Generating a Plane Wave in the Near Field with a Planar ...inside.mines.edu/~rhaupt/journals/Microwave Journal 2003.pdf · The plane wave appears at such a low level because half

September 2003 Issue: Technical Feature

Generating a Plane Wave in the Near Field with a PlanarArray AntennaThis article presents a method of generating an approximate plane wave in the near field using aplanar phased array antenna. Optimized complex array weights are found using a genetic algorithmand a least squares solution. The genetic algorithm (GA) proves to be a more practical approach thanthe least squares solution. Computer results with the GA show that the synthesized approximate planewave has significantly lower field variations over a desired plane wave region in the near field thanan equivalent uniform array.

by Randy Haupt, Utah State Univers ity

From: Vol. 46 | No. 9 | September 2003 | Pg 152

Measuring far field antenna patterns requires separating the transmit antenna and the antenna undertest (AUT) by a large distance in order to minimize the field amplitude and phase variations across thetest aperture. When the AUT is many wavelengths across, the far field distance becomes quite large.Thus, in order to measure the antenna pattern of an electrically large aperture indoors, techniques likea compact range, near field scanning and antenna focusing are necessary. These approaches allow foraccurate prediction of the far field pattern even though the measurements are taken in the near field.The field variations across the AUT become smaller as the separation distance increases.

The IEEE defines the far field of anAUT in terms of the maximum phasedeviation across the AUT. For non-low sidelobe antennas the maximumphase variation is /8 radians, and thecorresponding far field distance isdefined by

where

D = the maximum extent of theaperture

= the wavelength

The greatest phase variation for a rectangular aperture occurs at one of the corners, and D is thediagonal of the AUT. As D gets large and/or gets small, the minimum Rff increases. The phase and

amplitude variations across the AUT are highly correlated. These correlated variations result in patternvariations close to the main beam; far from the main beam, however, the pattern is relatively

undisturbed.1

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Antenna arrays have been used to generate a planar field in a test volume for electromagnetic

susceptibility testing. Hill used a constrained least squares approach developed by Mautz3 to create anapproximate plane wave in the near field. The results show a reasonably flat (within 3 dB) amplituderesponse for small linear arrays of line sources. The phase response is not shown and implementingthe source constraint is limited. This approach was experimentally applied to a seven-element array of

Yagi-Uda antennas at 500 MHz.4 The results were very encouraging. In another study, a circular array

was synthesized to create a plane wave at the interior of the array elements.5 Scanning of the planewave is demonstrated. For further background information on creating a plane wave in the near field,

the reader is referred to J.E. Hansen.6

A more recent method was developed to create a plane wave in the near field from a linear array of

line sources7,8 using a genetic algorithm. This approach combines the ideas of array focusing and acompact range. The location and weights of an array of line sources were found that approximate aplane wave at a desired location in space. The optimized amplitude and phase ripples across the AUTare much less than those created by a uniform array with the same number of elements. The optimizedapproximate plane wave is a significant improvement over a uniform array or a single line source. This

method also proved successful in experimental testing, as reported in Courtney, et al.9,10 The ideabehind these papers is to find the amplitude, phase and position of the elements that create a relativelyconstant field amplitude and phase along a linear aperture in the near field. A linear array can onlycontrol the field in the plane of the array but not the field in the orthogonal plane. The resulting plane

wave is robust in bandwidth and physical depth.8

This article presents results of generating a plane wave using a planar array instead of a linear array.The complex array weights are found using a least square solution and compared to the results foundusing a genetic algorithm. This technique significantly reduces the separation distance between thetransmit antenna and the AUT.



Fig. 1 The planar transmit array that generates an approximate plane wave in the near field.

FormulationThe mathematical expression for the transmit planar array of isotropic point sources shown in Figure1 is given by

where

Nel = number of elements in the array

wn = anejpn = complex weight of element n



k = 2 / = wavelength

Rmn = distance from element n to the field point (xm,ym,z0) on the plane wave

z0 = distance from array to the plane wave

The 1/Rmn factor produces small amplitude deviations across the desired plane wave area some

distance from the array. On the other hand, the exponential factor creates large amplitude and phasedeviations across that same planar area. The planar array lies in the x-y plane; the plane wave region isalso in the x-y plane but z0 away from the array. The array elements are in a square lattice. Samples in

the plane wave region are also on a square grid.

