The diurnal motion of a synchronous satellite necessitates mechanical tracking when a large aperture, high gain antenna is employed at the earth terminal. An alternative solution to this tracking problem is to use a self phased array consisting of a number of fixed pointed elements, each with moderate directivity. Non-mechanical tracking and adequate directive gain are achieved electronically by phase coherent summing of the element outputs. The element beamwidths provide overlapping area coverage of the satellite motion but introduce a diurnal variation into the array gain. The optimum element beamwidth and pointing direction of these elements can be obtained under the condition that the array gain is maximized simultaneously with the minimization of the diurnal variation

Hodge, D. B.

Theobold, D. M.

English

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' 	 7	 1
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` By	 THE OPTIMIZATION OF SELF-PHASED ARRAYS FOR DIJRNAL MOTION
TRACKING OF SYNCHRONOUS SATELLITES
D.M. Theobold and D.B. Hodge
i
i
The Ohio State University
ElectroSdence Laboratory
Department of Electrical Engineering
Columbus, Ohio 43212
Technical Report 4299-5
June 1977
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TECHNICAL REPORT STANDARD TITLE PAGE
Reprt Ne.	 2 Govetnrnent Accesi,ien Me.
	 3 Rrciprtnt's Cotolog He
i 
	
^an^-^i1^tf!(0 	 5 -Report Dote	 _
THE OPTIMIZATION OF SELF-PHASED ARRAYS FOR 	 June 1977
DIURNAL MOTION TRACKING OF SYNCHRONOUS 	 4 6 Perforrnrna Otgar ilotron Cede
.SATELLITES
bX' Neobold and D.B. Hodge 	 8 Per formi ng Organ ization Report NoESL 4299-5
__4- - -
	
-	 _ -
	 -9. . Ptrbcnring Organization Marne owd Addttss
	 10 Work Unit Ne.
The Ohio State University ElectroScience
Laboratory, Department of Electrical 	 11 Controct or Grant No	 -
Engineering, Columbus, Ohio 43212	 NAS5-22575
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NASA, GSFC	 Technical Report
Greenbelt, Maryland 20771	 i11 Sponsoring Agancy Cade
E. Hirschmann, Code 951, Technical Officer i
1S. Sapleatttttery Na+ae
	 --- --
it
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The diurnal motion of a synchronous satellite necessitates
mechanical tracking when a large aperture, high gain antenna is
employed at the earth terminal. An alternative solution to this
tracking problem is to use a self-phased array consisting of a
number of fixed pointed elements, each with moderate directivity.
Non-mechanical tracking and adequate directive gain are achieved
electronically by phase coherent summing of the element outputs.
The element beamwidths provide overlapping area coverage of the
satellite motion but introduce a diurnal variation into the array
gain.	 '
The optimum element beamwidth and pointing direction of these
elements will be obtained under the condition that the array gain
is maximized simultaneously with the minimization of the diurnal
variation. Optimal design examples will be presented for arrays
consisting of two and four elements tracking several models of
diurnal satellite motion.
17. Kay w«ds (Salteted by AuMa(s}) 	 ^Is. Distribution Statement
Synchronous satellite
self-phased array
Satellite tracking
19. Soewity Claaail. (af this rapers) 	 20. Security Claraif, (of
	
page!
	 X21	 e. Me of Pogts	 122. Pric 'U	 i - _-g- -- - 	 U	  this	
19
"For sale by the Cltociallbouse (of Federal Scientific and Technical Information, Springfield, %uRtnia 22111•
11
TABLE OF CONTENTS
Page
I. INTRODUCTION
	
