300 MHz ARRAY USING PHYSICALLY SMALL
SLOT ANTENNA ELEMENTS
M.A.H. Ibrahim
and
J.A. M. Lyon
This document is subject to special export controls and each
transmittal to foreign governments or foreign nationals may
be made only with the prior approval of AFAL (AVWE), WrightPatterson Air Force Base, Ohio 45433

FOREWORD
This report, 1770-2-T, was prepared by the Radiation Laboratory
of The University of Michigan, Department of Electrical Engineering, 201 Catherine Street, Ann Arbor, Michigan 48108, under the
direction of Professor Ralph E. Hiatt and Professor John A. M. Lyon
on Air Force Contract F33615-68-C-1381, Task 627801 of Project
6278, "Study and Investigation of UHF-VHF Antennas." The work
was administered under the direction of the Air Force Avionics
Laboratory, Wright-Patterson Air Force Base, Ohio 45433. The
Task Engineer was Mr. Olin E. Horton and the Project Engineer,
Mr. Edwin M. Turner, AVWE. This report was submitted by the
authors in October 1969.
The prefatory material to this report, Sections 01, 02, and 03,
covers the general aspects of the utilization of miniaturized slot
elements in antenna arrays. The remainder of the report, Chapters I
through V and the Appendices, presents detailed background information
and an analysis of the coupling of loaded slot elements. Chapters I
through V and the Appendices have been written entirely by Dr. M.A.H. Ibrahim
and constitute his recent doctoral dissertation.
ACKNOWLEDGEMENT
The following individuals contributed eithez r analytical or experimental
effort for some parts of this report: Dr. C.J. Digenis, Y-P Liu,
D.G. Berry and B. Whitney.
This report has been reviewed and is approved.
/ OS<EOI A. DOMBROWSKI
t Colonel, USAF
/ hief, Electronic Warfare Division
ii

1770-2-T
ABSTRACT
The coupling between two loaded rectangular waveguides opening in an
infinite conducting ground plane has been investigated. The slots are equal
in size and placed so as to have the broadsides parallel. The analysis is
made in two steps:
i) the coupling between the apertures is derived;
ii) the effect of the backing cavities on the coupling is determined.
The reaction concept, a form of the "variational technique", was used in the
analysis. This method helped to simplify the mathematical work and provided
very reasonable results.
Substantial experimental work on the behavior of coupling versus frequency
and coupling versus slot separation was performed. In order to simplify the
experimental investigation the standard empty X-band waveguide (0. 9"xO. 4")
was filled with dielectric loading of Emerson and Cuming Stycast Hi-K dielectric
material. Materials having dielectric constants of 5, 7, 11 and 15 were used.
Most of the measurements for the loaded case were in the S-band range of
frequency (2. 6 - 3. 95 GHz). The choices of the S-band frequency range and the
dielectric constant were made to permit use of the anechoic chamber in the
first instance with reasonable accuracy and small waveguide in the second
instance.
An extensive comparison of coupling for the case of standard waveguides
with and without loading has been made and explanations are given for the observed differences. An iteration procedure has been used to determine the
effect of the electromagnetic interaction between the slots.
Detailed computer programs were developed for the analytical expressions.
Numerical results for many important cases were then obtained. IveiLhods and
results can readily be extended to other frequencies and other slot sizes.
iii

TABLE OF CONTENTS
PREFATORY MATERIAL 1
Section 01: Introduction 1
Section 02; Arrays and Interaction 2
Section 03: Conclusions 23
I INTRODUCTION 25
1.1 Approach to the Problem 26
II THEORETICAL ANALYSIS 28
2. 1 Assumptions 28
2. 2 Formulation of the Fields in the Different Regions 28
2. 3 Evaluation of U 36
2. 4 Iteration Procedure 50
2. 5 Consideration of the Short-circuited Waveguide Backing
the Slots 53
III NUMERICAL ANALYSIS 66
3.1 General 66
3. 2 Calculation of Reflection Coefficients at the Aperture 66
3. 3 Calculations of the Aperture-to-Aperture Coupling
Between Slots versus Frequency 70
3. 4 Coupling Aperture to Aperture versus Spacing 86
3. 5 Evaluation of the Cavity Effect 96
3. 6 Evaluation of Coupling 120
IV EXPERIMENTAL INVESTIGATION 121
4. 1 General Discussion 121
4. 2 Coupling Measurements 126
4. 3 The Far-field Patterns 148
V SUMMARY AND CONCLUSIONS 152
5. 1 Areas for Future Investigation 155
REFERENCES 157
APPENDIX A: The Reduction of Quadruple Integrals to Double
Integrals. 159
APPENDIX B: Examples of Computer Output Data on Coupling 162
iv

LIST OF TABLES
3.1 Relation between FN and f for ur=l, er=15, a=0.02286m,
b=0.01016m. 78
3.2 Relation between FN and f for,r=1, cr=11, a=0.02286m,
b=0.01016m. 78
3. 3 Relation between FN and f for,r=l, er=7, a=0.02286m,
b=0.01016m. 78
3.4 Relation between FN and f for /r= 1, er=l, a=0.07136m,
b=0.034036m (S-band). 79
3. 5a Relation between FN and f for Ur=1, er=l, a=0.02286m,
b=0.0106m (X-band). 79
3. 5b Relation between FN and f for,r=l, er=l, a=0.0157988,
b=0.0078994 (Ku-band) 79
3. 6 Cutoff Frequency vs Dielectric Constant of the Loading
Material; a=0.02286m, b=0.01016m. 80
v

LIST OF ILLUSTRATIONS
0. 2-1 Experimental Power Pattern (linear) of Ferrite Slot in Ground
Plane, H-plane Cut. 7
0. 2-2 Calculated Power Pattern (linear) for 3-element Ferrite Slot
Array (H-plane). 7
0. 2-3 Experimental Power Pattern (linear) for 3-element Ferrite Slot
Array (H-plane). 7
0. 2-4 Broadside to Broadside Coupling of Closely Spaced Rectangular
Slots Showing a l/r2 Trend. 9
0. 2-5 Far-field Coupling for Representative Angles. 10
0. 2-6 End to End Coupling of Closely Spaced Rectangular Slots Showing
a l/r4 Trend. 11
0. 2-7 Comparison between Experimental and Theoretical Results. 12
0. 2-8 Coupling of Air-filled S-band Waveguide Slots in Common Ground
Plane, 0. 5X spacing, Broadside to Broadside Arrangement. 13
0. 2-9 Coupling of Ferrite Loaded Slots in Conducting Plane, 0. 5X
Spacing, Broadside to Broadside Arrangement. 15
0. 2-10 Coupling of Ferrite Loaded Slots in Conducting Plane, 0. 5X Spacing,
End to End Arrangement. 15
0. 2-11 Interaction of Certain Chosen Groups of Slots Corresponding to
Three Radii; A, B and C. Polarization Aligned Left to Right. 18
0. 2-12 Returned Power Level to Feed Generator for Many Element Array
as Function of Number of Slots Used in Computing Interaction. 21
2-1 Arrangement and Slot Configuration. 29
2-2 Method of Images. 32
2-3 Source and Scatterer. 36
2-4 Coordinate Transformation. 47
2-5 Coordinate Transformation for Reduction of Quadruple Integral to
Double Integral. 48
2-6 Multiple Reflection between Transmitter and Receiver to Represent
Interaction between them. 52
2-7 Short-circuited Cavity. 55
2-8 Standing Electric and Magnetic Fields in Cavity. 56
2-9 Input Impedance to the Cavity at z=0. 57
2-10 Configuration of Waveguide when it is Closed at Both Ends. 58
2-11 Equivalent Circuit for Cavity Slot Antenna. 59
2-12 Resonance and Half-power Bandwidth. 64
3-1 Aperture Normalized Conductance and Susceptance, Er=5. 71
3-2 Aperture Normalized Conductance and Susceptance, Er=6. 72
3-3 Aperture Normalized Conductance and Susceptance, Er=10. 73
3-4 Aperture Normalized Conductance and Susceptance, er=ll. 74
3-5 Aperture Normalized Conductance and Susceptance, r=15. 75
3-6 Variation of Cutoff Frequency vs Permittivity of Loaded X-band
Waveguide.. 81
VI

3-7 Standard S-band Coupling vs Normalized Frequency. 84
3-8 Standard X-band Coupling vs Normalized Frequency. 85
3-9 Standard Ku-band Coupling vs Frequency. 87
3-lOa Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=2. 88
3-10b Aperture to Aperture E-Plane Coupling of Loaded Slots, er=3. 89
3-10c Aperture to Aperture E-Plane Coupling of Loaded Slots, r=4. 90
3-lOd Aperture to Aperture E-Plane Coupling of Loaded Slots, r=5. 91
3-lOe Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=7. 92
3-lOf Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=ll. 93
3-1g Aperture to Aperture E-Plane Coupling of Loaded Slots, Er=15.94
3-10h Aperture to Aperture E-Plane Coupling of Loaded Slots, r=20. 95
3-11 Broadside Coupling between Empty Slots (Separation Center to Center). 97
3-12 Aperture to Aperture Coupling vs Distance in Wavelength. 98
3-13a Resonance of the Cavity (Digital Plot), d=0.031, er=7. 105
3-13b Resonance of the Cavity (DigitalPlot), d=0.031, r=ll1. 106
3-13c Resonance of the Cavity (DigitalPlot),d=0.031, r =15. 107
3-14a Resonance of the Cavity (Digital Plot), d=0.1015, er=7. 108
3-14b Resonance of the Cavity (Digital Plot), d=0.1015, r=11. 109
3-14c Resonance of the Cavity (Digital Plot), d=0.1015, er=15. 110
3-15a An Expanded Digital Plot for the Resonance Peaks, n=l. 111
3-15b An Expanded Digital Plot for the Resonance Peaks, n=2. 112
3-16a An Expanded Digital Plot for the Resonance Peaks, n=3. 113
3-16b An Expanded Digital Plot for the Resonance Peaks, n=4. 114
3-16c An Expanded Digital Plot for the Resonance Peaks, n=5. 115
3-16d An Expanded Digital Plot for the Resonance Peaks, n=6. 116
3-16e An Expanded Digital Plot for the Resonance Peaks, n=7. 117
3-16f An Expanded Digital Plot for the Resonance Peaks, n=8. 118
3-16g An Expanded Digital Plot for the Resonance Peaks, n=9. 119
4-1 Anechoic Chamber Ground Plane. 122
4-2a Test Antenna Mount. 123
4-2b Arrangements for Loaded X-band Waveguides. 124
4-2c Arrangements for Standard S-band Waveguides. 125
4-3a Resonance Cavity and Dielectric Filling plus Coaxial to Waveguide Feed. 127
4-3b The 0.031m and the 0.1015m Dielectric Loaded Cavities. 128
4-3c Comparison between Loaded and Unloaded Cavity. 129
vii

4-4 Standard S-band Waveguide Coupling vs Frequency 131
4-5 E-plane Coupling vs Frequency for Two Slots Spaced 11. 4 cm 132
4-6 Coupling vs Frequency for Loaded X-band Waveguides, r= 11,
d=0. 031 133
4-7 Coupling vs Frequency for Loaded X-band Waveguides, Cr 15,
d=0. 1015 134
4-8 Gain vs Frequency of Dielectric Loaded Cavity 135
4-9 Anechoic Chamber Set-up for Far-field Pattern Measurements 136
4-10 Coupling vs Frequency for Loaded Cavity 138
4-11 Gain vs Frequency for Loaded Cavity 139
4-12 Coupling vs Frequency for Loaded Waveguides 140
4-13 Gain vs Frequency for Loaded Cavity 141
4-14 Experimental Set-up for Coupling with Point by Point Tuning 142
4-15 Coupling vs Frequency with Tuning both Transmitter and
Receiver at Each point 143
4-16a Coupling vs Frequency Point by Point, er=15 144
4-16b Coupling vs Frequency Point by Point, r =ll 145
4-16c Coupling vs Frequency Point by Point, er=7 146
4-17 Coupling vs Frequency Point by Point, xl=0. 03, Er=7 147
4-18a E-plane Radiation Pattern of a Loaded X-band Waveguide 149
4-18b H-plane Radiation Pattern of a Loaded X-band Waveguide 150
5-1 Explanation of Non-Uniformity of Separation of the
Resonance Peaks 156
viii

LIST OF SYMBOLS
a Slot dimension in H-plane
b Slot dimension in E-plane
d Cavity length
dI Aperture to feeding probe dimension in the cavity
d2 Probe to short side dimension of the cavity
E Electric field intensity
f Frequency
F Electric vector potential
FX, Fy F, Cartesian coordinates of F
FN Normalized frequency (FN = ka / 7r)
J Surface current in amp/meter
H Magnetic field intensity
k Wave number
ko Free space wave number
I Cavity length
M Magnetic moment
n An integer
n Unit vector perpendicular to the aperture
U Amplitude of the received illumination
v Velocity of waves in material medium
x Cartesian coordinate
XI Center to center separation of slots
A
x Unit vector in the x-direction
y Cartesian coordinate
Y Admittance
A Unit vector in the y-direction
z Cartesian coordinate
Z Impedance
z Unit vector in z-direction
ix

ao Attenuation constant
P Phase constant
y Propagation constant, or a variable of integration
e Dielectric permittivity
Co Free-space permittivity
er Relative dielectric constant or permittivity
rlT ]Second cartesian coordinate (source point)
First cartesian coordinate (source point)
X Wavelength
Xo Free-space wavelength
Xc Cutoff wavelength
Xg Waveguide wavelength
p Permeability
PQo Free-space permeability
wur Relative permeability.T Hertzian electric vector potential
T"' Hertzian magnetic vector potential
p First cylindrical coordinate
w Angular frequency
x

1770-2-T
PREFATORY MATERIAL
Section 0.1: Introduction
This report covers the work to date on a task devoted to the utilization of
loaded slot antenna elements in arrays. The objective of the task has been to
develop the advantages and criteria for the use of physically small slot antenna
elements in a 300 MHz array. It was recognized early that one of the chief advantages of such an array would be the possibility of reduced interaction between elements. A large part of this report is devoted to an analysis of loaded
slot elements and an evaluation of the coupling between such elements. The
reduced level of coupling can easily be ascertained from these analytical
methods for specificinstances through the use of an appropriate computer program. The detailed studies, including the essentials for programming on a
computer, are found in later chapters.
This report includes the study of simple three-element linear arrays.
The data obtained clearly show that with miniaturized slots the coupling involved
from one element to another is sufficiently small so that the interaction can
frequently be considered negligible. This fact has been deduced by comparing
the array patterns obtained analytically based on assumed element illumination
with experimentally determined patterns. It has been found that the experimentally derived patterns are extremely close to the theoretically calculated patterns. This means that the illumination of each element has been very close
to the assumed illumination neglecting interaction and as used in the calculations.
No modifications in the feed network were made in order to compensate for
the driving point impedance level of each element. For instance in the case
of the broadside three-element array, the feed network was arranged to give
three in-phase values of illumination of equal magnitude. Equal lengths of
lines were used so that the phasing and feed current magnitude were all the
same before each of the feeds was connected to a loaded slot element. After
the feeds were connected to the individual slots and the array pattern tests
made, it was found that the patterns so derived were very close to those
1

predicted by calculation. The spacing of the elements was very nearly the usual
half wavelength for a broadside beam.
Section 0. 2 of this chapter describes the nature of the interaction problem
among various elements of an array. This section also gives the experimental
results on simple three-element linear arrays.
Section 0. 3 indicates the conclusions obtained through the use of the information contained in this report. The remaining chapters contain the detailed
analytical work on the interaction of elements. Chapters I through V and the
Appendices comprise in their entirety the doctoral dissertation of M. A. H. Ibrahim
of which he is the sole author.
Section 0. 2: Arrays and Interaction
0. 2a: General Discussion
The interaction of elements in an array of elements can be expressed in at
least two ways. One is by means of the mesh equations for coupled antennas.
These mesh equations contain impedance parameters which theoretically can be
evaluated by means of appropriate line integrals. The most important impedance
for each element is the driving point of impedance. The driving point of impedance
itself is a function of the self impedance as well as each of the mutual impedances
which come into being through the interaction of each element on every other
element.
In the analytical work which has been done and is contained in Chapters II and
In, the analysis of the interaction of one loaded slot upon another has been on
the basis of a consideration of the electromagnetic field boundary value problem
Results have frequently been given in terms of the power level in dB received
at one antenna referred to the power level input at the other antenna. The
coupling then gives the power level of the receiving antenna as so many dB below
the power level of the transmitting antenna. The dB coupling level data can be
directly interpreted in terms of mutual impedance. However, thiisis not always
necessary. The main objective of the analytical work has been to show that
2

miniaturized slots are accompanied by much lower interaction than air-filled
slots at the same frequency and of standard size. On this basis, the work in
Chapters I through V is back-up or supporting information for the proposition
that miniaturized slots used in phased array antennas are accompanied by the
very considerable benefit of decreased interaction. In fact, the interaction
level is now sufficiently low to allow the frequently used compensation schemes
for the driving network to be eliminated or greatly simplified.
In the consideration of phased arrays the problem of driving point impedance becomes strikingly evident especially in the case of using the array
for transmitting. In general, power from each element may be coupled back
into every other element. If there is a relatively high level of interaction of
one element upon another, then it is necessary to have appropriate matching
techniques at every frequency and for every beam position. Obviously, it is
extremely difficult to obtain a good impedance match over a wide bandwidth.
It is even more difficult when matching must be good for wide ranges of scan
and for various scanning modes. A convenient measure of the need for matching
is obtained by considering an array of elements with respect to the total returned
power from all the elements normalized to the total power fed to the array.
This return of power to all elements will be a function of frequency and position
of scan for the array beam. The advantages of miniaturized slots would then
be associated with a major reduction of the returned power summed over all
elements over a useful range of frequency and over the desired range of scan.
There is some restriction on the bandwidth associated with a need to keep
the interaction low. This is brought out in the analysis given in the later
chapters. A possible bandwidth goal for phased arrays might be 15 percent.
It is less easy to generalize on a desirable scan angle. The requirements for
scan are very closely associated with the system requirements of the
antenna. Certainly for many anticipated uses it is desired that the scanning
angle from the normal to the array approach 90~. The indications given in
this report are that interaction may be so considerably reduced that the influence of scanning angle on the interaction, although present, is relatively
3

insignificant since the overall level of interaction is always small. This
fact will be explored in some detail in succeeding paragraphs.
The use of loading material in a slot in order to reduce the size of the
slot for a given operating frequency points to the possibility of an additional
advantage not mentioned heretofore. There are available high quality dielectric materials, some of which are of ceramic nature. Such materials can
be utilized to improve the hardening aspect of the antenna array, where
hardening is used to denote capability of withstanding the effects of nuclear
detonations. On the other hand, there are also useful materials for loading
such as the ferrites. Ferrite material also may be useful for hardening purposes. However, there is a temperature limitation on ferrite which should be
taken into full consideration.
0. 2b; Reduction of Coupling with Dielectric Loading
For miniaturized slots the nature of the reduction in coupling from one
element to another is due to the reduced area of aperture of each element
and to the impedance mismatch at the aperture associated with the loading
material. The reduced aperture means that a receiving element, or more
properly an element receiving energy from another element, is capturing
less energy. This advantage would be true for either linear polarization of
each element or for circular polarization for each element. However, in this
discussion, major attention is being placed upon linearly polarized elements.
Each element investigated in this study was a dielectrically loaded element.
The size of the element was so reduced that a slot formed by the open end of
an X-band waveguide when loaded with dielectric material within the waveguide
constituted a dipole slot suitable for S-band operation. This means that slots
2 13/16 " x 1 11/32" formed by the use of WR-284 S-band waveguide, will be
supplanted by loaded slots 29/32" x 3/8" formed by WR-90 standard X-band
waveguide filled with Emerson and Cuming stycast high-K solid dielectric.
Various values of relative dielectric permeability were used; 5, 7, 11 and 15.
4

The relative sizes of these slots are illustrated in Chapter IV (Fig. 4-2a)
The miniature slots as just described and as used in the experimental
arrangement utilized a short length of waveguide behind each slot. Each slot
was, of course, mounted in a common large conducting plane. In actual array
use, it is very likely that the many elements would also be mounted in a common conducting plane. The use of the common conducting plane provides a
3 dB increase in the directivity of each slot element. Furthermore, the plane
defines the physical field situation much more precisely than if the slot elements
are merely an arrangement of cut off waveguides without a common boundary
metal plane surface. Without such a plane surface, considerable uncertainty
is introduced into the effective radiative structure of each element. Currents
will tend to run back down the outside of the waveguide walls. For this reason
it is believed that the use of the common ground plane has been justified in the
experimental effort which is to be described.
It was convenient in undertaking the experimental work on the interaction
of elements to have the experiments performed at S-band. The analysis of
the interaction between two slots based upon an iterative procedure as described in Chapter II is applicable to slots at any frequency and with any size
as well as any type of loading material.
0. 2c: Experimental Ferrite Loaded Slot Arrays
The experimental work done on simple three-element linear arrays
utilized slots having ferrite loading and the size of each slot element was different from that described in the comparison of loaded and air-filled slots.
For these experiments on the three-element arrays the slot was 5"x2". The
depth of the cavity backing of each slot was 1 1/2". The center frequency of
each slot was 350 MHz. The loading of filling material of these slots was
ferrite EAF-2; the relative permeability of this ferrite was 6. 6, the relative
permittivity, 12. 6 and the magnetic qualify factor, 30.
As a preliminary in the experimental work it was appropriate to take the
radiation pattern of a typical ferrite loaded slot. For this purpose Slot No. 108
5

was mounted in a 5'xl-1/2' aluminum ground plane A radiation pattern taken
at 345 MHz is shown in Fig. 0. 2-1.
Use of standard procedures for the calculation of a broadside array of
three slots similar to No. 108 and fed equally both in magnitude and phase
resulted in the pattern shown in Fig. 0. 2-2. An array of three slots was then
fabricated with the slots having the magnetic fields collinear (slots arranged
end to end in a line). The experimentally determined radiation pattern is
shown in Fig. 0. 2-3. Since the patterns of these three figures are so similar
it was concluded that there was comparatively little interaction between slots.
This means that the illumination of the slots was not appreciably modified by
mutual coupling. The feed network to the three slots was made with three
identical branches, one being connected to each slot. Further work on the
same three-element array with uniform progression of phasing of the elements
also showed agreement between calculated and experimental patterns. Thus it
appeared that for ferrite loaded slots the interaction of coupling of slots was
relatively low. This indicates that the driving point impedance of each slot
in an array is relatively independent of scan angle.
0. 2d: Observed Coupling Effects of Dielectric Loading
Utilizing the information shown in Chapter III, it is possible to make some
simple comparisons which are helpful in evaluating the return power. For
example, in the consideration of a phased array of elements assume a requirement of 15 percent bandwidth. If two slot elements are made using the usual
air-filled S-band waveguide, it can be found from Fig. 3-11 (Ch. III) that the
coupling for a spacing of 0. 5X will be approximately -15 dB. On the other hand,
the same frequency band can be used with X-band waveguide if it is filled with
appropriate loading material such as with a permittivity of 7. 0. In Fig. 4-17
(Ch. IV), it can be seen that X-band waveguide so loaded as to be used for S-band
frequencies can provide for a 15 percent bandwidth with a coupling of the order
of -30 dB. It is observed that the use of dielectric loading material has made
the waveguide feed components somewhat frequency sensitive whereas the airfilled waveguide slots did not exhibit this degree of sensitivity. However,
6

