CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
THE ACTIVE REFLECTION COEFFICIENTS OF CERTAIN PHASED ARRAYS
.,
A comprehensive report submitted in partial satisfaction
of the requirements for the degree of Master of Science in
Engineering
.. by
Yosef Klein
The Thesis of Yosef Klein is approved:
B.
Hoerne1~
California State Un-iversity, Northridge
TABLE OF CONTENTS
Page
LIST OF VARIABLES .
v
LIST OF FIGURES
viii
LIST OF TABLES
xii
ABSTRACT
xiii
Chapter 1 - INTRODUCTION
1
Chapter 2 - ADMITTANCE MATRIX THEORY
5
Chaptcl~
3 - ARRAY RADIATION . . . . . .
3.0 General considerations and definitions
14
14
3.1
Radiated field from single aperture .
15
3.2
Radiation from an array of apertures
19
3.3
Realized gain patterns
23
3.4 Efficiency
Chapter 4 - f"'UTUAL
4.1
25
.
AD~1ITTANCE
MATRIX
General discussion
4.2 Mutual admittance two slots
26
28
Chapter 5 - COMPUTATIONAL METHODS
35
5.1 Admittance matrix . • .
35
5.2 Active admittance and reflection coefficent
41
5.3 Array gain patterns . . .
45
Chapter 6 - RESULTS AND CONCLUSIONS
52
REFERENCES
95
APPENDICES
A.
THE ROLE OF THE GENERATOR IMPEDANCE . . . . . . . . .
97
Page
B.
THE GREEN'S FUNCTION "G" • • • • •
100
c.
COMPUTER PROGRAM TO CALCULATE Y12
112
0.
COMPUTER
PROGRAr~
TO CALCULATE THE ACTIVE AotHTTANCE
iv
118
List of Variables
Number of waveguide rows
N
Number of waveguide columns
-E
E)ectric field vector
H
Magnetic field vector
Modal electric field vector
ii
Modal magnetic field vector
v
Column matrix of aperture mode voltages
I
Column matrix of aperture modes currents
v
Aperture modal voltage
i
Aperture modal current
a
Waveguide width (broad wall)
b
Waveguide height
(narrow wa11)
Circumferential unit vector in cylindrical coordinates
Unit radius vector in cylindrical coordinates
Waveguide aperture area
Diagonal matrix of the real part or the generator
admittance
Diagonal matrix of the generator admittance
Vn
Aperture modal voltage of the n-th waveguide
in
Aperture modal current of the n-th waveguide
Square root of the incident power in the n-th
waveguide
Square root of the reflected power in the n-th
waveguide.
v
1\ A
A
x,y,z,
Unit cartesian vectors
Pav
Average power delivered to the antenna
r
Reflection coefficient
Element excitation delay angle
8o
Scan angle
Yac
Active admittance
y+
Column matrix of the square root of
cident power from the generators
y-
Column matrix of the square root of the net reflected power to the generators
y
Admittance matrix of the array
M(r')
r·1agneti c vector current density (source)
r'
Radius vector from origin to a source point
A'
Area of the source aperture
r
Radius vector from origin to a field point
k
Free space wave number
~he
net in-
permittivity of free space
Radian frequency
Am
Free space electric vector potential
Zo
Impedance of free space (377 ohm), the ratio of
the electric to the magnetic field in free
space of a plane wave
n
Unit vector norma 1 to the sut'face
Ea
Aperture electric field
Array factor
p
Poynt·i ng vector
e
Co-latitude angle (Fig 3.1)
vi
G(8,¢)
Array realized gain factor
n
Array Efficiency
Y1,2
~1utua 1
Yi,j
admittance between two adjacent e1ements
of the array
t~utua1
admittance bet\t1een element i to j of the
Array
g(S,S,E:,R)
Dyadic.Green's function for a magnetic field on
conducting cylinder
s
Length of the geodesic betv.1een b:o points
B
Angle between the geodesic and the ¢ direction
e:
GTD distance parameter
R
Cylinder radius on a cylinder
Infinitesimal magnetic dipole moment
Infinitesimal magnetic field vector due to a
magnetic dipole moment dM
dy, dz
Infinitesimal cartesian displacements
1, m
row and column of a matrix respectively
Vi
Element of an ordered Toplitz Matrix
Dx, Dy
Center to center distance between adjacent waveguides in the x and y direction
Separation angle between adjacent waveguides
Input power
g(8,¢).
Aperture gain
vii
List of Figures
Figure 2.0
Waveguide array on a cylinder
Figure 2.1
Coordinate system fo_r wave_g_uide electric and magnetic field vectors E and H
Figure 2.2
Equivalent circuit for n-th waveguide aperture
represented by mode voltage v and mode current i
Figure 3.1
Coordinate system for a radiating aperture
Figure 3.2
Coordinate system for waveguide array of radiating apertures
Figure 4.1
An infinite cylinder with two slots showing
electric and magnetic field vectors t, Hused
to calculate Y
Figure 4.2
Coordinate system and electric field distribution for a circumferential slot
Figure 4.3
Coordinate system and electric field distribution for an axial slot
Figure 4.4
Two slots on the surface of a conducting cylinder
Figure 5.1
Dimensions of a WR-90 waveguide
Figure 5.2
Integration grid coordinate system and variable
notation
Figure 5.3
H field along a surface ray vs. distance from a
directed magnetic dipole for Theta=90 deg direction
on cylinder with KR~9.53
Figure 5.4
Array arrangement and variable notation for array
factor computation
Figure 5.5
Coordinate system and variable notation for a
cylinder in a uniform field incident at an
angle e0 .
F·igure 6.2
Array of 20 circumferential waveguides on a
cylinder
viii
Figure 6.3
Array of 12 axial waveguides on a cylinder
Figure 6.4
Realized gain for a 3x3 element array with
element separation kd=4.788, scan angle
broadside circumferential orientation H-scan
Figure 6.4a
Array factor for a 3 element array, element
separation kd=4.788 scan angle 0 deg
Figure 6.5
Realized gain for a 3x3 element array element
separation kd=4.788 scan angle 30 deg
circumferential orientation H-scan
Figure 6.5a
Array factor for a 3 element array, element
separation kd=4.788 scan angle 30 deg
Figure 6.6
Realized gain for a 3x3 element array element
separation kd=4.788 scan angle 60 deg circumferential orientation H-scan
Figure 6.6a
Array factor for a 3 element array element
separation kd=4.788 scan angle 60 deg
circumferential orientation H-scan
Figure 6.7
Realized gain for a 3x3 element separation
kd=4.788 scan angle 90 deg circumferential
orientation H-scan
Figure 6.7a
An~ay
Figure 6.8
Realized gain for a 3x3 element array element
separation kd=2.394 scan angle 0 deg circumferential orientation E-scan
Figure 6.8a
Array factor for a 3 element array element
separation kd=2.394 scan angle 0 deg
Figure 6.9
Realized gain for a 3x3 element array element
separation kd=2.394 scan angle 30 deg circumferential orientation E-scan
Figure 6.9a
Array factor for a 3 element array element
separation kd=2.394 scan angle 30 deg
circumferential orientat-ion E-scan
factor for a 3 element array element
separation kd=4.788 scan angle 90 deg
ix
Figure 6.10
Array factor for a 3x3 element array element
separation kd=2.394 scan angle 60 deg circumferential orientation E-scan
Figure 6.10a
Array factor for a 3 element isotropic
array element separation kd=2.394 scan angle
60 deg circumferential orientation E-scan
Figure 6.11
Array factor for a 3x3 element separation
kd=2.394 scan angle 90 deg circumferential
orientation E-scan
Fi gw~e 6 .lla
Array factor a 3 element isotropic array
element separation kd=2.394 scan angle 90
deg circumferential orientation E-scan
Figure 6.12
Array factor for a 21x1 element array
element separation kd=2.394 scan angle 0
deg ozial orientation E-scan
Figure 6.12a
Array factor for a 21 element isotropic array
element separation kd=2.394 scan angle 00 deg
Figure 6 . 13
Array factor for a 21x1 element array element
separation kd=2.394 scan angle 60 deg
Figure 6.13a
An~ay
Figure 6.14
Array factor for a 2lx1 element array element
separation kd=2.394 scan angle 90 deg axial
orientation E-scan
Figure 6.14a
Array factor for a 21 element isotropic array
element separation kd=2.394 scan angle 90 deg
Figure 6.15
Reflected coefficient vs. scan angle WR-90
waveguide circumferential orientation Hscan 3x3 array KR=100
Figure 6.16
Reflected coefficient vs. scan angle WR-90
waveguide circumferential orientation E-scan
3x3 array KR=100
F·igure 6.17
Reflected coefficient vs. scan angle WR-90
waveguide circumf~rential orientation E-scan
3x3 array KR=lOO
factor for a 21 element isotropic array
element separation kd=2.394 scan angle 60 deg
X
Figure 6.18
Maximal reflection coefficient vs. scan
angle WR=90 waveguide circumferential
orientation E-scan 5x5 array KR=lOO
Figure 6.19
Reflection coefficient vs. scan angle WR=90
waveguide circumferential orientation Escan 5x5 array KR=lOO
Figure 6.20
Reflection coefficient vs. scan angle WR=90
waveguide axial orientation E-scan 7xl linear
array KR=lOO
Figure 6.21
Reflection coefficient vs. scan angle WR=90
waveguide axial orientation E-scan 7xl
linear array KR=lOO
Figure 6.22
Reflection coefficient vs. scan angle WR=90
waveguide axial orientation £-scan 7xl
linear array KR=314
Figure 6.23
Reflection coefficient vs. scan angle WR=90
waveguide axial orientation E-~can 7xl
linear array KR=314
Figure 6.24
Reflection coefficient vs. scan angle WR=90
waveguide axial orienta.t-ion E-scan 2lxl
linear array KR=314
Figure 6.25
Reflection coefficient vs. scan angle WR=90
waveguide axial orientation E-scan 2lxl
linear array KR=314 element spacing 0.6 wavelength
Figm·e 6.26
Efficiency vs. scan angle of a 3x3 WR=90
waveguide circumferential orientation Escan KR=lOO
Figure 6. 27
Efficiency vs. scan angle of a 5x5 WR=90
waveguide circumferential orientation Escan KR=lOO
Figure 6.28
Efficiency vs. scan angle of a 7xl WR=90
waveguide axial orientation E-scan KR=lOO
xi
List of Tables
Table 6.2
Summary of analyzed array physical parameters keyed to the radiation properties
of the arrays
xii
ABSTRACT
THE ACTIVE REFLECTION COEFFICIENTS OF CERTAIN PHASED ARRAYS
by
Yosef Klein
Master of Science in Engineering
This study uses an admittance matrix approach to calculate
the active admittance and reflection coefficient of finite rectangular waveguide arrays on infinite
cy1inde}~s.
The admittance
matrix approach considers the array to be a multipart network
represented by an admittance matrix.
The elements of
thi~
matrix
are related to the mutual coupling between elements, which is
computed via the geometdcal theory of diffraction (GTD) on an
element by element bas-is.
In this case, each element of the ad-
mittance matrix represents the mutual coupling between slots on
a cylinder.
finally, the active admittance and the reflection
coefficient for each waveguide element in the array is computed
by tlri s method for various scan angles.
Once the active input
admittance of each element is computed, the radiation pattern for
various scan angles is then
obtain~d
Xi i i
for
normaliz~d
input
~ower.
Chapter 1
INTRODUCTION
. Phased array antennas are vital to the design and integration
of radar and ECM systems.
Modern design of both planar and con-
formal scanned arrays requires high performance in gain and side
lobe control.
Supplement to these performance requirements in
the design practice of modern phased array systems is wide instantaneous band\'Jidth, multiple main lobes and the ability to form
multiple nulls over the scan coverage.
