I> r,
lllltNN+ll ••21
..
rd=l. t:: .r.~ l,
!l Y•Z I I I I • '•·'1•1 I
Z~L·~Y/IGY•~Y·~··~,,
ZI~•-OY/I~Y•GY•~Y•AYI
lY~G•~1~TIZUL•l~L+ZI¥•ZJ~I
1H
,;r. l TE I 6, l 0 I
C~·1 T I'IJ!':
~:.-::a•~"•c•
lO
CC"iTI!'tJE
ST';P
E-.:>
Fr• EC. t 0 «ol I II t N.._•ll t Zl I II t N.'ll +1 I t X tl ,L tll Y tZ YAr,
•.,2~
APPENDIX B
EXAMPLE OF PULSE EXPANSION PROGRAM
71
72
INPUT - Card data for Pulse Expansion Program (typical).
o.q~o
o.o
OoO
Oo4
Oo60
Oo60
Oo2
,
~.ooo
o.o
Oo4
o.,
Oo60
Oo20
Oo6
•
311
-<'·~
2·5
-2.5
-2.5
i'o!i
•2o5
i'e5
2·5
•2o5
2o5
2o!i
-2·'
15
2
2
15
2
2
73
Computer output of dipole example.
OU'I'PUT
_____N_
Q,O
!),•)
_,,.,1)'\CJ(I
l,<,o,\11)1)
I.:J'JijllQ--;>,;;.,;},)oi-· .:/.,11l)r·---.'.I1Jt•17
(),IJ
0,0
I),Q
-0-.~.-1~'1\
-.•.rr.:J·>o ..._~t.•!,n
o.o
o.o
n.o
o._,,:•.l~
-I, r, ill')\)
-l.\1,.-.1,·/
·-:,);1 I I
0.0
0.0
(j,l')
II,()
?,,,,.,~,Hl
-7.1'·'·1•'1
J,o')'),)i;'()-;:-l:.>·i,·,;~----\,1.! (d.
7
I,.()•,IV·(I
-1."r•''fJIJ
-1.'~'"\,~'1
t .•• uOI·O
··-r.Q1''J"'
-,J,II·'·'·7
e.,,,·o··r.
-0.11.1·'·7
-G.l.llH••1u
---,;-:u;i0'oin-::i·,-,i,-{;7,J'-:J;r\-)J(J(i-
l"·'~'O;JIJ
n.~ . u·1U'~
.
C1.•.~l-----~•Q
0,0
1),1)
.-,-,---·c,·:-1')
-r•.sv··,.-,o
o._l'
-n.IMhi·--·o.•J
o.tu.f.7
o.o
7.r'"lv·~o----o-:•.1-i>i,-:..~,:-··~H3
~-.,q'lP0--'),1'·'·'·'/
_ \_ __ ,,,, .. -·---·--
1),111'11
_____ 1.!.}'~'\"'-~-
n.o
u.o
o.n
o.o
n,j ______ c.Q_
(),·1}1'1'3---0.'>rJ>10tJ---O.O-----ll.O
'·''~~r.1
tl.t4'!31l
n.o
ll.O
'l,O
n.3
---iT.- ,-;-;-!i"tc--,-,,h'TI•r--i~ .:·r~)-;;-;J--,-;-1;;; ,:7'·-r,·:n------o~o----ri;o
tz.wH•,,o
1,11•'·'·7
1.:J~~3
t.'ioi-:JOO
r.o
o.o
o.o
- - 1 3 • .JJUl;,,---l.<oOJ,'j(l ----J,(.t.t.b7 -- i .B !13) ·-·o,,-;·----o.o
u.u-·
Jio,ll'I!'PJ
\,11,1'1!
I~'. •!•!0'1~--:,. !f,f
:',l(,l,f.(,
;>,<1(;•JQQ
0,0
Q,U
,;7 ----:; > :n-ij -- -?, o,llUO')
0. 0------ ·.
ci. i1
0,0
0 ~0
11, 33 1 J J
'),,,,,,
o~-J'iH.l--
r.,,1311
··'l.-]31~,-~'·"'',3'
o:•. '3-j-
•J.:n,,-,
o.
