This document analyzes how shock environments affect the main beam characteristics of phase array radars. A numerical model is used to simulate the shock event and calculate the transient displacements of radar array elements. These displacements are then applied to statistical and numerical models to determine how they impact key beam parameters like gain, sidelobes, beamwidth, and pointing error. The results show the displacements, especially in the Z direction, degrade beam performance and could compromise radar operation depending on its intended use. Both the statistical (Ruze) and numerical (Monte Carlo) models are evaluated and found to provide reasonably accurate predictions of the beam variations induced by the shock-induced array distortions.

1. 1
Analysis of the Variation in the Main Beam Characteristics of
Phase Array Radars when introduced to a Shock
Environment.
Nicholas J. Manzi, Dung N. Tran, Eugene Ngai
Raytheon IDS
Sudbury MA
Abstract- Phase Array radars are designed to tight
tolerances in order to meet exact design constraints
placed upon the characteristics of their main beam.
A numerical and statistical approach will be utilized
to investigate the variation of the following key
parameters; Gain degradation, Side-lobe
perturbation, Beam width variation and Beam
Pointing Error (BPE). Their subsequent effect on
normal operation of the Radar will be looked at in
detail in this paper.
INTRODUCTION
Three-Dimensional Phased Array Surveillance Radars
are used to search, detect, and track aircraft, missiles,
unmanned aerial vehicles, and heli-copters at long
ranges above the horizon. These Radars are comprised
of numerous radiating elements that are arranged in a
detailed pattern. This detailed design determines many
of the resulting characteristics of the Radar, specifically
Main Beam Intensity, Width and Orientation. Any
measured variation in the of the array will cause the
Radar to perform differently then originally specified.
Since these radars are used in a variety of applications
they are exposed to the variety of environments that the
radar must be designed to withstand. One particular
environment of interest is a shock environment. A
shock event results in a pressure pulse, which will in
turn excite vibrations in the array face. These vibrations
lead to subsequent displacements that will alter the
detailed design of the Array face.
The best representation of this event is created by an
modal transient analysis using MSC/NASTRAN. In this
model each element on the array face exposed to the
shock event pressure is loaded with a triangular
pressure pulse of a given magnitude and duration. This
will accurately predict the dynamic response of the
array face structure subject to the shock event.
Taking the transient displacement data provided by the
MSC/NASTRAN modal solution and applying it to the
radar’s array element spacing will result in a different
array pattern, and consequently a different radiation
field then originally specified. Variation of the
following key parameters: Gain degradation, Side-lobe
perturbation, Beam width variation Beam Pointing
Error (BPE), and Mono Pulse BPE and their subsequent
effect on normal operation of the Radar will be
explained in the following sections.
RADIATED FIELD
The Radiated Field at any given point P(x,y,z) can be
calculated by summing over all array elements [1] and
is given by:
Fig.1 Visual Description of summation over all elements.
( ) ( )rkjaf
r
e
rE
i
ii
jkr
!!
!" #
$
exp,)( %& (1)
Where
( )!",if
is the array element pattern and ai is
the complex excitation. Defining the wave number k
!
and element distance r
!
! as:
)ˆcosˆsinsinˆcos(sin
2
ˆ zyxrkk !"!"!
#
$
++==
!
zzyyxxr iii ˆˆˆ ++=!
!
This leads to the following expression
( )!
"
#
cos
2
iii zvyuxrk ++=$%
!!
(2)
where