It is crucial to consider the mutual coupling between elements
of the array when designing phased array antennas, since mutual
coupling causes unexpected iobes, null filling 0,2] and error in
direct'ing the array.
~lutual
coupling alters the input impedance
of the isolated array element so that it becomes scan and location
dependent [1] and also alters the individual isolated element
radiation pattern.
This phenomenon complicates the analysis of
the array because the radiation equation can not be decomposed
into an array factor and a reference element radiation pattern.
,IThe
pur·pose of this study is to develop a simple mathemat·ical mo-
del for the analysis of a N x M rectanglular
array~of
mutually
coupled open ended rectangular waveguides on a cylinder.
t'1any analysis methods used in the past have included the ussumption that for sufficiently large periodic arrays, the centrally
1
2
located elements can be considered as if placed in an infinite
array.
Borgiotti [1] ·chose an infinite periodic array of equally
spaced waveguide elements and based his analysis on the periodicity of the array.
In his analysis, the tangential electric and
magnetic fields jn the radiation half space are expanded into
an infinite Floquet series [1, Equation 6 & 10].
This series is
then equated to an infinite waveguide modal expansion over the
waveguide aperture [1, Equation 8,9,10], (continuity of electric
and magnetic field over the aperture).
An error is then intro-
duced to the field continuity equations by truncating the infinite
F1oquet series and modal expansion into N terms.
This truncation
er-r· or is minimized by orthogona 1i zing the Nth order sys tern in
function space [1, Equations 11 & 12].
The solut·Jon of the ortho-
gonolized set of equat-ions results in the input admittance of an
array element (Y;n) as a ratio of N to N-1 ordered determinants.
Hessel has introduced the concept of eigenexcitation for periodic arrays [2,3].
The eigenexcitation is defined as a uniform mag-
nitude progressive phase excitation.
The eigenexcitation forms
an orthonormal vector basis (function space) for all possible excitations of the array.
Based on Floquet•s theorem for uniformly
spaced arrays [2, Equation A1], the array problem is reduced to
a unit cell for which the input admittance is found.
method vJas used
by
A similar
Shapit-a [ll-,5] to obtain the input impedance·
of arrays mounted on cyl·inders and curved surfaces [5, Equations
3
20 & 21 .
To overcome the difficulties encountered when using the methods
of Borgiotti, Hessel and Shapira in finite and nonuniformly spaced
arrays, the method presented in this study uses an admittance matrix approach.
This method considers the arr·ay to be a multipart
network represented by an admittance matrix.
The elements of
this matrix are related to the mutual coupling which is computed
via the geometrical theory of diffraction (GTD) on an element
by
element basis.
In this case, each element of the admittance
matrix r·epresents the mutual coupling bet\veen slots on a cylinder.
Finally the active admittance (Yac) and the reflection coefficient (r) for each waveguide element in the array is computed
by this method for various scan angles.
Once the active input
admittance of each element is known, the radiation pattern for
var·ious scan angles is obtained for normalized input power
(forced excitation).
Steyskal [6] has used a similar method in
which he calculated the scattering matrix for a finite array of
circular waveguides on a cylinder.
He obtained the scattering
matrix from the admittance matrix by standard matrix transformations [6, Equation 5] and the elements of the admittance rnatrix
were also calculated with the aid of a GTD formulation.
The
derivation of the Green's function for the coupled magnetic fields
bebieen tli'Jo
slots in the bJO studies is different.
The analys·is
used here used the work of Kouyoumjian [7] \vhose GTD for·mulation
4
is based on Kellers original work.
Steyskal •s Green's function
was expanded into Hankel functions approximated by Airy functions,
where as the Green's function used in this work is expanded into
Fock type
Ait~y
functions [8].
I
I
:
I
I
I
Chapter 2
AmHTTANCE MATRIX THEORY
The equations for the active admittance and reflection coefficient for an arbitrary element of a waveguide array can be developed using TEM mode transmission line theory to define mode
voltages (Vn) and currents (i ).
n
amplitude and phase of
t~e
These modal quantities represent
electric and magnetic field vectors
in the n-th waveguide respectively, and are calculated as a function of the input power.
This study considers an MxN (M rows, N columns) waveguide array
on a conducting
to
11
cylind~r
as shown in Figure 2, and is limited
the one mode approximation 11 , i.e. the feeding waveguides
as wel"l as the radiating apertures, support only the TE1o mode.
The fields in the waveguides and the aper-tures can therefore
be r-epresented by (Figure 2.1)
=
ev
(2.la)
H -·
hi
{ 2.1 b)
--e
z~1/ab
E
COS( ny /a)
(2.1c)
-h - ~~1/~- COS(;y/a)
(2.1d)
1 = { exh·dA
J
(2.1e)
=
A
6
FIG 2.0
.
Wavegufde array on f1n1te cylinder
7I
R
...
z,
•
_,.
-
FIG 2. 1
.....
- - -P"
....
. _..yP
v.
H
Coordinate system for waveguide
electrfc_and_magnetic field
vectors E , H..
8
where the modal electric and magnetic field vectors
equal for all waveguides.
e and
h are
They are normalized quantities by vir-
tue of Equation 2.le.
The amplitude and the phase of the modal electric and magnetic
field vectors, are represented by the column matrices V and I
respectively.
These matrices can be treated as voltages and
currents in an equivalent TEM transmission.
Let the following
column matrices represent the mode voltages and currents in the
MxN port TEM transmission line network:
I
n
=
1,2,3 ... MxN
(2.2a)
n
= 1,2,3 ... MxN
(2.2b)
Matrices V and I therefore also represent
th~
modal amplitude and
phase of the electric and magnetic fields in the waveguide array.
When a power source is applied to the n-th transmission line
of Figure 2.2, the voltages and the currents in this arbitrary
selected transmission line are [9]
2 ~I +
= Ygvn
2 ·~JC\}n-
-- -Y 9*v n
g
n
+ in
+ in
(2.3)
(2.4)
9
---fbo.
in
+
Yac{fi)
FIG 2.2·
: Equtvalent circuit for. n-th waveguide
a.pertur~e represented by mode voltage
vn
and mode cur rent
1n
10
where,
In+ -
Square root of the net incident power from the
generator.
= Square root of the net power reflected
In-
by the
generator:
G
9
= Real part of the generator admittance.
Y
9
= The generator admittance.
These forward and
reflect~d
current waves are related to the
power in the nth transmission line by
(In+)2-(In-)2- = (In+)(In+)* + (In-)(In-)* = RE(vnin*)
,(2.5)
where the definition of the average power is given by
P
=
(2~6)
RE(v ni n*)
Since Equation 2.6 represents the net power delivered to the nth
antenna aperture, In+ is the square root of the net incide~t
power and In- is the square root of the net reflected power, (as
defined previously).
Therefore, by definition, the reflection
coefficient for the nth aperture is
I·
I
i
_ _j
11
- Y *V (a ) + i (a )
=
9
n
n
(2. 7)
Yg Vn(a) + in(a)
The overall array is scanned to angle
8
0
off broadside by
generating and increasing (or decreasing) phase delay which
changes by angle a. for each array element.
In this case, In+
is given by
(2.8)
To define the active admittance for this array, the right hand
side of Equation 2.7 is divided by vn(e 0 ), i.e.,
=
-Y 9* + in(e 0 )/Vn(e 0 )
(2.9)
where the active admittance is given by
(2.10)
Note here that, in order to stress scan angle dependence, variables depending on a have been interchanged with 80 •
This defini-
tion of the active admittance (Equation 2.10) leads to a reflection coefficient which can be written as
(2.11)
12
Calculation of the reflection coefficient of Equation 2.11
requires that the active admittance, Yacn' be available, Equation
2.3 and Equation 2.4 can be recast in matrix form as
2 ~I+ = YgV + I
2
~Ig
I
= YV
-Y *V + I
--
g
(2.12)
(2.13)
where
(2.14)
Column matrices V and I have been defined in Equation 2.2 with
Gg = Diagonal matrix of the real part of generator
admittances.
Y = Diagonal matrix of generator admittances.
9
Y
= Admittance matrix of the array.
Equations 2.12 through 2.14 can now be solved for V to obtain
v =
2VG:", (Y + Y)-1 I+
g
g
(2.15)
The active admittance of each waveguide aperture in the array is
obtained by substituting the elements of V from Equation 2.15
and the elements of I from Equation 2.14 into Equation 2.10.
This
value of the active admittance is then used to calculate the reflection coefficient of each aperture by use of Equat·ion 2.11.
13
For maximum power transfer between the generator and the aperture,
the generator admittance must be chosen to make the reflection
coefficient of Equation 2.11 zero, that is
(2.16)
=
A perfect match however, can be obtained only at one specific
scan angle which can be arbitrarily selected (see
~ppendix
A).
In this study, the match condition is selected at broadside.
The
broadside match condition from Equution 2.16 is therefore given by
=
In practice, control of v9 is obtained by inserting a matching
network between the generator and the antenna aperture.
Chapter 3
ARRAY RADIATION
3.0 General considerations and definitions
As mentioned in· a previous section, this analysis uses a
one mode approximation for the aperture field distribution.
This
section will show that under this approximation the following
conditions hold:
1.)
The pattern multiplication principle applies hence,
the array and the aperture radiation properties can
be calculated in b1o separate calculations.
{Equa-
tion 3.18)
2.)
The array field can be calculated from the circuit
parameters developed in Section 2. (Equation 3.17)
'
i
I
I
iI
------ ------·~----- -- ---- . J
14
15
3.1 Radiated field from a single aperture
The expression for the electric vector potential at point P
associated with a magnetic current density is given by [10]
where Figure 3.1 shows the coordinate system and
M(r
r'
1
)
=
Magnetic vector current density source.
= Radius vector from the origin of the coordinate
system to a source point.
-r
= Radius vector from the origin of the coordinate
system to a field point P.
k
= Free space wave number k = 2n(;.._.
A
= .Free space wave length.
A
= Area of aperture.
dA
= Infinitesimal subarea of the aperture.
Eo
--
Free space dielectric constant =
8.854·10-12 Farad/Meter.
16
z
p
y
FIG 3.1
X
: Coordinate system for a radfating
aperture
17
The far zone electric and magnetic fields [10] for this electric
vector potentials are given
= -jwAm
(3.2)
E = Z0 Hxr.
(3.3)
H
where
w
= Radian frequency.
Z0 = Free space impedance (377 ohm).
(3.4)
For the open ended waveguide case the equivalent magnetic current
density [10] is given by
M(r•) = 2Ea(A•)xn
(3.5)
By substituting Equation 3.5 into Equation 3.1, the expression
for the electric vector potential becomes
The radius vector to a source point can be expressed as
(3. 7)
and for r >> r•(far field condition), Equation 3.7 is reduced to
,. _,
==
r - r·r
(3.8)
18
Substitution of Equation 3.8 into Equation 3.6 gives the final
expression for the electric vector potential .
..
19
3.2 Radiation from an Array of Apertures
The vector potential (electric) associated with the j-th element
of the rectangular array shown in Figure 3.2 is given by
Amj
=
s 0 exp(-jk 0 r)/2rrr jEa(A'j)x~exp(-jk 0 ~·;:'j)dA'j
Aj
(3.10)
For the coor·dinate system in Figure 3.2, the aperture field is
given by
(3.11)
~Jhere
ej is g·lven by Equation 2.1c.
Substituting Equation 3.10
into Equation 3.11 results in the vector potential for the j-th
element given by
Amj
= s 0 exp{-jk 0 )/2rrr vjJ;jX~jexp(-jk 0 ;.;j')dA'j
A·J
(3.12)
and the potenital associated with the entire array is then obtained by adding the vector potentials of all array elements,
i'. e.,
(3.13)
or more explicitly,
20
I
FIGURE 3.2 : coordinate system for waveguide
array of radiating apertures
21
The order of integration and summation of Equation 3.14 can be
interchanged and the vector potential for a reference element can
also be defined as
Equation 3.14 can therefore be rewritten
a~
(3.16)
The transition from Equation 3.14 to Equation 3.16 the surface
of the array is flat, i.e. the radius of the cylinder is large
and all normals to the apertures are parallel
-
Section 5 will show that the error introduced by this assumption
is small enough to validate its use.