Fig. 2 Field amplitude in a 2 x 2 quadrant in the desired plane wave region due to a 6 x 6 uniform array at 10 .

Consider a 6 x 6 element transmitting array with elements arranged in a square lattice with spacings dx

= 1.0 and dy = 1.0 . These large spacings reduce mutual coupling effects, making the point source

model a better approximation to reality. Optimized spacings previously reported7 were on the order of



a wavelength or larger. The desired 4 x 4 l plane wave region is z0 = 10 away from the

transmitting array. This distance is considerably less than the far field distance of Rff = 64 . Moving

this plane wave region to Rff = 64 would result in a phase variation of 22.5° and amplitude variation

of 0.33 dB. If the array weights are uniform, then the amplitude distribution at the plane wave region isshown in Figure 2 and the phase distribution in Figure 3 . These plots are the first quadrant of theplane wave region with the other three quadrants being symmetric about the x- and y-axes. The point(0,0) is the center of the plane wave region and is where the four quadrants meet. The maximumphase variation across the plane wave region is 60° and the maximum amplitude variation is 8.6 dB.These variations far exceed the IEEE standard and are unacceptable for most measurements. The nexttwo sections present two approaches to minimizing the amplitude and phase variations in the desiredplane wave region.

Fig. 3 Field phase in a 2 x 2 quadrant in the desired plane wave region due to a 6 x 6 uniform array at 10 .

Approach I: Least SquaresEquation 2 can be put in matrix form Ax=b, where A is the Green's function matrix, x the complexelement weights and b the field values at the plane wave. The resulting matrix equation is given by



When M = N, a direct solution is found; otherwise, when M > N (more field points than elements), aleast squares solution is found. Since the exact field values at the plane wave are unknown, theamplitude is assumed to be one and the phase zero. Once the weights are found, they are normalized.

Solving Equation 3 (given the same array configuration as in the last section) yields weights thatproduce a very flat amplitude and phase field distribution over the plane wave region. In fact, using M= 81 sample points in the quadrant results in weights that produce a maximum amplitude variationacross the plane wave region of 0.003 dB and a phase variation of 0.02°. These amazing results are ata cost, though. Unfortunately, the plane wave has extremely low amplitude as can be seen in a plot ofthe expanded region about the desired plane wave, as shown in Figure 4 . Note the very flat amplituderesponse in the desired plane wave region; the level is at approximately -70 dB compared to the +5 dBfor the uniform array. This very low level would place the plane wave in the noise level of anexperimental setup. The low amplitude, coupled with significant scattering from nearby objectsilluminated with high amplitude fields, makes this approach impractical.

Fig. 4 Outside the 2 x 2 quadrant, the field amplitude increases dramatically when the optimized weights are found via the



least square approach.

The plane wave appears at such a low level because half the array weights are out of phase with theother half; in other words, it is a difference pattern with a null in the main beam. In the far field, the1/Rmn terms are replaced by a constant 1/R that can be factored out of the summation. In addition,

symmetry can be imposed on both the amplitude and phase of the array weights in the form ofcombining symmetric phase terms using Euler's identity for cosine. In the near field, however, Rmn is

not a constant and the amplitude of symmetric phase terms are not the same, so symmetry cannot beimposed. Consequently, numerical optimization must be used to find an acceptable solution.

The least square constraint originally proposed in Mautz3 and used to generate near field plane waves

in Hill2 was tried. It requires the source weights to have the constraint

where C is a constant. Unfortunately, the Green's function matrix has a large condition number (on the

order of 1010), so when the Hermitian matrix is formed from the Green's function matrix (see the

procedure in Mautz3 and Hill4), the condition number is even higher (on the order of 1017). Thiscondition number invalidates any results obtained with double precision arithmetic. Thus, anotherapproach is needed for this planar array configuration.



Fig. 5 GA performance over 600 generations.

Approach II: Genetic AlgorithmThis optimization problem is of sufficient complexity that a local optimization algorithm, such as theNelder Mead down-hill simplex or conjugate gradient, does not find an adequate solution.Consequently, a GA is used to explore the multi-modal landscape of the cost function. One advantageof the GA is the ease with which constraints are added to variables. In this case, the array weights arebound by 0 n 1 and 0 Pn 2 . Analytical-based methods like least squares require an

extensive effort to add constraints and the constraints are very limited. The GA also does not have toworry about the condition number of the Green's function matrix. In addition, imposing symmetry onthe array weights about the x-axis and y-axis eliminates the possibility of creating the plane wave in anull of the field pattern. Unlike least squares, the GA can also optimize array element spacing in alldirections and can be used with experimental data. Thus, the GA could be used in the actualexperimental setup and adjusted for environmental effects.