\133_-'),·ll:n·,
0, ~. 13 l1
·------------------------------------------- --------------·---r~
•u
li'/1\Vr
611, ---- .. --
f. '''lrlfl6---o·. i, r, 1~ 7
' • .., '··~('I
f).'·'·., \I
- - - i. ·-'·~6:,-,--u. ,.- ,,,-,,;,' • ',1'· f•IJ
··---- J.-, t:fJ
). 1?!
- - - - • • 1 •.
0 •'•''' 7r1
1: , ) - - ( } ••• · , · ;
')f)
').
~·)tr--·n.
~·,
•.ur ·\..-.
'·, ''"''·
) • ~ 'f r, l (j
I) • '.· "\:' (',1,
---f:~:)n-lr.--,;:·;~4ti4-
I. ,.,U!)cf
u • ..,,, .. " ~
1.:1<,~~
'),Sl~\1
•.
'\.~1~~2
· - - i ;-,·r:n y,J--1'): '"',r.. -;,,;
,.·~~J
---r;-4-4'i;'.;;;-(l, f, I O•f?
c .•:')': 1:\
..• r:1;'4'·
··--o. ()•),..-4~~
fl.,. 1 ~-'·'·
......;. vt·''·~
-'i'·· ,, •'·?0
l~.'l'1t.-~,,
\\.tll~'';
'''·7•''t.~
-JI.~c.7'•'?
·-;;·;~i(,~;T-
o,r.l'l7_
1:.r1 l'·'•
-v-. u~~.,-1·-i!--·u-~lq·.,., ~ ·- :. o. 1! u f 4-:..·L 'l.-., ~~ 7,y-:~.1'0~'11
-n.nv··:>"'
~~~rnt 1'~ ·· -0.''·)"•?1
.J.C'C',:";·7.
-O.P·l'•''"'
-~ ~r:-4·0·~--
.:.-t:: c........-;;_~(i:tic-;-.rt -t-..J ·1-;r~.·,; i Lli·, ~· i ~)-c~-1-·l·i-~7.·1·! t.-1
\'.!:"'·1::'1
(J, d\1.'1· 7
-!:l.c•.>'.tt.
-'). (•0:; 71\
n,n0;4~
-').~1144
o.~,~·"l"'·
·o. rw.t..
~
o.n0~~~
tl.(;n?J'l -:.·J.·uull«t~--,J .• (hL'f't'
~~n·•Ja~~~o:'?''..'•
.,.wn~
l-
7t.,l'·'·71
11.1~ 1 1.1
·7-,,'l'·'·'·'l
it·l.or·.t.') - )1.z~r;l')~
f\t,.Tl t?.')
n.n·,f ,. "6.4., ,,1 c.·>. ':!7't ..,n l J,t.r,Ol7t..''
tl.rl'. ,..,,,
'''· ~'2~()1
77•'•\'d''• 171.7.'.•?'· r
CJ.f\ll1P"" lflC. '1f)~''j7
tJl•.t,nt,4'). 1'•1. 1r,'".Jt.
l),rqq;
l\.,)11.,1
-::u.o.;t,'•"
.,.t:r,,~,1
n1~4l'li.''
1\."•'•""l'•
-·(,P ;.,,, .,.
-o.<•olf,z
tt7.7'•:,P'l 1'•'··~'•"").4 l.-i''·'41l'l
I J 7. \(. 7(,(, .I (,It oOIJllr, f'71 u. 'I> i ;:r,
l17.~P~74
1~1.471olD
Zl~.~~~~~
1'•'1.J'1!•Cit, 'Jl'.••:!'l'•-1.::!. 2"-?.:>:''t·"'
11>2.0l'l'i' 7V•,t.1'l)t, ~""•l17H
n,•JO\Iol,
o.l)·'ilt;.; rg.-e-9!>i?ib'';:n"9?TtT;-4Tl~il-
r
I
!
I
I
I
I
I
74
OUTPUT - Computer output for dipole with parasitic
element.