1
II. GAIN OPTIMIZATION
	
3
III. NUMERICAL DESIGN
	
4
IV. CIRCULAR DIURNAL MOTION
	
6
V. STRAIGHT LINE DIURNAL MOTION
	
9
VI. ADDITIONAL EXAMPLES
	
13
VII. SUMMARY AND REMARKS
	
13
REFERENCE
	
19
t
i	
__
i i i
I.	 INTRODUCTION
The design of an earth-space communication link which utilizes
a synchronous satellite at centimeter and shorter wavelengths presents
the problem of choosing an antenna and tracking method. Although a
synchronous satellite is considered to be "stationary" with respect to
an observation point on the earth, it experiences East-West perturbations
due to earth oblateness and solar radiation and North-South perturbations
resulting from solar and lunar gravitational forces. These motions
appear to an observer at a fixed point in the earth as a diurnal (perio-
dic over twenty-four hours) variation in the apparent look-angle which
projects on a plane normal to the propagation path as an approximate
ellipse with eccentricity from zero to one or as any of a number of
other closed paths. This "diurnal motion" must be tracked if narrow-
beamwidth antennas are chosen for a communication link.
Narrow-beamwidth antennas offer high gain where a large system
gain-bandwidth product is desired. However, their large apertures and
the necessity of mechanically tracking diurnal motion impose high cost
on the antenna and mount equipment. In addition, a failure in the
mechanical system or the control electronics disables the communication
link entirely. A single element of moderate aperture offers an alter-
native tracking method. Such an antenna may be fixed pointed if its
beamwidth is wider than the satellite motion. However, its gain will
be low compared to a large aperture tracking antenna and hence such a
system would be susceptible to precipitation fading. An alternative
to a single large aperture tracking antenna or a single moderate aper-
ture fixed antenna is to utilize a large number of small aperture ele-
ments in a self-phased array. (Figure 1).
A self-phased array antenna provides high gain in a particular
direction by phase coherent summing of the outputs of many small aper-
ture elements, each having wide beamwidth. The array may be fixed
pointed and non-planar, employing the cohering electronics to track
diurnal motion. Although a large array eliminates the need for a track-
ing mount and large aperture antenna, the associated increase in the
number of front ends and element processing circuitry may more than
offset any savings. However, if one element fails, the signal-to-noise
ratio degrades gracefully instead of catastrophically.
In contrast, this paper examines the characteristics of a self-
phased array system consisting of a relatively small number of elements,
each employing an antenna of moderately large aperture and narrow beam-
width (Figure 1).
The small array is designed by choosing an elemental beamwidth
which approximately covers the expected diurnal satellite motion.
Adequate system margin is achieved by adding a sufficient number of
such elements. This compromise design sim)-lifies the circuitry required
for an array with a large number of elements, eliminates the need for
71
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mechanical tracking, and permits graceful degradation in signal-to-noise
ratio with the loss of an element. However, because beamwidth is on the
f	 order of the diurnal satellite motion, the amplitude of the received
signal will be a function of time (with a daily period) and a function
of the geometric factors determined by the stabilization and perturbations
of the satellite orbit. This paper addresses the question of optimizing
the choice of antenna elements and their pointing in order to achieve
maximum system gain margin simultaneously with the minimization of the
diurnal variation.
1I.	 GAIN OPTIMIZATION
Array directive gain is defined as
G a maximum radiation intensity
average radiation intensity
for the array phase-locked to a source within the angular limits of
acquisition of the antenna-receiver system. As the satellite undergoes
its diurnal motion, Gp will vary with time as the array electronically
tracks the satellite. Consider a typical diurnal gain variation as
shown in Figure 2 and define minimum directive gain as
GM = Min	 (Gp(T)t
24 hrs
Fix the number of antenna elements at N and define the half-power ele-
mental beamwidth in elevation as :)e and in azimuth as Yee. Let the
beamwidths be equal and normalize to the largest angular variation of
the satellite motion, .r (refer to Figure 3):
tie- - se - B
	
