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AIR FORCE REPORT NO. SAMSO-TR-7 3-136
AEROSPACE REPORT NO. TR-0073 (3450-16) -2
A73 01458
Cl
_,
5026242
Predicted Radar Cross Section of Thin,
Long Wires Compared with Experimental Data
Prepared by J. RENAU and M. T. TAVIS
Electronics and Optics Division Engineering Science Operations
72 OCT ~~
Reentry Systems Division
THE AEROSPACE CORPORATION
Prepared for SPACE AND MISSILE SYSTEMS ORGANIZATION
AIR FORCE SYSTEMS COMMAND LOS ANGELES AIR FORCE STATION
Los Angeles, California
APPROVED FOR PUBLIC RELEASE: DISTRIBUTIONRmJJiNE~Q UBRA~Y.
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~-~ir Force Report No. o J.-.!2. AMS 0- T R- 7 3 - 1 3 6
Aerospace Report No. I [j_R-0073(3450-16)-2
PREDICTED RADAR CROSS SECTION ·oF THIN, LONG WIRES
COMPARED WITH EXPERlMENTAL DATA.·.
Prepared by
J. Renau and' M. T. Tavis ·
Radar and Power Sub,division Electronics and Optics Division Engineering Science Operations
72 OCT 01
~;;_ntry Systems Division THE ~ROSPACE CORPORATION . . i El Seg~ndo, California
Prepared for
SPA~~:\~D MISSILE SYSTEMS ORGANIZATION ------"' ~ FORCE SYST;EMS COMMAND
LOS ANGELES AIR FORCE STATION Los Angeles, California
Approved for public relea$e; distribution unlimited
FOREWORD
This report is published_ by The Ae.rospace Corporation, El Segundo,
California, under Air Fore~ c.ontra~t No. F04701-72-C-0073. This report
was prepared by the Electronics and Optics Division, Engineering Science
Operations, at the request of Concepts and Plans Group, Reentry Systems
Division.
This report, which documents research carried out from December 1971
through September 1972, was submitted for review and approval on
31 January 1973 to Ronald L.' Adams, 2nd Lt, USAF, SAMSO (RSSG).
The authors acknowledge with gratitude the assistance of Mary Barling,
John Kohlenberger and Dr. William Helliwell who aided in preparing the
computer-generated figures used in this study. The authors also thank
Dr. Duclos for his backing of the study.
• W. Capps, Director Radar and Power Subdivision Electronics and Optics Division Engineering Science Operations
. ·'
Approved by
G. R. Schneiter, Director Program Definition
· Concepts and Plans Reentry Systems Division :be'velopment Operations
Publication of this report does not constitute Air Force approval of
the report's findings or conclusions. It is published only for the exchange
and stimulation of ideas.
ii
~~ · 6Rona:idL Adams, 2nd Lt, USAF ECM/Pen Aids Project Officer System Engineering Directorate Deputy for Reentry Systems
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ABSTRACT
In order to determine the validity and accuracy of Radar Cross
Section (RCS) predictions for thin wires, the predictions of the closed-form
expressions developed by Chu, Tai, Van Vleck, et al., and Ufimtsev have·
been compared with carefully measured backscattered RCS vs angle of
incidence for various length thin, long, cylindrical conductors. Further,
the prediction of an open-form numerical analysis based on the Source
Distribution Technique and programmed by M. B. Associates as the BRACT
computer program was also compared with the experimental data.
It was found that ( 1) the BRACT computer results agree with experiment --- --~~-------··- -- ·-
so well (within ±1 dB for all reliable data) that it may be used with great con-
fidence for any length thin wires and can be used as reference data for com
paring with the predictions of the closed-form solutions; (2) the results of
Chu are not accurate except at broadside incidence; (3) the results of Tai
compare favorably with data up to k.e values of about 17 if corrections are
made to the approximate formulas to correct for broadside incidence; (4) the
results of Ufimtsev compare well with experiment for all kt values considered;
and (5) the results of Van Vleck, et al., appear to be very accurate for all k1
values considered except for near end-on incidence. In a separate report to
be published soon one of the authors of this report (M. Tavis) has shown that
this deficiency is due to numerical approximations in the theoretical
expressions.
iii
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CONTENTS
ABSTRACT ......... .
I.
II.
III.
IV.
A.
B.
c. D.
INTRODUCTION
WIRE DIMENSIONS AND RCS DATA
THEORET~ALEXPRES@ONSANDTHESDT
COMPUTER PROGRAM .......•.......
A Review of Tai's Theoretical Expressions and Comparison of Predictions with Data ....... .
Predictions Based on Van Vleck's, et al., Expression and Comparison with Data . . . .•...
Ufimtsev' s Solution
SDI Technique
CONCLUSIONS
REFERENCES
APPENDIX A .
v
iii
l
3
5
7
10
12
15
17
19
A-1
l.
2.
3.
I
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
FIGURES
Measured Backscattered RCS for Thin, Metallic Straight Wires, kQ = 4. 44, Circular Polarization
Measured Backscattered RCS for Thin, Metallic Straight Wires, ki! = 9. 2, Circular Polarization
Measured Backscattered RCS for Thin, Metallic Straight Wires, kQ = 11. 7, Linear Polarization
Measured Backscattered RCS for Thin, Metallic Straight Wires, kQ = 13, Circular Polarization
Measured Backscattered RCS for Thin, Metallic Straight Wires, kQ = 17, Linear Polarization ..
Measured Backscattered RCS for Thin, Metallic Straight Wires, kQ = 34. 8, Linear Polarization.
Measured Backscattered RCS for Thin, Metallic Straight Wires, kQ = 45.7 .. , ........... .
Comparison of Tai' s Predicted RCS with Measured Data, kQ = 11.7 .........••............
Comparison of Chu' s Predicted RCS with Measured Data, kQ = 11.7 ...................... .
Comparison of Tai' s Predicted RCS with Measured Data, kQ = 4~44 ...............•........
Comparison of Chu' s Predicted RCS with Measured Data, kQ = 4. 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of Modified Tai' s Prediction with Measured RCS Data, kQ = 11.7 ........................ .
Comparison of Modified Tai' s Prediction with Measured RCS Data, ki! = 13 ........•..•.••..........
Comparison of Modified Tai' s Prediction with Measured RCS Data, ki! = 9. 2 ...........•.•...........
vi
.
.
.
.
.
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21
22 ·I 23 I 24 I 25
26
27 I 28
I 29
I . . . . 30
I . . . . 31
. . . . 32
. . . . 33
. . . . 34
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I FIGURES (cont. )
~
t 15. Comparison of Modified Tai' s Prediction with Measured
RCS Data, kP = 4. 44 .............•........... 35
l, 16. Comparison of Modified Tai' s Prediction with Measured RCS Data, kR = 17 .......•..••.........••.. 36
I 17. Comparison of Modified Tai' s Prediction with Measured
RCS Data,· kQ = 45. 7 ........................ . 37
18. Comparison of Van Vleck's Predicted RCS with Measured
I Data, k2 = 4. 44 . ............................ • 38
19. Comparison of Van Vleck's Predicted RCS with Measured
I 20.
Data, k2 = 9. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Comparison of Van Vleck's Predicted RCS with Measured
I 21.
Data, k£ = 11.7 ............................ . 40
Comparison of Van Vleck's Predicted RCS with Measured Data, k2 = 13 ............................ . 41
I 22. Comparison of Van Vleck's Predicted RCS with Measured Data , k£ = 1 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
I 23. Comparison of Van Vleck's Predicted RCS with Measured Data, kQ =45. 2 ........•.................... 43
I 24. Comparison of Ufimtsev' s Predicted RCS with Measured Data, kQ = 4. 44 ................•.........•. 44
I 25. Comparison of Ufimtsev' s Predicted RCS with Measured Data, kQ = 9. 2 . . . . • . . . . . . . . . . . . . . . . . . . . • . . 45
I 26. Comparison of Ufimtsev's Predicted RCS with Measured
Data, kQ = 11.7 ......••................... 46
2 7.
I Comparison of Ufimtsev' s Predicted RCS. with Measured Data, k2 = I 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
28. Comparison of Ufimtsev' s Predicted RCS with Measured
I Data, k2 = 1 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
I I vii
I
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
1.
FIGURES (cont. )
Comparison of Ufimtsev' s Predicted RCS with Measured Data, kQ = 45. 7 ........................... .
Comparison of BRACT Calculated RCS with Measured Data, k Q = 4. 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of BRACT Calculated RCS with Measured Data , ki = 9 . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . ~
Comparison of BRACT Calculated RCS with Measured Data, k Q = 1 1 • 7 • • • • • • • • • . • • • • • • • • • • • • • • • •
Comparison of BRACT Calculated RCS with Measured Data, kQ = 1 3 . • • . • . • • • • ; • • • • • • • • • . • . . • •
Comparison of BRACT Calculated RCS with Measured Data, kQ = 1 7 . . • • • • • • • • . • • • . • • . • • • • • • • • •
Comparison of BRACT Calculated RCS with Measured Data, kQ = 34. 8 ......................... .
Comparison of BRACT Calculated RCS with Measured Data, kQ = 45. 7 . . . . . . . ....
BRACT Calculated RCS, kQ = 157
Van Vleck vs 1.Himtsev Predicted RCS Values, k£=157 .....•......•..........•..
Comparison of the Generalized Van Vleck Formulation with Van Vleck's original Expression, k1 = 13 ••.•••.•.•..
TABLE
Experiment Parameters .......................... .
viii
49
50
51
52
53
54
55
56
57
59
61
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I. INTRODUCTION
Interest in the backscattered Radar Cross Section (RCS) of thin,
long, cylindrical conductors has led several organizations including The
Aerospace Corporation to obtain RCS measurements vs angle of incidence
for various length thin, long, conducting wires.
Having assembled such carefully measured RCS data, it was logical to
use the more commonly known closed-form theoretical expressions published
in the literature on this subject for comparison purposes. The relatively
simple analytical expressions used are those derived by Chu (unpublished
but discussed by Van Vleck, et al., in Ref. 1, and used in Ref. 2), by
Van Vleck, et al. (Ref. 1), and Tai (Ref. 3),':' and the known expressions
derived by Ufimtsev in relatively recent publications (Refs. 4 and 5).
The purpose of this technical report is twofold. The first purpose is
to compare the predictions from the four theoretical expressions with the data
over a wide range of length-to-wavelength ratios in order to establish which
theoretical expression possesses general validity for predicting all the experi
mental data considered and to within what accuracy. The second purpose
is to compare with the data the RCS predictions obtained by using a computer
program specifically designed to yield the RCS of thin, not necessarily
straight, conducting wires.