If the summation ·in Equation
3.16 is defined as
F(e,¢)
= r.v.
exp[-jk ;.(~·--~· )]
j J
o
J
e
(3.17)
then Equation 3.15 can be written in 11 array factor 11 form i.e.
-
Amt · = F(e,¢) Arne
(3.18)
In this case, Ame is the vector potential associated with the
reference aper·ture and can be found by GTD [7] or by classical
methods [10], and the F(e,¢) function is kno'fm as the array
22
factor.
(The same array factor will be used in Section 3.3 in
the expression for the gain of the array.)
23
3.3 Gain Pattern
The basic definition for gain pattern is given by (antenna
elements assumed lossless)
Power density per unit solid
Realized angle in direction (e,¢)
Gain = 4rr
Total incident power on the array
(3.19)
The power density per unit area in the far field in the (8,¢)
direction is given by the derivative of poynting vector P where
P
= .5Re
JE x Fi*
dA
(3.20)
and
l-dPj
=
I . 5Re (-E x -*
H ) I
(3.21)
dA
The power density per unit solid angle in this direction is
dP
dD
=
(3.22)
(3.23)
The total normalized input power to the array is the sum of the
incident power at each input to the array, i.e.
(3.24)
24
and by substituting f from Equation 3.3, Equation 3.22 becomes
d"P
(3.25)
d0.
Also, H from Equation 3.2 yields
(3.26)
and since
w
Am e
(3. 27)
F(e,¢)
(3.28)
then by substituting Equations 3.26and 3.24 into Equation 3.18
gives
(3.29}
Ga·in
By definition, the radiation pattern of an
arbitt~ary
element is
(3.30)
then
IF(8,<P)I
Gain
=
g(e,¢)
(3.31)
25
3.4 Efficienc_y
The efficiency of a scanned array can be defined by [10]
Ga(e,¢)
n
=
Ga ( o,o·)
<
1
(3.32)
and essentially computes the ratio between the power radiated
at scan angle (8,¢) to the power radiated at broadside (e=O, ¢=0}
scan.
Other choices are also applicable as long as the denomin-
ator in Equation 3.32 is taken at the scan angle for which the
array is "scan matched".
'
'
Chapter 4
MUTUAL ADMITTANCE
~1ATRIX
4.1 · General Discussion
A general expression for calculating the elements of the admittance matrix will be derived in the following section.
By
definition, an element of the admittance matrix Yij is the ratio
of the current in element i to the voltage in element j with all
other voltages except Vj set to zero.
In this case, all elements
except the i-th are therefore covered with a conductive screen
and the admittance matrix has reduced the multiaperture cylinder
problem in to one containing only two slots; one slot is represented by a magnetic current source (voltage). and the current
(magnetic field) is calculated over the second slot (Figure 4.1).
A GTD solution is especially well adapted in solving for Yij
and the problem of calculating the magnetic field due to a magnetic current source on an infinite cylinder has been solved by
Kouyoumjian [7]. Both his work and that of others [11,13] is
used in the following paragraphs.
?F.
27
F!G 4. J.
An infinite cylinder with two slots
electrtc and magnetic field vector
E1 ,H 1 and E~ ,H2 to calculate ~ 2
28
4.2 Mutual Admittance - two slots
Mutual admittances are usually identified with the non-diagonal
elements of the admittance matrix while the diagonal elements
are usually referred to as self admittance terms.
One expression
for the mutual admittance between two slots is given by [14]
(4.1)
where, (Figure 4.1)
A2 = Aperture surface area of slot 2.
dA2 = Unit normal aperture 2.
Hl =
~1agneti c
field when slot 1 is excited with voltage v1,
and slot 2 is covered by a perfect conductor.
E2 = Electric field when slot 2 is excited with voltage v2,
and slot 1 is covered by a perfect conductor.
From Appendix B, the expression for the magnetic field, H , on
1
slot 2 is given by (Figure 4.1)
ill
=
f
A1
9·Ml (Q' )dA 1
(4.2)
29
FIG 4.2
:
Coordinate system and electric field
distribution for two coupled
circumferential slots on a cylinder
30
where,
g
is the dyadic Green•s function for the magnetic field
H1 at slot 2, produced by a unit infinitesimal magnetic dipole
located on slot 1.
For the one mode approximation, the TE 10 fields in slot 2 are
(4.3)
The electric field distribution for a circumferential slot of
Figure 4.2 is given by
{4.4)
For the axial slots of Figure 4.3 the electric field distribution
is given by
(4.5)
Substituting Equation 4.3 into Equation 4.1
{4.6)
Using the derivation g·iven in Appendix B, the mutual admittance
for two circumferential slots is given by
(4. 7)
31
\
\
-9,
FIG 4.3
:
Coordinate system and electric field
distribution for two coupled
axfa1 slots on a cylinder
32
and for two axial slots the expression for the mutual admittance
is
(4.8)
where, as defined in Appendix B
g¢ (S,S,~,R)
= Scalar Green's function for circumferential
slots.
g2
(S,s,~,R)
=
Scalar Green's function for axial slots.
where in (Figure 4.4)
S ~
Geodesic
distance between two subareas on the slots.
= The angle of s to the cylinder circumference.
E = GTD distance parameter.
R = Cylinder radius.
Note that:
1.)
The scalar Green's function in Appendix Bare noted as
g(~,Hb,Ht),
but since Hb and Ht are functions of R,E,S ·
the notation was changed here to g(S,S, B,R}.
33
.....
.... -- 1.
b.
c1rcum~erantia1
slots
on developed cy1fnder
n. three dimensional
view
FIG 4.4
:
c. axfal slots on
developed cylfnder
Two slots on the surface of a
co·nduct 1ng cy l i ndor
34
2.)
The scalar Green•s function can be interpeted as the
mutual coupling of two infinitesimal subareas belonging to slot 1 and 2.
Chapter 5
cmWUTATIONAL METHODS
5.1 Admittance Matrix
The admittance matrix is calculated on an element by element
basis; i.e. all possible two waveguide combinations are selected
and the mutual admittance calculated for each pair.
The wave-
guides used in this study are assumed to be standard WR-90 waveguides, which are used as a common radiating aperture at X-band
(8200 Mhz to 12400 Mhz).
shown in Figure 5.1.
The dimensions of this waveguide are
Two orientations with respect to the cylin-
der axis are considered: axial and
circumferential~
waveguides are arranged with their broad
wal~
i.e. the
parallel to the main
axis or to the circumference of the cylinder respectively as
shown in Figures 6.2 and 6.3.
Also, two cylinders are used:
one
a 16 wavelength radius and the other a 50 wavelength radius.
The
calculations of the elements of the admittance matrix are performed at 9000 Mhz and, at this frequency, the WR-90 waveguide
supports only the dominant TE10 mode.
Equation 4.7, which is used
to calculate the mutual admittance is repeated here for clarity
(Figure 4.4).
36
t
b::0,4..
I
I
~ o.s•
.,
a:o.9
--..1\!Qlili!fl!f.""--~--...,.----,-.·-
FIG 5.1
:
-"ill-
.
1.0 ----~
Dimensions of a WR-90 waveguide
37
Y1,2
= ::
f f
dA 1
Al
dA 2COS (1r/a •yl )COS(1r/a. y 2 )g ( S,
~, €,R)
A2
(5.1)
An integration grid is defined in Figure 5.2, where the two
circumferential slots are separated at a distance of y0 and z0 ,
in they and z axis respectively.
Each slot is divided into
p x q sections across each waveguide, thus generating subareas
dy 1 dz1 for slot 1 and dy 2 dz 2 for slot 2. Subarea Q' of slot 1
can be considered as an infinitesimal magnetic source with magnetic
field intensity at Q in slot 2 given by
(5.2)
The entire contribution of all subareas in slot 1 at Q in
~lot
v1ill generate a magnetic field \'lith an intensity of
H1
=
To find
v ,
EE
P1Ql
COS(rr/a ~y )COS(rr/a •y )g(S,P,E,R)dy dy
1 1
2 2
1 2 (5.3)
1 2 ~ the total contribution of all Q' sources over the
entire area of slot 2 must be summed to obtain
y, 2 = ___::.f___ E E E 2: dy dz dy dz COS(rr/a 1.y )COS
1
.1 '
1 1 2 2
a1b1a2b2 q1p1q2P2
(5.4)
2,
38
z
l
!
._
Yo
z
{3\
__y__
FIG 5.2
Integration grid coordinate system
and variable notation
39
For this equation, values for the geodesic
arc lengths are
given by
(5.5)
Angle B between tbese geodesics and the cylinders circumference
are given by
(5.6)
and, the GTD distance paramenter
E
becomes
(5.7)
For circumferential slots {waveguides), the Green's function
of Equation 5.3 is given by (Equation 4.7)
'( 5.8)
and for axial slots, the Green's function of Equation 5.3 given
by (Equation 4.8)
(5.9)
The accuracy of calculating mutual admittance Y1,2 by numerical
integration is strongly dependent on grid density, i.e. the number
of subareas p x q.
The program of Appendix C uses p=7 and q=2,
and gives a total of 14 subareas; this number of subareas is sufficient to accurately calculate v1, 2 since when p=l4 and q=4,
40
showed a change of only 0.5 percent.
Also, the running time
increased three (3) times due to matrix size.
q=2 division therefore appeared to be adequate.
The p=7 and
Note, the pro-
gram of Appendix C is general and may be used to calculate any
rectangular slot
~onfiguration
on any cylinder.
41
5.2 Active Admittance and Reflection Coefficient
Equations 2.10 and 2.11 of Section 2 provide a basis for computing the active admittance Yac and the reflection coefficient
r versus scan angle
00
.
However, before Equations 2.10 and 2.11
can be used, Equations 2.12 through 2.14 must be solved for I and
V.
The relevant expressions are given by
(5.13)
=
( 5.14)
I
= Yv
(5.15)
v = 2 Gg (Y g + v)-lr+
(5.16)
The knowns of Equations 5.13 through 5.16 are Y1,2 (admittance
matrix elements computed by Equation 5.4) and r+, the vector of
forced excitations (varies with scan angle).
The generator ad-
mittance matrix v9 is also unknown at this point. One solution
for Yg is given by Equation A.6 of Appendix A. However, this
approach was found to be
ative" procedure
~;-Jas
impl~actical;
instead, a different "iter-
used to find Yg in each array considered here.
Solutions via this procedure converge sufficiently within two
42
iterations and lead to a satisfactory solution of Equations 5.13
This iteration algorithm is described as follows:
through 5.16.
1.)
Set
v9 =Vii* in equation 5.16, where Vii is a dia-
gonal matrix containing only the diagonal elements of
the admittance matrix.
The * in this case represents
11
11
complex conjugates.
2.) Solve Equations 5.16 and then 5.15.
3.)
Calculate Vac via Equation 5.13 and also obtain
f(8 )
0
via Equation 5.14.
4.)
If
r
>
0.1, set
v9 = Vac*(
=0) and repeat steps 2
through 4 above.
5.)
If r < 0.1, save the last values of
Vg , Vac , V and I.
.
A computer program given in Appendix D uses the above algorithm
to solve Equations 5.13 through 5.16. This program takes advantage
of the block Toplitz property of V [15 16].
5
A matrix is said to
be centrosymmetric Toplitz if it is in the following form:
43
y
t
Y1l,Y12'Y13. · .Y1n
v0 ,Yl'Y 2... Yn
Y21•Y22•Y23···Y2n
Y1,YO,Y1 ... Yn-1
Y31•y32'y33· .. Y3n
Y2,Y0,Y1 ... Yn-2
=
=
(5.17)
Where Yi are the elements of the n-th order matrix Yt and in this
case are given by Equation 5.4.
