I
i!
ii
i,
i
]!
11
THE VIVALDI ANTENNA AS AN
ELLIPSOIDAL REFLECTOR FEED
i;'
FOR MICROWAVE HYPERTHERMIA
!|
by
i
i John Michael Vanderau
1 B.S., Ohio State University, 1979
l[
1
A thesis submitted to the
Faculty of the Graduate School of the
(I
University of Colorado at Denver
j! in partial fulfillment
l[
i'' ,
of the requirements for the degree of
Master of Science
|j Electrical Engineering
i,
jj 1992
I ('**'**$ -
This thesis for the Master of Science
degree by
, John Michael Vanderau
has been approved for the
Department of
Electrical Engineering
by
Edward Wall
February 5th, 1992
Date
jl
Vanderau, Jphn Michael (M.S., Electrical Engineering)
I
The Vivaldi! Antenna as an Ellipsoidal Reflector Feed for
i
Microwave Hyperthermia
Thesis directed by Professor Jochen H. Edrich
ABSTRACT
The
paraboloidal
ellipsoidal reflector, unlike the more common
reflector, possesses two focal points, lending
itself to [thermal imaging and hyperthermia treatment
i1
h
applications. Whereas typical antenna feeds are relatively
i .
I,
narrowband! and present significant aperture blockage to the
i
reflector, jthe Vivaldi antenna is a broadband antenna that
presents a
plane of
physically small cross-section to the aperture
the reflector. Two trial antennas were
in
!i
|i
designed add built and their performance ascertained. The
j1
narrow beamwidth, high sidelobe and backlobe amplitudes
typical of | S the Vivaldi antenna restrict its application to
|i'
ellipsoidal .reflectors with longer focal lengths.
This abstract accurately represents the content of the
candidate's
thesis. I recommend its publication.
Signed
Jochen H. Ed rich
IV
CONTENTS
CHAPTER j
1. ! Project Goal...............
i
2. Theory of the Vivaldi Antenna
3. | Design of the Vivaldi Antenna
4. j Summary and Conclusions ...
APPENDIX.!!,.............................
i,
REFERENCES..............................
i1
. 1
. 5
19
42
48
51
Chapter 1
Project Goal
An ellipsoidal reflector, with its dual foci, lends
!L
M
itself to hyperthermia treatment and imaging thermography
i *
i.
applications by its ability to detect energy from or deliver
energy to a secondary focal point. Milligan [1] has analyzed
such an antenna system developed by Edrich [2],[3], but since
i i
j i
the electromagnetic energy is focused to or from a
secondary
(field) focal point located coaxially to the
primary feed focus, a significant portion of the microwave
i
energy can! be blocked by the reflector feed, as illustrated
in figure 1i!
1
The diameter of the reflector is 35.8 in. The distance
from the vertex of the ellipsoid to the primary feed focus is
I
18 in., subtending an angle of 126. The distance from the
vertex of the ellipsoid to the secondary field focus is 29.25
in.
Figure 1. Ray paths
illustrate aperture
blockage.
An additional con-
straint is the operating
bandwidth of the current
reflector feed. To obtain
a broad bandwidth, mul-
tiple feeds are used and
by neccessity are offset
from the primary feed
focal point, leading to a
degradation in spatial re-
solution and sensitivity.
2
A desirable feed would offer a multi-octave
i'
bandwidth l! while simultaneously minimizing aperture
blockage. A particular interest is the capablity to operate
from 3.7 to 4.2 GHz and 8.0 to 12 GHz. In addition, 31 to 33
GHz is of interest, however, available resources were
;l
incapable of operating in this frequency band. Consequently,
the direction of this research was to develop an acceptable
K
antenna feed covering the lower two frequency bands.
, i
Complementing the previously cited attributes, a
desirable feed for this reflector would also be of short
i
longitudinal length (there must be sufficient clearance
between the two focal points to accommodate both the
antenna feed and the target region of interest). Other
desirable traits would be a 126 beamwidth between first
nulls, maximizing illumination of the reflector while
3
ll'
simultaneously exhibiting small spillover losses and
negligible side- and back-lobe radiation; the latter two
,|1
traits being a requirement for good spatial selectivity.
