Synthesis of a Plane Wave in the Near Field with a Planar Phased Array
abstract. Generating a plane wave in the near field is important for antenna
measurements in confined spaces, This paper shows how to create an approximate
plane wave in the near field from a planar array of isotropic point sources using
two approaches, a least square fit and a genetic algorithm, to find the complex
amplitude weights.
1. Introduction
Illuminating an antenna under test (AUT) with a plane wave is imponant when
measuring antenna patterns. The usual approach is to separate the transmit
antenna and the AUT hy a large distance in order to minimize the amplitude and
phase variations across the test aperture. Sometimes separating the two antenna
by the far field distance is unreasonable. A method was developed to create a
plane wave in the near field from a linear array of line sources [I], [2] using a
genetic algorithm. This method proved successful in experimental testing as
reported in [3] and [4].
This paper investigates the synthesis of a local plane wave in the near field of a
transmit planar array. llnlike the previous reponed work, this paper uses a planar
array instead of a linear array. In addition, the complex array weights are found
using a least square solution and compared to the results found using a genetic
algorithm.
11. Formulation
The mathematical expression for the planar array ofpoint sources in Figure I is
eiven bv
where
Ne,=number of elements in the array
w,
=complex weight of element n
k=2nll
l =wavelength
0-7803-7846-61031117.008 2 W 3 IEEE
792
R,, =distance from element n to
(xm,ym,z)on the plane wave
z =distance from array to the plane wave
planar array
Figure 1. The planar array generates a plane wave over a prescribed region.
The I/R factor contributes little to amplitude deviations across a planar area some
distance from the a m y . On the other hand, the exponential factor creates
tremendous amplitude and phase deviations across that same planar area.
Consider a point source that radiates equally in all directions from the origin. The
amplitude and phase are constant on any spherical surface having the point souree
as the center. The far field of an antenna is the distance at which the antenna
radiates an approximate plane wave across the test aperture. In the far field, the
source must be 2 0 ' 1 1 away from the approximate plane wave in order to satisfy
a maximum phase deviation of id8 radians.
111. Approach I: Least Squares
The synthesis of a plane wave can be formulated as matrix problem Ax=b, where
A is the phase matrix, x the element weights, and b the field values at the plane
wave. Converting equation ( I ) into mamx form yields
When M=N,then a direct solution is found, otherwise, when M>N,a least s q u m s
solution is found.
Figures 2 and 3 show the resulting amplitude and phase field distributions over a
2mx2m area (dashed square in figures) as synthesized by solving (2) for the
793
complex weights. The array is a 5x5 planar array with a square element lattice
and an element spacing of Ih in both directions. The plane wave is 5m from the
aperture and the frequency is 500 MHz. There are 16x16 field points in the plane
wave area. In the desired plane wave region, the amplitude and phase deviations
are quite small. Unfortunately, the plane wave occurs in a null of the antenna
pattem as can be seen in’ Figure 2. Solving (2) for the complex m a y weights
always results in a phase distribution that forms a null in the field amplitude.
Noise and multipath would certainly destroy the plane wave quality of this
approach in a practical implementation.
xinm
‘Figurk 2~FiAdamplilude distribution
‘in dB 5m from the array found by
salving (2).
Figure 3. Field phase distribution in
degrees 5m from the array found by
solving (2).
IV. Approach I 1 Genetic Algorithm
A CA can also be “Sed to find the optimum array weight for (2). One advantage
of the GA is the ease with which constraints can be added to variables. The
objective function used in this minimization is given by
where a&*- = the complex field at the desired plane wave location. Figures 4
and 5 show the resulting approximate plane waves synthesized by a 16x16
dement array with 12 element spacing in the x and y directions. The maximum
amplilude deviation is about 3 dB and the maximum phase deviation is about6.5”
194
xinm
Figure 4. Field amplitude distribution Figure 5. Field phase distribution 5m
5m from the m a y found by solving (2) from the m y found by solving (2)
using a GA.
using a GA.
V. Conclusions
The least squares approach consistently found an approximate plane wave in a
null of the near field. On the other hand, the GA found a near plane wave like
field at the desired location in the near field. This approach to generating a plane
wave in the near field is quite useful for measuring large antennas.
Bibliography
[I] R.L. Haupt, "Generating plane waves from
a linear array of line sources,"
2001 AMTA Conference, Denver, CO, Oct 01
[2] R.L. Haupt, "Generating a plane wave with a linear array of line sources,"
IEEE AP-S Trans.,Dee 02.
[3] C.C. Courtney, D.E. Voss, R. Haupt, and L. LeDuc, "The theory and
architechwe of a plane wave generator," 2002 AMTA Conference, Cleveland,
OH, Nov 02.
[4] C.C. CoUrmey, D.E. Voss, R. Haupt, and L. LeDuc; "The measured
performance of a plane wave generator prototype," 2002 AMTA Conference,
Cleveland, OH, Nov 02.
795