These elements are dependent only
on the differences
i
= jl - ml, (1 rm-Js, m columns)
If the elements of this Toplitz matdx Yt{y 0 ,y 1 , .. ·Yn) are
considered as sub-matrices and also have the centrosymmetric Topliz matrix pr·operty, then Yt is said to be
11
block Toplitz.
11
The block Toplitz matrix often occurs in array problems and
other integral equations when solved numerically and is attractive for two reasons:
1.)
From the symmetry of the block Toplitz matrix as is
depicted in Equation 5.17, one row or column of the
matrix is sufficient to define the entire matrix.
44
2.)
It can be inverted by an efficient algorithm [15,16].
The rectangular waveguide array described in this analysis has
a block Toplitz admittance matrix where the submatrix
Yo
repre-
sents the 11 self admittance 11 for waveguide elements and y , re1
presents the mutuals between one waveguide and all adjacent rows.
In general, the Submatrix yi represents mutuals between elements
of all rows separated (i+l) rows.
This block Toplitz property
is obta·ined in the waveguide problem if the apertures are numbered
sequentially as shown in Figures 6.2 and 6.3.
As an example of
simplified data entry, consider an 8X8 array where the admittance
matrix is of order 64 and contains 4,096 elements.
However,
utilizing the block Toplitz property, input is required for only
64 different elements, (one rovJ of the matrix).
45
Array Gain Pattern
5.3
The computer program which is given in Appendix D computes the
gain patterns Ga for a uniformly excited array (II+!
different scan angles.
= 1) at
The expression used for Ga is given by
F(8,¢)
Ga
=
=
F(e,¢) I N
(5.18)
where
{5.19)
For a two dimensional array as shown in Figure 5.3 the position
vector to an element can be decomposed into its• x andy coordinates, i.e.
(5.20)
Where (Figure 5.4)
Ox
- Adjacent element separation x direction.
Dy - Adjacent element separation y direction.
n
-
Integer count of element position along x coordinate.
m
= Integer count of element position along y coordinate.
46
z
y
X
.
/~
. X~
m
FIG 5.3
•
• •
•
• n
Array arrangment,varfable notation
for array factor computation
47
and hence Equation 5.19 becomes
F(e,<P)
=
l: V exp[-jk ;. (nDx·~ + mDy·y)l
m,n m,n
o
j
=
(5.21)
=
Thus the final expression of the array factor for a two dimensional
array is
F(e ,cp)
=
l:
m,n
Vm,nexp tjk 0 (SIN8COScpnDx + SINeSINcpmDy)]
(5.22)
Note that the voltage of element Vj in Equation 5.10 is now denoted as Vm,n to symbolize the position (mDx,nDy) in the array,
and the count notation in Equation 5.19 was one dimensional and
is a two dimensional count notation in Equation 5.21.
Hence
the summation notation of Equation 5.22 can be expressed as
=
F(e,cp)
l: L:
mn
Vm,nexpfjk 0 (nDxSINeCOScp + mDySINeSIN<P)]
(5.23)
Consider an array scanned in the axial plane of the cylinder
(say the y,z plane of Figure 5.4), then
<P
=
rr/2
(5.24)
48
and the artay factor is given by
F(e,¢)
=
(5.25)
IE Vmnexp(-jk 0 mDySIN8)
·mn
An array which is scanned in the circumferential plane of the
cylinder (x,z plane of Figure 5.4) has
¢
= 0
(5.26)
and the array factor becomes
F(e,¢)
( 5. 27)
Samples of calculated gain patterns for these cases are given
in Section 6.
The voltages Vmn of Equations 5.25 and 5.27 are
calculated via Equation 5.16 and substituted into Equations
5.25 and 5.27 (program of Appendix D).
Equations 5.20 through
5.27 assume a cylinder of a large radius (R/A
>>
1).
In this
case, the gain factor for this array is actually calculated
with sufficient accuracy by replacing the large cylinder with an
infinite planar ground plane.
The small phase error introduced
into the forced excitation vector results in acceptable error
in calculating the gain (mostly in the low side lobe region).
An example of the size of error which results from this approximation, is now illustrated.
Figure 5.5 shows a circumferential
cut through an 8x8 array with incident field E,H arriving at
49
I
FIG 5.4
,_J
-
E)H
I
/
Coordinate system and variable
notation for a cylinder in a uniform
field incident at angle
50
angle e from normal.
The angular separation $ 0 between two ad-
jacent elements in this case is given by
$0
=
Dy/R
(5.28)
and hence, the de)ay angle between these adjacent elements is
given by
= 90 - (100 - 0)/2 -
a
(Q0
-
0)
=
Q0
-
0/2
(5.29)
The phase delay between these two adjacent elements is therefore
(5.30)
A planar array of the same overall dimensions has a delay between
adjacent elements given by
.(5.31)
The scan direction for both arrays is obtained by adjusting the
phase angle between each element, and the pointing error is then
$ 0 /2
degrees.
A sample calculation for the array used in this
study is given as follows:
Case 1 : Circumferential Orientation
Cylinder radius P.
-
Element separation Dy
A.
=
1.312 inch
16A.
= 1 inch
(5.32)
51
R
~
16A
~
2.997
¢
~
Dy/R
~
(1/20.997)57.3
{5.33)
~
2.73 deg
(5.34)
In this case, the phase error causes a scan error of 2.73/2
=
1.36 degs which can be considered negligible for the arrays of
this size (small).
Case 2 : Circumferential Orientation
Cylinder radius R
~
50A
Element separation Dy = 1 inch
Following the previously calculated steps,
¢
= .875 degs
(5.35)
which leads to a scan error of .437 degs which is also
negligible.
Chapter 6
RESULTS AND CONCLUSIONS
The purpose of this study was to develop a simplified method
for analysis and design of phased arrays on cylinders.
To demon-
strate the use of the equations and methods previously given, calculations of various array parameters for two basic types of arrays
were made, a circumferential slotted array (Figure 6.2) and an
axial slotted array (Figure 6.3).
The array factors for these arrays are shown in Figure 6.4
through Figure 6.14.
Figures 6.15 through 6.25 display results
for the active reflection coefficient (r) and Figures 6.26 through
6.29 show the calculated efficiency for the same cases.
mary of these results is given in Table 6.1.
A sum-
This table tabulates
the calculated parameters for the various cylinder sizes (radius),
types of scan (E or H), element separation and number of array
elements.
This table also, keys these results to the two array
configurations (Figures 6.2 and 6.3).
All graphical results are
displayed versus scan angle from broadside.
To observe the effect of mutual coupling, the array factors
given in Figures 6.4 through 6.14 are best compared to the array
factors of non-coupled isotropic elements scanned to the same
direction off broadside as (Figures 6.4a through 6.14a) the actual
\<Javegu-ide array.
The array factor for an array of isotropic elements
52
53
TABLE
1.
Realized Gain
2.
Reflection Coefficient
3.
Efficiency
6.1
54
~R
,
'J
"e:t
..,
~Qs.,
'l
1\
\S
-----~
FIG Bo2
:
Rrr~y
of 1& cfrcumfar0ntial wavaguids
en :a cyl i
nd~r
55
~R
0,4"
0.9''
s '
1.o•
F'IG
S.~
••
'9
to
Array cf 12
cy11ndsr
7
8
11
La.
axi~1
w~voguide~ o~
a
56
with progressive phased delay between adjacent elements is given
by [10] (displayed in Figures 6.4a to 6.14a)
Fi(e)
= 1/p
SIN(Py/2)
(6.1)
SIN(y/2)
where
(6.2)
P = Number of array elements.
The waveguide array factor is given by (Equation 5.23 repeated
here)
F(8,~)
=
~ ~Vm,nexptjk 0 (nDxSIN8COS~
+
nDYSIN8SIN~u
(6.3)
This comparison in Figure 6.4a through 6.14a. is consistent
since the results of Figures 6.4 through 6.14 are for the array
scanned only in one major plane.
The results in Figures 6.4
through 6.7 show the nulls at the expected angles and the sidelobe
levels do not appear to be raised from the expected levels, as
the number of elements of the array is small.
The presence of
the grating lobe in these figures is not surprising, since the
first grating lobe should appear at 18 degrees.
For theE plane scanned arrays (Figures 6.9 through 6.11),
the element spacing in the scan plane is smaller than for Figure
6.4 through 6.7 and hence the higher side lobe level is observed.
57
The side lobe levels which are expected to be down approximately
9.5 db, are down only 6.5 db (Figure 6.11) from the main beam.
Also, the infinite nulls expected at 62 degrees (Figure 6.10a)
in Figure 6.10 and at 50 degrees (Figure 6.11a) in Figure 6.11
are filled in.
In Figures 6.12 through 6.14 (21 element linear
array), theE-scan shows a large increase in side lobe level
(approximately 8 db) and nearly all nulls are filled in (Figures
6.12a through 6.14a).
In Figure 6.15, the reflection coefficient
of the 3x3 H-scanned array is displayed and consistent with the
array factor of Figures 6.4 through 6.7; there are no extreme
reflections in the array.
For the same array scanned in the E
plane (Figures 6.16 and 6.17), the reflection coefficient reaches
0.74 in elements 4 and 6 of the array.
This phenomenon could
explain the null filling observed in Figures 6.9 through 6.11.
Figures 6.18 through 6.24 show high reflections for the larger
arrays at scan angles approaching endfire.
In the 5 x 5 E-scanned
array the reflection coefficient is between o.5 and n.B.
P1n increase in cy1indet"' radius also increases the reflection
coefficient.
This is due to stronger mutual coupling and is con-
firmed in Figures 6.21 through 6.24 where the cylinder radius has
been increased approximately 3 times.
The reflection coefficient
in this case increases from 0.7 to 0.8 for the center element and
from 0.9 to 0.95 for the 6th element.
An inct·ease in the array size is not necessarily accompanied
58
by an increase of element reflections, because of mutual coup-
ling phase cancelations.
This is confirmed for the 21 and 7
element arrays (Figures 6.21 through 6.24).
An increase in element spacing causes the reflection coefficient to drop from 0.9 to 0.6 in Figure 6.25.
A reflection peak
also occurs at a scan angle where the grating lobe appears.
Figures 6.26 through 6.28 show the efficiency of ·the array
versus the scan angle based on Equation 3.29 of Section 3.30.
These figures can be used to give a simple estimate of over all
radiation performance of the array.
In conclusion it has been shown that phased array mutual coupling
and reflective coefficients for certain finite conformal arrays
can be successfully analyzed using the techniques derived here.
These techniques have been programmed for a digital computer and
results shown here give good agreement with expected and previously published results.[l8]
59
\\
\
'\
\
~
~~~~~~~~~~~~~~~~~~-J
-90
-S0
.
•
-30
0
30
SB
rea11:z~,cl gt'iin f'or a 3X3 ~H.\Vatjufde
~lem~nt •~paratton kdM4.7SB scan
brondnid~
H- scan
90
array
angle
60
Fi
array factor for a 3
el~mant ~epar~tion
b ro ti\dt~d dll:)
ml~men~ array 9
kd~4.7BB scan nngle
61
Ga
L... '-~ 1 ,,
-90
FIG 8.5 :
realized gain for a 3X3 wavagutda array
element separation kd~4.78B scan angle 30 deg
ctrcumferantial orientation H-scnri
62
gro.ttng lobe
:;t;?'