, |
The Vivaldi antenna described by Gibson [4] was
i.
thought to1; be a viable candidate to enhance the overall
performance of the ellipsoidal reflector antenna system.
Its attributes and its application are fully discussed in the
following chapters.
i
4
Chapter 2
Theory of the Vivaldi Antenna
The Vivaldi antenna is a quasi-frequency-independent,
planar antenna structure. It is a travelling wave antenna
which exhibits an end-fire pattern. Being a quasi-frequency
r
independent; antenna, its performance (in particular, its
beamwidth) is essentially constant with frequency.
r
The nature of the Vivaldi's frequency independence
I
can be seen by considering that, at a given frequency, only a
small section of the exponential will radiate efficiently.
i '
At transversal spacings less than X/2, the energy is tightly
5
bound to the conductors. When the spacing approaches X/2,
the energy, more readily couples to the radiation field. As
i>
the frequency changes, the wavelength is scaled and a
ii
different section of the antenna radiates, it too, being
scaled while maintaining the same relative shape.
i'
The directivity of the antenna, as with other
11
travelling-wave antennas, is characterized by an
11
approximately linear dependence on its LAc ratio, where L
I -
is the longitudinal length and A,c is determined by the
spacing of the elements at the tips (Xc/2). Typical numbers
for directivity are given by
D 1OLAc (2.1a)
3Xo
6
I
Gibson's research not only indicates that the E- and H-
plane beamwidths are approximately equal at a given
I.
frequency, but that both the E- and H-plane beamwidths
i
remain essentially constant for 3 < A,c/A. < 12, where the
maximum separation between elements (tip separation) is
kc/2. I
The equation that describes the exponential taper of
the Vivaldi iantenna is [5]
i;
y =
+ 9 e+x
2
(2.2)
where y is the lateral displacement between the radiating
r,
elements and the axis of the antenna, g is the feed point
gap, x is the longitudinal displacement along the antenna
7
from the feed point, and a is the exponential scaling factor
(a > 0). Tlie geometry is depicted in figure 2.
+Ac
4
1 *~*s**sSs|sÂ£i|s|SÂ£|s|||||s|!|||s|||||s|
Figure 2. Coordinate
system. j:
While heralded as a
broadband, frequency-
independent
antenna,
there are (as with log
periodic antennas, for
-Ac example) practical band-
width limitations deter-
mined by the minimum
(feed gap) and maximum
(element tip) separations. The lower frequency limit is
found froml, the relation
kc
ymax = i 4
(2.3a)
8
II
where Ac is the wavelength at the lower cut-off frequency
!
and ymax is the maximum lateral displacement off axis.
; i
Similarly, the expression
^ = ^ (2.3b)
2 4
'i
defines the upper cut-off frequency where again, g is the
separation 'at the feed point.
i
The analysis of the Vivaldi antenna proceeds as
follows: ! First, the finite-sized antenna elements are
i.
assumed to be infinite in extent, that is, not truncated
above and below the regions where |yl < nor beyond the
regions where x < 0 and x > L. With this assumption, the
i
antenna problem is reduced to finding the field distribution
!i
within an infinite tapered slotline. Once the electric field l
l
i
9
within the1 slotline has been determined, an equivalent
, i
magnetic current source M is defined by the relation
i
M = -n x Es
(2.4)
;l
where n is a unit vector normal to the E-field in the slot.
i1.
i
The superscript "s" is used to differentiate between the
I!
field that exists within the slot (the near field) rather than
the far field pattern radiated by the Vivaldi antenna.
The radiated far fields are computed by taking the
11
Fourier Transform of the magnetic source distribution in
the aperture of the slot using the relations
!
(2.5)
10
l
and
I.
I!
! I
11
j
E = -1V x F
E
(2.6)
Figure 3. | Staircase
approximation of the
exponential taper.
i.
11
The exponential
taper of the slot is
modeled as a "staircase,"
as depicted in figure 3.
