AMLYSLS or A PARALLEL ARRAY
or weveeumz 0R CAVITY-BACKED-
RECTANGULAR SLOT ANTENNAS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UfiIVERSiTY
SATNAM PRASAD MATHURJ
1974

  
     

  

L I B R A R Y
Michigan S t2 ‘56
University

This is to certify that the

thesis entitled

ANALYSIS OF A PARALLEL ARRAY OF WAVEGUIDE
OR CAVITY-BACKED RECTANGULAR SLOT ANTENNAS

presented by

33 tnam Prasad Ma thur

has been accepted towards fulfillment
of the requirements for

Ph.D . degree in Electrical Engineering
and System Science

 

 

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ABSTRACT

ANALYSIS OF A PARALLEL ARRAY OF WAVEGUIDE
OR CAVITY-BACKED RECTANGULAR SLOT ANTENNAS

By

Satnam Prasad Mathur

The circuit and radiation properties of a finite parallel
array of N transverse, waveguide-backed, rectangular slot antennas
are investigated in this thesis. A theoretical analysis is based
upon a rigorous integral equation approach, and the theoretically
predicted results are compared with experimental measurements.

An array of flush-mounted slot antennas consists of N
parallel, transverse slots cut in an infinitesimally thin ground plane
of large lateral extent and backed by a rectangular waveguide or
cavity. Two modes of array excitation are considered: first by an
incident dominant-mode wave in the backing waveguide and secondly
by impressed driving currents maintained at the centers of one (or
more) slots by a system of input currents.

In the theoretical investigations, a system of N coupled,
Hallén-type integral equations for the N unknonw slot field distribu-
tions in the apertures of the array elements, maintained by either
type of excitation, are formulated. These equations are solved
numerically when the array is excited by an incident, dominant-mode
wave in the backing waveguide and by an approximate analytical method

(new, extended King-Sandler array theory) when the array is driven by

a system of input currents.

SATNAM PRASAD MATHUR

Numerical results for either type of array excitation in-
dicate (for appropriate slot and waveguide dimensions) the existence
of a slow wave along the array aperture. It is also demonstrated
that the phase velocity of the traveling wave along the array aperture
can be controlled by varying the width of the backing waveguide (if
the array is excited by a dominant-mode wave) and that scanning of
radiation beam is consequently possible. No such beam scanning can
be achieved by variation of waveguide width when the array is excited
by a system of input currents, indicating a behavior similar to that
for a Yagi-Uda slot array cut in a ground plane in an otherwise un-
bounded space. Phase velocity of the aperture field along the array
is also sensitive to slot lengths and element Spacings. Optimum slot
dimensions to achieve endfire radiation of maximum directivity with
reasonable side and back lobe levels are determined.

Impedance characteristics of the array are studied for both
types of excitation. For the array excited by an incident wave, the
SWR in the driving waveguide is relatively insensitive to backing
waveguide width while it is quite sensitive to slot parameters
(dimensions and Spacings). Impedance of the Yagi-Uda parasitic,
waveguide-backed slot array behaves essentially as that of a comple-
mentary dipole array except for a shift in resonant length of the
driven element.

An experimental investigation of slot field distributions,
the distribution of aperture field along the array, the SWR in the
transmission system exciting the array, and the array radiation field
is carried out. Excellent agreement between experimental results and

those of the numerical, analytical solution is obtained for the array

SATNAM PRASAD MATHUR

excited by an incident waveguide mode. It is indicated that the

new, approximate parasitic array theory for waveguide-backed slots
predicts accurate results for arrays of narrow slots having deep

backing waveguides or cavities.

ANALYSIS OF A PARALLEL ARRAY OF WAVEGUIDE
OR CAVITY-BACKED RECTANGULAR SLOT ANTENNAS

By

Satnam Prasad Mathur

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Electrical Engineering and System Science

1974

ACKNOWLEDGMENT

The author wishes to express his sincere appreciation to his
major professor, Dr. D.P. Nyquist, for his guidance and assistance
throughout this study. His active participation in the project and
his willingness to discuss problems as they arose made working with
him a rewarding experience. Thanks also go to Dr. J. Asmussen and
Dr. J.S. Frame for their many suggestions and technical advice on
problems encountered in the research. He wishes to thank other members
of his guidance committee for their time and interest in this study:
Dr. B. Ho and Dr. K.M; Chen.

The research reported in this thesis was supported, in parts,
by Division of Engineering Research, Michigan State University.

Finally, special thanks go to Mr. Marc Butler and Mr. R. Kotsch

for their help and encouragement throughout this study.

ii

Chapter
1

2

TABLE OF CONTENTS

INTRODUCTION ....................................

FORMULATION OF BASIC THEORY FOR THE WAVEGUIDE-
BACKED SLOT ARRAY ...............................

2.1 Geometry and Statement of the Problem ......
2.2 Integral Equations for the Slot Field
Distribution ...............................
2.2.1 Magnetic Field in the Interior
Region (y > O) ......................
2.2.2 Magnetic Field in the Exterior
Region (y < 0) ......................

2.3 Conversion to a System of Hallén-Type
Integral Equations for Slot Voltage

Distributions ..............................
2.4 Radiation Field Maintained by the Slot
Array ......................................
SLOT ARRAY EXCITED BY DOMINANT BACKING-WAVEGUIDE
MODE: NUMERICAL SOLUTION .......................
3.1 Introductory Remarks .......................
3.2 Simplification of the System of Hallén-Type
Integral Equations .........................
3.3 Numerical Solution Using Pulse Functions
and Point-Matching .........................
3.3.1 Numerical Evaluation of the Various
Integrals ...........................
3.4 Radiation Field Maintained by the Slot
Array ......................................
3.5 Input Impedance to Backing Waveguide .......
3.5.1 Numerical Evaluation of I .........
3.6 Numerical Results ............... 4 ..........

iii

10

19

23

27

32
32
33
34
37
43
44

49
SO

Chapter

4 SLOT ARRAY EXCITED BY IMPRESSED CURRENT:
APPROXIMATE ANALYTICAL SOLUTION .................

4.1 Introductory Remarks .......................
4.2 Extension of King-Sandler Dipole Array
Theory to the Waveguide-Backed Slot Array ..
4.2.1 Properties of the Kernel and
Resulting Approximations ............
4.2.2 Alternative System of Difference
Integral Equations for the V,(x)
4.2.3 Analytical Approximations forISlot
Voltage Distributions ...............
4.2.4 Reduction of the System of Integral
Equations to a System of Algebraic
Equations ...........................
4.3 Evaluation of Various Y—Functions ..........
4.3.1 Evaluation of Y and Y
(for hk s io/4)d533 ........ iii? .....
4.3.2 Evaluation of Y ................

4.3.3 Evaluation of Y ................
4.3.4 Evaluation of Y . .................
4.3.5 Evaluation of Y .................

4.4 Radiation Field Maintained by the Slot
Array ......................................
4.5 Numerical Results ..........................
4.5.1 Investigation of Several Special
Cases ...............................
4.5.2 Five and Ten-Element, Waveguide-
Backed, Yagi-Uda Slot Arrays ........
4.5.3 Ten-Element, Waveguide-Backed Slot
Array ...............................
4.5.4 Frequency Dependence of a Yagi-Uda
Slot Array ..........................

5 EXPERIMENTAL INVESTIGATION OF WAVEGUIDE -BACKED
SLOT ARRAYS .....................................

Introductory Remarks .......................
Anechoic Chamber and Experimental Set-Up ...
Measurements on Waveguide-Backed Slot Array
Excited by a Dominant-Mode Incident Wave

5.4 Measurements on Yagi-Uda, Waveguide-Backed

UlUlUi
“NH

Slot Array . ................................

6 SUMMARY AND CONCLUSIONS .........................
REFERENCES . .................................. . .............
APPENDIX I .................................................
APPENDIX II ................................................

Page

66
66
67
68
76
78
79
84

84
88
9O
93
94

108
127

137

140
141

148

154

Table

3.1

3.2

4.1

5.1

5.2

5

.3

LIST OF TABLES

Page
Backing waveguide SWR (S-slot array) for various
slot lengths and Spacings (a = 0. 6x0 , b = 0. 3x0 ,
F1 = F2 = 0, n= 2Ln(4h/e) = 10 0.6) .............. 63
Driving waveguide SWR (IO-slot array)h for various
backing waveguide widths (b= 0. 3x0 , =O.22xo,
F1 = P2 = 0, fl = ZLn(4h/e)= lO. 6)0 .............. 65

Comparison of various Y-functions and input
impedances for a slot antenna (new theory) and
its complementary dipole (published results) .... 106

Experimental and theoretical driving waveguide

SWR (8-slot array) for two backing waveguide

widths (b= 0. 3A0, h = O. 22),o , F1= F2

n= 2Ln(4h/e) = 8.1) ............................ 155

Experimental and theoretical SWR's maintained by
a lO-element, waveguide-backed, Yagi-Uda slot
array (a = 0.6xo, b = 0-3k0: h1 = 0.25xo,

h2 = 0.24xo, hD = 0.22io, (A2)ref = 0.25xo,

(Az)dir = 0-33K0, F1 = -1.0, F2 = 0.0) on its

50-ohm coaxial exciting system .................. 164
Experimental and theoretical SWR maintained by a

lO-element, Yagi-Uda slot array (G1 = G0,
h = 0.25A0. h2 = 0.24i0, hD = 0.22Lo.

1
(A2)ref = 0.25xo, (Az)dir = 0-33A0) on Its SO-ohm
coaxial exciting system ......................... 164

LIST OF FIGURES

Figure Page
2.1 Geometrical structure of a waveguide (or cavity)
backed parallel array of N rectangular slots .. 8
2.2 Illustration of principle of superposition for
the determination of interior magnetic field .... 12
2.3 Geometry for radiation field calculation ........ 28
3.1 Integration subdivision and matching points for
numerical solution of the integral equations .... 35
3.2 Voltage distributions in the elements of a
5-element waveguide-backed slot array ........... 51
3.3 Amplitudes and phases of slot voltages in the
elements of a 5-element array for various slot
Spacings ........................................ 54
3.4 Amplitudes and phases of slot voltages in the
elements of a 5-e1ement array for various slot
lengths ......................................... 55
3.5 Amplitudes and phases of the slot voltages in the
elements of a 5-element array for various backing-
waveguide widths ................................ 57
3.6 Dependence of E—plane (¢ = ~n/2) radiation field
patterns of a S-element slot array upon width
of its backing waveguide and slot length ........ 58
3.7 Amplitudes and phases of slot voltages in the
elements of a lO-element array for various
backing waveguide widths ........................ 60
3.8 Dependence of E-plane (¢ = -fi/2) radiation field
patterns of a lO-element slot array upon width
of its backing waveguide ................... ..... 61
4.1 Magnitude of Kkk vs kol(x-x')\ ................ 72
“1,0 u .
4.2 Kki (x,x ) vs kolx-x \ ........................ 73
4.3 Admittance of a single cavity—backed slot as a
function of cavity depth ........................ 101

vi

Figure

4.4

Slot voltage distribution in a single cavity-
backed slot for various cavity depths (com-
parison with Galejs' variational solution) .....

Dependence of the front-to-back ratios of the
radiation field patterns for a two-element
parasitic slot array upon the slot spacing .....

Comparison of amplitudes and phases of slot
voltages with those of currents in a comple-
mentary dipole array ...........................

Comparison of E-plane radiation patterns for a
8-director Yagi-Uda slot array with those of a
complementary dipole array .....................

Voltage distributions in the elements of a
5-element Yagi-Uda slot array ..................

Amplitudes and phases of the slot fields in the
elements of a 5-element Yagi-Uda slot array for
various director Spacings .. ....................

Dependence of E—plane (¢ = -n/2) radiation field
patterns of a 5-element Yagi-Uda slot array upon
element Spacings ................... . ...........

Amplitudes and phases of the slot fields in the
elements of a 5-e1ement Yagi-Uda slot array for
various director lengths .......................

E-plane (m = -n/2) radiation patterns for a
5-e1ement Yagi-Uda slot array for various
director lengths ...............................

Input admittance to the driven element of a
Yagi-Uda parasitic slot array as a function of
its half length hzl),o .........................

Amplitudes and phases of the slot fields
(voltages) in the elements of a lO-element Yagi-
Uda slot array for various director Spacings

E-plane radiation patterns of a lO-element
Yagi-Uda slot array for various director
spacings . ......................................

Amplitudes and phases of the slot fields

(voltages) in the elements of a lO-element
Yagi-Uda slot array for various director lengths

vii

Page

103

104

107

109

110

112

113

116

117

119

120

122

124

Figure

4

5

.17

.18

.19

.20

.21

.22

.23

.24

.25

.1a

.1b

.2

E-plane radiation field patterns of a ten-
element Yagi-Uda slot array for various

director lengths ................................

Amplitudes and phases of slot fields in the
aperture of a ten-element slot array for various

director element Spacings . ......................

E-plane radiation field patterns of a ten-
element slot array for various director
Spacings .......................................

Amplitudes and phases of slot voltages in the
elements of a lO-element slot array for various
backing waveguide widths ......................

Dependence of E-plane (m = -n/2) radiation field
patterns of a lO-element slot array upon width
of its backing waveguide ......................

Amplitudes and phases of slot fields in the
aperture of a twenty-five elements, waveguide-
backed, slot array ............................

E-plane radiation field pattern of a twenty-
five element, waveguide-backed slot array ....

Frequency dependence of the E-plane radiation
field pattern of a ten-element, waveguide-backed
Yagi-Uda slot array ..........................

Input admittance to the driven element of a
ten-element, Yagi-Uda waveguide-backed slot
array ........................................

Photograph of slot array cut in ground plane
and mounted in anechoic chamber (with near-
field probing system of dipole receiving
antenna) .....................................

Photograph showing close-up view of the slot
array and the coaxial near-field probing

system used to measure the aperture field along
the array and the slot field distributions

Photograph of various microwave instrumenta-
tion (and part of the backing waveguide system)
used in making measurements on the waveguide-
backed slot array ............................

viii

Page

125

128

130

131

133

134

136

138

139

142

142

143

Anechoic chamber and block diagram of
experimental set-up ............................

Comparison of theoretical and experimental
amplitudes and phases of slot voltages in the
elements of a 8-e1ement array for various
backing waveguide widths .......................

Slot field distributions in the elements of a 8-
element, waveguide-backed, slot array ..........

Theoretical and experimental radiation patterns
for a 8-element slot array with two different
backing waveguide widths .......................

Comparison of theoretical and experimental
amplitudes and phases of slot voltages in the
elements of a ten-element Yagi-Uda slot array

Theoretical and experimental radiation patterns
for a ten-element, Yagi-Uda slot array .........

Comparison between theoretical and experimental
relative amplitudes and phases of slot voltages
in the elements of a ten-element, waveguide-

backed, Yagi-Uda slot array ....................

Slot field distributions in the element of a
lO-element, waveguide-backed, Yagi-Uda slot
array ..........................................

Theoretical and experimental radiation patterns

for a ten—element, waveguide-backed, Yagi-Uda
slot array .....................................

ix

Page

1 44

150

152

153

157

159

160

162

163

CHAPTER 1

INTRODUCTION

A theoretical and experimental investigation on the circuit
and radiation properties of a finite, waveguide-backed slot array is
performed. The slot array consists of a system of narrow slots cut
in a thin conducting ground plane of large lateral extent and backed
by a waveguide designed to support the dominant TElo-mode wave. The
slots are cuttransversely through the broad wall of the waveguide,
which can be arbitrarily terminated at each of its ends. Two methods
of array excitation are investigated: 1) an incident dominant-mode
wave in the backing waveguide and 2) an impressed current between the
edges of one or more slots at their centers. Such slot arrays are
popular for aerOSpace applications due to their inherent adaptability
to flush mounting.

(1)

It has been demonstrated by Burton and King that a slow wave
can be excited along a parallel array of appropriately spaced, resonant-
length slots cut in a large, thin ground plane. Since such a slot

array is the complementary antenna to a parallel array of cylindrical
dipoles, the existence of a slow wave along the slotted ground plane

is also suggested by the results of investigations by Mailloux(2)’ (3)
on the excitation of a traveling surface waves along long Yagi-Uda
dipole arrays. It has also been demonstrated by Coe and Held(4)

that a similar slow wave aperture field is obtained when each individual

slot is backed by a rectangular cavity.
1

If the slot array is backed by a waveguide on one side of the
ground plane, it may be possible to control the phase velocity of the
traveling wave along the array aperture by adjusting the phase velocity
of the fast, dominant wave in the backing waveguide. It has been
demonstrated by Hyneman(5), using an approximate variational method,
that, for an infinite array of closely Spaced transverse slots, two
useful means of radiation pattern control involve either adjusting
the transverse geometry of the backing waveguide or varying the element
Spacing in the Slot array. It is, therefore, possible to excite a
slow wave in the array aperture and consequently achieve endfire
radiation. By appropriately controlling suitable array parameters or
the backing waveguide dimensions, some degree of control can be
achieved over the phase velocity of the traveling wave aperture field
with a consequent capability to scan the direction of the radiation
beam. This phenomenon is studied in detail in the present research.

Elliott(6) investigated (using an approximate variational
technique) a serrated waveguide of infinite length where infinitesimally
Spaced transverse slots are cut into, and extended completely across the
broad waveguide wall of variable thickness. Elliott concluded from
his study of the serrated waveguide that the fundamental mode in the
waveguide and along the array aperture has a complex propagation con-
stant, which is insensitive to frequency but modestly sensitive to
wall thickness. It was also found that the phase velocity of the
fundamental mode wave along the serrated waveguide can be varied by
controlling the serration width to Spacing ratio or serration length

(which is equal to the broad wall width).

'(7)

From the experimental investigations by Kelly and Elliott
of the same serrated waveguide it was confirmed that the phase
velocity can be effectively controlled by varying the wall thickness.
However, thick-walled serrated waveguide presented practical problems
in its construction, while it was found that varying serration
width to spacing ratio was ineffective insofar as the variation of
complex propagation constant in the serrated waveguide was concerned.
In summary, only the thin-walled, serrated waveguide provided least
mechanical problems in its construction while a reasonable variation
of leakage (attenuation) was achieved as the length of serrations
was varied (while phase constant remained relatively insensitive to
such variations). Among effective methods for wide control over phase
constant (phase velocity) were found to be the variation of serrated
waveguide width and the properties of its dielectric loading.

In light of the above observations, it was decided to study
more rigorously the radiation and circuit properties of a finite
parallel array of N waveguide-backed, rectangular slot antennas
cut transversely through the infinitesimally thin broad wall of a
backing waveguide. It is assumed that the Slots are backed by a
waveguide of variable width "a" which is terminated by arbitrary

reflection coefficients F and F2 at its two ends. It is further

I
assumed that the array is excited either by a dominant-mode incident
wave in the backing waveguide or by means of impressed (input) currents
at the centers of any number of the array elements. A rigorous
integral equation technique is used to determine the slot fields in

the elements of the finite array. In terms of these slot fields,

the circuit and radiation characteristics of the array are determined

for various slot parameters and backing waveguide dimensions to
optimize the array design and to determine the feasibility of scanning
the radiation beam from endfire to off-endfire. A set of experi-
mental measurements is made to confirm the theoretical predictions.
Using a technique Similar to those reported by Galejs(8)’ (9)
for the analysis of single, cavity-backed slots, a system of N
coupled, Hallén type integral equations for the N unknown Slot
field distributions in the apertures of the array elements is
formulated as indicated in Chapter 2. These equations are solved

(10)

both numerically by Harrington's moment method and analytically

(approximate) by a modification and extension of the King-Sandler

(11) to the waveguide-backed slot array. The

dipole array theory
radiation field of the array as well as its circuit properties can
be readily calculated in terms of the known Slot fields.

The numerical solution for the slot fields in the aperture
of the array excited by a dominant-mode-incident wave is indicated
in Chapter 3. Numerical results presented in Chapter 3 for five and
ten element arrays indicate the existence of a slow wave aperture
field for an optimal set of array parameters, with resultant endfire
radiation. The feasibility of scanning the radiation beam of the
array by varying the phase velocity of the traveling wave in its
aperture through variations of the backing waveguide width is demon-
strated.

Chapter 4 develops an approximate, analytical solution for
the slot fields in the array excited by impressed 6-function currents

at the centers of one (or more) of its elements. Results presented

in Chapter 4 indicate the existence of a slow wave along the array

aperture, and hence the existence of endfire radiation, for an Optimal
choice of array parameters in five, ten and twenty-five element arrays.
The new, extended King-Sandler array theory also predicts accurate
results for an array of slots cut in a ground screen immersed in other-
wise unbounded free-Space (complementary to a dipole array) and for a
single slot backed by a rectangular cavity; the new predictions agree

well with the published results by Galejs(8)

and by King, Mack and
Sandler<12). It is also pointed out in Chapter 4 that this new array
theory does not predict accurate results for an array of wide slots
or for array backed by a shallow waveguide (or a cavity).

Experimental results are compared with corresponding pre-
dicted theoretical results in Chapter 5. It is found that the numerical
solution predicts accurate results (agrees well with experiments) for
an eight-element, waveguide—backed slot array excited by a TElo-mode
incident wave. However, the new theory for a waveguide-backed, Yagi-
Uda slot array excited by an impressed current predicts accurate
results (agrees relatively well with experiment) only for an array
of narrow slots.

The circuit properties of the slot array excited by a dominant-
mode-incident wave in the backing waveguide indicates a reasonably
good match between the antenna input terminals and the standard wave-
guide circuitory which is employed to excite the array i.e., the SWR
is relatively low. As the slot lengths exceed their resonant lengths,
the SWR in the backing waveguide increases sharply. As the width of
the backing waveguide is reduced (to vary the phase velocity of the

aperture field) so its dominant mode approaches cut off, the SWR also

rises, although not as sharply as might be anticipated.

If the slot array is driven by an impressed current, the
input admittance to the driven element passes through a resonance
when the length of that element lies between 0.22 kc and 0.23 x0
for both five- and ten-element arrays. The ten~element Yagi-Uda
slot array diaplays a band width of approximately 400 MHz for a
design where its input admittance passes through resonance at the
center frequency of 3.0 GHZ.

A brief summary as well as conclusions are included in

Chapter 6.

CHAPTER 2.

FORMULATION OF BASIC THEORY FOR THE
WAVEGUIDE-BACKED SLOT ARRAY

2.1 Geometry and Statement of the Problem:

In this chapter a system of Hallén-type integral equations
will be developed for the voltage distribution in the elements of
a parallel array of N waveguide-backed, rectangular Slot antennas.
The final expressions for E-plane and H-plane radiation fields of
the array and its circuit properties are then calculated in terms
of these slot fields.

The geometrical structure of the array of N waveguide-
backed slots is shown in Figure(2.1). N slots of width 26 with
centers at x = g are cut parallel to the x-axis in a large con-
ducting ground plane at locations 2 = 21, z = 22, z = 23,...,

z = zk,...,z = 2N. The slots are backed by a rectangular waveguide
of width "a" and height "b" which is terminated by reflection co-
efficients FI and P2, reSpectively, at z = 0 and z = c. The

kth element of the array has length 2h and is center-driven by

k

an input current Ik; an approximately one-dimensional Slot field

Ekz(x) is excited in its aperture by I and its coupling to other

k

elements of the array. When the array is excited alternatively by

an incident dominant-mode wave in the backing waveguide, then I = 0,

k
for all k. In section 2.2, a theory encompassing both modes of

excitation is presented.

round
backing lane /
Ewave guide

2: z=c
x-O I l '
26:“! I
l
/ ' / X/ ii / / : /
I I
' D. e I
1"1 :/ I1 12 2h. Ii / N :FZ
TB 10 : Elz Ezz(x) E1 (x) ENux)
mode I l
wave F / I / / / / I
eel--- ........ +-----—-----/
:rx=a /
1 2:21 2:22 z:z 2:3

 

a. Top View of waveguide (or cavity) backed slot array.

r—Zhi “—1 ground plane

 

y=0 x=0 x:a

(6:60: ”:1; 090:0)

 

 

 

b. Side view (in a cross section at z=zi) of waveguide (or cavity) backed slot
array.

Figure 2.1. Geometrical structure of a waveguide (or cavity) backed parallel
array of N rectangular slots.

2.2 Integral Equations for the Slot Field Distribution:

The system of integral equations for the electric field dis-
tributions in the rectangular slots is based upon the boundary con-
dition for the tangential components of magnetic field at their

jwt

apertures. A harmonic time dependence of the form e is presumed
throughout the development. Let Ho(x,y = 0-,z) be the magnetic
field at a point just outside the aperture in the exterior region,

y < 0 and Hi(x,y = 0+,z) be that at a point just inside the
aperture in the interior region y > 0, then the boundary condition

for the tangential magnetic field at the aperture of the kth slot

requires that

:3)

x Lilies = o+.z> - 1'1?ij = 0'.z>] = E:<x.z) (2.1)

where h 9 is the unit normal vector to the aperture pointing
from the exterior region (y < 0) into the interior region (y > 0),
and R:(x,z) is the impressed electric surface current maintained
in the kth slot. In all expressions, subscripts refer to a specific
element in the array.

It is assumed that each slot is in general driven by a current
Ik of angular frequency w flowing in a wire of infinitesimal thick-

ness at the center (x = g) of the Slot. Therefore, R:(x,z) can

be expressed as
“e _ _e _ fl
Kk(x,z) — z Ik6(x 2) (2.2)

where 6(x - %) is the Dirac delta function.

The scalar component equations from expression (2.1) are

10

+ o _ - _ _§
0 .2) - HkX(X.y - 0 ,z) - Ik6(x 2)

“12((x,y

(2.3)

i _ + _ o '

For narrow Slots with hk.>> e and 23/),O << 1, where N)
(9) . (13)

is the free-space wavelength, it is well known that
Hi(xy=0+z)~0 and Ho(xy=0-z)%0
kz D 3 k2 3 a

The basic boundary condition to be satisfied at y = 0 therefore

takes the form:
H (X 0 Z) - H (X 0 Z) —" I 0(X - 8/2) . (2.4)
kx ’ ’ kx ’ ’ k

2.2.1 Magnetic Field in the Interior Region (y > 0):

The EM field in the interior of the backing waveguide or
cavity can be regarded (by linear Superposition) as having two dis-
tinct sources of excitation, namely:

(i) the EM field radiated into the interior region through
the aperture of the kth slot with its impressed surface current
"Ki(x.z), (k = 1,2,...,N) and

(ii) the EM field of an incident dominant-mode wave in the
backing waveguide. It is assumed that the backing waveguide supports
only the TE10 dominant-mode as a propagating wave.