* Equations (21) and (22) given by Tai are expressions averaged over the polarization angle. In order to recover the expression before the averaging process, multiply Eqs. (21) and (22) of Tai by 8/3 cos44J. Moreover, there is a misplaced bracket in the logarithmic term of these equations. For this reason, the corrected forms will be given in the text of this report.
1
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II. WIRE DIMENSIONS AND RCS DATA
In May 19 68, an RCS experiment on thin wire configurations was
conducted at the Radar Target Scatter (RATSCAT) Division at Holloman Air
Force Base, New Mexico. This experiment was directed by J. W. Curtis
and L. Martinez of The Aerospace Corporation. A thin, straight wire was
one of the configurations measured under this effort. The radar frequency
chosen for the straight wire experiment was 450 MHz (radar wave
length X. = 2/3 m). Because of practi~al limitations, yet desiring to
achieve a large value of kQ (where kQ = 2rr Q I A and 2Q = L = total length of
wire), a copper wire with a total length of 2.48 mwas chosen, i.e., kQ = ll. 7.
The radius of the wire was "a" = 4 X 10- 4 m ( 15. 8 mil) or ka = 2rra/ A = 3. 78
X 10-3
• The linearly polarized electric field vector of the radar was chosen - -in the plane formed by k and J. so that cos ljJ = 1. The wire located in the far
field of the radar was mounted on a very low RCS support (styrofoam holder)
such that -40 dbsm cross section could be measured with a signal-to-noise
ratio of about iO dB. The calibration measurements performed on known
size spheres were accurate to within 1 dB of the predicted RCS. During the
RCS vs angle measurement, the angular accuracy was about± 1 deg.
Subsequently, the measured RCS values of more pieces of differing length
wires became available, and the seven almost randomly selected data for
presentation in this report are a good representation of reliable RCS
measurements over a wide range of kQ values, where the radius "a" of each
wire is such that ka << l.
The measured backscattered RCS data in decibels relative to a meter
square (dbsm) vs aspect angle (90 deg at broadside) for the differing length,
thin, metallic straight wires are shown in Figs. 1 through 7. * The accuracy
of the RCS measurements is within about 3 dB at the peaks and degrades at the
.),
"'In Fig. 7, the angle 8 is zero at broadside and 90 deg at grazing incidence.
3
nulls of the RCS patterns based on calibrating spheres. The errors in the
Avco data (Fig. 6) are larger. The angular accuracy is about ±l deg. The
pertinent parameters of each experiment are given in Table l.
Fig. No.
l
2
3
4
5
6
7
Table l. Experiment Parameters
Source of Data
Lincoln Lab. (Ref. 6), performed by Sigma Inc. , Fla.
Lincoln Lab (Ref. 6)
SAMSO/ Aerospace Corp., performed at RATSCAT
Lincoln Labb (Ref. 6)
Univ. Michigan
Avcoc
M. B. Associates, performed at Sigma Inc., Fla.
Wavelength, m
0.227
0.227
0.666
0.227
Set at l. 00
0.69
Set at l. 00
Polarization a Transmit Receive· ka
Circular Circular 4. 2 X 10-3
4. 44
Circular Circular 4. 2 X 10-3
9. 2
Linear Linear. '3. 7 8 X lO- 3 ll. 7
Circular Circular 4.2Xl0- 3 13
Linear Linear 3.95Xl0- 2 17
Linear Linear 9. l X 10- 3 34.8
Linear Linear 0.22 45.7
aCircular Transmit and Receive RCS will be 6 dB below that of Linear Transmit and Receive.
bLincoln Labs data appears to be somewhat in error up to the first null.
· c Avco data is several decibels in error due to experimental difficulties.
4
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III. THEORETICAL EXPRESSIONS AND THE SDT COMPUTER PROGRAM
Tai's expression(Ref. 3, Eq. 21) for the backscattered RCS from long
thin wires with k.t » 1 and ka << 1 when the wire is nonresonant (namely k 1 is nrr
not equal to Z' n = 1, 2, 3, •.• ) is
X 1
[l / co;29
sin(2kicos9) . 4a cos s1n
- l - cos(2kQ )cos(2kQcos e)] 2
sin2ke ( 1)
The expression derived by Chu (unpublished) for the same conditions and
. referenced by Van Vleck, et al. (Ref. 1), also given in Ref. 2, is
(2)
where
o- = radar cross section
a = radius of wire
2R = total length of wire
Qnx = natural logarithm of x
RnY = 0.5772 orY::::: 1.78
k .2rr = A
5
e = aspect angle ( e = 90 deg at broadside incidence)
l\J = polarization angle, defined as the acute angle between_!he _ incident electric field vector and the plane defined by k and £
P(l\J) = cos 4
l)J for the case of transmitted linearly polarized waves and received in the same direction; for random polarization,
2rr
p = 2~ J cos 4
l\Jdl\J =~. 0
2 P( .l') COS l\J t • 1" • • h 1 f • 1 't' = 2 ransm1t 1near, rece1ve r1g tor e t cucu ar
P(l\J) = ! transmit right or left circular, receive right or left circular
To test the validity of the theoretical expressions for thin, long wires
of infinite conductivity, the parameters of the copper wire used at the RA TSCAT
Center were inserted in the expressions Eqs. (1) and (2), with P(l)J) = 1, and the
predicted eros s sections a:( 9) vs e were obtained. The predictions from
Tai 1 s expression are plotted in Fig. 8, where the experimental data are also
shown for comparison. The predictions obtained from Chu' s expression,
together with the same data, are plotted in Fig. 9. In both figures, the ordinate
is the absolute backscattered eros s section in dbsm vs the aspect angle with
respect to the wire.
The results of Figs. 8 and 9 indicate that for kl = 11.7, Tai 1s expression
predicts the correct number of lobes in the RCS vs angle pattern and, within
a few decibels and degrees in position, the magnitude of the predicted RCS
agrees with the data except near the nulls. Chu 1 s expression yields results
that agree very well with the data only near broadside (aspect angle near
90 de g), but leads to very erroneous results at all other angles. Chu 1 s
expression for all the other wires discussed in this report led to similar
erroneous results, except at broadside incidence.
6
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A further test of Tai 1 s and Chu 1 s expressions is obtained by using the
parameters of Fig. 1 and Table 1. The predicted values from Tai's
expression are shown in Fig. 10 and those of Chu' sin Fig. 11. In both
figures, the data have also been plotted for comparison purposes. It is clear
that Chu' s expression for broadside is the more accurate one .
A comparisonofEqs. (l)and (2)for the aspect angle 8 = 90deg reveals
that Tai' s expression at broadside reduces to:
1 p ( lJJ)
7T ~;t + [ en(Y~a)J 2
and Chu' s expression reduces to:
P(l);)
(kQ _ 1 ~ cos2kQ)
2
sm2kQ (3)
(4)
Equations (3) and (4) at 8 = 9 0 deg are identical except for the second
term in brackets in Tai's expression. Indeed, it will be shown below that the
correct asymptotic (ki » 1) equation that one obtains from Tai 1 s expression at
8 = 90 deg, i.e., broadside, is Eq. (4) and not Eq. (3) as published.
A. A REVIEW OF TAl'S THEORETICAL EXPRESSIONS AND COMPARISON OF PREDICTIONS WITH DATA
According to Ref. 3, Eq. (26), when a linearly polarized wave with the
electric field in the plane of incidence is incident broadside ( 8 = 90 deg) upon a
thin, metallic wire, of infinite conductivity, the radar cross section CJ" is
given by
(5)
7
where
go - 2(sinx - xcosx)
y = 2cos2
x(-1 + cos2x- jsin2x) 2L(2x)}
·a + j2cosx(xcosx - sinx) (2n4 + n
X = kR = 2rr2 22 = total length of wire A. ,
n = 2En~ a
a = radius of wire
L(y) -ju
1 - e du = Ci(y) + jSi(y), for any variable y
u
Gin(y)= fy 0
1 - casu du = Qn(Yy) - Ci(y) u
y
Si(y)
Ci(y)
Si(y)
Therefore,
= 1. 78
=[ sinu d -- u u
0
siny· z for y >> 1
. y .
z TT cosy for y >> 1 -z- y
L(y) :::: Qn(Yy) + j; for y » 1
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1T When x » l and x ~ n2 so that xcosx >> sinx, inserting from the above ·
definitions, and neglecting higher order terms, one finds that
2 2 2 go 4x cos x -- ~----~--------------------~~~~~------------------------~ 2 2 . . 2
2cos x(-1 + cos2x- jsin2x) + j2xcos x{ln4 + n- 2in2"{x) + 21Txcos x
Since
Rn4 + n - 2Qn2'Yx = -2 Rn('Y~a)
then
2
and
go 2x
---y;; - 1T - j [2 R n ("~a)] for x » 1
;z =~~ =(!) (;)2 + x2 l
[ Q n ("~a) J 2 = 1T (; ) 2
(6)
Equation ( 6), derived from Ta1' s own general expre s sian, is identical,
at 8 = 90 deg, to Chu's expression, Eq. (4). At this point it was obvious that
the complete general expressions of Tai 's derivation should be used rather
than the asymptotic expression given by Tai. However, simplicity of his long
wire expression is appealing from a practical point of view, since the general
expressions are rather lengthly. In order to preserve the simple expression
of Tai (Ref. 3, .Eq. 21) a simple arbitrary functionf(e) is chosen that modifies
the second term in brackets of Tai' s Eq. ( 1), and modifies it only near broad
side so that, for k 1 >> 1, it properly reduces to Eq. ( 4) or ( 6). That is
j(e) = 1 _ (sin (2kicos8) )2
. 28 2kQcos8 sin ' for kR » 1,
- 9
where f(8) = l for all values of 8 except for 8-90 deg, where f (8 )- 0 near
broadside. The new expression for the RCS of a non-resonant thin, long, -
metallic wire may be written as
l P(4;)
TT (;)
2 + [Qn('{k~sin8)] 2
X l [l ; co:28 sin(2kQcose) _ f(e) l- cos(2k~)cos(2kRcos8)] 2
4 cos sm2kQ sin 8
(7)
Predictions of the RCS based on Tai's modified expression, Eq. (7),
for the wires described in Table l,. are shown in Figs. 12 through 17 together
with the appropriate data. As can be seen, the modified Tai's expression
for the RCS of thin, long, nonresonant, metallic wires predicts the RCS vs
aspect angle for 4. 5<kQ < 17 within a few decibel over the entire aspect angle
variation, except near the nulls. However, for kR~45. 7, the predictions
vary by more than 10 dB from the measurements. Since the modified Tai's
expressions are simple, they may be used for quick and fairly accurate
estimates for the kQ values given above.