(2)
a	 ^r
With N fixed, GM may be plotted as a function of B in order to find the
maximum of GM. For the optimization problem, GM will also be a function
of the points g of each element axis in elevation and azimuth (e i and
Bp i for the it^ element). The maximum of GM will be called the optimum
minimum directive gain, GM:
GM(B,N,ei,si) = Max GM (B,N,o i , ^i )	 .
A numerical technique was used to determine B and the e i and s i for
various closed-path models of satellite motion.
3
E.?
Gp(TI
GM
Ilk
o	 a	 12	 19	 24
T ( HRS
Figure 2. Minimum directive gain GM.
III. NUMERICAL DESIGN
Diurnal motion may be approximately modelled by an ellipse with
a major axis of ,% degrees and eccentricity E between Zero and one.
where
Minor Axis
E
^)
(`Major Axis
l	 A Gaussian antenna pattern P i (,, ,,p-)	 for the i th element, defined as
A	 ti	 2
-3[
is used as an acceptable representation of a	 realizable narrow-beam
element. pattern.	 The directive gain for a single element may be shown•
to be, approximately.
i
F
A
i
t
a
i1
of
CL,
 DIURNAL MOTION(#80.880)/l
r	 ^J
8	 ANTENNA
/	 PATTERN
J
CL ANT. AXIS
	 •^+;.8.81 )
Figure 3. Geometric definitions.
5
01,
G	 41253
	 [1] ,	 (5)
De 0 e0
where the half-power beamwidths, ('e and Vi e , are given in degrees.
For N fixed and the diurnal motion restricted to an ellipse of eccen-
trici^y E and major axis , degrees in the S direction, o set of GM(B.
N,e i ,^i) was evaluated for a given beamwidth B. The evaluation was
performed at five minute time increments along the ellipse in order
to find the minimum directive gain G M (B,N,0i,i,_ir). The optimum GM was
obtained as the pointing of all element axes ( ^ i and -ri) was varied.
The examples which follow show GM plotted versus B, which varies from
0.2 to 2.0 in 0.1 unit increments. The same plots also present the
corres ponding optimum element axis angles f'i and si as a function of
B. Also noted on all plots is Max{G ) which is the highest gain real-
izable for a particular ellipse E a 	 number of elements N.
IV.	 CIRCULAR DIURNAL MOTION
Figures 4 and 5 illustrate the optimization )f GM for two and
four antenna elements, Mespectively, where the diurnal motion is de-
scribed by a c..J a (E-1). For N =2 (Figure 4) both elements must be
pointed at the center of the circle and have beamwidth 0.85a in order
^
o achiev the maximum G'	 Similarly, maxim m G' for four elements
Figure 5^ is achieved w1b center pointed e^emeNts of beamwidth 0.85,,.
A tentative conclusion from the numerical model for circular diurnal
motion is that maximum optimal directive gain is achieved with center
pointed elements of beamwidth 0.85.,, independent of N. The proof is
easily shown.
Given the elemental pattern of (4) and directive gain of (5),
let N beams be centered on the diurnal motion path (cai= ^4,i=0). The ith
element voltage is
P•/20
v i = 10	 (6)
and the array coherent output is
PA - 10 1 0910 1(v 1 +v2+...+v N )(v l +v 2
+...+v
N)l 	 (7)
For the circular case, the minimum directive gain is constant at any
point in time, so let
e=0 and t-a/2
^l
I1t
t
!l
(8)
6
a
7T► 	 z	 f
N=2
E=1.00
D
0
0.00
m
0
mO
O ^
0A
N
0
tlN
O
N
`N
w
Im
O
0 . 0	 0.5	 1.0
	 1.5
BEAMW IOTH /a