The slot fields are repre-
sented as a sum of
weighted basis functions
and conservation of
power at each step
junction is enforced as
the necessary boundary
condition.
11
JanasWamy [6], [7] has computed the transverse
it
electric field in the slot. He expands the transverse E-field
component1 as a finite summation of weighted basis
functions as
i
Ey Â£ anr> (2.7)
The basis function chosen for the approximation is
li
i;
(2.8)
where Tzn are even-ordered Tschebyshev polynomials of the
first kind, \ W is the maximum tip separation (W = 2ymax
i
where ymax is given by equation {2.3a}), and y is the lateral
12
displacement off axis (see figure 2). Although many
I
different basis functions could be selected for the analysis,
Janaswamy states in [6] that choosing equation (2.8) for the
basis function affords a more efficient numerical
computation. The transverse electric field Ey is given by
M2 is the finite number of basis functions employed and the
1'
other quantities are as previously defined.
j (
Since, conservation of power P across each step
I
junction is I a postulate of the analysis,
1
13
I
Vp-|Ey(0)|- W
= constant
(2.10)
Ey(cc) is the Fourier Transform of EÂ£ Z {, is the charac-
teristic impedance of the ith. section of the slotline, and
the expans|on coefficient of the first basis function of the
l
itb. section! is given by a\ (associated with the dominant
mode of the slotline; the higher-order coefficients aj, n>2,
are associated with evanescent modes occurring at the step
junction). The Fourier Transform quantity Ey(a) results
from formulating the problem in the spectral domain, a
technique introduced by Itoh [8].
i
i'
The ,,spectral domain technique uses the Fourier
r
Transform of the usual space domain quantities to reduce a
I
system oft coupled simultaneous integral equations to a
14
simpler system of algebraic equations. This method is
briefly outlined in the Appendix.
i
i
Letting Ey(0) = ai and including the phase factor of
i>
the wave propagating down the slotline, equation (2.9) is
i
written as
where once again the superscript / is used to denote the isb.
section of the staircase approximation to the slotline taper.
i
From equations (2.11) and (2.4), the equivalent
magnetic current is therefore
15
I
The z-direction is, referring to figure 2, directed into the
n
page. Janaswamy [6] then evaluates the far field to be
l
E = tee + 4E$
(2.13)
Ee and E$ are the sums of the individual contributions from
each stepped section of the slotline, that is,
N
Eo= Â£ El (2.14a)
N
(2.14b)
i=1 I
I
16
i
Without derivation, the final
jl
11
far fields is
I1 .
result expressing the radiated
Ee = Ea(koWicos 9)
|c-i^L[F*(uL)-F*(ui)1
sin 8 I Vc' sin 0
+ pe+i kndfCub-F(uj) j l
Vc' + sin 9 *
(2.15a)
and
4 = EL(0){- :e'Jtoc- -[sin + (F(pj,)e*jv" F(p^e*M)
'c' + COSU
+ sjn |V2(cM) IF* (qi,)- F* ($]]
. jj ^JfcocL r|n ^F(p},)e-Jv-F(p^e-Jv')
|C' COS <|)
- sin iV2(c'+1)[F*(4)-
(2.15b)
where the Various quantities in equation (2.15) are given by
d = (^e.)
^ insertion
k ~Â¥
a-o
I uU= 0>
Uh,(= k^/d+sin 9)
Vhf= kox^/d+cos )
i kox'h./c'-cos <|>)
I' qh,/-=koxUcLl)
V i,/=koxili/(d+1)
Ph,/-* koxj, /l+cos) (2.16)
and xj and ){, are the lower and upper coordinates of the itii
section. 'The function F(Â£) is the Fresnel Integral.
Reiterating, the sum of all the individual E Â£ and E^'s define
!i
the total field and are given by equations (2.15a)-(2.15b).