In general, excitation is provided by the simultaneous applica-
tion of an incident TE10 wave and the impressed surface currents
E:(x,z); in practical implementation of the array, however, only one
or the other of the two excitations is utilized.

The interior magnetic field can be expressed as the super-

position of the magnetic field of the TE10 wave incident at z = 0

11

with arbitrary reflection coefficients T and F2 at z = 0 and

1
z = c reSpectively, and that radiated by the aperture fields of the

kth slot (which involves the impressed surface current R:(x,z)).
AS illustrated in Figure (2.2b), which considers the backing
waveguide with a load ZL at z = c corresponding to reflection

coefficient F2 and a TE10 wave incident at z = O, the interior

field consists of traveling TE waves prepagating in the i_z

10

directions.

 

It can be shown<14)that the transverse fields for such a
wave are:
h h
2JB 2 -jB Z
d _ “h 10 10
Et — E e1(l + er )
and E Ah zjahoz _jB:Oz (2.5)
H=—°-(’éxe)(1-1"e )e
t Zh l 2
10
ZL - Z2 -2jB:c
where F = (————————9e e
2 h
ZL + Z
1
2h kogo . d
10 - h — wave impe ance for TE10 mode,

810

2 2
310 ‘ \/‘o- (fi/a) - phase constant of TElO mode,

 

£0 = 120n = intrinsic impedance of free Space,
211

k = w/h e = —” = free-Space wave number,

0 O 0 A0
and E0 = arbitrary amplitude constant for incident

TE10 mode.
8: and h: = -%—-(%x3:) are the dominant, transverse mode fields
Z
10

with the assumed normalization condition

1.2

s-s- -
381 S2

 

 

 

 

 

 

 

   

I S I = conducting boundary
inc. 4!. -._a_._..LA
TE r l I
10 1 k. S I *— load
W TE, TM ' . TE, TM 1‘2 2
mode MODES I h = -§ ' MODES L
.wave —. I S1 82' —'.
e L
l I
z = O z = 21 z = z2 z = c
(a) General Problem.
I
inc. F2
1‘1 TE10 —'>
mode
wave
2 = 0 z = c
(b) Field maintained by incident (c) Field maintained by slot
TE10 mode only. fields in aperture 88.

Figure 2-2. Illustration of principle of Superposition for the
determination of interior magnetic field.

Ic.s.(e X hl) . st = 1

and can be expressed as

ah A 0 fl
e1 - -y ab Sin(a )

 

ah . «h A 2
“1 = (TE-Hz er> = x h sing—DE)
Z10 ab 210

Expressions (2.5) can now be rewritten as:

 

 

 

-+ 22:0 216:0” 'jaioz 11x 3
Et(x’z) = -y EO ab (1 + er )e sin(;—) g
(2.6)
h h
a 2ja z -je z
a (x,z) = a E 2 (1 - 1“ e 10 )e 10 snug)
t o h 2 a
abZ10 J

As illustrated in Figure (2.2c), the interior field excited
by the slot fields in the apertures of the array elements (which
are ultimately related to the impressed surface current R:(x,z)
in the kth slot) will consist of both TEq as well as TMq modes beyond

the cross-sectional planes at z = 21 and z = 22' In general,

(14)

these fields can be expressed by modal expansions as:

E: A E +B
2 E q q q q]
q
-o - 4+
= +B
H E [Aq H q Hq] for z 5 21
q
and
-e —rI' —o—
E=ZECE+DE]
q q q q
q
-o -—H- -o-
H = C H + D H ... for z 2 z
EEC: q <1 <11 2

where summations over q include all TE as well as all TM modes,

l4

 

 

 

and the modal fields are(14)
:Yz
—r{- —o -o q
E— = e x + e x, e
q tq( ,y>_ zq( m
I z
4!» 1 ,. _. Yq
H—= +--ZXe x, +h x, e
q L_Z q< y) zq( m
q
Yq is the propagation constant of the qth guided wave mode. Re-
flection coefficients qu and qu are defined as
F = 33.: amplitude ofggth mode wave incident upon 2 = 0
lq _ Aq amplitude of qth mode wave reflected from z = 0 2:0
and
F = 31 = amplitude of qth mode wave incident upon 2 = c
2q Cq amplitude of qth mode wave reflected from 2 = c 2:0

 

Therefore, the modal expansions can be rewritten as

and

s a- 4+
E = A E + r E
E q[ q 1q q]
for z s z (2.78)
H [E‘ + r fi+ 1
= A
2 q q lq q]
q
-0 —a+ -o-
E = c E + r E
E q[ q 2q q]
for z 2 z (2.7b)
a .4+ 4- 2
H = 2 c [H + r H
q q q 2s q1

Let the following surfaces be defined:

closed surface consisting of waveguide boundaries and
transverse cross-sectional planes through 2 = 21 and
z = 22.

aperture surface consisting of the slots which comprise

the array elements.

cross-sectional surface through 2 = 21.

15

82 = cross-sectional surface through 2 = z

2.
The boundary conditions to be applied at the waveguide boundary

surface S - S - S2 are:

l
n X E(?) = 0 ... in the surface S - S1 - $2 - Sa outside
the aperture (on the conducting boundaries)
fi X E(?) f 0 ... in the surface S3 of the array (on the
aperture surface)
Making use of the Lorentz Lemma and the normalization con-
dition

J‘ (3 xfi)-’éds=1 ...forallq
c.s. q

 

 

q
where c.s. = any waveguide cross-section, it can be shown(ls) that
constants Aq and Cq are given as
+ .

I( > + F2 1‘ >
and

1H”. I(+>

C .... zqr I“ 13 £11) (2.9)
q (1C1 zq

where aperture integrals have been defined as

+ A -0 -o+
I(-) =£n - (ES XH(-))ds
q q
a
and E8 is the slot field in the array aperture and h =-9.

At points in the aperture of the narrow slots, the slot

u—ps
field E can be approximated by the one-dimensional expression

Es=v E E:(x,z)

16

+
The integrand of 1;") can be rewritten as
. 48 46+) . . 8 -(i9 3 . —(i)
n - E X H —' = - ° 2 E X H = E (x ° H
( q > y ( z q ) z q )

therefore the aperture integral becomes

+y 2
1(1) = +Li Es(x',z')h (x',0,z')e q dx'dz'
q '_ z Xq
a
or
N :y z'
+
I(-) = + 2 E? (x',z')h (x',O,z')e q dx'dz' (2.10)
q _i=1 12 xq
is

where Sia is the aperture of ith slot in the N-element array.

The x-component of magnetic field can now be expressed as

Y 2 'Y z
= v t _ q q
HX(X.y.Z) 2 Aqhxq(x .2 )L e + que ] ... for z s 21
“Y 2 V z 2.11)
= ' ' q _ q
Hx(X:Y’Z) 2 thxq(x :2 )[e que ] ... for z 2 22
The normal mode magnetic fields are<14>
_ __l___ 3m n nn . nfix may
(bx)nm — (k ) -—E-(;—) Sin(-;—) cos( b ) ... TE Modes
c nm abZ
nm
2.12)
_ 1 6men UNI . nnx HEEL
(hx)nm - zi—y- -—E—'(g-) Sin(-;—) cos( b ) ... TM Modes
c nm abZ
nm
where
2 2
(kc)nm = Ynm.+ k = cut off wave number of nmth mode,
1 for m = 0
em = , and
2 ... for m i O
h,e ,
Z = wave impedance of (TE,TM) modes.

nm

17

The above expressions of (hx)q satisfy the ortho-normalization

condition:

--0

fl q 0 for p f q
“ - X h ds = 5 =
4:13.12 (ep q) p l for p=q

(p,q) are general indices which include all TEnm and TMnm modes.
If relations (2.9), (2.10) and (2.12) are used, then equa-

tions (2.11) can be expressed as:

2 .
a (x,y,z) = —1—- —a§-+ k3 Es(x',z')Gi(x,y,z,x',z')dx'dz' , (2.138)
x wuo BX z
a

where the interior Green's function G1 is given by

 

 

 

 

 

i 1 m m 6m
G (X.y.z,X'.Z') = — Z 2 _
ab n=l m=0 Ynmtl (r1)nm(r2)nm]
. ass - nfiX' aux. .
51n( a )Sin( 3 )cos( b )fnm(z,z ) (2.13b)
with
w \z-z'l w <z+z'> y (mu
. '= nm _ nm _ nm
fnm(2.2) e (I‘l)nme (1‘2)nme
+ Ynm‘z-z" 2 13
(P1)nm(rz)nm ( o C)
N
and in summary i = 2 g ,
i=1 .
a 1a
2 2 2 2 2 .nn 2 mfi 2
= -|- = - — -—-
(kc)nm k0 Ynm kx +’ky (a ) + (b )
2 2
Ynm ’ for ko < (kx + ky )
2 2
for k0 > (kx + ky )
(F1)nm and (F2)nm must be specified for various assumed terminations.

l8

Expressions (2.6) along with equations (2.13) result in a

4.
total interior magnetic field at aperture surface y = 0 as:

2
i g I a 2 S I I i I u c u
Hx(x,z) mu 2 + k0 g Ez(x ,z )0 (x,z,x ,2 )dx dz

 

 

0 8x a
h h
2jB z ‘15
2
+ Bo h [1 - (r2)loe 10 3e 10 sum?) (2.14s)
abZ
10
and equation (2.13b) gives
1 1 on an em DTTX
G (X.2.X'.2') = — 2 _ 810(—
ab n=l m=O Ynmt1 (r1)nm(r2)nm] a
sin<9§-"-)f m(2.2'). (22141))
N
In equation (2.14a) i can be replaced by 2 i where Sia
i=1
is

is defined as the aperture of ith Slot

Sia = the slot surface = a - h S x s g_+ hi

2 i 2

2i - e s z s 21+ 6 .

Therefore, the interior magnetic field just inside the aperture of

the kth slot at y = 0+ is

 

H1;(x,0+.2) = 4— L+ {2181.2ij .z'>c (x,z,x .z')
wuo 5x2 i= =1
h
213: “is Z
102 10 fix
dx'dz' +Eo [1- M2)10 ] e sin(;‘9

abZ lO

. for (x,z) E Ska. (2.15)

19

2.2.2 Magnetic Field in the Exterior Region (y < 0):

(16) a solution to Maxwell's equations

It is well known that
for the EM field E(?), E(?) at any point in (otherwise unbounded)
free Space can be expressed in terms of its values on a closed sur-

face S which encloses all the electric sources which maintain the

field as

i563)

-f{jwe0[fi X fi(?')]G(?.?') + [a - E(?')3vG(I—',?')

+ [3 x E633] x ve(?,‘r"))ds' (2.16)

Rd?) -§{Le x HG») x vc<'r‘.'£-") - JweOL’fi x E<?')]c<?.?'>
S
+ 1.3 - fi<?'>]vc<?.?')}ds' (2.17)

where G(?,F') is the Green's function for unbounded free-space

-jko\? - P‘l

 

and h- is the outward directed unit normal vector to S. Jackson(16)

has demonstrated that when the closed surface S assumes the form
of an infinitesimally thin (plane) pancake shaped surface of infinite
lateral extent which separates two half-space regions, then the EM
field in either half-space can be expressed in terms of fields which

are maintained on its plane boundary So as:

E63) = Y La x E(?')] x vc°(?,?')ds' (2.18)
0

20

a?) = -5[ {-jweotfi x E’(?')]G°(r.r> - [a . WW]
0
vc°(¥,?')}ds' (2.19)

when h is the unit normal vector to the half-Space boundary So
which is directed into the field region of interest and Go(?,;')
is the free-space Green's function for a half-space

-jk \? - P'
e 0 \

2n\; - F"

 

6°63?) =

The boundary conditions at the surface 30 in the y = 0

plane are:

a x E(¥') # o
... in the aperture surface S of the
h . fi(;.) # 0 array, and a
a x E(?') = o
in the plane conducting surface 80 - Sa
fi . H(?') = 0 outside the slot apertures.

The integration over surface So in equation (2.19) therefore
reduces to an integration over aperture surface S (in the y' = 0
a

plane) of the slot array as

'fi(¥) = -£ {-ngcja x E(?')]c°(?,?') + [a - fi(?')]vc°(?,?')}ds'. (2.20)
a

In the array aperture surface Sa’ 6 = -9 and ds' = dx'dz'

and for narrow slots

(2.21)

21

From the Maxwell equation

V X E = ‘onfi 1
BE
3 - it’d“) m —-1—--?- . (2.22)
““0 ax

Substituting equations (2.21) and (2.22) into equation (2.20) yields

Hx(x,o',z) = -£ {jmeo ES z'(x ,z W: (x, z,x' ,z ) + Lin—11:0; ,z')

a—<;°(x, z ,x' ,2 ")}dx dz' for (x,z) 6 Sa (2.23)

ax
-jko\j4x-x')2 +-(Z-z')2

e

2fi\/Yx-x')2 +(z-z')2

where

 

 

Go(x,z,x',z') =

 

It is readily verified that

:; Go(x,z,x',z') = ' g T GO(X,Z,X',Z')

2
a—§'Go(x,z,x',z')= -a-—---GO (x, z,x' ,z ')
ax BXBX

and the boundary condition on the slot field at + h requires that

E.
2 -' k

S I = I =
E20“! :hk,z ) 0 . (2.24)

hum

Equation (2.23) can consequently be written as

1+3 a/2+h1

z
1J1I1{a_T'ES(X .Z ) 5"G0 (x, z,x' ,z ')
2i

0 - -
H (X.0 .2) =-
kx -e a/2 -hi ax 12

ll ['12

2
+ koE:(x',z')Go(x,z,x',z')}dx'dz' ... for (x,z) E Ska. (2-25)

22

Upon integrating by parts with respect to x' and making

use of equation (2.24), equation (2.25) can finally be expressed as

:x.(x 0 .Z) = ‘L [5.7+ k2 o] 2 i E:Z(X',2')G°(X.z,X'.Z')dX'dZ'
wuo 5x i=1 ia

for (x,z) E Ska. (2.26)

. . i + o - .
Substituting Hkx(x’0 ,z) and Hkx(x,0 ,z) from equations

(2.15) and (2.26) into boundary condition (2.4) yields

'J—'[a-—-+-k2 o] 2 ziE E? (x',z')Gi(x,z,x',z')dx'dz"+
muo 5x2 i=1ia 12

8219202 -ja:02 nx
[l- (F2)10 ]e sin(;—) +

 

2
;&—’[a—§-+ RD] 2 E:z(x',z')Go(x.z.x',z')dx'dz' = Ik6(x - 8/2)

 

ia
or
2 2 N s
[a-§-+ RC] 2 g Eiz(x',z')G(x,z,x',z')dx'dz' = -jwuo Ik6(x - a/2) +
ax i=1 ia
h h
215 z ‘15 Z

, 2
quoEo h [1 - (r2)10e 10 ]e 10 sin(§5) ... for (x,z) 6 3kg

abZ

(2.27)

where it has been assumed that e = so and p = “o for the interior

region (y > 0) and

i
G(x,z,x',z') = G (x,z,x',z') + Go(x,z,x',z').

h

2 233202 'jaioz
Let g(z) = bzh [1 - (F2)10e ]e . Equation (2.27)
210

can then be more conveniently written as

 

23

2 N
2 s
LL+ k ]{ 2: E. (x',z'>c(x.z.x'.z'>dx'dz'} =
8x2 0 1:1; 12

is

-jkogoik5(x - a/2) + jkogoEog(z)sin(g-§) for (x,z) e ska. (2.28)

Equation (2.28) is the basic integro-differential equation
for the unknown slot electric fields, E:z(x,z), in individual elements
of the array. These slot fields are excited by an incident TElo-mode

wave of amplitude Eo as well as the impressed current Ik at the

center of the kth slot.
2.3. Conversion to a System of Hallén-Type Integral Equations for
Slot Voltage Distributions:

Integro-differential equations (2.28) can be converted to
a system of Hallén—type integral equations for the unknown slot field
distributions. There are distinct advantages associated with utiliza-
tion of the Hallén-type integral equations:

(i) direct numerical solutions are relatively easily
obtained using the Hallén formulation, and

(ii) an extension of the King-Sandler dipole array theory
can be deve10ped for the waveguide-backed slot array. The latter
leads to an approximate analytical solution for the slot field dis-
tribution that results in a great reduction in computation time.

If a function Ak(x,z) is defined as

.21/2+}:i zi+e
Ak(x,z) E 2 I E:z(x',z')G(x,z,x',z')dx'dz' (2.29)

i=1 a/Z-h. z.‘e
i 1

2

then equation (2.28) becomes:

2
a_2 _. , .3
(6x2 + ko)Ak(x,z) - -Jkog01k6(x a/2) + jkogoEOg(z)Sin(a ). (2.30)

24
A complementary solution to the inhomogeneous differential equation

(2.30) is obtained as

c
Ak(x) — A cos kox + Azsin kox (2.31)

1

where A1 and A2 are arbitrary constants. Since the particular

integral for an equation of the form

d2
—-1+ay=f(x)
dx
is given by x
1
YP(X) = E'j f(s)sin a(x - s)ds,
0

then equation (2.30) can be shown to have the particular integral

‘K I jk C E 8(2)
A11:(x,z) = 79—15 sin ko\x - a/21 - ° °2° aim-El).

Y
10
Another term involving sin kox is also obtained in the particular in-

tegral. However, sin kox is already included in equation (2.31) and

therefore it is neglected in the expression for the particular integral.

The general solution to the equation is thus

- c P
Ak(x,z) — Ak(x) + Ak(x,z)
= Alcos kox + Azsin kox + All:(x,z). (2.32)
Since E:z(x',z') is symmetric about x' = a/2, i.e.,

s , _ s
Eiz(a/2 - g, z ) - Eiz(a/2 + g, 2'),
it follows, by definition of Ak(x,z), that
Ak(8/2 ' g, Z) = Ak(a/2 + g: 2).

This requires a relationship between A1 and A2 and leads to a

final solution to equation (2.30) of the form:

25

Ak(x,z) = 0 cos ko(x - a/2) + A:(x,z). (2.33)

k

In terms of definition (2.29) and solution (2.33), a new

system of pure integral equations is obtained as

2
a/ +hi Zf+e

 

N
2 ES (x',z')G(x,z,x',z')dx'dz' = C cos k (x - a/Z)
iz k 0
i=1 a/Z-h. z.-e
1 1
J'C J'k Q E 3(2)
0 . _ _ o o 0 fix
- —E—-Ik31n ko\x a/2\ 2 sin(;—) (2-34)

V10
. for (x,z) E Ska and k = l,2,3,...,N

This is a system of Hallén-type integral equations for the
unknown slot field distributions. It must be remembered, however,
that in arriving at expressions for interior and exterior magnetic
fields, H:x(x,0+,z) and H:x(x,O-,z), respectively, use was made of
the assumption that the slots are thin and narrow having hk >> 3
and koe << 1; the longitudinal component of the slot field, E:(x,z)
could therefore be neglected. For wide slots in which E:(x,z) is
not negligible compared to E:(x,z), it is probably not possible to
reduce the resulting system of coupled integro-differential equations
to a Hallén-type system.

It is permissible, for the assumed narrow slots described
above, to use a quasi-static field approximation for the z-dependence
of E:(x,z). The electric field in a slot of width 26 cut in a
(8 )

thin, conducting screen of infinite extent has the approximate form

(obtained by a conformal mapping technique)

V1(X)

 

(2.35)

S

 

2 2
n e - (z-zi)

26

where Vi(x) is the voltage distribution along the ith slot. This
expression is consistent with the definition for the voltage dif-
ference between the edges of the ith slot. According to the usual

definition for voltage difference an identity is obtained as follows:

z - z +
i e s Vi(x) i 6 dz
V1(x) = -J Eiz(x,z)dz =

.. I
. - 2 2
z£+e 2.1 e \/[e _ (2-21)

Using the above approximation for E:z(x,z), equation (2.34)

 

 

 

= Vi(x).

becomes a system of integral equations for the voltage distributions

along elements of the slot array as follows:

 

 

 

N 8/2'1'111 Zi+€ V.(X')
E I I 1 G(x,z,x',z')dx'dz'
.= _ - 2 2
KO /2
= Ckcos ko(x - a/2) - —§— Iksin ko\x - a ‘
ngkO . TTX 2 3
- Y2 Eog(z)31n(;—) ... for (x,z) E Ska and k — 1, , ,...,N.
10 (2.36)
Let equation (2.36) be satisfied for k = l,2,...,N at
centers zk of the kth slot (this reduction to a one-dimensional

system of integral equations is possible since the z-dependence of
E:(x,z) has been specified approximately in terms of the square-

root edge singularity factor) and let the kernel for the system of

integral equations be defined as

G(x,zk,x',z')

= .( . '. '
2 ' 2 Kk1 x x z )
fl\/C - (z - zi)

K(x,x',z') E

 

 

then

27

 

Na/Z-l-hizifi;
u 1: It: _

2 i l Vi(x )Kki(x,x ,z )dx dz Ckcos ko(x a/2)

i-l IZ-h '3 .
ii

3Eillik /2 j°°s 111"- £ k=123 N

- 2 k8 n o‘x - a \ - Y2 og(zk)s n(a ) ... or , , ,..., .
10

(2.37)

Equations (2.37) are a system of N quasi-one-dimensional,
coupled, Hallén-type integral equations for the slot voltage dis-
tributions, Vi(x), i = 1,2,3,...,N. Once this system of equations is
solved, either by applying numerical methods or by extension of the
King-Sandler dipole array theory, the radiation fields maintained
by the slot-voltage distribution along the array aperture as well as
the input impedance to the antenna can be calculated. Solutions to
the system of integral equations will be obtained in Chapters 3 and
4 by numerical and approximate analytical methods, respectively. In
the next section, the E-plane and H-plane radiation fields are de-

termined in terms of the voltage distributions Vi(x).

2.4 Radiation Field Maintained by the Slot Array:
Let the origin of Spherical coordinates be located as indi-
cated in Figure(2.3). ; is the position vector from the origin to

any point in space, while ? is the position vector locating any

source point in the array aperture. The electric field at any point

in space is then determined from equation (2.18) as
E6!) = -i [a x E‘(?')] x vc°(?,?')dx' . (2.38)
a

It is noted that

28

.soHuwasuamu vHon :oHuwaku pom hhuoaomo

 

.m.~ ouswam

 

29

I

F5
é
A
a:
BL
t:

VGO (:3-13‘...) -'

 

. 1 + jkoR e JkoR
where K = (f - f'), R = \f - f", and E = %’- Subject to the

usual radiation zone (kOR >> 1) approximations

. for radiation fields,

such that for points in the radiation zone

 

 

_ _. -A."°' _. A _.

jkoR Jko(r r r ) Jkor jk (r-r')
" e A e A e O
R-—-—-- 3'r = r

2nR an an

then
-jk r a
0—9—9 jke 0 jk (’f'r')
VG (r,r')~ -'r—-9—-—-—e 0
an

For points in the half-space y < 0, then fi = -§, and at

source points in the array aperture
-O-O A S
E(r')w z Ez(x',z')

such that the radiation field becomes (for koR >> 1)

-jk r
e 0
2hr

8 jk (23')
i Ez(x',z')e o (a X %)dx'.

EH?) .. -jko

Since i = E sin 9 cos ¢ +’§ cos 6 cos ¢ - & sin ¢

30

while E = i sin 9 cos ¢ + y sin 9 sin ¢ + 2 cos 9, the radiation

field takes the form

-jk r N
flr—O . e O A . S ' '
E (r)¢¥ Jk. ----(a cose cos ¢ +’9 Sin ¢) 2 I E (x ,z )
o 2flr _ 12
i-1 S.
1a
jk (x' sin 9 cos ¢ + 2' cos 9)
e ° dx'dz'. (2.39)
Since koe << 1, then
jk (x' sin 9 cos ¢ + 2' cos 9) jk x' sin 9 cos ¢ jk 2 cos a
e o ‘w e o e o i

and using equation (2.35) to express E:z(x',z') in terms of the
voltage distribution Vi(x) in the ith slot, the radiation field

assumes the form

-jkor . N jk z,cos e
(& cos 9 cos ¢ + 9 sin ¢) 2 e O 1
i=1

 

—Or u...
E (r) ~ jko 2m.

Zifa a/2+h1 V (x') jk x' sin 9 cos ¢
I 1 e 0 dx'dz'. (2.40)

zi-e a/2-hi fi\/€2 _ (z'-zi)2

The integral over 2' evaluates to unity as

 

2 +6
1 dz!

I
21—3 nfz - (z'-zi)2

such that equation (2.40) takes the final form

= 1

 

at a e-jkor A N jk zicos e
E (r)iv jko an (¢ cos 9 cos ® + 9 sin ¢)iEIe

 

a/2+hi jk x'sin 9 cos ¢
I Vi(x')e 0 dx' . (2.41)
a/2-hi

31

It is convenient to specialize the above expression for the

radiation field to the principal planes, namely the E—plane

(¢ - -fl/2) and H-plane (¢ = 0), to obtain
'Jk 1' 2+h
..r .. - Jkoe o N jkoZ.Cos 9 a/ i
EE(r) N -9 T 2 e 1 I Vi(X')dX.
i=1 a/Z-hi
. E-plane radiation field (2.42)
and
-jk r
flr q jk e o N jkozicos e a/2+h1 jkox' sin e
E (r)~&————cos e 2 e V,(x')e dx'
H 2‘“ i=1 ilz-hi 1
(2.43)

. H-plane radiation field.

It is clear that once the slot voltages, V1(x'), have been
evaluated by solution of equations (2.37), the radiation fields are

readily calculated from eXpressions (2.42) and (2.43).