1
.s
·r
t
0
FIG
G~5a
:
array factor for ~ 3 Glam~nt ~rray ,
element ~0par~tfan kdoo4.788 ~can nngl~
30 drag
63
grating lobe
"""
Ga
1
I
NH~l fXt'u:l gafn for a
elarnant ~eparatian kdm4.7BB
cfr~umf®rent1al orientation
tiG G.G
:
w~vaguicla
scnn ~ngl® 80
3X3
H-scan
array
deg
64
grat.ing lobe
Ft
L....&....L. .,. .L
se
11-
t
e
• ..1-t.....l
se
--F"IG-s-;;-s u.-:----nr-r\:'l.y--F'~o-t.o!'"---to f"_l\_3---c-e)J_@rnent u r· ray ,
~lem~nt
60 deg
~~paratton
kdm4.788
~can
~ng~l®~.--------~~-
65
e
L..... -a.lr,
-Sfl
-S0
FIG G.?
:
el®m~nt
I
0
r6alfzed gain far a
30
3X3
$~p~ratfon kd"4~7B8 ~can
90
array
angle 90 d~~
elern~nt
c f rcurri-'lH'en·t 1 al or 1~ntat ton H-sct:tn
66
F'1
30
FIG S .. ?a
:
60
50
array factor for ~ 3 tll tlffi~n.t &rt~~y ,
o 1t.'lmcnt ~ti"#JHU'n·t; f on kdr~t>4 .. 798 $tHm nng 1e,
sa
dttg
67
Ga
.7
.s ~
L...L...J~~u.A.o..lo..d.....a-~~b....'
-90
-60
riG 6. S
:
real
f ::~:ad
gain
ftH·
a 3X3
UHlv~gui d~
elament •~paration kd~2~394 scun angl~
c f rm.amf'ea~~)ni; fa 1 tH"' f ent~t 1on E:-za:an
rn
arr-ay
de~
68
t!G S .. Ba :
array facto!"' for n 3 el('}ment arrU}' ,
~Gparatfon kd~2.394 soan nngl0
brtHltht f d€9
elamflnt
69
Ga
1
.9
T
SLLif.t -6.3 Db
+
L~._t.
-SB
FIG 6.8
e.. t
'1-• L•
-sa
:
t
I.,,.,.,_:::
.. a_•' •
-aa
I
I
r~~1iz~d gain for a
alom~nt sep~ratfon kd•2.394
cfrcumfa~enttal ort~nt~tfon
e
L
1 " L,,
'
30
.... J
,, 1 • !.
60
w
90
3X3 wav®gutde array
sc~n
ang10
E~~cnn
sa
d~g
70
rt
1
.7
9
1.... ... t "-·J.d.t...J. "'"' L."'
....Sti!
-60
~.L.~·,.......~.........,-~.l....t-~.J-.a...J
-30
nrr~y
B
30
G0
90
factor for • 3 element array
e 1ement.
30 d~g
f~etntt" e\it ion
kdwh!2 .. 394
~c ~n
1
tUi'tf
le
71
Ga
SLL.,. -G .. 5 Db
i
j
I
1~ :
rt')-a! t~~d gain -f~r· a
el~m$nt sap~r·utfon kcl~~.S£~
c ft'"ctnnf'~f't'INt i ~ l or f errt~:t f on
FIG 6 ..
!
W:tiV~guicl® ~r-ttt'!y
~can ang~~ fl~ dGg
e:--~~an
3X3
72
Ft
1
I
array factor far a 3 element array ~
kd~2.394 •can angle
~l®mant •®p~ratton
S0 clt>Jg
73
p '
G~
1
.s
.a
,_.
~
ff
t
~
l
+
.?
+
/
J
l
....
•
;'
i
f
!
t
i
.I
!
I
i
'
f"!Jd
S.~i.:
t·~~lf:r~d ff~df'l .ftH" li 3)(~i W~Vftguh;le ttr-ray
~lt'l~ll!lf1t s.-.~par~tfc:;n kcl~2.:Hl4 .~a~rt. ~.ng!~ 30 d~g
ctruuru1eren~tat artant~tton E-~can
74
~l...l.~!-
-90
-se
~r
~
ray -¥ artc·te:n.. of't) r-
i ®ment
90
dtlfj
~~p~r-~t f
tt 3 ~ 1~fg:tHr~.;
on
array
Ju~Bf.!'2 ... 394 wo~n
11
~mg' ~
75
Ga
-90
-60
-30
0
30
60
90
FIG 6.12 :
realized gain foi~ a 21X1 waveguide array
element separation kd=2.394 scan angle 0 deg
axial
orientation E-scan
76
f:tG tL12fi :
ttrn.ly
•fill~1.t.:~f•
f~r
a 21
@l~m~nt
urr~y
~
w1ament aeparatfan kd"2.394 scan angle
k:r·oa~hd d~
77
Ga
1
.9
.8
.7
.6
.5
.4
.3
.2
--~
~~-J-t..-l-L--L.......l..-w.--'---'--~~.J:;...._.~--'--l--&.~""---~-J.._
-90
-60
-30
0
30
FIG 6.13 :
realized gain for a 21Xl element array
element separation kd=2.394 scan angle 60 deg
axial
orientation E-scan
78
FIG S.l3tt :
arr-ay of&e·ttH• f<:tt' a 21
. ~h:r:~~nt
Sla d~g
$~JHH'atton
&1\i)ffi~t~i..
l<d,..x£L.3B4
art•;:]!.y ,
~;oan
~ngi@
79
Ga
1
.9
.8
.7
.6
.5
.4
.3
.2
e
L
I
-90
I
I
!
--l.....&.....
--60
FIG 6.14 :
-30
0
30
60
90
realized gain for a 21Xl waveguide array
element separation kd=2.394 scan angle 90 deg
axial
orientation E-scun
80
rt
fiG B.14a :
array fac~or for e 21 ~1ament arr~y ,
&1emunt sup~ratton kd~2~394 acan angle
Sta
t~arrg
81
Reflection
coefficient
1
.9
.8
3
.7
6
.6
s
.5
center element (5)
.4
~~
.3
edge element (1)
.2
------------------......._.__,____,__,~
0
10 2 0
I
30
I
I
L
I .
I
I
40 50 6 0
'
scan ang 1e
L
70
L
--1-..!--L-+-1
80
9 0 100
FIG 6.15; reflection coefficient vs.scan angle
WR-90 waveguide circumferential
orientation H-scan 3x3 array KR~100
82
Reflection
coefficient
e1ement 4 or 6
.7
t---:.3
t:
---+---6--i
8
7
9
.2
•1
t
scan angle
.._.......--L.
(1
I
I
<
I
I
L
I
I
I
l _.........J..__ ,J
I
&
..J.----1--1
10 20 30 40 50 60 70 80 90 100
FIG 6.16; reflection coefficient vs~scan angle
WR-90 waveguide circumferential
orientation E-scan 3x3 array KR=l00
83
Reflection
coe·ff i c f ent
edge element (3,9)
.7
.~--
.6
center element (5)
.5
-----
.4
.3
~~~~~-+-4~~~~.~-L-L
0
10 20 30
40 50 60 70
l
80
l
l
I
I
90 100
FIG 6.17; reflection coefficient vs.scan angle
WR-90 waveguide circumferential
orientation H-scan 3x3 array KR=100
84
Reflection
coe·tf f c f ent
1
.9
.8
/'
.7
8-th element
.6
.5
I
.4
_/
.3
•2
•1
l
0
I
I
I
I
I
10 20 30
FIG 6.18;
/
/
..
scan angle
l
I
.!..-L~-1
40 50
60
I
70
I
I
80
I
l
I
I
90 100
maximal reflection coefficient vs.
scan angle WR-90 waveguide circumferential
orientation
E-scan 5X5 array KR=100
85
Reflection
coef'ffcfent
1
.9
1
2
3
4
5
6
7
8
9 10
1 1 12 - 13 14 15
16 17 18 19 20
21 22 23 24 25
.8
.7
.6
center element (13)
.5
.4
.3
edge element (15)
.2
•1
scan ang1 e
&.-.-L...-&.---.L..-&..... I
I
'
I
I
I .
I
I
I
I
I
I
I
I
I
10 20 30 40 50 60 70 80 90 100
FIG 6.19;
reflcctfon coefficient vs. scan angle
HR-90 waveguide cfrcumferentfal
orientation E-scan 5X5 array KR~100
86
Reflection
coefficient
1
.9
.8
.7
~
.6
6-·th
.4
.3
fl
.2
•1
~--..+-L-1.
0
FIG 6.20;
'
I
&
I
I
l
~-th
e 1 ement
scan angle
'
L
I
. I
L
~-L-4.-1
10 20 30 40 50 60 70 80 90 100
reflection coefficient vs. sc~n angle
WR-90 wavegu1de axial orientation
E-scan 7xl linear array KR=100
87
Ref 'I ect ion
coe-F'f 1 c f en·t
1
.9
.8
.7
7
6
5
4
3
.s
.5
7-th element
F-~
.4
.3
:1
element
.2
•1
scan ang 1e
l
I
l
'
I
' ......1- "
L
I
-L...+.-1.-...~-..L-&......-l
10 20 30 40 50 60 70 80 90 100
FIG 6.21;
reflectfon coefffcfent vs. scan angle
WR-90 waveguide axial orientation
E-scan 7X1 linear array KR=100
88
Re·f 1ect 1 on
coe.fffcfent
1
.a
.7
.6
.5
.4
center element
7
6
5
4
~~
2
1
--
.3
angle
FIG
H~22;
reflection coefficient vs. scan angle
HR-90 waveguide axial ortentatton
E-scan 7X1 lfnear array KR=314
I
•
89
Reflect fen
coe·f'f 1 c f ent
.5
7-th element
.4
~
.3
---
1-st element
__·~
.2
.
.__
•1
&..---1~-..........t.
0
FIG 6.23;
10 2 0
~I
&
I'
I
I
l
l.l
30 40 50 6 0 7 0
.
I
I
l
scan angle
l..a-.J
80 90 100
reflection coefficient vs. scan angle
NR-90 waveguide axfal orientation
E-scan 7X1 linear array KR~314
90
Reflection
coa-f•r i c f ent
.8
.7
.6 ~
K5
.. 5
4
3
.4
2
1
.3
center element
.. 2
scan ang 1e
1---'---1'- r--.4---4-~r....-41-----4-'-~~~
10 20 30 40 50 60 70 80 90 100
FIG 6.24;
reflection coefficient vs. scan angle
WR-90 wav~gufda axial orientation
E-scan 21X1 linear array KR=314
91
R~_,lcc'tion
coefftct~nt
.5
.4
21-at element
.3
.. 2
•1
:can angle
~~~~~~~~~~~~.~~~~·
1m 20 aa 40 sa sra 70 ea
m~
tee
cc~.ff 1e hmt. v~. $Cnn t'J.t'lg hl
WR-Se u;~vegu h"'hl ~x 1 31 cH" t ant.rr~ ton t::-t.Jc;~n
21x1 array KRm180 0.8 wave1cn~th ~paofng
f--IG S. 25; raf 1elC't hm
92
efficiency
1
.9
.8
.7
.6
.5
.4
.3
.2
1
•
L .·
0
FIG 6.26;
_,_;can
angle
10 20 30 40 sa -s0--70--aa----s-e---niJ0______ ------ --'-L-o-l
1
•
'
1
'
1
1
Efficiency vs. scan angle of a 3X3
WR-90 waveguide cfrcumfertfal
orientation E-scan KR~l00
93
efffc1ency
1
.9
.7
.s
.5
.4
,3
.2
•1
scan angle
L~
0
FIG 6.27;
·l
I
l
.&"l
I
I
I
l
I
J.
I
..L....&........l.-.L-.-1.......&..--.J
10 20 30 40 50 60 70 80 90 100
Efficiency vs. scan angle of a 5X5
WR-90 waveguide c1rcumfertfal
or1enta~ion E-scan
KR~100
94
ef'f1cfency
1
.9
.7
.s
.5
.. 4
:: ~
•1
f
L
0
FIG 6.28;
scan ang l a
I
I
I
1......&. l
I
I
I
1
'
I .'
