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.

( ) ( )rkjafr

erE

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(sin2

ˆ zyxrkk !"!"!#

$++==

!

zzyyxxriiiˆˆˆ ++=!

! This leads to the following expression

( )!"

#cos

2

iii zvyuxrk ++=$%!! (2)

where

2

!"

!"

sinsin

cossin

=

=

v

u

Adding in beam scanning parameters Us and Vs, gives:

( )!"

#cos)()(

2isisi 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 iiizyx <<!! , gives:

( )!"

#cos)()(

2isiiisiii 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:

21),(01

!"= eGG #$ (7)

And G2 is the attenuated error free pattern: 2

2

2

2

2

2 !"#

$%&'

(!"#

$%&

=)

*

)*

Cu

eC

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.

( )( ) !!! "

"

dAOA

c #$

$=

1

( ) ( ) ( )dtttT

AT

o

!""!

!!

" +#

= $#1

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

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

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.