350 (o 10o.
-10o 1o 9 350
FIG. 0. 2-1: EXPERITIENTAL POWER PATTERN (LINEAR) OF FERRITE
SLOT IN GROUND PLANE, H-PLANE CUT.
350' ~ lo' 350: 10~o
3-00
04 0,,100 I20 3' o 3 0 10 0.350-0..
FIG 0 2-2: CALCULATED POWER PATTERN (LINEAR) FI 0. 2-3: EXPERIMENTAL POWER PATFOR 3-ELEMENT FERRITE SLOT ARRAY (H-PLANE) TERN (LINEAR) FOR 3-ELEMENT FERRITE
7 SLOT ARRAY (H-PLANE).
7 SLOT ARRAY (H —PLANE).

increased isolation can be obtained using the smaller slots and the loading
material over a useful frequency bandwidth. In order to make this comparison
still more apparent, attention is called to Fig. 4-4 (Ch. IV) which shows the
experimentally determined coupling for S-band air-filled slots. From this
curve it is observed that for a frequency of 3. 0 GHz, the coupling is approximately -17 dB.
It is interesting to note that the coupling between the air-loaded slots can
also be determined from previously published reports (Lyon et al, 1963; 1966).
A number of graphs from these earlier reports are introduced here as Figures
0. 2-4, -5, -6 and -7. Figure 0. 2-4 shows that for a spacing of 0. 5X, two
broadside slots would have a coupling level of -14 dB. This figure was based
on X-band but is valid at other frequencies, if the slots scale exactly with
frequency. Figure 0.2-5, when applied to the same physical situation, would
yield about the same result. For this case, 0=0~. Read along the vertical axis
for a separation of 2X, the value of -26 dB. Then for a separation of 1X, add
6 dB. Then for a separation of 0. 5 X add still another 6 dB. The total is:
026 dB + 6 dB + 6 dB = -14 dB. The graphs in Fig. 0. 2-5 were also obtained
for X-band but are applicable to other frequencies, upon scaling. This presupposes the size of the slot has been changed proportionally to wavelength.
This of course is the case for standard size waveguides, and it is true in
comparing S-band waveguide in size with X-band waveguide. Figure 0. 2-8
shows coupling for S-band slots at various frequencies but at a spacing of
0.5 X.
The frequency bandwidth associated with dielectrically loaded waveguide
slots is quite different from that with ferrite material loading in a waveguide.
This presupposes that the ferrite material has approximately equal values of
relative permittivity and permeability. This being the case, the addition of
such material does not change the susceptance at given points of discontinuity in
the waveguide and the waveguide fittings as much as a dielectric material. A
dielectric material tends to increase the capacitive susceptance at points of
discontinuity whereas the ferrite material involves increases of both kinds of
susceptance and thus there is compensation.
8

f - 9.20 KMC
X = 3.26 cm (1.283")
5 kg= 4.67 cm
a * 0.4"
b = 0.9"
10 \ ^\ THEORETICAL
3n \ ^^ EXPERIMENTAL
~ 20
z
o 25
30
35.1.2.3.5.7 Ik 2 3 5 7 lox
d
FIG. 0. 2-4: BROADSIDE TO BROADSIDE COUPLING OF CLOSELY SPACED
RECTANGULAR SLOTS SHOWING A 1/r2 TREND.
9

-25
~- ~ ~. EXPERIMENTAL
THEORETICAL \ /
f 9.20 KMC
\~~* ^~X = 3.26 cm (1.283")
~_. 2 \ * ^s.~~Xgr 4.67 cm
-30 a = 0.4"
\s. \ b = 0.9"
EXPERIMENTAL- \
THEORETICAL - -—'- N
-35* \\
~rd \ \
* * i -' - i d = 2 X
50 I X ~.,, I I \.
8 -40
0O 10' 20* 300 40~ 50~ 60~ 70~ 80~ 90~
FIG. 0. 2-5: FAR-FIELD COUPLING FOR REPRESENTATIVE ANGLES.
10

20
25-
30
-- *1 \ —THEORETICAL
35
d
40 -o
40- 0
0
EXPERIMENTAL -
4550-
5S- ---—.- - I-I I 1 1 1 t I I I I I I iii.1.2.3.5.7 IX 2 3 5 7 IOX
d —
FIG. 0. 2-6: END TO END COUPLING OF CLOSELY SPACED RECTANGULAR
SLOTS SHOWING A 1/r4 TREND. f = 9.20 GHz, X= 3.26 cm
(1. 283"), Xg= 4. 67 cm, a = 0. 4", and b = 0. 9"
11

~N^^ <>~~~~A &Experimental
} Theoretical
*, 2
a = 45~
c (db) =90
a =90~
f = 9.20 KMC
X = 1,283 in.
at = b = 0.9 in.
bt = b 1 0.4 in.
trJX~~\ rta80
o
I _ I Ii. I I I tI It0 0 Id I I I/ I I I I I I I
0.1 0.2 0.3 0.4 0.5 0.6 7 08t9 I d/x 1.5 2 3 4 5 6 7 8 1(
FIG. 0. 2-7: COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS.
12

-10
Coupling
(dB)
o15
20 I
2.5 3.0 305 4.0
Frequency (MHz)
FIG. 0. 2-8: COUPLING OF AIR-FILLED S-BAND WAVEGUIDE SLOTS
IN COMMON GROUND PLANE, 0. 5X SPACING, BROADSIDE
TO BROADSIDE ARRANGEMENT.
13

0. 2e: Observed Coupling Effects of Ferrite Loading
In the consideration of the advantages of miniaturized elements in arrays,
some of the experimental work was done with dielectric loading material merely
for convenience. Other parts of the experimental work used ferrite material.
The type of ferrite used was EAF-2 characterized by a relative permittivity
of 12. 6 and a relative permeability of 6. 63.
The contention that a ferrite loaded slot is less frequency sensitive than
a dielectric loaded slot has been borne out by experimental evidence of other
studies (Adams, 1964). One of the slots used in this current study (5"x2"xl 1/2"),
has a bandwidth of 19 MHz based on a VSWR of 3. A comparable dielectrically
filled slot of the size 12" x 3" x 5" has a bandwidth of 10 MHz on the same
VSWR basis
Additional experimental measurements have been made for the coupling
from one ferrite loaded slot to another. These coupling values have been obtained for spacings of approximately 0.5 X and are values for the slots being
end to end and also for the slots broadside to broadside. The data are plotted
in Figs. 0. 2-9 and 0. 2-10. It is observed, as expected, that the broadside
to broadside level of coupling is somewhat higher than the end to end coupling.
However, both levels of coupling are very much reduced from what they
would be if air-filled slots were used. Again, in this case the comparable
values for air-filled slots would be -14 dB (Fig. 0. 2-4) for broadside to
broadside coupling and -20 dB (Fig. 0. 2-6) for end to end coupling for this
spacing. Note that the experimental data on S-band slots as shown in Fig. 0. 2-8
shows -15 dB for broadside to broadside coupling. In an array of slots, the
end to end coupling is relatively insignificant.
0. 2 f: Array Returned Power
One of the important measures for an array of many elements is the total
returned power from all active elements. For instance an array of many
elements may be fed with illumination to give a desired steer angle. Then for
a given position of the beam a measure of the returned power coming from all
14

0
-I0
-20- 30
- 40
o -50
-60
-70
Frequency (MHz)
I I I I I I i I I I
320 325 330 335 340 345 350 355 360 365 370
FIG. 0. 2-9: COUPLING OF FERRITE LOADED SLOTS IN CONDUCTING PLANE,
0. 5X SPACING, BROADSIDE TO BROADSIDE ARRANGEMENT.
0-10
-20-30
0
_-40
50 -
-60
-70Frequency (MHz)
I I 1 1 I I I I I I
320 325 330 335 340 345 350 355 360 365 370
FIG. 0. 2-10: COUPLING OF FERRITE LOADED SLOTS IN CONDUCTING PLANE,
0. 5X SPACING, END TO END ARRANGEMENT.
15

of the elements can be made. A total returned power of 10 percent or less
would be considered reasonably good. A returned power of 10 percent would
mean the returned power is 10 dB below the incident total feed power. If
there were 100 elements in an array, each of them coupling at the level of
-30 dB to each of the others, and then all of the returned powers in each of
the elements added together in going back to the feed network, the returned
power level in the feed network would be 10 dB above the incident feed power
to one element. However, this would be -10 dB below the total incident feed
power to all elements. Obviously, it is assumed that each element is fed the
same power as every other element, and also the phase of each element is
the same. This would correspond to the beaming being broadside. Such
crude calculations tend to be conservative. Fields are assumed to be in phase
and, therefore, additive.
0. 2 g: Interaction of Elements in an Array
Consideration of the interaction of every element of an array on every
other element can assume somewhat sophisticated levels. At the time of
writing this interim report, the ultimate in detailed analysis of this interaction
has not been accomplished. However, methods are available for this. If, in
the future, additional effort is required on this project, it is expected that it
will assume the form of detailed machine calculation of this interaction problem.
Two significant sources of information on this general analytical approach are
Lechtreck (1968) and Amitay et al (1968)
Approaching the problem of interaction in a relatively simple manner
corresponds to the method which is outlined in the following paragraphs. For
this purpose, interaction is considered on the basis of an array with all elements
equally illuminated and in phase. This corresponds to having a broadside beam
for the array. Of course, as the beam scans, the interaction will change.
However, having evaluated the interaction in terms of returned power for the
broadside beam, it is possible by comparing with results of others (Butzien,
1968) to get a reasonable assessment of interaction and to predict the change in
16

the interaction level due to having miniaturized slots. Miniaturized slots using
dielectric loading will give a somewhat different level of interaction from
miniaturized slots using ferrite loading, as will be discussed in succeeding
paragraphs.
0. 2 h: Examples of Returned Power for Full-Sized Air-Filled Slots
The returned power compared to the input power per single slot is considered. Figure 0. 2-11 indicates a two-dimensional array of rectangular slots
with uniform separation of 0. 5X. Assume that all of the slots are fed with the
same power level. Also, assume that each slot is in phase with every other
slot.
The returned power to slot No. 0 is considered. Referring again to
Fig. 0.2-11, the contribution due to the slots inside circle "A" is as follows:
(1) Slots 2 and 4: The coupling between 0 and 2 or 0 and 4
is of the order -14 dB. This can be obtained by looking at
Fig. 0.2-4. Therefore, the coupling to slot 0 is of the order
-14 +10 log 2 = -11 dB.
(2) Slots 1 and 3: The coupling between either 0 and 1 or 0 and 3
is of the order -20 dB (see Fig. 0. 2-6). Therefore, the power
received by slot 0 from elements 1 and 3 is of the order:
-20 + 10 log 2 = -17 dB. Then the power received by element
0 from all four slots in circle "A" can be computed.
antilog - 1.7 = 0. 01995
antilog - 1.1 = 0. 07943
sum = 0. 09938
10 log 0. 09938 = -10. 03 dB power received
from all four slots.
The contributions due to the slots in a larger circle "B" can be found as follows:
(3) To calculate the power received by slot ), take all the slots
enclosed by circle "B". The contributions due to 5, 6, 7 and 8
must now be added to the contributions found for slots inside
"A". The separations of these slots from slot 0 is:
2' x 0. 5X = 0. 707 X.
From Fig. 0. 2-7, the coupling level between each of these
slots (5, 6, 7 and 8) and slot 0 can be obtained for a = 450;
17

3 B B I D
/ / \
u/ o/ n- D60' \
/; \ \ v4 / I ]
\4 \ \ 8 \
^B^3\ 80 ^3EL 70/
FIG. 0. 2-11: INTERACTION OF CERTAIN CHOSEN GROUPS OF SLOTS
CORRESPONDING TO FOUR RADII A, B, C AND D.
POLARIZATION ALIGNED LEFT TO RIGHT.

the value of this coupling is -25 dB. Therefore, the total of
the contributions from the four elements is:
10 log 4-25 - -19 dB
antilog- 1.9 =.0127
antilog- 1.003 =. 0998.0127+. 0998 =. 1125
10 log 0.1125 = -9. 5 dB.
The -9. 5 dB indicates 11. 25 percent of the incident power per
slot is returned to slot 0 considering all slots in circle "B".
The contributions due to the slots in the still larger circle'C" can be found:
(4) For the enclosed elements in circle "C", the additional
elements will be 9, 10, 11 and 12. The received power from
elements 9 and 11 is obtained from Fig. 0.2-4. Each of these
has a coupling to slot 0 of -20 dB.
10 log 2 -20 = -17 dB contribution from 9 and 11.
The coupling from slot 10 or 12 to slot 0 can be found from
Fig. 0. 2-6 to be -32 dB.
10 log 2-32 =-29 dB contribution from 10 and 12
antilog -2. 9 = 0. 00125.
Then the total power level from all elements enclosed in circle
"C" can be found:
10 log (0.07943+0.01995 + 0.0127+0.01995+
0.00125) = 10 log 0. 13328 =
-8.75 dB power received from all
elements in circle "C".
The contributions of all elements within circle "D" may be obtained now by
including elements 13, 14, 15, 16, 17, 18, 19 and 20:
(5) Slots 14, 15, 18 and 19 are spaced 2. 23X from slot 0 and the
angle a = 63020' using Figs. 0. 2-11 and 0. 2-7. Interpolation
on Fig. 0. 2-7 will yield -31 dB of coupling each slot to slot 0.
10 log 4 - 31 = -25 dB.
For elements 13, 16, 17 and 20 the spacing is still 2. 23X but
the angle a is 26030'. Again, using Fig. 0.2-7 the coupling
of each slot to slot 0 is -44 dB.
19

10 log 4 - 44 = -38dB
antilog - 2. 5 =. 003145
antilog - 3.8 =. 000137
10 log (0.13328+0.003145+0.000137)
= 10 log 0.136562= -8.65 dB power received from
all elements in circle "D".
The results of the interaction for various numbers of nearby elements considered are shown in the graph of Fig. 0.2-12. As shown, the returned power
of an array of many elements (say, 1000) will be on the order of -8. 65 dB as
judged by consideration of the first 20 elements surrounding any one element.
0. 2 i: Returned Power for Dielectrically Loaded Slots
As examples of returned power for air-filled slots it is to be noted
that the two great contributions are from the two nearest slots
which couple broadside to broadside with the slot under consideration.
For air-filled slots this coupling level is -14 dB. For dielectrically loaded
slots the corresponding level is -30 dB, (see Fig. 4-17, Ch. IV). Although
complete information is not yet available it appears with good certainty that
for bandwidths of approximately 15 percent or less, the returned power for
an array of dielectrically filled slots will be substantially less than for airfilled slots.
0.2 j: Returned Power for Ferrite Loaded Slots
Recent experimental data on the coupling of ferrite filled slots are shown
in Figs. 0.2-9 and -10. The lower levels of coupling would then result in a
substantially lower returned power level. Reading from the figures for a
frequency of 350 MHz and spacing of 0. 5X the broadside to broadside
coupling is -35 dB; the end to end coupling is -37 dB. It is interesting to
observe that these two coupling values are very close. This contrasts with
the unloaded slot case with corresponding values of -14 dB and -20 dB.
Again, making use of Fig. 0.2-11, the returned power to slot number 0
is considered, assuming all slots are ferrite loaded and have the above
coupling behavior. The contributions due to the slots inside the circle "A"
are as follows:
20

0 -8I
A C
4 elements 8 elements 12 elements 20 elements
FIG. 0. 2-12: RETURNED POWER LEVEL TO FEED GENERATOR FOR MANY ELEMENT
ARRAY AS A FUNCTION OF THE NUMBER OF SLOTS USED IN COMPUTING
INTERACTION. Array has all elements illuminated with same phase and
magnitude. Spacing of elements is 0. 5X and the coupling level is in terms
of input power level to any one slot.

(1) Slots 2 and 4: the coupling between 0 and 2 or 0 and 4 is of
the order -35 dB. This can be obtained from Fig. 0. 2-9 for
ferrite loaded slots. Therefore the coupling to slot 0 is of
the order
-35 dB + 10 log 2 = - 32 dB.
(2) Slots 1 and 3: the coupling between either 0 and 1 or 0 and 3
is of the order -37 dB as can be obtained from Fig. 0. 2-10.
Therefore, the power received by the slot 0 from elements
1 and 3 is of the order
-37 dB + 10 log 2 = - 34 dB.
Then the power received by element 0 from all four slots
in the circle "A" can be computed:
antilog -3.2 = 0. 00063
antilog -3. 4 = 0. 00040
sum = 0. 00103
10 log 0. 00103= -29.9 dB
This last value is the power received in terms of the incident power level to one
slot. The contributions due to the slots in a larger circle "B" can be found:
(3) To calculate the power received by slot 0 take all the slots
enclosed by circle "B". The contributions due to 5, 6, 7 and 8
must now be added to the contributions found for slots inside
"A". The separation of these slots from slot 0 is:
I x0.5X = 0. 707X
Since tests have not been made for the dependence of coupling
of ferrite loaded slots on separation in echelon, assume the
most conservative dependence of 6 dB per octave of distance
as for broadside to broadside alignment (this is well established).
This is conservative since 12 dB is the value for end to end
alignment.
0. 707X
20 log 70 = 20 log. 414 = 3 dB
0. 5X
The coupling level between each of these slots, 5, 6, 7 and 8, and
slot 0 can be obtained as:
-35 - 3 = -38 dB.
Note that -35 dB has been chosen arbitrarily rather than -37 dB,
again being conservative. Therefore the total of the contributions
from the four elements is:
22

10 log 4-38 = -32 dB
antilog - 3.2 = 0. 000629
antilog - 2. 99 =0.00102
sum = 0.001649
10 log 0. 001649 = - 27.9 dB.
This -27. 9 dB is for all slots 1 through 8. It indicates that 0. 16 percent
of the incident power in a slot is returned considering all slots in circle "B".
This contrasts with 10. 99 percent for air-filled slots.
The contributions of slots in the still larger circles "C" and "D" can
readily be included by similar calculations. In this way it is found that the
returned power level using contributions from all slots in the "D" dircle
is -26. 9 dB. This value -26. 9 dB means the returned power per element from
20 elements surrounding this element is 0. 2 percent of the incident power fed
to each slot.
Section 0. 3: Conclusions
The work covered in this report has considered a simple linear array of
three elements. The experimental radiation patterns of this array indicated
a low level of interaction of elements. This low level of interaction is attributed
to the use of miniaturized slot elements. The miniaturization has been accomplished
by either ferrite or dielectric loading. Such loading imposes some bandwidth
restriction on the elements but this constitutes no real limitation for the usual
application of phased arrays. Of course, the loading material in slots creates
an additional material interface which is of importance both as to the interaction
of the elements, and the bandwidth. The following chapters give considerable
analysis of the slot-to-slot interaction with loaded slots. The net result of the
work reported here is that the use of smaller loaded slots reduces element
interaction; the driving point impedance of each slot in an array of other active
elements remains more nearly constant than for unloaded slots. It has been
shown that the use of ferrite loaded slots rather than air-loaded slots at a given
frequency can result in a returned power level of 0. 2 percent instead of 13. 6
23

percent. This example was based on conservative calculations for an array
of 21 elements. This great reduction in returned power or level of interaction
would carry over to much larger arrays using ferrite loaded slots.
References
Adams, A. T. (1964), "The Rectangular Slot Antenna with Homogeneous Isotropic
Loading, " The University of Michigan Cooley Electronics Laboratory
Technical Report No. 147 (5549-7-T).
Amitay, N., P. E. Butzien and R. C. Heidt (January 1968), "Match Optimization
of a Two-Port Phased Array Antenna Element, ". IEEE Trans., AP-16,
No. 1, 47-57.
Butzien, P. E. (1968), "Antenna Element Match Design for Planar Phased Array
Antennas, " Proceedings of the 18th Annual USAF Symposium on Antenna
Research and Development, 22 pp.
Lechtreck, L. W. (1968), "Effects of Coupling Accumulation in Antenna Arrays, "
IEEE Trans., AP-16 No. 1, 31-36.
Lyon, J. A. M. and R. M. Kalafus (March 1963), "Aerospace Antenna Coupling
Factor Interference Prediction Techniques as Related to the Electromagnetic
Compatibility of Different Systems, " presented at the Contractor's Conference
on Electromagnetic Compatibility Research, Wright-Patterson Air Force
Base, 3 pp.
Lyon, J. A. M., R. M. Kalafus, Y-K Kwon, C. J. Digenis, M. A. H. Ibrahim and
C-C Chen (1966), "Derivation of Aerospace Coupling-Factor Interference
Prediction Techniques - Final Report, " The University of Michigan
Radiation Laboratory Report 6633-1-F, AFAL-TR-66-57, AD 483051,
UNCLASSIFIED, 334 pp.
24

Chapter I
INTRODUCTION
The interest in the problem of coupling between two loaded slots backed by
loaded waveguide cavities arose when a study was initiated on the use of arrays
of miniaturized slot antenna elements. The interest was based upon prior work
of the writer on the coupling between unloaded slots and also by the writer's
early association with the design problems of antennas of reduced size. The
problem of electromagnetic coupling between loaded slots is considerably more
complicated than the coupling between unloaded slots. The problem can be
considered from a physical viewpoint as a microwave network of considerable
complexity due in part to material interfaces. Such a picture is
helpful in the consideration of incident and reflected waves at various locations
in the physical arrangement.
A thorough search of prior literature was made in three directions:
i) A survey of the literature was made for information on loaded rectangular
cavities including the characteristics of these antennas as individual elements:
aperture admittance data were sought. There were several papers on this
subject such as Adams and Swift and Hatcher. There were some reports
on the properties of rectangular waveguides covered by dielectric or plasma
19 13 1
slabs as in Cockrell and Galejs. The report by Adams was studied very
carefully. He used the variational technique to obtain the normalized conductance
and susceptance. Some slight changes in his formulas were made and then used
as indicated in Section 3.2 of Chapter III. It was possible to obtain the aperture
reflection coefficient using simple relations involving normalized conductance and
normalized susceptance.
ii) Information was sought on the radiation from empty rectangular waveguide
openings in a ground plane and also coupling between such guides. There were
several papers and reports in this area in addition to work in which the writer
25

had participated at an earlier date. Some of these are; the final report by
Archer and Hardie, Galejs and Mikenas and Mayes. Both Galejs and
Mayes have used the reaction concept in their formulations. Some reports
from The Ohio State University such as Jennetti, Svoboda, OSU 0 21
have some discussions on an empty or unloaded cavity-backed rectangular slot
antenna.
Finally, the report of Lyon,et al, where integral equation formulations
were used to predict the coupling of empty slots flush mounted in a ground
plane proved helpful.
iii) A search in the literature for information on coupling between loaded
slots proved fruitless. There were no papers which have treated the coupling
between loaded slots except one recent paper by Wu which appeared in the
March 1969 issue of Radio Science. The paper describes a theoretical investigation
whereby an integral equation method is used for an analysis of coupling between
parallel plate waveguides loaded with dielectric plugs. This present investigation is different in several ways. The analysis is completely different from
Wu's since in the present investigation there is a short-circuit at the bottom
of each cavity. Also in this investigation, using rectangular waveguide, both
theoretical and experimental results have been achieved. The resonance
phenomena in this investigation has been explained very carefully; Wu16 did
not do this but indicated more information was needed.
This literature search was continued at the same time the experimental
work was being conducted on the coupling of loaded slots with cavity backing.
In the meantime several papers on the reaction concept such as Rumsey5
12
and Richmond were read carefully and the reaction concept was applied in
the theoretical analysis.
1. 1 Approach to the Problem
The coupling was sought by first obtaining the coupling between loaded
slots on an aperture-to-aperture basis and then to ascertain the effect of the
26

backing cavity later. From these two analytical steps the total coupling can
be obtained. An iterative procedure has been used to compensate for the interaction between the two slots especially when they are very close to each other in
spacing.
Some of the graphs in the text will show that the effect of interaction is more
pronounced in the case of near field spacing. Interaction is of much less
significance for large spacings of slot elements. The situation closely parallels
the inductive coupling of two wire circuits.
27

