# Performance Comparison between Nonidentical  Segmented Exponential Concave and Nonidentical  Segmented Exponential Convex Serrated CATRs

Authors: T. Venkata Rama Krishna; P. Siddaiah; B. Prabhakara Rao
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

This paper presented a theoretical and numerical assessment for nonidentical segmented exponential- (NISE-) convex and NISE-concave serrated plane CATRs by changing number of serrations. The investigation was based on diffraction theory and, more specifically,  on the diffraction formulation of Fresnel. The compact antenna test range (CATR) provides uniform illumination within the Fresnel region to the test antenna. Application of serrated edges has been shown to be a good method to control diffraction at the edges of the reflectors. In this paper, the Fresnel fields of NISE-convex and NISE-concave serrated CATRs are analyzed using physical optics (PO) technique. The PO analysis is applied in this paper for plane reflector serrated CATR only. The same analysis is applied to any type of reflector. In this paper, lens-based reflector is not considered. It is observed that NISE-concave serrated CATR gives less ripple and enhanced quiet zone width than NISE-convex.


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## Body

## 1. Introduction

The
CATR provides a relatively new method of measuring the far-field
characteristics of microwave antennas. The test antenna is illuminated by a
collimated beam of energy from a paraboloid. The linear dimensions of the
reflector must be at least three to four times that of the test antenna in
order that the illumination on it approximates a plane wave. This technique
leads to far field pattern simulation and requires smaller distances than in
conventional methods. A CATR can be situated in a normal anechoic chamber. Due
to the good indoor performance, the method is time- and cost-effective and the measurements
are performed in real time. Rigorous
analysis of the CATR is needed in order to obtain a realistic assessment of the
test zone fields. Some factors which should be taken into account are as
follows:(i)
nonideal system components, such
as tapered feed pattern and mechanical inaccuracies of the reflectors;(ii)
nonideal environment as due to
reflections from walls and obstacles in the anechoic room;(iii)
stray radiation level;(iv)
reflector surface accuracy; and(v)
edge diffraction at the CATR
reflectors that contribute to scattering at the edges of the reflector, thus
modifying the far-field pattern.
The fifth factor has a large impact on the performance of the test-zone field. The methods to
reduce edge diffraction are as follows.(i)
One method is
creating an incident field with an amplitude pattern that falls off sharply at
angles corresponding to the edges of the reflector. Such a feed pattern is not
easy to create over a large range of frequencies and it reduces effective test-zone
size.(ii)
Another method is to use of microwave absorber at the reflector edges.(iii)
The third method is to use of small flanges around the aperture
edges.(iv)
The fourth method depends on serrating the edges of the
reflector in order to redistribute the diffracted rays in many directions away
from the quiet zone area.
Application of serrated edges is a good method of controlling the diffraction effects at the
edges of the reflector. The way in which the reflector surface is terminated
affects the electromagnetic properties of the radiating system. The signal
reflected by the parabolic reflector is basically a plane wave which exists
over the extent of the surface. This reflected field abruptly stops at the
surface termination. Since the electromagnetic fields must be smooth and
continuous, there will be strong diffracted field emanating from the terminating
edges. This field then interferes constructively and destructively with the
plane wave causing amplitude and phase variations of the field illuminating the
test antenna or scattering body and degrades the performance of the range. This
diffracted signal is one of the major contributions limiting the use of the
compact ranges.The serrated edge
termination is intended to form a transition between the reflector surface and
free space. In performing this function, it is designed to diffract energy away
from the target zone. However, a finite edge is composed of corners which
diffract energy in all directions including the target zone. The use of
serrated edges is tantamount to creating corners which can also diffract into
the target zone. Each of these corner diffracted fields will be relatively
weak. Appropriate designs of serrations ensure effective cancellation of these
diffraction components in the test (quiet) zone. It has been shown
theoretically that the application of serrations minimizes distorting
diffraction effects in the Fresnel region of a CATR [1, 2].


