The objectives outlined in the original proposal for this project were to create a well-documented computer analysis model based on the finite-difference, time-domain (FDTD) method that would be capable of computing antenna impedance, far-zone radiation patterns, and radar cross-section (RCS). The ability to model a variety of penetrable materials in addition to conductors is also desired. The spiral antennas under study by this project meet these requirements since they are constructed of slots cut into conducting surfaces which are backed by dielectric materials

Beggs, John H.

Luebbers, Raymond J.

Penney, Christopher W.

English

NASA Technical Reports Server

NASA-CR-192237
Finite Difference Time Domain
Modeling of Spiral Antennas
PROGRESS REPORT
Covering the period of
August 15, 1991 to March 25, 1992
for
Grant NAG-l-1332
submitted to
,O
r._ r0 o
t ,.- _0
m U
Z _ 0
r,j
0
Mr. Fred Beck
GCD, M/S 490
National Aeronautics and Space Administration
Langley Research Center
Hampton, VA 23665-5225
and to
Captain Doug Havens
AFEWC/ESAS
San Antonio, TX 78243-5000
submitted by
Christopher W. Penney, John H. Beggs
and Raymond J. Luebbers
The Pennsylvania State University
Department of Electrical and Computer Engineering
University Park, PA 16802
submitted on
April I5, 1992
https://ntrs.nasa.gov/search.jsp?R=19930009397 2020-03-17T07:35:08+00:00Z
I. Introduction
The objectives outlined in the original proposal for this project were to create a well-
documented computer analysis model based on the Finite-Difference, Time-Domain (FDTD)
method that would be capable of computing antenna impedance, far-zone radiation patterns, and
radar cross-section (RCS). The ability to model a variety of penetrable materials in addition to
conductors is also desired. The spiral antennas under study by this project meet these
requirements since they are constructed of slots cut into conducting surfaces which are backed
by dielectric materials.
II. Background
The modeling approach chosen for this effort is the based upon the Finite-Difference,
Time-Domain (FDTD) method. FDTD is a numerical, volumetric method of solving Maxwell's
curl equations in the time domain. It is especially useful for modeling geometries which involve
complex shapes and/or dispersive materials. The spiral antennas under study by this project
contain curving surfaces and dispersive backings and thus FDTD is well suited for this problem.
The reader is assumed to have a basic knowledge of the FDTD method, however several papers
detailing the basic theory are provided [1]-[3].
There was considerable interest in frequency independent antennas in the 1950's which
led to the development of the spiral antenna. The original idea of an equiangular spiral antenna
was proposed by Rumsey [4]. He suggested that an antenna that was specified only by angle
could be independent of frequency. A change in frequency would lead to a change in angle, or
rotation about the spiral. His original proposal was for a spiral with arms of infinite length.
Future work done by Dyson [5] and Curtis [6] found that arms of one and one-half to two turns
(or 540 ° to 720 °) were necessary for an operation. The bandwidth ratios possible for these
antennas can be calculated using a simple formula relating the outer diameter of the spiral to the
separation distance of the arms at the center of the spiral [7].
There are two designs of antennas under study by this project: one with spiral arms
formed by thin slots cut into a conducting disk, and one with expanding spiral arms. These two
styles of spirals are shown in Figure 1. Each arm of the antennas is described by two equations
for the radius, one for the inner surface of the arm and one for the outer.
For the spiral slot design, the equations for the radii of one arm are as follows.
rl = roe_ (1)
r2 = rl + y (2)
where ro is the initial radius, ¢ is the angular position, and y is the thickness of the slot. The
parameter a is the flare (expansion) rate described by
a - z (3)
tana
where ct is the angle between a radial line and an arm of the spiral as shown in Figure 2. The
second arm can be formed by rotating the first arm by 180 °. The equations for the inner and
outer radii of the second arm are
re = r3 + 7. (5)
The equations for the radii of the arms for the second type of spiral are shown below.
For the first arm, the inner and outer radii are
rl = roea_ (6)
and
r2 = roeaC, - 8>. (7)
For the second arm, the radii of the first arm are rotated 180 ° as shown below.