Chapter II
THEORETICAL ANALYSIS
2. 1 Assumptions
In order to formulate the problem there are several assumptions which
are appropriate. The exact solution of this problem is very complex. Some of
these assumptions are going to be listed at the beginning and the rest will be
mentioned at the point where the assumption is made. To simplify the analysis,
it will be assumed that only the dominant TE10 mode is propagated at both
transmitting and receiving waveguides. The assumption is justified by the
experimental work, as we shall see later. Of course, for operation above the
cutoff frequency there may be more than one mode propagating. It will be
assumed at the start that the scattered field from the receiving slot is not going
to affect the excitation at the transmitting slot but compensation for this will
be made later in the analysis, by means of an iterative technique. Also it will
be assumed that the apertures are equal in size with the broad sides parallel
as shown in Fig. 2-1(a).
2.2 Formulation of the Fields in the Different Regions
The proposed method of solving the problem is to assume an aperture
illumination at the transmitting slot and then solve to obtain the scattered field
inside the receiving aperture. The analysis is that of a three region boundary
value problem with Regions I, II and III as shown in Fig. 2-l(b).
Region I
The fields are formulated in general as shown in the equations:
-- 2 *
E= V(V- t) + X A E X -jWPu Vx X (2.1)
H =V(V V 7r)+k2 r-+jwe Vx ri (2.2)
28

y, r)
Trans. Rcvr
0 b xi x2 x,:
(a)
z
z
II
Ground Plane
(b)
FIG. 2-1: ARRANGEMENT AND SLOT CONFIGURATION
29

where
k2=u2 E c (2.3)
7T is the Hertzian electric vector potential
and
" is the Hertzian magnetic vector potential.
From the vector identity,
Vx Vx L= V(V- L)-V2L (2.4)
and knowing
V2L+k2 L =0, (2.5)
then E and H equations can be rewritten in the form
E= Vx VxT -j Mu Vx;r (2.6)
H= Vx Vx'+ jw E Vx F. (2.7)
For E =0, then set wr = r^=z"= 0 and substitute in (2. 6) and (2. 7) obtaining
EI and HI in the waveguide which are:
E = - ij Vx (7 r Y) (2.8)
and
H =VxVx (7r* ) (2.9)
I y
where the subscript I is taken to denote the fields in Region I. Therefore,
there results
-jz jpz+Re )sin o mry nrx mn z
=(e - - A sin cos - e
I a mn a b
x
m= n=o (2.10)
where the prime on the double summation indicates that (1, 0) term has
already been removed and where the subscript x is used to denote the xcomponent of the field,
3 1- ( where f > f (2.11)
V (2 C11)
30

f is the cutoff frequency,
C
v- (2. 12)
and
R is the reflection coefficient of the dominant mode at the aperture.
Region II
Region II is the free space upper hemisphere over the ground plane
(see Fig. 2-l(b) ). The method of images is going to be used to derive the fields.
To formulate the equations of the fields in Region II take
a b
-- b f ~ e-jkp
F = I Ir M ds (2.13)
e
0 a0
From Fig. 2-2 an application of the equivalence principle yields (b) from (a)
and according to Huygen's principle the result as (c) is obtained and this is going
to be used in (2.13). Therefore
- 1 a b - eko
F= 2E x n - dx dy,
where M is the magnetic moment vector
P = /(x)2 +(y-7) 2+z2 (2.14)
and
F is the electric vector potential.
Since for the problem at hand
Exn Exz = -ExY (2.15)
then
a b
i- ejkop
F=F =F - EJ - dxdy) (2.16)
and
F =F = 0, (2.17)
x z
31

z=0 z=0 z=0
E,H I E,H E,H t E,
zero'
I fields..
Il | Ms=Exn I Ms=2Exn
sources I I
matter I
I _ at. _
n n t n
(a) (b) (c)
FIG. 2-2: METHOD OF IMAGES

where Ex is the x-component of the field at the transmitting aperture or, in this
case, the aperture illumination.
The fields in Region II could be represented, in general, by these equations:
E= - Vx F-j A + V(V A) (2.18)
0 J Eo
and
H =VxA -jwe F+ —- V(VF). (2.19)
The radiating source in this case is the rectangular slot denoted by TR and
located in an infinite ground plane (see Fig. 2-1). Since only a dominant mode
is going to be assumed then the aperture illumination can be given as:
E 0) =0 (1+R) sin - x (2.20)
where a
where R is the reflection coefficient of the dominant mode at the aperture and
e, rl are the source coordinates as shown in Fig. 2-1(b). Therefore from
(2.20) and (2.16)
a b2 e-jko (x-) +(y-rl) +z
F= - I J J 2(1+R)sin a e ddn2 (2. 21)
4 4 ^x-?)+(y-7) 2+z2- (2.21)
where x, y, z are the field points of the coordinate system and
k = t i. (2.22)
From eqs. (2.18) and (2. 19) by setting A = 0, then
E=-VxF, (2.23)
and
H = -jE + F + - V(V F ). (2.24)
o jL4~
Therefore
33

- aF aF- aF aF
\ y A YJA yA y
E= —X +z -J=X -— Z
_ az ax_ az ax
and
a b -JkoP
E1I= a f r 2(1+R)sin ( ) 3 (-jp-l) dr (2. 25)
X 00 p
which is the x component of the field in Region II. Notice that E O=0at z=0 or the
tangential component of the electric field is zero over the ground plane (perfectly
conducting). Also the other components can be obtained inthe same manner
For the H field we can write
a b
Hay = - 4 j ( o ClY e -d de
1 k a 2 (7r7)/ e -jkop
H= — I 2(1+R)sin( -)2 e d-jp d (2.26)
HII 4 rjw axay a 2(l+R)sin(T ) e d.d.)
1nd 1 I r f 0 r7 P
+ 2= (l+R)sin ) e — d d (2.27)
4r \ a p
and a b
a2 7T) eJkoP
H1 ____ 2(1+R)sin(-) d d) (2.28)
II 4ir jc azay\ a p
z ~
Region III
In Region III again it is to be assumed that the only mode existing is the
dominant mode. Therefore as in Region I:
E (x, y, z)= U sin a e p, (2.29)
IIx_ a
where U is the complex amplitude of the electric field at the receiving aperture
34

and from Maxwell's equations,
aE
a =-/ jwHy. (2.30)
y
Therefore:
_H- 1 a (UsinYeJZ)
Hm j4Lu az a
and
HO =- Usin yej. (2.31)
III to a
y
At the surface of the ground plane or at z = 0,
H _ - Usin( —) y (2.32)
y
where as before
B =- - ( 2 ) (2.33)
v
or
27rf L fc 2 =(1 2 1 2 v/X 2
hx~f f h X v/X
or
r 2
) 2 / ) (2.34)
where X is the wavelength in the material medium. Therefore, from (2. 33)
and (2. 32),
H -U _ sina y (2. 35)
III 2a a
where
jL is the permeability of the material
e is the permittivity of the material used
C = COer
= Jolr a
35

co, e are the free space and relative permittivity of the material
r
respectively, and
Mo',r are the free space and relative permeability respectively.
The objective now is to use the formulations of the fields in Regions I, II
and II to obtain an expression for U the complex amplitude of the electric field
at the receiving apertures corresponding to a given excitation of the transmitting
aperture.
2. 3 Evaluation of U
To obtain an expression for U there are several ways, including an integral
equation formulation or variational technique. For this analysis the reaction
concept of Rumsey is going to be used since e it simplifies the formulations.
As stated by Rumsey, J(c) generates the same field as "g" on the source free
side of S as shown in Fig. 2-3, where g is the source, J(c) represents the
surface distribution of the electric current which is induced on the scatterer by
g, and S is the scattering body.
J(c) S
Source
S >
Scatterer
FIG. 2-3: SOURCE AND SCATTERER.
36

For the specific problem at hand g is the same as the transmitting aperture with
a given illumination and the scatterer is the receiving slot.
The boundary conditions with the tangential magnetic fields at the receiving
aperture states:
A _ A A -
ZxHi= ZxH +ZxH (2.36)
in s I'
where
H is the scattered field inside the waveguide,
in
H is the scattered field in the half-space over the ground plane,
and HII is the field due to the illumination of the transmitting aperture.
Also
A _
ZxH =CZx Hin (2. 37)
S in'
and A _ 1 A
and ZxH = ZxH (2. 38)
in 1+C II
For the free space case C is taken to be unity as in Lyon. Therefore,
A - 1 _
in 2 ZxH (2. 39)
Also from the reaction concept which states
(a,b b= fff (b) dJ(a)-H(b) d K(a), (2. 40)
V
where
V is the volume which contains the sources,
J (a), K(a) are the electric and magnetic current densities of source "a",
and
J (b), K(b) are the electric and magnetic current densities of source "b".
It is also helpful to note that the reaction formula has the properties of a scalar
product.
The reciprocity theorem of all the sources that can be contained in a finite
volume is expressed by
(a,b) =:b,a). (2.41)
37

The linearity of the fields is implied by the identities,
<a, b+C> = <a, b> + <a, C>, (2.42)
and
<A a, b> = A a, b> = <a, A b>, (2.43)
where the notation "A a" means the "a" field and source are multiplied by the
number A. Also
<x,a> 4x, C>, (2.44)
where
x is the test source.
Equation (2. 44) expresses the condition that "a" and "C" should "look"
the same to an arbitrary test source "x". Choosing the test source to be "a"
itself, then (2. 44) contains "a" as the assumed value of sources and C is the
correct value of it.
<a, a> = <a, C> (2. 45)
from (2. 40), (2. 41) and
<a, C> C,a>= a f-J(c) E iidS (2.46)
S
From the boundary conditions it is known that
j= x H. (2.47)
Therefore
A
J(c) = -H11 x, (2. 48)
t
where the subscript t is to denote the tangential component at the aperture.
From Eqs. (2. 46) and (2. 48) there results
(a, C> = fI HIk EItdS, (2.49)
and
<a, a> f J(a) Et dS (2. 50)
S
38

in which J(a) is the current assumed when the scatterer is removed or
J(a)= -H + H (2. 51)
I It
t
where HMit is the tangential component of the magnetic field in the Region III at
15
z = 0. Following the same procedure as given by Rumsey, let
a=Uu. (2.52)
Then apply (2. 50) and (2. 52),
< a, a> = Jf (-HI t+HI ) (U EDut) dS (2. 53)
S
where
E t= sin -Y. (2. 54)
IIIut a
From Eqs. (2. 45), (2. 49), (2. 52) and (2. 53),
u u m En ^u dS=U J2 (- +H
U J HII EudS=U f (H +HmuIIt)EIIIutdS
S t t S
Simplifying the above to get an expression in U,
2U I HIItEIutdS=U JHInU Emut dS
S S
or
2 H E dS
UV- — ^7 —------- S (2.55)
ff Hmniu dS
S t t
where
E =sin T at z
IL u, atz=
H =- sin -, at z = 0,
IIIut /u a
S is the area over the aperture of the receiving slot, and
3 is defined as before in Ea. (2. 34).
39

A special case of the above would be that of free space slots and it can
be analyzed as follows.
Again
<a,a> =/f(-Hit+Hit)UEmutdS (2.53)
where
Em ut=sin a x
UIt a
From (2. 39), which is
A - 1 -
ZxH = ZxH
in 2 II
2
and using C = 1 as in Lyon then in this case
Hin = HI
r H =2H. (2.56)
II III
t t
Substituting in Eq. (2. 53):
<a, a> = J(-2HIt+Hit) UEm ut dS,
or
<a, a> = -U IfHIItEEut dS
However
HI =UHI
t t
Therefore
<a, a>=-U2 /H uEmu dS (2.57)
S Illt It
from Eqs. (2. 45) and (2. 46).
40

Therefore
-U IfJ(C)EI dS = -U2 H UII EudS
S ut S t t
or j,
JJHIE EI dS
U =. (2.58)
((H E -dS
II utEI II utdS
This is for the free space case where u = u and e =, where S is the receiving
0 o
aperture area, and
E = sin z= 0
IIIu a
H sin, Z = 0.
IIIu tqL a
To obtain the value of U it is necessary first to obtain HIt. From (2.27), again
write
a b -jkaP
H =- ( 2(1+R)sin 2 dd
II 4w a p
y 0
2 a b -jkop
k p
47-r jwj0Ay2 I |2(1+R)sin --- ded7, (2.27)
where
P x-()2+ y-r) 2+z2
To simplify (2.27), consider the second term on the right hand side,
a b -jkp
L = a2 ( 2(1+R)sin l e dd)
Jy2 4a 41

Interchanging the order of differentiation and integration and substituting
a2 a2
ay2 ayarj
a y2
yields
a b e2 -jkop
7rr 7a e
L= - 2(1+R)sin r ( ) dedrl. (2.59)
a ayarl P
Integrating once by parts with respect to rl in the manner
ju(x)v'(x)dx=u(x) v(x)-Jv(x)u(x)dx.
Let u7u=cos,
Let u = sin U' - -cos,7
a a a
e-jkp a a e-jkop
v = (, and v' =a (P) a
ay p aray p
Hence
_ b-jk p a a b -jk p
L=-J 2(1+R)sin r dr + 2(1+R)cos -(e )ddr,
a P a " ay P
or
L= a 2(l+R)cos - (- ) dedrl (2.60)
a I a ay p
Therefore, Eq. (2. 27) reduces to
a b -j
1 r e "jkop
14 j~ a J 2(1+R)cost1(e- ) djdr7. (2.61)
4u2+Oi w a I aay p
42

Next, return to evaluate U using (2. 61). At z = 0,
P = P' = x 2+(yt-r)2. (2. 62)
Then the numerator of Eq. (2. 55) can be written in the form
Numerator =
{ a a ab -jkop
a 4 I a p
1 a b(1+R)s r) a'
2 - f I 2((l+R) )sin(- sin-ye ddrndydx
x2 a a b.-jk p'
1 w ffr raab7nry e k t
2(T f ff 2(l+R)cos-sinT s(e-,)ddrdydx.
4jwuo a I a a y p'
Then simplify the second part on the right hand side of (2. 63) by means of integration by parts as before.
Take ry r ry
u =sin-, u'= - cos-,
us a a a
e-jkoPt and ejkop'
v=- aynd v'= -
P' ay pa
43

Let
a a b jkop''I= f fJI 2(l+R)cos( 2U1 )sin ( dy da e d
x 0 a 0 0
Then integrate by parts
I=| ff(2(l1+R)cos sin aY e dedridx
Sfa pt
X2 a a b -jkop
- a J 1; 2(l+R) cos 77 os Ty e, ddrldydx.
a Ja a p'1
Again the numerator can be written as follows
Numerator =
x2 a a b -jklp'
jWE _k?Ty e
2 -J 2( +R) sin2 sinry e_ ddrndydx
47r a a p'
11
2 X2 a a b -jk pt
1 7Tr2y
+ 4j. w 2 | I 2(1+R)eos- cos - ddrd —yddddx
4?rjwp a 2 a a P
Some simplification yields:
Numerator =.2.a a b jwto (i+R) -jkp,
o |n| [n ry+ (1+R) i rl i y e
0sin- sin -- COS — cos — dg drtdydx
JoL r a a 2. a a P'
0 & 0a j (.a
(2. 64)
44

Use the trigonometric identity below:
7r Ty al a7r
ClCSa a osa a a a- 2sinl- si -+
C aos - cos aYC an a 2y CL-C2)cos (y-r))+(Cl+C2)cos a (y
C1 and C2 would be defined as:
and
w7 (1+R)
C = r l (2.65)
and ~ (1+R)
C -. (2.66)
2 j 7
Finally U can be rewritten in the somewhat simpler form:
x2 a a b -jkoP'
U= 2~C l+C2 cos" (p —adrjdydx,
2C 3 IO r f j1C C2)cos a (y-r')+(C+C2)Cs a (Y+r
(2.67)
where
C3 iHiut Eit dS
px2 a T.a.27.y
- | 1-( ) sin dydx
2 a a
x *'0
12
=2V _2a (x2-xl)
In Eq. (2. 67)
C= - 1- ^ - (2.68)
since from Fig. 2-l(a), x2-x1 = b. X in this case is the wavelength in material
medium corresponding to a certain operating frequency.
45

Since this integration of (2. 67) could not be obtained in a closed form, computational
techniques are needed.
From the point of view of computational techniques a quadruple integral is
impractical to use. Therefore, the aim here is to reduce Eq. (2. 67) to a simpler
form. The simplification is going to be achieved by means of transformation of
coordinates. First change the limits of integration for x from x1->x2, to the
new limits 0 -* b using a new variable x'. This corresponds to a shift of axis
as in Fig. 2-4.
The new variable x' is defined by:
Xl = X-X1
Thus
dx' = dx.
Also, take
21' 2 2
p'= W~x-)2+(y-r)2 = /x,+x1 -Ky-r)2
(x+x) 2+(y-r)
To reduce (2. 67) to a double integral which is practical in numerical calculation, another coordinate transformation is now performed.
y-I7 =, x'-~ = a,
y+r77= a+, and x' + = b+
The above are equivalent to a rotation by 450, translation and magnification by
eI1 (see Appendix A and Fig. 2-5). Notice that
e o ~)'-jkk /(o4xl)2+'
f(%y, u v)= (CC2)cos 7+(C1+C2)cos (a+)
a a ^(,+x )2+,y2
This simplifies to
v){(l, 2co~(lC2co ) - jk /(^+x1)2+^2
f(%y, Ua, V)=((C-C2)coSa-(C )cos2+2, (2. 69)
\ 4. ~ - a- - " - - 91 _ a - //. \2.92'
where f is an even function in y, jI and v.
46

y Y'
Trans. Rcvr
_ _. ___.._
x10 x2 x'
FIG. 2-4: COORDINATE TRANSFORMATION

\ y y 1ft' ~
71l, Y2
\ ^0
FIG. 2-5: COORDINATE TRANSFORMATION FOR REDUCTION OF QUADRUPLE
INTEGRAL TO DOUBLE INTEGRAL.

Then:
a b -y (b- -c)jk -
III I I~c^-C~cos I(C +C )COS7]- dvd/-tdcydy
u=_ ) lW oi (-L) 1 C2 a 2 aed
~-~-~ 0 a (b-a)
2C3 a 0 ~a-7 (b1a) -jkV V(+x1) 2-jk2'
+ I c cC ) F cos7W-(C+C )cos eAh dzdijudd7
1f fo J-(b) -2 a 1 2 a 4
(2.70)
Performing integration in both v and,u the expression for U can be reduced to
f a b _-jk pt d
( 2(b-ca) l-C)(a-)cos7 -(C1+C) sin a e p dcd.
3 a 0 _jk p'
aCT a ny~e0
+ j2(b4'A)-C )(a-y)cos- y —(C+C )sin-t dardy. (2.71)
1 2 a e 2 P'
This is for X < 2a,
p' =vkc+xl)2+~,
(1+R) |7
u j-z J k-o
ko(l+R) O
C2-
and
ab _ _ ab 2/ 2
C3=- 2-2 2 1 — ab
R is the aperture reflection coefficient.
A further simplification can be made:
49

Consider:
a 0 -jk p'
Il=1 2(b+)C )-C2) (a- ) 4cos 7 (C ) dody
a b -jkoP"
= 2(b-)[ C 2)(a-y)cos 7 t -(Cl+C2)sin j e (2.72)
fC 2 aa 7 p2"
where
Ptt =(xl-f)Y+Y 2
Therefore, Eq. (2. 71) reduces to:
ab a, e-jk t -jll a
1 r^ f ^V1 ""-ko p'
u= J C C2)(a- cos-'Y (CC2)sina ]y(b -) + e- d
-o.C a 1 2 a PEc0IIa 7
(2. 73)
2. 4 Iteration Procedure
It was assumed at the beginning of the analysis that the illumination of the
transmitting aperture is not affected by the scattered field from the receiving
aperture. This is a very good assumption if the two apertures are sufficiently
far apart. The assumption is not justified for very close spacings of the
aperture.
The interaction between a transmitting and receiving antenna as a result of
scattering can be described as follows. If a transmitting antenna is considered
completely isolated and matched to its line, then when a receiving antenna is
introduced into the field of this antenna there will be a scattered field as can be
seen from Eq. (2. 36). This scattered field from the receiving aperture when
intercepted by the transmitting slot, in turn gives rise to a wave transmitted down
the feed line causing a mismatch. Also the transmitting slot gives rise to
scattered fields that again will be partly intercepted and partly scattered at the
50

receiving aperture. Thus the interaction is due to a multiple scattering and
absorption process as can be represented by Fig. 2-6.
An analysis corresponding to the above explanation can be made. From Eqs.
(2. 73), (2. 65) and (2. 66) then:
/iC>\ E = (1+R) l+R)
C 2r(1+R) o (I(+R) R ~1
C1 2 jcooa2 j R)L a2 j~r.'
C -C2=(1+R)C4,
and
C +C2=(1+R) C,
where
toe
C = -~ (2. 74)
4 jwj a2 jT
and
C5 = 22 + j~ (2. 75)
5 jwtAa 2 j 7
1 b -jkoj' _jko, JkoP"
U=(l+R) 5( 4 (a-y)coss aIC sin (b-a) C-(b ) + e dody=(I+R),
Ua+R f E4 i ae e -j0ip"
(2.76)
where
f 4(a -jkoP' -, e o
F= 4 E(a-,y) cos- P'y- - C5sin +(b-a) { Ip + - dady. (2.77)
00
Now assume that the magnitude of the aperture illumination is (1+R). Then
the received aperture illumination will be (1+R)F for the first iteration. For the
second iteration there will be (1+R)F3 added to the results for the first iteration
and so on. Therefore the received aperture illumination is:
UI= (l+R)F [l+F +F,4+F6+...,
or
UI=(1+R) F2 (2. 78)
51

This is the upper hemisphere in the problem
Incident wave
Received Waves
Trans. V Rcvr
FIG 2-6: MULTIPLE REFLECTION BETWEEN TRANSMITTER AND
RECEIVER TO REPRESENT INTERACTION BETWEEN THEM.

where UI stands for the modified U. This iterative process has been described
rather simply due to the fact that the reciprocity property has been used. The
factor F of coupling is the same from transmitter to the receiver as from the
receiver to transmitter. The absorbed power or energy in the transmitter due
to scattering from receiver has been ignored since it is small compared to the
original excitation.
2. 5 Consideration of the Short-Circuited Waveguide Backing the Slots
2. 5. 1 General Discussion
The problem under study uses a short-circuited waveguide section at both
transmitter and receiver. The shorted waveguide section is fed by means of
a coaxial-to-waveguide probe. Consider first the effect of the cavity alone as
shown in Fig. 2-7.
The arrangement of the axis for formulation is taken as in Fig. 2-1 except
for taking z to be positive in the reverse direction. Note that the input impedance
seen at z=0 or at the aperture of the homogeneously filled waveguide with material
described by 1l and e1 and shorted at ""l is:
Zg1(0) Zgltanh 7l I, (2. 79)
with the subscript "g" referring to the field in the guide and the subscript "1"
referring to waveguide Number 1. Also,
1'gl agl+ gl' (2. 80)
and
3 =27r/Xgl (2.81)
~g= = 2 /gl
where X is the guide wavelength in the filled guide. If the attenuation
gl
could be neglected, Eq. (2. 79) could be rewritten
2,K
Zgl(O)=Zgl tan i (2.82)
ggl
where
Zg1=.. =...._ (2.83)
gl l-(f/Z /l-(fc/f2853
53

y
lAl, El
Short
0 - -. z
FIG. 2-7: SHORT-CIRCUITED CAVITY.