Reference [6] provides a sample graph of Vivaldi antenna E-
II
and H-plane patterns based on these equations and
experimental measurements. Reasonable agreement
between computed and measured patterns in [6] is apparent.
i
18
I
Chapter 3
Design of the Vivaldi Antenna
I! 1
As indicated in Chapter 1, the design criteria dictated
a lower cut-off frequency of 3.7 GHz. For this particular
I;
application,! 3.2 GHz was chosen for the cut-off frequency,
which afforded a 500 MHz "guard band." The intent of this
il
500 MHz frequency margin was to ensure that the radiated
wave would couple reasonably well to free space before
11
striking the open circuit discontinuity presented by the
i .
tips of the radiating elements of the waveguide structure.
The maximum tip separation is l
l
19
d = 2 | ymax |
= Xc/2 = c/2fc (3.1)
= 11.803 in/nsec + (2 x 3.2 GHz)
= 1.84 in
r
i'
As was also stated in Chapter 1, a 126 beam width
was desired for this application. Reference [4] provides a
i
design curve of experimentally determined beamwidths vs.
i
the Xc/X. ratio. This ratio is the longitudinal length at the
lower cut-off frequency fc in wavelengths.
i
To npaintain a reasonably constant beamwidth versus
frequency, a selection of a longitudinal length of at least
2XC is required (Gibson's design curve shows the E- and H-
plane beamwidths coinciding at greater than or equal to 3Xc
for a rotationally symmetric beam, but to obtain a wider
20
I
r
beamwidth while maintaining near-rotational symmetry, a
choice of 2Xc will suffice). The design graph shows an E-
i:
plane beamwidth of about 60 and an H-plane beamwidth of
roughly 65j for a longitudinal length of 2Xc. Consequently, a
I
length of 3.5 in. (1.9Xc) was selected. A short length also
maximizes the available clearance between the reflector
II
feed and the secondary focal point.
The 60 beamwidth was recognized as a risk factor
since under-illumination of the reflector by the main lobe
i'
and the {first sidelobes (which exhibit a 180 phase
difference relative to the main lobe) being captured within
the aperture of the reflector would degrade the sensitivity
of the overall antenna system. However, the prospects of
i
the Vivaldi presenting minimal aperture blockage and wide
bandwidth prompted research to continue.
21
I
'l
Table 1 summarizes the physical aspects and expected
performance characteristics of the initial design.
Materials ii 0.006 in. brass
Type'feed Coax-to-slotline
ii Longitudinal length 3.5 in.
i1 Maximum (tip) separation 1.84 in.
| Feed gap distance 0.050 in.
Equation of taper y = .025e+l-2282x
Expedited directivity
D 10log(l0L/A,c) 13 dB
11 Expected E- and H-piane
Beamwidths 0) i: 60 and 65
Table 1. Design characteristics of antenna #1.
lFrom Gibson [4]
22
Antenna patterns were measured on this Vivaldi
antenna at 4.0 GHz, 8.0 GHz, and 12.0 GHz, and are presented
as figures i 5-7 on subsequent pages. The beamwidth
11
performance is summarized in table 2, which follows the
i'
antenna patterns.
The sensitivity and spatial resolution of the integrated
reflector/Vitaldi feed were also measured. Figure 4
l
depicts the' method used. The performance of the integrated
antenna system can be summarized as follows:
ii
(1) The power received from a thermal point source was
reduced by about 8 dB when using the Vivaldi as compared
I'
to a conical horn.
(2) When moving a small neon light source from the
I
periphery of the dish to the vertex, it was observed that
only about the inner 1/3 of the dish area was illuminated
23
Ellipsoidal
reflector.r
by the Vivaldi antenna.
neon
(3) The backlobes were
light
found to be significant,
Vivaldi 1
antenna;
source
as evidenced by varying
0^ thermal
point
source
the location of the
thermal point source
Figure 4. Neon light and
about the secondary focal
thermal point sources used
to evaluate performance. point.
j
The unsatisfactory performance of this Vivaldi
antenna can be attributed to several factors. The first
i
sidelobes, ait roughly -10 dB from the main lobe and falling
within the aperture of the reflector (recall that the first
sidelobes are 180 out of phase from the main lobe),
n,
seriously degraded the sensitivity of the overall antenna
system. The substantial backlobe radiation caused the
24
I,
H-plane
Figure 5. E- and H-plane antenna patterns at 4.0 GHz.