r3 = roea(_ - n) (8)
and
r4 = roe_C_- _ - s_. (9)
The parameters here are the same as for the spiral slot antenna with the addition of the parameter
6 which is the rotation angle between the inner and outer surface of an arm. The rotation angle
is typically chosen as 90 ° since this creates a symmetrical case which is known as self-
complementary. The self-complementary spiral has matching impedance between the metal arms
of the antenna and the cut-outs from the antenna. This condition leads to the best pattern over
a broad frequency range.
III. Modeling Approach
FDTD allows objects to be modeled in several different coordinate systems, however it
is most convenient to use the cartesian system. This leads to three-dimensional cells composed
of materials specified by the modeler. There are two possibilities for modeling the spiral
antennas: full cell in which the minimum cell size is the same as the slot width, and subcell in
which the slot size is much smaller than the minimum cell size. The full cell approach is more
readily implemented because it uses a full FDTD cell to model the smallest features of the
antennas (in this case the slots). This approach forces the cells to be smaller than required by
the basic FDTD constraints on cell size so that only relatively small antennas can be
accommodated within available computer memory. A subcell model allows for larger FDTD
cells since it models the arms of the antenna as structures that are smaller than the minimum cell
size. The subcell method requires a modification of the FDTD equations and the geometry input
program. The procedures followed to construct a full cell model of a spiral antenna using an
FDTD code written at Penn State are discussed below. The procedures for subcell modeling are
discussed in the future work section of this report.
The first step taken in full cell modeling of a spiral antenna in FDTD cells is to set a
planar circle of FDTD electric field components to perfect conductivity. The spiral arms of the
antenna are then formed by resetting the field components of certain cells in this disk to free
space. These cells that form the arms are found by calculating the radius of the arm at a given
angle using either equations 1, 2, 4, and 5 or 6, 7, 8, and 9, depending on the antenna type. The
calculations of the radius are performed for all angles from zero to the maximum angular rotation
of the spiral which is determined by the number of turns specified by the programmer. If a cell
lies between the inner and outer radius of the arm, the electric field components of the cell are
set to free space.
The antennas have two different types of dielectric backings. The first type is composed
of layered slabs of different complex permittivity, the second type is formed by a single dielectric
slab with a cone cut from its center. Either of these types can be created in FDTD cells by
setting the electric field values of the cells inside the dielectric to the corresponding values of
permittivity, permeability, and conductivity. The backing for the dielectric with the conical
cutout is shown in Figure 3.
The geometry-input code is written so that the user is only required to specify the design
parameters of the spirals, the program will construct the antenna in FDTD cells. The parameters
that must be specified by the user are: the radius of the antenna, the expansion rate of the arms,
the center separation between the arms (the initial radius), the number of 360 ° turns in each arm,
the dielectric backing style, and the type of antenna. Once the type of antenna is determined,
the user must then specify either the slot width or the arm rotation angle.
IV. Progress to Date
Initial specifications for the spiral antennas were provided by Mr. Ed Utt of Wright
Laboratories at Wright-Patterson Air Force Base, Ohio. These specifications were for prototype
spiral slot and self-complementary antennas of 2 inch radius. Using this information, a geometry
input program was written using the FORTRAN programming language. This program was then
integrated into the Penn State FDTD computer model. In the geometry input program,
subroutines were written to perform the commands which construct the spiral antennas in full
FDTD cells. The user of the program needs only to fill in the parameters of the antenna
mentioned in part III above to construct an antenna in FDTD cells. All of the design parameters