## 2. Evaluation of Fresnel Field of Serrated Edge Reflector by PO Analysis

The analysis of the Fresnel field of a square aperture
with NISE serrations is applied  using the method of PO. In this paper,
four different shapes of serrations are used in four sides. This analysis is so
general that it can be applied to any serration geometry. This paper presents a
gist of the analysis of NISE-convex and NISE-concave geometries shown in Figures1
and 3. The
Fresnel diffraction formula which gives the x-polarized field over an arbitrary
plane (z = constant) in the Fresnel region is [3, 4](1)Ex(x,y,z)=−jk2πze−jkz∫−∞∞∫−∞∞Eax(x′,y′)×ejk{(x′−x)2+(y′−y)2}/2zdx′dy′.
It will be a
laborious task to find an analytical expression in a closed form for the
Fresnel diffraction pattern of an aperture with these serrated edges. Hence,
recourse is taken to decompose the aperture area S into three parts S1,
S2, and S3, such that S=S1+S2−S3 (see, Figures 2 and 4). A quasianalytical expression can now be derived for the
Fresnel field [5–9]. The NISE-convex and NISE-concave serrations described by
the boundary functions h+(x′),h−(x′) and g+(y′), g−(y′) are expressed as Fourier series of width-modulated
exponential with rate of rise “ai.” The serrated edges are described
by the functions h+(x′),h−(x′) and g+(y′), g−(y′), and Eax(x′,y′)=E0 for (x′,y′)∈S.Figure 1
Square aperture reflector with NISE-convex serrations.Figure 2
Decomposition of apertureS into three parts S1, S2,
and S3.Figure 3
Square aperture reflector with NISE-concave serrations.Figure 4
Decomposition of apertureS into three parts S1, S2,
and S3.Now, (1) can be rearranged as
(2)Ex(x,y,z)=−jE02e−jkz(I1+I2+I3),where(3)I1=kπz∫−h=−b0/2h=b0/2ejk(y′−y)2/2zdy′∫g−(y′)g+(y′)ejk(x′−x)2/2zdx′,=kπz[F(t+)−F(t−)][F(s+′)−F(s−′)];I2=kπz∫−w=−a0/2w=a0/2ejk(x′−x)2/2zdx′∫h−(x′)h+(x′)ejk(y′−y)2/2zdy′,=kπz[F(s+)−F(s−)][F(t+′)−F(t−′)];I3=kπz∫−w=−a0/2w=a0/2ejk(x′−x)2/2zdx′∫−h=−b0/2h=b0/2ejk(y′−y)2/2zdy′,=kπz[F(s+)−F(s−)][F(t+)−F(t−)];t±=kπz(±h−y);s+′=kπz(−g−(y′)−x);s−′=kπz(−g+(y′)−x);s±=kπz(±w−x);t+′=kπz(−h−(x′)−y);t−′=kπz(−h+(x′)−y);F(s)=∫0se−jπr2/2dr=the complex form of the Fresenel integral.


### 2.1. Fourier Series of Width- and Height-Modulated NISE-Convex Serrations

The serrations
described by the boundary functionsh+(x′),h−(x′) and g+(y′), g−(y′) are expressed as a Fourier series of width- and
height-modulated nonidentical segmented convex function. The Fourier series
expansion of h+(x′),h−(x′) and g+(y′), g−(y′) is
given by
(4)h+(x′)=h−(x′)=g+(x′)=g−(x′)=C01+∑n=−∞∞Cn1e(jnω/T)t,C01=1T[2p1(t0+t)+ta(1−e−ap1)+t(p2−p1)],Cn1=1T[−2taa2+n2ω2+2(t0+t)nωsin(nωp1)+2t1(p2−p1)(p1nωsin(nωp1)+1n2ω2×(cos(nωp1)−cos(nωp2))−p2nωsin(nωp1))+2te−ap1a2+n2ω2(acos(nωp1)−nωsin(nωp1))].


### 2.2. Fourier Series of Width- and Height-Modulated NISE-Concave Serrations

The serrations
described by the boundary functionsh+(x′),h−(x′) and g+(y′), g−(y′) are expressed as a Fourier series of width and
height modulated nonidentical segmented concave function. The Fourier series
expansion of h+(x′),h−(x′) and g+(y′), g−(y′) is given by(5)h+(x′)=h−(x′)=g+(x′)=g−(x′)=C02+∑n=−∞∞Cn2e(jnω/T)t,C02=1T[t(p2−p1)+2t0a(eap1−1)],Cn2=1T[−2taa2+n2ω2+2t1(p2−p1)×(p1nωsin(nωp1)+1n2ω2×(cos(nωp1)−cos(nωp2))−p2nωsin(nωp1))+2t0eap1a2+n2ω2(acos(nωp1)−nωsin(nωp1))].


## 2.1. Fourier Series of Width- and Height-Modulated NISE-Convex Serrations

The serrations
described by the boundary functionsh+(x′),h−(x′) and g+(y′), g−(y′) are expressed as a Fourier series of width- and
height-modulated nonidentical segmented convex function. The Fourier series
expansion of h+(x′),h−(x′) and g+(y′), g−(y′) is
given by
(4)h+(x′)=h−(x′)=g+(x′)=g−(x′)=C01+∑n=−∞∞Cn1e(jnω/T)t,C01=1T[2p1(t0+t)+ta(1−e−ap1)+t(p2−p1)],Cn1=1T[−2taa2+n2ω2+2(t0+t)nωsin(nωp1)+2t1(p2−p1)(p1nωsin(nωp1)+1n2ω2×(cos(nωp1)−cos(nωp2))−p2nωsin(nωp1))+2te−ap1a2+n2ω2(acos(nωp1)−nωsin(nωp1))].