25
Figure 6. E- and H-plane antenna patterns at 8.0 GHz.
i,
26
H-plane
E-plane
Figure 7. E- and H-plane antenna patterns at 12.0 GHz.
27
! 1 Expected 0) Measured
! i 4 GHz 8 GHz 12 GHz
E-plane '
beamwidth 11 60 indeter- 32 27
, minate (2)
H-plane
beamwidth 65 indeter- indeter- 50
i i minate (2) minate (2)
Il
Table 2. Beamwidth characteristics of antenna #1.
1 Expected performance based on Gibson's
experimentally determined beamwidth design curves [4],
2The ripple on the antenna patterns made it difficult
to determine whether each maximum near boresight is a
bonafide lobe or whether the ripple is due to multipath
effects froijn the antenna column, mixers, collet, standard
gain horn mounting bracket, etc.
28
spatial resolution to suffer as well, because significant
i
amounts of energy could be detected from directions and
points in space other than that of the field focal point.
To correct the resolution and sensitivity deficiencies
i'
I
of the first Vivaldi design, a second design iteration was
i1
undertaken.! This involved modification of both the
transverse : and longitudinal dimensions of the Vivaldi
antenna.
I
I
Considering the first deficiency, it was thought that
an insufficient frequency guard band might be causing the
i i
high backlobe radiation. That is, since the transition from
'i
travelling-wave modes to standing wave modes is not an
II
abrupt phenomenon, the cut-off frequency may have been
selected too high, allowing a significant standing wave
29
]
i!1
pattern (versus the desired travelling wave) to occur and
causing a dipole-like behavior in its radiation pattern.
i
M
If th|s were true, a wider aperture would permit the
i
radiation to more readily couple to free space by not
encountering reflection at the tips of the antenna. This
l_
would enhance the spatial resolution of the overall antenna
r
system since less extraneous energy would be entering the
r
feed via back lobe reception.
i
Addressing the second cited deficiency, it was
I,
thought that a shorter longitudinal length would improve
jt
the antenna performance by decreasing its directivity
(recall from equation {2.1a} that the directivity D is given
by D 10log{10LAc}), thereby increasing the beamwidth of
I
the main I6be. If a significant portion of the first sidelobes
30
could be forced to fall outside the capture area of the
reflector, less phase cancellation would occur and overall
r
11
system sensitivity would be improved. A second antenna
! i
i {
incorporating these ideas was fabricated. Its construction
11.
I;
properties are presented in table 3.
As before, antenna patterns were measured at 4.0, 8.0,
12.0, and Jalso 10.0 GHz. These patterns are presented in
I'
i
figures 8 to 15. Antenna beamwidths at three of the four
frequencies are summarized in table 4 for comparison with
the first design. A comparison of the patterns show little
improvement, if any, between the second generation
antenna arjid its predecessor. Based on these patterns,
!
h
similarly poor results in sensitivity and spatial resolution
were expected, thus making the effort to perform these
tests hardly worthwhile.
i
31
I1
I i
Materials 0.006in. brass
Type feed Coax-to-slotline
! Longitudinal length 2.8125 in.
l! Maximum (tip) separation 3.1875 in.
, Feed i gap distance 0.050 in.
| Equation of taper y=.025e+i.4773x
Expected directivity
1 1 | D 10log(10L/X,c) 9.5 dB
j| Expected E- and H-plane l
beamwidths 70 and 180
Table 3. Design characteristics of antenna #2.
i
i '
32
I
I1
1 ! ExpectedO) Measured
' 4 GHz 8 GHz 12 GHz
E-plane
beamwidth i1 60 90 90 90
H-plane | i
beamwidth jl 65 140 ~~ o o 90~~
Table
4.
Beamwidth characteristics of antenna #2.
The performance of the reflector using either of the
I
antennas discussed in this chapter was deemed to be
ji
unsatisfactory for this application. No further studies were
undertaken
and the effort was consequently abandoned.
T Footnotes of Table 2 apply.
33
Figure 8. E-plane pattern at 4.0 GHz.