The real parts of the fields in the waveguide are:
Re(E )=Usinwt sin 2 (z- (2. 84)
and
Re(Hx)= - cos t cos -(z-)]. (2.85)
Re( g g
The standing electric and magnetic waves in front of the short-circuit are displayed in Fig. 2-8 in terms of Xgl the guide wavelength for Section 1.
The impedance z (0) which is the input impedance at z=0 varies as shown
in Fig. 2-9. It is noticed from Fig. 2-9 that the impedance is zero at z=which is at the shorted end of the waveguide. It varies from -oo to +oo at
X-kgl/4 and then returns to zero again at t-Xgl/2 and so on. The pattern
shows resonance points at n Xgl/2 where n = 1, 2, 3,..., from the shorted
end and antiresonance at points n Xgl/4 where n=l, 2, 3,..., from the shorted
end.
It is to be remembered that these resonances and anti-resonances do not
take into consideration perturbations due to the material or the perturbation
due to the feed probe into the waveguide. These effects can be included as
was done by Adams1 and will be mentioned later.
For more physical insight, it is to be noted that if the waveguide is tuned
to a multiple of a half wavelength, then in this case if a short is placed on the
open end of the guide, as in Fig. 2-10, at z=0, nothing will be changed inside
the guide. The wave pattern in this special case can satisfy the boundary conditions at both ends. On the other hand for the case of any arbitrary length I,
a phase mismatch results between the standing wave building up in front of the
shorted end at z=e and that required by the boundary at z=0. Destructive
interference takes place between the incident waves and the reflected waves at
the aperture (z=0) and therefore the amplitude of the standing wave decreases.
Note that the impedance expression (2.79) remains the same since the only requirement is that a wave entering from the aperture at z=0 is totally reflected at
55

E H
Short
FIG. \ \ / / 2G I
z
FIG. 2-8: STANDING ELECTRIC AND MAGNETIC FIELDS IN CAVITY.

Inductive
jx
} I v - Reso ance
-Xgl \-(3Xgl/4) IQ -(Xgl/4)
O. JCapacitiv \ -(Xgl/2
-jx
-L... AntiResonance
FIG. 2-9: INPUT IMPEDANCE OF THE CAVITY AT z=0.

short end or z=1 and thus superposes with the returning wave to form a standing
wave pattern.
FIG. 2-10: CONFIGURATION OF WAVEGUIDE WHEN IT
FIG. 2-10: CONFIGURATION OF WAVEGUIDE WHEN IT IS
CLOSED AT BOTH ENDS.
2. 5.2 Feed Perturbations
Perturbation of Metal Probe: The equation for the correction of the frequency of resonance is
w-w 2xV 2 2 2 1
wo =abd os d k2a2 (2.86)
where
r is the resonance frequency obtained without perturbation
V is the volume of the probe
and
dl, d are as shown in the simplified equivalent circuit of a cavity slot
antenna (Fig. 2-11).
a and b are the dimensions of the cross section perpendicular to the z-axis
as shown in Fig. 2-1 at the beginning of the analysis.
58

Aperture Probe
G B — Short
GA BA G 0 - Bp
co I
t —--- d! ---.... —---— 3 -
FIG. 2-11: EQUIVALENT CIRCUIT FOR CAVITY SLOT ANTENNA.

Material Perturbation: The equation of material perturbation compensation
is given by:
W-to 2Vlrk 22 2 - d 7rd
o 1 k a-7r Am Cos2 1 e+.s 2 1 (2. 87)
abd 2 2 d, d
O ka
and for dielectric material, r =1 degenerates into
o 2V A~ 2 1
3- sin2 (2. 88)
W abd e d
where
V1 is the volume of a cylindrical hole in the material,
and
ao is the resonance frequency without perturbation.
It is to be noted that the resonance frequency of the cavity corresponds to the
resonance frequency for the coupling. This fact will be shown in a subsequent
chapter on experimental work. The Smith chart can also be used to obtain
frequencies as was done by Adams.
2.5.3 Resonance Frequencies and Bandwidth
As can be seen in Section 2. 5.2 where are several ways to obtain the unperturbed resonance frequencies. The resonance frequencies could be obtained
from the equation
Xg
= n 2. (n=1,2,3,...). (2.89)
Another way to do this is by consideration of Eq. (2. 79) which is:
Z l(O)= Zg tanh g.
gl ) gl gl
Then
1 1
Ygl(0 () Z(= zg coth (gl+ igl) (2.90)
Zgi Z g l (0) zl60
60

where
Ygl(0) is the input admittance to the cavity,
Z g = — 2 as defined before,
81 f
and
3 J # i - [-( ]. (2. 92)
Then
a
gi n -1 (2. 93)
Pgl [/l+tan26' +lJ
where tan 6 is the loss tangent of the material and
d.E
e" =' -je". (2.94)
From (2. 91), since
O o4/'oo
then
agl 42Per r {/1+tan26 -1} [r-f)2] (2. 95)
gl o/2 r r — ]
where
wr and e are the relative permeability and permittivity of the material.
For the case of dielectric material with a low loss tangent, Eq. (2. 95) can be
rewritten as:
gl o/tan6 [ ce) 2
gi V T k 0-Mer 4 f
since
k La ~JI
k (O / reC (2.96)
61

Therefore
a = (ktan6/2) -(). (2.97)
gl
Now Eq. (2. 90) becomes
1 cos(agl +gl )
Y(0) = -z
zgl sin( gl +jgl)f
but
/cosh2 a -ecos2 3 +sinh2ot a sin2 s
|coth(oa+j|3) = / g 22 gl12 gl2 (2.98)
sinh a -lcos j3 1+cosh a -lsin 2 1I
gl gl g l
Therefore
e e f f cosh2 a glcos23 jl+sinh2a % sin2 -glB
1Y 2(0)1 1_(02 c 02 g gggl gl gl 1 ]
|y20 ) L 2 ~o/r g1 —--- 2. 2. 2.(2.99)
1lA01Ar (I sinh a 2l cos glf +cosh a glsin 2glf
Since the energy delivered to the waveguide varies with I Y (0) | then by computing Eq. (2. 99) for various frequencies it is possible to obtain the unperturbed
resonance frequencies and also the bandwidth as will be shown in Chapter llI.
To approximate, roughly, for the bandwidth of a cavity with low loss
material in it and for small phase deviation A from resonance, insert in
Eq. (2. 99) the approximate forms;
2 2 2
sinh a -gl (agl)' cosha o -g 2 1
gl gi' g1,
sin2 Pbl -* A2 and cos23gla - 1.
Then
Y(O) 01 1- -2 2 1 (2.100)
2 odr f 2 + 2(a( 1)
gl
But for resonance, the approximation condition is that should be an integral
multiple of the half wavelength or,
62

a nX a
_ gi 2 g gn?r. (2.101)
gl 2 g
Consider that IY(0)2 falls to its half value, when the section is detuned from
resonance by a change in phase.
A z ag1
gl
+ n?r (2. 102)
2 -g
This is equivalent to a change in the resonant length by T A Zh/2. iut
AZh X A. ag
__h - hgl 41 -+ S1 I (2. 103)
2 27 2 - g
Alternatively, the phase shift can be produced by changing the input frequency
from its resonance point f to f +. Therefore
o o 2
A Z Af a
h h n _ 2gl
h 2 (2. 104)
fo gl
From another viewpoint Eq. (2. 104) indicates that the bandwidth for the case
using low loss material is approximately equal to the loss tangent (see Fig. 2-12
for symbols defined above).
2. 5. 4 Waveguide Wavelength Xg
In general there are multiple modes in the waveguide with dielectric material
filling of relative permittivity Er.
The general formula for waveguide wavelength is:
n o, (2. 105)
no n Xo 2
TrET ( 2a )
63

1.0
Ir3 "I.0. 5
rT- -A li "0-. AAfh
I] I f.S f
o
FIG. 2-12: RESONA NCE AND HALF-POWER BANDWIDTH.

where
X is the wavelength of the TE mode,
no no
X is the free-space wavelength, and
0
a is the dimension of the broadside of the waveguide.
For example, for the assumption of the dominant mode TE1 in the section filled
with the dielectric material,
Xo
XlO-r)g /
f is the cutoff frequency of the filled guide and
c
f =(2. 106)
c 2a s T
65

Chapter III
NUMERICAL ANALYSIS
3. 1 General
Since some of the formulation was not obtainable in a closed form, extensive use was made of The University of Michigan's Computing Center
for the evaluation of these integrals. The IBM 360/67 digital computer has
been used in most of these calculations.
The Fortran programs for each case were written and compiled using
the -Fortran compiler and both the Fortran programs and the object programs were stored in the IBM 360/67 memory. After the specific program
was cleared of errors and made operational, the Fortran program was destroyed from the file leaving only the object program, thus releasing storage
space. Then the object program was available for either a'batch job' or for
MTS (Michigan Terminal system through teleprinter).
3.2 Calculation of Reflection Coefficients at the Aperture
The first program was written to find the reflection coefficient, R, at the
aperture. Since there was already an analysis of this in the literature, rather
than duplicate that of Adams, a careful examination was made and some
slight variations were introduced.
The formulas which have been used in computations of the normalized
conductance and susceptance are:
222 2 a b
2/r(ka r) C
2N,2 j J2' - (b-) 3r(a —y)cos -L+a 1 sin ]
o X a bk a -?r Ja I 1
sin(ko / 2a)
-2 — dody (i)
66

22 a2 -a b
B 2/lr(ko af C) el Y]
Y 2(b-) 22 (a-y) +a - sin r
Yo
o r a b k2a2,-r2 E a
cos kVT2
d d y (ii)
Since both (i) and (ii) have a singularity at the origin, a small rectangle was
considered with sides of ratio Q to the original dimensions of the slot, or,
namely, Q a and Q b.
An approximate formulation was then developed for that area and these
were:
Cl C
22 2 -2 5.57+
2G 2 (koa b 1 D 1_ Q4_
Yo 7rk2a2_7r2 a'r r Q3( ) 4
+ 2.06Q-.616 Q (iii)
B r(koa - a b lb 1 a 2
Yo 02 ka-2 2 2b 2 2 a b
C1
( —1)
3 D
f 22 2 2 g b 22 2 1
-+ -(2r +k a )+ g- (k2a -7r2)
/a, b2,a 21 bd 2 2
(d-l)+ —( d-1 -T (2 a -7r Q
+a2 b 2 a
(iv)
-1 b
where 0, = tan -
0 7
f= log tan (-+ -)
67

01
g = log tan -
jb2 I
d = - +1 = sec 0
2 2
C= ka +iT
and 2 2
D = k a -7
1 o
The change was mainly in Eq. (iii) where 1/ JA was used rather thanV/Er.
The contribution for the small area around the origin was calculated by (iii)
and (iv) and was compared with the polar coordinate integration on the same
area. A good agreement was obtained. The rest of the integration was performed
by cartesian integration using Eqs. (i) and (ii). The two results were added together to give G/Yo and B/Yo
The change in Eq. (iii) has helped to stabilize the integration with respect
to both the parameter Q and the number of divisions to be taken in using
Simpson's rule. An example from the computer calculations is shown below
using Adams' formula (2. 54) and the corrected one (iii). Using Eq. (2. 54),
Adams, with other equations to calculate G/Yo, B/Yo gave these results.
Case (a): For b/a = 0. 444, Lr=3. 0, Er=3. 0, NS=30, NL= 50 where NS and
NL are the divisions for the integrations and Q = 0. 05,
G = 0.2542843 B = -0.9999304
G' = -0. 1218292 B' = -0.9656234
1 1
G2= -0. 3980840D-01 B = -0.5975625
2 2
Then FN =1. 100000, G/Yo = 0. 9264673D-01, B/Yo= -2. 563116,
where G and Gt are for the integration outside the small rectangle around the
1 1
origin. Similarly, B and Bt give the susceptance for outside the small rec1 1
tangle. G2 and B2 are for the integration over the small rectangle around
the origin, and FN is the normalized frequency.
FN = ka /. (3. 1)
Note: -0. 3980840D-01 - -0. 03980840
68

Case (b): For the same parameters as in Case (a), except for the
change Q = 0. 01;
G1 =0. 1476497 B1= -2. 064409
G' = -0. 2789158D-01 B' = -0. 3865364
1 1
G = -0. 1711596D-02' B = -0. 1254613
Then 2
FN=1. 100000, G/Y=O0. 1180465, B/Yo=-2. 576407
Case (c): For the same parameters as in Case (a), except for Q=0. 1;
G1 = 0.3695354 B = -0.2567721
G' = -0.2024414 B' = -1. 190663
~~~~~~1 1
Then G = -0. 1437410 B = -1.118208
Then 2 2
FN=1. 100000, G/Y=O0.2335299D-01, B/Yo=-2. 565642
Note that corresponding to the change in Q, there is a significant change in
G/Yo. This should not be so distinct. However, using Eq. (iii)
along with the others, the computer gave these results.
Case (1): For b/a = 0. 444,'r= 3. 0, er = 3. 0, NS = 30, NL = 50 and Q=0. 05.
G1= 0.2542843 B = -0. 9999304
G'- -0. 1218292 B' -0. 9656234
1 1
Then G2 = -0. 1326947D-01 B2= -0. 5975625
FN=1. 100000, G/Y =0. 1191857, B/Yo=-2. 563116
Case (2): For the same parameters as Case (1) except Q = 0. 01
G1= 0. 1476497 B = -2.064409
G' = -0. 2789158D-01 B' = -0. 3865364
1 1
G2= -0. 5705321D-03 B2= -0.1254613
Then
FN=1. 100000, G/Yo=0. 1191876, B/Yo=-2. 576407
-0. 1711596D-02 - -0. 001711596.
69

Case (3): For the same parameters as Case (1) except Q = 0. 1,
G1= 0. 3695354 B1= -0. 2567721
G= -0. 2024414 B' = -1. 190663
1 1
G2= -0. 4791366D-01 B2= -1. 118208
Then
FN=1. 100000, G/Yo=O. 1191803, B/Yo=-2. 56542.
Note that G/Yo and B/Yo are more stable with the change of Q. Also, in
choosing Q to be very small, both results come very closely together. Also,
a variation of the number of the divisions for the integration has been made
according to formula (iii) rather than (2. 54), Adams1, and thus gave these
results.
For b/a=0.444, Er=10. 0, Pr=10. 0, NS=30, NL=50 and Q= 0. 05,
then FN = 1. 5 gives G/Yo=0. 1134798D.01 and B/Yo=-4. 186671.
Keeping all the parameters the same as before except NS=100 and NL=100, then
FN = 1. 5 gives G/Yo=0. 1134785D-01 and B/Yo= -4. 189243.
These indicate that there is no error of 2. 5 percent in the value of G/Yo
which has been claimed when Eq. (2. 54) of Adams1 was used.
The reflection coefficient can be obtained from G/Yo and B/Yo:
1R = G/Yo +j B/Yo (3.2)
Using the computer, several values of Cr were considered as well as different
values of the ratio b/a as shown in Figs. 3-1 through 3-5.
3. 3 Calculations of the Aperture-to-Aperture Coupling Between Slots versus
Frequency
3. 3. 1. General Discussion
A computer program has been written to calculate the coupling between two
parallel slots in an infinite conducting ground plane. Since the formulas to predict
the coupling as shown in Chapter II could not be obtained in a closed form they
70

b/a.8
YO.2.02
__.02 __.._____________ —---.-.12-,b/a. -. -_ _,,,,,,, /-.27,,444;,,/ /
I / "7 - — I.1 / 15 1
FOR l./ ~,/r/ ---'/.41
B,,,;','"k/
I i''I'///''
II I' I Ii
I/ I
i~'i ~ ~ Normalized Frequency FN
-.85..-. —-_ —----- - --— + —-- -.4-
1.10 1.30 1.50 1.70 1.90
FIG. 3-1: APERTURE NORMALIZED CONDUCTANCE AND SUSCEPTANCE
FOR r=rl. 0, Cr=5.0, — (B/Yo) — (G/Y)
71

b/a
-^..8
-.07
b/a. 6
Yo' - -
-06g9/''a. - -.444
- -. /./. —'.5.37 —/ /\.7I
__ l[ / 6"'. 8 — d.
-'I.,,
—..0.'.- __ldl''' --.' J w1
*..../..
~ CO f. /N /
-.0 82 - I,,0, /,
I,
67 fii',
nil
2 hg
/l / //l'I/ //
1.10 1.30 1.50 1.70 1.90
FIG. 3-2: APERTURE NORMALIZED CONDUCTANCE AND SUSCEPTANCE
FOR.0 c=7., — (B/Y -(G/Y)
I / 8~',Ss72
I I /' / II
I' Ill!I I(I
I I t,
orirl,'::,:' I
1.10 13 1
72

b/a
G
02. 1 4
r.02 - 2-i - -.., ~ ~ ~ ~ 0 o...
-*13".'~~~~-&.''. Z ——.. --- i -. ~, / /so
-.91 / v'. /f,
-.29" 3 - v O,
-- /' -. ",.444
60/-" / / /
I / //
-1 I Z,,,
9 I f'Il
-.6 0 1.3 15 1/7 1.9 /0
I gI II
-/.7//._//.7
#fI!
/ / 8
"I I II/
If
I /
1 11',
i1 F1
I 3i
I I,
-.91 qII
/' i,
ss
~'r~ ~ Normalized Frequency FN
1.10 1.30 1.50 1.70 1.90
FIG. 3-3: APERTURE NORMALIZED CONDUCTANCE AND SUSCEPTANCE
FOR Pr=1.0, erlO=10. O, - - - (B/Y), - (G/Y).
73

b/a
G e r-s-. 0 - -.: —-- --. - -- - -, _ 4
444
W.3.1
15s 0 -. - i'..i
-.6- -,X0 11'
-..1............,0, ",,.' 5'
I /,* -.31' /'/-
# ~ ~</'/ \ ".7
I - - - // /' \ —- *9
/-/79,',';"- O,~
01.11 SO. 944
I I"
-1 lN m eFeey, III
/s I st t7
F O gill
ig
-.95 If
FOpr1. 1 30 1. 5 0 1. 70 1( g0

b/a
G A9 8
0 6
444
~~~~~~~~~~~~~^-. 02 -
-.18" b/a 1.,-' ^-'"- -'': 5.
P-5. - -,V -'
-.98~ - -, - z -9
J. O' Ol", O
r.
r w
- f I II
I R, -
I / i
ill',fit
=.50' 301'.-50 1.70 1.901
-- 75
i;
1.1 II_ |iI
1.10 111
+''/,,/''75

were evaluated by numerical calculation using Simpson's rule of integration.
The program was devised at the beginning to calculate the coupling versus
frequency (normalized frequency, FN = ka/7r). The parameters which were
considered variable in this case were the dimensions of the aperture'a' and'b', separation of the slots from center to center, the permeability and
permittivity of the material loading. The object program was stored in the
memory of the IBM 360/67 in such a way that the input data required by the
program were a, b, ur, Er, x1 (separation of center to center) and the
range of frequencies over which the calculations are required. For example,
if the frequency range is between a normalized frequency FN1 = 1.1 (which is
just over the cutoff frequency of the guide), and FN2= 2.0 (which is the upper
range of the frequency), then the program would call for repetitive calculations for the desired frequencies. The desired frequencies can be chosen with
a uniform separation DFN (divisions of FN). The cutoff frequency corresponds to the normalized frequency of 1.0.
For example, if it is desirable to make calculations at intervals of 0. 1,
from 1. 1 to 2. 0 (normalized frequency), then DFN = 0. 1. Then the number
of repetitive calculations = FN2-FN1/DFN = (2.0- 1.1)/0.1 = 9 and
FN2-FN1
NFN= +1. 5
DFN
where NFN is an integer,
FN = FN1 + (NFN-1),DFN,
where - is the Fortran multiplication symbol and NFN takes the values of 1
up to 10.
The relation between the normalized frequency and frequency depends
on the size of the guide. It also depends on the parameters of the material
loading.
76

Since
FN = ka/7r (3.3)
where k = 2/X (3.4)
and 2- Fo' f (3. 6)
0 x 2 0f
Then
f = GHz (3. 7)
2aV r" r
The cutoff frequency
0. 3
f = — ~3 GHz,(3.8)
c 2a, r r
where a is taken to be in meters, and FN is the normalized frequency
which is dimensionless. A scale can be obtained between FN and f for different values of waveguide parameters and material loading. Consider the
case of dielectric loading where M.= 1, Er= 15, a = 0. 02286 meter and
b = 0. 01016 meter (the dimensions of standard x-band waveguide). The cutoff
frequency in this case from Eq. (3. 8) is
0.3
f -= ~3 = 1.6942 GHz.
2x 0. 02286 x i. x 15.
In the following tables, correspondence between FN and f in the S-band
range is shown for different parameters of material loading.
77

TABLE 3.1: r= 1, r=15, a = 0.02286m., b=0.01016 m.
ka
FN =- f (GHz)
1.0 1.694
1.1 1.864
1.2 2.033
1.3 2.202
1.4 2.374
1.5 2.541
1.6 2.711
1.7 2.880
1.8 3.050
1.9 3.219
2.0 3.388
2.1 3.558
2.2 3.727
2.3 3.897
TABLE 3.2: jr=l, r= 11, a=0. 02286m., b=0. 01016m.
ka
FN =- f (GHz)
1.0 1.978
1.1 2.176
1.2 2.374
1.3 2.572
1.4 2.770
1.5 2.968
1.6 3.165
1.7 3.363
1.8 3.561
1.9 3.759
2.0 3.957
TABLE 3. 3: r=l, Cr=7, a=0. 02286m., b=0. 01016 m.
ka
FN =- f (GHz)
7T
1.0 2.480
1.1 2. 728
1.2 2.976
1.3 3.224
1.4 3.472
1.5 3.720
1.6 3.968
1.7 4.216
78

TABLE 3. 4: /,r=l, Er=l, a=0. 07136m., b=0. 034036m. (S-band)
ka
FN= - f (GHz)
1.0 2.102
1.1 2.312
1.2 2.522
1.3 2.733
1.4 2.943
1.5 3. 153
1.6 3.363
1.7 3.573
1.8 3.784
1.9 3.994
2.0 4.204
TABLE3. 5a: /r=l, Er=l, a=0. 02286m., b=0. 01016m. (X-band)
FN=ka f(GHz)
1.0 6.562
1.1 7.218
1.2 7.874
1.3 8.530
1.4 9.186
1. 5 9.843
1.6 10.499
1. 7 11. 155
1.8 11.811
1.9 12. 467
2.0 13. 123
TABLE 3.5b: ur=l, er=1, a=0.0157988, b=0. 0078994 (Ku-band)
FN=ka f (GHz)
7r. 1 10. 444
1.2 11.393
1.3 12.343
1.4 13.292
1.5 14.242
1.6 15. 191
1.7 16.140
1.8 17.090
1.9 18.039
2.0 18.989
S-band (2. 6 - 3. 95 GHz); X-band (8.2 - 12.4 GHz); Ku-band (12. 4 - 18. 0 GHz)
79