which must be set by the user are located in the same section of the code and are thoroughly
commented. The geometry input code for full cell modeling has been completed and verified
by graphically displaying the antennas created.
Research is currently being conducted to develop a subcell FDTD model which will allow
larger antennas to be modeled. Accurate wire antenna impedance calculations using a subcell
modeling method based upon Faraday Law contour modifications to the FDTD equations have
been demonstrated [8].
V. Future Work
In order to test the validity of the FDTD calculations, the full cell modeling method will
be verified by comparing the results with measurements of the radiation patterns, impedance and
RCS which will be obtained from the liturature and possibly from measurements taken at Wright
Laboratories. The FDTD code will then be modified to allow for subcell FDTD equations and
a subcell geometry input program will be developed. This will allow larger antennas to be
modeled in FDTD cells. The results obtained using the FDTD subceil method will then be
verified.
,.
,
.
°
,
7.
8.
References
K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's
equations in is.tropic media," IEEE Trans. A. & P., vol. AP-14, pp. 302-307, May 1966.
A. Taflove and K. Umashankar, "Finite-Difference Time-Domain (FD-TD) Modeling of
Electromagnetic Wave Scattering and Interaction Problems," IEEEA & P S,c. Newsletter,
pp. 5-20, Apr. 1988.
A. Taflove, "Review of the formulation and applications of the finite-difference time-
domain method for numerical modeling of electromagnetic wave interactions with
arbitrary structures," Wave Motion, vol. 10, Dec. 1988.
V. H. Rumsey, "Frequency independent antennas," 1957 IRE National Convention Record,
pt. 1, pp. 114-118.
J. D. Dyson, "The equiangular spiral antenna," IRE Trans. on A. & P., pp. 181-187,
April 1959.
W. L. Curtis, "Spiral Antennas," IRE Trans. on A. & P., pp. 298-306, May 1960.
J. D. Krauss, Antennas, Second Edition, McGraw-Hill, New York, 1988.
L. Chen, T. Uno, S. Adachi, and R. J. Luebbers, "FDTD method analysis of a monopole
antenna mounted on a conducting rectangular box", Accepted for presentation at IEEE
AP-S International Symposium, Chicago, Ili., July 1992.
Figure la: Spiralslot antennamodeledin full FDTD cells.
Figure lb: Self-complementary spiral antenna modeled in full FDTD cells.
a_
p :ke
• ' " i _.I ! , //\ )_, z
t I II '1, ),: "I I . an:tan Q
/ / /\
Figure 2: The definition of the angle a. This is a conical projection of the planar spiral slot
antenna.
• ll,l,l.il,l.a,i, i
"llllilmll
I ...... I:
' lzl|l!
......i!"::'Llll |![111 IIII
.... itr,,!11 I1|IIII |!11
1111111111
L.lillllill
Illllllll
II1111111
_,111111111
[1111|111:1
_! I I I I I I1 1
_.111111111
i i i i i I 1 i ! i .
[.111 [_ IIIII Ii
.... ::"'J'" ;_ ''"ti:"
_l"lll:l|I,, . ,,,,,,I",,
_lll|lll II111111,,,_l| ,,
......... |,, :fill .... .
_ I I 11 111 I I11 I J 1 J[_l[J_
]1 [1111|11if":: ::: .........."fill||,, i: .........
_1 J J J J J J J ] [ _ J J J J I ] J J [ J J J J I.J ].)
........... 'ii.....,....""'..... .....
:1 I I I !11 Iltl
ll[llllll IIIIIIl]Lllllll[lllq[IIIIII!11 I!11111
[111111111 tlllLIIliIII-IIIIIIIB
_IIII11111 IIII!1II IIIII1_111111
I,I [l]_] i[lllll,l[lllllllllll.l-I'l.l-l'
1 1 I I I ! i I I I I I I I I [ I I I I ! I 1 I I i I I I I i I
II{l:llll|lllllllllll-
:|::::ll: i_ |,,,, ...................
llllll:l| |"':l .....................IlliliilllllL]iiI|11
' ,11111111[11|1111111,111111
i_::::II:",, j .............. _,,,, ,,
.............. Hll, 1 I111I11 IIIILIII-I
[ILLIIIII I Ill IIIlllllllllllllll]
....... 11 ,,,1,,,, ,,,,,,,,,, ...... H
:!i-IIii111 i; lllilll-Illll,l.lllllll'i-II.lll :I !i
I,I-IIIlIllllIIIIlI{]IlJILIIILII_ILllllllllllllllll Illlllll
__l:|'l||:l:
,...Jti.................. .....
......................... lll:lll .... :_llllllilllllllllllll I| II IIII
._.Jillllllllllllllllllllillllillll
!.l,l,lilllllllillillllllllllllllllll[* ii1.11111111111111111,1-11|1111111
iiii11111111111111111111111111111
I.IIIIII11 II II I|
1||1I11 |1II[ i I I
..... :l .... fill llllll:111111IIiil 11
!,lllllll|illlllilllll|llllllll]lll
IIiiiiiiiiiiii11111111111111111111
I-I11111111111111111111111111111111
i-iIIIIIIIIIIIIIIi11111111111111111
!llllllil]lllll [JIH._ILIIJLiI_IIIJ[J-
-IIii1111111111111 I1'111 IIii
Figure 3: Cross-sectional cut of a full F'DTD cell model of the conical dielectric backing.


Finite difference time domain modeling of spiral antennas

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server