CHAPTER 3

SLOT ARRAY EXCITED BY DOMINANT BACKING-WAVEGUIDE MODE:
NUMERICAL SOLUTION

3.1 Introductory_Remarks:

In this chapter a numerical solution to the system of
integral equations (2.37) is discussed. Only one mode of array
excitation is considered, namely, that where the impressed field
in the array aperture is maintained by a TE10 mode incident wave
in the backing waveguide. The impressed current (coaxial current
generator) mode of array excitation is considered in Chapter 4.

The point matching method, a Special case of the method

(17) is applied to reduce the system of integral equa-

of moments
tions to an algebraic matrix equation. The voltage distribution
in the ith slot of the array, Vi(x), is first expanded in a series
of appropriate functions, after which the integral equations are sub-
sequently point matched to reduce them to a system of linear algebraic
equations for the coefficients in the expansion. Numerical processes
of integration and matrix inversion are applied to calculate the
expansion coefficients; the series for Vi(x) is then summed
numerically to reconstruct the voltage distribution. All of the
numerical operations are implemented on a high-speed digital computer
(CDC 6500 system).

Some simplifications to the system of integral equations

(2.37) are discussed in the next section, while sections 3.2 and 3.3

32

33

deal with the formulation of the basic matrix equations. Expressions
for radiation fields are derived in section 3.4, while section 3.5
deals with the input impedance to the backing waveguide. Numerical

results are discussed in section 3.6.

3.2 Simplification of the System of Hallén-type Integral Equations:

Since it is assumed that I 50, equations (2.37) can be re-

 

 

 

 

 

k
written as:
+
N a/Z-I-hi 21 3
'El I I Vi(x')Kki(x,x'z')dx'dz' - Ckcos ko(x - a/2)
i
jkOgo TTX
- 2 Eog(zk)sin(;-) ... for k — 1,2,3,...,N. (3.1)
V10
The field components of a TE10 mode incident wave, for
1
the given geometry (Fig. 1) of the backing waveguide are given as( 4)
h h F
-'E 22 -'
H = J O (:9) 10 cos(E§)e JBIZ
2 go Za ab 8
h
'18 2
H = E sin("—x)e 1 P
x 0 b2 a
a 10
h
2Z '13 z
_ _ 10 fl 1
By - EO b sin(a )e J (3.2)

 

all other field components being zero. An inSpection of equations
(3.2) reveals that Ey and Hx components of the TE10 wave are
symmetric (even functions) about the point x = a/2. Recall also
that the integral equations for the V1(x) are based upon the

boundary condition (at y = 0) for Hx; the incident field component

of H: can be regarded as the impressed field which excites the

34

slot fields. Thus the induced field (voltage) in the slots must
also be symmetric about x = a/2 since the slots are all

symmetrically placed about that point. That is
Vi(x = a/2 - u) = Vi(x = a/2 +'u) ... for i = 1,2,3,...,N. (3.3)

Utilizing property (3.3), equation (3.1) becomes:

N 8/2+hi Zi+e

Z " V.(x')L (x,x',z') +'K .(x,a-x',z')]dx'dz'
i=1 a/2 21-3 1 Kki k1

'jC k

= o o . E5
- Ckcos ko(x a/Z) -—--2-——-Eog(zk)s1n(a )
Yio
. for k=1,2,3,...,N, (3.4)
G(x,zk,x',z')

 

where Kki(x,x',z') =

 

n\//2-(Z'-zi)2

G(x,zk,a-x',z')

2
"ff-(2'41)

3.3 Numerical Solution Using Pulse Functions and Point-Matching:

 

 

K-ki(x ,a-x',z ') =

Let the upper-half of the ith slot be partitioned into Mi

rectangular subsections as shown in Figure(3.1) M, is any integer
1

and the partitioning of the ith aperture is described as:

hi
Ax -_ 3
i Mi
8 —
xim—Z -Axi(m- g)’m—1,2,3,000,Mi.,
Ax. Ax.
= ' o -_1- ——1
(Ax)im the 1nterval. xim 2 s x < xim + 2 . (3.5)

Define a set of pulse functions fim(x) as:

35

 

(AZ)11 (AZ) “12:1. (M) i,
2
—— 7 * * * x = 2 + h1
AxMi I I . I I xMi
“(M-m I I . I I I x(M-1)i
' "I 7| ‘I .
I _. _, I
I —_I TI —I I
I ...... I
I I I C I I ' C (Mm. xm)
I _, | 2
I I I I m = 1,2,...,M
I . . are Matching
I ...... I points for Pulse-
I I . I I I Function solution.
I _. l
'1??? .
__.'_ _. I
I
(“>31 I . I I X31
(”021 I I . I I X21
I I I a .
— —X = '2-
l--—l--fl
zi-e zil 21(E_;;_) iP Z +6
Figure 3.1. Integration Subdivision and Matching Points

 

 

 

 

 

 

 

 

 

for Numerical Solution of the Integral Equations.

36

1 ... for x E (Ax)im

fim(x) = (3 .6)
o ... for x é (Ax)im_-

Expression (3.6) defines a set of pulse functions to be associated
with the ith slot aperture.
Let the slot voltage distribution in the ith aperture be

approximated by a pulse function expression as

3
P“

Vi(X) “

m

"M

I aim fimm . (3.7)

where the aim are unknown expansion coefficients. Substituting
expansion (3.7) into the integral equations (3.4) and point-matching
the integral equations at the set of points (xkn,zk), k = 1,2,3,...,N
and n = 1’2’3’°"’Mk’ which locate the center of subsections de-
fined by (Ax)kn, the system of integral equations (3.4) are reduced

to a system of linear algebraic equations

‘M

 

N i -jI;k m:
k“ - -é ____0_0. kn
.2 2 aimIim Ck cos ko(xkn 2) — 2 E0g(zk)sin( a ),
i=1 m=l v
10
.. for k=l,2,3,...,N,and n=l,2,3,...,Mk, (3.8)
where
kn . . .
Iim = field contribution at the kth slot, nth subsection

due to sources in the ith slot, mth subsection,

or
Z +e
Ikn B i [K ( I I) + K ( _ I 2.)](1 Id l (3 9)
im I I ki xkn’x ’2 ki xkn’a x’ x 7" ’
zi-e (Ax)im

37

Equation (3.8) can be further simplified by enforcing the

boundary condition for Vi(x) at x = §'+ hi’ namely,

E- =
Vi(x - 2 + hi) 0

which implies that aim = 0 for m M . Equation (3.8) then

 

i
becomes:
M -l
2 g a Ikn - C cos k (x - g') = jgokb E g(z )sin(flxkn)
_ im im k 0 kn 2 2 o k a
i-l m=l Y
10
. for k = 1,2,3,...,N, and n = 1,2,3,...,Mk . (3.10)

Equation (3.10) is a matrix equation for the aimf The system must
still be point matched at subsection n = Mk. In the latter matrix
equation, the number of equations and unknowns are equal. If, for
example, each of the N slots is partitioned into M subsections
such that M1 = M for i = 1,2,3,...,N, then the total number of
equations is NM. The number of unknown coefficients aim is
N(M-l) while there exist N unknown constants Ck such that the

total number of unknowns is also equal to NM.

3.3.1 Numerical_Evaluation of the Various Integrals:
Various integrals involved in equation (3.10) are given by

equation (3.9) as

214's
kn I I _ I I I I
Iim i - IAX.) [Kki(xkn’x ,z ) + Kki(xkn’z x ,2 )]dx dz ,
i e im
where
Gi( z x z ) + Go( 2 x' z')
xkn’ k, 9 xk : k: 3

 

 

2 I
n e (2 Zi)

38

and

i . . o
G (xkn,zk,a x ,z ) +'G (x

"\jéZ - (z'-zi)2

In order to bring out the nature of singularities that will

kn.zk.a-X'.Z')

 

I I _
Kki(xkn.a x .z )

occur in the evaluation of 1:2, consider the following two cases:

(a) Case 1 # k: In this case source-point and field-point are

in different slots; as a result Go is only a slowly varying

function of 2' during the integration from 2' = z - e to

i

z' = 21 +‘g, and G can be regarded as approximately constant

with respect to z' and equal to its value at z' = z . That

i
is

kn ' I
+ - '
I. R‘I [C(x ,zk,x ,zi) C(xkn,zk,a x ,zi)1dx

Z.+e
dz'

1
X
inc 2 , 2
1 n s - (Z '21)

 

 

However, the integration over 2 evaluates to unity, while
for sufficiently small Axi (large Mi) the x' integral can

be approximated, and

k

n
Iim ~ [C(xkn’zk’xim’zi) + C(xkn,zk,a-Xim,zi) ]Axi. (3.11)

(b) Case i = k: This case corresponds to the situation where
source-point and field-point both are in the same aperture.
Here G0 can vary significantly during integration with reSpect
to z' and a more accurate approximation is needed. Let the

ith slot be partitioned along the z-direction as indicated in

39

Figure(3.1). Let the maximum number of subsections be P, then

 

 

26
AZ - P
= - +' - , = 1,2,3,...,P,
zip 31 e (P $>AZ P
= ' . _ E S A_Z_ 2
(Az)ip the interval. zip 2 z 5 zip + 2 . (3-1 )
With the above partitioning scheme, the expression for I1m
becomes:
I I _ I I I I
Ikfl R5 2 “I. [C(xkn’zk’x ,Z ) + C(xkn’zk’a x ,2 )]dx dz .
km = I ' ' 2 2
p 1 AX )kln (AZ )kp ”f _ (zl_zk)

(3.13)

Basically, two types of singularities can occur in evalua-

tion of expression (3.13); one when 2' = 2k i.€ which corresponds
P+l
to the cases of p = l or p = P, and a second when p = (‘5‘)
and m - n. In the first type (the square root edge singularities),
the denominator of (3.13) approaches zero while the second kind in-
volves a Green's function singularity where G0 becomes infinite.
These singularities are all integrable, however, and the improper
integral (3.13) is convergent. There are, thus, four Special cases
for the evaluation of (3.13):
. P+l

(1) Cases excluding__p = 1,,p = P. (p = _2- and m = n): In this
case there is no singularity in the integrand of (3.13) and Kki

varies slowly over the region of integration such that for sufficiently

small Axk and A2 (large ‘Mk and P)

40

[K (x ,x',z') +'K (x ,a-x',z')]dz'dx'
{Ax.)km IAZ.)kp kk kn kk kn

R5 [Kkk(xkn’ka’zkp) + 18(k(xknSanxklnl’zkp)]AzAXk ' (3'14)

(ii) Case4_p = l (edge singularity): In this case the variation
of G over (Az')kp can be neglected, but the edge singularity
must be integrated analytically:

I S (k’m’n) R, ‘I‘ [K (X :x'IZ') + (X ,8‘X',Z')]dZ'dX'
p {Mka (A2,)kp kk kn IS£k kn

R

I
’ , , +
LG("kn 2k x 21(1) C(x

.a-x'.zk1)]dz'dx'
(AX')km

kn’zk

z-dfiz
k dz!

XI
zk'e n\jé2 - (z'-zk)2

The 2' integration evaluates to

 

 

z -g+Az
Ik dz' =
zk"a “\fiz - (2'-zk)2

Therefore,

 

 

:IlI-I

II -é£_
[2 + Arc Sin(e 1)] .

1 l . Az
_ -
sp(k,m,n)Ie [2 + Arc Sin(e 1)][G(xkn,zk,x] ’zkl)

+ C(an’z ,a-ka,zk1)]Axk . (3.15)

k

(iii) Case 4p = P g(edge singularity): In this case the variation
of G can be ignored, but the second edge singularity must be

integrated:

41

I [K (x ,x',z') + Kk (x ,a-x',z')]dz'dx'
y I kk kn k 101
IAX )km (AZ )kp

I _ I I
Is I ' [C(xkn,zk,x ’zkp) + C(xkn,zk,a x ,zkp)]dx
AX )km

2+3
k dz'

2139'“ "f - (23-2192

As before, the z'-integration evaluates to:

 

2 +3
k dz!

I .. 2
Zk+6 AZ TI\/€2 _ (zl_zk)

ll
IH

 

n_ . _A?-.
[2 Arc Sin(1 e )]

l
dlI—I

[lg-i- Arc sin(A-z- - 1)],
e

and approximated integral becomes:

 

.1. l 45-
Sp(k,m,n) ms[2 +1T1 Arc sin(e 1)][G(xkn’zk’xkn’sz) +
C(xkn,zk,a-ka,zkp)]axk. (3.16)
1"”. I
(iv) Case p = -E- and m = n (Green 3 function singularity):
+
First, it is to be noted that when p = 251' and m f u,

there are no singularities in the integrand of expression (3.13),

1111.

and its evaluation is given by expression (3.14) with p = 2 .

In the case when p = Egl' and m = n, then there is
singularity in Go, but the square root edge singularity term re-

mains relatively constant during the integration over (Az')k and

approximately equal to its value at x' = ka, z' = zkp such that

42

Sp(k.m.m) R: LI [C(x .X',2') +

Z
we I I km’ k

G( ,a-x',z')]dz'dx'

ka,zk

or

Sp(k,m,m) R5 “:TI Go(x

e ,zk,x',z')dz'dx' +
(AX )km (AZ')kp

km

1 i
— + - .
e [G (x] ,zk,xl ,z ) C(x] ’zk’a x ,z] )1Az AX]

The improper integral (whose integrand is singular at

x - ka, z' = zk)

o
(ka

J2

{ G ,zk,x',z')dz'dx'
AX')km AZ')kp

 

ka-I-Axk/Z zk-l-Az/Z e-jngékm-x')2+(zk-zI)2
= j‘ I dZ'dX',
ka-Axk/2 zk-Az/2 {E/kam-x')z+(zk-z')2

 

can be evaluated, as shown in Appendix I, to

GO(X I I I 2 0 k 2

 

{Ax')km (Az')kp

Ax Ax Ax
z k 2 k z z 2
L2 1n(—AZ + /1 + (A—Z’S) ) + —2 1:1(“*—Mk + 1 + (LAxk) )1.

giving the final expression for Sp(k,m,u9 as:

 

Ax Az jk Ax Ax
SP(k’m’m) N we {11 E 2 Axk 1!"(AZ 1 + (A2 ) ) +
1. AL M. 2 i -
Az 1n(Axk + + (Axk) )3 + G (ka’zk’ka’zkp) + C(ka’zk’a "km’zkpn'

(3.17)

43

11:: as defined by expression (3.9) can now be numerically
computed using expressions (3.11) through (3.17). Matrix equation
(3.10) can subsequently be solved (by numerical matrix methods) for
unknown expansion coefficients aim' This consequently leads to

the numerical determination of slot voltage distributions Vi(x),

i = 1,2,3,...,N.

3.4 Radiation Field Maintained by the Slot Array:
It was shown in section 2.4 that the radiation field main-

tained by the slot array is given by

 

ar d e'Jkor . N jkozicose a/2+-hi
E (r):a jko an (@ cos 9 cos ¢ +’9 sin ¢) 2 e I Vi(x')
i=1 a/Z-hi
jk x'sine cos ¢
e ° dx'. (2.41)

Use of symmetry condition (3.3) leads to the following modification

of equation (2.41):

-jk r
-vr _, Jk e o A jk a/2 sinecoscb N jk 2 cos 9
E (r)vw ——%-;-- (@ cos 9 cos g +’e sin ¢)e 0 2 e o i
n i=1
a/2+hi
x Vi(x')cos[ko(a/2 - x')sin 9 cos ¢]dx'. (3.18)
a/2

Substituting the pulse function expansion (3.7) for Vi(x') in
terms of (now determined) coefficients aim yields for the x'-

integral the following eXpression

a/2+hi a/Z+hi Mi
I Vi(x')cos[ko(a/2-x')sin 9 cos ¢]dx' esj' Z aim im(x')
a/Z a/2 m=1
Mi
x cos[ko(a/2-x')sin 9 cos ¢]=~ E aimI cos[ko(a/2-x')sin 9 cos ¢]dx'

m=1 (AX')im

44

where for sufficiently small Axi (large Mi) the latter integral

is readily approximated such that

a/2+h1
I Vi(x')cos[k (a/2-x')sin 9 cos ®]dx'
o
a/2
(Mi
‘¥ mil aimCOS[kO(a/2-xim)sin 6 cos ¢]Axi.

Finally, the expression for the radiation field becomes

-jk r
_. 1k 0
f(r) R3 2110 e r

A jk a/2 sin ecos ¢
(9 sin ¢ + & cos 9 cos ¢)e

N jk zicos 9 Mi
xze° z

_ cos[ko(a/2 - xim)sin 9 cos ¢]Ax1. (3.19)
i=1 m-l

im

Expression for the radiation fields in the two principal
planes, namely the E-plane (¢ = -n/2) and H-plane (¢ = 0) are

obtained from equation (3.19) as

-jk r M
fir q . jkoe o N jkozicos e 1
EEO) N -9 T >Ee 2 crimAxi
i—l m=l
. for E-plane radiation field, (3.20)
and
-jk r
jk e o jk a/2 sin e'N jk 2 cos 9
“r” A 0 O Oi
E(r)~¢ cos 9e 2e
H 2nr _
i—l
Mi
2- ... - .
x mil aimcos[kb(a/ xim)cos e]Axi for H plane radiation field

(3.21)

3.5 lnput Mance to liacking Waveguide.
The circuit properties of the slot array, described by the

input impedance to backing waveguide; this impedance can be evaluated

45

at an arbitrarily located terminal plane. In this analysis, the
input impedance is defined at the z = O crossectional plane of
the backing waveguide (Figure 2.1)). The impedance of the antenna

is therefore defined as

Ei
zin =- :5ng , (3.22)
( x 10 2:0
where
(E3!)10 = the y-component of the total interior electric field
associated with TElO mode wave,
(11:)10 = the x-component of the total interior magnetic field
associated with TE10 mode wave,
and
zin = the input impedance of the slot array at z = O.

In section 2.2.1, the EM field in the interior of the back-
ing waveguide was calculated using the principle of linear super-

position. Employing the same principle (E1) and (Hi)10 can

y 10

be written as

i _ inc 8 W
(Ey)10 ' (By )10 + (Ey)10 A
i _ inc 3
(Hx)10 — (Hx )10 + (Hx)10 (3'23)
J

 

where superscripts "inc" and "s" indicate, respectively, the incident
T310 mode field in the absence of the slots and the TElO mode
component of the EM field scattered by the slot array (which was

expressed as a modal eXpansion over all TEq and TM.q modes).

1 i
With the above definitions of (By)10 and (Hx)10, expression

46

(3.22) becomes

 

inc 3
(E ) +‘(E )
z, = - aX 10 y 10 . (3.24)
in inc

3
(H ) +01)
x 10 x 10 2:0

A procedure similar to the one followed in determining the
interior magnetic field excited by the slot fields in the apertures
of the array elements can be used to evaluate (E:)q, where Bub-

script q includes all possible TE and TM scattered modes. It

can be shown that such a procedure will yield for E8'
Y

 

 

co no 6m
3
E (x,y,z) = ES 2'0! ,2) Z Z _
y a {..- _1 m_ _0 MI (rpmupml
nTTX “fix' mbl "111'. Y“ "1(2 z )
sin(-;—)sin( a cos( )cos( )[en + (1‘1)nm
-Y (2+z') 'Y (2+2') "Y (Z-Z')
e “m - (r2)nme “m - (r1)nm(r2)nme “m ]]dx'dz'
. for z s 21. (3.25)

(E3)10 is obtained from equation (3.25) by retaining only the

n = l, m = 0 component term in the double summation. Recalling

that for m = 0, em = 1, (E3)10 is given as

9 _ ___ l
(E y)10 - sin(a w); E8 (x' ,z '){ ab[1 _ (F H)10(r >10] sin(fi —)

13h (Z'Z) “'th (2+2) 16h (2+2)
° [e 10 + (r1)loe 10 ' (P2)1oe 10

-j620(z-z') '
- (F1)10(F2)loe ]]dx dz' ... for z s 21. (3.26)

47

With the above result for (E:)10 and equation (2.6) for (E;nc)10,
the expression for (E;)10 can be written as
h h h
, 22 ZjB z -j8
1 10 10 10 fix
=- ———-— + _.
(Ey)10 Eo ab [1 (T2)10e ]e sin(a ) +
expression (3.26) ... for 2 S 2 . (3.27)

1

The corresponding expression for is determined

i

by using Maxwell's equation for curl ‘E, namely

which in component form yields

 

 

i_ ___Ea_iw
H
x =ijo 52 EV 5
Hi - T1 a--Ei . (3.28)
quo ax y ‘J

The first of expressions (3.28) along with equation (3.27) leads

to the final expression for (11;)10 as

 

h h h
22 ZjB z -j5 z
i _ 1. __I9 _ 10 10
(Hx)10 ' Zh Eo ab [1 (r2)10e 19
1
"1+1 sin(—)
108
h h
. jB (z-Z') -JB (2+2')
X sin(ߤ-){e 10 - (F1)10e 10
161;,sz -je*1‘0(z-z') ' '

.. for z s z . (3.28)

48

Substituting expressions (3.27) and (3.28) into equation

(3.22) and simplifying, the following expression for zin results

h 1 + (r2)10 - A IA

 

 

 

Zin = 210 1 _ (F2)10 + B IA 3 = antenna impedance at z = 0
input plane, (3.29)
where
1+(T)
A = 1 10 ,
E 2abZhO[1 - 10(1‘2)10]
1-(T)
B= 110 ,and
h
E 2abZ10 [1 - (F1)10(P2)10]
h . h.
"JBZ 332 I
= if if z(x' ,z')[e 10 - (F2)10e 10 ]sin(ߤ-)dx'dz'.

A special case of interest is that where there are no slots
. . . S . .
in the structure. In this Situation Eiz will be identically zero

implying IA will be zero. Thus it is found that

zh L1 + (P2)10

Zin=10 1 " (r2)10 ’

 

which is the well known result for an arbitrarily terminated wave-
guide. This provides a check on the derivation of expression (3.29).
If the quasi-static field approximation, as given by equation

(2.35), to the slot field E:z(x',z') is used, then

 

 

h
V .(x) I 'jB z.
I = sf sinch-ne 1° -
1= 1 2 a
is 11:/32 - (z'-zi) h
jeloz'

(r2)10e jdx'dz'. (3.30)

49

3.5.1 Numerical Evaluation of IA:

The z'-integrals in expression (3.30) can be evaluated
analytically while x'-integrals must be computed numerically in
terms of the pulse function expansion for Vi(x) using the same

technique described in section 3.3. With the x-partitioning scheme

described by expression (3.5) and expansion (3.7) for Vi(x),

 

 

 

 

 

 

 

. h I h .
a/2+h. z+e 'Jeioz jB10z
N 1 1,er 1- [e - (r2)loe ]
I = z j v.(x')sin(—)dx'j‘ dz.
A i=1 a/Z-h 1 a z -e 2 , 2
i i 11‘ e - (z -zi)
or
. h I h I
-JB 7- JB 2
M +
N 31 "XI zi 6 [e 10 '(rz) 108 10 1
1 = 2 z z a, sin(—-—)dx'j' dz'.
A i=1 m=l lm(Ax') a z - 2 2
im 1 e ngjé - (z'-zi)
However
2 + +' h z' h
i 6 8.1610 i43.51021 h
I dz' = e J (a e),
Z - 2 2 O 10
i e n\Jé - (Z'-zi)
where JO(B:06) is the Bessel function of first kind and order
zero with real argument 8203, therefore
h h
N -j5 z je z
___ 10 i _ 10 1 h
IA 21:1[(e (r2)10e >Jo(eloe)]
M.
1 fix]:
mil aim Sin( a m)Axi . (3.31)

Equation (3.31), along with expression (3.29), completely

determine input impedance Zin

50

3.6 Numerical Results:

In order to solve for the slot voltage distributions, the
input impedance of the array and its radiation field, a computer
program was developed and implemented on a CDC 6500 system.

As indicated by matrix equation (3.10), the order of the
matrix involved is (NMi) X (NMi) if all the Mi are equal. The
unknowns involved are the N(Mi - 1) expansion coefficients aim
and the N constants Ckk' For a lO-element array, for example,
and with Mi = 10, a matrix of order 100 X 100 results. It,
therefore, becomes clear that initial runs must be made for say,
N = 5, thus reducing the matrix size and resulting in a consider-
able saving in computation time. It was found that for Mi < 10

the evaluation of various integrals involved in the calculation
kn

of Iim as well as the convergence of the numerical solution
might not be satisfactory. Results were thus obtained first for
a S-element slot array with M1 = 10 to determine the optimum
array dimensions, i.e., the element lengths and element spacings,
that give rise to optimal end-fire radiation from the array.
Figure (3.2) shows the relative amplitude and phase dis-
tributions of voltage in the element of a 5-element waveguide
backed slot array. Various dimensions are identified on the top
left hand corner of the figure. In all cases, the double Fourier

1
series, used in the evaluation of G was truncated after “max

and mmax terms (the total number of terms summed is, therefore,

equal to n X m ). The element lengths are fixed at
max max
$7.. 0.22 and element spacings at ‘%5 = 0.1. The backing wave-
0 0

guide is matched at both the input and output terminals

51

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52

(F1 = P2 = 0), and the width of the slots is Specified by the para-
meter 0 = 24n(4h/e) = 10.6. It is interesting to note that the
numerically computed voltage amplitude distributions in the slot
elements closely approximate sinusoidal variations along the slot
axes; this suggests that an extension of the King-Sandler dipole

array theory(11)

might be applicable to this problem. Maxims of

the amplitude distributions in the slot elements also display a
standing wave character. Slot number 1 has maximum induced voltage,
slot 3 has minimum voltage, slots 2 and 4 have identical distributions
of intermediate amplitude, and slot 5 has an induced voltage of
amplitude nearly equal to that in slot 1.