I
..
I
I
L &..-J
10 20 30 40 50 60 70 80 90 100
Efficiency vs. scan anglo of a 7X1
WR-90 waveguide axfal orientation
E-scan KR...,100
REFERENCES
1.
Borgiotti, Giorgio V., ~~~1odal analysis of periodic planar
phased arrays of apertures", Proceedinfl..Qf_ the IEEE,
Vol. 56, No. 11, (November 1968).
2.
Hessel, A., ·~1utual coupling effects in circular arrays on
cylindrical surfaces - Aperture design implication and
analysis .. , Phased 1rrays Antenna, (Artech House:
r~1assachusetts 1972 pp. 273-91).
'
3.
Sureau, J.C. and Hessel, A., 11 Realized gain function for
cylindrical arrays of open ended waveguides .. , Phased
~\!ray Antennas, (Artech House: r~assachusetts 1970.
4.
Shapira, J., Felsen, Leopold B, and Hessel, Alexander,
"Ray analysis of conforma antenna arrays
IEEE
_tran~~ctions on antenna_~j- propagation, Vo~p-·25,
No. 5, (September 1977).
11
11
,
5.
Shapira, Joseph, "Surface ray analysis of mutually coupled
an·ays on variable curvature cylindrical surfaces",
Ray analysis of conformal antenna arra~, (Polytechnic
Institute of New York, PH.D. June 1974 .
6.
Steyskal, Hans, "Analysis of circular waveguide arrays on
cylinders", IEEE transactions on antenna and propagation,
Vol. Ap-25, No. 5, (September 1977).
7.
Pathak, Probhakar H. and Kouyoumjian, Robert G., An analysis
of the radiation from apertures in cur'ved surfaces by
the geometrical theory of diffraction .. , Proceedina of
th~)EEE, Vol. 62, No. 11, (November 1974}.
8.
Fock, V.A., "Electromagnetic difft·action and propagat·ion
problems:, (New York; Pergamon, 1965).
9.
Kurokaltla, K., "Po\'Jer waves and the Scatter·ing
IEEE Transaction on r:iicrowave Theory and
Vol. MTT-13, No.4~ (July 197g-;.-
10.
11
Collin and Zuckel', Antenna
New York
95
Them~y,
Part 1,
~·1atriX ,
Technique~,
11
f~cGraw-Hill:
96
11.
Lee,
Shung-~Ju and Sa fa vi -Na i ni, Safi eddi n, "Approximate
asymptotic solution of surface field due to a magnetic
dipole on a cylinder", IEEE Transactions on Antennas and
Propagation, Vol. Ap-56, No.4, (July 1978).
12.
Lee, S.H., "GTD solution of slot admittance on a cone or
cylinder", El ectrornagneti c Lab, Dept. of El ec. Engr.
University of Illinois, Urbana - Champaign, 1977
(Contrac~ No. N0019-77-C-0127).
13.
Lee, S. ~J., ~~~iutual admittance between slots on a cylinder
or cone 11 , Electromagnetic Lab, Dept. of E1 ec. Engr.
University of Illinois, Urbana - Champaign, 1977
(Contract No. N0019-77-0127).
14.
Borgoitti, Giorgio V., "A novel expression for the mutual
admittance of planar radiating elements", IEEE Transaction on Antennas and Propagation, Vol. Ap-16, No.3,
Tt~ay 1968) .
15.
Pries, Douglas H., "Toplitz f,1atrix: its occurrence in antenna
problems and a rapid inversion algorithm", IEEE Transactions on antenna and propagation, (March 1972), Vol.
Ap-19.
16..
Sinnot, D.H., "r•iatrix Analysis of 1 inear antenna arrays of
equally spaced elements", IEEE Transactions on Antenna
and propagation, Vol. Ap-20. (May 1973).
17.
Marcuvitz, N., Waveguide Handbook, (McGraw-Hill:
1951).
18.
Zaghlovo, Amir and r~acpphie, Robert, Analysis of Finite
phased arrays of narrow slots using correlation matrix
method", IEEE Transactions on Antennas and Propagation,
Vol. Ap-27, No. 2, (March 1979).
19.
Burnside, t4.D., "Principal plane patterns analysis of an aircraft antenna", (short course notes given in Ohio State
University).
20.
Morse, Philip M. and Fesbach, Herman, Methods of Theoretical
f!!L~ts:s, Part 1, McGrav1-Hill: New York, {1953).
New York,
11
APPENDIX A
THE ROLE OF THE GENERATOR H1PEDANCE
98
Appendix A
THE ROLE OF THE GENERATOR IMPEDANCE
The matched condition for
an
arbitrary input power is obtained
if the reflected power is zero, i.e.
r- = o
(A.1)
where, 0 is the null matrix.
Or more specifically, substitute
Equation 2.14 into Equations 2.13 and 2.12
(Y - Yg *)V
=
(A.2)
0
(A.3)
A.2 has to be valid for all possible excitations, i.e. for
various r+, this is possible for V ~ 0, only if,
Det(Y - Yg*)
= 0
(Det represents determinant)
(A.4)
But, A.2 is valid if the matrix (Y-Y *) has at least two depen--
9
dent rows or columns.
Since Y is the admittance matrix of an
arbitrary array the rows or columns are mutually exclusive, thus
for· Y,
Det(Y) /:
0
(A. 5)
and· hence,
Det(Y - Yg*)
/: 0
(A.6)
99
Subtraction of a diagonal matrix Yg from Y does not alter the
property in Equation A.4.
The conclusion is that Equation A.2 is false, because it does
not exist for V ~ 0, therefore, the assumption in Equation A.l
(I-
=
0) that a
p~rfect
match can be obtained for all excitations
is invalid; instead the following argument will show that a perfeet match can be achieved only at one particular scan angle.
Let us substitute Equation A.3 into Equation A.2 and set the
excitation matrix to be,
I+
==
I +
(A.7)
0
I 0 + is the excitation required to scan the array to a specific
angle e0 •
'(A.8)
Equation A.7 is the match condition for scan angle 00 and it has
to be solved for
v9 . v9 in this case will be the required gen-
erator admittance for a perfect match at scan angle 00 .
Solving
Equation A.7 is a tedious task, however, as shown in Section 5.2,
a simple numerical method of setting
angle can be used.
v to match at some specific
9
APPENDIX B
THE GREEN S FUNCTION "G"
1
101
Appendix B
THE GREEN'S FUNCTION "G"
Consider Figure 8.1, which shows a conductive curved surface
on which an infinJtesimal magnetic dipole moment is positioned
at point Q'; at another point Q on this surface, the magnetic
field (Green's function) has to be calculated.
Based on the GTD, the major contribution to the magnetic field
at this point will be through the ray traveling on geodesics
along the curved surface, {Figure B.l).
Pathak and Kouyoumjian [7] obtained an expression for the surface
current (magnetic field on a conductor) due to an infinitesimal
magnetic dipole moment on a cylinder.
Based on their derivation
and others [11,12,13], the dyadic Green's function for an infinites·imal dipole on a cylinder is given by [11], (Figure 8.2)
-
g(Q)
Ito
A
A
lt.
= dM·(b'bHb + t'tHt)
(8.1)
\vhere the magnetic current density associated with an aperture
is given by
-M(Q I)
and the infinitesimal magnetic dipole moment is given
-
dt~
- M(Q')dA'
(8.2)
by
(8.3)
102
shadow
FIG lJ .. l
region
••
Ray
contour~
on
~
a magnetic current
cy1inder due to
~ourca
103
z
"t
/
"'b
,..,
n
,.
n
,.
t
riG B.2
:
Coord1nute ay~t®m for ray
on a cy1 inder
geode~tcs
L----~---~-·---~-~·----- . ·- -
104
" n,"
t,
"
The tangent, normal and bi norma 1 at Q,
=
b
(Figure 8.2)/
A
.1-l
1..
'
n"
"bl
I'
The magnetic
The tangent, normal and binormal at Ql.
=
fi~ld
at point Q (Q
j
Q
on the curved surface
1
)
is given by [10]
H1 ( Q)
=
f g•M(
QI ) dA .
(8.4)
A1
When expanding the dyadic expression for the Green s function
1
H1(Q) becomes [20, pp 54, 55]
H1(Q) =
f [M (
QI
) •
bJb
I
Hb +
[M (Q t ] t
I ) •
I
Ht} dA
I
(
8 . 5)
A
Substitution of Equation 8.5 into Equation 4.6 becomes the expression for the mutual admittance for any arbitrary two slots
on a curved surface, i.e. (Figure 8.3, Q s to aperture 1)
1
v1 , 2
:
-:
JJe
2
1
x
[CMrb 2 Jb 1Hb
+
(M1 .t2 Jt 1Ht]dA1 .dA2
A1 A2
(8.6)
Note: M{Q
1
)
=
M1
105
FIG 8.3
:
Coordinate system and ray geodesics
for two cfrcumferentfal slots on a
cylinder
106
For two circumferential slots (Figure 8.3)
(B. 7)
and
(8.8)
In the ray coordinate system (Figure 8.3),
o1 and z2 are
given
by
(8.9)
A
z2
=
A
A
. (8.10)
SINSt 2 + COSSb 2
and hence, the expression for the
~utual
admittance can be dev-
eloped as follows:
II
II.
[(M 1SINSHb)b 1 + (M 1COSsHt)t 1]dA 1dA 2
Y1 , 2 " -:
1
~ ~ e 2M 1 (SINs~ 2 + COSsb 2)x(HbSINsb1 + HCOSs~ 1 )
A1 A2
X dA1dA2
y
1,2
" -:
1
~
J
A1 A2
2
2
ezM1 (HbSIN s + HtCOS s)dA1 dAz
(8.11)
107
For two axial slots, we have
(8.12)
Following the previous development the mutual admittance for two
axial slots is
(8.13)
Let
(8.14)
and
(8.15)
be the scalar Green's function of a circumferential and axial
slot respectively.
The longitudinal and transverse components of the scalar Green's
(8.16)
108
p
Hb(S,E:,R)
(v(E:)
=
2
+ (1-2j/ks) U (E:) + j("\"2kRt)- /3 U'
(d](j/ks)G(s)
(B.17)
and
= k2Z0 exp(-jks)/(2rrjks)
G(s)
The surface ray parameters will now be defined.
(8.18)
The arc
length of the geodesic is (Figure 8.2)
s
:::
(B.19)
The curvature parameters of a surface ray are
Rt
= Radius of curvature in the longitudinal direction
of the surface ray (t direction).
Rb -
Radius of curvature in the transverse directidn
of the surface ray {b direction).
Their value is calculated from the radius of the cylinder by
2
R
t
=
s
(8.20)
Rb
= R/SIN 2s
(8.21)
R/COS
A plane tangent to the infinitesimal magnetic current source
divides the external region to the conductive cylinder into,
an illuminated region, shadow region, and a boundary region which
.
109
is extended into both the illuminated and the shadow region.
These three regions are defined by the distance parameter
c.
along the surface ray between the source and the field point,
and is given by
{8.22)
The regions are defined as follows:
E
= 0
E
<
1
Defines the transition region.
E:
::=:
1
Defines the shadow region.
Defines the illuminated region.