2. 2
!"
!"
sinsin
cossin
=
=
v
u
Adding in beam scanning parameters Us and Vs, gives:
( )!
"
#
cos)()(
2
isisi zvvyuuxrk +$+$%&'
!!
(3)
Taking expression (3) and substituting back into (1) and
completing the summation will produce the Radiated
Field for the undisturbed array pattern for any given
beam scan parameters Us and Vs. The next step is to
account for the array panel distortion introduced by the
Shock Event. This is accomplished by redefining (xi, yi,
zi) as follows:
ii
iii
iii
zz
yyy
xxx
=
!+=
!+=
(4)
Applying this to (3) for iii zyx <<!! ,
gives:
( )!
"
#
cos)()(
2
isiiisiii zvyyvyuxxuxrk +$+%+$+%&'(
!!
(5)
Taking expression (5) and substituting back into (1) and
completing the summation will give the Radiated Field
for the excited array pattern for any given beam scan
parameters Us and Vs.
STATISTICAL APPOARCH TO THE RADIATED
FIELD
An alternate approach to handling this surface
distortion created by the shock event was developed by
[2] and compiled in [1] Skolnik’s, Introduction to
Radar Systems. The Radiated Field is expressed as:
21),( GGG +=!" (6)
Where G1 is the attenuated error free pattern:
2
1
),(01
!"
= eGG #$ (7)
And G2 is the attenuated error free pattern:
2
2
2
2
2
2 !
"
#
$
%
&
'
(!
"
#
$
%
&
= )
*
)
*
Cu
e
C
G (8)
For a reflector distortion application, one has:
!
"#~4
21 =$=$=$ (9)
Where !~
is the standard deviation of the reflector
panel distortion measured in the same units as ! . This
distortion deviation will produce the phase front
variance
2
!
. The phase front variance
2
!
is assumed
to be Gaussian. Finally, in equation (8) C represents the
correlation interval of the surface error and !sin=u .
When applying Ruze’s formulation, equation (1), for
array surface distortion a modification is required to the
phase-front variance
2
!
, which yields:
2
2
1
~2
2
1
!
"
#
$
%
&
'(
)
*+
(10)
2
2
2
~2
2
1
!
"
#
$
%
&
'(
)
*+C
(11)
Equation (10) conforms with the standard treatment in
[3]. Equation (4) incorporates the correlation interval of
the surface panel error as expected. The correlation
interval can be approximated as C ≈ 0.8 τc
(12)
(13)
This is illustrated in Fig. 2
F
ig. 2 Correlation interval.
From (10) and (11), the relative magnitude of the
scattered field due to the array panel surface error can
be readily deduced. From the scattered field, general
characteristics (gain degradation, side lobe change,
beam pointing error and beam width change) of the
perturbed radiation can be derived.
( )
( ) !!! "
"
dA
OA
c #
$
$
=
1
( ) ( ) ( )dttt
T
A
T
o
!""
!
!
!
" +
#
= $
#1

3. 3
For a circular symmetric beam, it is easy to realize that
the beam-pointing errors are symmetric and continuous
at the bore site. The fact that the beams are circular
symmetric and the bore site beam-pointing errors are
only dependent on z-directed tolerance constrain the
beam-pointing errors to be invariant at bore site.
ARRAY GEOMETRY AND DISPLACEMENT
DATA
The shock environment was modeled in NASTRAN to
produce a modal transient solution. This solution
provides the displacements resulting from the induced
vibrations from the over pressure on the array face. A
generic 64 by 56 element step circular array with a best
practice 2! element spacing was implemented to
complete the analysis. The displacement data provided
by the MSC/NASTRN (Fig. 3) modal solution needed
to be interpolated from the provided NASTRAN array
to fit the actual radar array spacing. Data filters were
then applied to filter out any disconnects resulting from
this interpolation.
Fig.3 NASTRAN provided Z-Displacement vs. Time
Z-displacements were orders of magnitude higher than X
and Y- displacements and dominated the analysis. The
above graph shows the time history Z-displacements and
identifies points of interests that will be looked at in detail
later in this paper. Taking these displacements and
applying them to the statistical and numerical methods
presented above gives the characteristics of interest
mentioned above.
RESULTS
Looking at the two times of interests (.03 and .06
seconds) called out in the will give a good idea of what
kind of affects this shock environment will have on
overall operation above. Looking first at the spread of
the Z-displacement across the array face (Fig. 4).
As stated in the Ruze’s theory one would expect that
the beam pointing error is dependant upon the deviation
(spread) of the displacement as opposed to the actual
magnitude of the displacement. Performing the
summation over all of the array elements (equation
1and 5)and looking at the bore site will give a different
beam patterns compared to the original beam patterns.
Fig.4 Z-displacements for t=0.03 sec and t=0.06 sec
The influence of the smaller X and Y-Deflections can
be seen in subsequent U and V-Scan Pointing Error.
First looking at the Y-deflection and the V-Scan(Fig. 5)
Fig.5 Y-displacements and resulting BPE V-scan for
t=0.03 sec and t=0.06 sec.
one can see, as slight as it may be, that the larger spread
creates a larger beam shift at various U and V locations:
The same can be said for X-deflection and the U-Scan.
Fig .6 shows precisely this. From this one can deduce