··· --o:u-···- -- u:o ---· ---:;·~\lico--i .·~iiJoo
-2.50000
-"J.~3)H
··-i'..lb067
-7.UUUtiU
0.0
0.0
0,0
0.0
o.o
u.o
-.l..lt.&bl
-l,!JJ33·~
),_Oilt..UO___-l.PiiB __ :-I•LI>t.'>l .. -l.'JUiJUO
... uuuvll -1.50CHlO -I.JH\3 -l.lt·M.l
!i.uuuuu -l.l1,1lt>7 -l.OUutJU .._.-_U.IlJBJ
t..Guvoo-·.:.o·.tl.lJJl ·:u.t.t.t.67 -u.5uouu
7,\IUUUU .-I!.'JJUOO
-U,J)HJ _:-_0.11,{,1,7
H.UGO~u
-O.l6LL7
-u.uUJUO
O.lt.Lt.7
9.ouuu~
o.l&bt.7.
o.JJ333
_o.5uuou.
lU.UJOUU
O.~Jul.lU
o.t.t.Lt.7
O.ll.lH3
ll.UJUuD
O,R\133
l.OOJUU
1,1(,1,(>7
--..j2-.I!·JUOO-l .lb<>L 1
l • .13 l 3J
I. 50JO~
lJ.u·~\lUU.
1.~.uuuu
l.&t.t,<>l
l.AI133
1 ... ~JOOO
1.B)JJ)
2.UOJ00
i.lbbbb
l,llllt'OU
-·-···i.lll.iUuO
0.0
0.0
0.0
O,U
l.lt..bt.7
2.313Jl
i.5Juuo
O.i'u<H·U
o.lllJJo
-i'..r•JJoJO
~.~Juoo
2
lt.,Ot10'JU -;>.•,oouu .. _:7.31.1D -.l.lt.t..bl
1 7.UJtJUu····-l.I•Jt,t.7 -;.u;,uuu···:I.UlJJl
I~ .UUUOO
-l,ll H ]I .. -I ,t.t,ul>l
-1. ~U•iUO
1~.ououu
-l.~lOJu ~I.JJJJJ
-l.lhh67
O.IUOclJ
-l.l~bb7
-I.U~JU0
-0.~11]3
21,.J .. uuu·
-O.O.lJH
-IJ,L:.•,t.7
-o.~_,uuil
-o.'Jo~uo
-~.11111
-u.l~~67
-Oolbt..t.1
O.lt.bb7
-u;cuuuo-u.lbt.t.l
7?.\Jvouu
---2l.OuUlJiJ
l4.0duJO
z~.uJOOt
u.~Jouo
2L,uUOOO
U.UJlJ)
Zl.uuuu~
t.thht.l
0 • .70000
u..'ullUU
L• .l.uuuu
O,lJGUll
30.JuJoo
l.~ouuo
2.l(.t.t>7
U.lOvvO
l.u.'~U(,
. 0. '"' lll7
t:; • ~ '> J I l>
1. V'•Oe\l
O. 4
1.u1~~o
o.~t~~7'>
I~ ~t.
u.zouvu
o.;Juuo
C.2UUU0
U.lOJUJ
0.2CUUC
C,lJJUJ
Q,j))JJ
O./uooo
u.?ovuo
u.21Juou·-.. o.1uuuc
O.ZOJUJ
iJ. 3HB
C.lOOt.IU
o • .l.l3H
0.33})3
0,/0UOJ
~.?OGOC
t.20UUO
u.zouuo
o.;uuuo
O.ZUJUO
o. J3H3
0, 33JH
O.HHl
l.CUJUO
t.3JJl3
_l,ILohl
O.lOJUO
O.?OJUO
o.2uuuo
o. 70000'
b.zoouu
c.zuuoo
0.3BH
l.~oooo
l.t.t.~L7
l.OJ133
u.zuooo
z •.Jl,l'H_ .... 7..!>tJJOO
U,JO.III•
v·•l0$&1.