B. PREDICTIONS BASED ON VAN VLECK'S, ET AL., EXPRESSION AND COMPARISON WITH DATA
The expression derived by the above authors (Ref. l) was one of the
earliest in this field. Even though S. Hong (Ref. 4) claims _Yan Vleck's study
to only be applicable to short wires, our experience has been that Van Vleck's
expressions are of much greater general validity than given credit. This can
be seen in Figs. 18 through 23, which are the predictions obtained when the
parameters of the wires of Table 1 were used for the predictions. The
10
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cross section expression of Van Vleck, et al. (Ref. 1), is reproduced
here for easy reference. Note that these are Van Vleck's approximate
formulas.
IT= 4rrP(~) I (F' + F") sin 2qQ + 2(G' + jG") cos(qR) fsin(q + 13 )Q q . Lq+13
+ sin(q -13HJ + 2(H' +jH") sin(qR) [sin(q +13)Q - sin(q -f.>)£] 2 (S)
q-13 . q+l3 q-13
where
P(~) = asgiveninthetext
A. s-2' = 2 log -- I. 154 · e rra
2G' ljJ((3Q)
= ljJ
2(13R) + Z
2(13R)
TT 2G" --2 S1 f
Z ( 13R) = lJJ
2{13R) + Z
2(13R)
2G"
lJJ(13R-rr/2) rr 2H" = · z z -z~
ip (13R-rr/2) + Z (13R-rr/2) 2H'
ljJ ( y) = - ( S1' - ~ ) X co s y + ( rr I 4 ) sin y
Z(y) = (1 /2) (loge (4132) + 0. 577] sin y - (rr/4) cos y
11
A = -{l/2)[log (13Q)) + 0. 712 e .
q = 13 cos e
13 2 TT /'f...
'A. = radar wavelength
2Q = total length of wire element
a = radius of wire element
e = angle between line-of-sight from the radar to the wire element and the axis of the wire element
j = J-1
As can be seen from Figs. 18 through 23, the expressions used by Van
Vleck, et al., when compared with the data, give good RCS results for all the
cases shown except for end-on incidence. The difference between data and
predictions in all cases is within 4 dB at the peaks and within 2 to 3 deg in
look angle. The accuracy in the nulls is not quite as good. Tai and Van Vleck
are about equally as accurate at the lower k£ values.
C. UFIMTSEV 1S SOLUTION
The RCS of long, thin wires as given by Ufimtsev (Ref. 5) is taken
from Ref. 4 and given below. The predictions based on these equations are
compared with data in Figs. 24 through 29.
l. THE CASE FOR 8 ~ TT /2
4 I 2 4 cos q,. s(e)l
12
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where
S(e) = - s1n - · x.n . 4(e) n [ i ] 2
Yka sin2
(e/2)
+ ikL2cose 4(e) n [ i J e • cos 2 . x.n 2( e)
Yka cos 2
+ eikL( 1 + COS e ) 2 1· 4(8) ,1, n [ i J • • s1n - · '+' • x. n 2 - . (e) Yka sin 2
_ cos 4 ~~)·l!J+·Qn[ i (e)Jl Yka cos 2
+ cos8. £n (-i-)· [eikL2(ljJ )2 + eikL2(1 +cos e)(lj! )2 D Yka + -
_ 2 eikL(3 +cos 8 ).y; .y;.y; J - +
y = 1.781
ljJ = i lT- 2 Q n (Y ka)
Qn ( i2~L2) - E(2kL) e -i2kL Yk a
2 . ,~, ilT - £n(Y q±) '+'± = ---~-i-2k_L __ )---(~2=k~L~q~±~)~-~--i-2-q--~k~L---
£n 2 2-E 2 2 .e ±k2a 2 k a k a
q ± = (k2a)
2 ( 1 =F cos e )
E(y) ~ .[y c~stdt+iiy s in t d t - i 1T I 2 t
n = 1-l!J2
· i2kL
e
13
2.
where
a = radius of the wire
L = total length of the wire
e = the angle between the propagation vector and the wire
4> = the angle between the E vector and the wire
THE CASE FOR 9 = rr /2
2 (} ( e = rr /2, 4>) ;x. 4
= cos 4> X Is I 2 lT
S = i;; _ ikL (\ii)2
E(2kL) + A - 1/2
2A2
A 2
+~ [ lfJ _ .en(if2),l ,1, A 2 4 Yka J '+'
ikL e
-2 + 2(lfJ) X Qn (_i__) (ei2kL _ lfJei3kL)
DA2 Yka
l 2i ) A = Rn\Yka
lJJ = lJJ ± ( e = rr /2)
As can be seen from Figs. 24 through 29, Ufimtsev1 s re sitlts compare
very well with data for all kl values considered. However, the expressions
are complicated and the modified Tai may be used easily for the smaller kl
values. The location of the peaks and valleys compares well with the data
and the accuracy of the predictions is comparable to that predicted by
Van Vleck.
14
I I I I I I I I I I I I I I I I I I I
I I I I I I I I I I I I I I I I I I I
D. SDI TECHNIQUE
The SDI technique has been used by M. B. Associates to develop a
computer program designated as BRACT (Ref. 7). The BRACT program uses
the numerical solution of the thin wire integral equation to solve the complete
electromagnetic scattering problem for arbitrary wire structures. For these
arbitrary scatterer s, the program sets 1up the structure matrix relating the
incident field to the resulting induced currents and calculates the induced cur
rent distribution on the scatterer. The currents thus obtained are used to
calculate the scattered fields. BRACT is designed for execution on the
Control Data Computing System and is coded in FORTRAN IV as released under
the SCOPE version 3. 1. 2 operating system.
This program was used to calculate the RCS of long wires, using the data
presented in Table 1. The results are compared with experiment in Figs. 30
through 36. As can be seen, the calculated values for all values of kR are
nearly on top of the data except for the two cases, kQ = 13 and kR = 34. 8
where the differences are about 4 dB. It is known that the data for kl = 34.8
is in error, and it is believed that the data fork£ = 13 is also in error, since
every theoretical technique applied gives more error for these two cases than
all other cases. BRACT was also used to calculate the RCS of a wire with -4
kl = 157, A = 2 em, and a radius of 1. 52 X 10 m («A). The results are
presented in Fig. 37.
In order to prove the accuracy of the Ufimtsev and Van Vleck closed
form expressions for very long wires, the RCS of the long wire k£ = 157
was calculated using the expressions of these authors and the results are
given in Fig. 38. The results obtained by BRACT and the results obtained
by the closed-form expressions are nearly identical, being about 1 dB dif
ferent at the peaks and following the dips very closely, although Van Vleck's
expression predicts much lower null values than the other two cases.
15
I I I I I I I I I I I I I I I I I I I
IV. CONCLUSIONS
The comparison of. the results of the calculations with measured back-.' ·, :'..,...
scattered data from var,ious' leng~h, 1thin, .. cylindrical ~onductor s shov,;. s th~t, of the four analytical expressions, Chu' s exp~~ssi~n yi~ids fair R.CS·~~ree~ • ment with the data only at broadside incidence. The modified Tai's pre
dictions at all angles are within a few decibels of the RCS data for wires of
Q 2nQ 4. 5 < k < 17, where kQ =-A-, 2£ =total length of wire, and A= wavelength.
For larger values of kQ, Tai' s expression yields predicted values which differ
from the data by as much as 10 dB. Van Vleck's expressions lead to predic
tions that are within a few decibels of the RCS data at all angles greater than
20 deg from end-on, and for all thin wires tested, namely kQ > 4. 5. Even
end-on, the predictions can be used as an estimate if it is noted that the RCS
should approach zero as e- 0. The predictions from Ufimtsev' s analytical
expression yield results that are no more accurate than those of Van Vleck
(except near end-on incidence) and in the nulls. Except for end-on incidence,
Van Vleck's approximate expressions are as good or better than all others if
one is willing to let the RCS approach zero at end -on by using the shape of
the last lobe in the pattern as a guide. In Ref. 8 it is shown that the deficiency
in Van Vleck's theoretical predictions at end -on are associated with approxi
mations as published, and when the general expressions given by Van Vleck
are used, the deficiency of the theoretical predictions at end -on disappear.
In appendix A these expressions are given and in Fig. 39 a comparison of
the calculation using these expressions and Eq. ( 8) is shown for ke = 13.
The BRACT computer program, whose results are most rigorous,
agrees with the data within l dB and ±l deg. The analytical forms take about
one minute of computer time to yield results for a given wire for all angles of
incidence, while the SDI technique, competitive with the time used by the
programmed analytical forms, takes a little longer, depending on the length
of wire. Moreover, the general usefulness of BRACT becomes evident
should one bend or distort the wire under consideration. Then, as expected,
17
the analtyical expressions for the RCS do not apply to the new configuration,
whereas the BRACT computer program is flexible enO\~gh so that for the
most complicated configuration of a distorted thin, metallic wire, accurate
results for the RCS may be obtained within a few minutes.
18
I I I I I I I I I I I I I I I I I I I
I I I I I I I I I I I I I I I I I I I
1.
2.
3.
4.
5.
6.
7.
8.
REFERENCES
J. H. Van Vleck, F. Bloch, and M. Hamermesh, "Theory of Radar Reflections from Wires or Thin Metallic Strips," J. Appl. Phys. !.§., 274 (March 1947).
G. T. Ruck, et al. , Radar Cross Section Handbook, Vol. 1, Plenum Press, New York- London (1970).
C. T. Tai, 11 Electromagnetic Back-Scattering from Cylindrical Wires, 11 ·
J. Appl. Phys. 23, 909 (August 1952).
S. Hong, Scattering Patterns and Statistics of a Long Wire, Lincoln Lab Technical Note 1967-55 (December 1967).
P. Y. Ufimtsev, "Diffraction of Plane Electromagnetic Waves by a Thin Cylindrical Conductor, 11 Radite Khnika i Electronika ]_, 260 (1962), English translation, p. 241.
M. Rockowitz, Static Patterns of Dipoles, Lincoln Lab Project Report PA- 179 (March 1971 ).
RCS Computer Program BRACT Log - Periodic Scattering Array Program, M. B. Associates, San Remon, Cal., Project No. 00041 (January 1969).