4. 4
that the overall Beam Characteristics are mainly
influenced by the larger Z-displacements and then are
skewed to the left or the right by the spread of the X
and Y-Displacements.
Fig.6 X-displacements and resulting BPE U-scan for
t=0.03 sec and t=0.06 sec.
NUMERICAL (MONTE CARLO) VS. SATISTICAL
(RUZE) COMPARISON
Comparing results obtained by the two different
methods will provide valuable insight on how accurate
Ruze’s statistical approach calculates variations in the
key radar parameters mentioned above. Fig. 7 shows
BPE as a function of time for the two different
methods. Both approaches take their calculations at the
bore sight of the array.
Even though the Ruze calculation neglects any
contribution from the X and Y-distortions it seems to
align quite well with the predicted radiated field
Fig.7 BPE (Ruze vs. Monte Carlo) vs. Time
calculations that do account for these distortions.
Another contributing factor for the slight difference in
the calculations is that when making the Ruze
calculation one must assume a purely Gaussian
distribution in the given distortion. These results seem
to prove that the Ruze approximation is a very accurate
approximation for pointing error results. This clearly
flows down to the Beam Width (BW) variation
parameter (Fig.8). Plotting this Parameter vs. time gives
pretty much the same relation as before. One can see
that the Ruze approximation is more conservative than
that of the Monte Carlo.
Fig.8 BW change% vs. Time
However when it comes to Gain and Side Lobe
Degradation the opposite seems to occur. The results
seem to agree even more but in these cases the Monte
Carlo Calculation seems to be more conservative. This
can most likely be attributed to the consideration of the
X and Y- distortion. Even though the Z-displacement is
much larger than the X and Y displacements the
Fig.9 Gain and Side Lobe degradation vs. Time

5. 5
combined shift of the two seems to result in a greater
destructive interference, this correlates to a greater
degradation in the main beam and its subsequent side-
lobes. The results can be seen in Fig 9
CONCLUSIONS
Statistical and numerical representations of the
distortions associated with the shock event seem to
match up fairly well. For these calculations a Typical
S-band frequency was implemented. One can easily see
that these characteristics are highly dependant upon the
chosen frequency. Raising the frequency to higher
bands will result in higher errors and have a greater
effect on the normal operation of the Radar. This being
said, overall performance of the radar can be
compromised when introduced to this type of
environment. How much so would depend on the
intended use of the radar.
ACKNOWLEDMENT
We would like to thank the entire MED department in
particular Kevin Eagan and Joel Harris for their effort
and support on this task. We also like to thank Kevin
Cassidy for his hard work in getting this task funded
and completed. N.J Manzi would like to thank Dr.
Robin Cleveland for his great introduction to arrays and
beam characteristics in his Acoustics II class taught at
Boston University, without the foundation built in that
class he would never of been able to contribute to this
task. N.J. Manzi would also like to thank Guy
Thompson II for sharing his technical expertise in a
variety of applications relevant to completion of this
task.
REFERENCES
[1] M. I. Skolnik, Introduction To Radar Systems.
New York, NY: McGraw-Hill Book Company,
1980.
[2] J. Ruze, Antenna Tolerance Theory- A Review,
Proc. IEEE, vol.54, pp. 633-640, April, 1966.
[3] A. Bhattacharyya, Phase Array Antennas, pp 467,
2006 Wiley Inter-Science.