The function U and V and their derivative U' and V' are the
known Fock functions [8] for complex t, real c and given as
follows:
1
= -Jdzexp{ tz-1/3z3)
-y;1
{8.22)
3
w2{t) =-v:rr·Jdzexp(tz-l/3Z ) = W1*(t)
{8.23)
r2
The integration contour r 1 goes from 0 to
Arg z
=
oo
along the line
{8.24)
-2n/3
The integration contour r 2 goes from 0 to
oo
along the line
110
Arg z = 2rr/3
(B.25)
U and V are given by
V(E:)
(B.26)
U(s)
(B. 27)
(B.28)
The residue series representation of the Fock's functions for
s
~
0 are
v(E) =
U(s)
exp[-jTI/4-Vrrs
%
-1
00
(t' ) exp(-jst' )l
n=1
n
nj
%
-1
E(t' ) exp(-jstn)
= exp(jrr/4)2vrrc n=1
n
L:
%oo
v.1 (t:) =
exp(rr/4)2VTis
V' (E)
0.5exp(-jrr/4}-Vns-~ L:
L:
n=1
00
n=l
exp(-jst'n)
U'(E:)
(B.30)
(B.31)
exp(-jst'n)
1
=
(B.29)
(1-j2 t'n)(t' )
n
(8.31)
(B.32)
111
For small positive s, s
>
0 the value of the Fock's functions
and their derivatives are given by
V(s)
~ 1-~/4exp(jn/4)s~
+ 7j/60s 3 + 7yrr /512 exp
(-jn/40)s ~-4.141 ·10- 3 •s 6
(8.33)
U(s)
~
3
.
1 +fi /4exp(jn/4)s ~ + 5j I 12s 3 +5/TI/64
exp(-jn/4)·s~ -3.701 • 10-2. s 6
(8.34)
V(s)
~ 1 + /Tr/2exp(jn/4)s~- 7j/12E: 3 + 7/rr /64E:% +
4.5 • 10-2 • s6
(8.35)
V'(s)
~
3/81n exp(-j3rr/4) E:
2.485 • 10- 2 - € 5
% + 7j/20E: 2 + 63/'IT /1024E: v:2
-
(8.36)
U'(s)
~
3/4/IT exp(-j3n/4) + 5j/4E: 2 + 45/IT /128exp
(-jn/4) - 2.221 • 10-1 • E:5
(8. 37)
Where (t) and (tn)'s are zeros of W2 (t) and
and they are tabulated in [8].
w2 •(t),
respectively,
APPENDIX C
COMPUTER PROGRAM TO CALCULATE Y12
112
113
START
MUTUAL ADMITTANCE
COMPUTATIONS
SET INTEGRATION
SUB AREAS TO 14
.
CALCULATE
N'S FUNCTION
AIRS OF SUB AREA
.;;>---f!lilili'!
PRINT
RESULTS
FOR
y12
USE SMALL ARGUMENT
EXPANSION FOR THE
FOCK FUNCTIONS
USE THE
SERIES
REP RES EN
TATION
FOR THE
FOCK
FUNCTION
PRO~RAH
FIE:..O
14/llt
PROGRAM
REAL K
l
OPT=l.
fl~LOCINPUT,OUJPUTJ
REAL Ki<.
COMPLEX HFrtl~ZYl2DtZSUH
C0;1PLEX
l HF I
COMPLEX JJ,HB,HT,V,Vl,UtJZ,VZ,G
COMPLEX Al,A2,Al
COM~ON /CF/V,U,Vl,V2,U l
COMMON AltA2,A3,JJ,F
REAL KRT,KR8,H 1 KS
10
REAL TN(lO),TNPl(l))
/DATAl/TN,TNPI
~OMMON
DATA TN /2.33811,4.08795,5.5215b,b.78671,7.99417,
$9.02265tl0.04017,ll.008~2,ll.93oO?,l2.8l878 I
OATA TNPI/ l.Oa79,3.2420 t4o81010•6Qlb331,7.37218,
*8.48849,9.53545,l0.~27bbtll.4750&,t2.38~79
PRINT
~
zo
5,T~
(lX,*TN•,lOF8.5J
FO~MAT
PRINT b,TNPI
b FORMAT tlXt*TNPI*,lOF8.5a
Kf<.=lOO ..
PA1:4.•ATAN{l.)
JJ""ChPLX(O.,l .. J
Al~SORTtPAit*CEXP(JJ*PA!/4.)
l5
A2=SORT(PAIJ¥CEXP(-JJ*P41/4.)
A3=SQRTtPAII•CEXPl-3e*JJ*PAl/4~i
3J
THETA=O.
FHI-=2 .. 74
F=9.
-RON=PAI/160.
K=l.•PAI/1.3123
OEG .:180./PAI
ZO=O.O
DO LtOO
II:l,j
fHl=l9.18
IFtlO.EJ.O.OOl FHI=2.70
00 lOU I=l~2
YO=KR*Frli•RoN
IP1.:;;7
IP2=2
/K
IPJ=-7
l P"t=2
All=0.4
A22:::Q.,4
811= .. 9
822==.9
OZl=A 11/ I P2
OY1=6ll/ IPl
jJ
Oll.=--A22/lP4
OYZ=B22/lP1
ZLl=-All/2.-DZl/Z.
Yll=-Bll/2.-DYl/2.
lL2~-A22/2.-DZ212~
YL2=-B22/2.-DYZ/2o
1SUt1=0.,
00 lG
l
~l=l,IP2
l=lLl~Dll*Nl
I
PROGRAM FIELD
74/74
115
Dt' T ::r: 1
FIN
~.t.b
N2,..ltlP 1
Yl==YLl+DYl*N.2
DOlO
00 30
N3=l,IP~t
l2=ZL2+.0Z2.•N3
00 ItO N4=lt1P3
Yl=YL2+0Y2•N4
T~Sl =K~(ZO +Z2-ll)
TKS¥ =K•tYO +Yl-Yl.
KS=TKSI•+z + TKSY~•2
K~=SURTtKSI
THETA ;ATAM2lTK5Z,TKSY)
·IFtABStSIJHTHETA) a .LT.O.OU
HtETA=O.Ol
IF(ASS{COStTHETA~).LT.O.Oll
T~ETA=l.5b
lJ
KRT=KR/COSlTHETAl~•z
KRB;KR/SINtTHETAJ~~l
M=t0.5•~RTI**tl./3.t
P=M'~<KS/i(RT
If (P.LT .. 0.7)
CALL FOCK IP t
1'>
~0
fU 1
TO 2
CALL FOCKl(P)
GtJ
1
Z
RTRB=KRT/K~B
HB=-ttl.-JJ/KS)+\1 -J/KSHZ.
+JJI(SJRT{2.
t'~<(RJ)t<t-(2./3
i*V2 +JJ/(S~RTt2.J*<RT)*~(2./3.)*RT~8*UZ)•GtKSt
HT=t JJ/KSJ*(V + (l.-2.•JJ/KSl*u •JJ/(SGRJt2.)•(RTl~*
*
*U2 )'i<G{ KSt
AHB =CAtlStH5 t
f.b
AtH= CABSHtTJ
c
.- CALCU:..ATiuN
THIS
HFHI
IS
:HO•tSlHlTHETl))*•~
FOR
CI~CUMF
SLOT
+iT•t:~SlfHETA)l*•l
ZSUH=ZSUM+CUS(PAI/Bll•Yl)*:OSlPAI/jZZ•Y~)*ifHl
COI'H
It)
9,)
H!U~
30 CONTINUE
20 CONI HIUE
10 CO fiT I NUE
ZYlZO:ZSU~*DZl*DYl•DZZ•DYZ*(-2.)/)0RTiAll•A22~bll~bZ
ZY120=ZY12D*o.45~=4
XMAG :CABStZYlZD)
PHASE =ATAN2(AlMAGtZY12J),REAL{LYllJ))•JEG
9j
DB=
20.*ALOG10tXMA~)
PRINT
150
l50~ZO
f0KMATtlX 1 •ZO=*t2Xtf8.3l
lOv
PRINT 200
200 FORMATilOKt*Yl2 FO~ ZO=O *l
PRINT l02 9 lY12D,XMAG,PHASE,DS,Fil
lOl FORMAT (lX 7 *Yl2•tlX,(El~.4tlX,*J*,~l2~41,4X,*MA~Yll*,
*
10')
203
PHASE*t2X,F7.2,*D3*,2X,f7o2,*F10*,LX,F7.2l
PRHH 203.,riFHI
FORMATtBX 7 *HFHI~,4X,{F7.l,lX~•Jt,l~~F7.L)}
FHI=FHI +2.74
100 CONTINUE
llJ
ZO==l0+0 .. 5
400 CDI'HlN\JE
S TOt'
ENu
SJ.lkOUTINE FOC<
116
D?T=l
14/7ft
SJBROJTINE FOCX(X)
l
REAL TNC10)y
TNPI(l~)
COMMON Al,A2~A3,JJ,F
,PA!
COMMON /CF/V,U,Vl,V2,UZ
COMPLEX AltA2tA3,JJ,VtJ,V2,Ul,U2,Cl
COH"ON /DATAl/ TN,TNPI
fl:::.:SORT(X)
V=O ..
U==O.
li.J
v 1=0.
VZ==O.
U.2==0 ..
Cl=CEXPfCHPLXtJ.O,-PAI/3.)1
00 20 i'ol:::l,lO
15
ZTrt=TlH tH*Cl
ZINPI==TNPI(NJ*Cl
C3=CEXPfCMPLXCO.O,-X)•lTS?Il
C4=CEXP{CMPLX(O.O,-X)*lTN)
V='V+C3/ZTNPI
U=U+ C4
Vl= Vl•C3
V2 +(
V2=
l.O-CMP_X{0.0,2.~X)•lTN~It•CJ/ZINPI
U2=(l.O-C1PLX{0.0,2.•Xt3.t*lTNl*L~
20
CONTINU~
V=V{l;AL•fl
3J
U=U*Al•F3t2.
Vl=Vl*A't*F3*2.
Vl=V2*0.5*A2/Fl
Ul=U2f"fl\\A1*3.
RETURN
END
+U2
.
FTN 4 .. t
1
117
74/7't
SJlROUTINE FOC<l
rTN 4.6
SU3ROUfiNE FOC~l(~l
COMMON /Cf/V,U,Vl,Vl,Ul
COMMON Al,A2,A3,JJ,F
,PAl
COMPLEX AltA2tA3,JJ,V,J,Vc,Ul,JZ,:l
V=l.-O.l5*~l*P*•l.5+JJ•7./b0.*?~+3•7.•A2/5i2.
••P••4.S-4.141E-3*P~•o
*b
U=
l.-Al*0.5*?**l.5+5./l2.~JJ*P*~3+5./64.¥A2*?•*4•5
•*P.tc•b
1)
Vl=
V2=
l.+0.5*Al•P+•l.5-1./l2.*P•~3-7./64.*~2•P•~4.5~4
3.*A3/8.•P+•J.5+7.*JJ/20.*P••2+63./l024.•A2•P••
•.f -l* p * *5
Ul= 3.*A314.•P•*0.5+5.*JJf4.*P•*2•45./128.*A2~P**3•5
l-2.221E-l*P**5
15
RETURN
ENfJ
.·
APPENDIX D
COMPUTER PROGRAM TO CALCULATE THE ACTIVE
118
AD~1ITTANCE
119
START ACTIVE
ADMITTft.NCE
COf~PUTATIONS
READ Y
SET CONSTANTS
r-
[ liN!~~ J
,-=r_, .
I
SCAN
~
FROM
90 D-EG--•_
___.
. __., .l
CGr1iPUTE:
ACTIVE ADr>'liTTANCE
REFLECTION
COEFFICIENT
_j
_§8_lli_P~~TJERNS .
-L--·---
.
.
END
L_.____
120
1't/7lt
PRJGRAM AR~AYllNPUT,OUTPUTt
DIMEN51UN AG(25),Z~C25),AV~t25t,ZV~i25),F(37) 9 L(50),
COMPLEX Yt25,25l,Yfl(5,5,5),YGlL5t~5J,YYGt25 9 Z~),YY~
COMPLEX TESTC25,25ltVNtZ5),!G(2j)yi{25Jtll(25J,YAC(L
COMPLEX C(25),S{5),fl(37),JJ
COMPLEX YGU25)
CDK=4.7d8
J.
PAI=~t.•ATAKtl.)