M. T. Tavis, Van Vleck Revisited: The RCS of Thin Wires, Aerospace Corp. Report No. TR-0073(3450-12)-1 (September 1972).
19
I I I I I I I I I I I I I I I I I I I
- . - ___ __j_ - ______ L ___ - -· -_ __l___ -
01 POLE LENGTH 32. 0 em FREQUENCY 1320 MHz
l_ ____ _L _____ -
ki = 4. 44 -10 ka = 4. 2 x 1o-3-
'A = 0. 227 m
-20
E
I (\ en .a "0
/ .. -30
I V)
u £t::
-40 I
-50~----~-----r~---+-----1
0 30 60 90
ASPECT ANGLE, deg
Figure 1. Measured Backscattered RCS for Thin, Metallic Straight Wires, ki = 4. 44, Circular Polarization ( 9 0 deg corresponds to broadside)
ZI
' I
1 I 1· Dl POLE LENGTH 66. 3 em FREQUENCY . 1320 MHz
I 1 ki = 9. 2
-10 ka = 4. 2 x 1o-3 -
A. = 0. 227 m
-20 ( ~·
\ ; E ! \ UJ ..c "0 -30 .. V)
u ~
-40
-50 1-1----4----ll----1---1--+--____:_---1
0 30 60 90
ASPECT ANGLE, deg
Figure 2. Measured Backscattered RCS for Thin, Metallic Straight Wires, kl = 9. 2, Circular Polarization (90 deg corresponds to broadside)
22
I I I I I I I I I I I I I I I I I I I
-------------------0
~
-10
-20
E Cl>
..c:::l "'CC
-en c.,:)
. D:: -30 N w
-40
-50
/~ Jr\ !'
(\ I .. 1\ I\ -~ ... .A
" (\ '
l" I!
~ \ 7 '
.. .
~ )
TOTAL LENGTH {21) = 2. 48 m RADIUS = 4 x 10-4 m RADAR WAVELENGTH = 2/3 m
k.l = 111.7 I I
330 0 . 30 60 90 120
Figure 3.
ASPECT ANGLE, deg
Measured Backscattered RCS for Thin, Metallic Striaght Wires, k£ = 11.7, Linear Polarization (90 deg corresponds to broadside)
Jr\ I
•'
.
150
I I I
DIPOLE LENGTH 93.5 em FREQUENCY 1320 MHZ
I I kl = 13
.,.10 · ka = 4. 2 X lo-3 -)... = 0.227 m
-20 I
E t ~ ., en
..a "'C 1\ .. -30
l I ' ~ ' V)
u 0:::
\
. 0 30 60 90.
ASPECT ANGLE, deg
Figure 4. Measured Backscattered R.CS for Thin, Metallic Straight Wires, kl = 13, Circular Polarization (90 deg corresponds to broadside)
24
I I I I I I I I I I I I I I I I I I I
I
I I I I I
10
I I I E 0
en ..0
I "'0
.. V)
u
I 0:::
I -10
I I -20
I I I I ~--
I
) I I I kL = 11
ka = 3. 95 x 10-2
A. = 1 m
~ r ~
" I
n n ~ ~
" ~ N ~
~ - ·-·-·--- ____ .., ____ --- . - ···--· --· -
174 162 150 138 126 114 102 90 78 66 54 42 30 18 6 0
ASPECT ANGLE, deg
Figure 5. Measured Backscattered RCS for Thin, Metallic Straight Wires, k.l = 17, Linear :Polarization (90 deg corresponds to broadside)
25
E en .a .,
.. V)
u ~
10
5
v 0
;
-5
-10
-15
90 78
I I I k.l = 34.8 ka = 9. 1 x 10- 3
A. = 0.69 m
(\
I \
' ~
" ft
~ ... .._ '""
... - ...
66 54 42 30 18 6 0
ASPECT ANGLE, deg
Figure 6. Measured Backscattered RCS for Thin, Metallic Straight Wires, k1 = 34. 8, Linear Polarization (90 deg corresponds to broadside)
26
I I I I I I I I I I I I I I I I I I I
----------------~--
E en .0 ~
.. V)
u ~
N -...)
30
20
10
0
-10
-20
-30 90
1 I I I I EXPERIMENTAL MEASUREMENTS k.i = 45. 7 ·· (sigma· incorporated reflectivity range)
ka = 0.22 . -A. = 1m
..
"
' ~
n
80
1\ / ft (\ 7 \ I n h n ~ n " n (~ y
~
70 . 60 50 40 30 20 ASPECT ANGLE, deg
Figure 7. Measured Backscattered RCS for Thin, Metallic Straight Wires, kl = 45: 7, Linear Polarization (90 deg corresponds to broadside)
1\ \
'
10 0
ASPECT ANGLE, deg
Figure 8. Comparison of Tai' s Predicted RCS with Measured Data, kQ = ll. 7
28
I I
'
I I I I I I I I I I I I I I I I 1.
I I I I I I I I I I I I I I I I I I I
-50 .. 1:;·
l : : ; : i il : : ! :
-60
i :,.! E:.;T !.1. ~ I i.f
[ i (! ; ~ . ! ' : I ... ' ·\"1 ; lin ; ' H1 ! : ' ' t ;:·r : : :·!
i t t i I ' j tli1 ! ' ! I
I i ! ! rfh 81> I . Ill ; i i I i ' ~rl l I'' ;
! !·J •ll
20 40 60 80 100
ASPECT ANGLE, deg
Figure 9. Comparison of Chu' s Predicted RCS with Measured Data, kQ = 11. 7
29
;11i ill! 1'!1u!1Jiu1/:l !11
1'· i1iiii-Hi:[tt'l1·: !J!+i 111 !1
111
1/IJ: !ill r·;! 1:1: ~·: r::: ;::: ;::;~·~~ ~~n :-!~. itt' J Ld U. . ~ Ll.).. ~m· . . ~,.L ,;-:.. _, . :1-r:-:t-:-fh+i ++w. "~-.:. : h ,;-, *"~ .;..,_·~ ; .. c ; ~ ... :
::; i l : !: ! i j i 1 I ] IIi 11111 lj:!: i il . '.! I · · ll j! Jjl ifj, i i 1 i! i I 1 ;! ·;!: j '; 1 ; ::.: I; j, i':: ; i 1. :: 1 ! ; :; !1
:: 1ii' d1! li' fl:t 1ill 'il !ill'! 1iil !.: ,ill :tii •!1: lilt 1:!: ;[!' :!ii lij! 11; il. ~~ -20 ,.,,''I! ill' , .•• ,,, 'i\ ., .. , .. 'i·· '' ·I I •' j'l ,,, ''I' . ;! 1 ' ' '•' ''I ·!~' liij 1 l!';,lijl''l'il:j·!'·lil.:.t1 .::i!·l·l ... l·•li.l!'t:•:.!.' l'''lli'l.;j'''l·.,,,
! 1
: •• : : I 1.' ! ,I ,' ,: i i , . ; ! ; ,I , i i l 1 : ; ,! 1. i ~.:.... ' · · ; , ! l ! : ; 1 : :,' l :. ! 1 • 1 1 i · · ' : I ; · • ' ' · ' ' 1 i I I 1 l ' :I -- .;..;.;..;++i-!''+h~+H· -'--'-1. +':P _:_ ·~-c-:-L!- ·. ., , +H:h+:'
d-~! l;ii lill1lll! ;ij; ;!;;.~:~~ rqi ;;: ~: ~ili \111 il 1-· ! ·-··1:!.,1~.: ~!.-',:: .. ,; .. fl.::,i ;·:.~~.--l .. :.t .. ;,i~ ~-·. ··.1: !,[J·· ~~-~ ··' m fOJ il··· •.• ~, , ..•. ! 1 L· ••. -f.--~~-,-*,..,...,H-+1!-+ti+ft'-r-r~m-'-rl i ;; ,:1, !i'' itti'l: ,,:,~· ~~ ~u·· 11.,,,· :~·~.; ,:"~.~:~ ::u il : ~. . 11:.: '!': 1
::. ·~n· ,,., . , II ·I , ljJI . It II. F•: I' I',, II r . ','' I . . ': , ... ".' ! '.; :. i, '.:.' •.' ,· H+ ;.tl-L '+!- : ~ . . . . 1.4 . I • IIi I'. I I'' f-l..: . ..:.. .. :.,_" . . ' .. ! I; . ''
I '! :1 ';I :1 :rl·! tl :I; i!l.'l~l·l I: I I,,. I: I'! \'II IIi I i: I~:-.:_!:: I il! '' ·,:I: iii : '! ~ l.Li.l.i. 0:.1:: i: I: !,, : ..• ·~·'I :, !. ;, ·, : ii 1 ;11 ! !tl:::~~~lliwi :'1 !:• '!:1 ,:!, i'~l'\'ili '11' ,; i 1' 1: :L!.i~~;li :;:;
-80
f! ii ~~ .. , r
i ~-4-
-100 0
; i ' i i I ' ' l!ii ; !I
~ '.~ :
100
ASPECT ANGLE, deg
Figure 10. Comparison of Tai's Prediction with Measured RCS Data, k£ = 4. 44
)0
I I I I I I I I I I I I I I I I I I I
I I I I I 1--
1: I I
I I I I
I I I I
-20
,..30
-40
-50
-60
-70
-80
-90
:+-t: ....
-100 _;ii! )FL~ i 0 20
i I ' ! li.H l ' ; '
_, l I ' ; ,,
' ' ' i ' ' i I ' i ; ! l ' I ;
I ' i I i ! ' 40 60
;
.. . !: ·t:: --4·-f---+..;-:-+~ '". ~ l;; .i.~+i -~1 ;+
:: :: :-: -~ • i 'I " . . ; : .. ; :! i ~! l; ·! H· i ~: j :; l t i':-
' ! I ;·:. : ! : ~ I ' ~ : . :
80 100
ASPECT ANGLE, deg
Figure 11. Comparison of Chu' s Predicted RCS with Measured Data, kQ = 4. 44
31
·-..