Figure (3.2b) indicates an essentially uniform progressive,
delay or lag in the phases of (successive) slot voltages along the
array aperture. The light line on the figure indicates the shift
(lag) which would occur along the array if the wave in its aperture
were a traveling wave propagating at the Speed of light. This
light line gives an indication whether the actual wave excited along
the array aperture is a fast or Slow wave; a slope greater than
that of light line indicates a larger phase constant, it corresponds
to a traveling-wave aperture field with a phase velocity slower
than the speed of light while a lesser slope indicates the existence
of a fast-wave field. It can be identified from Figure (3.2b),
therefore, that a wave traveling very nearly at (or, on the average,
slightly slower than) the Speed of light is excited. Since endfire
radiation is eXpected when the phase velocity of the aperture field
is less than or equal to the speed of light, this array is capable

of maintaining such a radiation field. This is confirmed by the

radiation pattern presented later in Figure (3.6).

53

Figure (3.3) shows relative amplitude and phase distributions
of slot voltages in the aperture of the same 5-element slot array
considered in Figure (3.2), except that now the effects of various
element Spacings, A2, are considered. From Figure (3.3a), the
standing wave nature of the slot voltages (aperture field distribu-
tion) is apparent. For example, when A£'= 0.05 the array aperture

0

length is 0.20 lo and the distribution indicates approximately a

quarter-wavelength standing wave pattern. Similar conclusions are

A2 A2

reached for the array with - = 0.1 and -— = 0.2.
x0 x0
From Figure (3.3b), it is apparent that spacings of

A5 = 0.05 and %5 = 0.1 yield slow wave aperture fields along
0 o
the array, and thus an end fire radiation, while 95 = 0.2 gives

0
rise to a fast wave and consequent off-endfire radiation. Thus a
spacing of '%E = 0.1 is identified as an Optimum element separation;

0

smaller spacings will create difficulties in the construction of
such an array.

Figure (3.4) again indicates amplitude and phase distribu-
tions of slot voltages against axial location of the slots with
element half-length, h, as a parameter. Again, a standing-wave
aperture field pattern, which is indicative of large reflection
near the terminal end of the array is demonstrated by Figure (3.4a).
Figure (3.4b) indicates a slow wave aperture field along the array

for %-’= 0.24, 0.22 and a wave at nearly the Speed of light for

o

%- = 0.20. For an element half-length of %—'= 0.26, a fast wave
0 o

is indicated. The optimum value for half-length %-* appears to

o
be roughly 0.22. Shorter slot lengths or lengths greater than

resonance lead to fast wave fields, while lengths close to resonance

result in a strong standing wave pattern along the array aperture.

54

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56

It can, therefore, be concluded that for array parameters

of %£'= 0.1 and h—-= 0.20 a slow wave aperture field will be

0
excited along the array.

Figure (3.5) shows the relative slot voltage distribution

(in amplitude and phase) vs axial slot location,'&§, along the
o
array aperture with the backing waveguide dimension "a" (its

width) as a parameter. Figure (3.5a) indicates the amplitude dis-

tribution of slot voltages against fig , guide widths of i— = 0.505,
0 o
0.55, 0.6 and 0.7. The value %-'= 0.50 is the cut-off dimension
0

for the incident TE10 mode wave, at which point its phase velocity

becomes infinite, so a fast-wave aperture field is expected for

§—'= 0.505 (cut-off was avoided to prevent numerical difficulties

0
from occurring in the computer program). As the phase plots of

Figure (3.5b) indicate, a slow wave field can be excited along the

array aperture with i‘ = 0.70 and 0.60 while a fast wave exists
0
with §—'= 0.505. §-’= 0.55 results in excitation of a wave along
0 o

the array that travels perhaps slightly faster than the speed of
light. End fire radiation can thus be expected with the backing

waveguide dimensions set at ‘%- = 0.70 or 0.60 with %T-= 0.30.
o o
Off-endfire radiation is expected for %-'= 0.505 or 0.55 and
o

b
'- - 0.30.
A

o
The above conclusions are confirmed by Figure (3.6) which

indicates the radiation field in the E-plane (¢ =-§) of a 5-

element Slot array with a—' as a parameter. The upper half of

X
o
the figure concerns an array having h—-= 0.22 backed by wave-
0
guides width §T-= 0.505 and 0.60. It is observed that even though

0
the radiation patterns are rather broad (due to small number of

57

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58

b = O 3 X $2 = Zln(4h/€) n = m = 5
O “lax max
r1 - r2 = 0 0 = 10, 6 . PM =
AZ = 0. 1 X0 h/6 = 50.0 MM = 10
90°

 

 

h = 0. 22 A
relative OT .
of radia ion . v ' ‘
field
a = 0. 6 A
o. 26 x0 /
30°
h = 0. 24 )‘o
/
0. 20 A /
° /
60°

 

 

Figure 3.6. Dependence of E-plane (¢ = - Ir/Z) radiation field

patterns of a 5- element slot array upon width of
its backing waveguide and slot length.

59

array elements and consequent short aperture electrical length) a

beam-scanning capability is achieved by changing if' from 0.6 to

o
0.505. The lower half of Figure (3.6) shows the dependence of

E-plane radiation field patterns upon parameter h—- with §-’= 0.60.
Endfire radiation is achieved for 2r-= 0.20 and 0.24, whil: for
h-'= 0.26 the main beam is in the off-endfire direction of

efi~ 65°. This is to be expected since Figure (3.4b) predicted

a slow wave along the array for %-= 0.20, 0.22, and 0.24 but a
0
fast wave for h—'= 0.26.

With optimum array dimensions %-’ and fig- now determined

0 o
as 0.22 and 0.1, respectively, by studies on the S-element array,

an investigation of a lO-element waveguide backed slot array was
initiated. Again, the possibility of achieving a beam scanning

capability by controlling the backing waveguide dimension %-

0
was considered. Figures (3.7a) and (3.7b) indicate the amplitude

and phase distributions of slot voltages (field maintained in array

aperture) against the axial slot location for backing waveguide
widths of i— = 0.505, 0.55, 0.60, and 0.70. The large reflection

0
at the terminal end ofthe array persists as expected for an array of

finite length and the aperture field distribution shows a significant
standing wave component. Figure (3.7b) indicates the excitation

of a slow wave when ir'= 0.70 and possibly for §-'= 0.60. A
o 0
fast wave is excited along the array with §-'= 0.505, and 0.55.

Ac

Figure (3.8) confirms these predictions by indicating an

end fire radiation pattern for ir'= 0.70 and near end fire
0
radiation for a:- = 0.60. However, for L = 0.55, the maximum
0 o
of the radiation field shifts to approximately 45 degrees off-

O

60

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61

b = 0.3K h = 0.22X n =m =5
0 0 max max

T1 = I‘z = 0.0 $2 = Ztn(4h/€) PM = 5

AZ = 0.1710 = 10.6 MM = 8

 

)0

1800 I _:
relative

amplitude
of radiat ion
field

 

Figure 3.8. Dependence of E-plane (4) = - Ir/Z) radiation field
patterns of a 10- element slot array upon width of
its backing waveguide.

62

end fire while for i— = 0.505, an almost broadside radiation

0
pattern is achieved. The beam scanning capability with respect

to h-' variation was not investigated for the lO-element array
0
since the half-length h can only be changed electrically (by

operating at different frequencies) and results in a simultaneous

change in the element Spacing fig.
0

The input impedance Zin to the backing waveguide is com-

puted at the arbitrarily chosen input terminal plane 2 = 0; the
standing wave ratio (SWR) in the driving waveguide is subsequently

calculated in terms of the normalized input impedance zin = Zia/2:0.

Table (3.1) indicates backing waveguide SWR for a 5-element array with

various slot lengths and spacings. It is observed that to minimize
SWR optimum slot spacing is %E'= 0.1 while the optimum slot half-
0

length is %—'= 0.22. These dimensions are identical to those

0
determined earlier for optimum excitation of a slow-wave in the

aperture.
It is interesting to note that even for very closely spaced

slots ($5 = 0.05) the SWR rises only to 6.90, which indicates a
o

reasonably good match is maintained between the backing waveguide
and other waveguide circuitory that is employed to excite the array.

The SWR increases rapidly to a value of 17.0 for %-= 0.26, a

0
slot half-length much in excess of resonant length; earlier results

also indicated a rapid decay in the aperture field distribution
for this case. It appears that the wave in the array aperture has
become evanescent and the slots simply constitute a strong reactive

discontinuity which produces a TE reflected wave of high

10
amplitude. Such slot lengths are, therefore, to be avoided in the

63

 

h/I. 0.2 0.22 0.24 0.26

 

SWR 1.79 1.20 3.70 17.0

 

 

 

 

 

 

 

a. Variable h/koforAz : 0.1Ko

 

Az/xo 0.05 0.10 0.20

 

SWR 6.90 1.20 2.02

 

 

 

 

 

 

b. Variable Az/ko for h = 0. 2210

Table 3.1. Backing waveguide SWR (5-slot array) for various slot lengths
and spacings (a=0.6)\o, b= 0.3161“ =I‘2=0,SZ=21n '

l
(4h/e)=10. 6).

64

design of waveguide-backed slot arrays. Table (3.2) indicates driving
waveguide SWR for a lO-element slot array with various backing
waveguide widths, a . This lO-element array utilizes the optimum

o
array dimensions, a5 = 0.1 and %—'= 0.22, obtained by studies

0 o
on the 5-element array. It is observed that to minimize SWR the

optimum backing waveguide width is i— = 0.70. This agrees with
o

the results indicated in Figure (3.8) for E-plane radiation field

pattern for this lO-element array.

A number of additional numerical results are compared

with similar experimental results in Chapter 5.

65

 

a/x o. 505 0. 55 0. 6 0.7

 

SWR 7.7 1.85 1.77 1.71

 

 

 

 

 

 

 

Table 3.2. Driving waveguide SWR (IO-slot array) for various backing
waveguide widths (b = 0.3). , h = 0. 22ko, P1 = F2 = 0,
Q=2£n(4h/e)=10.6). °

CHAPTER 4

SLOT ARRAY EXCITED BY IMPRESSED CURRENT:
APPROXIMATE ANALYTICAL SOLUTION

4.1 Introductory Remarks:

 

An approximate analytical solution to the system of integral
equations (2.37) is presented in this chapter. Only the impressed
current (coaxial current generator) mode of array excitation is
considered here; the other mode of array excitation (by an incident
TE10 mode wave) was considered in Chapter 3.

It should be noted here that a slot array, cut in an in-
finite ground screen situated in unbounded Space and excited by
current generators located at centers of the slots, constitutes a
complementary problem (byBooker's extension of Babinets' principle

18
( )) to that of a dipole array excited by voltage generators at

centers of the dipoles. This is the Special case of the present
problem when the waveguide backing is removed (mathematically G1

is replaced by GO). Since the dipole array problem has been
successfully treated using the King-Sandler(12) diPO1e array

theory; an extension of this theory is applied to obtain an approximate
analytical solution for the waveguide backed slot array. Section

4.2 outlines the application of the King-Sandler (two term) dipole
array theory to the present problem, resulting in an algebraic

matrix equation for the coefficients of distribution functions which

describe the slot voltages in the array aperture.

66

67

Section 4.3 deals with the evaluation of various functions
encountered in the matrix equation. All the integrals encountered
in evaluation of these functions are evaluated either in closed form
or by approximate numerical methods. Section 4.4 develops expressions
for the radiation fields in terms of the approximate aperture field
distributions. A computer program was developed to implement all
of the numerical operations on a high Speed digital computer (CDC

6500 system). Numerical results are presented and discussed in

Section 4.5.

4.2 Extension of King:Sandler Dipole Array Theory to the Waveguide-
Backed Slot Array:
Since it is assumed that no dominant-mode incident wave
exists while the impressed currents are non-zero, then E0 = 0
and Ik # 0 such that equations (2.37) can be rewritten as:

2+ +
8/ hi. 2i 3

 

 

N
Z V.(x') ,(x,x',z')dx'dz' = C cos k (x - a/Z)
i=1 a/2-h A. 1 K'” k °
i i
o
_ _ - ... = 2 00-, 9 0
j 2 Ik sin ko‘x a/Z‘ for k 1, ’3’ N (4 1)
where
G(x,zk,x',z')
Kki(x ,X',Z') = (4.28)
2 , 2
n e ‘ (z '21)
and
G(x,zk,x',z') = G1(x,zk,x',z') + Go(x,zk,x',z'). (4.2b)

In view of its definition in expressions (2.29) and (2.35), the

function Ak(x) (the left-hand-side of integral equations (4.1))

68

can be written as

N 8/2'1'1'11 zi+g
A (x) = 2 v (x') " .(x,x',z')dx'dz' (4.3a)
k i=1 £l2-h. 1 %,-e Kkl
i i
or
a/Z-I-h,
N 1 ._
A (x) = 2 v (x') .(x,x')dx' (4.3b)
k i=1 i/Z-hi i Kk‘
where
zi+e
R#i(x,x') = I Kki(x,x',z')dz' . (4.3c)
2,-3

4.2.1 Properties of the Kernel and Resulting,Approximations:

The King-Sandler dipole array theory is based on the particular
peaking and non-peaking properties of the kernel which appears in the
system of integral equations for the dipole array. It is imperative,
therefore, that an investigation of the real and imaginary components
of kernel Eki’ Re(R#i) and Im(Rki), be carried out before attempting
an extension of the King-Sandler array theory to this problem. The
case of primary interest is that where i = k, that is when the
source point and field point are located in the same slot. Equations
(4.2) and (4.3) suggest that the variation of R#k(x,x') with x'

be studied with x fixed. Rewriting the expressions for Kkk(x,x',z'):

i ' | o g I
G (x,x ,zk,z ) +'G (x,x ,zk,z )

 

 

I I =
Kkk(xIx .2 )
‘llz ( _ z,)2
n a 2k
0 ' . e'jkoRkk
G (x,zk,x ,z ) = W-

kk

 

2
Rkk -J(x - x')2 + (2k - z')

69

 

          

and
i 1 °° °° '3
G (x,z ,x',z') = -' Z _ Sin(fln
k ab n=1 m=0 Ynm[1 (r1)nm(r2)nm]
fnm(zk,z')
I -Y m‘zk-z'1 -ynm(zk,+ z')
fnm(ZkIZ ) = - (T1)nme
Ynm(zk + z') Ynm1zk - 2'1
- 0‘2)an - (P1)nm(r2)nme . (4.4)

Since an array of thin Slots cut in an infinite ground

plane is complementary to an array of thin strip dipoles (and

)(1),

ultimately to an array of equivalent cylindrical dipoles

(19) it is sufficient to investigate the behavior of

Gi(x,z ',z') with respect to x' and z . The Go(x,zk,x',z')

k,x

Greens' function is complementary to that encountered for the strip

dipole array, for which an equivalent array of cylindrical dipoles

with appropriate equivalent radii can be found; the kernel for the

latter system has been investigated in detail by King and Wu(zl)-

Gi(x,zk,x',z') as given by equation (4.4) is a function of
I I

variables x and z , The z'-dependence can, however, be analytically

integrated out since

....
2kg

ca 6

1 a m nfix
= — 8 2 (—)S in(—)
k's ‘W/Z _zl)2 ab n=l m=0 Ynm a

f+ez e-Ynm12 k - z"

k-g :J2_ _z,)2

for (Fl)nm( = (F = 0. (4.5)

G1(x,,zk,x' ,2 ')dz'

 

 

 

 

nm
. g-
in( a

 

k2)nm

70

For the purpose of numerically investigating the properties

of K the backing waveguide is assumed to be matched at both ends

kk’
such that (F ) = (T ) = 0. The behavior for non-zero reflection
1 nm 2 nm

coefficients is qualitatively similar. As shown in Appendix II

 

 

Zk+e e-Ynm12k - z '1
I dz'
Zk-e ‘n\/Lz - (zk - z')2
[10(anme) - Lo(anme)] ... for Ynm a anm.+ jO
[Jo(6nm€) ‘ jHo(8nme)] ... for Yum = 0 + jBnm (4.6)

where

I (a s) = Modified Bessel function of zeroth order and real
argument ahme,
L (a e) = Modified Struve function of zeroth order and real

ar ument
8 C1an I

Jo(Bnme) = Bessel function of zeroth order and real argument
Bums, and
' Ho(Bnme) = Struve function of zeroth order and real argument
Bnme'

With expression (4.5), the kernel Rkk(x,x') can be written

88
+ +
_- Zk e _ ‘-1 2k s o
I = K I I I = I I I I
lfi‘kbnx ) j‘ kk(x .x ,2 )dx Kkk(xIx ) +j' lfidgxm .2 )dz
zk-e Zk‘e

where the superscripts "i" and "0" refer to the inside or the out-

side components of the kernel, and

71

 

-—1 1 °°
(XIX' = - 2‘. 2 (—)sin(—)si Mn? )
Kkk ab n= —1 m=0
[10(anme) - L 0(anme 3)] ... for Ynm = anm + j0
[JO(Bnme) - jHo(Bnme)] for Ynm = 0 + jenm (4.7a)

and
Go(x,zk,x'z')

o
( 3 '3 ' = '
Kkk x x z ) 2 ' 2
fl 3 - (zk - Z )

Using the complementarity principle, the radius of an equivalent

 

 

(4.7b)

dipole is found to be g, and, therefore, R:k(x,x') can be approximated

 

 

 

 

as
zk+e -jko \/(X'X')2 +(3/2)2
I K:k(x,x',z')dz' —Kkk(x, x ')N /32 fie e .(4.7c)
zk'e 2nx/(x-x' ') 22+(I.;/2)

Equations (4.7) lead to the following approximate expression for
R#k(x,x') for the purpose of studying its behavior with respect to

x':

Zkk<x.x'> ... E§k(x.x'> + fi’dgxnc')

. 'fio ‘-i
Figure (4.1) shows the behaVior of \Kkk(x,x')\ and \Iq‘k(x,x')\
against ko‘x - x'). It can be seen that as x' ~»x, both
1EER1 and [ikk1 have significant peaks. It is also noted that
—1
1Kkk‘ follows 1Kkk1 closely if e - 10/200 with nmax mmax 100.
Other points on the figure Show effects of variation of e from
xo/400 to 10/60. There is no significant change in the shape of

the curve. Figure (4.2) shows Re(R:k(x,x')), IuK§:k(XIX'))I

72

 

 

 

 

kk

1000 -
| I I I I I I I I :1
I ._
‘ -_
A IEik(x,x')I, e-‘AO/ZOO, “max = __
"i mmax = 100
A \Kkk(x,x')\, e=xO/200, nmax=mmax=_13
o Iiik(x,x')I, sac/400, nmax=mmax=lb
100% D IEIER(X’X')I’ €=)\O/6O, nmax=mmax=mr
E -. _ I d
—_ _- 8 IK:k(x,x )I, e = xO/zoo :-
x I
as —- , 4
"' ._ A ..
u A
O
13 a '-
10.0 "- A ‘ 2 g -
D
A D g 8 D g
D 8 ~ D I:
an o z A
a - 8
0
A8 I: ‘
D [3
1,0 I I I I I I I I I I
0 0.3” 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0
ko\(x - x )I
Figure 4.1. Magnitude of E vs k0|(x-x')I.

73

o
...xuix m> FxJCOMHM .Né 0.5m?”

TXIuLOx

H.N w... m4 NJ . ad o.o m.o 0.0

0.7

 

 

I/ .ixaleH._ llIuI.
\ IIII\\|I.(IIMIII
use:

 

 

o‘ I!

 

 

 

74

“-i ‘i
Re(Kkk(x,x')) and Im(Kkk(x,x')) vs kon - x'I. Again, as
x a x', the real parts of both Rik and Rik have significant
peaks while the imaginary parts of both E: and IE1 remain
k kk
relatively constant.
This behavior of Ekk(x,x') leads to the following
expressions for Re[Ak(x)] and Im[Ak(x)]. According to the de-

finition for Ak(x), its real and imaginary parts are

N a/2+hi
Re[A (x) = A (x) = z V.(x')—I (x,x')dx'
k 1 RR i=1 £l2-h 1 KkiR
and
E N a/2+h1 _
Im A (X) = A (X) = 2 V (X') (x,X')dX'
where

Kki(x,X') = KkiR(X.X') + JKkiI(X.X')-

Since the kernel EkkR(x’x') is quite small except at or near

x' = x, where it rises to a large value, while E#1R(x,x') is

always relatively small for i i k, then it is clear that slot
voltage Vk(x') near x' - x is primarily significant in determin-
ing the value of AkR(x) at x. In other words

N a/2+hi

A (x) = Z V.(x')_ . (x,x')dx'
kR i=1 a/2-hi 1 KklR

a/2+hi
[(1-5. )
1k £l2-

N

is i: Vi(x')E#iR(x,x')dx' + sikYkkak(x)] (4-88)

1 h.
1
where

0 ... if 1 + k

ik

75

and YkkR is a proportionality constant.

Due to the non-peaking nature of EkiR for i i k, the
remaining integrals in equation (4.8a) are relatively small in
amplitude with functional forms which can be approximated analytically.
Through extensive numerical computations, King and his co-workers
(12) have demonstrated that the functional forms of each of these

integrals is approximately proportional to cos ko(x - a/Z). AkR(x)

can, consequently, be approximated as

N
R’ - - 2 +
AkR(x) 1:1(1 aik)vkiRcos ko(x a/ ) skiwkkkvk(x) (4.8b)
where the Yki are constants.

Integrals involved in AkI(x) can be approximated due to the

non-peaking property of kernel Ekil for all values of i and k.
(12)

Again, King and his co-workers have demonstrated that a simple

analytical approximation for Ekil is valid for antennas with half

lengths satisfying koh1 s gfl. Subject to this approximation for

Ekil’ it follows that each of the integrals in the expression for
AkI(x) is approximately proportional to cos ko(x - a/2). Con-
sequently, an approximation for Ak1(x) which is valid for

511

kohk S Z-’ is given by

AkI(x)%¢ AkI(a/2)cos kko(x - a/2) (4.8c)

where AkI(a/2) is, of course, a constant.

Expression (4.8b) is essentially the proportionality exploited
by King-Sandler dipole array theory. As indicated earlier, the con-
tribution to AkR(x) in equation (4.8b) from terms with i i k is

small compared to the term for i = k. Thus to a good approximation

76

However, Vk(x = a/2 + hk) = 0 while A = a/2 + hk) is small

kR<x
but non-zero. It is therefore apparent that a more accurate pro-

portionality is
Vk(x) a [Amos - AkR(a/2 + hk)],

such that Vk(a/2 + hk) = O. Evidently, a better approximation

than equation (4.8b) is

N
AkR(x) - AkR(a/2 + hk) ... 1:10 - 5

k
ik)deiRFox(x) + bidekkRvk(x) (4.9a)

k 0
where Fox(x) — cos ko(x - a/2) - cos ko(hk)' Similarly, a good

approximation for the difference form of AkI(x) is

AkI(x) - AkI(a/2 + hk)4~ AkI(a/2)[cos kko(x - a/Z) - cos ko(hk)].(4.9b)

4.2.2 Alternative System of Difference Integral Equations for the

Vi(x):

 

It is now necessary to formulate an alternative system of
integral equations for the Vi(x) in terms of the difference
ARI!) - Ak(a/2 + hk)'

From equation (2.33) it is found that

= _ .2
Ak(a/2 + hk) Ck cos kohk j 2 IR sin kohk . (4.10)

Subtracting equation (4.10) from equation (2.33) gives

Ak(x) - Ak(a/2 + h = Ck[cos ko(x - a/2) - cos(kohk)]

k)

C
- j 52'1k[sin kOIX - a/ZI - sin kohk]° (4°11)

77

Solving equation (4.10) for C leads to

k

l

Ck -.— ———_cos(kohk) [Ak(a/2 + hk) + j --2—— Ik sin(kohk)],

or

jéo

c=——————
k cos(k ohk)

1
[Uk + 5E sin(kohk)] (4,12)

where Uk has been defined as
u =-—1—A(a/2+h)
k jg k k
0
2+
1 N a/ hi _
-'_“ Z V.(x') (a/Z +'h ,x')dx'
jgo H= i/Z-h, 1 Kki k
Substituting expression (4.12) for Ck in equation (4.11):

Jgo I

_ _________ .;§ .
Ak(x) - Ak(a/2 + hk) — cos(kohk) {2 Sln k0(hk - Ix - a/Z‘) +

Uk[cos ko(x - a/2) - cos kohkj}

or
jg Ik Mk
Ak(x) - Ak(a/2 + h k)= ——;(E:fi—;2 [— 0x(x) + U kF: x(x)] (4.133)
where by definition
k _ .
MOx(x) — Sln ko(hk - Ix - a/ZI)
F:x(x) = cos ko(x - a/2) - cos kohk . (4.13b)

Definition (4.3) for Ak(x) yields the integral repre-

sentation

a/2+-hi

IIMZ

1/2‘r v i(x )dei(x,x')dx', (4.14s)
a/
hi

where Ehk1(x,x') = E’ki(x,x')i - Kki(a/2 + hk,x').

78

Equations (4.13) and (4.14) lead to an alternative system
of integral equations for the Vi(x) in terms of the difference

Ak(x) - Ak(a/2 + hk)’ as:

N a/2+hij§01k
151 L“: v (x' )de i(x,x')dx' = W [—M: xx()+ ”1.1ng(")]° (4.15)

4.2.3 Analytical Approximations for Slot Voltage Distributions:

The next step is to make use of the approximations contained
in equations (4.9) in order to deduce a simple analytical approxima-
tion for the slot voltages Vi(x). With equations (4.9), it is
clear that the integral in equation (4.15) can be approximated as
follows:

2
a/ +h1

1/2‘Ihvio:)de1(x’x')dx'w[dekRvk(X) + Q F: x(x) + w a: x(x)] (4.16)
a/

"I452

where Q and W are constant coefficients and H:x(x) is defined as

H:x(x) = cos $51.00: - a/Z) - cos koh (4.13c)

k'
Substituting approximation (4.16) into integral equations

(4.15) and solving the resulting equation for Vk(x) it is found that

 

 

 

JQO Ik k jQ cos(k0 h k) k
V15") " ‘i’dkchos(kohk) [i— Mox(x) + (Uk + go k)Fo x(x)
jW cos(kohk) k
+ Co Hox(x)].