Note that FN=1. 0 corresponds to the cutoff frequency and the Table 3. 6
indicates how the cutoff frequency depends upon the values of Er of the dielectric
material for a fixed waveguide with transverse cross section of a=0. 02286,
b=0. 01016 meters (standard X-band waveguide).
TABLE 3. 6: wr = 1. 0, a=0. 02286m., b=0. 01016m. (X-band)
er fc (GHz)
1.0 6.5617
2.0 4. 6398
3. 0 3.7884
4.0 3.2808
5.0 2.9345
6.0 2.6788
7.0 2.4801
8.0 2.3199
9. 0 2. 1877
10.0 2.0750
11.0 1.9784
12.0 1.8942
13.0 1.8199
14.0 1. 7537
15.0 1. 6942
16.0 1. 6404
Therefore the cutoff frequency for standard X-band waveguide filled with
dielectric material versus Er is as shown in Fig. 3-6.
The coupling was calculated without the iterative procedure of Section 2. 4.
To compensate for the interaction between the two slots, especially if they are
very close to each other, the iteration factor was included in the evaluation.
On this basis the coupling was obtained. As expected the effect of the interaction
becomes more noticeable if the two slots are very close to each other in wavelengths. Of course, the separation in wavelength depends on the separation of
the guides in the physical layout. For a given physical separation it also
depends on the operating frequency. For example, if the physical separation were
kept constant and the operating frequency were swept higher, in effect, the separation
80

8.0
6.0
C O f c
(G Hz)
4.0
2. 0..|11 13.15 Er
FIG. 3-6: VARIATION OF CUTOFF FREQUENCY VS PERTTY OF LOADED XBAD
WAVEGUIDE, a=O. 02286m., b=0. 01016 m.

in wavelengths of the two apertures increases. Also it was noticed that the
value obtained with the iteration converges to the one obtained without iteration
for large separation of the slot antennas in wavelengths.
To formulate the coupling from an assumed illumination of the transmitting
aperture and to evaluate the corresponding illumination of the receiving aperture
consider the Poynting vector P
P=ExH,
P = Re (Ex H.
av
Therefore, the average received power in the receiving slot is:
b a
Pav=Re f (Usiny x) x (- - U'sin -Y) dydx
Sav I a CL a
b a
=-Re r f lu2 sin2 7 dydx,
where the negative sign indicates that the power flow is in the negative direction
of the z-axiL shown in Fig. 2-1.
ab U12
av 2 wu
where
f3=2r 2
Similarly, the transmitted power can be obtained.
2IU2
C (coupling in dB) = 10 log10 - =10 log10 amplitude 2
T amplitude of TR aperture ilum. I
glo0 amplitude of TR aperture illum
82

For the calculations the amplitude of the aperture illumination was assumed to
be 1+R.
Several curves of coupling versus frequency were evaluated for both empty
and loaded operation of the waveguide cavities. Of course, it should be noted
here that this is the coupling from aperture to aperture; the effect of the backing
cavities is not taken into consideration at this point but will be later on.
3.3.2 Empty Waveguide Case
For the case of empty waveguide the effect of the cavity is less pronounced
than in the case of loaded cavity. This is because in the case of the empty cavity
the bandwidth is wider than in the case of the dielectric loaded cavity.
The bandwidth for a low loss material is approximately equal to the loss
tangent. The graphs obtained for empty or air-filled cases show very good
agreement to the order of a fraction of dB with the theoretical counterpart as
in Lyon et al 2. Also it is within approximately 1 dB of the experimental
results as will be seen in Chapter IV. But for the case of the loaded slot, the
effect of the cavity has to be added and then the comparison can be made. The
coupling versus normalized frequency for the case of empty S-band waveguide of
dimensions a=0. 07136 m., b=0. 034036 m., is shown in Fig. 3-7. The separation
of the two slots (center-to-center) is 0. 075 m., and the range of variation of
frequency is from 1. 1 to 2. 0 on a normalized scale. The relation between the
normalized frequency and frequency in GHz is given in Table 3. 4.
In Fig. 3-8 the coupling versus frequency is shown for the case of the empty
X-band waveguide of dimensions a=0. 02286 m., b=0. 01016m., for a separation
of 0. 114m., and a range of frequency between 1. 3 and 1. 9 normalized frequency
(see Table 3. 5 for corresponding real frequencies). Notice in both cases that
the coupling pattern drops with an increase of frequency according to a 6 dB/
octave frequency characteristic. Also, the differences between curves, with
and without interaction between the two slots, decreases as the separation between the two slots increases (see Appendix B ).
83

-15
-16
-17
-18
v, -19
I -20
U -21
-22
-23
1.1 1.2 1.3 1.4 i5 16. 7 1.8 1.9 2.0 FN
FIG. 3-7: STANDARD S-BAND COUPLING VS NORMALIZED FREQUENCY,
a=0. 07136 m., b=0. 034036 m., xl= 0. 075 m.,,r=l. 0, er= 1. 0

-28
-29
-30
| -31
Q -33
-34
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN
FIG. 3-8: STANDARD X-BAND COUPLING VS NORMALIZED FREQUENCY,
a=0. 02286m., b=0. 01016m., xl=0. 114m., Ar=1. 0, er= 1. 0.

Then the coupling for a pair of Ku-band slots spaced at 0. 114 m. from each
other was studied. The coupling was calculated at different normalized frequencies(see Fig. 3-9). The results agree with Fig. 4-6(b) in the chapter on
experimental investigations.
3. 3. 3 Dielectric Loaded Case
Several graphical curves have been evaluated for standard X-band waveguide
with dimensions a=0. 02286 m., b=0. 01016m., filled with dielectric material of
er= 2, 3, 4, 5, 7, 11, 15 and 20 (see Fig. 3-10, a-h ). Notice that x1 = 0. 013m.,
corresponds to two slots put next to each other or metal-to-metal contact. It
is very interesting to note that the coupling in this case does not follow the
6 dB/octave of frequency trend as in the case of empty slots. The variation
depends on the material loading parameters. It is to be noticed that the lower
the permittivity number, the more the variation is restricted; in some cases it
is about constant, as in Fig. 3-9. For the higher electric permittivity the
curve goes to a minimum with increasing frequency and then rises. For a
large portion of the frequency band the variation is confined to about 2 dB. Also
since the coupling level is low in this case the iteration process has little
effect (see Appendix B). Of course, it should be remembered that this coupling
is aperture-to-aperture and does not take into consideration the resonance
cavity which is feeding it.
3. 4 Coupling Aperture to Aperture versus Spacing
3.. 4. 1 General Discussion
The computer program of Section 3. 3. 1 was next modified to permit
variation in frequency for a fixed spacing and to obtain the corresponding
coupling level. Also for a fixed frequency the program provides for variation
in spacing between the slots and then the evaluation of the corresponding coupling
level. Therefore, upon supplying the proper data, the computer will evaluate
coupling versus frequency or coupling versus distance according to need.
86

-34
-35
co' -37
0
-38
-39
1. 3 L:4 11.5 1.6 1.7 1. 8 1.9 2.0 2. FN
FIG. 3-9: STANDARD Ku-BAND COUPLING, xl=0. 114m., VS FREQUENCY,
a=0. 015988m., b=O. 0078994m.,,ur=1.0, Cr=1. 0.

-11
-12
u -15
1.1 1.'2 1.3 1 4 15 1:.6.7 1.8 19 2'0 FN1
FIG. 3-10(a): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS
a=0. 02286m., b=0. 01016m., x1=0. 013m.,r=l. 0, Er=2. 0.

68
Aperture Coupling (dB)
00', 0' C..
oh
O
tO
00i
- 3
X,
vo 0
P U)
o P I
m o \
tl I 1
r d

-17
-18
-19'. -20
bJ
-21
0
-22
~ -23
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN
FIG. 3-10(c): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS
a=0. 02286m., b=0. 0 -016m., x1=0. 013m., ir=1. 0, er=4. 0.

-18
-19
-20
^ g> -21 * < 4
o -22
0
-23. -24.
-'1 ~............. __ L 1. _ ~,,;, ___
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FN
FIG. 3-10(d): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS
a=0. 02286m., b=0. 01016m., x1=0. 013m., ur^=1. 0, er=5. 0.

-19
-20
-21
P -22
X\ =fl -23
-25
1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 17 1. 8 1. 9 2. o 2 1 FN
FIG. 3-10(e): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS
a=0.02286m., b=0.01016m., xl=0. 013m.,,r=l, er=7.

-19
-20
-21
-22
0 -23
-25
-26
-27
-28
1. 1 1.2 1.3 1:4.5 1.6 17 1.8 9 2.0 2.1 2.2 F
FIG. 3-10(f): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS
a=0.02286m., b=0.01016m., x1=0.013m., r=l, Er=11.

Aperture Coupling (dB)
CD NC 10 I IS N M I3! N
I0 CD o0 J C) Cr1,4 X IN 1- 0 CD
CO
6 t -
0
0 l_3 - A
M ~ W
O C
0 0
oo t /
M
r /

-1
-20o
-21 \
-22
-23
E -24..g -25.C? \
0
o -26
-27-
-28
-29 4
-30
-31;
1.1 ^+2 its it f'5 rt I^T I^S 1*^ 1^ t _4 i6 N
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2:2 2.3 2:4 2.5 FN
FIG. 3-10(h): APERTURE TO APERTURE E-PLANE COUPLING OF LOADED SLOTS
a=0.02286m., b=0.01016m.,x=0.013m.,.=l, er=20.

The increment in spacing for calculations was taken to be smaller in the
range from 0. IX up to 2X, namely every 0. 1X to show in more detail the effect
of interaction between the slots which is more pronounced for closer spacings.
From a spacing of 2X up to 10X it was varied in steps of 1X. Also the
modified program included the program of Section 3. 2 as a subroutine which
could be called upon to evaluate the aperture conductance and susceptance; also
the aperture reflection coefficient can be obtained as shown in the examples of
the computer output given in Appendix B.
3.4.2 Graphs
3. 4. 2(a): Figure 3-11 which is drawn on semi-logarithmic paper represents
the coupling in dB versus spacing in wavelength. (see also Appendix B ).
As has been mentioned before the effect of the interaction between the two
slots is noticeable for closer spacings. For slots further apart in wavelength
the differences due to interaction is very small. This curve agrees within a
fraction of a dB with Fig. 4-4 of Lyon et al2.
3. 4. 2(b): For the loaded case of aperture-to-aperture coupling it is shown
in Fig. 3-12 that the approximation of 6 dB/octave of spacing is quite good
Notice in this case spacings can be smaller than 0. 4X as in Fig. 3-11; for
this case the parameters of the guide are a=0. 02286m., b=0. 01016m., and
it is filled with dielectric material with relative permittivity of 11. 0. The
operating frequency in this case is S-band; therefore it was possible to go down
to 0. 1 X0, where Xo is in the free-space wavelength (Fig. 3-12).
3. 5 Evaluation of the Cavity Effect
Due to the loaded cavities which are backing both transmitter
and receiver slots, as shown in Fig. 4-2 of Chapter IV, the resonance
and anti-resonance points become very distinct. These waveguides
can be considered as a five-sided cavity as was explained in Chapter II.
An approximate way for rapid calculations of the resonance and antiresonance points can be made according to the analysis in Chapter II.
96

For resonance points:
Q 2 =n-g (n= 1,2,3,... )
where Q is as defined before (length of the cavity) and X is the waveguide
wavelength.
For anti-resonance:
x
I = n-, (n= 1,2,3,... ).
A formula has been developed to predict the resonant frequency; likewise another
one for anti-resonance could be developed.
For resonance:
x,
Q=n 2
but
O__ 1 3 x 1010
X P i/UrTTU'' 2
41-( ) c
3x 10 1
n- =n
2 ^
Finally
2
fres!L54n + f2 GHz (n=l, 2, 3...) (3.9)
res. J — r j
where Q is in meters,
1 0. 3 0.15
f _1 0.3 0.15 GHz, (3.10)
2a4T' 2a 7! a Prcr
where a is in meters. Therefore
0.15 n2 1
f 0.15 In2 + 1 (3.11)
res. 2 ^ a2
99

It is to be noted in the empty or unloaded cavity,r=1 and er=l. Therefore
Eq. (3.11) becomes
f = 0.15 +- (n = 12,3,...) (3.12)
res.'2 2
a2 a
By comparing (3. 11) and (3. 12) note that the resonance frequency is determined
by the broadside length of the aperture'a' and the depth of the cavity. It also
depends on the parameters of the material loading.
Another way to find the resonance of the cavity and at the same time be able
to predict the bandwidth is by means of Eq. (2. 79) in Chapter II which is:
Z(0) = Zg tanh (alg +j )
or
Y(O) = coth (1 a+ j g),
and
= 1 i ~osh2a lg cos2 31g +sinh a Slg sin 2lg]
Y2(0)1 __lg =g Fg Ig' j ~ I 2 2 i 2.-2 -- 2 —
Z1 2 sinh a S cos 31 +cosh a1 g sin 1
Ig| ig 1Ig lg Ig9
This last equation is proportional to the power received. A computer program
has been written for this equation to evaluate IY(O)2 versus frequency. The
output of the computer is best presented by means of the plots shown in Figs. 3-13
to 3-16. These graphs were plotted directly by means of the University of Michigan
Digital Plotting System. An example for calculating the resonant frequency corresponding to the half-wavelength method is given below.
Example 1
For a = 0. 02286m., b=0. 01016m., -= 0. 031m., Er= 7. 0, = 1. 0,
f = 2. 48008 GHz.
c
100

For n = 1
0.15 1
fres. + = 3. 08163 GHz.
I1res./2 ~ a2
For n= 2
f2 res = 4. 41935 GHz.
Example 2
For a=0, 02286 m., b=0. 01016m., e=0. 031m.,,r= 1. 0, er = 11. 0
f = 1.97842.
C
For n = 1
fl res. = 2. 4583 GHz.
For n= 2
f = 3. 5254 GHz.
2 res.
Now consider the perturbation formulas applying to this case. For
the metal probe and Lr=l, Er= 11.0,
diameter = 0. 1143 cm. (metal probe)
projected length inside the guide = 0. 47498 cm.
d = 2. 48792 cm d2= 0. 60706 cm
d = 3. 09498 cm.
Here dl, d2 and d are as shown in Fig. 2-11 of Chapter II.
Then
W-wo 2V r 2 2rd1 2 1
-- -+- Icos - cos2
030 abd k d 2 s
Now take
f = 2. 4583
then 2 x f
ten k k = 3x010 E = 17. 0674876.
101

Now 2
V = 7r (da) x length = 0. 0060943 cm3
and
2V 2x 0. 0061 2 2 2
abd 2. 286x1. 016x3.10 o2(29 39)- 072286)2 os44
(17. 07 x 2. 286)2
= 0. 0017 [sin2(19. 39)-0. 00645 sin2(54. 69]
= 0. 0017E0. 33199)2-0. 00645 x (0. 816)2]
= 0. 00018 (this is very small).
For the material filled hole or cylinder around the probe consider it to be
filled with polystyrene having a dielectric constant of 2. 54 and at the range of
frequencies considered.
W)-o 2V Ae 2 s id
"^ ax _sin -1
abd e d'Jo
diameter of the cylinder = 0. 55118 x 102 m.
length of the hole = 1.016 x 10-2 m.
In the above A e is negative or positive corresponding to changing to a lower
or higher dielectric constant respectively. The volume is easily found to be
V da. =0.2423 cm3.
Also
2V
-- = 0.0675.
abd
Also
A _ 2. 54 - 11
-=ks - 11 = -0. 7691.
e 11
Therefore
W-W
= -0. 0675 -0. 7691 x cos2 (54. 69) = 0.0173435'o
The total perturbation = 0. 0175 or 1. 75 percent.
102

Therefore fl should be = 2. 4583 (1+0. 0175)= 2. 501 GHz. For the other peak
the material perturbation is the same but the probe perturbation changes because
of the factor k in the formula. The metal probe perturbation at 3. 5254 GHz is
now found:
0. 00017 x. 33199)2 9. 8596 x (0. 816)2'o L(24. 48)2 x2.286
Then for 3. 5254 GHz the total perturbations are 0. 017522. This is not much
different from that at 2. 4583 GHz. Therefore,
f2 = 3. 5254 (1+0. 017522) = 3. 587 GHz.
Notice that these values are very close to the experimental measurements
(see Fig. 4-7 in Ch. IV). There is good agreement considering the errors in
measuring the dimensions and the experimental errors of instrumentation.
Also, there might be some variation due to a slight change in er with a change
of frequency.
Example 3
For a = 0. 02286m., b=0. 01016m., I = 0. 031 m., ur = 1. 0, r = 15. 0
fc = 1. 6942 GHz (cutoff frequency).
Using Eq. (3. 11) the resonance frequencies are:
For n = 1
f = 2. 105 GHz
1 res.
For n = 2
fre = 3. 01899 GHz
2 res.
103

Example No. 4(a)
For a = 0. 02286 m., b= 0. 01016 m., I = 0.1015 m., jur=l. 0, er=15.0
the following corresponding resonant frequencies using Eq. (3. 11) are:
n = 1 f= 1. 7368
n = 2 f2 = 1.8583
n = 3 f3 = 2.0448
n = 4 f4 = 2.2804
n = 5 f5 =2.5516
n = 6 f6 = 2.8482
n = 7 f7 = 3.1631
n = 8 f8= 3.4913
n=9 f9=3.8293
n = 10 f0 = 4. 1750
Example No. 4(b)
For a = 0. 02286m., b=0. 01016m., 1=0. 1015m., lp=1.0, r=11. 0, the
resonance frequencies using (3. 11) are
n = 1 f1 = 2.027
n = 2 f2 = 2.169
n = 3 f3 = 2.388
n = 4 f4 = 2.663
n = 5 f5 = 2.979
n = 6 f6= 3.325
n = 7 f7 = 3. 694
n = 8 f8 = 4. 077
Example No. 4(c)
For a = 0. 02286m., b=0. 01016m., Q=0. 1015m., p,=1. 0, Er= 7. 0, the
resonance frequencies using (3.11) are:
n = 1 fl = 2.54
n = 2 f2 = 2. 72
n = 3 f3 = 2.99
n = 4 f4 = 3.338
n = 5 f5= 3. 735
Note that these resonance frequencies agree with the digital plots shown.
104

Co
0o
0 3.08 4.418'2 48 2.98 3.47 3.97 4.46'4.96 5. 46 5.95
Frequency (GHz)
0o
Co
a=0. 02286m
d= cavity length, 0. 031m.
n=l n=2 Ar=1 0
0. I I E =7.0
o
C1 I - ka/yr)
So
bD
0'.4
0 \J
Normalized Frequency (FN = ka/7r)
~- -- t. ~:...j m. I I. J I, ~' — \ —— I ——.'1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40
FIG. 3-13(a): RESONANCE OF THE CAVITY (Digital Plot).

or
) 2.46 3.53
o S. 4.. I
u1[98 2.3 2.77 3.17 B.56 3.96 4.35 4.75
Frequency (GHz)
o n=1 n=2 a= 0. 02286m.
d= 0. 031m.? - - ) 1 A tr = 1 -
- er=11 0
o0
0
0
do.
0
CT IsNormalized Frequency (FN = ka /r)
0,I 4.'1o0 1.20 1.40 1.60 1.80 2.00 2.20 2.40
FIG 3-13(b): RESONANCE OF THE CAVITY (Digital Plot).
FIG 3-13(b)- RESONANCE OF THE CAVITY (Digital Plot)

0o
o
2.105 3.02._ _ 4 - -. — _t. -4 +- - _ — - - ___ + -
"I 69 2.03 2.37 2.71 3.05 3.39 3.73 4.07
Frequency (GHz)
Co
so=^~~~~~~~~~~0.~ ~~a=0. 02286m.
d=0. 031m.
/r=l. 0
e=15.0
o n=1 n=2
cq
~ 0o I \
0
L)
~; L + Normalized Frequency (FN=ka/7r)
1.00 1.20 1.- - 1.60 1.80 2.00 2.20 2.40
00FIG. 3-13(c) RESONANCE OF THE CAVITY (Digital P 2.20 2.
FIG. 3-13(c): RESONANCE OF THE CAVITY (Digital Plot).

C3
C)
C — 4 - ---- ---- --- - ----- - -— 4 — ---- ------ ------ ---- ---
2 148 2.98 3.47 3.97 4.46 4.96 5.46 5.95
Frequency (GHz)
-- n=3 n=4
~C> 8~~~~n=5 n =6 n=7,+^"~~~~~'|~~~ | + | at a=0. 02286m.
d=0. 1015m.
=r 1. 0
r~~~~~~~~~~~~,~ ~~-l —
~7Er=7.O
1 00 1.20 1.40 1.60 1.80 0.00
0
C) INormalized Frequency (FN = ka/7-)
11.00 1.20 1.10 1.60 1.80 2.00 2.20 2.140
FIG 3-14(a): RESONANCE OF THE CAVITY (Digital Plot).

C>
", 98 2.37 2.77 3.17 3.56 3.96 4.35 4.75
Frequency (GHz).. n=3 n=4 n=5 n6 n7
o n=6
C1 n=7
o | l l | a=0. 02286m.
d=O. 1015m.
/r= 1. 0
er =11.0
0I
co
bJ
0
C0
o Normalized Frequency (FN = ka/7)'1.00 1.0 1. 40 1.60 1.80 2.00 2.20 2.40
FIG. 3-14(b): RESONANCE OF THE CAVITY (Digital Plot).

o
Co
CD
0 ----- t - ---- --- -4-. - -4 ----- -4 —- ------ + -
1 69 2.03 2.37 2.71 3.05 3.39 3.73 4.07
Frequency (GHz)
o n=3 n=4
n=5
o11=3 114 1 n15 n=6 n=7 n=8
a=0. 02286m.
d=0. 1015m.
C' Ad II Ad I Ic II II Er=15. 0
CD
1.0 1.20 1.40 1.60 1.80 2.00 2.20 2.40
0
Co
Co
C'
CI
1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40
FIG. 3-14(c): RESONANCE OF THE CAVITY (Digital Plot).

c)
o'2 0.25 2i0.32 2i0.39 210.46 210.52 210.59 210.66 210.73
Frequency (GHz) f x 102
~ | \ a=0. 02286m.
X / \ d=0. 031m.
CM(vj"N~~~~~~~~ / \ Mr=1.0
r= 15.
-0 ~/ \ n=l
c0
C
o
0 —\
o\
g / Normalized Frequency FN x 102
124.10 124.14 124.18 124.22 124.26 124.30 124.34 124.38
FIG. 3-15(a): AN EXPANDED DIGITAL PLOT FOR THE RESONANCE PEAKS.

cl
0
^ —---- 1 —-— + —-------— ~ —---------- I -+ --— 4 —---— + —----— 4 —---- -— + —
3)1.57 301.66 301.74 301.82 301.91 301.99 302.08 302.15
Frequency (GHz) f x 102
0 / \ a=0.
0\- a=O. 02286m.
d=0.031m.../ \ r= 1.0
r=15.0
o / n=2
c oo: 0
C1. 1
0 /
Co! Normalized Frequency FN x 102
- -- ---- - ------- - --— I — I
178.00 178.05 178.10 178.15 178.20 178.25 178.30 178.35
FIG. 3-15(b): AN EXPANDED DIGITAL PLOT FOR THE RESONANCE PEAKS.

