38-7
EFFECTS O F ERRORS IN THE RESISTIVE TAPER ON T H E
FARFIELD SCATTERING PATTERNS OF STRIPS
Randy L. Haupt’
Dept of Electrical Engineering
United States Air Force Academy
Colorado Springs, CO 80840
Valdis V. Liepa
Radiation Laboratory
Dept of Electrical Engineering and Computer Science
The University of Michigan
Ann Arbor, MI 48109
Introduction
Resistive tapers that lower the sidelobes of or place nulls in the scattering patterns of strips have practical limitations. Amplitude and phase errors in the aperture distribution of an antenna cause the far field pattern t o differ from its theoretical no- error antenna pattern. Similarly, errors in the resistive taper of a strip create errors in the current density, which in turn creates errors in the desired scattering patterns.
This paper examines the effects of random errors in the resistive taper on the induced current densities in the strip and the resulting errors in the scattering patterns.
Modeling the Erron
The normalized resistivity of a strip (7) is given by o(x)=70(4
+ w
(1) where qo is the desired resistivity, 6 the perturbation, and x the distance along the strip. These resistive errors produce errors in the strip current density given by
J(x) =
+
C(X)
( 2 ) where J is the strip current density, .lo is the strip current density when there are no resistivity errors, and C is the perturbation to Jo due to the resistivity errors.
Substituting (1) and (2) into the Epolarized (electric field parallel to the edge of the strip) integral equation for the strip results in rlo(x)C(~)
+
$-I:aC(x’)H!i2)( klx-x’l)dx’ = -6(x)Jo(x) (3) where 2a is the width of the strip, k is the propagation constant, and x and
X’ are in terms of wavelength. This equation may be numerically solved for C.
The radar cross-section (RCS) of the strip with resistive errors is given by
LEEE
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U
=
(10
+
Uerr where uo is the no-error RCS and uerr is the RCS due to errors alone. the no-error and error currents into the equation for the RCS produces
(4)
Substitute uE($) =
-a
Jo(xr)ekxrcosddx’
+ A t n= 1
Cnjkxncosd12 ( 5 )
The deviations in the scattering patterns may be isolated by completing the square in
( 5 ) . aE(q5)
=
+
The first term in (6) is the RCS due to the desired resistive tapers ( u o ) . The remaining three terms (uerr) are due to the errors in the taper. When the errors are small, the fourth term has a much lower average level than the other terms.
Generally, the last three terms combined are much smaller than the first. The exception is in the nulls and low sidelobe regions of the RCS patterns. There, the level of the error terms may exceed the desired RCS.
Itesults
Random resistivity errors are assumed to follow a normal probability density function with a mean of 0 and a standard deviation of ukq0(x), where uk is a constant between
0 and 0.3. The plots for J, q, and
U are shown in Figure 1. The error terms, 6(x),
((x), and uerr, are plotted in Figure 2
for a uk of 0.3 and a strip 4 wavelengths wide.
Conclusions
Random errors in the low sidelobe resistive taper cause predictable errors in the induced strip current density and RCS. The large errors in resistivity also occur where the strip current density is smallest, thus mitigating their effects.
Reference
R.L. Haupt, “Synthesis of resistive tapers to control scattering patterns of strips,”
Ph.D. dissertation, The University of Michigan, 1987.
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-
ERRORS
ERRORS
-.
EXRORS
-
ERRORS
2.0 -2.0 0.0
X IURVELENGTHSI
-2.0 0.0
X BIRVELENGTHSI b. Normalized resistivity on strip a. Magnitude of induced strip current density, $o=90'
-.
ERRORS
ERRORS
0 n
30
(p
0
!M
60
(DEGREES1
30
0
0
I
0 30 60 50 c. Bistatic scattering pattern $o=90' d. Backscattering pattern
2.0
Figure 1. Error and nwerror plots calculated from the Epolarized integral equation when the strip is 4X wide, has a low sidelobe resistive taper, and has random errors with ~ ~ ~ 0 . 3 .
866

z a
9
-
- 1 N T E C R R L E O U A i I O N
D
0
-2.0 0.0
X IUAVELENGTHSI a. Magnitude of surface current, do=90', due to errors in the resistivity
I
4
-
- I N T E C R R L EOURTION
2 0 -2.0 0.0
X (URVELENGTHSI b. Errors in normalized resistivity
20
-
I N r E G R R L E O U R T i O N
0 30 60 p, IOEGREESI c. Bistatic scattering pattern bO=9O' due to errors in the strip current density
90 0 30 60 p, IOEGREESI d. Backscattering pattern due to errors in the strip current density
90
Figure 2. Isolated %polarized error contributions when the 4 X wide strip has a low sidelobe resistive taper and random errors with ck=0.3.
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