The above expression for Vk(x') suggests the following zeroth-order

approximation

Vk(x)Iw‘VkV(x) +-VkU(x) +'V (x) (4.17)

kD

79

 

with
k W
ka(X) - VkVMox(X)
__ k g
ka(x) - VkUFox(x) , (4.18)
V (x) = V Hk (x)
RD kD ox J
where VkV’ VkU’ and VkD( Ere con)stant coefficients to be evaluated.
1 , 0
King and his co-workers Q refer to the array theory

using equation (4.17) as an approximation to Vk(x) as the "three-
term theory". It is also well known that the "two term" approxima-
tion to Vk(x) obtained by dropping VkD(x) .1eads to results of

good accuracy for arrays with element lengths satisfying kohk s n.

The latter two-term voltage approximation will be utilized in the

investigation of the waveguide-backed slot array from this point on.

4.2.4 Reduction of the System of Integral Equations to a System of
Algebraic Equations:
A differential equation for the difference Ak(x) - Ak(a/2+hk)
is obtained by adding ~k:Ak(a/2 + hk) to both sides of equation

(2.30) (with E0 = O) as

2

L. 2 _ 2

(3x2 + ko)[Ak(x) - Ak(a/2 + hk)] - -onkoIk6(x - a/Z) - koAk(a/2+hk).
(4.19)

If Ehk1(x,x') is Split into real and imaginary components as

(x,x') + 1'15 (x,x'),

._ ' = _
de1(x’x ) deiR dkiI

then

80

Na/2+h1
[Ak(x) - Ak(a/2 + hk)]= z I v i(x' )Kdk i(x,x')dx'
i= 1 a/Z-hi
N a/2+hi
= 2 [Vi VM: X(X') +‘V (X' )] [K (x,X')
1=1 {IZ-h: 1UP 0“ dki R
_ l
+ jdeiI(x,x )]dx . (4.20)

Based upon the peaking property of EdkkR and the non-peaking
nature of EdkiR (i i k) and deiI’ the various integrals can be

approximated in terms of simple trigonometric functions as demonstrated

(12), (20).

by King et al. The results of these approximations

are as follows:

N a/2+hi
1:1 gnh Vi VM: x(x' )% ki(x, x ')dx' as VkVdekRM :x(x)
N i
+ jvka dkkIFo kx(x) + 121(1 ' bik)ViVYdeiFox(x) (4'21”)
N a/2-l-hi N i
,(x,x')dx"d Z V, Y F (x) . (4.21b)
i=1 a/2_11.1Vi-UFC31‘C(X del 1:]. 1U Uki OK

The various approximately constant Y functions in equations

(4.21) are defined as follows:

81

s 4
ydkkRoa/Z) for bk 110/
dekR = /
dekRm/Z + hk - xo/a) for hk > x0 4
a/2+h
J" k 1‘1 )E ( '>d
Y (x)= M x x x x
dkkR “1:00 a/2_hk ox dkkR
a/2+h
1 k
= v :
dekI k M; (x ) dek1(a/2,x )dx (4.22)
F (a/Z) a/Z-h
ox k
a/2+’ni
1
Y = ---—- M1 (x' )% k1_(a/2,x')dx' for i f k
dei F1 (a/Z) a/Z-h ox
ox i
a/2+h
1 1F :15 ')E </2 '>d'
Y = ---- x a ,x x
dUki F1 (8,2) a/2_h dki
ox i
Substituting approximations (4.21) into equation (4.19):
2 2 N
( 2 + Ron 2 ViVEde iR Mox(x)6ik +ijkiIFox (”51k
ax i=1
N
+ (1 ’ éideVkiF 01cm] + 1:111V1UYdUk1Fox(x)}
2
= - - 2 - ,
jkogolk6(x a/ ) koAk(a/2 + hk). (4 23)
Noting that the differentiations on M:x and th can be carried

out as

82

2
a 2 k
( 2 +’ko)Mox(x)

-2k06(x - a/2)cos(kohk)
ax

52 2 k 2
( 2 + ko)Fox(x) -ko cos(kohk)
and that Ak(a/2 + hk) can be evaluated in the form
a/Z-I-hi
,'_ I I
“/2411! vi(x )Kk1(a/2 + hk’x )dx
hi

Ak(a/2 + hk) —

I
IItdEZ

I
fllfl Z

_ .2
i 1[V1VYVki+ viUYUki] (4 4)

where

2
a/ +111

1 I _' I I
= ii/2-hi M 0X(x )Kki(a/2 + hk’x )dx

a/2+h.
1

1 I _ I I
wUki = j Fox(x )Kki(a/2 + hk,x )dx , (4.25)
a/Z-hi

equation (4.23) becomes

N

— 2
1El{viv[2koaikwdkm6(x a/ )cos(ko h ) + jk: 5 1deki I cos(kohi)

2
+ ko(1 _ 61k)Ydei cos(kohi)] + Vi UERZ OYdUki cos(kohi)]}

N

= jkogoIk6(x - a/Z) + k2: +av

WEVIV Vki iUYUki] ' (4'26)

Equating the coefficients of 6(x - a/Z) on both sides of

equation (4.26) leads to

 

_ 0 k
VkVdekR °°S(kohk) '

or

=jCo 1k
V , for k = 1,...,N, (4.27)
RV: ZYSRER cos(E‘hk)

 

83

while equality of the remaining terms in (4.26) yields

N
i:31{V1V[(l - 61k)Ydei cos(ko h 1) + jb ”k dki Icos(kohi)]
N

+ viUYdUkiCOS(kohi)] = iEIEViVYVki +'V1UYUk1]'

Rearranging the above equation finally leads to the desired matrix

equation for the ViU as
N N
iflvi UEYdUkicos(kohi) ' YUki] = 1:1V1V{YVki ' [Yde1(1 ' 61k)
+ jydkiléik1cos(kohi)} ... for k = 1,2,3,...,N (4.28)
or in matrix form
where
= - W
QVki YVk1 [Y Yde1(1 61k) + ijIkfl k3°°S k0 hi
and ?
QUki = YdUkiCOS kohi ' Yuki J ° (4'30)

 

[QU] and [@V] are (N X N) square matrices whereas [Vv] and
[VU] are column matrices of order (N x 1). The elements V1V

of [VV] are known from equations (4.27) while the elements ViU
of [VU] are to be determined by inversion of equation (4.29).

Once all the Y—functions as defined in expressions (4.22) and (4.25)
are evaluated, the [QU] and [Qv] matricescan be computed and

equation (4.29) can be solved numerically for the unknown elements

of [VU]. Formally, the ViU are given by

[VU] = [IU1'1[§V]LVVJ . (4.31)

84

With equations (4.27) and (4.31), the approximate expressions

for the slot voltage distributions can be written as:

 

v<> MIL k (h I /2\>
x N sin - x-a
k Zdechos(kohk) o k
-1
+ {[Qu] [Qv][Vv]}k[cos ko(x-a/2) - cos kohk]. (4.32)
The driving-point admittance Yk of the kth slot is defined
as
Ik

The corresponding input impedance to the slot is = l/ka

zk
4.3 Evaluation of Various Y-Functions:

An inspection of equations (4.2), (4.3), (4.22), and (4.25)
reveals that there are, basically, two types of integrals involved
in the evaluation of the various Y-functions, one with Gi(x,zk,x',z')
in the integrand and the other with Go(x,zk,x',z') in the integrand.
The integrals involving G1 will be evaluated analytically while
those involving G0 will be evaluated numerically, using the same

x and z partitioning scheme as described in Chapter 3 (Section

 

3.3.1).
4.3.1 Eggluation of dekR and. dek1 (£g£_ hk S xo/4):
Equations (4.22) indicate that
dek = dekR + ijkkI (4'34)
or
dek = EFL-— Re(Idkk) + j Tl— Im(Idkk)] , (4.35)
be(a/2) Fox(a/2)

85

where

2+h + .
a/ k zk e Sln k.o(hk - \x'-a/2\)

I =j‘
dkk
- - 2
a/2 hk zk ’3 11\/e:2-(zk-z')

 

{Gi(a/2 ,ZRIX' ,Z')

 

o i
+ G (a/2,zk,x',z') - G (a/2 + hk,zk,x',z')
o

- G (a/2 + hk,zk,x',z')}dx'dz' (4.36)
Equation (4.36) indicates that Idkk involves two types of integrals;
one with Go in the integrand, denoted by Izkk’ and the other
with G1 in the integrand, denoted by Idkk' Using the fact that

. . . . o .

Sln 1:0(hk \x a/2\) in the integrand 1n Idkk IS an even func
tion of x' about x' = a/2 and using the same partitioning scheme

as described by equations (3.5) and (3.12) (see Fig. 3.1), I can

0
dkk
be written as

‘M

2

k p I sin k.o(hk - x' + a/Z)
" {
m=1 p=1 AZ')kp (Ax')km “J62

 

 

O , 0 I I
Idkk'¢ {G (a/2,zk,x ,z )

- (zk-z')

+ Go(a/2,zk,a-x' ,z') - G°(a/2 + hk,zk,x',z') -

- c°(a/2 + hk, zk,a-x',z')}dx'dz' (4.37)
where
2
ka+Axk/ 2 zkp+Azl
I -—D , and { = J:
(Ax)km ka-Axk/Z AZ)kp ka'AZ/Z

Integral Izkk can finally be expressed as

M
k p
O
T k - U k 4.38
Idkk~ mil p31 E mp( ) mp( )] ( )

where

T

mp<k>=

UmP(k)

86

/’

(Ax 'Az) sin ko (hk .+ a/2)

ka
)w/z - (zk-zkpz)

+ Go(a/2,zk,a-x

k

[Gown/2.24k )

 

,xhn’zkp

km’zkp)] . excluding p = l, p = P,

l . g2 0
[0.5+F arc sm(e 1)]sin k0(hk ka + a/2)[G (a/2,zk,
o
ka,zkp) + G (a/2,zk,a-ka,zkp)]Axk ... for p = l and

p = 2 only.

(2 sin k h 2 Ax AK
2
o k‘)L'Jk .7122 ‘A—+ ELM zk +/1 + “3239)

o 2
1T2

2 +
+2.11k ”(ngFK/I +0.25%?) )1 for p =13—l
k k

 

 

 

 

 

\\ and m = 1. (4.398)

p

 

(Ax AZ) sin k0 (hk- ‘+ a/2)
k ka [Go(a/2 + h ,z

k k’ka’zkp)
)e'\/2 Z)2
2k?

0
+'G (a/Z + hk’zk’z-ka’zkp)] ... excluding p - l, p = P.

P+l _
(p = 2 and m — Mk)

 

       

 

 

 

= 'l . A5 _ . - o

[0.5 +TT arc Sln(€ 1)]31n ko(hk ka'+ a/2)[G (a/2 + hk’zk’

o
ka’zkp) + G (a/2 + hk,zk,a-ka,zkp)]Axk ... for p = l

and p = P only
Ax

Sln(ko TE) 0 z ZAxk

[fl AxkAz G (a/2 + hk’zk’a-ka’zkp) + 2

TT 6
AK Ax A2
2 z \/r z 2 k
+j1 + 4(—Az“) ) + Akan(A—-2Axk+ 1 + 0'25(Lc1xk) ) 311° 2 ]
P+l _
. for 2 and m — Mk. (4.29b)

 

87

These results are obtained using the same approximation techniques
described in Section 3.3.1 of Chapter 3.

Integrals Idkk can be evaluated analytically in closed

form, and are given as

i

= m ‘ 9—11 _ ' DE.
1dkk )3 [3111(2 s1n(2 + kxhk)]Dnan(k)Ynm(k) (4.40)

l m=0

IIMB

n
where k = (11) while
X 8.
6

D = m ,
nm nab ynéfl - (T1) (r2)nm]

nm

(4.41s)

 

2+h
a/ k

Ln(k) = i/2_h sin k.o(hk - \x' - a/2\)sin kxx'dx'
k

Zkosin(t2-IE
= (cos kxhk - cos kohk)’ (4-41b)

2 gn_2
[ko'(e)]

 

and (in terms of fnm(z,z') defined in Chapter 2)

Zkfe f (z ,z')dz'
nm k
Ynm(k) ‘I
Zk-e Jéz - (2k - z')2
-20rnmzk

("{[Io(anm€) - Lo(anme)] - (Fl)nme Io(anm€)

 

 

zanmzk
- (F2)nme Io(anm€) + (Fl)nm(r2)nm[lo(anm€)

Ynm(k) = + Lehman} if yum = anm + 10

< -2je z
n{[Jo(Bnme) - jHo(Bnme)] - (F1)nme Jo(8nme)

Zjenmzk
- (r2)nme Jo(anme) + (Fl)nm(r2)nm[Jo(anme)

(4.4lc)

 

L+ jHo(Bnm€)]} if ynm = O + jBnm°

88

(JO,HO) = Bessel and Struve functions, respectively of zeroth order
and real argument;
(IO,LO) = Modified Bessel and Struve functions, reSpectively of
zeroth order and real argument.
Appendix II demonstrates the details of the z'-integration involved
in obtaining equation (4.4lc).
Equations (4.38) through (4.41), when substituted in

expression (4.35), determine Y and, from expression (4.34),

dkk

Y and Y

dkkR dkkI'

4.3.2 Evaluation of Ydei:

’ .22 '
Equations (4 ) defines expreSSLOn for Ydei as
a/2+h,
Y = ‘_—l———— 1 Mi (x')E' (a/Z x')dx'
dki ’

dei F1 (a/Z) a/2-h OX
ox i

This definition can be rewritten as

 

 

 

 

_ _ -l = _ -1 o i
Ydei — (1 cos kohi) Idei (1 cos koh) (Idei + Idei) (4.42)
where
o al2+hi zi+€ sin ko(hi - \x' - a/2}) o
= I I
Idei I/Z j 2 {G (a/2,zk,x ,z )
a -h. z.-e J ,2
1 1 n e - (zi - z )
- G°(a/2 + hk,zk,x',z')}dx'dz' (4.438)
and
1 a/2+hi zi+e sin k0(hi - \x' - a/2\) i
ll ' I
Idei [/2 I {G (a/2,zk,x ,z )
a “h. 2,-6 2 ' 2
1 1 n\/e - (zi - 2)
- Gl(a/2 + hk,zk,x',z'))dx'dz'. (4.43b)

89

Again, 13Vki involves the function M:x(x') in its
integrand which is even about x' = a/2. Also, since i # k, 2'
can be taken to be approximately 21 for the purpose of performing
the x'-integration, that is, Go is a slowly varying function of
2' (good approximation for narrow slots, koe << 1). It is there-

fore possible to write equation (4.43a) as

2+3 a/Z-i-h.
O “I1 dz. 1 ° 1((h -X'+8/2)
Idei z_ 2 2f” ““01

i e n\Jé - (zi - z') a

[G°(a/Z,ZRIX'.zi) + G°(a/2.zk.a-X'.zi) - 60(8/2 + hk.zk.X',zi)

 

_0 _I I
G (a/Z + hk12k9a x ,zi)]dx

The z'-integration evaluates to unity as follows

 

 

z.+€ I- )
1 dz' 1 . (z zi +3
f = E'arc Sln -——;T—-1 = l,
- 2 2 ’6
216 MA - (zi-z')
d 1° b -
an dei ecomes.
a/2+hi
0 R5 - _ I 0 I I
Idei {/2 Sln ko(hi x + a/2)[G (a/2,zk,x ,z )

o , - o ,
+ G (a/2,zk,a-x ,zi) G (a/Z + hk’zk’x 121)

o
- G (a/2 + h ,zk,a-x',zi)]dx'

k

Using the same x-partitioning scheme described in equations (3.5),

o
Idei can be calculated numerically as
M
o i
g o - 2 o
Idei mil Sln ko(hi xim + a/ )Qm(k,1)Axi

where

9O

, _ o o _
Qm(k,1) - G (a/2,zk,xim,zi) + G (a/2,zk,a xim’zi)

- 60(3/2 + h ’Zi) - Go(a/2 + h ,a-x ,zi). (4.44a)

k’zk’xim k’zk im

. i I ,
In evaluating Idei’ the fact that for i # k, 2 1V zi

insofar as the z'-dependence of G1 is concerned (i.e., fnm(zk,z')

o i 0
I3 fnm(zk’zi))’ leads to the followxng evaluation of Idei'
1 a m nn nn
Ideii§ “El m20n[sin(§—)-51n(§— + kxhk)]Dnmfnm(zk’zi)Ln(i) (4.44b)

where kx and Dnm were defined earlier while

 

-Ynm‘zk-zi\ -Ynm(zkfzi)
fnm(zk’zi) "’ e ‘ (F1)nme
y (2 +2 ) y ‘z -z.\
- (r2)nme Hm k i + (r1)nm(r2)nme nm k 1 (4.44c)
and
2k sine-fl)
1. (1) = 0 (cos k h - cos k h.) . (4.44d)
n 2 nn 2 x i o 1
[kc-(T) ]

Equations (4.44) together with equations (4.43) and (4.42)

completely determine Ydei°

.. .2
4 3 3 Evaluation of Ydukl

According to definition (4.22), Yduki is
a/2+h,
__ -1 1 .
Yduki - (l - cos kohi) I [cos kb(x -a/2) - cos kohi]
a/Z-h
i
_ I I
dei(a/2,x )dx .

This equation can be expressed in the form

91

i

l 0
Y [Iduki + Iduki]

(1 - cos kohi)‘ (4.45)

duki =

where
a/2+h. Z,+€
1 i

Iguki = I I [cos ko(x'-a/2)-cos(kohi)][Go(a/2,zk,x',z')
a/2-hi zi-e

- G°(a/2 + hk,zk,x',z')]dx'dz', (4.46)

and
a/2+h. Z,+e
1 1

i = '_ _ i l I
Iduki I I [cos ko(x a/2) cos koh1]{G (a/2,zk,x ,z )
a/Z-hi zi-e

- Gi(a/2 +Ih ,x',z')}dx'dz'. (4.47)

k’zk

Since the shifted cosine function in the integrand of IgUki is

an even function of x' about x' = a/Z, then, using the same

0 and 1°

numerical techniques discussed in evaluating Idvki dkk’

equation (4.46) can be approximated as

M
i
o
Idukiie (Axi)m§1[cos ko(xim-a/2)-cos kohi]Qm(k’i) ... for i f k, (4.48s)

Mi P
o

N ' _ ' =
Iduki g z [Tmp(k) Ump(k)] ... for 1 k, (4.48b)
m-l p=l
where Qm(k,i) is defined in equation (4.44a) while T$p(k) and

I o
Ump(k) are given by.

Tép(k) -

and

U$p(k) =

r

b

F

 

-

€92

(Axk)(Az) [C08 kb(ka-a/2)-cos kohk]

" / 2 2
6 - (zk-zkp)

+ Go(a/2,zk,a-x

 

O
[G (a/2,zk,ka,zkp)

km,zkp)] ... excluding p = l, p = P,

(p = Eil' and m = 1)

l . 45 _ _ -
[0.5 + w are Sin(e 1)][cos ko(xkm a/2) cos kohk]

[Go(a/2,z p) + G°(a/2,zk,a-x

k’ka’zk km’zkp) 1A"k

. for p = l and p = P, only,

 

2 -jkOAXI(Az A_Z_ ZMk Ax 2
"26 (1 - COS kohk)[——-§-——-+ 2 Ln( A2 + ’1 + 4(5) )

 

z 2
+ Mkw<§°z+ 1 + 0.25(%:—1:) )1

+
. for p = 25; and m = 1 only (4.48c)

 

 

(AK )(AZ) [cos k (x -a/2)-cos k h ]
k 0 km 0 k [Go(a/2+h ,z ,x ,z )
e ' (zk-zkp)
o
+ G (a/2+hk,zk,a-ka,zkp)] ... excluding p = 1, p = P,

+
(p = 251' and m = Mk)

[0.5 + i are snug?- - 1)][cos ko(ka-a/2)-cos kohk]

0 O
[G (a/Zfik’zk’ka’zkp) + G (a/2+hk,zk,a-ka,zkp)]

. for p = 1, p = P only

1 Ax o
-7- [cos ko(hk--§k)-cos kbhk][nG (a/2+hk,zk,a-ka,zkp)AxkAz
118

z 24xk ‘q/""‘7;r"
+ 911“ A2 )+ 1 + 4052-152) + Aka(2A:x_ +\/;+0.25(2£_)2)
Ax Az k
1“,, g ] for p =1? and m =M‘k' (4.48d)

 

 

 

93

' ’5
Again, noting that for i i k, fhm(zk,z ) fnm(zk’zi)

in the integrand of expression (4.47), the closed form evaluation
i

 

 

 

of IdUki is
f -
2 ; 2nko[kosin(kxhi)cos(kohi) kxsin(kohi)cos(kxhi)]3 31n(flfl
2 2 nm 2
n=1 m=0 k (k - k )
x o X
f (z z )[sin k (a/2 + h ) - sin(Efl)]
nm k’ i x k 2
i
1ka1 =( for 1 IE k (4.49a)
; a 2ko[kosin(kxhk)cos(kohk)-kxsin(k6hk)cos(kxhk)] 1 (EH
3 2 2 8 n 2 )
n=1 m=0 k (k ' k )
x o x
. HE =
DnfiYnm(k)[Sln kx(a/2 +hk)-sin(2 )] ... for i k (4.49b)
K

where Ynm(k) was defined in equation (4.4le).
Equations (4.49) and (4.48) along with equation (4.45)

completely Specify YdUki'

4.3.4 Evaluation of YVki'

Equation (4.25) defines YVki to be
a/2+hi
= y sin ko(hi-\x'-a/2\)Kki(a/2 + hk,x')dx'.

Y

This definition can be eXpressed alternatively in the form

0 i

 

 

YVki = YVki + YVki (4'50)
where
O a/Zfili zi+€ Sin ko(hi_‘xl_a/2\) o
Y = I G (a/2+h ,z ,x',z')dx'dz'
i i e TTJe -(zi-z')

and

94
2
i a/ +hi Zi+€ sin k (hi-‘x'-a/2\) i
Y = ° C (a/2+hk,zk,x',z')dx'dz'.

Vki
- - 2 2
a/Z hi 2i 6 \j; -(zi-z')

Comparing these expressions for Y:

 

i
ki and YVki with expressions

(4.43a) and (4.43b), reSpectively, it is clear that the evaluation

for Y3 and Y; are given by certain terms from equations

ki ki
(4.44). Therefore, by inSpection of the earlier results for i # k

M

° = Ax k 1 k (h - + /2 [o°( /2+h

YVki imilk n o i xim a ) a k’zk’xim’zi)
O
+ G (a/2+hk,zk,a-xim,zi)]} ... for i # k (4.51a)

and

1 <13 m
YVki = nil mEOfl 31n kx(a/2+hk)Dnmfnm(zk,zi)Ln(i) ... for 1 i k. (4.51b)

For i = k, a comparison with eXpression (4.36) yields the following

o 1
evaluation for Yka and Yka

Mk 1»

O

Yvkk = E g Ump(k) ... for i = k, (4.52a)
m-l p-l
and
1 00 CD
Yka = “31 mgosin kx(a/2+hk)DndLn(k)Ynm(k) ... for 1 = k, (4.52b)

where express1ons for Ump(k), D m’ Ln(k), and Ynm(k) are given

n

by expressions (4.39b) and (4.41), respectively.

4.3.5 Evaluation of YUki'

Equations (4.25) define Y
a/2+hi

a '- - -
YUki [cos ko(x a/Z) cos kOhiJKki(a/2+h
a/2-hi

Uki as

I I
k,x )dx

9S

 

 

 

 

An equivalent expression for YUki is
Y = Y1 + w° (4 53)
Uki Uki Uki °
where
i a/2+hi zi+€ [cos k (x'-a/2)-cos k hi] i
YUki = I o o G (a/2+hk,zk,x',z')dx'dz'
- - 2 2
a/Z hi 2i e n\/; - (zi-z')
and
o al2+hi zi+e [cos ko(x'-a/2)-cos k.h ] o
YUki = g o G (a/2+hk,zk,x',z')dx'dz'.
a 2'11 2 -6 2 | 2
i i 11' e - (zi-z )
Comparing the above expressions for Yski and Yski with expressions

(4.46) and (4.47), respectively, it is clear that certain terms

from equations (4.48) and (4.49) constitute the evaluations for

o
Uki

earlier results

Y and Yéki’ reSpectively. Therefore, by inapection of the

 

 

M
i
o
(Axi)m§1 [cos ko(xim-a/2)-cos kohi][G (a/2+hk’zk’xim’zi)
o o .
YUki - + G (a/2+hk,zk,a-xim,zi)] ... for 1 # k (4.54s)
”k r
z r, U“: (k) ...for i=k (4.541))
m=1p=1 P
and
r' 2 ; 2nk0[kxsin(kohi)cos(kxhi)-kocos(kohi)sin(kxhi)]
2 2
n=lm=0 k(k -k)
x o x
211 m
sin(2 +k.xhk)sin(2 )Dnmfnm(zk’zi) ... for 1 ¥ k (4.55s)
1
3 2 h -
YUki m m ROEExsin(ko k)cos(kxhk) kocos(k6hk)sin(kxhk)]
E Z 2 r
n=1 m=0 k (k - k )
x o x
m+k 9.11 =
L'sin(2 xhk)sin(2 )Dnm‘Ynm(k) ... for i k. (4.55b)

 

96

With equations (4.35) through (4.52), all the Y-functions
are now completely evaluated. Equation (4.27), then, yields VkV’
k 8 1,2,3,...,N, while expressions (4.30) for the elements of
[6U] and [QV] can be calculated in terms of the known Y-functions.
A computer program was developed to numerically calculate the
elements of [Vv], [QV] and [QU] and subsequently solve numerically
the matrix equation (4.92) to generate the elements of [VU]. Know-
ing [Vv] and [VU], the slot voltages Vk(x), k = 1,2,3,...,N,
can be found using equation (4.17). In the next section, the

radiation fields maintained by these slot voltage distributions

are determined.