r'-lt 1 i ;'!j:L;i: .:11 ,T" ;1 1---- icr!~l~ li :::L~\:,LJ; ;.rt-lrtli: Htt·!:!JL::-:: ;r--:- .c-c- i;·; ', ,,, Tl"ir ,;~r; :f ttfL 1+if it±T ·ret"+:;; r~n ~~ Tt': ll':s:ld~ sfJ.l~~fft-±tL g lt£1 -~i-f' ±W-iit± ;:rl
r!-Lii!J 1! -: t l'tlil r VLJ' ~--- -im- 1- t:b __ : -_nt.t ~_;rc: ji ! :t:-r:tt_' -I i j_t'· 1-_l+t'[t'-]trtl' --ffj-1':;;: :ll-,-1 ;t'!:f!J! J.i!·_-I:T, Fit; :: :.i : f-J i If-it''' '-I 1-,- ' '-+-·- ,_,_ --!!>-·j-' -: I jlli 1-j rf-1- ' ---+ >·r I ii-I "i'' 1•1' -80 '-IT ·-·r-··' 1 ·1-· lt · •·1 ; -'"---:--~.-t__l_ H~ !-:. )-J,}J .--Ill u ~l Tb.J.: -~-;-·---~-'- ··-• iH•· L ,. ':1
0 20 40 60 80 1 00
ASPECT ANGLE, deg
Figure 12. Comparison of Modified Tai' s Prediction with Measured RCS Data, ki = 11. 7
32
I I I I I I I I I I I I I I I I I I
I I I I I I I I I I I I I I I I I I I
Figure 13. Comparison of Modified Tai' s Prediction with Measured RCS Data, kR = 13
E en :8
-30 1::1 L'jJ 1 !LJ1'i! 1111 +·11 [CiiHI'Hij·'~:r-:::i - ,i'll~illl'll ~r-tr ltl !ltr! jJIJJJ!_. 1-It1 jt!"ltilt! II I ,. 1i I I I - ' ' i I I !ill " .I I'! I ! i! ' I :t i i IEJ !n_ ~Iii ' -! ! I - ! ~JH ,__ :-Hi t.;J I
!): i i ill ~ 11 jli II i ! ill i I !I I II i 1: il i I L J ' i i ll i ~ : 11!!1 ;_II fli j I i l ! lfi It i 1; II! !II It !II II ! : i Ill! Jill IIlii
I 1
1 *~;I :it~ .1;; 1.~ It! f1' ~ ;j i !lt 1 1 ' nl ·! 1 !~ -~r·;r !·'I 1.1·1i!) ... ll ··!! ., ·!! ; .. 1\'1-i~l ll!il :::~: :tli lrl-1· i!!i l_ii_! !ill:l ·~~!II i'l 'il· ::k: :ii!IJ:il ~~ ~~ ~~~~lilt: ~~~!1 !!il It: 1-H jijli •• ! ; 1-+1 1! 1 1:; ·'ltl ! I. i ' . j··tl' :! ' · r ·r, i 1. i 1 1111 ~ 1 ·l~ . 11 ·: 1 • ~ 1-i 1 l-n . t·1· 1 i 11 1
:!ii 11:1:: i!li W1 ·,: ·~: !i:~~- ill; :: 1
· 1li i : i::l 1ii 111 jrl!t ),_, 1
ibj :li11 l11l::i: ,1 I JHLj:H~ -40 :J!j I' : '·' ,~r:' I •'11''!-l'h-iL •1 . -: , .. , r • : i_l_- !~ t··· -~ll :!-t.\.'_r, ~ !t··-! ·tnT
r't' ';l:! :.: u:: ::!!'tit li!ihii f:t, ;i i i: \it:ltl-11! ib 1111 :.;1: 1::: i'1·-H rtq ··fi~·J'-JF+HT.-.;:, ~-· !'di --:·Ti-l-L., .. , ,,-, 11·! ... 1 : -·l -· .. -!:'I" -:-. :rr· :· ~· .. - ·-~ -.--~ 1l+
Figure 14. Comparison of Modified Tai' s Prediction with Measured RCS Data, kR = 9. 2
34
I I I I I I I I I I I I I I -I I I I I
I I I I I I I I I I I I I I I I I I I
j i ! ' i ' I '
! ' ! ! t ;
-90 I
' i ' ' ' f ' ; i ! l ' ! '
ASPECT ANGLE, deg
Figure 15. Comparison of Modified Tai's Predicted RCS with Measured Data, ki = 4. 44
35
kl= 17 ka = .3. 95 x 10-2
20
ASPECT ANGLE, deg
Figure 16. Comparison of Modified Tai's Prediction with Measured RCS Data, kQ = 17
36
I I I
I I I I I I I I I I I I I I
I. I I I I I I I I I I I I I I I I I I
E ., :8 .. tJ a:::
F~gure 17.
ASPECT ANGLE, deg
Comparison of Modified Tai' s Prediction with Measured RCS Data, kQ = 45. 7
-40 --1--- -~- - -- ~ --- I .. -~-r- _c r -1: : -~v~ ~t-~---- +-- ,~ ~--+ --~C •••• _
1 -_~_;~---_· .... 11--~ ___ - ;,-~- _---ri_--_-_-'-_ ~~-------_ ;4lJJ-.--- cc.: L -~ ~b~~J~
:1: . [{! ... -: :::: "·• :.: ·;. ::::. : l
r-:1 i.. :, ' ...
-60 T-:--:--l'- c-f--'-'- '-- -- ·-:--:-,-1-:-'-1-t ----cr---h-:--'-ft'-1--c-' :...,..: b . .c:.._ • +-:-'.-+: .·'-'-, .+, :.;.,-,+1 _--'-*,.,.:..;
. :· .. :i:!
---tc--t-:-'--1-·-. ~--~ ~-~ ';+- ~~f-... ,., .· ::' _ .. ,·
::.-f. ',-: . -~~~c-..:-'..:-. I···
1--'-1--'--if--'--l-'-' -1. __ 1_..:-' ~- ~':- ~'-. __:_c. . ! :- ·-r.---:+'-.-" +' _:-c. :b:-:-1--:-:+ .,-1---+· '-'-+'7+-=1-:---'--70 T .: • . . . . : : : sf.. . . . .. . ... . . ': '! ''' II
0 20 40 60 80 1 00 ASPECT ANGLE, deg
Figure 18. Comparison of Van Vleck 1 s Predicted RCS with Measured Data, kQ = 4. 44
38
I I I I I .I I I I
I I I I I
I I I I I I
I. I I I I I I I I I I
-40 ::' jj I! i! w ! ![ l :! !! : J! l i! i r ! i i! !! i 1 r ! l i; i ! q i; ! ; i i I u I :. ! nn j i . i it' 11T ' ·Uri Ji, iii~ ::l[ :!i~ iU: i!ii1ilil !li· li ' ·; i 1 fii HH T i_ 'Jrb' i· -'i:if i 1'~. }I h±i :iH 1-j'l 1111 i;jlJi[li:ii ±( .: I J 1!~_1/ 1if] c', IWt:J.ii ilJ1
1'. l:jjl 1! •·• • lj.j, • ,_. • , .. ~· • ~- .. ...,.:; H! . .._ J ~ jl •• •!, -ul .. -1 1:r. -d~ I!! ·t.f:tj
' r TG~ Uil ~!'llil' 1::.&:-'{UTtl_ :j ' ' ·· ·· .lf" irH i t ':i:H ::r -51.1 .• ,, .. • F' .. 1r111 IfH' r ,,., .. , .:tt. ·• .. ;Jt:Jf:f!it+t:l'tt.r.lt~t.' ·I···+ ''til:ii'l!:l .u j!Ji
-!-. r+-+- --4~ :t.!:! .tt.U 1n-r 1 ,... .r:;-r <~ i :it.. t:tt
1<1 !t~·f i+!t :Ht ~,·J: !rf: 'Jil h'1·: t H /ii :·1. wl: 1.111 :;; 1ift 1•11 Itt. n•t T' ~~-···· .rn. l ,~ 1\ 1, \P- I t\i" ~-ttti fl. HI '
1·r Ht w:H Ftf 1'-11iHol+JT 1>-1•• p:rut:;:r. Hf tfi! ,:It ·1iH i: ltt :· · '. 1~1, -70 ·!+· ,,.., ,,. . • .,'tt.iii ll jl '" ·t\! .
0 20 40 60 80 ASPECT ANGLE, deg
100
lr:
; 1
+
II -:r '.' . . Rl
Figure 19. Comparison of Van Vleck:' s Predicted RCS with Measured Data, k2 = 9. 2
39
E "' .0 'U
-10 .,; u «
ASPECT ANGLE, deg
Figure 20. Comparison of Van Vleck 1s Predicted RCS with Measured Dat.a, ld ::-: 1 l. 7
40
I I I I I. I I I
I .I
I I I I I I· I I I I
I I I I I
I I I I -I
I I. I I I I
I
v; u IX
' . '
40 ! . --· : .,· : : .· ·: 'I . ! , . . '!' t---f---- -- ~ ~- _:_+-'-· ---· j - _L~ --+-- - cc:hc --- - - '-~r-.·_~·_j; 1-j....:l '--1---.J..-.:-'-4--. ---~· .:..;.4.;....:' '4' .;....:. 4. ~-+-'-1-i:.;,.'+' ...:....t-1'.;..:..+.:..-.~.;.:....+;_;' .j..!.,..,..;. ~~;_;.~~-·-H
301-:~:~:·-~·-~:(44:,-.;....:.'-'---,•'4 ·~·· ~:~.J.:.c{ ,.,., ' ' · =3~. kl = 13 ka = 4. 2x1 o-3
>.. = 0.227 m
:~~ ~~I.
~ cr. .:i_! POLARIZATION: TRANSMIT CIRCULAR :. ·: :
RECEIVE CIRCULAR ~~f-:-':;j :" . I
ASPECJ" ANGLE, deg
Figure 21. Comparison of Van Vleck's Predicted RCS with Measured Data, kR = 13
41
20
< ' : I : : : : ; ~ j i : : ; : : ; ' t ' u ' : : ; : ;_: : : : ; i ~ I ~: l ; :: it : ! il : : : i : ; i f : !·F it:J _40 ?~< cJ:: · iU ;::_: ;1:; !:.: :~· ;:: :,; i/: ti
1; rJ:i ;1:
1 IJ!I;b::::r:!Ft:n
'·.' .. _',- :_·· ... ·:-.1, •· '::: ·,'_,_:_: !: · ,,,· · , ,: : ,, i,',.:,ii,d:,';'l!_-:1,~ •. ;,,: l,l,\,H.•J,:i. !,1,!1: :_:,,Jl !.l_.t;',·
1-+~-Ci..,.~~'...L':J.''+.:·:;;) ii · ;· 't ~t lr!.r I·· 111·~1 -~;n ···rd ~·~- "~"L----,- -11 ·:··~· : __ ':: 1 ··,: ::_, :, ',•.;,, :_,· :_ :.' ;_ ,_' ;_ :_. '_
1 • 1 • I rL, i : :-:; ~: 1: :1
1 ~ ; : i ·1' U r. !,.11,1, 11! li !J•ni! 1•:;1 :_~:.Hi<;_: !,•; t. i1:~~ ;- :-t ; ~-i .; I 1 . 'Lr~-~ I ttl· 11 ~. t 1 1~1 -l+t - -~ 1 t 1-, +! r 1 ;-i ;~: . ~-· r +t-~-· -r-!-t
Figure 22. Comparison of Van Vleck's Predicted RCS with Measured Data, kR = 17
42
I I I
I I I I I I ·I I .I I. I I I I I
I I I I I. I.