Fig. 6 Field amplitude (in dB) using a GA over a 2 x 2 quadrant of the desired plane wave region that is 10 away from a6 x 6 uniform array.

The objective function used in this minimization is given by

where Em are the complex field samples from the desired plane wave region. Values of the objective

function, or cost, generally range between 0 and 2. The first term is the maximum phase deviationnormalized to . As long as the plane wave area is not too large, this term stays less than one. Thesecond term is the normalized maximum field amplitude. These terms can be weighted to achieve adesired emphasis on either the field amplitude or phase. In this example, they are weighted equally.



Fig. 7 Field phase (°) found using a GA over 2 x 2 quadrant of the desired plane wave region that is 10 away from a 6 x6 uniform array.

The continuous parameter GA used a population size of 80 with 1 percent mutation rate, 50 percentcrossover rate, single point crossover, and ran for 30,008 function evaluations. Figure 5 shows theGA's progress over 600 generations. Most of the work was completed in about 40 generations and theresult was essentially attained in less than 100 generations. The solid line is the best cost in thepopulation at each generation, while the dashed line is the average cost of the population of 80chromosomes. Fluctuations in the average population cost are due to mutations.

The results are quite good. The maximum field amplitude variation across the desired plane waveregion is 0.75 dB, as seen in the contour plot of Figure 6 . The maximum field phase variation acrossthe desired plane wave region is 32.4°, as seen in the contour plot of Figure 7 . These field variationsare a significant improvement over the same field variations due to a uniform array. The fieldamplitude is 60 dB higher than the least squares solution. Figure 8 is a plot of the field amplitude overan extended area about a quadrant of the plane wave. Unlike the field amplitude resulting from the leastsquares approach, this amplitude drops off outside the 2 x 2 plane wave quadrant. Table 1



displays the optimum array weights found by the GA.

Fig. 8 Outside the 2 x 2 quadrant, the field amplitude decreases when the optimized weights are found via the GA.

It is important to investigate the sensitivity of the optimization result to changes in separation distanceor frequency. The array weights were optimized for a frequency of f0 and a separation of z0 = 10 .

Figure 9 shows the maximum amplitude and phase variations over the plane wave region as a result ofmoving the transmit array in z. The greatest harm to the plane wave is done by decreasing theseparation distance, because both the amplitude and phase variations increase. Increasing theseparation distance produces greater amplitude variations but less phase variations. The decrease inphase variations results from z0 moving closer to the far field. These results indicate that the

separation distance in an experimental setup at 1 GHz could be a few centimeters in error withoutmuch impact on the plane wave. Figure 10 shows the maximum amplitude and phase variations overthe plane wave region as a result of changing the frequency, f0. The maximum deviations are much

sharper for changes in frequency. There is a clear minimum for maximum amplitude deviations at f0.

The minimum of the maximum phase deviations occurs at 0.98f0. Phase variations increase above the

center frequency because the far field distance Equation 1 increases as the wavelength decreases. Iftesting occurs over a bandwidth, then the optimization should be done at the highest frequency insteadof the center frequency.



Fig. 9 Maximum amplitude and phase variations as a function of the distance between the transmit array and the AUT.

Many different optimization runs were done for various transmit antenna configurations, separationdistances and plane wave region sizes. Good results were obtained for element spacings between 0.5and 1.5 . Increasing the plane wave size or decreasing the separation distances increases the fieldvariations across the desired plane wave region. Figure 11 is a graph of the best cost found by a GAfor the 4 x 4 plane wave region at a distance given by z0. A single point for the uniform transmit

array is plotted at z0 = 64 for comparison. Figure 12 is a plot of the optimized cost vs. the width of

the plane wave at z0 = 10 . If the transmit antenna were a linear array, then the maximum extent of

the plane wave region would be 4 . Since the plane wave region is square, the maximum extent is adiagonal that is 5.7 wide. There is a trade-off between field amplitude and phase variations that canbe exploited by weighting the terms in the cost function. Thus, a smoother field amplitude is possibleat the expense of more variations in the field phase, or visa versa.