4.4 Radiation Field Maintained by the Slot Array;
It was shown in section (2.4) that the radiation fields
maintained by the slot array in two principle planes, namely,

E-plane (m = -fl/2) and H-plane (m = 0) are given by

-jk r
«r 4 A Jkoe o N jkozicos 9 8[24-111
Es“) ‘6 '9 ‘77—— : e I “and".
n i=1 a/2-hi
. E-plane radiation field (2.42)
and
-jk r
_,r _, JR 8 ° N jk zicos e a/zfii jk x'sin e
EH(r);e & 02 cos 9 E e o I Vi(x')e 0 dx'
"1' 1=1 a/Z-hi
. H-plane radiation field. (2.43)

Substituting the approximate analytical expression for V1(x')

from equation (4.17) recognizing that coefficients V1v and V1U

have now been determined, the x'-integrals in the above radiation

97

field equations takes the form

a/2+hi
£/2-h {ViVSin ko(hi \x a/2\) + ViU[cos ko(x a/2)
i
- cos kohi]}dx' ... E-plane radiation field (4.56a)
and
a/2+hi
I I
£[2-h {vasin ko(hi \x a/2‘) +-ViU[cos ko(x a/2)
i
jk X'sin 9
- cos kohi]}e 0 dx' ... for H-plane radiation field (4.56b)

Noting that

 

a/2+hi W
-2
- '- ' = —— -
I/z h sin ko(hi \x a/2\)dx k (cos kohi 1),
a - i o
a/2+h
1 2
I cos k0(x'-a/2)dx' = E-’sin koh" and >
a/Z-h o 1
i
a/2+-hi
J‘ cos k h_dx' = 2h. cos k h, (4.56c)
a/Z-h o 1 1 o 1
i
J
then equations (4.56a) and (2.45) yield a final-expression for
at «
EE(r) as
-jk r
o N jk z cos 9
EE(r) m 9 LP: 1218 [Viva cos kohi)

+ V1U(sin kohi - kohicos koh1)] ... for E-plane radiation field. (4.57)

Noting further that

98

 

 

 

 

a
a/Z‘I'hi jkox'sin e -2 ejko 2 Sin 9
I sin k (h - \x'-a/2\)e dx' = (-—)
o i k 2
a/Z-hi 0 cos 9
[cos kohi - cos(kohiSLn 9)] (4.58)
and
a/2-I-h1 jk X'sin 9
I [cos ko(x'-a/2) - cos k hi]e 0 dx'
a/2-h o
i
jk §sin e
- -2 e O 1rrcos(k h )sin e sin(k h sin e)
- k { ZfiL o i o i
0 cos 9
sin(kohisin 9)
- sin kohicos(koh131n 9)] - sin 9 }, (4.59)
euqations (4.56b) and (2.44) yield for E;(?)
- k r jk 9; sin 9
«r d 1e 1 o e o 2 N jkozicos e
EH(r):~ -¢ nr cos 8 2 e [Viv[cos koh1 -

i=1

cos(kohisin 9)] + ViU[cos(kohi)31n e sin(kohisin e) -
. . 2 .
sin kohi008(koh1810 9) + COS(kohi)31n(k6hisin 9)(cos 9/31n e)}]

. excluding a = 0, n and (4.60)

E.
2 .
Equation (4.60) becomes singular for e = 0, n and E: For these

values of e, the x'-integrals must be revaluated. Therefore,

for 9 = 0, equation (2.43) assumes the form

-jk r
fir‘d jkoe o N jkozi a/Z-I-hi
A __——_ - .-
EH(r) m a) 2m 5, e J‘ {vivsm ko(hi \x a/Z‘)
i-l a/Z-hi

+ ViU[cos ko(x'-a/2) - cos(kohi)])dx' ... for 9 = O.

99

Using the results of expressions (4.56c), the above equation becomes

-jk r

EEG-3 a -3 1971—:— 11glejk°zi[vw(cos kohi - 1) + ViU(kohicos kohi -
sin kohi)] ... for e = O. (4.61)
For 9 = n/Z, equation (2.43) gives
E§(?) = o ... for e = n/2. (4.62)
For 9 = n, equation (2.43) yields
fir d A jkoe-jkor N 'jkozi a/2-l-hi
EH(r)¢¥ e --3§E:-- .2 e I {Vivsin ko(hi - ‘x'-a/2\) 4-
1=1 a/Z-hi
ViU[cos ko(x'-a/2) - cos kohi]}dx'. (4.63)

Using expressions (4.56c), the final form of E;(?), for e = n

becomes
-jk r
o N -jk z
"r"__nle Oi _ .
EH(r) — ¢ nr igle [VVi(1 cos kohi) +-VUi(31n kohi
-kohi cos kohi)] ... for e = n. (4.64)

Equations (4.57), (4.60), (4.61), (4.62) and (4.64) com-
pletely determine the radiation field maintained in the principle

planes of the slot array.

100

4.5 Ngggrical Results:

In order to solve for the slot voltage distributions, the
driving point admittance of the array and its radiation field, a
computer program was developed and implemented on a CDC 6500 system.

As indicated by matrix equation (4.29) the order of the matrix
involved is (N X N). For a lO-element array, then, a matrix of
order of (10 X 10) results. This is to be compared with the matrix
of minimum order of (100 X 100) that must be inverted to obtain
adequate accuracy in a direct numerical solution (Section 3.6). Based
on the results of section 3.6, ten x-partitions (Mk = 10) and five
z-partitions (P = 5) were found to yield adequate accuracy in the
numerical solution of the various Y-integrals while the double
Fourier series truncated at “max = mmax = 10 is found to provide
sufficiently accurate results.

4.5.1 Investigation of Several Special Cases:

In order to check the accuracy of the extended King-Sandler
theory for a waveguide backed slot array and to test the computer
program, several special cases were considered and the numerical
results compared to those of previously published research.

Figure (4.3) indicates the admittance Y0 = C0 +jBo of a
single cavity-backed slot as a function of the cavity depth. These
theoretical results of the new, extended King-Sandler array theory
are compared to Galejs' theoretical (based upon a variational

(a)_

approach) and experimental results Various dimensions are
listed on the upper right hand corner of the figure. It is observed
that for shallow cavities, the Go (and Bo) values do not compare

well, however, for bl),o = 0.2 and 0.5, both Go and B0 compare

1...
0
1

O

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102

fairly well with Galejs' results. A decreasing cavity depth makes
Bo approach an inductive short circuit (B04 ~m). Resonance which
is defined by B0 = 0, occurs at a cavity depth bl),o = 0.3. These
deviations are to be eXpected since, for shallow cavities, the
voltage distribution in the slot cannot be accurately predicted (8 )
using the simple sine (M:%(x)) and shifted cosine (F:¥(x)) dis-
tribution functions.

The inaccuracy of the voltage distributions in a slot backed
by a shallow cavity is confirmed in Figure (4.4), which shows the
relative amplitude distribution of slot voltage for a single cavity
backed slot with various cavity depths as a parameter. It can be
observed that the new theoretical results based upon the modified
and extended King-Sandler theory compare quite favorably with those
predicted by Galejs' variational solution for bl),o = 0.27 and 0.5.
However, for shallow cavities (b/Ao = 0.005 and 0.05) the new array
theory does not accurately predict the voltage distributions. The
inability to accurately describe the voltage distribution in a narrow
slot backed by a shallow cavity in terms of simple trigonometric

(8). It is evident that the

functions was also observed by Galejs
new array theory is limited to slot arrays with backing waveguide
having depths of b 2 0.25 x0.

Figure (4.5) demonstrates another Special case consisting of
a two element parasitic slot array. Front-to-back ratio of this array
is plotted as a function of the element-spacing, Az/Ao, in the figure.
It is observed that with a parasite element length of hZ/x,o = 0.25

this element behaves like a reflector, while for hz/xo = 0.20 the

parasite behaves like a director for 0.05 s Azl),o s 0.25, and as a

of slot voltage

max

relative amplitude V(x - a/2) /V

103

 

 

 

 

 

a = 0.6 K e = Zl = 0.001 A0
c = 0.05 A h = 0.3 1
o 1 0
1‘1 = r2 -1.0 n = 2 Ln(4h1/e) = 14.2
1.0 r I r I I I
;/:E.— A\ extended King-
Z" [5‘\ \\ Sandler theory
0.9 \A A \ ‘
\ \ \ A\ G lj ' 1a
a e 3 var -
0 8' \ \\ A \ A tional solution _
A A \ \ b
_ \ \ A T = 0'5 _
0.7 \ \\ \ o
\ 0_3\7 0.5
0.6- A\ A -
\A \ \ 0.005, 0.05
\
0.5- \ A \A \A _
0.005
\ \005 0.27 \ \
'\ \ -
0.4L \ A A \
[i \ \
0.3— \ A \ \ _
\\ \ \
\ \ \
0.2- [S\ A§\\ \s —
A
\ \e— ‘A \ \
0.1— \ A/AA \A\ \A \-
A\ [S/ [5“ ‘\ \
0.- I I l I \A/I 4 L 1 175 \
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
location (x - a/2)/h1 along the slot
Figure 4.4. Slot voltage distribution in a single cavity-backed

slot for various cavity depths (comparison with
Galejs' variational solution).

1(34

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(nag) 01:91 xoaq-on-nuozg

105

reflector with 0.25 < Az/xo s 0.40. This indicates that with a
driven element of near-resonant length hl/xo = 0.24, the director
length should be shorter than its resonant length with Az/xo s 0.25
while the reflector length should be greater than its resonant length.
It was pointed out in section 4.2 that a slot array cut in
an infinite ground screen situated in otherwise unbounded space is
complementary to an array of strip dipoles and ultimately to an
array of cylindrical dipoles. Consequently, the results for a single
slot cut in an infinite ground plane and surrounded by free space
on both sides (G1 = Go) should give the same results as those for
the complementary cylindrical dipole radiating in free-space. Table
4.1 shows such a comparison for a slot with half length h = 0.125 10
and fl = 10.6. Numerical results for the cylindrical dipole were

taken from the book by King, Mack, and Sandler (12).

It is observed
that the complementarity between a slot antenna and an equivalent
cylindrical dipole is well established by numerical results of the
slot theory developed here.

Figure (4.6) demonstrates a comparison between results for a
ten-element (8-director) Yagi-Uda array of cylindrical dipoles and a
complementary ten-element slot array studied using the new slot array
theory with the backing waveguide removed. Figure (4.6) shows the
relative amplitude and phase distributions of slot fields (comple-
mentary dipole antenna currents) in the elements of a 8-director
Yagi-Uda array. The results for a Yagi-Uda array of cylindrical
dipoles are taken from King, Mack and Sandler (12) (abbreviated

as "KMS results" in the diagrams). An excellent agreement between

result for the complementary Yagi-Uda slot array and the complementary

106

 

 

 

 

 

 

 

 

 

 

 

 

slot antenna complementary
quantity (new theory) dipole (published
results)
YdR 6.8646 6.9307
- . 2 - ,
YdI 0 l9 4 0 1934
Yv(h) 0.8202 - 0.8222 -
j0.5166 10.5189
w (h) 0.4542 - 0.4583 -
” 10.2670 10.2682
Ydu 7.1853 - 7.2607 -
i0.09915 10.0963
3.1804+j92.5250
Zin(ohms) 3.2022 + (after con-
j9l.488 version)
Table 4.1. Comparison of various Y-functions and

input impedances for a slot antenna
(new theory) and its complementary
dipole (published results).

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108

dipole array is obtained; this confirms the accuracy of both the
analytical formulation of the new slot array theory and the computer
program developed to implement it. Figure (4.7) indicates the
associated E-plane radiation patterns, and again a good agreement
between results for the two complementary arrays is obtained.
4.5.2 Five and Ten-Element, Waveguide-Backed, Yagi-Uda Slot Arrays:
A five-element, waveguide-backed Yagi-Uda slot array (one
reflector and three directors) was investigated initially to ascertain
the optimum values for the various array parameters (with minimum
expenditure of computer time). Figure (4.8) shows the relative
amplitude and phase distributions of voltages in the elements of
the 5-e1ement Yagi-Uda slot array. Various dimensions are identified
at the top left-hand corner of the figure. The impressed currents
Ik = 0 for k = 1,3, 4 and 5 while for k = 2, IR = (l + jO)
amperes. The half-length for director elements, h3 - h5 is fixed

at 0.20 x0 while h = 0.25 lo (reflector) and h

1 = 0.24 x0

2
(driven element). The Spacings between reflector and driven element
as well as between the driven and director and all adjacent director
elements is Az = 0.25 ID. The backing waveguide is matched at the
output terminals (F2 = 0) while it is short circuited a quarterwave
length in front of the reflector element (F1 = -l.0). The width of
the slots is specified by the parameter Q = 2Ln(4h2/e) = 10.6.

The voltage amplitude distributions (Figure (4.8a)) in the
slot elements of near resonant length closely approximate sinusoidal
variations along the slot axes, which is expected since the slot

voltage is described by shifted cosine and sinusoidal functions.

Maxima of the amplitude distributions at the centers of slot elements

109

= = .25 = 2 = .
G G hl o 5 x0 (I Ln(4h2/e) 10 6
= .2 - = .2
hz o 45 x0 h3 hm 0 0 x0
(Az)dir= 0.3 x0 (AZ)ref = 0.25 x0

 

    
 

 

 

9
1800 1 % IL 'L 00
0.2 0.4 0.6 0.8 1.0
relative amplitude of
radiation field
9

 

O

120

 

W KMS results

—0— -O——O— new slot array theory

Figure 4.7. Comparison of E-plane radiation patterns for a
8-director Yagi-Uda slot array with those of a
complementary dipole array.

110

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111

also display a standing-wave character. Slot number 2 has maximum
induced voltage at its center, slots 1 and 3 have almost identical
amplitude distributions of intermediate amplitude while slots 4 and

5 have nearly identical distributions of minimum amplitude, slot 5
having a slightly greater induced voltage than slot 4. This is
expected, since slot number 2 is the driven element with slots 1

and 3 situated symmetrically relative to slot 2. Slot 5 has a greater
induced voltage amplitude than slot 4 since the array is a finite
one, and reflection of the wave in the aperture of the array is
expected at its terminal end, thus resulting in a standing-wave field
distribution in the elements along the array aperture.

Figure (4.8b) indicates an essentially uniform, progressive
delay or lag in the phases of (successive) slot voltages along the
array aperture. As before, the light line on Figure (4.8b) gives an
indication whether the actual wave excited along the array aperture
is a fast wave or a slow wave; since a slope greater than that of the
light line indicates a larger phase constant, it corresponds to a
traveling-wave aperture field with a phase velocity slower than the
speed of light while a lesser slope indicates the existence of a
fast-wave field. It can be identified from Figure (4.8b), therefore,
that a wave traveling slightly slower than the speed of light exists
along the array aperture. Thus, endfire radiation is expected from
this array configuration, as is confirmed by Figure (4.10).

Figure (4.9) demonstrates relative amplitudes and phases of
slot voltages in the aperture of the same 5-e1ement Yagi-Uda slot
array considered in Figure (4.8), except that now the effects of

various element Spacings, A2, are considered. From Figure (4.9a),

112

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113

a = 0.6 to

b = 0.3 10 h1 = 0.25 to

F1 — ‘1'0 h2 = 0'24 1° (Az)ref = 0'25 ko nmax = mmax
1‘2 = 1.0 113 - h5 = 0.20 "o n = 2Ln(4h2/e) = 10.6 PM = 5

90°

180 TL

relative amplitude
of radiation field

 

0(Az)dir 0.15 ID
120

 

Cl (A2)

‘ 0.20 K
0

dir

A(A2)dir = 0-7-5 A0

C(42)“r = 0.30 x0

A(A2=)dir = 0.35 "o

Figure 4.10. Dependence of E-plane (¢ = -n/2) radiation field
patterns of a S-element Yagi-Uda slot array upon
element Spacings.

 

114

the essentially decaying, traveling-wave nature of the slot voltages
(aperture field distribution) is apparent. Slot 5 again has a greater
induced voltage amplitude than slot 4 when the element Spacings are
small, resulting in a short array aperture, due to reflections at the
terminal end of the array. For large element Spacings, for example,
(Az)dir/).o = 0.30 or 0.35, which results in a large array aperture
slot 5 has lower induced voltage amplitude than slot 4, indicating
smaller terminal-end reflections.

Figure (4.9b) indicates a progressive phase delay (lag) in
the elements along the array, and all values of (Az)dir give rise
to a slow wave aperture field. Endfire radiation is, therefore,
expected for all values of (Az)dir, and in order to identify an
optimum director-element separation, the radiation field of the array
must be investigated.

Figure (4.10) shows the E-plane radiation field pattern of
the same five-element, Yagi-Uda slot array investigated in Figure
(4.9). All the radiation fields are end-fire in nature as expected.
It is observed from Figure (4.10) that for (Az)dir/),O = 0.15 and
0.20, the side lobe amplitude is quite small with maxima of those
lobes occurring at approximately 110 and 100 degrees, reSpectively.
The mainlobe beam width, however, is relatively large due to the
fact that the array has only five elements, one reflector and two
directors. Major lobe beam width is decreased as the number of elements
in the array is increased. For (Az)dir/),o = 0.30 and 0.35, the main
lobe profiles are essentially the same as those discussed above, but
the side lobes are now more pronounced while their maxima lie in the

broadside direction. If (Az)dir/),o = 0.25, the major lobe is

115

identical to that for (Az)dir/),o = 0.30, these are sharper patterns
than for (Az)dir/),o = 0.15 and 0.20. The side lobe for the
(Az)dir/),O = 0.25 array has its maximum along the broadside direc-
tion and its amplitude is smaller than those for (Az)dir/},o = 0.30
and 0.35. The back lobe radiation (at 9 = 180°) is observed to be
quite small for (Az)dir/ko = 0.20 and 0.25. Therefore, it appears
that an optimal director element separation for this five element
array is (Az)dir/),o = 0.25.

Figure (4.11) studies the same five-element array discussed
in connection with Figures (4.9) and (4.10) except that in this case
the director lengths (hD) are being varied from 0.18 ko‘ to 0.22 x0
while (Az)dir/).o = 0.25. Figure (4.11a) indicates the relative
amplitudes of the slot voltages versus the slot location (AZ/x0)
or number. For hD/xo = 0.18 and 0.20, the relative amplitudes
are as expected, i.e., maximum field (voltage) is excited in slot
number 2, but for hD/xo = 0.22, the maximum slot field occurs in
element number 4. This is due to the fact that for near-resonant
length slots with hD/xo = 0.22 a standing wave field is set up
along with the array aperture. A similar phenomenon is observed in

(20)

connection with Yagi-Uda dipole arrays as the director lengths
approach resonance. However, the array with shorter directors

maintain aperture fields having an attenuating traveling wave nature.
Figure (4.llb) shows the absolute phases of the slot voltages along

the array versus slot location (Az/xo), and it can be observed that

each value of hD/xo results in a slow-wave aperture field and expected

end-fire radiation. To find the optimum director length, the E-

plane radiation field patterns for various values of hD/xo, as shown

in Figure (4.12) must be considered.

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117

a B 0.6 10
= . = . = .2
b 0 3 1o hl 0 25 0o (Az)ref 0 5 “o
r1 = -1.0 h2 = 0.24 4o (Az)dir - 0.25 1o
r2 = 0.0 n = 240(4h2/e) = 10.6 PM = 5
90°
60°

 

 

 

 

 

 
   
  
    

 

b ‘ e
:1 -
.0 13“ 1
€.I.0 0.2 0.4 0.6 0. 1.0I o
180° I—a‘N‘UI I I I?
relative amplitude
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0
O O
150 . 30
CJhD = 0.18 10
O
120 6°
IShD = 0.20 10
90°
InhD=0.22),o

Figure 4.12. E-plane (m = -n/2) radiation patterns for a S-element
Yagi-Uda slot array for various director lengths.

118

It is observed from the radiation patterns Figure (4.12) that
for hD/xo = 0.20, the main lobe is reasonably narrow while the back
lobe remains quite small in amplitude. For hD/xo = 0.18, the main
lobe becomes quite broad, while for hD/xo = 0.22 the back lobe
radiation is singificantly increased in amplitude. In all cases,
however, endfire radiation is obtained, as expected. It appears, then,
that hD/Ao = 0.20 is an optimum director length for this 5-element
Yagi-Uda slot array with waveguide backing.

With the optimum parameters of (Az)dir/).o = 0.25 and
hD/ko = 0.20, the circuit properties of the waveguide-backed, 5-
element, Yagi-Uda slot array are investigated. Figure (4.13a)
demonstrates the variation of the input admittance Yo to the
driven element of the array as a function of electrical length
hzl),o of the driven element. It is observed that both the input
conductance (Go) and input suceptance (Bo) increase monotonically
with increasing hZ/xo for 0.15 s hzl),o S 0.35. The resonant length
of the driven element is found to be h2 = 0.225 10 for this array.
Input admittance to the array depends upon its driven element length
in a manner very similar to the dependence of impedance of a single,
isolated slot as a function of its length. In the range hz/ko = 0.20
to 0.25, the array admittance passes through the resonance and pre-
sents a relatively low input conductance which can provide a near
match to practical transmission lines.

The study of a ten-element, waveguide-backed, Yagi-Uda slot
array proceeds in an identical manner to that just discussed for five
element array. Figure (4.14) indicates the relative amplitudes and

absolute phases of slot voltages in the elements of a lO-element

119

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121

Yagi-Uda slot array as a function of its director Spacing, (Az)dir/xo.
The array dimensions are listed at the top left-hand corner of the
figure. These dimensions are essentially equal to those used for
the S-element array, except that nmax = mmax = 10 instead of 20.
This reduction in number of terms retained in the double Fourier
series Greens' function G1 was made in order to save computation
time without sacrificing too much accuracy.

Figure (4.143) shows the decaying, traveling-wave nature of
the array's aperture field, consisting of the slot fields excited in
the array elements. The "end-effect" of a finite array is observed
to result in a standing wave component near its terminal end; slot
voltage amplitude in element number 10 is larger than that for slot
number 9 in all cases. The relative amplitudes of slot fields along
the array decay more rapidly for greater values of (Az)dir/xo. This
is understandable, since a large value of (Az)d1r/)\o corresponds
to an array aperture of greater length, and hence greater attenua-
tion of the aperture field as it travels down the array.

Figure (4.14b) demonstrates the absolute phases of slot fields
(voltages) versus axial slot location, (Az)dir/xo. It is observed
that all values of (Az)dir/).0 will excite a slow wave in the array
aperture. Larger values of director spacing result in lower phase
velocities but a high rate of aperture field decay, which results in
a degraded E-plane radiation pattern for that case as shown in
Figure (4.15).

Figure (4.15) compares the E-plane radiation field patterns

for various values of (Az)dir/).o in the lO-element array. It is

interesting to compare these plots with those of Figure (4.10). It

a = 0.6 X
o
b = 0.3 A
0
F1 = -l.0
F2 = 0.0

  

180

l
r

   

150

Z§(Az)d1r=O.ISAO

O (mm-0.2%

Ib(Az
bur-0.25%o

D (Az)dir'=0.30),o

Figure 4.15.

relative amplitude
of radiation field

122

h1 = 0.25 “o

h2 = 0.24 Io

h3 - hlo = 0.20 I0

0 = 2Ln(4h2/e) = 10.6

0

9O

 

 

l

‘f’

 

l

 

90

  

60

E-plane radiation patterns of a lO-element Yagi-Uda
slot array for various director Spacings.

123

is immediately obvious that main lobe beam width is much sharper
for the lO-element array than for its S-element counterpart, as one
would eXpect it should be. Radiation field (Eeplane, ¢ = -fl/2)

plots for (Az)dir/xo = 0.15 and 0.20 are almost identical as far as

their main lobes are concerned, even though for (A2) 0.15

dir/A0

a minimum at e = 600 is sharper than that for (Az)dir/x

o 0.20.
The back lobe for (Az)dir/Io = 0.20 is of smaller amplitude than
that for (Az)dir/),o = 0.15. As (Az)dir/1o values are increased
to 0.25 and 0.30, the main lobe becomes broader, the minimum at

e = 600 becomes less sharp and the back lobes become larger. It
appears that a value of (Az)dir/),o = 0.20 will provide an optimal
E-plane radiation pattern, and the mechanical difficulties associated
with the construction of closely-Spaced slot arrays will also be re-
duced. Therefore, (Az)dir/),o = 0.20 is the optimum dimension for
director Spacing in the lO-element array. It is different from that
optimal dimension which was obtained for a 5-element array, though
not significantly so.

Figure (4.16a) indicates relative amplitudes of slot fields
(voltages) against slot location, Az/xo, for various values of
director lengths, hD/Ao = 0.18, 0.20 and 0.22. Curves for hD/xo =
0.16 are identical in shape to that for hD/xo = 0.18 and there-
fore this case is not plotted in Figures (4.16) and (4.17). The
relative amplitudes for larger values (approaching resonant length)
of hD/xo are greater than those for hD/Ao = 0.16 and 0.18. Again,
the decaying, travelingdwave nature of the aperture field as des-

cribed by relative amplitudes of slot fields along the aperture is

apparent from Figure (4.16a). For near-resonant-length directors

124

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125

 

 

 

 

a - 0.6 to bl = 0.25 lo (Az)ref a 0.25 to
b - 0.3 ha h2 = 0.24 lo (Az)dir - 0.20 A0
F1 - -l.0 h3 - h10 - hD nmax ”‘mmax = 10
r2 - 0.0 n = 2Ln(4h2/e) = 10.6 PM . 5
90°
120° 50°
150° 30°
6
o 0.2 0.4 0.6 0.81. o
180 i 1 % J. I I }
relative amplitude of
radiation field
6
30°
A bl) = 0.18 x0 600

o» hD I 0.20 A0

a hD = 0.22 x0

Figure 4.17.

  

 

90

E-plane radiation field patterns of a ten-element
Yagi-Uda slot array for various director lengths.

126

with hD/xo = 0.22, however, the relative amplitude distribution is
quite irregular with relatively little decay and a large standing-

wave component similar to the situation which was observed for a 5-
element Yagi-Uda slot array.

Figure (4.16b) indicates a progressive phase delay along the
array aperture. It is observed that for hD/xo = 0.18, a fast wave
is excited along the array aperture and off-end fire radiation would
be expected in this case. For hD/xo = 0.20 and 0.22, slow waves
are excited and consequently end-fire radiation is expected for these
cases. Due to the significant standing-wave component in the aperture
field for directors of hD/xb = 0.22, a significant back lobe (back
radiation) is expected in the array's radiation pattern.