C)
C)
- i- -— + — -— + —F-+- -+ — - -— 4 —----.. —— +- - --- - -4 -- -
1q4. 15 204.24 204.32 204.4 1 204.49 2 201.58 201.66 204.75
Frequency (GHz) f x10
o
Co
a=0. 02286m.
b=0. 0106m.
d=0. 1015m.
o /r \ =1.0
-- ~ / ~\ Er=15. 0
n=3
bD CM
o / \
0 \
C)
CMU
o Normalized Frequency (FN= ka/T7)
- ___________FN x 102'120.50 120.55 120.60 120.65 120.70 120.75 120.80 120.85
FIG. 3-16(a): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

C)
Co??7.70 227.77 227.84 227.91 227.97 228.04 228.11 228.18
Frequency (GHz) f x 102
o a=0. 02286m.."' b=0.0106m.
d=0. 1015m.
r/=1. 0
r =15.0
n=4
o
s\
o.3
C).
/ Normalized Frequency (FN=ka/7r)
=~irI ~ I I _+ + _ +FN x 102
cu. - i —----- 4 --'4-I ---- -—. - --.... —- 4 -. --------
134.40 134.44 134.48 134.52 134.56 134.60 134.64 134.68
FIG. 3-16(b): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

0o
Co
i4h. 81 254.89 254.98 255.06 2.15 255.23 255.32 255. 40
Frequency (GHz) fxl02
0C
I~U~~~~~~~~~~t,"~ \ ~a=0. 02286m.
b=0. 0106m.
d=0. 1015m.
/ \r=1.0
C/ \ E=515. 0
_ i-/ \ n5
i'r
0 \
)
O
o Normalized Frequency (FN= ka/ir)
^/O~~~~ ~~~~FN x 10
(1J 1 4 - 4_ | __ — 4 4 A 4
150.40 150. 45) 150.50 150.55 150.60 150.65 150.70 150.75
FIG. 3-16(c): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

C)
0 - - - ---------- ----- --- -- -------- - --- ---
"2 4.16 284.54 284.63 284.71 284.80 284.88 284.97 285.05
Frequency (GHz) f x 102
a=0.02286m.
o b=0.0106m.
d=0. 1015m.
1.0
Er 15.0
C)^~ / \ ~~~~n=6
Cn~~~~~C
0
C I
0)
CD
C)
0 \
0 /Normalized Frequency
C0 /FN x 102'167.90 167.95 168.00 168.05 168.10 168.15 168.20 168.25
FIG. 3-16(d): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

0
-— t —--- ---- 4 -4 i-t — --- -4 4 4 —— t- -4 4 4 —C -~-. —
5.97 316.04 316.11 316.17 316.24 316.31 316.38 316.I
Frequency (GHz) f x 102
0
0
a=0. 02286m.
b=0. 0106m.
CDP/~~ \ d=0. 1015m.
I \ er=15.0
Cq ~~~~~~~~~~~~~~~~~~n=7
~~~~
br"
0
U,
CL ~~~~~~~~~~~Normaliz-,al Frequency FN x 102
M- / V
fi~~ —t- /-t- \-f —-3- -~ —------ ~- ---— t- -t —— t-
o Normalized Frequency FN x io2'leI.50 1854 185.58 186.62 186.66 186.70 186.74 18.78
FIG. 3-16(e): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

0
-. -+ — f —— _ —-f — — I+- ----- --- --, — —,
318.814 348.91 348.97 349.04 349.11 344918 349.24 349.31
Frequency (GHz) fx 102
Co a=0. 02286m.
0~"~~~~~~~~~~~~~~~ ~~b=0. 0106m.
d=0. 1015m.
A=1.0
r
^' — ^E 15. 0
o Cr^l5*O
~ol~~~~~ X \ ~~~~~~~~n=8
00 ~ 0
bfD
0
O ~~~~~~~~~Normalized Frequency FNx io2
a*~ ~ ~~~~~~~~~ -- / 4
~I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
" 1 "/
Co
0 / ~~~~~~~Normalized Frequency FNx102'205.90 205.9'4 205.98 206.02 206.06 206.10 206.1m 206.18
FIG. 3-16(t): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

CD
c::. c- 1. I — A —--— f- - -— 4 — -— + —--- — +t — — + —— f- ----- I I-I —----— ~~ —---
3 2.72 382.79 382.86 382.93 382.99 383.06 383.13 383.20
Frequency (GHz) f x 102
C1
o) a=0. 02286m.
b=0. 0106m.
d=0. 1015m.
r = 1. 0
o E =15.0
0
r
n=9
CTI
0
0
0
C)
U-.
0\
CA
09
0 Normalized Frequency FN x 102
-4 i 4 —---- -+ -t —~ —-- — 4- - 1 - -- - -4- - - - f- - 4 -4 --'225.90 225.94 225.98 226.02 226.06 2S6.10 226.14 226.18
FIG. 3-16(g): AN EXPANDED DIGITAL PLOT FOR RESONANCE PEAK.

Example No. 5.
For a= 5t, b =2", x = 3/2 ", r = 6. 63 and ~r = 12. 6,
fc = 129.2246 MHz.
Using Eq. (3. 11) the resonance frequencies are:
n = 1 fl = 449.715114 MHz
n = 2 f2 = 871.135869 MHz
n = 3 f3= 1.298692 GHz
3. 6 Evaluation of Coupling
From Sections 3. 3 and 3. 5 the coupling of two loaded guides opening into
a perfectly conducting infinite ground plane can be obtained. From Eq. (2. 99)
in Chapter I which is:
2 2 2 2
2 1 [cosh a, 9I cos -1 I+sinh ag Isin 3gQ
y(O)| ig lgIg g
Z sinh 2 lg2 cos 2l +cosh 2a sin I lg J
1Zlg 12 Lsmh ~lg lg 9l " J
the power received is proportional to this factor. Therefore by multiplication
of this equation by an appropriate factor and adding to the coupling from aperture
to aperture in dB the total coupling can be obtained. Also the resonance frequency
and the theoretical bandwidth can be obtained. The effect on the resonance frequencies due to the material perturbation and metal probe perturbation might be
taken into consideration as in Example No. 2
120

Chapter IV
EXPERIMENTAL INVESTIGATION
4. 1 General Discussion
All the experimental work was performed in an anechoic chamber. The
chamber dimensions were 50' x 30' x 15', with an aluminum ground plane,
12' square by 1/8" thick mounted in the center of one of the 30' walls (see
Fig. 4-1). The nine individual sections of the ground plane were joined by
2" wide aluminum tape to form an electrically continuous surface at microwave frequencies. The center section has a removable 2'x3' section for the
test antennas. Within the 2'x3' part there was a removable circular disk of
14" diameter. Inside the disk there was a rectangular hole of dimensions
10" x 3" in which the slots could be mounted. Also, there was an additional
fixture to make the X-band slots to fit in the mounting as shown in Fig. 4-2.
The ceiling and the floor were covered with B. F. Goodrich HV-4,
vinyl-covered, four-inch hairflex microwave absorber. The rear 30' walls
and the 50' walls are covered with VHP-18, pyramidalabsorber. The center of the
rear wall was covered with VHP-26 absorber. The chamber had been lined with
interlocking aluminum foil sheets to provide a known uniform termination for
the absorbing material and to shield the room from external signals.
The frequency range of the chamber is 500 MHz - 50 GHz and has a nominal
reflection coefficient, for normal incidence, of -70 dB for X-band and higher
frequencies,tapering off exponentially to -40 dB for lower frequencies.
The accuracy of coupling measurements was approximately within the
range of + 1 dB (i. e., for the swept frequency use). For the case where the
swept frequency generator was not used, both transmitter and receiver were
tuned by means of a double-stub tuner, and the accuracy was a little less than
the former (i. e., within + 2 dB ).
121

"~~~~~~~~~~&:"~~~:~j*:@* ~~ ~s-~z:.. 424K''K'4"'~~~~~~~~~~~~~~~~~' ~;:s''4444~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~''::=:
K" 444....~~~~~~~~,, ~~~~,**,~~~94//'*~~~~~'~4w 94/2 ~ ~:::
K> 42'> 4K'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'d~":; 9 /'' K #t K+ 4' A
4~~~~~~~~~~~~~~~~~~~~~~O~:LSi./> K'.7 {~~~~~~~~~~~~~~~~~~~//4 t4 /4 7 4~~~~~~~~~~~~~~~~~ 4~~~~~~44444K' >4>A4'2 44..,.~~~~~~~~~~~~~~~~~~~~~~;;;~:::-::::
>494444""~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:j~-::::::::
~~~~~~~~~~~~ 4<~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:
KK'4K44K"K'K'K'KKKKK>KK'K44'K'%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~h:::-~i;"-:::'#''4K4K4
B~~~~~~~~~~~444>7
122~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'_'::::::

T:EST ANTENNA MOUT.
(Comparison soale is in inhes. )
123

-;~-~~~-
sj:i S~xZk:
iili-
RsXB —LJ
i O':iae:.1,-:~~~
e:~i~a:-z:::ni:Ds9i5li~~:ij
ac.::~a,:;~-,:j~:~C~~IOE:$gi:.: —~:- ~l p:_,aIi-::::::::r::::
ij::::::-:::::::-~
Bi:::::~:-:-::::::::
-x:m~~~w,;i~:~:.: a::s:al:i:t-:'::::::::::::::::::i-_:ai~fRPil4-I: —''ifi\i::-::~;:'::::::i~::: —_i
I; —::::::
d-i
B)i,:'-di-iP-r::::-:-::::::: ::-::-::-: —-:-::;,:-r;:_-i
rl: I 9:i-:r:::::i:-:;:-:::-:::I:::-::::::-:rl: —:: —:: I:::: _:'ia: —:
I:i::_:::::-::;:.;:j-~::::;-:- -:-:-.:;::-:_:_-:;::::i:::::: ::ii:-:l:b
bi_;i:i;i:::i-i _i-:-::-:::,.:::::::-:.: -:-i:-.-::
B1
"-ia;isk-~:;bi
b:::
a,,,,,i-,,,
i~
"1ID
I lii
a x
".
~c —
C:i:3
n:j~:s~,;1~""j'"'
r-r.;:,:::::-a?:
~:r iBE-"
is
i~~::~~ ~:i~~
Z
D
-~:::
-:-,,,,.~/i:-.:.:...j,,,,,,,,,,,
F]EQ. 42(1: AgIhbNGEMENTS Fgt LePDED XHND IAVE6ES.
(Csunprison nc3le is i irnes. )
%24

."VIG 42(c- RRNGEMENTS FOR STANDA~RD S-~BAND'WAVEGUIDES.
oparison scale sinchesb
125

4. 2 Coupling Measurements
4. 2. 1 Swept Frequency
Most of the experiments in this section were made in the S-band frequency
range, or, more specifically, the measurements were taken between 2. 0 and
4. 0 GHz. Also some experiments were carried out for swept frequencies in
the X-band range of 8. 2 to 12. 4 GHz. For the X-band measurements, it was
found that in feeding a loaded guide with X-band signal there is clear indication
that there is more than the dominant mode propagating. This was observed
from the shape of the radiation patterns. This was to be expected since the
X-band range of frequencies is far above the cutoff frequency of the loaded
guide.
At an early stage of the experimental work a dielectric powder was used
to load the X-band waveguide. It was then realized that it was very sensitive
to any movements. By moving the transmitting antenna from one position to
another, it seemed that the granules of the dielectric filling moved around.
For this reason it was difficult to get a repeatibility in the graphs. A substitute
to avoid this problem was a solid dielectric material which is machined to fit
the size of the X-band waveguide. Emerson and Cuming's Stycast High K solid
dielectric was used. The manufacturer gives a loss tangent for this material of
0.0008. Different values of the dielectric constant for the material were used
such as K' = 5, 7, 11 and 15. The material was machined to fit the cavity
and also a hole was drilled for the probe of the feed as shown in Fig. 4-3(a).
Two different waveguide lengths were used, d=0. 031m and d=0. 1015m as in
Fig. 4-3(b) to demonstrate the effect of the cavity length on the resonant
frequencies.
Also some measurements were performed for the cases of standard waveguides at Ku-, X- and S-band frequency ranges. It is interesting to note that
the coupling obtained experimentally for these cases is within a fraction of
one dB from the corresponding coupling obtained from the analytical formulations;
126

~i:~::s-g,'~~ _
nN:i:5":'"-:-;pi'dl;~Q
-i~
s:::::,::;,:
-3:s9i:
~~ —- s;c;;aa=I*"L~-ln-s- —-L:~531
%pi.:.a-~;-~a.~i;~-.';i
Izi:ljI-i —~i'; 4!:"i;
" U
"
iw i a -~.
a"p
x=:
Ff
i:i:
5- -""
i Y
-Y9:s
-":
n ~
L
h :*s
~z
I
a
gL k
"*B I
" D """) Rls-b
FH, 4 —3ga: TgE E8jbCE 9 QnN P$8 taEECTXG FHLls
27

128

A ~~&A,,.,,,,,t' ~ 7,,7<,.,,,.,
K ~ ~ 4947. ~~4
/~~r'4747'~ ~ ~ ~ ~~~~~~~~~~~~~~~~c
1<4*"'r
"7,~

compare Figs. (4-4, 3-7), (4-5a, 3-8) and (4-5a, 3-9).
Figure 4-4 is the coupling versus swept frequency between
2. 00 GHz and 4. 00 GHz for a pair of S-band waveguides spaced 0. 075m apart,
center-to-center. The upper curve of the graph is a measure of the power level
from the transmitter obtained by connecting the two coaxial lines; in a way it is
a measure of power level. The lower part is the coupling level with the two
apertures flush mounted 0. 075m apart and in the ground plane of the anechoic
chamber. Notice that the coupling follows approximately the 6 dB/octave trend
as frequency is varied. Also when the coupling was taken versus spacing for a
fixed frequency it showed 6dB/octave of distance confirning the analytical results.
Figure 4-6 shows the coupling versus frequency for an X-band waveguide
cavity of length 0. 031m loaded with Stycast High K solid dielectric material
machined to fit. In this particular case the material dielectric constant was
11. 0. The level curve represents the level of the transmitted power through
the coaxial lines from the transmitter to the receiver, as indicated in the experimental set-up shown in Fig. 4-14, with a 25 dB attenuator added to the
circuit. Note that the coupling resonant frequencies are approximately the
same as those predicted in the analytical work. The small error may be due
to the measurements of the guide length which might not be very exact. Also,
errors in the measurement equipment may exist. It is believed that
the split at the top of the resonance is due to the fact that the transmitter and
receiver cavities do not have exactly the same length. Therefore, these two
peaks represent the resonant frequency for each individual cavity. For that
reason plus the occurrence of multiple reflections inside the cavities, the 3 dB
or half-power bandwidth is wider than was expected theoretically.
Figures 4-7 and 4-8 are the coupling and gain respectively versus swept
frequency for a cavity of loaded X-band waveguide with dielectric constant of
15. 0 and the dimensions of the cavity 0. 02286 x 0. 01016 x 0. 1015 meters.
The gain curve was taken with the set-up for far-field pattern measurements
as shown in Fig. 4-9. The standard antenna which was used on the boom was
130

Coaxial power level or 0 dB level
-10 _
-20
Frequency (GHz)
I,....* — I _, 9I., A, A1 a _
2.5 3.0 3.5 4.0
FIG. 4-4: STANDARD S-BAND WAVEGUIDE COUPLING VS
FREQUENCY. x1=0. 075m.
131

-25
-30
0
-35 _________..__________
8 9 10 11 12
Frequency (GHz)
(a)
-30 dB (Calibration)
*-30
~-35
13 14 15 16 17 18
Frequency (GHz)
(b)
FIG. 4-5: E-PLANE COUPLING VS FREQUENCY FOR TWO
SLOTS SPACED 11.4 CM.
(a) Standard X-band Rectangular Waveguide.
(b) Standard Ku-band Rectangular Waveguide.
132

2.0 2.5 3.0 3.5 4.0
Frequency (GHz)
n=2
-25 dB level
-25
-35
n=1
bIf
0
-45
-55
FIG. 4-6: COUPLING VS FREQUENCY FOR LOADED X-BAND
WAVEGUIDES, Ir=1.0, er= ll., d=0. 031m., x 1=0. 03m,
a=0. 02286m., b=0. 01016m.
133

2.5 3.0 3.5 4.0
-15.... |
Frequency (GHz) 9
n=8
t=7
n=6
-25 t S
-35
bn
0
-45
FIG 4-7: COUPLING VS FREQUENCY FOR LOADED X-BAND
WAVEGUIDES, /,r=l. 0, Er=15. 0, d=0.1015m., xl=0. 013m.,
a=0. 02286m., b=0. 01016m.
134

2.0 2.5 3.0 3.5 4.0,,,.. -, I - -- I. -- I
Frequency (GHz)
Supply power level through coaxial lines
-35
3r=. r=5. 0- d=. 1015 m.
^~~~~~~~3

FIG 4-9CHAMBERLD
PATTEN MEASUREMENTS.
136

an AEL No. ARN101A (1.0-11.0 GHz) log-periodic type. The boom was fixed
in its position and the frequency was swept between 2. 00 - 4. 00 GHz. Note
that the coupling resonances approximately coincide with those resonances of the
gain of the antenna. Also, the dynamic variation of the coupling without matching
is about 35 dB and the peaks are lower as the frequency approaches the cut-off
frequency which is due to the mismatch at the feed.
Similarly the coupling and gain graphs versus swept frequency were obtained
for dielectric constants of 11. 0 and 7. 0 which are shown in Figs. 4-10 - 4-13.
4. 2. 2 Coupling Versus Frequency with Point by Point Tuning
There was no simple way to match the input port of the transmitting antenna
and output port of the receiving antenna with swept frequency. A circuit had
been constructed to take measurements while the frequency is varied point by
point; a double stub tuner was used at both the transmitting and receiving ends
as shown in Fig. 4-14. For each frequency the stubs were adjusted for maximum power received.
The experiments in this part were also conducted on the 0. 1015m. and
0. 031m. cavities. Also different values of dielectric constants were used as
in the swept frequency. In Figs. 4-15 - 4-17 the zero dB power level represents the normalized coaxial power level. The line representing zero power
level is not shown. It represents the transmitted power minus the power lost
in the circuit from the transmitter to the receiver when the coaxial lines are
connected as shown in Figure 4-14. The upper curve is the power level
as in the coaxial case but it also includes the antennas with the two apertures
connected together thus forming a cavity. The surface on the aperture was
machined to be smooth enough such that the mating surfaces would not act
as a discontinuity. Also the mating periphery was covered with foil tape
such as to prevent power leakage and to form an electrically continuous surface
between the guides. If this is done perfectly then the arrangement is like a
cavity filled with dielectric material and there is an input at one end and output
at the other end.
137

2.0 2.5 3.0 3.5 4.0
Frequency (GHz)
n=7
-25 dB level n=5 n=6
-25
n=4 I
n=3
-45
-55
FIG. 4-10: COUPLING VS FREQUENCY FOR LOADED CAVITY,
r=1.0, er-ll. 0, d=0. 1015m., xl=0.013m.,a=0.02286m.,
b=0. 01016m.
138

2. 0 2.5 33.0 3.5 4.0
Frequency (GHz)
Supply power level through the coaxial line
-25
139
-35
P4
-45
FIG. 4-11: GAIN VS FREQUENCY FOR LOADED CAVITY,
Par=1. 0, Cr=11.0, d=0. 1015m.

2.0 2.5 3.0 3.5 4.0
Frequency (GHz)
n=5
-35
n=2
FIG. 4-12: COUPLING VS FREQUENCY FOR LOADED WAVEGUIDES,
ur=l. O, Er=7.0, d=0. 1015m., xl=0. 013m.
140

2.0 2.5 3.0 3.5 4.0
-25 i Supply power level through coaxial lines.
-35
i -45
FIG. 4-13: GAIN VS FREQUENCY FOR LOADED CAVITY,
ur=l. 0, er=7. 0 and d=0. 1015m.
141

Alfred Sweep Attenuator requency
Oscillator AT104 Meter (1- TransModel Empire4Ghz) mitter
650 Devices H-P
Mlodel 536_
RG-9
Coaxial Cable
Coaxial Line
(Reference)
RG-9
Double Stub Tuner
Wide Range Re- Crystal
c,, ~~~~Crystal |
ceiving System Detector R
De~tector }.
S-A Model 402 P Receiver
HP Model
(Scientific-Atlanta
423 A
FIG. 4-14: EXPERIMENTAL SET-UP FOR COUPLING WITH
POINT BY POINT TUNING.

-10
n=2
T R
-~20 ~ Kn=1 J >C 0a=O. 02286m.
b=O. 01016m.
d=O. 031m.
- - / ^ x1=0. 03m.
0r.Lrl. 0
I C X V /a L r=l. 0' -30
-40. \/ ~ ~ Frequency (GHz)
2.5 3.0 3.5 4.0
FIG. 4-15: COUPLING VS FREQUENCY WITH TUNING BOTH TRANSMITTER AND
RECEIVER AT EACH POINT. (x) Coupling with x1=0. 03m., ( ) Power
Level through Cavity formed from Transmitter and Receiver.

-5
-10
-1n=05 a=O. 02286m.,n=7 L b=O. 01016m.
-20 d=O. 1015mI
30
-40
Frequency (GHz)
2.5 3.0 3.5 4.0
FIG. 4-16(a): COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling with x=0. 013m.,
() Power Level through Cavity formed from Transmitter and Receiver.

-10
-20
2.5 3.0 3.5 4.0
n=2
0
-30
x
a=O. 02286m.
b=0. 01016m.
d=0. 1015m.
Xl=0. 013m.
=l1. 0
-40
Frequency (GHz)
2.5 3.013 3. 5 4.0
FIG. 4-16(b): COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling vs Frequency with
x1i0. 013m., (o) Power Level through Transmitter and Receiver forming a Cavity.

-10
n=5
]Ert A /A- \1-~a=O. 02286m.
n=3 nb=O. 01016m.
l2.5Frequencd=0. 1015m.
x1=0.013m.
-202.5 3.0
Er=7. 0
rC
n=l
P L t T a R c
Ca -30C, X X
X
-40
Frequency (GHz)
2. 5 3.0 3.5 4..0
FIG. 4-16(c): COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling with x1 (spacing)
= 0. 013m., (.) Power Level through Transmitter and Receiver connected as
a Cavity.

-10
a = 0. 02286m.
b = 0. 01016m.
ll n=5 d= 0. 1015m.
n-40: x 0. X03m.
- 2.50 n=3.0,r3.5- 14.0
with x(separation) = 0. 03m. () Power Level through Transmitter and7. 0
Receiver con=2
-30
x
-40
X X
Frequency (GHz)
2.5 3.0 3.5 4.0
FIG. 4-17: COUPLING VS FREQUENCY POINT BY POINT. (x) Coupling vs Frequency
with xl(separation) = 0. 03m., (-) Power Level through Transmitter and
Receiver connected as a Cavity.