Fig. 10 Maximum amplitude and phase variations as a function of frequency.

ConclusionThis article presents a new way of generating an approximate plane wave region in the near field usinga planar array. Previous approaches used only a linear array. A least squares approach produces animpractical implementation. Adding a constraint to the least squares solution raises the conditionnumber of the Green's function matrix to an unacceptable level. The GA design produces anapproximate plane wave with small amplitude and phase variations. It is possible to increase theseparation between the transmit array and the approximate plane wave region without much decreasein performance. Decreasing the separation distance, however, dramatically increases the amplitude andphase variations. The amplitude and phase variations increase as the frequency moves away from thedesign center frequency. Increasing the frequency increases the variations more than decreasing thefrequency.



Fig. 11 Cost associated with an optimized transmit array for a 4 x 4 plane wave at a separation distance of Z0.

Although this problem was modeled with isotropic point sources, the GA works with any form of costdata. This data can be in the form of the output from a mathematical function or an experimentalmeasurement. As such, the point sources could be replaced by better antenna models or experimentalmeasurements and the GA would still find an optimal solution. When the idealized point sources inreferences 7 and 8 were replaced by an actual experiment in references 9 and 10, the GA was foundto produce comparable results.

References1. E. Brookner, Practical Phased Array Antenna Systems, Part 2 , "Antenna Array Fundamentals,"Artech House Inc., Norwood, MA 1991.2. D.A. Hill, "A Numerical Method for Near-field Array Synthesis," IEEE Transactions onElectromagnetic Compatibility , Vol. 27, No. 4, November 1985, pp. 201-211.3. J.R. Mautz, "Computational Methods for Antenna Pattern Synthesis," IEEE Transactions onAntennas and Propagation , Vol. 23, No. 4, July 1975, pp. 507-512.4. D.A. Hill, "A Near-field Array of Yagi-Uda Antennas for Electromagnetic-susceptibility Testing,"IEEE Transactions on Electromagnetic Compatibility , Vol. 28, No. 4, August 1986, pp. 170-178.5. D.A. Hill, "A Circular Array for Plane-wave Synthesis," IEEE Transactions on ElectromagneticCompatibility , Vol. 30, No. 1, February 1988, pp. 3-8.



6. J.E. Hansen, Spherical Near-field Antenna Measurements , Chapter 7, "Plane-wave Synthesis,"Peter Peregrinus Ltd., London, England 1988.7. R.L. Haupt, "Generating Plane Waves from a Linear Array of Line Sources," Antenna MeasurementTechniques Association Conference , Denver, CO, October 2001.8. R.L. Haupt, "Generating a Plane Wave with a Linear Array of Line Sources," accepted forpublication in IEEE Transactions on Antennas and Propagation .9. C.C. Courtney, D.E. Voss, R. Haupt and L. LeDuc, "The Theory and Architecture of a Plane-waveGenerator," Antenna Measurement Techniques Association Conference , Cleveland, OH, November2002.10. C.C. Courtney, D.E. Voss, R. Haupt and L. LeDuc, "The Measured Performance of a Plane-waveGenerator Prototype," Antenna Measurement Techniques Association Conference , Cleveland, OH,November 2002.

Fig. 12 Cost associated with an optimized transmit array for a separation distance Z = 10 and a square plane wave regionhaving a width W.

Randy Haupt received his BS degree in electrical engineering from the USAFAcademy, his MS degree in electrical engineering from Northeastern University,his MS degree in engineering management from Western New England Collegeand his PhD degree in electrical engineering from the University of Michigan. Heis currently a professor and the head of the electrical and computer engineering



department at Utah State University. His research interests include geneticalgorithms, antennas, radar, numerical methods, signal processing, fractals andchaos. He has published numerous journal articles, conference publications andbook chapters on antennas, radar cross-section and numerical methods, and is co-

author of Practical Genetic Algorithms (John Wiley & Sons Inc., New York, NY 1998). Haupt haseight patents in antenna technology and is director of the USU Anderson Wireless Center.

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Generating a Plane Wave in the Near Field with a Planar ...inside.mines.edu/~rhaupt/journals/Microwave Journal 2003.pdf · The plane wave appears at such a low level because half