Figure (4.17) shows the E-plane (m = -n/2) radiation patterns
for hD/xo = 0.18, 0.20 and 0.22. A slightly off-end fire radiation
pattern (maximum occurs at e = 200) is indicated for hD/xo = 0.18.
For hD/x0 = 0.20, end-fire radiation is achieved and main lobe beam
width is much narrower than that for hD/Ao = 0.18. The side lobe
with a maximum at e = 800 is also much lower in amplitude than
that for hD/ko = 0.18. The radiation pattern for hD/xo = 0.22 has
an even narrower main beam, but it has two significant side lobes
with maxima at 9 = 350 and 800 in addition to a back lobe of
significant amplitude. Thus a director length hD/ko = 0.20 is
identified as an optimum parameter for the lO-element, waveguide-
backed, Yagi-Uda slot array.

Figure (4.13b) indicates the variation of input admittance,
Y0 = G0 + jBo, to the excited element of a lO-element, Yagi-Uda slot

array with (Az)dir/),0 = 0.20 and hD/Ao = 0.20 as a function of

127

its driven element length. A comparison of this admittance with
that for an isolated slot, in an otherwise unbounded space

(0 = 10.6) shows the two admittances are quite similar. However,
the resonant length (hzl),o at which B0 = O) has shifted from

Sch 3 1.57 for the single, isolated slot (24)

to Boh2 = 1.415
(hzlxo = 0.223) for the lO-element, waveguide-backed, Yagi-Uda
slot array.

4.5.3 Ten-Element,_flaveguide:§agked Slot Array;

In this section, a ten-element waveguide-backed slot array
similar to that studied in the previous section is investigated with
all elements of equal length, element number 1 as feed point, and
all remaining elements acting as directors. Such a slot array
simulates to some degree, the array investigated in chapter 3 when
the slot array was excited by means of an incident TE10 mode wave

travelling in the backing waveguide. A similar unbounded slot array

(1). A

has been investigated experimentally by Burton and King
large waveguide-backed array with a large number of director elements
will be studied further in this section. The beam-scanning capability
of this array as achieved by varying the backing waveguide dimension
al),o will also be studied, and comparison will be made to the results
obtained in chapter 3.

Figures (4.18a) and (4.18b) indicate the relative amplitudes
and absolute phases of slot fields in the aperture of a ten-element
slot array as a function of location along the aperture for various
director Spacings (Az)dir/xo. Relevant dimensions for the array

are given in the top left-hand corner of the figure. The first

element of this array is assumed to be excited by an input current

128

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129

I1 = (l + jO) amps.,while all the remaining parasitic elements act
as director elements. (Az)12 is the distance between first and
second elements of the array. Figure (4.18a) indicates a strong
standing-wave nature of the aperture field, which is indicative of
large reflections near the terminal and of the array. Figure (4.18b)
indicates a slow wave aperture field along the array for all values
of (Az)dir/Ao. In order to determine the optimum value of
(A2)dir/),o for this array, the E-plane (m = ~n/2) radiation field
patterns for various director Spacings must be investigated as is
carried out in Figure (4.19).

Figure (4.19) shows that for small values of (A2.)dir/Ao
such as 0.1, 0.15 and 0.20, there exists a large amplitude back lobe
in the radiation patterns. For (Az)dir/ko = 0.1, the main beam is
quite broad. It is interesting to note that in Chapter 3, for a
similar lO-element array excited by a propagating TElo-mode incident
wave in the backing waveguide, (Az)/),o = 0.1 was found to be the
optimal element spacing. Therefore, even though the array is similar
in both cases, the mode of excitation significantly changes the array
behavior and its optimum element spacing. For (Az)d1r/),o = 0.25
and 0.30, the radiation pattern main lobes are sufficiently narrow
while their back lobes are relatively small. It is observed from
Figure (4.19) that an optimal choice for director Spacings is
(Az)d1r/),O = 0.30.

Figure (4.20) shows the relative amplitudes and phases of
the aperture field in the elements of the same array, except that
in this case the backing waveguide width is varied from a = 0.505 A0

(near cut off) to 0.70 10. Figure (4.20a) demonstrates large

130

- . - =- .22 .... =
a o 6 "o hl hlo o "o nmx mmx 10
b - 0.3 "o n - 2LII(4II/e) PM =- 5
r1 - -1.0 - 10.6 11 = 1 AMP
r2 - o.o (Az)12 = 0.25 Io 12 - 110 = 0 AMP
900
O
2
1 0 60°

 

 

 

 

6
180° = l I ‘U” IL 5 J. I 0
relative amplitude . /
of radiation I \/ ,,
field ‘ , ’\\ ’
i ‘b e

150 30°

0(AZ) d 1L130:2) =

 

dir
04%, 0.30),o 0
‘(MJ ' 120° 60
A(A2) - dir
dirD 0.2510 0
0.15;.o (A2)dir - 0.20;,o 90

Figure 4.19. E-plane radiation field patterns of a ten-element
slot array for various director spacings.

131

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132

amplitudes of slot fields with a strong standing wave component in
the elements of this array for a = 0.60 x0 and a = 0.70 x0. This
may be because a large dimension "a" results in a smaller decay rate
for evanescent modes in the backing waveguide, thus contributing to
evanescent mode coupling between elements and greater induced fields.
However, for waveguide dimensions of a/xo = 0.505 and a/xo 3 0.55,
the amplitudes of slot fields along the array are low to moderate
with a greatly reduced standing-wave component. Since for
al),o = 0.505 the dominant-mode wave in the backing waveguide is near
cut off, the evanescent-mode coupling is reduced. Figure (4.20b)
demonstrates a progressive phase delay in the aperture field. All
values of al),o appear to result in excitation of a slow wave, thus
end-fire radiation is expected for each of these cases.

Figure (4.21), which indicates the radiation field patterns

in the E-plane of this array, confirms the above conclusions. For

0.70, the back lobe amplitude is quite large.

the array with all0

In the case of a/Ao 0.505 and 0.55, the back lobe amplitudes are
reduced but remain significantly large. For a/xo = 0.60, both the
major lobe beam-width and the side (and back) lobe amplitudes are
optimized to reasonable values. Beam width of the major lobe is
very insensitive to variations in a/xb. Figure (4.21) illustrates
that beam scanning cannot be achieved (as was the case when this
array was excited by a traveling T310 mode incident wave) by varying
a/xo. Significant control of back radiation is, however, achieved
by variation of a/Ao or (Az)dir/xo (Figure (4.19)).

Figure (4.22) indicates the relative amplitudes and phases

of the aperture fields in the elements of a twenty-five element,

133

 

 

b = 0.3 k n = m = 10
O max max
F1 = -l.0 h = 0.22 xb PM = 5
1"2 = 0.0 n = ZLIIUIh/e) = 10.6 (A2)1 = 0.25 "o
(A2)dir = 0.30 90 I1 - l AMPS I2 - 110 = 0.0 AMPS
90°
0
120 600
0
150° 0
fl a - 0.505 "0 e
,/ 0.55 lo
.2 O O O 0‘
180° 1 ‘r : 1‘ 0.00IL 0:4 0%6 0? 1 00
relative amplitud-
of radiation '\~ \_\~“
field \\\\ '_ 0.60 x
o
9
o
0.70 *0 30

150

60

 

 

90°

Figure 4.21. Dependence of E-plane (m = ~fl/2) radiation field
patterns of a lO-element slot array upon width of

its backing waveguide.

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135

waveguide-backed slot array. The array parameters are shown in the
top left-hand corner of the figure. All the elements have the same

half-length, h - h

1 25 = 0.20 AD. This particular dimension for

element half length was chosen to avoid exciting a very strong
standing wave component in the aperture field of the array. The
width of the backing waveguide is a/Ao = 0.60 while the optimal
dimension for Az/xo was chosen to be 0.30. Figure (4.22a) in-
dicates an attenuating aperture field amplitude along the aperture
of the array. The standing wave (small amplitude) character of the
slot field (voltage) distribution is apparent from Figure (4.22a).
This is indicative of significant reflection of the travelling-wave
aperture field at the terminal end of the array.

Figure (4.22b) indicates a progressive phase delay in the slot
fields along the array aperture. It is observed that the slope of
the phase distribution plot for the array is nearly equal to that of
the light line, indicating an aperture field with a phase velocity
equal to the speed of light. Thus, an end-fire radiation pattern is
expected.

Figure (4.23), which indicates the E-plane radiation field
pattern for the array, confirms the above conclusion. The main lobe
is sharper than that for a ten-element array (Figure (4.19)). The
first minimum is 10.5db down and occurs at e = 25°. A large side
lobe is observed at e = 50°. The back-lobe has a (nearly) constant
relative amplitude of 0.33. The previous conclusion that a small
standing-wave component in the aperture field distribution (relative
amplitudes of slot fields) results in a baCk lobe of small amplitude

is thus again verified.

136

 

a = 0.6 A
o
b - 0.3 k0 h1 - h25 = 0.20 AC
c = 8.0 "o a = ZLn(4h/e) = 10.6 (02) = 0.30 kc
F1 8 -1.0 11 = l AMPS n = m = 10
max max
F2 = 0.0 I2 - I25 PM = 5
120° 60°
150° 30°
9
o 0.2 0.4 0.6 0.8 1.! o
180 I I I H: I I I I .0
relative amplitude of
radiation field
9
o
150° 30
120° 60°

 

Figure 4.23. E-plane radiation field pattern of a twenty-five
element, waveguide-backed slot array.

137

4.5.4 Frequency Dependence of a,ngi-Uda Slot Array:

Finally, the radiation and circuit properties of a ten-
element, waveguide-backed, Yagi-Uda slot array are investigated as
a function of its excitation frequency. The results of this section
are used to ascertain the operating bandwidth for this array.

Figure (4.24) indicates the E-plane radiation field patterns
of this array, with all the array dimensions indicated in meters.
It is observed that the radiation field deteriorates very rapidly
as frequency is varied from a center frequency of 3.0 GHz to 2.6 GHz
or 3.4GHz. The pattern beam-width of this array is identified as
approximately 400MHz.

Figure (4.25) indicates the variation of input admittance
to the driven element of the same Yagi-Uda slot array as a function
of its excitation frequency. This admittance curve has the same

behavior as determined by Nyquist and Mathur (25)

by a direct
numerical solution. It is observed that Bo increases monotonically
in the frequency range 2.0GHz < fI< 3.7GHz and drops to a lower
value for frequencies greater than 3.7GHz, while G0 has a peak.at
3.200Hz and again a sharper peak at 3.8GHz. In the frequency range
2.80Hz < f < 3.20Hz, the input admittance passes through resonance,

and presents a relatively low input conductance which can provide a

near match to practical transmission lines.

138

a . 0.06 m h1 = 0.025 m o = ZLn(4h2/e) = 10.6
b . 0.03 m h2 = 0.024 m “max . mmax - 10

c - 0.4 m h3 - hlo - 0.02 m PM.= 5

r1 - -1.0 (Az)ref . 0.025 m

r2 - 0.0 (Az)dir - 0.020 m

 

 

relative amplitude of

radiation field

I I l J
l I I I

0.2 0.4 0.6 0.8 1..

       

180

 
 
   

3.2

3.4 i 0

60°

 

O

90

Figure 4.24. Frequency dependence of the E-plane radiation field

pattern of a ten-element, waveguide-backed, Yagi-
Uda slot array.

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CHAPTER 5

EXPERIMENTAL INVESTIGATION OF WAVEGUIDE-BACKED
SLOT ARRAYS

5.1 Introductory Remarks:

 

In this chapter, the results of an experimental investigation
on the waveguide-backed slot array are presented and in some instances
correlated with the theoretical-numerical solution described in
Chapters 3 and 4. Typical relative amplitude and phase distributions
of the slot fields (at x = a/2) in the array aperture, E-plane
(m = -n/2) radiation patterns and input impedances were measured
for both the Yagi-Uda type slot array (excited by impressed current
of coaxial line feed) and the array excited by a dominant mode in-
cident wave in the backing waveguide. Similar measurements were made
for a ten-element Yagi-Uda slot array with backing waveguide width
at its maximum (a/Ao = max.) and the waveguide (interior) side of the
array covered by microwave absorber to simulate the theoretically
complementary strip dipole (and hence equivalent cylindrical dipole)
array.

Section 5.2 describes the anechoic chamber and experimental
set up. Section 5.3 presents the results forzuleight-element wave-

guide backed array excited by a TE mode incident wave and compares

10
these results with theoretical ones obtained using the numerical

solution described in Chapter 3. Section 5.4 presents the results

for a ten-element, waveguide-backed, Yagi-Uda array excited by means

140

141

of impressed currents and compares these results to the theoretical
ones obtained using the approximate analytical solution presented in
Chapter 4. Section 5.4 presents a Summary of the experimental and
theoretical results and attempts to explain the discrepensies that

exist between them.

5.2 Anechoic Chamber:§9d Experimental Set-up:

The experimental arrangement consists basically of an anechoic
chamber constructed with an aluminum image or ground plane forming
one of its walls. Experimental measurements are made upon the slot
array which is cut in the ground plane backed by a waveguide of
adjustable width, and subsequently radiates into the chamber. The
purpose of the anechoic chamber is to simulate a free half-space
environment. A photograph of the experimental anechoic chamber and
the slot array with its near field probing system is shown in Figure
(5.1). Figure (5.2) diSplays a photograph of instrumentation involved
for making various measurements in this experiment. Figure (5.3)
indicates the complete experimental set up, including the anechoic
chamber and the inter-connection of all the instrumentation which
was used. The backing waveguide was provided with movable side walls
such that the width "a" of the waveguide could be adjusted as shown
in Figures (5.2) and (5.3).

The anechoic chamber (dimensions of 8 ft. wide, 6 ft. height,
and 6 ft. deep) was constructed from appropriately covered wooden
frames. Its interior was completely covered by an aluminum ground
plane on one wall and with B.F. Goodrich type VHF-8 microwave absorbers
covering the remaining three walls as well as the floor and the

ceiling.

131

 

Figure 5.1a. Photograph of slot array cut in ground plane and
mounted in anechoic chamber (with near-field I whing
system and dipole receiving antenna).

 

F. AC '
igI e 5.1b. Photograph show1ng close-up view of the slot arrav
and the coaxial near-field probing svstem used to
measure the aperture field alon th -
. e arra
slot field distributions. 8 Y and the

 

 

 

 

I 'II! III-III! I III

 

Figure 5.2.

Photograph of various microwave instrumentation
(and part of the backing waveguide system) used

in making meaSurements on the waveguide-backed
slot array.

144

 

 

 

\\\__ microwave

 

 

anechoic
absorbers I
chambe r \ I i:
dipole receiving 6 ft
antenna .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

movable probe
1 KHz ‘g///f———
array
modulation l IZ///r——-aperture
t _waveguide 4 '
R F feed amplitude 50“
. . {-1 [‘2 ‘ detector "” load
Oscillator
adjustable TE
de backin 10
coaxia mo g VSWR
waveguide «OF——
V feed ’6 indicato
17* l-ilai
500 slotted line section
+
. i .42 .
VSWR 20db directional
indicatod coupler
‘ R.F.
! amplitude Phase
detector ‘ shifter Oscillator
500
load <03H7 l KHz
dulation

 

 

 

 

Figure 5.3. Anechoic chamber and block diagram of experimental set-up.

145

Figure (5.2) indicates the waveguide system that is used to
excite a TElo-mode wave in the backing waveguide and provide termina-
tions at its ends as well as the coaxial system used to excite (main-
tain an impressed current in the driven element of) the Yagi—Uda
type slot array. The latter arrangement is also shown (in its view
from inside the anechoic chamber) in Figure (5.1b). As shown in
Figure (5.2), two ninety-degree waveguide bends are used at either
of the terminal-ends of the backing waveguide, followed by two wave-
guide tapers to reduce the waveguide dimensions from (9.525 cm. X
4.445 cm.) I.D. for the adjustable backing waveguide to (7.2136 cm. X
3.4036 cm.) I.D., the latter being the standard S-band waveguide
dimensions. This particular backing waveguide size allows appropriate
guided wavelength adjustment for its T310 dominant mode wave and the
center frequency of its band for single, dominant mode propagation
is approximately 2.5 GHz. At the terminal end of the array, the taper
is followed by a standard S-band matched load (to effect a reflection
coefficient of F2 ='0), while at the initial (input) end the taper
is preceeded by a S-band waveguide slotted-line section with a wave-
guide probe; the input end of the slotted section is connected to an
R.F. Oscillator.

In the case of coaxial system excitation, as shown in Figure
(5.3), the R.F. Oscillator was connected through a 50-ohm coaxial,
slotted-line to a SO-ohm adjustable airline followed by a section of
50-ohm solid-jacketed cable (micro-coax). The SO-ohm solid-jacketed
cable passes through the ground plane (through a hole just below the
backing waveguide) and along a channel cut on the inside of the

ground plane which terminates adjacent to the center of slot number 2.

146

The center conductor of the solid-jacketed cable extends across this
slot at its mid-point. The outer conductor of the coaxial calbe lies
in (and is soldered to) the ground plane channel while its center
conductor passes transversely across the center of the slot and is
soldered to its opposite side to make a good electrical contact. The
details of the coaxial feed are indicated in Figure (5.1).

Figure (5.1) also indicates a coaxial, near-zone, electric-
field probe that can be moved horizontally as well as vertically to
measure the relative amplitudes and phases of the slot fields (voltages)
as well as the slot field distributions in individual array elements.

A movable, resonant-length receiving dipole was provided to
monitor the radiation field (E-plane) maintained by the slot array,
and was located a radial distance 106 cm from the center of the array
(which is approximately 8.83 ID at 2.50 GHz). The length of the
adjustable, coaxial air-line was adjusted such that the total line

I
length (solid-jacketed cable plus air-line) was approximately ‘9

2
such that its input impedance (measured by using conventional slotted
line techniques) was approximately equal to that of the driven element
of the array. A block diagram of the equipment arrangement used in
this experimental set up is shown in Figure (5.3).

The slot lengths, widths, and spacings in the experimental

lO-element array are as follows

h1 = 3.0 cm w = 28 = 0.4 cm (Az) = 3.0 cm
h 12
h2 = 2.88 cm —2 = 14.4 cm
3 4h2
h3 - th = 2.64 cm a = 2Ln(-;—) = 8.1 (Az)pq = 4.0 cm

...-I

for p f 1, q #

147

while the cross-sectional dimensions of the adjustable backing wave—
guide are a = 5.28 to 8.9 cm and b = 3.6 cm.

An excitation frequency of 2.5 GHz (correSponding to a free-
Space wavelength of IO = 12.0 cm) was used throughout the experiment.
The width of the backing waveguide was varied manually (utilizing
adjustable feed screws attached to the movable plates). Electrical

dimensions corresponding to the physical ones given above are there-

fore
= = = = .2
h1 0.25 X0 w 26 0.3333 k0 (Az)12 0 5 lo
h2
h2 = 0.24 *0 ‘2' = 1.2 A0 (Az)pq = 0.33 A0
4h2
_ = . = L —' = . ... ,
h3 hlo 0 22 I0 0 2 n( e ) 8 1 for p # 1 q + 1

a = 0.22 A0 to 0.74 Io, b = 0.3 Io.

The standing-wave-ratio (SWR) on the transmission system (wave-
guide or coaxial) driving the slot array was measured for both types
(incident waveguide mode or impressed current) of array excitation;
input impedances were not explicitly obtained. Since the input
terminal plane of the waveguide-excited array has an arbitrary loca-
tion, the input impedance is not well defined and therefore only the
input SWR was measured for comparison with the theoretically predicted
results. The impressed currents which excite the Yagi-Uda array are
maintained by the relatively long 0% 2.66 10) section of solid-jacketed
cable which is very lossy at 2.5 GHz. Due to the difficulty of
determining the electrical length and the attenuation and phase con-
stants of this lossy line section with sufficient accuracy, it was

found nearly impossible to transform the measured impedances looking

148

into the line section to those antenna input impedances at the input
point of the driven array element. Only the SWR on the 50-ohm air-
line slotted section which was terminated by the input to the lossy
feeder line are therefore, presented here and compared with the

theoretically predicted values.

5.3. Measurements on Waveguide-Backed Slot Array_Excited by7a
Dominant-Mode Incident Wave:

An eight-element, waveguide-backed slot array (usingedements 3-10
of the experimental set up described in the last section with elements
1 and 2 covered by conducting tape) excited by a TElO dominant-mode
incident wave was investigated experimentally. The slot voltage dis-
tributions, the relative amplitudes and phases of the slot voltages
in the elements along the array aperture, and the E-plane radiation
patterns were measured. Measurements on the ten—element experimental
slot array were not made since the lengths of the elements 1 and 2
were different from those of elements 3-10 and the presence of the
coaxial feeder cables in slot number 2 of the array (which is to be
used in the Yagi-Uda slot array measurements) will significantly load
that slot and affect its field distribution. Therefore, to achieve
an eight-element array of equal-length slots, the first two elements
were covered by conducting aluminum tape to reduce their aperture
fields to zero. Slots 3-10 of the experimental set up now forms the
required 8-element, waveguide-backed slot array with each slot of half-

length h/Ao = 0.22 and an element spacing of (A2)“,o = 0.33. The
A

.2

30’

thickness parameter is Q = 2&n(éh) = 8.1.
6

width of each slot is at an Operating frequency of 2.5 GHz; the

149

Figure (5.4) indicates the relative amplitudes and phases of
slot voltages in the elements of the 8-element array for various
backing waveguide widths "a"; both measured experimental results and
the results obtained analytically by the numerical solution (as out-
lined in Chapter 3) are presented. It is observed from Figure (5.4a)
that the relative amplitudes of measured slot voltages (dashed lines)
in slots 1 through 5 agree quite well with the theoretically predicted
results (solid lines) for a backing-waveguide of width a/Ao = 0.6.
Near the end of the array, the comparison is relatively poor. This
can be explained in two ways. First, in the experiment, when
a/Xo‘< 0.74 the movable plates that vary the backing-waveguide width
present a discontinuity at the input and output ends of the array,
while in the analytical solution both F1 and F2 were assumed to
be zero. Secondly, the numerical results from the analytical solution
are not absolutely accurate since they were computed using only 10-
partitions along the slot axes and only five terms were retained in
each of the double Fourier series when evaluating 01. The standing-
wave due to the terminal-end effect is, however, apparent in both
experimental and theoretical results. For al),o = 0.505, the agree-
ment between the theoretically predicted and experimentally measured
amplitude distributions is relatively poor although the general trends
agree. This is probably due to the fact that the TE10 mode in the
backing-waveguide is near cut off and its reflections at the dis-
continuities mentioned above becomes critical.

Figure (5.4b) indicates very good agreement between the

theoretically predicted results and experimentally measured ones for

the relative phases of slot voltages in the elements along the array

150

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151

aperture. It is observed that for a/)\o = 0.505 a fast wave is
excited in the’array aperture while a Slow wave is excited when

a/xo = 0.6. This confirms the feasibility for varying the phase
velocity of the traveling wave aperture field by adjusting the wave-
guide width. The Slow wave for a/xo = 0.6 is expected to produce
endfire radiation, while off-endfire radiation is expected for the
fast wave obtained with a/xo = 0.505.

Figure (5.5) indicates a comparison between experimentally
measured and theoretically predicted slot field amplitude distributions
for several individual elements of the array. They are observed to
compare very closely, being essentially sinusoidal distributions in
each case.

Figure (5.6), which indicates the theoretical and experimental
E-plane radiation field pattern for all,o = 0.505 and 0.6 confirms
the above predicted nature of the radiation fields. Again, the agree-
ment between theoretical and experimental results is very good for
a/Ao = 0.6 although the endfire radiation is obtained, the lobe
at 0 = 00 is not the main beam of the radiation pattern, the latter
being at e = 480. This behavior is due to the fact that in the
experimental set up (Az)/),o = 0.33 is relatively large. In Chapter
3 (section 3.5) it was demonstrated that to achieve endfire radiation
from the main beam, an optimum slot Spacing is (A2)“,o = 0.1. There-
fore, an off-endfire main beam radiation for (An/k0 = 0.33 is expected.
The experimental slot Spacing of (A2)“,o = 0.33 was originally chosen
for the Yagi-Uda array design, and could not be readily modified. It
was also pointed out in Chapter 3 that by changing the width of the

backing waveguide (a/xo) it is possible to achieve beam scanning.

152

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S2 = 2m(4h/e)
= 8.1
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150°

“O-O-O-O- a

-A-A-A-A- a

Figure 5.6. Theoretical and experimental radiation patterns

relative

amplitude 0 f

radiation field

120

0.6A
0

0. 505 A0

--<a~..€L

 

experimental

data

 

‘ o z ,I .
“=§!!!5?IL4 0.6
\

    

for a 8- element slot array with two different
backing waveguide widths.

" 0.8 1.

154

This is confirmed in Figure (5.6). The main beam shifts from 9 = 480

° (700 for experimental results) for

for a/xo = 0.60 to 0 = 80
a/xo = 0.505. It should be noted that near cutoff (a/xo = 0.505)
the E-plane radiation pattern has the same shape for both in numerical
results and experimental measurements even though an angular Shift is
evident between the two patterns. This shift is likely due to the
difficulty in precisely adjusting a/xo experimentally; a slight
error results in a modified aperture field phase velocity.

Table (5.1) investigates the circuit pr0perties of the eight-
element, waveguide-backed slot array with dominant-wave excitation.
It is observed that the SWR produced by the array load on the driving
waveguide is not excessively high, even at a/xo = 0.505 where the
width of the array's backing waveguide places it near cutoff for the
dominant TElO mode wave; a reasonably good match (at the input terminal
of the array) is therefore indicated between the driving waveguide
and the antenna System for various widths of the backing waveguide
(scan angles). As mentioned earlier, the input impedances to this
array depends upon the choice of input terminal plane, and are thus
somewhat arbitrary. For this reason, only the driving waveguide SWR's
computed from the corresponding input impedances are presented. A
favorable comparison between theoretical and experimentally measured

SWR'S is indicated.