Therefore the curve in comparison with the zero dB level represents the
power lost in the transmitting and receiving cavities. It is to be noted that the
level of the power lost shown by these curves is a little high because of the discontinuity at the mating surfaces in spite of all the precautions which were
taken. The difference is probably within the range of 1 - 2 dB. The lower curves
of the graphs represent the level of coupling for various separations, cavity
lengths, and dielectric constants.
It should be observed that the differences between these graphs and the
corresponding graphs with swept frequency are:
i) the level of coupling is higher in the case of tuning,
ii) the dynamic range of variations is less than before and it is around 20 dB,
iii) the resonance points are about the same with or without tuning.
4. 3 The Far-Field Patterns
The arrangement in Fig. 4-9 was used to measure the E-plane pattern which
is with 0=0~; 0 varies approximately from -90~ to +90~. The O-plane is perpendicular to the plane of the aperture at y = a/2 and parallel to the x-axis
(see Fig. 2-1). The reference antenna is AEL No. ARN101A (1 - 11 GHz) logperiodic and moves in a radius of 5'6".
The H-plane patterns were also measured. For this case 0=90~ and the
reference antenna moves in a plane perpendicular to the aperture at x= b/2.
The variation of 0 is between -90~ and +90~.
Different values of frequencies were considered and both E- and H-plane
patterns were drawn. A representative graph is shown in Figs. 4-18(a) and (b).
There was no noticeable variation in the shape of the curves with different
values of frequency in the S-band range except for the level of the gain. The
variation of the gain with frequency was shown in Figs. 4-8, 4-11 and 4-13.
The differences between these patterns and those of a free-space S-band
slot in a ground plane are:
148

, tt irj.....- I. i
@10 f 1 * 4F i +, _iTi Hl I I j
I 4. 14i- iiii i4[. 4! -i 4 i 11...... [' r!''!''l t- - t; 1:_i:-:iL li:~ L I i-::i-: I I "1
-90 -72 -36 0 36 72 90
0 (degrees)
FIG. 4-18(a): E-PLANE RADIATION PATTERN OF A LOADED
X-BAND WAVEGUIDE. f=3. 495 GHz, er=15. 0, ur=l. O
149

it:l l I
iI i j i, W 1, I,,,.... I!' I;' I I ~ 1 I I,' I: r I I
10
4 _.... l il_ _ _ - I
X-BAND WAVEG.IDE. f3.495 GHz, Er=15. 0 r=l. 0
X-BAND WAVEGUIDE. f=3.495 GHz, r. 5,
150

i) the E-plane pattern is almost a circular pattern, more so than for the
corresponding unloaded case,
ii) the H-plane pattern has the sidelobes somewhat higher in level than the
corresponding unloaded case,
iii) the gain is lower than in the unloaded case due to the losses in the
dielectric and the mismatch at the aperture.
151

Chapter V
SUMMARY AND CONCLUSIONS
The preceding chapters describe a theoretical, numerical and experimental investigation of the coupling between slot antennas with the
broadsides parallel; the slots were loaded or unloaded and placed in an infinite
conducting ground plane. The two slots were backed with a material loaded
cavity filled with homogeneous isotropic material. The investigation does not
claim credit for development of new antennas since both loaded and unloaded
slots have been introduced before in several papers and textbooks as mentioned
in the Introduction. Instead use is made of existing antenna types and an investigation is made of the coupling for both cases. The results of the two cases
are then compared. The investigation does not claim an advantage of loaded
slots over unloaded ones or vice versa, since each has its own merits and
drawbacks. Therefore it is up to the designer of the system, after taking into
consideration the characteristics of each, to decide which would be the best
choice for his system. For example, if a system has limited space and can
tolerate the loss of some of the power it might be a worthwhile compromise
to choose the loaded slots. As was seen in the investigation it was possible to
use a standard X-band waveguided loaded with dielectric material for S-band
frequency operation. It was also noted that the radiation patterns for this
arrangement were rather similar to the ones obtained from the standard S-band
slots.
A serious drawback of the dielectric loaded waveguides is the frequency
sensitivity as shown in both coupling and gain curves versus frequency.
No experimental work has been done to obtain the coupling versus frequency
for ferrite loaded slots but it is expected that the resonance phenomena would
be less pronounced in this case. The coupling between the apertures with
material loading did not follow the usual trend for the unloaded case which is a
152

drop of approximately 6 dB/octave of frequency. For the loaded slot case the
behavior of the coupling between the apertures depended mainly on the parameter
of the loading material. It was rather interesting to see that the coupling versus
frequency was almost constant with frequency for low values of dielectric constant.
For the higher dielectric constants the coupling changed more drastically. It
varied from minimum to maximum as shown in Figs. 3-7 to 3-10 (Ch. nI). The
variation of coupling with spacing followed the 6 dB/octave of distance characteristic.
The phenomena of resonance was investigated and a method to predict the
resonant frequencies and the half power bandwidth was obtained. A good agreement between experimental and theoretical results was obtained. It was noted
that resonant frequencies occurred when the cavity length was such that
= n (kg/2 ) where n is an integer.
It is interesting to note that if it is possible to maintain Xk constant or
approximately so corresponding to its value at the resonance peak, then it would
be possible to have a wider bandwidth and reduce the frequency sensitivity. To
look into the requirement to do so, consider Eq. (3. 10) (Ch. III) which is:
0.15
f = G Hz, where a is in meters,
c a/^
a
also
f2
or
0. 3
X 0 3 - in meters,
P/rCrf2 (0.15 )2
where f is in GHz and a is in meters.
Then for Xg to be constant irrespective of f implies that
UrCrf2-( 0 1 ) = constant.
153

If the material loading is to be taken specifically to be dielectric or ur=l then
This indicates that one way to have a wider bandwidth is by using a material
with a dielectric constant inversely proportional to the square of the frequency
over a certain band of frequency range. The existence or the physical realizibility of such material needs further investigation.
Another outcome of the investigation was that, as mentioned in the previous
chapters, the resonant frequencies depend on several factors which are; 1) the
length of the cavity, 2) the broadside length of the slots, and 3) the material
paremeters. Therefore if two slots are spaced close to each other on a ground
plane the mutual coupling can be controlled. For example, in order to decouple
the two slots, the parameters affecting the resonance frequency could be chosen
in such a way that the resonant frequency of one antenna coincides with the antiresonant frequency of the other antenna. For this case each antenna is acting
as a frequency filtering device to the power which is intercepted from the
other antenna.
It is very interesting to note that the spacing of the resonance peaks is not
uniform over the frequency scale. It is smaller nearest the cutoff frequency and
becomes wider away from it. This was observed both experimentally (Fig. 4-7 and
4-16) and from digital plots of the resonance points ( Fig. 3-14(a)+(c) ). This can
be explained by considering Eq. (3. 11) which is
Squaring both sides there results:
2
2 0.15 rn2 1
154
r E~ ~ ~ ~ ~ ~ ~ ~~~~~~(.1
r+-

or f2
res n2 1
0.15 t Q2 a2
or
f2
res n2
0.15 2
( D I n = 1. (5.1)
a -Er l/ a
Equation (5. 1) can be put in the form
2
f2 2
res n 2
i2 - = 1, (5.2)
A A2
1 2
where
0. 15
A: 15 (5.3)
a 4Ar^
A2 = /a, (5.4)
with a and x in meters. Equation (5. 2) is a hyperbola as shown in Fig. 5-1.
From the figure it can be seen that the resonance peaks are not uniformly spaced
but become more nearly uniform at higher frequencies or at the asymptotic part
of the curve. Also, from Eq. (3. 10) (p. 75) it can be seen that A represents
the cutoff frequency for the specific parameters Cr., e, a of the loaded cavity.
5. 1 Areas for Future Investigation
1) Experimental investigation of coupling for ferrite loaded slots.
2) Investigation or search for material to widen the bandwidth.
3) Study dielectric loaded slot array.
4) Study ferrite loaded slot array.
5) Design wide frequency band tuning arrangement for the feed.
155

A2
Asymptote n = - fres
A2
n
%24 __ f 6_ f6res
FIG. 5-1: EXPLANATION OF NON-UNIFORMITY OF SEPARATION OF
THE RESONANCE PEAKS.
A
FIG.~~~~~s \-:EPAAINO O-NFRIYO EAAINO
THE_ REOAC PEAKS.N

REFERENCES
1. Adams, A. T. (March 1964), "The Rectangular Cavity Slot Antenna with
Homogeneous Isotropic Loading, " The University of Michigan Cooley
Electronics Laboratory Technical Report No. 147 (5549-7-T).
2. Lyon, J. A. M., R. M. Kalafus, Y-K Kwon, C. J. Digenis, M. A. H. Ibrahim
and C-C Chen (April 1966), "Derivation of Aerospace Antenna CouplingFactor Interference Prediction Techniques: Final Report, " The University
of Michigan Radiation Laboratory Technical Report 6633-1-F, AD 483051.
3. Ramo, S., J. R. Whinnery and T. Van Duzer (1965), Fields and Waves in
Communication Electronics, John Wiley and Sons, New York.
4. Von Hippel, A. (1959), Dielectrics and Waves, John Wiley and Sons,
New York.
5. Lewin, L. (1951), Advanced Theory of Waveguides, Iliffe and Sons, Ltd.,
London.
6. Levis, C.A. (August 1956), "Variational Calculations of Impedance Parameters of Coupled Antennas, " The Ohio State University Report 667-16.
7. Kaplan, W. (1959), Advanced Calculus, Addison-Wesley Publishers.
8. Harrington, R. F. (1961), Time Harmonic Electromagnetic Fields,
McGraw-Hill Book Company, New York.
9. -- (June 1968), "Techniques for Integrating Solid-State Circuitry into
Antennas, " The Ohio State University Electroscience Laboratory Report
2142-16.
19. Jennetti, A. G. (November 1965), "A 1 Gc Cavity-Backed Slot Antenna
Field, " The Ohio State University, Contract AF33(657)-10386.
11. Svoboda, D. E. (September 1966), "Correcting for the Pattern Distortion
Effects of Mutual Coupling in Antenna Arrays, " The Ohio State University Report 2142-6.
12. Richmond, J. H. (November 1961), "A Reaction Theorem and its Applications to Antenna Impedance Calculations," IRE Trans., AP-9,
pp. 515-520.
43. Galejs, J. (February 1965), "Self and Mutual Admittances of Waveguides
Radiating into Plasma Layers, " Radio Science, 69D, pp. 179-189.
14. Mikenas, V. A. and P. E. Mayes (July 1967), "Analysis of a Log-periodic
Cavity-slot Antenna Using Three Port Network, " University of Illinois
Technical Report AFAL-TR-67-109, Dept. of Electrical Engineering.
15. Rumsey, V. H. (June 1954), "Reaction Concept in Electromagnetic Theory,'
Physical Review, 94, No. 6, pp. 1483-1491.
157

16. Wu, C. P. (March 1969), "Numerical Solutions for the Coupling Between
Waveguides in Finite Array, " Radio Science, 4, No. 3, pp. 245-254.
17. Archer, D. H. and G. S. Hardie (December 1963), "Investigation of Coupling
Between Closely Spaced Antennas and Development of High Insulation
Techniques, " Raytheon Space and Information Systems Division,
Santa Barbara, California
18. Swift, C. T. and D. M. Hatcher (April 1968), "The Input Admittance of a
Rectangular Aperture Antenna Loaded with a Dielectric Plug, " NASA
Technical Note D-4430.
19. Cockrell, C. R. (October 1968), "Higher-order Mode Effects on the
Aperture Admittance of a Rectangular Waveguide Covered with Dielectric
and Plasma Slabs," NASA Technical Note No. D-4774.
20. -- (November 1964), "Techniques for Integration of Active Elements into
Antennas and Antenna Structure" Ohio State University Electrical
Engineering Department Interim Engineering Report 1566-15.
21. -- (September 1967), "Techniques for Integrating Solid State Circuitry
into Antennas, " The Ohio State University Department of Electrical
Engineering Interim Technical Report 2142-12.
158

APPENDIX A
THE REDUCTION OF QUADRUPLE INTEGRALS TO DOUBLE INTEGRALS
For the numerical computations to reasonable in cost and programming
for the computer calculations, it was necessary to reduce the quadruple integrals to double integrals. This was performed by means of coordinate
transformations. It consisted of a rotation of axis, magnification and
translations.
Consider the general coordinate transformation;
1 = r7 cos 0 +y sin 0
1 = -rl sin 0 + y cos 0
If we choose 0 to be 45~, then
1
= - (-r + y)
1 2- +
Take
A2 = p1
2 = 1 (Magnification)
T2 = 71
Then
2 = (7 + y)
T2 = ( -rl + y)
Performing translation along P2 such that the origin would be at the center
of the rectangle as shown in Fig. 2-5 (Ch. I). Therefore
= -2 -a
or
p + a = ~2 = r + y (A. 1)
159

Let
7=2
or
= y - r7 (A. 2)
Due to the magnification
1
dr)dy = 2 d/d dy
Therefore the integration:
fa a (a-)1 (a-+))
f f(y, r)d dy= f(, 4d- fr f( y)d 7dy (A. 3)'0'"0 (a-7) - a (a+y)
If f(p, y) is even in ju and y then:
f(-,M, Y):=f(, -7) =f(, 7Y)
Therefore
a a a a-y
fI f(y, r)drldy=2 f f(u, y)ddy
Similarly
b b b (b-co) 1 0 (bi)
ff f(, x)ddxT =' I f(-, v)dvdvr+ f f(a, zdvdc (A. 4)
Wu<) *'0 -(b-ua) b T4-or)
also if f(cr, v) is even in cr and v then (A. 4) reduces to
b b bb-c
~~0bf(, z')d~dxt=2 f()dvdo
In general
160

b aa 1 b a (a-) (b-c)
f(x,y,), r)ddrndydxt=, I f(y,, A, z)dvd/dody
<O )<) 00 <0 -(ra-y) -(b-co)
a O (a-r" (b-h)
+4 t-( a-) ( b) f(v, /, a, y)dz34dody
-0 *b -(a-) -(b+-)
+ ~L b (a+)) (b -r)
4 fr f(v, PL, CT, jdv,?dod
4-a'O -(a+y) (b —)
1 0 0 ~(a+y) (b-ib)
+4 I f f ( f(v, IA, o, y)dvd/Adody.
-a (a+7) (b+o)
(A. 5)
In Chapter II f(y,, o, v) is an even function in y,,, v. Therefore:
b aa b a b a-Ty (b-o)
f rf6 ff(x,'y, n, )d, rdRdydx = f(7, r., cr, v)dv d doudy
O^ )o "o o'o -(b-cr)
a a.a-y (b+-)
o+ bJ 0 (bJ ) f(7y,co, v)d7/ddC dy. (A6)
161 (b+ <)
161

APPENDIX B
EXAMPLES OF COMPUTER OUTPUT DATA ON COUPLING
B. 1 Coupling Versus Frequency
B.1. 1
For an empty S-band waveguide for x1=0. 0341m., x1=0. 075m, A a,
B - b, EPSR * er, FMUR s Pr, XI separation center-to-center, UC =
coupling without interaction, and UCI = coupling with interaction.
B. 1. la:
A 0.71360000-01 8 a 0.346.36n00D-0n 0 0.00001')-01
EPSR = I.000000 n FMU a 1 *000000
NA 0 NI 3 0 3 NS - 30 NL 51)
FOR XI a.3410000D-01
F-N UC tUCI
1. In -I l. 339^ -9.97155
1.on0 -1. 099 - 2.137R3. 300 -13.16771 1 3. 13351
1.00 -13.71613 *1A3.76355
1.500 -14.13574 -14.25094
1.60 -14.51635 - 14.6423
1.700 n 14 90229 -15.1055R
I.) -r60 -15.31,60 -15.53R67
1 900 -15.7"103 -15.999A4.000n -16.30)64 -'165013
162

B. 1. lb:
A =.71360000-01 R = 0.34036000-01 =- 0.50000000-01
EPSR 1.0000n0 FMUR - 1.000000
NATI 30 NRI = 30 NS 30 NL = 50
FOR XI 1 0.7500000D-01
FN UC UCI
0131 D9. 0r)0) 15 - 114^13 -512 735
I.On - 17 165t1 -17.074q6
I - 1 19?79Tq =q~R.O?q.^nn -1. 7?16- 1.~60'33
1 500 - 9 133 I - 9.09728
I.6no - 19 54507 - 19.541
1i. 70 -1991373 -19.96331
1.0n -20n.9591 - O.37?n
1.QO -P 02 7Ct9 -n.,77q57. 000 -P21.1621q -21 20163
B. 1. lc:
A = 07136000D-01 B = 0.34036000-01 t 0O5000000O-01
EPSR = 1 000000 FMUR s 1.000000
NAI = 30 NRI= 30 NS 30 NL = 50
FOR Xl * O.2075000
FN UC UCI
OR I)F
1.100 -q3.5791 -23.26242
1.?nn -P5. 34572 -5 32054
1.300 -26 33617 -26.3435R.400 -26.95072 -26.96499
500 -27. 40R1I -27 3961.1
1.600 -27. R1R2 -27.79544
1.700 -2R 17953 -28.19244
1. rOO -2. 56F64 -2R 56822
1.900 -28 98639 -28 97585
2.000 -29.44464 -29.44963
163

B. 1.2
For an empty X-band waveguide with spacing of xl=0. 013m., xl= 0.02286m.,
0. 114m.
B. 1.2a:
A = 0.2e6n0 )-n F3- 0o.1016000no-01o 0.50000Ononnn)c;R - 1.000o000 FMUtR = 1*000000
NAI = 30 NFI = 30 NS = 30 NLt 50
FOR X1 0.13000001)-01
FN IJC JC
1.100 - 1.67733 -I 1.7S54
1?0 - i 3. -Rno - I 13.74 57
1.300 -14.45179 -14.64503
1.,00 -14.94471 -15.17159
1.50n -15. 930n - 5..3526
1 00 - 1 5.5 605 -1 5.2277.700 - 15s.637 -16.07R59
I. RO -16.16473 -16.33n55. 900 - 6.49024 - 6.59960
2.000 -1 6. 5484 - 6.90338
B. 1.2b:
A = 0.2* q6000D01 R = 0.10160000-01 0 = 0.5000000D-01
EP.R = 1. 000000 FMUR = 1.000000
NAI = 30 N91 = 30 NS = 30 NL = 50
FOR XI = O.22860000-01
FN L.C UCI
r13 DR
1.100 -15. 1675 -15.25937
1.200 -17.416429 -17.12500.0nn -1. n707 -17.96615.400n - 1.59560 -1q.47595
1.5n0 -1.96A40 -1R.R6764
I. Ar)o -4 9.27 11 6 - 19.22250
1.7n0 - 9.55607 -19.*56723
1.q00 -19.94716 - 9.90658
1.900 -20.16037 -20.24134
2.000 -20.50626 -20.57900
164

B. 1. 2c:
A 0.2289600())-0-)1 8 0 10160000 —01 Q = O,'SOOOOnnO)-
SP s- 1.00n00nnnFMI = 1.000000
NAIT = 0' NR I = 30 NS = 30,NL 5
FO XI = 0.1140000
F'N IJC UCI
)B3 1) H
10nn -9?21674 -?q.03(O
I.0.) -30. 254R4 -30 *6302
30nOq 1 1R.. -'31 18221
1.A0 -31. 740R -31 74599
1 500 -32. 1719 -32.12186
1 600 -32 44.31 -32. 44q5.700n -3?.73573 -32.73111
1.09 -33. n133 -33.03565
1.r90 - 33 34674 -33 34273
2 000 -33.69305 -33 69674
B. 1. 2d: Standard Ku-band waveguide coupling versus frequency output.
A = 0.1579SRO0-01n 0.79994000-02 0 = n.5000000D-01
EPSS = 1.00000n FMIJUR 1.000.000.
NAI = 30 NFI = 30 NS 5 30 NL - SO
FO0 Xi - 0.1140000
FN IJUC lJCI
O)R DR
1.o00 -32.996R2 -3.. 99475
I 30r)d -341. 02'749 - 34. 02494. 4nn -.34. 69 629 -34. 69953
1*500 -S3.l212i -35.2139R
I.An00 -35.67041 -.35.66919
I1.700 -36.11913 -36.11730
1 900 -36.59391 -36.59572
I.9no -37.054 -37.09514?2r.0 -37.63593 -37.63455. 100 -38.*4509 -389.24580
165

B. 1.3
For a standard X-band waveguide filled with dielectric material. These
parameters are common for the three tables which follow: a=O. 02286m.,
b=0. 01016m.,,r= 1. 0, separation center to center = xl=0. 013m.
B. 1.3a: cy = 7.0
FN UGC ICI
OR OB
1.100 -19.92142 -19.96417
I.?n0 -P?. 74857 -22.75982
1.300 -24.04925 -24.04954
1.400 -24.6473 -24 61905
1.500 -24. 7698 -24.77725
I.600 -24.70965 -24.69682
1 700 -24.49765 -24. 4R233
1.RO0 -24.21448 -24.19,717
1 900 -23.9813 -23. 7930
2 000 -23 57103 -23.55122
2 100 -23.24631 -23.22612
2.200 -22.931'5 -22.91165
B 1.3b: er= 11.0
FN UC UCI
) B 01
1.100 -19.R3601 -19.91287
1.200 -23. 0505 -23 23635
1.300 -25. 1408 -25.20028
1.400 -26.47726 -26.48592
1.500 -27.30095 -27.30501
1.600 r27.76787 -27.76875
1.700 -27.96353 -27.96200
1.800 -27.96154 -27.95808
I.900.-27.82349 -27.81837
2.000 -27.59736 -27.59080
2.100 -27.31824 -27.31042
2.200 -27.01053 -;7.00160
166

B.1.3c: ~r=15.0
FN lJC UG I
I 100 -19 42214 -19.5154%. 200 -22.7672 -22 91717
1.300 -25.02914 -25.05214
1.400 -26. 5214 -26. 6652
1 500 -27.77097 -27.7RP05
1.600 -2. 64666 -28 65255
1~ 700 - 9 rt61 5,' -'9. ^2649
1,P00 -29.64q76 -29. 5036
1.900 -29 R4291 -29. 4302
2.000 -29. *q097 -29 R7991
2.100 -29 79956 -29.79730
2.200 -29.63168 -29 62845
B. 1.3d:
A 22q6nnnr)-01 R 0 101 60000-01 Q = 0,50000000-01
F.PSR 20. o F00000M IMIJ 1.Onno00
NAT. 30n NRI = 30 NS = 30 NL = 50
FOR XI = 0*1300000n-01
FN.JC IJC I
OR O) R
1.100 -19.31 11 -19 13669
1.200 -22.47551 -2P 52.64
I..n -0.6 3^q9 -24. 6171
1./40 -26. 9247 -26.3107q.500 -27.60q13 -27 620?
1. 00 -2?. A750 -2R. 6954
1.700 -?9.56q57 -29.57504
1.R00 — 30.2699 -30.27354
1.900 -30.79920 -30.0227
2.00 - 31.16974 -31. 17161
167