u. u v H. u
IJ, 4'17L S
o.~JtJ35
._.uc£17
O. S I 'if,~
u.OUluG
1.17~(,0
·u.SJO<J~
v.out<>J
1,.lOI.tUU
- - f.l25LU
0.:0~3?3
1.2'>000
0.5~4~3
1.30000
0.!>6142
·----'·..3.!'.~.9~ _o._~o_cz
C.ZJU\JO
u,SJI33
I. IUUUll
- - l o l l SOU ---O~S967l
(). lOtJJ ~~
u.~~JJJ
1. I .l ~'Jlo
1.1'>000
---r.-z·, sou-· o: 57h-13
0,'\HH
U. 1J IH
O • .lnH
u,L6hu7
o.OOlfJ('
o. 1 •d'l',
0.~4224
1:.20000
o.zuuou
C. 70\Jclll
0.13133
u:··'>ll
I. OOGllC
U,JBH
O.JllH
O • .lH.lJ
0.333B
u.3HH
o.;uuuo
o.70Ju0
--.l<i:ouooo-1:ti J JJ:i--2·. ooojo--1. .1 •>~>'t.t. - -o. lvou\J· ··-···u. 2ouoc ·
2~.ooo~o
0,3333)
0.0
0.0
0.0
o.o
o.o
0.0
0,0
o.o
o.o
O,Q _________ 0.0
O.H333
0.0
U,O
O.J31l3
o.o
o.o
o.o
o.l3JJ3
0.0
0.0
O.o········0,3.l3H
0.0
0.0
0 0
0 33JH
:.i. 0 -----·0.0.. --·---o:o ·--·--0: lH33
u.o
o.o
o.o
o.nJH
0,0
0.0
0.0
0,3}JJ3
_u.u
o.u
o.o
0.11133
1~.uuuoo
~O.OJUUO
0.0
v.OOl Jll
o.v0llU
tl.zouoo
o.2Jvoo
SliSI'T
u,UIH·7
u. ~; 1 'I '•"
u.ul41'':
o. u 1., '' 'J ·
u.rJ \q•J~
-u. ~ 113/
O, 3JH3
O.HH3
3
l11.2<.:>1~:-b•l.lbt..74
<J. 1 c., 'I e 1 -4 'l. ~ ~ 3'• o
?.~t..711 l-J.4.eJt,)tl
_ u. 421 ;n ~·
1. t• 3A9~
u.u$1:.'1'1;
;?.u2u'J2 /t..,'•t-vl'•
u.v4uurl;
u. '>'I<J41 ,
O,lJIH
O.ZOU\)0
I ~-~·. ~ ~ ~ ~ 1~Ill: qt~l,q ~
0, vI l'o'l I
O.HH3
c.zoooo
llro!4G
[IH Ill
I~
u. 0 l I
c.zoouv
~ '). 'J
7 ·~ .. .,.
7t1, obI 41
su. 7l'db
£4,'1~l.l8
l.t.'l)/3
:-u. ~~bd~
2'/.UI&IY
-U,(i'l)J
u.JI'1Lii
S.l'lL'Iu~ S\.,73'117
51.JJ~t>2
-0.013441 o.ul.l~''l
R.'!luJ.i'
73.HB'>&
7J.8'>1'l4
-0,01031 1 v,Oiu40 t ll.79b41
<.5.307Ul
9b. ll>£22
!-u.JObJ7
u.oud4b'l 11>.74171 117.u4~'lu 118.lJ115
j
J.u\JI051-0.0J70'.>
o.uo71l
20.74'171 13:.1,82Uil
·o~c5J<:i4- -:;u·.i.i;J~>o7_ 1 _ o.vv~>1.,1 .2i,.e6'ltJ2 16u,llt.Jo~
.O.UO>Hl5
::-u_,OJ~32j__O,_Ou_539l2'l.15'ab2 183.30876
0.00017 '-0~0047ij O.OO .. ltl ·33.1>1102 l06,3058.1
. ~'.f:l_OO!)._i.:_ll!~~4_2_3 _ ~•.o_~io~'J
31J.4923Ci 23ll.,JL306
l40,3L229
lt.Z.774iJO
lt;'>.bll78
209.0)')4-8
.!3~!2116~
75
OUTJ'U'I'
o.o
t.Ooooo
7. 0(1000
],00000
4.00000
~
• OOQ(JQ
,, • 0'.1000
7.00000
6, nrJOOO
'1, O•JUUO
I o. o·JlloO
11.00()00
IZ.OO<lOJ
11.00000
-?.50000
-.?,)t,t./,1
-I.GJ11J
-7.J1)J)
-?,O<JJ(10
-l.~hh67
-I,!:;.){Hl•J. -1,3HJ3
-1.111~7
-I.O::JOOO
-o.c U3). -o,1.1.1,t>1
-0.50000 -0,33JJ)
-0,16667 -o.oo~oo
0.1~667
0.13133
o,.-,'HlO'l 0,1.!•61·7
0.~)311
),00000
J.lb667
J,lll33
-7.)6667
-1.01333
-1.50000
.. -lolb1•67.