5.4. megsurements on Yagi-Uda, Waveguide-Backed Slot Arrgy;

In this section, a ten-element, waveguide-backed Yagi-Uda
slot array is investigated. The array of ten transverse slots (cut
in a large ground plane and backed by a waveguide) consists of a

driven element, one reflector and eight directors. The reflector

155

 

 

 

 

a/ko 0.505 0.6
(SWR)EXP 6.3 1.45
(SWR)THY 5. 0 2.1

 

 

 

 

Table 5 .1. Experimental and theoretical driving waveguide SWR (8-slot

array) for two backing waveguide widths (b - 0. 3x0,
_ -o, 52: 21n(=4h/€) 8.1)

P

1

=1"

2

h=0.22x ,
O

156

length is hllxo = 0.25, the driven element length is hZ/xo = 0.24,
while all directors (slots 3 to 10) have the same length hD/xo = 0.22.
The reflector is situated 0.25 he in front of a Short-circuited end
of the backing waveguide (F1 = 0) and 0.25 X0 in back of the driven
element, while the spacing between the driven slot and all adjacent
directors is (Az)dir/),o = 0.33. The end of the waveguide at the
terminal end of the array iS terminated in a matched load. By removing
the backing waveguide, this array becomes complementary to a ten-element,
Yagi-Uda array of equivalent cylindrical dipoles. The width parameter
(Q) for Slot 2 is equal to the value a = 2&n(4h2/e) = 8.10.

Figure (5.7) indicates the experimentally measured and theo-
retically calculated (new array theory as described in Chapter 4)
results for a ten-element Yagi-Uda Slot array radiating into free-
space on either side of the ground plane. Significant array parameters
are Shown in the top left-hand corner of the figure. Figure (5.7a)
compares the relative amplitudes of slot voltages against slot number
for experimental data (dotted lines) and theoretical results (solid
lines). The theoretical results are obtained for both 0 = 24n(4h2/e) =
10.6 and 8.10. It is observed that the experimental data (0 = 8.10)
and the theoretical results (for n = 8.10) do not agree well, while
theoretical results for n = 10.6 follow the experimental ones
relatively well. Since, from the results presented in Chapter 4 (com-
parison with published King-Sandler theory) it is clear that the new
array theory predicts accurate results for narrow slots (0 = 10.6),
it can be concluded that the new array theory cannot accurately pre-
dict the array behavior for wider slots (0 = 8.10, for instance).

This is due to the fact that the peaking properties of the kernels

157

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158

in the system of integral equations (upon which the approximate array
theory is based) no longer exist with the behavior described in Chapter
4. It is therefore expected that the results of experimental measure-
ments will not correSpond well with the theoretical results for

0 = 8.10 but might compare more closely with those for n = 10.6.

Figure (5.7b) indicates a progressive phase delay for the
field along the array aperture. It is seen from the figure that a
slow wave is excited in the array aperture for all cases and that
experimentally measured and theoretically predicted results are in
good agreement. These aperture fields should lead to an endfire
E-plane radiation field pattern.

Figure (5.8) indicates the E-plane radiation field pattern
of this array. AS noted above, the experimental and the theoretical
results (for n = 10.6) agree very well while for n = 8.1, the
theoretical results predict a backfire pattern with a main lobe
maximum at 0 = 1800 rather than at 0 = 0°. This can be explained
by the fact that due to the excessively slow aperture field wave and
its particular amplitude distribution the phases of the slot voltages
excited in the array are such that they produce a field maximum in
the e = 1800 direction instead of at e = 0°. This assertion is
confirmed by a Simple calculation. Again, the non-validity of new
theory for wide Slots with n = 8.1 and its relative accuracy at
n = 10.6 is confirmed.

Figure (5.9) indicates the relative amplitudes and phases of
slot fields along the same Yagi-Uda slot array backed by a waveguide
of width al),o = 0.6. All other array parameters are the same as

those of the array described above. The agreement between experimentally

159

 

 

 

 

h1 a 0'25 A0 (Az)ref . 0'25 X0 nmax ' mmax = 1°
h2 - 0.24 k0 (1112)dig = 0.33 20 PM.= 5
h3-h10=0.22),0 G =G MM=20
o
90
120°
150°
0
180° I I
relative amplitudes
of radiation field
0
Q=8.l
150'

-4)—-<}- experimental 120°
data

-4l——1F-fl = 8~1

 

theory

Figure 5.8. Theoretical and experimental radiation patterns
for a ten-element, Yagi-Uda Slot array.

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161

measured and theoretically predicted amplitude distributions (for

n = 10.6) in Figure (5.9a) is not very good; the deviation near the
end of the array can be explained by reflections from the waveguide
adjusting plates as discussed in section 5.3. Figure (5.9b) indicates
the relative phases of slot voltages against slot locations. It is
observed that experimental and theoretical results agree well and
indicate a slow wave along the array aperture.

Figure (5.10) indicates a comparison between experimentally
measured and theoretically predicted slot field amplitude distributions
for several individual elements of the array. They are observed to
compare quite closely, being essentially sinusoidal distributions in
each case. Some discrepancies (for example in the plot for slot
number 1) that exist between theoretical and experimental results are
probably due to unavoidable experimental errors.

Figure (5.11) compares the experimental and theoretical E-
plane radiation field patterns for the above array. The agreement
between theory and experiment is quite good. This further confirms
the conclusion that the new array theory accurately predicts the array
behavior for narrow slots (0 2 10.6) while for wider slots (0 = 8.10)
the theory fails to predict accurate results.

Finally, the circuit properties of the Yagi-Uda slot array
are investigated in Tables (5.2) and (5.3). Experimental and
theoretical values for the input admittance could not be compared
directly due to uncertainties in the values of the attenuation and
phase constants of the lossy coaxial feeder line and its exact
electrical length. Since the theoretically predicted admittances for

the Yagi-Uda array without waveguide backing were found to agree with

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150° 30°

120° 60

_<3_cy._.experimental
data

 

.—o——o——- theory, 0 = 10.6

Figure 5.11. Theoretical and experimental radiation patterns
for a ten-element, waveguide-backed, Yagi-Uda
slot array.

0

O

(D

(D

 

1.! I... {Iii I

Table 5.2.

Table 5.3.

164

 

 

 

Q
(SWR)THY 14.0 10.5
(SWR)EXP 7.0 -

 

 

 

 

 

Experimental and theoretical SWR's maintained by
a 10-e1ement, waveguide-backed, Yagi-Uda slot
array (a = 0.6 NO, b = 0.3 A , h1 = 0.25 A ,

'-'-' .2 -"—' . :=
h2 0 4 A0. hp 0 22x0. (82)ref
(Az)dir = 0-33 Kb, F1 = '1-0, T2 = 0.0) on its

50-ohm coaxial exciting system.

 

 

 

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(SWR)THY 12.0 11.0
(SWR)EXP “ 9.25 -

 

 

 

 

 

Experimental and theoretical SWR maintaineg by
a lO-element, Yagi-Uda slot array (G1 = G ,

= 02 = 02 = .22 ,
h1 O S )‘O, hz. 0 4 x0, hD 0 X0

(A2)ref = 0.25 lo, (Az)dir = 0.33 K0) on its

SO-ohm coaxial exciting system.

165

12
published results from the King-Sandler( ) array theory (and experi-

ments) the experimental measurements were concluded to be faulty.
Thus only the SWR maintained on a 50-ohm coaxial air-line section by
the load impedance consisting of the input impedance to the lossy
feeder line was considered. These results for the array with and
without waveguide backing are presented in Tables (5.2) and (5.3).
For the reasons noted earlier, the theoretical SWR's for Q = 10.6
compare best with experimental measurements for a slot array with

a 3 8.1.

CHAPTER 6

SUMMARY AND CONCLUSIONS

A numerical and an approximate analytical method are applied
in a theoretical investigation of the circuit and radiation properties
of a waveguide-backed slot antenna array. The voltages excited in
elements (slots) of this array are determined as the solution to a
coupled system of integral equations by this numerical-approximate
analytical approach. In terms of these slot voltages the radiation
field maintained by the array as well as its input impedance (or
admittance) to the driven element are subsequently evaluated. An
eXperimental investigation of the waveguide-backed slot array was
also conducted. The results of the analytical solution and those
from the experiment compared favorably. It was clearly demonstrated
that the results from the approximate analytical solution, when re-
duced to several Special cases, compared favorably with previously
published results. A comparison of analytical results for the special
cases with the experimental data was again good.

A basic theory for the waveguide-backed slot array was
formulated in Chapter 2. The EM boundary value problem is formulated
theoretically in terms of a system of Hallén-type integral equations
for the unknown voltaged induced in the array elements (slots). In
deriving this system of integral equations, the slots were assumed

to be thin and narrow such that the longitudinal components of the

166

qu‘ —~

167

slot field could be neglected. The slot field was then related to
the slot voltages utilizing a quasi-static field approximation valid
for narrow slots. The system of integral equations takes into
account two modes of array excitation: first by an incident, dominant
TE10~mode wave in the backing waveguide, and secondly by means of
impressed currents maintained in one (or more) of the slots. The
general eXpressions for the radiation field maintained by the as yet
unknown slot voltages are presented in the last section of Chapter 2.

Chapter 3 presents a numerical solution to the coupled system
of Hallén-type integral equations for the array excited by the in-
cident TRIO-mode wave. By eXpanding the unknown (induced) slot voltages
in a series of pulse functions and subsequently point-matching the
integral equations, the unknown eXpansion coefficients are evaluated.
The input impedance to the backing waveguide was evaluated as the
ratio of total transverse electric field (incident and scattered by
the slots) associated with the TRIO-mode wave to the total transverse
magnetic field of the same wave at an arbitrarily located terminal
plane. The radiation field maintained by the slot array is evaluated
in terms of the (now determined) coefficients for pulse functions
series expansions of the slot fields.

Section 3.6 presents all the numerical results calculated
by the point~matching numerical solutions for the array with TElO-
mode excitation. These results include relative amplitude and phase
distributions of the slot voltages excited in the array elements as
well as the E-plane and H-plane radiation fields maintained by the
array for various array parameters. The circuit properties of this

array, as described by the backing waveguide standing wave ratio,

were also investigated.

168

It is found from the numerical results that voltage amplitude
distributions in the elements of the waveguide backed slot array,

excited by a dominant TE -mode wave in the backing waveguide, are

10
very nearly sinusoidal along the slot axes; this suggests the
feasibility of applying an extension of the King-Sandler dipole array
theory to this problem. The relative phases of slot voltages were
found to be a sensitive function of various slot Spacings, (Az)/xb,
slot lengths, h/Ao, and the backing waveguide width, a/xo. It can
be concluded from the results presented in section 3.6 that optimal
array parameters, to achieve directive end-fire radiation, were
a/xo = 0.6, (A2)“.o = 0.1 and h/xo = 0.22 for a slot width
Specified by 0 = 2Ln(4h/e) = 10.6. While one can achieve some
degree of beam-scanning capability by departing from various of these
optimal dimensions, it is found that by suitably changing ah,o the
main lobe of the E-plane radiation field pattern can be scanned
from a near broadside orientation for a/xo = 0.505 (near cutoff
for TElo-mode wave) to an endfire orientation for a/ko = 0.70. The
backing waveguide SWR was found to be minimized for h/ko = 0.22,
Az/xo = 0.10 and a/xo = 0.70.

Chapter 4 presents an approximate analytical solution for
the waveguide-backed slot array excited by impressed currents in one
or more of its elements. This solution is an extension of the King-
Sandler dipole array theory. For a ten-element array it is necessary
to invert a matrix of order (10 X 10), whereas in the direct numerical
solution described above a matrix of order at least (100 x 100) must

be inverted to obtain adequate accuracy for the same array. Thus,

utilizing the approximate analytical solution results in a considerable

169
saving in computational time. (For example, average computational

times for a lO-element, waveguide-backed slot array, using numerical
and new array theory are 528 seconds and 215.5 seconds respectively.)
This new approximate theoretical solution was tested by considering
several special cases. It was observed that this theory predicts
correct results for a single slot radiator (cut in an infinite ground
plane) with either unbounded free-space on both sides or a backing
cavity behind one side of the slot. Results for this case were com-
pared to those for a complementary dipole in free-space and previously
published theoretical and experimental results for a single, cavity-
backed slot. The new array theory also predicts the correct results
for an eight director (ten-element) Yagi-Uda Slot array (cut in a
ground screen with unbounded free-Space on either side), which is com-
plementary to a Yagi-Uda dipole array. It is also observed (by com-
parison with Galej's variational results) that this theory cannot pre-
dict correct slot voltage amplitude distributions for slots backed by

shallow cavities, as indeed one might expect to be the case.

The optimal dimensions for a five-element Yagi-Uda, waveguide-
backed slot array to produce a directive, endfire radiation field
were determined to be a director element spacing of (Az)dir/xo = 0.25
and a director half-length of hD/xo = 0.20 for a backing waveguide
of width a/xo = 0.6. For a ten-element Yagi-Uda slot array, the
above parameters are essentially unchanged except for (Az)dir/),o
which is now optimized as 0.30. This value of director element Spacing
compares quite favorably with the optimal Spacing of director elements
in a Yagi-Uda array of cylindrical dipoles. It was found that a
large standing-wave component was always present in the aperture field
of the array for relatively large director lengths, resulting in

significant back radiation (large back lobe in the E-plane radiation

F4313 uni-Pam AF l-‘na avvauv\

170

A ten-element, waveguide-backed Slot array with all elements
of equal length and the first element excited by an impressed current
was also investigated. This array was expected to simulate the array
investigated in Chapter 3. However, the optimal array parameters
were found to closely resemble those of the Yagi-Uda type slot array

rather than those of the array excited by a TE -mode incident wave,

10
indicating that the mode of excitation of the array significantly
changes its behavior and its optimum element spacings. It was observed
that a large standing wave component in the relative amplitude dis-
tribution of Slot fields along the array invariably results in a
large back lobe in its E-plane radiation field pattern.

Investigating the input admittance to the driven elements of
five- and ten-element Yagi-Uda slot arrays, it was observed that the
admittance passes through resonance when the driven-element length
lies between 0.22 ho to 0.23 x0. This resonant admittance presents
a relatively low input conductance, which can provide a near match
to practical transmission lines. The variation of input admittance
for a ten-element slot array is quite similar to that for an isolated
slot cut in a ground plane located in an otherwise unbounded Space.

The frequency dependence of the E-plane radiation field for
a ten-element Yagi-Uda slot array is investigated in section 4.5.4.
This investigation reveals that the E-plane field pattern degrades
rapidly as frequency is varied from a center frequency of 3.0 GHz
to 2.6 GHz or 3.4 GHZ. Therefore, the pattern banddwidth of the
array appears to be approximately 400 MHz. The dependence of input
admittance to the driven element as a function of frequency indicates

a resonance near the center frequence of 3.0 GHz.

171

In all the results summarized above it was noted that the
E-plane radiation field patterns have sharper main lobes (narrower
beam widths) as the number of array elements is increased. A twenty
five—element waveguide-backed slot array maintains a radiation
pattern having a much sharper main lobe and a relatively low back
lobe (if hl),o = 0.20) compared to the pattern for a ten-element
array.

Chapter 5 indicates a comparison between various experimental
results and theoretical predictions for both the dominant-mode-wave
excited, waveguide-backed slot array and the arrays excited by
impressed currents. The experimental and theoretical results for
an eight-element waveguide-backed slot array excited by a TElo-mode
incident wave agree very favorably. The analytical predictions
for the Yagi-Uda slot array do not agree as well with the experi-
mental results. This can be explained by the observation that the
new, extended King-Sandler array theory cannot predict accurate
results for wide slots (0 = 8.1) such as those used in the experimental
array.

In conclusion, the direct, numerical point-matching solution
to the system of coupled integral equations for the Slot voltages
appears to provide an accurate analytical means to investigate both
the radiation and circuit properties of waveguide-backed slot arrays.
The new, approximate solution for the waveguide-backed Yagi-Uda slot
array provides an efficient but somewhat less accurate means to
analytically examine its radiation and circuit properties. It was
found that beam-scanning can be achieved with a waveguide-backed

slot array excited by an incident, dominant TE -mode, by varying

10

the backing waveguide width. No such scanning capability can be

172

achieved with the Yagi-Uda, waveguide-backed Slot array excited by
an impressed current in its driven element. The new, extended King-
Sandler theory provides an analytical solution for a finite wave-
guide-backed slot array and also provides sufficient computational

efficiency to allow calculations for large arrays within reasonable

time limits.

REFERENCES

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

REFERENCES

Burton, R.W. and R.W.P. King, "An experimental investigation
of currents on a Yagi array of slot antennas on planar and
curved surfaces," IEEE Trans. Ant. Prop. £514, 4, 451-454
(July 1966).

Mailloux, R.J. , "Excitation of a surface wave along an in-
finite Yagi-Uda array," IEEE Trans. Ant. Prop. Ag;l§, 5, 719-
724 (September 1965).

Mailloux, R.J., "The long Yagi-Uda array," IEEE Trans. Ant.
Prop. QELLE: 2, 128-137 (March 1966).

Coe, R.J. and G. Held, "A parasitic slot array," IEEE Trans.
Ant. Prop. A§;1§, 1, 10-16 (January 1964).

Hyneman, R.F., "Closely-Spaced transverse slots in rectangular
waveguide," IRE Trans. Ant. PrOp. 43:1, 4, 335-341 (October 1959).
Elliott, R.S., "Serrated waveguide - part I: theory," IRE

Trans. Ant. Prop. 42:5, 5, 270-275 (July 1957).

Kelly, K.C. and R.S. Elliott, "Serrated waveguide - part II:
experiment," IRE Trans. Ant. Prop. M, 5, 276-283 (July 1957).
Galejs, J., "Admittance of a rectangular slot which is backed

by a rectangular cavity," IEEE Trans. Ant. Prop..ég;ll, 2,
119-126 (March 1963) .

Galejs, J., "Hallen's method in the problem of a cavity backed
rectangular slot antenna," Radio Science (J. Res. of NBS),

_6_?_1_)_, 2, 237 (lurch-April 1963).

Harrington, R.F., "Matrix methods for field problems," Proc.

IEEE .5_5_, 136-149 (February 1967).

King, R.W.P. and 8.8. Sandler, "The theory of broadside arrays,"
IEEE Trans. Ant. Prop. 9.13:2, 3, 269-275 (May 1964) .

King, R.W.P., R.B. Mack, and 8.8. Sandler, " Array of cylindrical
dipoles," Cambridge, Chapter 6, 181-232 (1968) .

173

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

174

Silver, 8., "Microwave antenna theory and design," Dover
Publications, Inc., New York, Chapter 9, 291-293 (1965).
Collin , R.E., "Foundations for Microwave Engineering,"
MbGraw Hill Book Co., Chapters 2 and 3, 11-143 (1966).

Hong, M.H., D.P. Nyquist, and KJM. Chen, "Investigation of
open-cavity radiators and backfire antennas part I: open-
cavity antennas," AFCRL report, AFCRL-70-0361, (July 1970).
Jackson, J.D., 'Classical Electrodynamics," John Wiley and
Sons, Inc., New York, Chapter 9, 268-308, (1967).

Harrington, R.F., "Field computation by moment methods," The
Macmillan Company, New York (1968).

Booker, H.G., "Slot aerials and their relation to complementary
wire aerials," J. IEE (London), 22, Pt. IIIA, 620-626 (1946).
King, R.WpP. and C.W. Harrison Jr., "Antennas and Waves: A
Modern Approach," The M.I.T. Press, Cambridge, Mass., Chapter
13, 657-688 (1969).

Collin , R.E. and F.J. Zucker, "Antenna theory part I," McGraw
Hill Book.Co., (1969).

King, R.W.P. and T.T. Wu, "Currents, charges and near fields
of cylindrical antennas," Radio Science J. of Res. NBS/USNC-
URSI, 622, 3, 429-445 (March 1965).

Watson, G.N., "A treatise on the theory of Bessel functions,"
2nd ed., Cambridge Univ. Press, Cambridge, Eng., (1952).
Abramowitz, M. and I.AJ Stegun, "Handbook of mathematical
functions," Dover Publications, Inc., New York, 495-499 (1965).
King, R.W.P., "The theory of linear antennas," Harvard Univ.
Press, Cambridge, Mass., 141-204 (1956).

Nyquist, D.P. and S.P. Mathur, “Analysis of a parallel array
of waveguide or cavity backed rectangular slot antennas," pre-
sented at the 1972 USNC/URSI-IEEE Spring Meeting,'Washington,
D.C., (April 13-15, 1972).

APPENDIC ES

APPEND 1X 1

APPROXIMATE EVALUATION OF AN IMPROPER INTEGRAL

The double integral

kafflxk/Z zkaz/Z e—jkbka
g __ I '
1 j‘ 1‘ 12,2 211ka dz dx (1.1)
"km'm‘k/Z zk

where

 

ka = <ka - x02 + (2k - 202.

and Axk and A2 are small quantities (i.e., koAxk1<< 1 and

kbAz << 1), is improper Since its integrand has a singularity when

x - ka and z = zk. This integral can be evaluated analytically

as outlined below.
Since kOka1<< 1 in this case, by expanding the exponential
into a power series and retaining only its leading two terms, equation

(I.l) can be approximated as

 

 

 

ka+Axkl2 213“” (1 - jk Rm)

I'd I I o j—- dz'dx'
x -Ax z -Az/2 ZWka
km k/2 k

or

Emmi/2 213W 2 Jko(Axk)(A2)

1m i dz'dx' - . (1.2)

211R :11

"km’Axk/z 16“" 2 k“

If the change of variables u = ka - x' and v - zk - z'

is made, equation (1.2) can be rewritten as

175

177

or

 

H
t\
:1 [N

"j ko(AXk) (AZ) Az Axk Ax
{ 4 +2 Ln[(Az)+ /1+(Az)]+
its), A5. + AL 2
2 mm) 1 + (Mk) 3}. (1.6)

APPENDIX II

EVALUATION OF DEFINITE INTEGRALS

Consider the integral

2 +3 tynm(zk+z ')
1 = j 8 dz' (11.1)

 

 

= + .
where Ynm anm janm’ and anm and Bnm are constants
Let the following change of variables be made,
(2' - 2k) = 6 cos 6.
then dz' = -e sin 9 d9, and equation (II-1) can be rewritten as
n eiynm(zzk +'€ cos 9)
Il—J‘ esine esinede
0
or
ifiv z n 1y 6 cos 9
I1 = e “m k g e “m d9 . (11.2)

Expression (11.2) can be evaluated for complex Ynm but here it
will be evaluated for the following two Special cases:
0 O , = ’ d
(a) yum pure real i e Bnm 0 an

(b) yum pure imaginary i.e., anm = 0.

Case (a): v

= a +'j0, then
nm nm

178

179

120 zk n fig 6 cos 9
I = e “m I e “m d9 . (11.3)
0
(22)

From Watson , it is observed that

11'
£ eix cos 0 dx = fl 10(X),

where 10(x) is the modified Bessel function of the first kind of
zeroth order with argument x. Thus, letting x = (anme), expression
(11.3) becomes

i?a z

11 = e “m k[n 10(anme)]. (11.4)

Case b : y

nm = 0 + jenm’ then

+'2 z n +' c S
I = e—J Bnm k e-Jenme. 0 9 d9 ° (II-5)
1
Again, by watson (22), it is observed that
11’
g 63" °°S 9 d6 = 11 .1000.

where Jo(x) is the Bessel function of the first kind of zeroth
order with argument x. It muSt be noted, however, that JO(X)

is an even function of x such that Jo(x) = Jo(-x). Therefore

n
+
8 e_jx cos 9 = n Jo(x)’
and expression (11.5) becomes
2:128 2k
= nm
11 e [n Jo(8nme)]. (11.6)

Consider the following definite integral

 

12 = i dz' , (11.7)

where = +- and and are constants.
Ynm Q'nm jBnm’ anm Bnm

Making the following change of variables

 

zk - z = 6 cos 9
then -dz' = '6 sin e de, and expression (11.7) can be rewritten as
c s
. ems . 91
= . ' d
I2 g s sin 9 6 Sin 6 0
or
fl/Z 1N 6 cos 9
12 = 2g e “m de . (11.8)
Expression (11.8) can be evaluated for the following two
cases:
(a) Ynm pure real i.e., Bnm = 0, and

(b) Ynm pure imaginary i.e., anm = 0.

Case (a): Vnm = anm‘+ jO, then

TT/Z in! 3 cos 6
12 '3 2£ e nm d9

n/Z n/Z
= 23 cosh(anme cos s)de‘i'Zg sinh(anme COS 8)d9o (II-9)

(23):

From the Handbook of Mathematical Functions

n/Z
2£ sinh(x cos 9)d9

fl L0(x), and

n/Z
2& cosh(x cos s)de

n 10(X).

181

where Lo(x) is the modified Struve function of zeroth order with
argument x and 10(x) is the modified Bessel function of the first
kind of zeroth order with argument x. Therefore, eXpression (11.9)

can be written as

12 = "[Io(°‘nm€) : Lo(oznme)] . (11.10)

Case (b): yum = 0 + jenm’ then

TT/Z flfinme cos 9

12 = 2 e de:

01'

n/Z n/2
12 = 2% cos(Bnme COS 9)de i ng sin(anme cos 9)d9. (11.11)

From Handbook of Mathematical Functions (23):

n/2
2f sin(x cos 0)d0
o

n H (x), and
o

n/Z
2f cos(x cos e)de
o

nJ(X).
0

where Ho(x) is the Struve function of zeroth order with argument
x, and Jo(x) is the Bessel function of zeroth order with argument

x. Using the above results, expression (11.11) can be written as
= + “H . 11.12
12 "[Jo(Bnm€) _ J c)(Bnm'sfj ( )

In summary,

+
Zke
11-£_€
k
and
+
Zke
12=1_
zke

182

 

 

 

" nm k _
I (anm€)'°° for Vnm‘anmrjo’

ijzanmzk

 

Jo(8nme)... for vnm=0+janm,

rn[10(a’nmc)iLo(anme)]... for ynm=a +j0,

nm

 

‘~n[Jo(Bnm€)ino(anme)]'°° for Ynm=o+JBnm°

”WU/N!

31