Figure 4. Array optimization for W2 and E-1.
2.0
0.5
	 1.0	 1.5	 O.0
7
0inr
0
V
AD %
V sp
N=4
E=1.00
O.S
	 1.0
8
E4 X X3
1.5	 2.00.00
m
0
0
U$
d
D 0
0.0	 0.5
	 1.0	 1.5
	 2.0
BEAMW I 0TH /ac
Figure 5. Array optimization for N=4 and E=1.
8
i^
t
t
so that
10 1og10 	10-31208	 N lt' 3!Z06 )	 .	 (9)
I(, il	 a1	 -`
The minimum (constant) directive array gain is then
Gp « 10 10910 41253	 + 10 log10 (N2 . 10-3r 1082)	 (10)
Find the maximum GM and the corresponding 8 by differentiating (10) with
respect to B and equating to zero
0# -1010910 e
	 +	 (i1)
8 , 0.83113	 (12)
and is independent of the number of elements. This value comparef
quite well with that obtained by numerical t,(-hniques for a circular
diurnal motion path.
V.	 STRAIGHT LINE DIURNAL MOTION
The limiting case of straight line (E=O) optimization is presented
in Figures 6 and 7 for N =1 and 4, respectively. The maximum Gay for the
axes remaining center pointed is obtained with S z O.85_; in bath figures.
This breakpoint for the straight line cases is simply a specialized case
of the ;roof for a circle. Equation (8) still describes Mir,G Q " because,
for the straight line case with center pointing, Go is alwa ys greater
than or equal to that value found for a circular path if diameter
Hence, ( 12) holds for straight line diurnal motion with N elements
center pointed.
For the straight line cases with the axis pointing optimi:'ed, the
maximum G' is obtained for the element axes separated and having beam_
widths of"the order ./N. For N-4 (Figure 7), it appears that the four
axes are equally spaced along the diurnal p^A h.
To obtaimi a more quantitative description of these cases, let the
axes of the buns be on the satellite path (i.e.. ^ i -0 and -,^^ y i,1 ) .
Restrict N to an even number and assume that the
	 correspondine to the
maximum optimal G^ are distributed symmetrically a out s =0
9
0
MWN/r1
iY
OV
i %
♦ ceV
N=2
E=O,00
r^
0
0.0	 0.5
	 1.0	 1.5	 2.0
w
0
0
ia:
O
0.0	 0.5	 1.0	 1.5
BEAMWIOTH/a
Figure 6. Array optimization for N=2 and E=0.
2.0
10
N=4
E=O .00
//^^
too a 8t _ 0
0.5	 1.0 1.5 2.0
T
SQ Z 39
0
1 Ul
0kI
ZON
Q
it
wwi
Q
i
I.Iw cj
0,0	 0.5
	 1.0	 1.5
	 2.0
BEAMW I OTH la
Figure 7. Array optimization for N-4 and E=0.
11
z )
From ttie ttl)meri aI illod`I. 'AS Nutlk' that the ltl\t 'in t`\ lI t'twt't'n Ad I ,itt'tlt
elements are Ntlual for the tlhtillldl k-.ist'
A Minimum in the diurnal pattern Of G w ll ottuI 'it ._••t) ht'r.iu"r of
syllvetr y .	 F rtu+l t lie ,j»untp t i till t h.i t ht',imw I d t h w i t t hi' tit •, hr i't•,ier
O.A. the main ttlntrihutitin to G at ;-i) comit', trom the t.ihVITIlt -' um of
t he twit idja_-t*nt beams	 Lb mt) (4) thrtiu,)h ('),
(	 Ol	 Its !Oki 1 0^^ l_ ' ; I
ii
t 10 lr ') 1t^ ^• ' ii)	 lt)	 1 .
0iff01 .01itiatt' (;51 with rt' pt'.t t 	 1, ,ill.( t'tlU,ltt' t0 .'('r'tl to k*+t.ii11 ttit,
Contii t i tin
t	 = 9 1	 to ..!41A	 1 t,1
1 1.eCun.i llliniuurm will t'i.'uI* ,i( ttit' ('11,)p0111t 01	 tti t • ,11 uVllnl	 t 
A.)ain, assume' that ttit' arra\ . Lill tV100tikill t 	 Gill at thi, paint i^ ,^ru+,irili
from the a.t),i.t'nt e'leme'nt ^i - N .'1.	 t l ,i1.) i•11 throukih
G ^ti y	 1	 l ti 1 0 110
+ 111 1ii±)1^1
	