-O.BJ133
o.o
0.0
o,qn11
1.16667
t.soooo
t.~ooou
1.666~7
1.~1111
l."l3D
2.16~~7
.?,00000
2.31113
2.11•1·66
.?.50(100
7.~0100
2.~0000
?, •>,IJOO
1o
)(,, 00000
0,20000
O,.?OD'lO
2, '•UUOO
o.o
~
r
0./:JUOO
o.zoooo
1. 'nJOOO
1.31133
I, 00000
0,61.1,(, 1
0. HJl3
I , )l,t.t, 1
o.n l1Jl
O.lUv()O
o. 71)11110
0,}01100
O,ZOOIJO
0.70000
0.7.0000
I '1, 0:10110
),lll~H
t.
71', O<IO::JO
I. '•:"JOO•J
1.16667
o.n
o.o
J,A}H)
l, I 61.1, 7
1?, 1)'!000
l.l. OIJJOO
('4. 00t100
0,0
7, I Lbl> 7
-2.'i0000
'll. 1)(1000
o.o
o.o
o.o
o.o
o.o
o.n
o.o
]
o. ?tJ<100
~•JlH)O
1.'>0000
1.13111
7,00000
Jl,OOIJ(I')
1 11. ')110'10
o.o
o.o
o.o
o.o__ _
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
0.0
0,0
0.0
0,0
-o.~oooo
I '•. 00000
0.70001)
2, '•OUOO
o.o
-0.16667
0.16667
0.50000
1~.00000
o.o
t>~.66
o. ':'•1000
0. 70CJ•JO
Ooi~61J7
0.?0000
0. ("JilUO
-O.JJ3H
-O.~OOJO
o.zot.oo
O.i''Jf·oo
-Q,bf,f-.~7
-O,fiHH
- 1•
-1.1~1>&7
-1.0DOQO
-1.3)}33
-1.~00JJ
-1.6~1.(>7
-1.633H
-?.00000
)1, 110000
-2. 333))
o.>oooo o.o
-.?,50000
37.. 00000 ·:2.50000-- -2.5()000
-2.1l·"'-'~
o.2;H:oo
-2.~ouoo
1'>.00000
2 ,, • 000~10
27.01000
21\,0,JOOO
2';,0JOOO
~O,I)UOOO
-·-·-
Computer outrut of folded dipole example of
Pulse
Expans1on
Program.
o.o
-7. •'-'OOO
.? • .-.oooo ·
. l
fll£0
I~ aoooo
1.07'>00
1. 0'•()00
1.07~·0·J
1.10000
t. 1 z~oo
1.1'1000
1.17~00
1.?0000
1.22500
1.75000
1.17~00
1~30000·
l.lZSOO
l.3't999
llH
o.~·1ooo
0.1(,(./,f
-o. 1~>1·'' 1
-o. •;oooo
-o.••Bn
(),11'1100
-I,RHB
-2.16667
l/IIAV(
tn•w
l:.o.oonoo
' -0, OlWOl
I
o. 4'·"n 1 -0.00')03
-o.ooao5
0 ·'• ~',40
O,lo 7ll•3
0,1, H?.l
0.41.~<19
-O.O::JJJ8
0 ·'• 1(,')1)
-0,00:.110
-0,00013
-O,OUJI7
-0.00021
-0.000?6
·o.~H
11
0.4'177f.