34
I
II
Figure 9. H-plane pattern at 4.0 GHz.
i
35
0
Figure 10. E-plane pattern at 8.0 GHz.
36
ii
I Figure 11. H-plane pattern at 8.0 GHz.
37
0
Figure 12. E-plane pattern at 10.0 GHz.
38
0
300
270
240
Figure 13. H-plane pattern at 10.0 GHz.
39
Figure 14. E-plane pattern at 12.0 GHz.
I
i,
i
40
I
I
0
180
Figure 15. H-plane pattern at 12.0 GHz.
41
Chapter 4
Summary and Conclusions
Ir
I I
An antenna presenting minimal aperture blockage in
1!
I
the aperture plane would be a prime candidate for selection
j;
I
as the feed for an ellipsoidal reflector system such as the
ij '
one used by Edrich [1]-[3]. Other desirable traits would be
]]
wide bandwidth (or, as a minimum, multi-band capabilities)
!i
j!
in conjunction with constant beamwidth over its operating
i
frequency range. The goal of this project was to determine
if the Vivaldi antenna presented itself as a viable candidate
for such a
reflector system.
42
Much of the literature indicates that the Vivaldi
i i'
antenna possesses these attributes. However, its
i
|f
beam width j was found to be too narrow for use in this
i1
jl
particular application. The desired beamwidth of 126 was
I' -
i!
r
not attainable. A beamwidth on the order of 40 is
j j
predicted in reference [4] until the overall antenna length is
shortened tp about 2XC, where the beamwidth begins to
broaden in both E- and H-planes.
Attempts to broaden the main lobe resulted in the
appearance
of significantly large sidelobes and backlobes.
Figures 5-7 and 8-15 (first and second generation antennas,
respectively) show that the amplitudes of the first
i
sidelobes were only about 10 dB lower than the main beam
j j
and that suppression of the remaining backlobes was not, if
! i
ij
at all, significantly better.
43
Experiments using a neon light test source emphasized
the debilitating effects a narrow main beam and large
1 [
sideiobes (at 180 phase relative to the main beam) had on
!
the antenna system sensitivity. Similarly, use of a thermal
point source as the imaging target exemplified the lack of
1'
I:
spatial selectivity.
|i
I'
Possible reasons for the Vivaldi antenna's
l!
unsatisfactory performance are:
| i.
ii
The balun network (see figure 16, next page),
l
i:
was designed as a X/4 balun at 4 GHz (3^/4 at 12 GHz).
i!
While simple in concept, relatively easy to construct, and
I
Jl
also providing a mechanism by which to support one of the
||
radiating Elements, it undoubtedly impaired performance
li
around 8 GHz because of the narrowband nature of this type
of balun. Acting as a k/2 length of transmission line at 8
44
I
I
ji
ii
GHz, the short circuit
condition of the balun/
coax feediine junction is
not transformed to an
open circuit at the feed
gap {as is characteristic
of balun lengths
Figure 16. The balun
is short-circuited
to the coacial line
providing {structural
support for the right-
hand radiating element.
Ii
[(2n+i)/2] Xj, but rather as
a short circuit (this
phenomenon is readily
apparent by rotating X./2
on a Smith Chart). This
permits unbalanced
currents to flow onto
the outer
feedline
and
jacket of the feed line, allowing
antenna to radiate.
the balun,
1
45
The antenna construction technique (a "home
workbench"]
project with limited tools available) was
probably too crude to expect robust performance at
microwave
frequencies.
The coaxial-to-slotline transducer approach [7]
i ]
i'
utilized here, in conjunction with the crude construction
j
techniques j cited above, is undoubtedly inferior to an etched
PC board approach utilizing a microstrip-to-slotline
transducer
[7], the latter being inherently balanced and also
more controlled during the fabrication process.
Typically, the Vivaldi antennas described in the
literature also exhibited rather substantial sidelobes and
backlobes.
This fact, however, was not highlighted as were
its attributes of wide bandwidth and constant beamwidth
j
with frequency (although too narrow for this application).