I I I I I I I I I I I I I
60 ,[:I}: !!i! ~:~[it !!l : i, F! 1 ::: !ili It: I IW! !!!i' I: [J 1 lW [ u Ill 1!1 lH \li:
1~:: ': .· · i 1 t ::!.', ;; i il!i:,·:cu ::r::u !l ~~.[ill_, ~ · :1 ::r ;\· ; ~~~ :: ;;1:H' ,,' r: • ';:i 11[! (, '~ lill [ll_ifr: 1\'!! \! IJ! fill\ U j~ it lH r li;l ' 1 :r:;n ii ·::: !c.· :' + vAN vLECK · · · · · · lili· ' ;
50 ··:; :::, ,·' . : :r:, ·. 0 DATA :.''!,' 1:,;_:_· ' • : ; I : ; ! ~ I • : ! I : I i ; ; I : : ~ :
::J 'li !•; :" ','; i kL = 45.2 I il!: :i 1i b~\~: I ii\U' i 1 :i: ,' ka = 0.22 11'' 11:. !" 1!::! 1!
1, r~[qj)jll:: ll: i >- = SET TO 1.00 m !Ji~ 1 .
40 tit!! : ';i 1 •ri; '
1 Ul 1 POLARIZATION: TRANSMIT LINEAR 11
1
, !1[1. '•!li' !i RECEIVE LINEAR i-! : i irtii i ! i i fll w I i I Hi ' ' rffim;:mTim:rr!:ttl:m:ttttttm ~!/t, ' ' ''! I I : HL 'Ui I
30 '' '
'i' jl
· r: i •. r~t; 1 1 ~ · , · , 1 1 ~ ' 0 :n; . , . 'I !.
u i: i I 1
I,
t1 liF! : • I:; I ' -20
' p! . ''L} i · .
-40 tit. 0 20
i '
40 60
• , I
Fr+ ' '
80 ASPECT ANGLE, deg
I ,
100
Figure 23. Comparison of Van Vleck's Predicted RCS with Measured Data, kQ -= 45.2
43
E en .a .,
.. V)
u ~
* UFIMTSEV
o~++H# o DATA I k.l = 4. 44
I+H-41-'-H-H-
I I ir'il ka = 0. 0042
! 111 l i : A. = 0. 227 m
; fl=·
It f~~
IJ..· : it r J l ~ L~ i-1
-10 · i1! 1 :!!:POLARIZATION: TRANSMIT CIRCULAR-;, I ,_RECEIVE CIRCULAR t+H+t+t++lt t1 · . Tt+'it'l t:+r- 1-11 1
-+· . :r.rtiL :H :~ U: I ' T':·l-'i : :+ F;:., ~r; 1 : :t · 1
fi ' : :l(i~ ~-- d::t ,:s--ic , • -•- "'l ii=H + ' ·.. .
' .... ;~f.!: -- · .. ' : ~!
-30
I
II
iII ' I I '
I ' l -40 I ·I
I '. ·I
ASPECT ANGLE, deg
Figure 24. Comparison of Ufimtsev's Predicted RCS with Measured Data, k£ = 4. 44
44
I I I I .I I I I I I I I I I I I I I I
I I I I I I I I I I I I I.
I I I I I I
E en
.Q "'0
.. V)
u 0::
20 40 60 eo 100
ASPECT ANGLE, deg
Figure 25. Comparison of Ufimtsev' s Predicted RCS with Measured Data, kQ = 9. 2
45
ASPECT ANGLE, deg
Figure 26. Comparison of Ufimtsev 1 s Predicted RCS with Measured Data, kQ = 11. 7
I I I I I I I I I I I I I I I I
I I
I I I I I I I I
E en .a ~
.. I
V)
u IX
I I I I I I I I I I
* 0
' : : l! ; r: ; :! ! : 1; , , ; . ; ! : 1;!: :! 1. II· ! 1 tiiJ I! · 1! -. llJ:! II!; 1 1 : 1, ·1, i: !· ·1)1 ••! ,,,, 'I• llji j·i·!l 'il, J!ll 1 • JIJJ l:j ·j·j!j,].;li ·HJ'II:.•I I·J· 1·: :+ ,. "! t!· ;•·;I ;,·_ ·, . ·!-i-1 I ·I; -·~; If ••• - -1+1 :. •t-"1: .. I ~-:~ .. I
:; :t:,:-r· 't.' .!Jill r:: -:-~:t·tf:ttl- i·!t!T:'r -1-'L! ·t'-li'r- J.H-u•:'tt f'lt:!-:lti_tJ: j::Ji It!~ " L: r ! .-! i-:-f. ''J !-i·i r + -1 ~ -j.l-lj '. 1- tH;-• ,-r-!-i tq. L. ; I t-1 i-1+1 .. " 1'1~·1[-1 I ;. ~-- ·r · r-•·r ~-~- · ;~. :·t·l+ ...J.,-t- ·II I H·t-H-t!- -~rr H~~ " '-Hi-HHH· .:+.··Hi-!.
UFIMTSEV [1:1*1- !-!- 4 l·:.t-r-h !' ··r ·i ..1... -,
:f:!:j·Li ±j:+f Li::i±frcr DATA
13 kl = [l:fiiit[ m+~+t~
ka = 4. 2 X 1 o-3
~~: tg}. ).. = 0. 227 m
10 ~tn-::~T.. POLARIZATION: TRANSMIT CIRCULAR - ·:r rWt - f8_:~~: RECEIVE CIRCULAR EmJJ.. ·rtt:·:J~~,ft~clf±!: Htj: dif r,+rFJ :FiJf ~rf:l: t;ti HH:bEl:rJ.i:!EJt:J:Hc"~- ' - +H++i~:t=~t~ :ttt-_--,'+:EF.-:r~+-tf+r t+it 1-r-;-1- H~+.J:,.r-r~ r,-·!- ++~-· ·:·1+·:-+;·i· ~q:f-.:t:;t+~tffi=F-R+ ::P~+:~Tl:t:p:~ X!:f~r-4-
20 40 60 80 100
ASPECT ANGLE, deg
Figure 27. Comparison of Ufimtsev's Predicted RCS with Measured Data, kQ = 13
47
"' V)
u 0:::
Figure 28. Comparison of Ufimtsev' s Predicted RCS with Measured Data, kQ = 17
48
I I I I I I I I I I
!
I I I I I I I I I
I I I I I I I I I I I I I I I I I I I
E en
..0 "0
ASPECT ANGLE, deg
Figure 29. Comparison of Ufimtsev 1 s Predicted RCS with Measured Data, kQ = 45. 7
49
0
-10
-20
-40
Figure 30.
40 60. 80
ASPECT ANGLE, deg
Comparison of BRACT. Calculated RCS with Measured Data, kR = 4.' 44
50
I I I I I I I I I -I
I I I I I I I I I
I I I I I I I I I I I I I I I I I ·I
I
E Ill
:8
:'
i·j 1
' . . ! I: : ;. j l.
• ;c; I li: 1' I ' · ' ! , 60 \-;j I ·Ill 1 • • , 'i 1
- iti k 1· i I. ' : Iii :il H' I . ,, ~~T' ;i! I
: . . . ! !' I ' !! ! I i ,, I ~: :;I 1IT . i l ~:: : I :;p : ; : • . : i
_70
1iJ:): · ... i! ':· :. · :: :: :: 111 1 , : r 1Hiii 1 ',iii ; [!i ::: •i· i£t:'::·,~ ·,··!!, :i:lH 1 1,,,Tr r·~~~ 1,;:ii:IT,1F i! I ii I' I . ' I' ' . i I. . . : . . n· ' ' rr;' ,_; ill r;:·· ii ,j;! ,' II . ' : I: I I I !: I I!;' ! I :: Iii :' 1:; !!'i :;!; :i: ,Ill 1; I ii~ Iii :Iii 1:1!1 1 11 iii 1!11 i {, i ·::~ 1 ;
-80 It+~ ..... ! ++i-'+./..f+++1-H+I+H++++I H+ll v BRACT l I '
,).: ~~~:~ I I • 1 kL1
= ~.~TA ll@ !I! :, 1 · : i 1 , ka = 4.2x1o-3 !,~J;
· !1[ ! ' ; · 1 I ! "A = 0. 227 m : i - 90
1 i ::
1•
1
·, :I POLARIZATION: TRANSMIT CIRCULAR :n,· :. 1 ·_ I ! 1 RECEIVE CIRCULAR i :
-100 0 20 40 60
ASPECT ANGLE, deg 80 100
Figure 31. Comparison of BRACT Calculated RCS with Measured Data, kQ = 9. 2
51
-30 ',, : .. I; .• I' ' I H ' u i:'t I'· '
' , '· ~r . f!!J· '£!
· , I • i;rq~~~ , ,, rc'' rH::!f 111
i ' ' !
i I i I ! I I I I li i ':iii . fl ' iii ;
-90 0 20 40 60 80 100
ASPECT ANGLE, deg
Figure 32. Comparison of BRACT Calculated RCS with Measured Data, kQ = 11. 7
52
I I I I I I I I I I I I I I I I I I I
I I I I I I I I I I I I I I I I I I I
ASPECT ANGLE, deg
Figure 33. Comparison of BRACT Calculated RCS with Measured Data, kQ = 13
53
ASPECT ANGLE, deg
Figure 34. Comparison of BRACT' Calculated RCS with Measured Data, kR = 17
54.
I I I I
' !
I I I I I I I I I I I I I I I
I I I I I I I I I I I I I I I I I I I
ASPECT ANGLE, deg
Figure 35. Comparison of BRACT Calculated RCS with Measured Data, kE = 34. 8
55
-10
E .. .D -20 "0
.,; u. a:.
Ul, C' -30
I F ·gure 36 Comparison of BRACT Calculated RCS with Measured Data, k1 = 45. 7 1 •
- - - -- --- - - --- - - - - --
-------------------H' 14 II,-_· +--H,.,--+·-I---+:...;_+-'-11r-~-11· ~rrl,"_:-1 ....:.:···+-t---'fJ-·~i'~,..,t-k-i, -r-· ~ ~i RE~uLrs ~ THE BR~CT coMPUTER PROGRA~1 ~,..-
-15 :I~~·· ~ +- .,.Jc .... I .. I I' -+-t--'t-t-:-'._·'+1- . •.·.·.·. -.~ .. OR. · ... '. kL .=,_ •. ·.'·.~"'.7.·r;c -... '.•""_:·,_. ·''·'· .. ~- .... ·-· r:--r--:-1 , · . ~r·- ·-r :•:·" L l. · · 1 ..