o.5oc3'>
O. '>I R94
·a. 57953
0.">4012
-o. ooo:n
_-0.00!)37
-0.00045
0.'561:'t0 -0.000'53
0.57189. -0.00063
"'o~~o;on
0.10000
0.70000
o.zoooo
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.o
o.b
o.o
u. :>oooo·
0.20000
o. ?t>ooo
0.70000
o. }0100
0, ?0.100
0.20000
o.:nHl
0,3HH
O.H.IH
O,H".ID
0,.131)]
o.Jnn
O.J~Hl
0,1HH
0,3BH
0.3l3H
0.33133
0,)))3)
0.331}3
O.JJJH
o.:nn3
0, 331H
0,)}"133
o.
lllJl
o. 13}33
O.ll"JOO ·
O,HHl
0,1313~
o.~oooo
o.zoooo
O,lJJH
0.2UJOO
0,3BH
O,JHH
0.'1:,\311
0,3BH
0.20000
0.70000
0.711000
0.70000
0.20000
C,7.0000
0.20000
-7.'5000:)
4
:.2.50000 -.--0.-20000
o: 10000
)!,(, ,,
7
-I.'>OJOO
StJSPT
0.00170
O,OIJ1PO
O,t>OJ 'II
0,00202
o.oon'+
o. 002?7
0.0(1241
0.00251>
0.00772
0.00290
0.0030?.
0,00331
0,00355
0.0038J
OoOO'tl4
0,;>\)IJOO
O.l'Jf100
Y_
~AC.
lllf AL
o. 70000
0.70000
o.zouoo
0.£0JUO
0. H ) ] )
0.20•>00
o.n~.n
o.ooooo
o:2oooo·
liMAG
o:oo110
-O,R088A-~HI.810~7
~87.01105
(1,0111
fll)
-5.1o0~3h-~~~-~9175
~~~.77~12
0. 0<.!1 '-II
-'1.~~~14-5?1,50342
511.~4106
o.oo;oz -ll. J'I7J7-4'14,01 343 '•'i".l'1~07
0,01)?14 -lb,AP606-4L6,002l0 4~1>.10786
o.uonr -l0,0172l-411,3U147 439,7bl96
0.00?41 -l3.00\)b4-413.7197.4 ~14.1">741
0. OOZ'\1, -25.bSAS7-1R9,)~430 38'1,99921
o. 00213
0,00291
0.00311
0,00333
0.003'>8
I
0,33JH
0.7J(J00
0,7JUl10
0,2JJOO
I
O.BJH
0,}0000
-o. 11·1·6 1
I!
-zq.o61lB-365.~4ll7 J&b,S27'~5 ··
-30.2027b-342.50b84 143.63569
-32o067AI-3ZO,Z3364 321.~357.1
-JJ.6l1'>6-Z?8,516ll 100,40454
~l4.9't953-Z77,Z9077 279,47192
O,OO:\A6 •35.6593R-7'>6.4h802 2Sg,9)'>06
0,00419 -JS,97485-Z35o9S52J 218.68192
I
I
I·
I
APPENDIX C
TRIANGULAR EXPANSION PROGRAM
I.
'
77
APPENDIX C
TRIANGULAR EXPANSION PROGRAM
IN'rRODUCTION
This computer program is the one used to solve for
the input impedance of the eight antennas described in
this report.
The computer program is a modification of
the program written by Strait 8 and was specifically
adapted to solve the two dimensional unsymmetrical folded
dipoles.
The program is written as a main program with
two sub-routines, one of which (CAL Z) is used to solve
for elements of the Z matrix which are·given by equation
14 in Chapter II.
The program is written in Fortran IV,
Level G, and has:been adapted for use with the IBM 370
batch processing.·computer.
The program is capable of handling any three
dimensional wire body configuration but the input format
is set up for two dimensional figures such as the flat
dipole antennas of this report.
The input format has
been modified in DO loop 50 which uses straight wire
segments and sub-:divides them to determine the point
coordinates required as input to the program.
Originally, the individual coordinates of each
point were input 1rather than allowing the computer to
solve for the point locations on each straight wire.
Caution is required when using the input format of DO
I
78
loop 50.
The final straight wire segment of each
designated wire must be extended one increment to allow
the program to calculate the final point on the
designated wire.
With reference to Figure.3.5, it will
be noted that the first wire has three straight subwires, the third straight wire has a dashed line and a
point extending one segment past the end of the wire.