B. 1. 3e:
A = -n.?o^600q-0e)-l = 01 O. 60ln)0-nl o = o.50n00000l)-0
no - n? r0r) n 1) oiM(JI o- 0 n so nonIO- o i
2S000000( FMUR 1.000000
NAI = 30 NI = 30 NiS 0 30 NL = 50
Fni0 X I -. 130n 0D)-01
FN IUC, UCT
1. l n - 1.50556 -11.20670
1* PO -1 3.1*SO4 -12*99226
1,o0 -10. 7 7? - - 3.6?3qR
I.4 00 -13.99R55 -13.7707
1.500 -13.96r525 -13.97640
1.h*O -13.93971 -140l1147
1*700 -13. R555 -14.022R5
I.*00 -13.q2937 -14.02949,
1.900 -13.7R094 -14 039R0.n000 -13.750r1 -14.05700
B. 1.3f:
A = O.q2RqAOOo-0)1 = 010160000-01 r 0.50000000-01
EPSR = 3.000000 KMUR 1.0000000
NA[ = 30 N1I = 30 NS = 30 NL =' 50
FOR XI 1 0.13000000-01
FN!JC i.iC
1) 13 0 P3,
*. 00 -15.16019 -b*.03304
1.0nn -1 6.1772 -16.72254
1. 00 -17.3177? -17.23417
1o400 -17.4155? -17.34109
1. 5nO - 1 7.34165 - 1 7.7 1
I* AOO -17.19165 - 17 14257
1.700 -17.011 3 -16.9R099
I R00 -! 69,41.2 - 16.81 70 C.900 - 16. 6432R - 6.66336.on0r -16.4760? -16.52637
168

B._1. 3g:
A= 0.?R60002nE)-01 8 = 0.10160000-01 n O.5o00o00D-Ol
4.00000O O MUR - 1 000000
NAI: 30,NI)I I 30 NS'=S 30 NL = 50
FOR XI = r013000000-01
FN.)I, IJ(
1)1 1.) F3
1.100n -7.4919? - 7.,44991.o0n - 19.34451 -19.3016
1.300 -1.Q1739 - 9.7RqR4
I.on) -?).01R99 -19.97.31
1.500 -19.91176 -19.q6559S
I.00 -1 9.706R -19. 66379. 700 -19.46463 -19.42311
1.00 - 9.20q11 -19.17239
1.900 -1R.95446 - 1.92724
2.n00 -18.71?12? 18.6961A'
B. 1. 3h:
A 0.qR60nn0)-n R a 0.10160rn-01 Q = 0.5000o000D-01
S —,, = 5,n0000onr) " sMI I OcnnlOc)
NAI.30 N:tI 0n NS F 30 NL = 50:'0R X I = 01300000o-01
FN UC UC, 100 -18.8711 - 1887?19
1.200 -1. 0/60 -21.03;350
1.3no -21.812q6 -21.79115
1.4n -. n0r0A0 -.1.'9q-29
1. 00 -R 1.934/78 -21.905 1
I.Aro -?.7?967 -21 69765
I.70 -.4 61 -1 -.42830
00) -2 I.1662 -2.13683
I.o -2. 75001 - 20. 3-4239. n0 -?0.5q375 -20.556q9
o. n0 -Po. 3054 -20.2670
2.an0 -n. 05) 019 -0n.03519
2~.3 - 9.q 10 -19.80O00
2. 1')O - 9.5q93 - 19.59373
2.500 -1 9.3366 - 19. 40432.,6nn - o. 0671 - 19..3530
p.7r00 -19.04364 -19.0595
P.3^o - l.q9901 - I.95541
2.9n0 -1*R77234' - 1.84271
3.000 - 186631 6 -1874688
169

B. 2 Coupling Versus Spacing
In the following these notations are common: A a a, B * b, EPSR a Er,
FMUR - r, GB = N1 and N2
where
N1=G/Yo and N2 =B/Yo
for the normalized frequency specified FN. Also,
R= N3 and N4,
where N3 is the real part of the reflection coefficient and N4 is the imaginary
part of the reflection coefficient. Also, L indicates the separation between
the two slots center-to-center in wavelengths. X1 indicates the equivalent
in meters of L for the specific case aL hand.
170

.4
C
00
C C! I
c -' — ct. ~' - r-. CV -C C'C. C c C'n'IV C, C". n r
C c c C r c C; r C e n c C
C vo Oj' " s-' *r %r C,,,' C C - r c in c F'. — t C. c c o
C _, ~ C ~ ~. o.. O ~..... ~. O O...... 4..
C C
c C\1 C —. c C P*r c cv,, J N Nc,, Lf M-l -cn,''- c, I r'
c- c' CC'C.C, C C CC r c C CC C~ -. n - r c c r
C C'.) c -X~, g T~ C C C; n C %'' q Ln \C r - W C. N CO)
C — <, —. C --,. Cv C, un Cv', C-,4;:- C, \Co C,. V - ro o C
C r I I i I i I i i I i i IC i i I i IC i i i I I I I
~ - o,-.-I
C C'0
C
~e C,C
*V ( C C C C C C C C C C C C C C C C C C C C CC C C C C
CI C C'.' C'C C,C C C C C C, C,'C C C
~zC C C- C, 1"- ^ CC-'Cc,, CCC C C0 C' C fC C C'
~ " ".' - C CC C, C C C C C IC. C C — rC C C C C C. C uCC-CC
C c rw y c < c c' c <, r" C:r ",. O- C r c:- _' t.C...' 0,' ". % - 0 0 0 0 0 0 0.,*,S 0. 0 r 0 T 0 ar ^ 0 - 0Ce
CI C 1 * a- - - -.. - v- - -. -' - -.......... C -- C r r -.
CC C C C C C CC C C- C C C c C C C C cCC C C
c'- 0'
II-c r --------
*v. r 1 - CccccC C CCC C cC C C C C C CCCC ccCC
*C C c C ccC- C. C c
C * C CCCCCCCCCCCC C CCCCCCCCCCCC C
CC.. CCcc C Ccccc CC c c Ccc C C C c C C C CCCC. C 0 CC C C'Cccc C C C c C C cc.) C C C C C CC"'. C CC C 0,- o,<' rnr ac - ---- -
1.- c - c.. - - - - - - - n m * - - - - - -C.c'
e -c t -... _ _ _ _. _. _. _. _ C_ _ c" q L.* r-. {:.c'***. —
C* Z C CCCCC

B. 2.2
The parameters are as in Section B. 1.1.
A = 0.2 RoS.60000-01 3 = 0.10160000-01 Q 0.50000r(lO-01
R'R = Io11.00 FMUR = 1. 00000
N AI nR N3 I = 30 NS = 30 NI = 50
FOR FN = 1.500000
G. 0n.30823470-01 -0.3090832 R 0.7810194 0.5343343
L XI uCl Il,
0. 10 0 r ). 1010907n01)- -3174 -3 *: 74057: 083
0.2000000 n.2021P1ID-O1 -31.24739 -31.24590
0. JOt))3)0)) 0). 30327;0-0 1 -33. 75363,33. 75455
0. no0or000r) 0. 40436290-01 -35;71 362 -'. S 71 77
0.5000c.000. 5054536)-01 -37. 37070 -37.37194
n., O non0 0l. S665^430-r1 -3. 793^A -3R.79316
0.7000000 0.70763510-01 -^0.03263 -4n.n017g.3000000 Or. 0725RO0)01 -41.1' 665 -41 ~ 2163
0.9n0nn00000 0.tnq00165-01 -42.10409 -i10
1.000000. 1010907 -42.99637 -42.98679
I. 1 n0'O 0.111199R.-43.7R97' -43.7.q97F
l.oinnnO 01.?13099 -44.V?6r -44.52659.300000 0. 1 31179 - -/5.0075 -45 *.P073
1.400000 0. 415 70 -45. 3965 -45. 3977.500000 0- 151 6361 -46.4 O955 -46.42974
1600000 06. 117z52 -46.9s~45 -46.99?/46
l.7C10000 O.171q5a2 -7.50 65 -47.50251
i. qnr00000 0. l19633 -47.9937q -47.,9936^
1.o00000n0.19o07?4 -4R. 45q7 -43.45393. 000000 0.20 1R4 - 4 o9053 -49. 900064?. 00000 0.11P905 -.31 -4099-9.32101
3.100000 0.3133912 -52. 6622 -5. 6R6P3
4.100000 0.4144720 -55.10325 -55.10o826
5. I00000 0.5155627 -57.00095 -57.00095
6. 00000. 6166534 -5. 55449 - 5.5544R
7. 00000 0.7177441 -59. 72~03 -59. R7203.1i00000 0.R81q349 -61*01590 -61.01590
9. 0000 0.9199256 -62.02657 -62.02659
10.10000 1.021016 -4"e-*-85 -62.93185
172

UNCLASSIFIE D
Security Classification
DOCUMET CONT ROL DATA R & D
(Security classification of title, body of abstract rnd indexinrg lnnotation must be entered when tile overall report Is cflssllled)
I. ORIGINATING.ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION
The University of Michigan Radiation Laboratory, Dept. of UN C LA S S IF I E D
Electrical Engineering, 201 Catherine Street, 2b. GROUP
Ann Arbor, Michigan 48108 ____
3. REPORT TITLE
300 MHz Array Using Physically Small Slot Antenna Elements
4. DESCRIPTIVE NOTES (Type of report and Inclusive dates)
Second Interim Report, Technical
5. AU THOR(S) (First name, middle i'i tlal. laa t namo)
Medhat A. H. Ibrahim and John A. M. Lyon
6. REPORT DATE 7,,. TOTAL NO. OF PAGES 7b. NO. OF REFS
August 1969 172 27
8a. CONTRACT OR GRANT NO. 9u. ORIGINATOR'S REPORT NUMBE.R(S)
F33615-68-C-1381 1 — T
1770-2-T
b. PROJECT NO. 6278
c- Task No. 627801 h. c TlHER REPO RT NO(S) (Anry other,numbers that may be as s lined
this report)
d. AFAL-TR'-69-299
10. DISTRIBUTI-ON STATEMENT
This document is subject to special export control and each transmittal to foreign governme nts
or foreign nationals may' be made only with prior approval of AFAL (AVWE), Wright-Patterson
Air Force Base, Ohio, 45433
11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Air Force Avionics Laboratory
Air Force Systems Command
Wright-Patterson Air Force Base, Ohio
13. ABSTRACT
The coupling between two loaded rectangular waveguides opening in an infinite conducting
ground plane has been investigated. The slots are equal in size and placed so as to have the
broadsides parallel. The analysis is made in two steps: 1) the coupling between the apertures
is derived, and 2) the effect cfthe backing cavities on the coupling is determined. The reaction
concept, a form of the "variational technique" was used in the analysis. This method helped to
simplify the mathematical work and provided very reasonable results.
Substantial experimental work on the behavior of coupling versus frequency and coupling
versus slot separation was performed. In order to simplify the experimental investigation the
standard empty X-band waveguide (0. 9"x0. 4") was filled with dielectric loading of Emerson and
Cuming Stycast Hi-K dielectric material. Materials having dielectric constants of 5, 7,11 and 15
were used. Most of the measurements for the loaded case were in the S-band range of frequency (2. 6 -3. 95 GHz). The choices of the S-band frequency range and the dielectric constant wer
made to permit use of the anechoic chamber in the first instance with reasonable accuracy and
small waveguide in the second instance.
An extensive comparison of coupling for the case of standard waveguides with and without
loading has been made and explanations are given for the observed differences. An iteration
procedure has been used to determine the effect of the electromagnetic interaction between the
slots.
Detailed computer programs were developed for the analytical expressions. Numerical
results for many important cases were then obtained. Methods and results can readily be exended ~to other frequencies and other slot sizes.
DD FO RM 1473 UNCLASSIFIED
So curit I',sIiic('hon -

Security Cla.ssification
| ~~~~14.~~K Y LWDIN'K A LINK B LINK C
K. WO DS.
______ ROLE WT ROLE W T ROLE 7WT
Antennas
Antenna Arrays
Electromagnetic Coupling
Coupling Reduction
Dielectric Loaded Antennas
I^ l'lr ltv ('!.ts, fi,.,,s ~

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1 Goodyear Corporation 1 K4S Industries Inc
Attn: Technical Library PO Box 1778
1210 Massilon Road Ann Arbor MI 48108
Akron OH 44315
1 Ling-Temco-Vought Corporation
1 Grumman Aircraft Engineering Military Electronic Division
Corporation Attn: Technical Library
Attn: Technical Library 1200 Jupiter Street
South Oyster Bay Road Garland TX 75040
Bethpage NY 11714

DISTRIBUTION LIST
Nr C oies Co ies mraCosies Compies
Litton Systems Inc 1 Martin Marietta Company
Amecom Division Attn: Technical Library
Attn: Technical Library PO Box 988
1140 E W Highway Baltimore MD 21203
Silver Spring MD 20910
1 McDonnell Douglas Aircraft
1 Lockheed Aircraft Corp Corporation
Electronic & Armaments Attn: Technical Library
System Office Box 516
Attn: Technical Library St Louis MO 63166
PO Box 551
Burbank CA 91503 1 McDonnell Douglas Aircraft
Corporation
1 Lockheed Aircraft Corp Attn: Technical Library
Missiles and Space Division 3855 Iakewood Boulevard
Attn: Technical Library Long Beach CA 90801
PO Box 504
Sunnyvale CA 94088 1 McDonnell Douglas Aircraft
Corporation
1 Lockheed Aircraft Corp Attn: Technical Library
Lockheed-Georgia Division 3000 Ocean Park Blvd
Attn: Technical Library Santa Monica CA 90406
South Cobb Drive
Marietta GA 30060 1 McDonnell Aircraft Company
Attn: Technical Library
1 Lockheed Aircraft Corp 2000 North Memorial Drive
Lockheed Electronics Company Tulsa OK 741.15
Attn: Technical Library
US Highway 22 1 Melpar Inc
Plainfield NJ 07060 Attn: Technical Library
7700 Arlington Boulevard
1 Martin Marietta Company Falls Church VA 22046
Denver Division
Attn: Technical Library 1 Mitre Corporation
PO Box 179 Attn: Technical Library
Denver CO 80201 Middlesex Turnpike
PO Box 508
1 Martin Marietta Company Bedford MA 01730
Orlando Division
Attn: Technical Library 1 Motorola Inc
PO Box 5837 Government Electronics
Orlando FL 32805 Division
Attn: Technical Library
8201 E McDowoll Road
Scottsdale AZ 85257

DISTRIBUTION LIST
r Copies Comanies Nr Copies Companies
1 North American Aviation Inc 1 Radiation Inc
Autonetics Division Systems Division
Attn: Technical Library Attn: Technical Library
3370 Miralonma Avenue Box 37
Anaheim CA 92803 Melbourne FL 32901
1 North American Aviation Inc 1 Radiation Inc
Attn: Technical Library Attn: Technical Library
International Airport 1755 Old Meadow Road
Los Angeles CA 90009 McLean VA 22101
1 North American Aviation Inc 1 Radio Corporation of Anerica
Attn: Technical Library Missile & Surface Radar
4300 E Fifth Street Division
Columbus OH 43216 Attn: Technical Library
Mail Stop 101-210
1 Northrop Aircraft Corp Moorestowm NJ 08057
Norair Division
Attn: Technical Library 1 Rantec Corporation
3901 West Broadway Attn: Technical Library
Hawthorne CA 90250 23999 Ventura Blvd
Calabasas CA 91302
1 Page Communications Engineers Inc
Attn: Technical Library 1 Raytheon Corporation
3300 Whitehaven Street Attn: Technical Library
Wash DC 20007 Santa Barbara Operation
PO Box 1542
1 Philco-Ford Corporation Goleta CA 93017
Aeronutronic Division
Attn: Technical Library 1 Raytheon Corporation
Ford Road Missile Systems Division
Newport Beach CA 92663 Attn: Technical Library
Hartwell Street
Bedford MA 01730
1 Raytheon Corporation
Equipment Division
Attn: Technical Library
40 Second Street, PO Box 520
Waltham MA 02154

DISTRIBUTION LIST
Nr Coies Coanies Nr Copies Copanies
1 Republic Aviation Division 1 Sperry Rand Corporation
Fairchild Hiller Sperry Gyroscope Division
Attn: Technical Library Attn: Technical Division
Department 61, IMezz 9 Mail Station V-119
Farmingdale NY 11735 Great Neck LI NY 11020
1 Ryan Aeronautical Company 1 Sperry Rand Corporation
Attn: Technical Library Sperry Microwave Electronics
5650 Kearny Mesa Road Division
PO Box 311 Attn: Technical Library
San Diego CA 92112 PO Box 1828
Clearwater FL 33517
1 Sanders Associates Inc
Microwave Division 1 Stanford Research Institute
Attn: Technical Library Attn: Technical Library
95 Canal Street Ravenswood Avenue
Nashua NH 03061 Menlo Park CA 94025
1 Scientific Atlanta Inc 1 Sylvania Electronics - West
Attn: Technical Library Electronic Defense Laboratories
PO Box 13654 Attn: Technical Library
Atlanta GA 30324 PO Box 205
Mountain View CA 94040
1 Sedco
Attn: Technical Library 1 Sylvania Electronics Systems
130 Schmitt Boulevard Attn: Technical Library
Farmingdale LI INY 11735 100 East First Avenue
Waltham MA 02154
1 HRB Singer Corporation
Attn: Technical Library 1 Teledyne Systems Corporation
Box 60, Science Park Attn: Technical Library
State College PA 16801 12525 South Daphne Avenue
Hawthorne CA 90066
1 Southwest Research Institute
Attn: Technical Library 1 Texas Instruments Inc
8500 Culebra Road Attn: Technical Library
San Antonio TX 78206 13500 N Central Expressway
Dallas TX 75222
Space Technology Laboratory
Attn: Technical Library
1 Space Park
Redondo Beach CA 90278

DISTRIBUTION LIST
Nr Cop es Com,,,)anies Nr Copies Government Offices
TRW Systems 3 AFAL (AVLE - Mr. Horton )
Thompson Ramo Wooldridge Inc Wright-Patterson AFB OH 45433
Attn: Technical Library
1 Space Park 1 AFAL (AVWC - r. Whrig)
Redondo Beach CA 90278 Wright-Patterson AFB OH 45433
Westinghouse Electric Corp 1 AFAL (AVTI - Mr. Greene)
Aerospace Division Wright-Patterson AFB OH 45433
Attn: Technical Library
PO Box 746 1 ASD (ASEP - Mr. Brooks)
Baltimore ID 21203 Wright-Patterson AFB OH 45433
Westinghouse Electric Corp 1 FTD (TDCES - Mr. Wiegert)
Defense and Space Center Wright-Patterson AFB OH 45433
Attn: Technical Library
PO Box 1897 1 FTD (TDEE)
Baltimore MD 21203 Wright-Patterson AFB OH 45433
1 Wheeler Laboratory 1 ASD (SEACC - Mr. Stahler)
Attn: Technical Library Wright-Patterson AFB OH 45433
Box 561
Smithton NY 11787 1 ASD (SEAEM - Mr. Mulligan)
Wright-Patterson AFB OH 45433
Government Offices 1 ASD (ASNPD-30); Wright-Patterson AFB OH 45433
USAF (AFCSAI)
Wash D C 20330 1 ASD (ASBED-20- Mr. Stecher)
Wright-Patterson AFB OH 45433
Hq AFSC (SCAP)
Andrews AFB 1 AFSC (SCE)
Wash D C 20331 Andrews AFB
AWash DC 20331
E WASL
Wright-Patterson AFB OH 45433 RADC (TMATA)
Griffiss AFB NY 134+2
1 RADC (E2CRA)
Griffiss AFB NY 13442
1 RADC (EICOT)
Griffiss AFB NY 13442
1 AFCRL (CRD - Mr. C. J. Sletten)
L G Hanscom Fld
Bedford lMA 01730

DISTRIBUTION LIST
Nr Cooies Government Offices Nr Copies Government Offices
1 AFETR (ETGCS) 1 Director
Patrick AFB FL 32925 Surveillance Department
Evans Area
Hq USAF (AFRDD - Lt Col Lieber) Attn: Technical Document Center
Wash DC 20330 Belmar NJ 08056
1 Hq USAF (AFXAI) 1 Commander
Air Battle Analysis Center US Naval Air Test Center
Deputy Director of Plans for Attn: WSST-54
War Plans Patuxent River ID 20670
Wash DC 20330
1 Conmmnding Officer
USARDVL (Attn: SIGRA/NAI)
Fort Monrmouth NJ 07703
1 Commanding General 1 Director
US Army White Sands Missile Range US Navy Electronics Laboratory
Attn: Technical Library Code 3220
White Sands 1NI 88002 Attn: Library
San Diego CA 92152
1 Commanding Officer
US Army Electronics R&D Activityl Conmander
Attn: SEI;S-ED US Naval Ordnance Test Station
White Sands NI 88002 Code 4021 (Mr. J. A. Mosko)
China Lake CA 93557
1 Department of the Army
Ballistic Research Laboratory 1 Commander
Attn: Technical Library US Naval Ordnance Laboratory
Aberdeen Proving Ground MD 21005 Attn: Mr. Dave Schriner
Corona CA 91720
1 Commanding Officer
Harry Diamond Laboratories 20 DDC
Conn Avenue & Vann Ness Street, NW Cameron Station
Attn: 240 Alexandria VA 22314
Wash DC 20438
1 AUL (3T-AUL-59-30)
1 Chief, Bureau of Ships Maxwell AFB AL 36112
Attn: Code 312
Main Navy Building 1 NASA Goddard Space Flight Center
Wash DC 20360 Attn: Antenna Branch - Mr. Lantz
Greenbelt MD 20771
1 Director
Naval Research Laboratory 1 NASA
Attn: Code 5200 PO Box 33
Wash DC 20390 College Park MiD 2074+0

DISTRIBUTION LIST
Nr Copies Universities Nr Copies Universities
1 Cornell Aeronautical Laboratory 1 Polytechnical Institute of Brooklyn
Attn: Technical Library Microwave Research Institute
Buffalo NY 14221 Attn: Teclnical Library
55 Johnson Street
1 Georgia Institute of Technology Brooklyn NY 11201
Engineering Experiment Station
Attn: Technical Library 1 Stanford Electronic Laboratory
Atlanta GA 30331 Attn: Technical Library
Stanford CA 94305
1 Johns Hopkins University
Applied Physics Laboratory 1 Syracuse University
8621 Georgia Avenue Electrical Engineering Department
Silver Spring MD 20910 Attn: Technical Library
Syracuse NY 13210
1 Lincoln Laboratories
MIT 1 University of Dayton Research
Attn: Technical Library Institute
Lexington IA 20653 Attn: Technical Library
300 College Park
1 New Mexico State University Dayton OH 45409
Head, Antenna Department
Physical Science Department 1 University of Illinois
University Park M1 88070 Department of Electrical Engineering
Attn: Technical Library
1 North Carolina State Urbana IL 61803
of the University of North Carolina
Attn: Technical Library 1 University of Michigan
PO Box 5275 Radiation Laboratory
Raleigh NC 27607 Willow Run
Attn: Technical Library
1 Northeastern University 201 Catherine Street
Attn: Dodge Library Ann Arbor MI 48104
Boston MA 02115
1 University of Southern California
1 Ohio University Electrical Engineering Department
Electrical Engineering Department Attn: Technical Library
Attn: Technical Library University Park
Athens OH 45701 Los Angeles CA 90007
1 Johns Hopkins University 1 University of Texas
Carlyle Barton Laboratory Electrical Engineering Research
Charles & 34th Streets Laboratory
Baltimore MD 22218 Attn: Technical Library
Route 4, Box 189
Austin TX 78756