I'H-"-:T"-t-'+-:-:-T-'-t-ti:rTc.; t-' cc--r----r:- . . . : l '\I : . +-:-'1-"-+'"-+-:-.. +.' +".-:-+-:±:,.:f-'.:.:.t'-=f:.:....r'+'c't-'-"-f"-tc:--t-. -:-:-" . r .. -:t ·.,.T-:-r--. :-j--. • : -1:-:-" F
-20 ·: : -':, i:i ::, ~~ j .: ...... : :. : •••. :\ ••.• .• • :· ••.••..• :•,, .::· .. . .. .... .. . • ..... ·::' •·•·• •: •••.•.••. · ... . ,•:::·::r• ······-·· ii"••·· ... IAIP'!"'Il;••·· :::·· ······:•·• r' ,, ... .
. . ... :~·::I~:: > , .• ··~~ .·· I .. : c] .: , ::,- . ., .... , ., , .... . ····· ....
-25~~--~··~···+~·3··~~-+.~;_ -.,+.~+-~.~ .. r.~.~~~··~:~~~~~--~~~~· ~--~.-.~:_--.+_x~·~~J~-~._.-*··~--~·+=r~J,··.~~~·+~~;·•4,+:.~t~:!~;:.=.=,_,r~7~~--"' . . .... ,;: . . .... .. :••.[! : ·, ·.~:: .... :, ': ": .... ·_•: :_•:_:~·.• . :· 1 1'1 -1 _. 1. . ·· :• '.•.'. r . ·:: :o. .. i .J ., ·. li_.: ' "·' "f:..cur\.~+--1
&:-4·-'++~a:,+.+-t--i.,..+:.::r-+:H~T~+-+-:-'+--·+~"'t-' -· '-+--__ •.· .... ··. :: L IT'_ : ·.'.',· .. T• ·.·-.· ... -~ ...•. ::. ::: :;.: ·:;: ~~iqj _::.: ·· · · .... 1. : ·.:)·· 7 t"'!J" • :a .'t:ll
-3o . .. . .. .1 .. ... . .. .. . ··· ·,1 ;• !i': . .., . •· •· 1
-· .. :: •••• ;:~: :::. . .. . .. .. ... . :·:+--r-'11-· .:-'+-.i..f--t'·,.._· r'c'''+' ·.:..:.,··r-··..,'·t .:..:.,"cr-'"--+"f'i----i \'-c.~~~ : ',·.• ' :~:: '::_:_ .•.·.~t:'--JJ ...... · ;,; .,.·: ·JJ>:: .... I I: \:c. :: : :. : :J:.
-40 • ·.:.;J~+.±+c-:-1'-':'f:--:"1-+-t'~''--t---f_l_•:·t·'-l<i,...:.:. '~:t' :.;.,::•t'••_'·t ·-+:-~:t--tir--·f· --+·· - r-- i - __ ,!_..,...... •---..- r-:--• c-- ... :---t-.::t-'· ·- ~-.-: 'I ! i T. i
H~~::;.p+-':'H. :.::.: .. + .. :.:: ... ·f'-. :-'f. -:f~-rt-J1 .~ r-.. --- --; -r-· ~.. ! - --~~--~-.t:-1 ~- __ ;! ____ L,"':l1
1 ,
... · ·•· • ·• ,." • · .. ·' 1 ..1 ·1, ,
I. I .i ... -· -.,.-,·- '-~~ ·--11 .... -+++--1-+ 1-'--F
.· ..
-45 ... • • . .....+:-:4b-.,.f'-t--b--+--:-t--·r ·-:-t: • --t -+- · · : . :i --l---1-i- ~L _ l:-c-f-"P.+-·-'1".:.':-.":Fb-:..: :· ..... + . :! ····t'· .. -_r -.. r-'1-· ..... -! I i I b ' ..• .• :::: :::: .
... ': ....... . - ...-.-!--:'-+..:'. ..., '; :: : , .. , ...... .
". :::; ::
Figure 37a. BRACT Calculated RCS, kl = 157 (She·et 1)
\.]l
00
E ,.
i -35r··~··r'··~~~~~~~~---t·~--t·~-t-~---~-·t•~:::+'~'~t'7~''~:~::+:·~·-+·:~·-+::~:·+·'~::+''+"+i:+.··+·-~-·+·~·+.··~~~+'~··+·~-~~~-~-~-~-~~~~-~-q-~IPJF;h~ .,; .. : ,.. .: . . .... ... . .. . .. . . . .. .... :: : ..... " . .!.: ·I :: ; .. ·- . . .. .
~ .... . . ..... ::'.' ·: ...... ' '.:·~ :· ·:·. :;:: :•:: '" :·:: ::: :'fr: .. ··:'•::- :, :· . ·:. ~- . ... .. "::J. ....... -
:~ ·. 'P .... ~- I iL · .4-. :•· n.:: .. '; ;;; '~ ,:;,i· . !
~1 ,::: ... · · · · .: '" c; ;:: i1l_! IT ,. : .. .,; : .•• rl:: : .. - : ::: l;j •; •:i!l!f : : ·~, ' . : ·.: "' -
-45 r. __ ~:ii:. -'l _____ ~r ___ -f:_-l_t'·. ;_t-•.. ~~---'-1-r-•. _.:~:_.!~;-:._•tt-tJ~Jr::'tl:~ ·•"-~':If-=~:"-~', ;~'' w, '*.; ~·~;·*'-*_ .. +· 'i"'lh-.':-+>~:.rn-"-41* ... *. *-_4 __ ~----~RH+.-·4; ;4,· ~:.::.J_: .;..: ~-f::::·:.:t._b· .. f.:,_-•_4:.:_ ~._._:_-~.-4-.~ _-""':.:.·.~~---•_·+·-···:_·.i.:_J ::1 !·"!' ._.. ;; ~~~~ ~i-~ : :; :::.; .. ; :::: ::~i ; :· .. :: ··- ... :-.
.... t-·· •••• .. .. . ·-·· ..... ,. : : : ~ : i . -- ..:r ·::.L: .
..
'I .:j. ~·~-~~-+·~·tt~·~·+· ~~-~-+H~-,+T~'~.~;:,~ii~'~~8~--~·~~-~b+
-55 tt-H....wt--+"-'tt+ ... c:t. 1-H:\--'+"-llr+.,J-H-+-:-:-_J.:..:_ .~+:.c"~.f:-:.,.f*!,....;.: ,+· .-'-:111-. ·..;.:;b.::-"-1. I ~,.:.+.J.+;+m;..:;~~Fl++..:.:..:+-+++...J..--'FH~~-:.:.:-·W
. .. . .... • I . :•: :·~: I l . . .
-60 u . j >; 67.5 69.5
t:: 71.5 73.5 75.5
' i;: .· . .
77.5 79.5 81.5
ASPECT ANGLE, deg
:!: ::.;i: ..... .
·r 83.5 85.5
·Figure 37b. BRACT Calculated RCS, k1 = 157 (Sheet 2)
87.5
•.:: :."f:.'.
···•, :::'·II:: 89.5
---··---- --------·····---------- --
I I I I I I I I I I I I I I I I I 1---
I
E en
.Jl "'0
Or-~~T-~~~~~~~~~~~~~~~~~~~---~~~----~--~------~--~----------~-----~L--~----: ____ • ____ fl-i __ ·_ ·~---J ·1-T ________ , ___ J ___ t ___ J ____ ~--+-------·- ~---L- : __ __!-: l . -~- -· ;----. l · - . • I l ! . i . i ! , . - i --, -- 1 --~- :
-+--+-·!--+:-· -+---~---+-----+----+- : , ··-----+--;- Ij f -- : . I' I I I .
~--- ____ , __ · ·------~·-:-:----·-- ~---- 1------- f--· --r-. t-:·:·+--~---i-----. -L--j--1-· .J--r-:-~----
-10 ~I I ! ··1---~)il!.· I.· :,:,···.:POLARIZATION= LINEAR ______ L ___ _L ___ _; ____ t_ __ ~-- ... i..·--. ~~nt ... l ' ! I:! • •··• ' ~ ~ i • -•--~- ~--~r-- e.----:- f-- ~~ •---~---• WAVELENGTH = 0. 0200000 m - t- ~- -t -+ ·:·-:- r---:- _ !- __ _
I . 1-- >---- -~WIRE RADIUS = 0. 0001520 m ----+----
1·----+------r' -".·---.!llf.;..--J-. ___ j._ ____ ~---i 1 1 . .
-20 > !j - f -:-~ ! - 1-- k.l = 157 .! -~t. -i-----l- i-- 1 ~ --;---
1 '" .· VAN VLECK CALCULATIONS AS---+---- ---t----1----i- ' :-- --+-----
1 I . INDICATED BY "+"SYMBOLS i • I l I . : - --}- --- !-- --+--- --1 f--·- 1-----1--------:--·---- -· -1------ 1-----~ f----. ------• 1-'--t----+-- -- - i - +--. t· ---- i·- --+- ~ --- --- -; ... -~----L ! 1 . • i . UFIMTSEV CALCULATIONS AS i 1 i ' . i 1 • r • · A · -! · DENOTED BY "*" sYMBoLs -----~---1~---~--r -f- .... .J ___ _ . --: a· . . . i , , .. I
-30 .__: ~- i - . :~ p fr+ ' ! -r-_ ----;--;-.• --•. ·· :----r---t-~·~--,---··-~-r~-L- Fr_::r~~ ~>--· .• -.:: -
f----i- -~--- -----HlH,-----~~-_.J+.-!!II+l j I 1 ,,-=-+--,---1-~--~+----+----t --;---_----+!_-- . ~--•--- ------,----·-+---+·- ;-- r·- ,-- r---+----i-- -~ -- ' l :, --
•• _ __._ ___ +-· --!-+K-....... --f.~+----1~------+-1~·-fo-E-1 li----'-Jf-411--l-tl-..--1:-..-+~---'~--+-·-+---!---~ ~--1-----i---4-----·'---· -~--- ·-t- -- ---r . . · i 'j I. · · . · ! j I ~---- ---1-, .. --'-------1-~.,.~-----114-8- ·-t -·-1~-----...... ~·~"..:II~W-M.---,.__,---II(-,adc-,---.-!--·--:-l·: · --t--· 1-:-:-.:_1-~·_;...:~. ~-: . :-- -+- ·· 1-- . :----- ~- --+-----
. I
·' ~;
J-.-.:...:..j------ 1- -- +-- -;
-40 v; --+-+-:+!--·-·Hfii-!---Hrl-t---IH---+f-~~·1~..!1-¥t-IJ..li-~'~---H--tliii-A-,..._.a.:..JI--&-·......_~h---l-..:. ' ! j • ~ j - 1- ---~ --- ... - ·--l---+-------,r-'A· ...........