JJ=CHPLXfO .. .,l.J
lJ
Al=O.O
TE==O ..
· RDN=PAI/180.
DEG=l.,/RDN
Kl=5
M2=~
c
INPUT DATA·
REAO•,(YGI(J),J~l,Z5J
READ•s(Y(l 9 Jlt,Jl=ltKl),{YlltJ2~tJ2:l,Kl!,(Yt3,J3),J
l'J
C
MA~
•(Y(4,J4),J4mltKlt,CY<5,J5),J5=l,KlJ
ELEHENT NO IN FIRST COLOH
NO==l+( K 1-U •Kl
PRlNT•,tYGl(Jt,J=l»25)
DO lll=l .. Kl
PRINT ~til,(Yl!I,J),J=l,Kl)
2.5
FOKMATClXt*Y*t12,5tE12.4,El2.4~t
-~
CONTINUE
1
C
MO
C
CO~VERT
ARRAY
IHMENS ION
Hu,.Kl¥Kl
DD
b
~
3S
DATA
CALL
3J
C
it 0
TO HATRIX FQRM
ARNGfYYlyKl,Y~
5 I= ltKl
PRINT b,I,CtYYl(I,Il,JlJ,Jl~l,Kl),ll=l5<1J
fORHATtlX,*YY*t!2,5(El2.4,El2.43i
CONTINUE
BUILD LDW£R PORTIO~
DO 10 Il.=l,H2
f13::.fi2-Il-t-l
DO
DO
20 12-,l,HJ
30 13=1,!<1
00
itO
*
14==1-,Kl
Y ( 1 3 + M2 Ol + I 2-2 ) , I 4 + M2
CONI HWE
30
CONTINUE
lO
CONTINUE
10
CONTINUE
PORTION
C fiLL UPER
00
00
~0
5J
C
MATRIX
-1 ) J = Y n L 1 2 , I 3 ' I 4 .)
Y
50 I5==l,MO
50 J5=l,H!J
Y(l5,J5)
DEFINE
NAfKIX
*( H
Y
3
Y(J5~L5)
MATRIX
YY~
DO oOio"'·'ltMO
D 0 6 0 Jb"' 1., r1 (J
YGtl6,JoJ~to.o1o.o~
IFt16.EO«JbJ
bO
tGiib,JoJ~YGl(lbl
YYG(l6,J6)=YG{lb,Jb) + VCI&,JbJ
YYGlfi6tJ6)=YYGli6,Jb)
CALL
CM!N'ItYYG,MDtLstl)
PROG.<AN ARRAY.
OPT== l.
00
121
500 KMT=l,l)
A-=-SltHAH•COX
DO
110 Kl=l,Kl
t.J
DO
110 Kl=l,NO,Kl
lG(K2+K3-lJ=CEXP(JJ•(K2-lJ•Al
110
C0 NT I NU E
CALL CEFLHT (YY~tiG,V,NJ,MJ,lJ
CALL CcFLHT(Y,V,IL,MO,MU,ll
DO bOO KI=l,MO
YAC(Kll=Il(KI)/VtKIJ
G(KIJ=(YAC(Kll-CONJG(YGL~I,~l))!/lYAC{KlJ+YGlKl,KI))
AGtKIJ=CARSlG(Kltt
·lG(Klt=ATA~2(AI~AG(G(KlJ),KEAL(Gl~lJI)*)EG
7iJ
VNlKll~z.•REAL(YG((!,Kl))/(YAC(KI)•YG{Kl,KIJ)•lG(KlJ
AVI'H K I l =CAS S ( VtH KIt l
ZVNlKIJ=ATAN2(AlMA~(VNtKilJ,REAL(Vi(Kl)))*lEG
bOO
15
CONTINuE
lVIT"'Al~OEi;
200
202
203
PRINT lOthlVIT
FORMAT (lX,•SCA~ A~GLE ; •,F7.31
PRINT 202,(AVN(J),J=l,MuJ
FORHATC1X,¥AVN*tl3=7.3//t
PRIKTZOJ,(ZVN{J),J=l,MOJ
fORMATflX.•ZVN(Jl*tl3F9.3/I)
PRINT
20~
FORMAT
204~tAG(JJ,J=l,HOI
ClX,*AG*wl3~7.3/l,
PRINT 205,{ZG(J),J=ltHOI
205
FOKrlAT
t~Xt*ZG•tl3F9.3//)
PRINT 206,(YACtJ»,J;l7MOt
350
FORMAT llXt*YAC• ,5tE12.~,~12.4l//l
00 350 HC=l.,Kl
StMCl=(O.O,O.O)
DO 400 MA:l,Kl
00 400 MB~l,NOtKl
400
S(MAJ~VN(H3+MA-l)
206
9J
+S{MAJ
Tf=-90o'~'RDN
00 -410 NS=lt37
ftHS)=li .. O
fHMS):{O .. o,o.o;
lOJ
420
DO 420 tH>l,Kl
fl(HS):fliMS) +
410
f(NS,=CAB5tFltHiJ)
TE~TE• 5.•RDN
207
FORMAT(lX~•PATTERN•}
208
500
PRINT 208~(f(M$),M~=l,37)
FORMAT (1X,l3F9.3//)
Al=Al+lO.*RDN
SC~C)•CiXPtJJ~(NC-L)*SlM(TE)•COXi
PRINT 207
10~
STOP
END
I
fTt~ 4 .. 6!
122
O?T=l
St.H>ROUTINE ARNG
l
SUB~OUTINE ARNG (YY~tKl,YA)
COMPLEX YAl25,25),fY5tKltKl,KlJ
C BUILD FIVE TOPLITZ MATRICES
C Hl NO OF BLOC~ T. ~AT~
00 10 H5 3 lt K 1
DO 10 I::.l1tKl
DO 10 J,.l,Kl
C Kl
NO
OF ELEME IN EA:H BLJCK
YY5{M5~l~J)=YAlH5,IAB5(l-J)+lJ
10
CONTINUE
RETURN
··£ 1>1 i.)
fTN 4 .. t
)-
123
SUjKOUTINE CEFLMT
CEF-HT tE,H,R,1tNtL)
COMPLEX EtN,H),H{i,L),RtN,L)
1
SU&~OUT1NE
00
OU
10
20
lJ
10
10
LL:;;:l.,L
NN-=l,N
R(NN,Llt=C~PLX(O.O,O.OJ
00
20
K=l ,L
OU
20
1=-l,N
Oil
20
J=l,H
Rti,KI=Etl,J)~H(J,KJ+~(l,K)
RETURN
END
SlHROUTH~E
124
CMINV
SUdROUTINE CMINV (~,N,L,Ht
COMPLEx Afl),Bl~A,rlOLO
DIMENSION lll),h{lt
l
1\iK:c-f'i
00 80
l'i.
2
l~i
NK .. NK+N
ltKJ=K
11(1(. )=K
KK=NK+-K
BIGA:r.::A{KK)
10
00 20 J:J(,.N
.IZ=ti*(J-lJ
·DO 20 l=lhN
I J:lZ -t-I
15
10 lftCABSt3IGA}-CABS!AiiJ)))
15 BIGA=AtlJ)
15,20~21
LUO=I
tl(K);;:;;J
2Jo
20 CONI I NUE
J=LUO
·-1FtJ-K3 35,35,.25
25 KI=K-N
DO 30 1"" 1,111
KI=KI+N
HOLD=-At KIt
J I -= K I-;{+ J
A{Kl )""At Jl)
30 A(Jlt""HOLD
35 I-=tHK)
!Fll-Kt't5,45,38
3;)
36
JP;::!'i'~<H-U
DO 40 J-=l,N
JK=NK+J
JI=JP+J
HOLO==-A( JK J
ACJK)=AtJH
itO A(J1i===hULO
45 lf(CABS,BIGA)) 48,~ot48
46
PRINT 900
900 FORHATtlHO,•HAT~IX IS SINGULAR•»
RETURN
46 DO 55 I=l•N
I f ( I -K t 5 (}, 5 5 '~ 50
50 IK==NK+I
A(lKt=ACIK)/(-B!GA)
5~
CIJNTINUE
DO 65 1=1,"4
lK=-i'iK+l
HOU>=A {1 K)
.I J:;: I-N
DO 65 J=l,N
I J"' l J+N
lftl-K$ 60,65,6J
60 IFlJ-Kl 6Z,65i6l
62 K J:::: I J-I ·JoK
A{1J)=HUL.DfA{KJt•AIIJJ
65
CONT lNik
9
.
125
SUoROUJINE CNINv
14/14
O?T=l
K J:::i,-N
DO 75 J=l11N
KJ=KJ+N
bJ
lf(J-K) 70,75t7D
70 AtKJ)=AIKJ)/BIGA
7~
CONTINUE
A{KK)=l/SIGA
80 CO:iT!NtJc
lOa K=K-1
..lf(K) 150.,150,1)5
105 l=LtK,
7J
IF(I-Kt 120,120,108
106
JO~N*f l<.-1 J
JR.:.N* U-1)
DO 110
J=l~N
JK=JQ+J
75
HOLD""'At JiO
JI::JR-t-J
. A{ JK)=-A( JI)
110 AtJ!):HOLD
120 J=tHK)
IFCJ-K~
100.100,125
12> KI=K-N
00 !30 I ""l,N
Ki=tU-+N
HDLD=Ail<. I J
85
Jl""i4a--K+J
At K 1) =-A [ J1 J
13) A CH) ""HOLO
GO TO 100
15!) RETURN
90
ENu
:.DJl745,-ol00963i <.OOl778,-o0008687) t.001835,-.0l08733) l.001778,-.ooua6a71
~.GOl692taOJ0008~221 (~Oul57o,.OOOl213J {.00i58lte000126~) l$001576,.0001213:
:vOJl544,-~0007536) (.001628~-.0007769) (.0Jl603,-.)006B4) t.OOl62o,-QOOv77~9J
:»DDl692,.3J0008~22l !e00154b,.ODD1213t l.OOl58l~uOJOl26Y) L.001576,.00012l3J
;9D01745,-.J009b3) (.D0177B.-.oooe&a7) luOD1335,-.o~oa733t l.OOl778,-.oooa5~7J
• l
ol59J~-02
.1551E-02
.9~45E-04
-.1056E-03
.2l6DE-04
.3887E-05
2
.2438~-03
-.b750E-03 -.9825E-04 -.~lB7E-O~
.1952~-04
-~1256~-04
3 -~4036~-03
.1478E-03
.!B74E-04
.1478E-03 -.lj63~-04 -~Z777E-04
' ~
.Zb85~-03
.l358E-03
.1179~-03
-.~943E-04
-.3~62~-0~
.2662E-04
5 -.7139E-04 -.2189E-03 -.143ZE-03 -.39lOE-04
.4304E-04
.~15~E-04
ri l
.15~JE-02
&1551E-02
.9545E-04 -.105&E-03
.l2BJE-04
.3887~-05
.~~45E-O~
-~1056E-03
el590f-02
.1~51~-02
~9)45E-04
-.1056~-03
~lZH
~2280E-04
.3B87E-05
.9545E-04 -01056~-03
.1590~-0l
.lj51~-02
.9~~
.~989E-DS
~B924E-05
.2280E-04
.3B87E-05
.9545E-04 -.1056~-03
.159
9ll33E-O~
-&b208E-05
.&989E-O&
.8924~-05
e2lB0~-04
o3d67E-05
.95~
·y 2
.l433E-03 -.6750E-03 -~9B2~E-04 -.6187~-0~
el95lE-04 -~1256~-04
-e~825E-04
-~blB7~-0~
.2438E-03 -~6750~-03 -.9325~-04 -.6l87E-04
~195
.l952E-D~
-.ll56~-04
-.9825E-04 -.&187~-04
~2438~-03
-.6750i-03 -.9dl
J
., ·-• '>C:.C
._oL
-· _/,] i1
7t=--:-0_'t