jlii-l: it)h`[1 .'-tN-1
t
1)lttt'rentiatlIlk)
	
w 	 1 rt' pv t t 	 1, an.i ll`.tll+t .0llattik i ll	 till('
tititiills
i '^ ( N"-iN+1) + .\(a-4N1 t.' - 0	 110,1
( 4
i
1
E
k
i
i°
t
4.
E
it
i
1
4
s 1
i
^i
1
1
1
1
e
0
e
The smaller root of (18) represents the distance between the axes of two
adjacent elements. For example, if W, a-0.59a and if N-4, A-0.27x.
Figure 6 indicates that maximum GA occurs when 2^-A E .65a and Figure 7
gives 2^p2-A-.30a. Using (16) and a $0.59a, one obtains B-0.35 ► and for
e-0.27n, B-.16a. This agrees reasonably well with the n umerical model
results of 0.4a and 0.2a beamwidths for N-2 and 4, respectively.
VI. ADDITIONAL EXAMPLES
Two cases of elliptical model results are presented in Figures 8
and 9 for N-2 and diurnal motion eccentricities EA0.75 and 0.25, respec-
tively. Both indicate that the optimal GM for center pointing is still
obtained with B-0.85u since, as in the straight line case, GO is always
greater than or equal to that value obtained for a circular path of
radius a12 with all elements center pointed. However, maximum optimum
gain occurs with the element axes separated and having narrower beam-
widths.
Similar results for N-4 are shown in Figures 10 and 11. Here,
note that a i separation is also required for E-0.15 and B less than
0.45m. The behavior is very similar to that of the elliptical cases
for N-2. Center pointing of all elements requires 8-0.85,a and maximum
optimum gain is achieved with axis separation and narrower beamwidths.
VII. SUMMARY AND REMARKS
An elliptical model of the diurnal motion of a synchronous satel-
lite has been presented. A self-phased, fixed pointed array of a small
number of antenna elements with parabolic gain patterns was proposed
as a means of tracking this motion. An optimization criterion of maxi-
mizing array directive gain with the simultaneous minimization of
diurnal gain variation was defined and applied to the model. Numerical
techniques lead to the conclusion that if all element axes are fixed
pointed at the center of the satellite path, the optimization criteria
are met with element beamwidths chosen to be 0.83 times the maximum
angular extent of the diurnal motion. Nigher gain is possible with
narrower beamwidths and axis separation, but the design requires that
the satellite's diurnal path eccentricity and inclination of the major
axis remain fixed.
A summary of the design results for E-0 and 1 isppresented in
Figure 12. Minimum directive gain is plotted versus Vr x angular extent
of satellite motion in radians x diameter of equivalent total aperture
in wavelengths. Using this common definition, a comparison may be made
with the cases of a single antenna element using mechanical tracking or
a single element fixed pointing at the center of diurnal motion. For
a given equivalent total aperture, a larger number of elements results
in a directive gain closer to the mechanical tracking case and always
13
0N
E = O i 7
as
s
A
i	 0•5	 1.0	 1.5	 2. 0
It
^i
In
NZ
Q
O
0.0	 0.5
	 1.0	 1.5
BEAMW I DTH Ja
Figure 8. Array optimization for N-2 and E-0.75,
14
2.0
N O&2
E=O,s
a.5	 1.0 14 !1!
a
wwa
ICD
0•0	 a•5	 1.0	 1.5
BEMMWIQTNla
Figure 9, Array optimization for N-2 and E-0.25.
15
*W\
ca
11•@
N=ll 
E=O.7S 
8 
0.0 O.S 1.0 t.5 a.o 
I~~~~ .. ~~ .. ~~~~~ 
i~--------------------------~~----------------------------~ o 
aN ~. 
1&-
o 
~----------~=============== ~~~-----------------------------o 
. 
~. 
ICD . 
. 
o_;-~~~~~~~~~~~~~~~ 
I I' • ~ I • • • • 0.0 0.5 1.0 1.5 2.0 
BEAHWIDTH/a 
Figure 10. Array optimizl·;on for N-4 and E-0.75. 
16 
N
E = O a 5
aits0-0.0 Oka.	 1.0
ca
Ino
0.1
Q
1.5 2.0
0
N
1@
u^
0
O
0.0
	 0.5	 1.0	 1.5
BEAMW I 0TH la
Figure 11. Array optimization for N=4 and E•0.25.
17
5.0
I0
In
InAP
P'4
goo
O!
z;z
I
I
in I	 I	 a a A a I III	 -- I 	 a I I A LA.WIV M
0.1	 1.0	 10.0
vrWr x ANGULAR EXTENT OF SATELLITE
MOTION (rod) x DIAMETER EQUIVALENT
TOTAL APERTURE %WAVELENUTMW
Figure 12. Comparison of mechanical tracking, fixed
pointing, and small self-phased arrays.
^^ i
18
r
11	 .^ 	 z
fi


The optimization of self-phased arrays for diurnal motion tracking of synchronous satellites

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