## 2.2. Fourier Series of Width- and Height-Modulated NISE-Concave Serrations

The serrations
described by the boundary functionsh+(x′),h−(x′) and g+(y′), g−(y′) are expressed as a Fourier series of width and
height modulated nonidentical segmented concave function. The Fourier series
expansion of h+(x′),h−(x′) and g+(y′), g−(y′) is given by(5)h+(x′)=h−(x′)=g+(x′)=g−(x′)=C02+∑n=−∞∞Cn2e(jnω/T)t,C02=1T[t(p2−p1)+2t0a(eap1−1)],Cn2=1T[−2taa2+n2ω2+2t1(p2−p1)×(p1nωsin(nωp1)+1n2ω2×(cos(nωp1)−cos(nωp2))−p2nωsin(nωp1))+2t0eap1a2+n2ω2(acos(nωp1)−nωsin(nωp1))].


## 3. Results and Discussion

The technique
presented here is best suited to the analysis of serrated reflectors commonly
employed in compact range systems for reduced edge diffraction. A square
reflector of aperture dimensions90λ×90λ is considered to be equipped with
NISE. Equations in Table 1 represent the Fourier series of NISE-concave and NISE-convex serrations.
Table 2 represents width-modulation factors and a number of serrations used for CATR. Table 3
represents height of the serrations. Equations in Table 1 have been used in
conjunction with (3) to evaluate
the Fresnel field at a transverse distance in wavelengths at a distance of z = 64 λ from the reflector aperture plane over the
line y=0, 0<x<90λ. An integration step size of 0.25λ has been
used. The Fresnel integral was simulated
using Matlab7.01 [10]. The Fresnel field is computed for the different values of
width- and height-modulation factors indicated in Tables 2 and 3. The relative
power in dB versus transverse distance in wavelengths with the space constant ai = 0.6
for exponential serrations is presented for different cases in Figures (5–7). This implies
less ripple at the centre of the quiet zone which is indicative of better
cancellation of diffraction effects.Table 1
Nonidentical exponential serration functions.
Nonidentical segmented concaveNonidentical segmented convexDefining
equationf(x′)=y1+y2+y3+y4,f(x′)=y1+y2+y3+y4,y1=tp2−p1(x′+p2),−p2<x′<−p1,y1=tp2−p1(x′+p2),−p2<x′<−p1,y2=t0e−ax′,−p1<x′<0,y2=t0+t(1−eax′),−p1<x′<0y3=t0eax′,0<x′<p1,y3=t0+t(1−e−ax′),0<x′<p1,y4=tp1−p2(x′−p2),p1<x′<p2,y4=tp1−p2(x′−p2),p1<x′<p2,wheret0=te−ap1.wheret0=te−ap1.Table 2
Width modulation factors for NISE serrations.
CASEPP1/PP2/PNumber of serrations1(a0/2)/112.56.04512(a0/2)/56.255.33322.5023(a0/2)/37.504.66715.034(a0/2)/28.1254.011.2545(a0/2)/22.501.5956(a0/2)/18.753.07.5067(a0/2)/16.072.666.42878(a0/2)/14.06252.05.62589(a0/2)/12.501.3335.09Table 3
Height modulation factor for NISE serrations.
t1λFresnel-zone field for90λ×90λ NISE-convex
and NISE-concave serrated CATRs for cases 1, 2, and 3.
(a)(b)(c)Fresnel-zone field for90λ×90λ
NISE-convex and NISE-concave serrated CATRs for cases 4, 5, and 6.
(a)(b)(c)Fresnel-zone field for90λ×90λ
NISE-convex and NISE-concave serrated CATRs for cases 7, 8, and 9.
(a)(b)(c)


## 4. Conclusions

This paper presented a performance evaluation of the NISE-serrated edge reflector with
rectangular aperture for different values of width- and height-modulation factors.
The quiet zone field of a90λ×90λ is assessed for different cases as
illustrated in Tables 2 and 3. Quiet zone width and ripple parameters are to be
considered as a-figure-of-merit factors. These two factors are the deciding
parameters regarding which serrated CATR gives better performance. From Figures
5, 6, and 7, it is observed that less ripple and enhanced quiet zone width are
observed in this NISE-concave serrated CATR than NISE-convex serrated CATR. It is concluded
that NISE-concave serrated CATRs give better performance. This NISE-serrated
edge CATR gives better quiet size and less ripples than triangular and on/off
rectangular serrated CATRs.



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