The coordinate of the imaginary extra point would be the
end coordinate for the end of the third straight wire of
designated wire No. 1.
COMPUTER PROGRAM
INPUT
Item
Description
1
NW
Number of continuous wires.
1-3
I3
1
NP
Total number of points.
4-6
I3
1
NR
Number of different radii.
7-9
I3
1
NSW
Number of straight wires.
10-12
I3
1
FREQ
Start frequency in GHz.
13-22
F10.5
2
xs
X-Coordinate of starting
1-10
F10.5
11-20
F10.5
point of first straight wire.21-30
F10.5
Card
Column
Format
point of first straight
wire.
2
XE
X-Coordinate of end point
of first straight wire.
2
zs
Z-Coordinate of starting
79
Card
2
Item
Description
ZE
Z-Coordinate of end point
of first straight wire.
2
NINC
Column Format
31-40
F10.5
41-50
I10
1-3
I3
4-6
I3
Integer number of divided
wire segments.
Repeat card 2 until NSW is satisfied.
3
LL(1)
Starting point number of
first continuous wire.
3
LL(2)
Starting point number of
second continuous wire.
Continue inputting the starting point number of each
continuous wire until NW is satisfied, with format
20I3.
L~
LR( 1)
Point number at which radius
change occurs.
1-3
IJ
Continue inputting the point at which radii changes occur
until NR is satisfied, with format 20IJ.
5
RAD(1) First radius encountered on
first wire.
1-14
E14.7
Continue inputting the next radii encountered, with
format 5E14.7.
In order to change the frequencies and interval over
which the antenna impedance is calculated, DO loop 30 must
be modified and the number of frequency steps IVI1
= 1, 9
modified to give as many frequencies as desired and the
85
LAIJI•LA.I .. AJI
LAILAJI •O.AJ
I' CJ•LAIJJJ 14tlt14
I Jl•Jl+N
t CONT"INUI
REtURN
END
',
86
~U9ROUTJNE CALl INtNloZI
Zl9999ltP~II2000!tUtUl•U2tU3tU4tU~tU6
CO~~ON XXI~O~ltXYI~OOitXZI~OOltTXI~OOlaTYI~OOJaTZISOOJtALI,OOI
COMPLfX
Cn~MnN Tl999!tTPI999lt8Ka~AD2110ltLI,OltLRI101
Dl~fNS!ON OCI20001
.
U•IO.•lol
PJ•Jol4l~93
[TAr3n.730'7
Cl•ol25/PI
C2•o2~/PI
Jl•l
J2•-2
DO l J•ltN
IFILIJll-JJ 3t4t3
4 J2•J?+2
Jl•Jl•l
3 JJ•IJ-11•4
J4•J3•1
J!>•Jio+l
Jb•J~+l
J7•Jl'>+l
K4•J2+1
K!>•K4+1
K6•Kl>+J
K7•Kb+l
~l•ALir.4)+.ALIKe,)
~~·ALI~6l•~LIK7l
TIJ4l•AL1t4l•o~IALIK4li~J
TIJ.,l•ALIY~l•IALIK4l•o~•ALIK~I)I51
TIJ6l•ALI~6l•IALIK7l+a~-ALIK6ll/$2
TIJ7l•ALI(71•,~•ALIK'7115Z
TPI Jlol•ALI>:411.51
TPI je,)aALIK.,l/51
TPIJ6l•-A~IK6)152
TPIJ7l•-A~IK7!152
J2•J2•2
1 (0.'0 IIIUE
ll3 •U•B>t • E'l A
Ut.•-UIBK•FTA
.
·'
!1>::2•11Y.•BICI2o
BY. 3:8K2•BK/3o
N9•0
N2•1
NO•l
N3•-2
DO 10 NS•ltN
l~ILIN2l-N51
11 K>t•l
12t1lt1i
N3•N3+2
N2•N2·•1
(,0 T 0 13
12 Kl(•3
DO 14 NF•ltNl
N4•NF+Nl
N!>•N4+Nl
N6•N5+Nl
DCINFJ•OCIN!>!
OCIN1tl•D,cJN61
i
J