46
Since 11 the relatively narrow beamwidth of the main
ji
lobe and the amplitudes of the first sidelobes limit the
I,
sensitivity of the antenna, as well as the backlobes
i
degrading the spatial resolution of the antenna, the Vivaldi
did not prove itself to be a satisfactory candidate in spite
ii
of its low { physical cross-section and wide bandwidth.
i
)l
Efforts to broaden the main beam conflicted with the
; |
requirement of suppressed sidelobe and backlobe levels.
11 '
j i
Consideration Of the results of the two designs investigated
here, in conjunction with the recorded studies found in the
j!
ji
literature, show that an acceptable compromise between
ji 1
these two issues could not easily be obtained.
Appendix
The Spectral Domain
The Spectral Domain Technique is a method whereby
the Galerkin Method is applied in the Fourier Transform
i
i,
domain. This technique is described in reference [8].
To quickly review the Galerkin Method, we wish to
determine an unknown source distribution responsible for a
known resultant field. We start by representing the source
distribution as a finite sum of weighted basis functions.
j
The basis functions can be as simple as pulse functions,
each spanning a sub-domain of the source region, or they
i
can be more complicated functions, such as a Fourier series
48
representation, which span the entire domain of the source
region. The coefficients (weights) of these basis functions
i
are as yet undetermined. This representation yields a
single equation relating the known field (by the appropriate
i!
Green's Function) to the source which contains "N" unknown
coefficients in its series representation.
The Galerkin Method creates an additional (N-1)
equations so we have "N" equations in "N" unknowns. This is
accomplished by multiplying the original equation by a
testing function. This process is performed "N" times using
"N" unique testing functions. The Galerkin Method chooses
the basis function in the original series representation as
its testing function, i.e., we multiply both sides of the "jft"
equation by the "jtii" basis function. Once we have created
"N" equations in "N" unknowns we can solve for the unknown
coefficients using standard matrix algebra.
49
The application of the Galerkin Method in the spectral
i;
domain comes into play when the NxN matrix contains
convolution integrals. These terms are readily represented
l!
as the algebraic product of the Fourier Transforms of the
two terms comprising each convolution integral. This is a
much simpler representation than the original convolution
integrals. ]| More importantly, there results considerable
savings in computation time versus the numerical
evaluation of N2 complicated integrals. That is why, for
certain classes of problems (like the microstrip and
slotline problems discussed in [6]-[8]), the Spectral Domain
Method is the preferred approach.
50
I
References
[1] T.A. Milligan, "The Use of an Ellipsoidal Reflector for
Hyperthermia," unpublished report dated 30 March,
1987, to Dr. Jochen Edrich, Professor of Electrical
Engineering at the University of Colorado at Denver.
i '
i
[2] J. Edrpch, et a/., "Imaging Thermograms at Centimeter
and Millimeter Wavelengths," Ann. New York Academy
of Sciences, vol. 335, pp. 456-476, 1980
j,
[3] J. Edrich, "Microwaves in Breast Cancer Detection,"
ji 9 9
European Journal of Radiology, vol. 7, No. 3, pp. 183-
193,1986
!;
[4] P.J. Gibson, "The Vivaldi Aerial," Proc. 9th European
Microwave Conf., Brighton, U.K., 1979, pp. 101-105.
i
[5] K. Frantz and P. Mayes, "Broadband Feeds for Vivaldi
Antennas," Antenna Applications Symposium,
University of Illinois, 1987.
ii
[6] R. Janaswamy and D.H. Schaubert, "Analysis of the
Tapered Slot Antenna," IEEE Trans. Antenna
Propagation vol. AP-35, pp. 1058-1064, 1987
51
[7] R. Janaswamy and D.H. Schaubert, "Dispersion
Characteristics for Wide Slotlines on Low-
Permittivity Substrates," IEEE Trans. Microwave
Theory Tech., vol. MTT-33, pp. 723-726, 1985
[8] T. Itoh, "Spectral Domain Immitance Approach for
Dispersion Characteristics of Generalized Printed
Transmission Lines," IEEE Trans. Microwave Theory
Tech., vol. MTT-28, pp. 733-736, 1980
I
j
52
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