j-+:--· ·_1--M--+--·-·~--·· -__..._•·_-_-:-~~:-1-l!HI'.IH-:_-J-'--~--N, ~-.q. · W4-111-iH--I--f.'I-I--H·II-• ... ~11'9J---fiiJ-~ l-"'~ H' •. ~----r~n~nl!l1tt4~~~~~~, 1f•, h11
1ll·1 ~~· i . .• ;
u .. - --!'--------1-- -
0::: T- .; -!----. 1---· - - --- .. , . - ~ - :---+- -
50 ! I ~ • l • : ' • . [ L.l i _ 1---+-: ·------!---+-----!-ll-r----+----1-----~~----l·~~.J~·:.-1 II-4<U.:......-K--1J .. ·~tt-Hit'-fl~-tt-tlrllt-ttillt-Jit-·ftHH·-IIHIHf·~.--tt,..._lt-:ll~ ............... .._ .• J..-8--..._... ·, .. r·-·r.·· -- I::;,>··---"----!---
---+-----,.___ __ ,,___ :·· r-J- .: 11 w''t ~!-i: --- h:- - ·t • ; :l :' • • ' . .. '
~--~-~-+--+--+..._t---hll-+--t~---+t:--~1-+. •_...:. "Hii--.;..t..::' . .:..t~ '' t:-:..JI+-,--li-:..JIJ;--I~_._........_u ... ...a.-+&-~._.._._~~ _._. ··~J.;.~;.-t--+-t ,._.. ___ ------ ',.IL__, ... ..._ ___ i, -·. f--- ·· . :-
: . I . • . : ' ' i : ' I . j i ,i I.· .. : : 1----- ---11------+-----+---.-+---t---W---+-----&-----~J..- +1.1---'--tw---,-.~-;--'-l-:......ii4~ .. 1W-4 .. ·~11r-:1~1F-1'_.~~1-11~-IF--II---I---J---t;l--~l-11i-=t--:-l-...:-+ -i------ -~ i · . - --- 1- --•- -
-60r-+-+-~-~~~f-·-h~~-~~+':~,~·-l_·· ~:~:~~~~;~r~~j~~~~~~~~:1~~~~~~~-..:.~--~~~~~-_...:.~ j<:•i . .•• _,1·. .i ., •• , '.;I··-; I' i . -:•· ·I· . ! i, 'IL
. : : 1 :I>':;;. II I' ..
1----+--+----f----+--d----t-----l----t---l"~-1--'""----'+, .,.,, .· .:...:--l:.l!f----1 i.:..j, :. l-q; 1!1--tl , ...... : I-!-"' i • ' -- -.:. 1-- !---- --- ~- '------ . ' -- ···- ------: :; ! ; : '' i i ': i ': I ': • . . .. - ':: . : i -- !--~- H :'
. • . , : :. .. , I. ; , I ' -----·J..;.n~-_4----l---+--1
- 70 ~-:---. +-.---~-l~---------~---+----ll---:··~=:=· ,_ -+. ~--_·---··· +-~l-_·_. --~·· -~~~-:-~:-,r-:_--~~. + ...... ·~--:..:..• -+. t--:-1...,. •. -'-_, •• --+ .. -t~-----~:~~~:~:=~::~:-~:~:~=+~ -_·_ ~1:-~:--_-· __ · __ ~~-~-:~~:--_--· --r+ . I - - , + :~ --~ ,~=>-- j~ : t-f--t·-f_-+· .. ·-: ·J-+1--'--:-' ___ '--·:1 -~:==+-1--.....;.--., .... ___ ·_-·+f--------.~ I-··"'"·.·~·· .. ;._ .• r.-. ,c.:.. -H-. ~; ·.._;_.--'-+-1 . .....;.' :-:--!: ..;.,..,...'+--. -'-. ~- -+---'-,-+--.. +----------f-'-----1-tr.--.-._-·:-f~-~:=:=·~~:-~:r------~j----·_-·--~-~·------l-----:"r'·:·-- __ -- ----lt.·_ -_ r,· ------+-·:·~- _----+'-P.,.:.... __ ~_-+· __ --. ~+--------.... -7 ... ~--------~--' •. · - . • · ~ . : f: : i : 1 • ~ • , , • • _ _ _ __IL • • ,
. . . . . I; ' . . . . --.. . ' : ::.I')' .:1::.·. -,: ,: - .. i:··· . . -!~------ I IJ ·t-· ! : 'I I' , : ;~ · . . : --~__:_- ._:_:_f.--- ! ......... _ '.... ----J--
;1' :: ,::;iJ;!lLUJtili:lHlll·., : :r•' ·::: .• • l:i,. l·' • -J: :: ··- .. l. \ ·. T ...:::. -so~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. _.~~~~~~~~~~~
T - . !
. --1- -l --+-
! -·---l'-'---+-·-- f-- ·+ ---
: , ..
0 10 20 30 40 50 60 70 80 90 100
ASPECT ANGLE, deg
Figure 38. Van Vleck vs Ufimtsev Predicted RCS Values, k.f = 157 59
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-10
H , . m . .· ... . . ,:r : : ~k1 ;;::, · : ;- : .. : H-\· ~ : :- r ' • ' .
l ! ~' i-t i f' : ::r ~;t !
H: t-~ .· ' . I i I E en .a
"'0 -20 .. V)
u ~
-30
: !J :
20 40 60 80 100
ASPECT ANGLE, deg
Figure 39. Comparison of the Generalized Van Vleck Formulation with Van Vleck's original Expression, kl = 13
(d
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APPENDIX A
The expressions for monostatic cross section from a generalized
Van Vleck (Ref. 8") theory would be
E1
::: sin 2q£ _(A+ B) sin ((3 +g)£_ (A_ B) sin ((3 - q)£ cos e 1 + cos e 1 - cos e
9 ::: 13 cos e
21T 13 - } ...
A ::: 2K cos g.£ - (neiq.R.
- (Eeii3.R. 2L cos 13£
B ::: 2 iK sin q.R. -.(Deiq.R.
2 iL sin 13£ - {Eeii3.R.
F ::: Cin 213£
+ Ge -iq£}
+ Fe -i 13£)
- Ge -ig£)
- Fe -i(3£}
E ::: F - Cin 413£ - iSi4131
G ::: -F - Cin 2(13 - q).R. - iSi2(13 - q).t
D ::: F - Cin 2(13 + q).R..- iSi2{13 + q).t
A-1
(A-2)
(A-3)
(A-4)
(A-5)
(A-6)
(A-7)
(A-8)
(A.:.9)
(A-10)
L = 2 1 o g 2
( 2 a£ ) + 2 1 o g 2 + ~ -. C in 413 £
sin 4 ~£ - 1 4fH
- i [si4fH +{cos 413£ -1)] 4(3£
K - 2log2 (:')+ 21og 2 +~-Gin 2(~ + q) £
_ C · 2( A _ ) _ sin 2( 13 + q) £ _ sin 2( 13 - q) £ m t' q £ . 2(13 + q)£ 2(13 - q) £
- . f S"2(13 + )l +5"2([3- )£+[cos 2 ( 13 + q)£ - 1] 1 ,. 1 q 1 q 2( ~ + q) £
+[cos 2( ~- q)£ - 1] I 2( ~ - q)£ J
-cosy dy = ~ ( -1)nx2
n y - L...J 2n( 2n) !
n=1 0
Jx co . L (-i}nx2n+1
Si x = 61ny Y dy = ( Zn + 1 )( 2n + 1) ! n=o
0
£ = length of wire
a = radius of wire
e -- angle between the propagation vector and the wire
-cp = angle between the E vector .and the plane formed by the wire and the propagation vector.
A-2
(A-11)
(A-12)
(A-13)
(A-14)
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I UNCLASSIFIED s Cl . ecunty_ - ass1 1cauon
DOCUMENT CONTROL DATA. R & 0
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(S•curlly rlaa81llr.•tlon of title, hndy of fltbllltac:t 11nd lruJ,.,dnf1 annotation mu11t ''• enterod wl1ttn tl1e ovttrsll ttlputl I• c-ln•tillled)
1. ORIGINATING ACTIVITY (Corporate author) 2B REPORT SECURITY CLASSIFICATION
The Aerospace Corporation Unclassified
El Segundo, California 2b GROUP
3. REPORT TITLE
I PREDICTED RADAR CROSS SECTION OF THIN, LONG WIRES COMPARED WITH EXPERIMENTAL DATA
4. DESCRIPTIVE NOTES (Type ol report and Inclusive dates)
I S. AU THORISI (First name, middle lnlllal, Ia at name)
I Jacques Renau and Michael T. Tavis
6- REPORT DATE 78- TOTAL NO. OF PAGES '1b. NO. OF REFS
72 OCT 01 66 . 8
1- ea. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER($) .
F 04 7 0 1 -7 2- C - 0 07 3 TR-0073(3450-16)-2 b. PROJECT NO.
I c. 9b. OTHER REPORT NO(S) (Any other numbers that may be assillnec' this report)
d. SAMSO- TR- 73-136
10. DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited
I· II. SUPPLEMENTARY NOTES 12. SPONSORING Ml Ll T ARY AC Tl VI TV
Space and Missile Systems Organization
I Air Force Systems Command Los Angeles, California
13.1 ABSTRACT
I In order to determine the validity and accuracy of Radar Cross Section (RCS) predictions for thin wires, the predictions of the closed-form expressions developed by Chu, Tai, Van Vleck, et al., and Ufimtsev have been compared
I with carefully measured backscattered RCS vs angle of incidence for various length thin, long, cylindrical conductors. Further, the prediction of an open-form numerical analysis based on the Source Distribution Technique
I and programmed by M. B. Associates as the BRACT computer program was also compared with the experimental data.
It was found that ( 1) the BRACT computer results agree with experiment 'so
I well (within ±1 dB for all reliable data) that it may be used with great confi-dence for any length thin wires and can be used as reference data for compar-ing with the prediction of the close-form solutions; (2) the results of Chu are
I not accurate except at broadside incidence; (3) the results of Tai compare favorably with data up to kl values of about 17 if corrections are made to the approximate formulas to correct for broadside incidence; ( 4) the results of
I Ufimtsev compare well with experiment for all k£ values considered; and (5) the results of Van Vleck, et al., appear to be very accurate for all k£ values considered except for near end- on incidence. In a separate report to be published soon one of the authors of this report (M. Tavis) has
I shown that this deficiency is due to numerical approximations in the theoretical expressions.
I DO FORM 1413
IFACSIMILEl UNCLASSIFIED. Security Classification·
UNCLASSIFIED Security Classification
u. :KEY WORDS
RCS of Thin Wires Analytical Expressions of RCS of Wires Wire RCS Computer Predictions of RCS Comparison of RCS Predictions
Distribution Statement (Continued)
. Abstract (Continued)
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UNCLASSIFIED Security Classification
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