Characterization of a Parabolic Reflector for Use in a Compact Range

Characterization of a Parabolic Reflector for Use in a Compact Range

William Archer

Governor’s School for Science and Technology

2013-2014

Mentor Signature

Dr. Roland Lawrence

Old Dominion University/Jefferson Labs, ARC Building

Abstract

Electromagnetic wave tests for materials and objects require researchers to consider both traditional spherical wavefronts as well as flat wavefronts. Flat wavefronts, or plane waves, are generated when a spherical wave is propagated a large distance away from the source, or feed. Traditionally, a far-field range is required for testing with plane waves. However, compact ranges are a phenomenal alternative, as they require much less space and a lower cost to operate; they can even be built in an indoor environment. Compact ranges consist of a feed and a parabolic reflector. The feed is placed at the reflector’s focal point. When a spherical wave is produced at the focal point, it will be reflected back as a flat wave. A parabolic reflector was found at a salvage yard with no specifications; normally the reflector’s focal length would be given by the manufacturer. In order to find out the focal point, the reflector must be modelled mathematically. This was done by making a physical grid and measuring ___ points. These points were then fit to a custom equation in MATLAB, which gave the focal point. The experiment was redone by measuring __ of points with an ultrasonic ruler. The measurement error for both was determined by running twenty-five Monte Carlo simulations and getting the standard deviation and mean for the best fit.

Introduction

The Department of Defense is researching ways to keep electromagnetic energy in the microwave region from leaking inside or out of objects such as gaskets and valves. This electromagnetic energy comes in the form of waves which hit the objects and pass through manufacturing defects or small holes used for mounting. Researchers illuminate the object with an antenna and measure the electromagnetic energy coming through the other side to determine where leakage may occur.

When an object is illuminated from a close source, spherical waves can provide unreliable measurements due to interference. When the spherical wave hits an object at leaking areas, the waves that exit through the other side may have a phase difference, which could potentially cancel one another out (Figure 1). This is a serious problem for researchers wanting to detect electromagnetic leakage. Therefore, in order to mitigate this source of measurement error as well as simulate the actual electromagnetic environment in which the object will be experiencing, plane waves can be used.

Wavefronts become flat as a wave travels a large distance away from its source (Figure 2) (Brown 2007). This flat wavefront is called a plane wave. An ideal plane wave has a phase difference of zero over the distances of interest, which allows researchers to accurately determine both points of leakage and its magnitude.

Far-field ranges can be used to generate plane-waves (Figure 3). However, far-field ranges require a large space, typically larger than 2mi, and much more energy to propagate the wave the required distance (Way, et al. 2000). This makes far-field ranges both scarce and expensive to operate. Shielding effect testing on far-field ranges is therefore limited to large objects, such as planes and other large vehicles. Smaller objects are much more difficult to test due to the expansiveness of the range.

Alternatives, however, exist; compact ranges are used for plane wave generation and can be built in a much smaller setting (Shields and Fenn 2007). They consist of a source for the wave and a parabolic reflector (Way, et al. 2000). The source, or feed, is placed at the parabolic reflector’s focal point (Chia, et al. 1995). The focal point is the place in which parallel rays converge; when a spherical wavefront is reflected back, it will be flat (Fitzpatrick 2007).

The parabolic reflector used in this study is 56 inches by 65 inches by 12 inches deep and was found in a salvage yard (Figure 4). There were no specifications accompanying the reflector, including the focal length, focal point, or an equation that mathematically models the surface. The effort of this research is to use location measurements along the surface of the parabolic reflector to find the focal point.

Materials and Methods

Measuring the Parabolic Reflector

In order to determine the focal point of the parabolic reflector, an accurate equation must be made to model the shape in three dimensions. Measurement points were made at ___ locations every 2.58 inches in the horizontal plane. These measurements were made with the help of a 62.5 in x 62.5 in grid which was placed over the parabolic reflector. With precision in mind, a jig was designed to make each point along the grid (Figure 5). This jig has a measurement error of ± .02 inches. Once in place, the depth at each point was measured using a ruler. The data were then run through a non-linear optimization algorithm in MATLAB and fit to the equation,

z=x+∆x2+ y+∆y24F,

where x,y, and z are the respective matrices, Δx and Δy are the distance from the vertex, and F is the focal length (Burnside and Gupta 2005). For a paraboloid centered at x = 0 and y = 0, the focal point is (0,0,F); thus, the focal point for the physical parabolic reflector is equal to (Δx, Δy, F) in a three-dimensional space.

Determining Alternate Methods of Measurement

The ability to make measurements quickly and accurately was also considered. The measurements were retaken at ___ points along the parabolic reflector with an ultrasonic tape measure. The previous steps were completed on these new measurements as well and then compared to the ruler measurements for analysis.

Determining Reasonable Error

The accuracy of the ruler was found to be ___ inches. According to manufacturer specifications, the ultrasonic tape measure was 99.5% accurate per inch. However the instrument will only display the nearest inch.

Human error was determined by taking measurements on multiple days in five shallow locations as well as five deep locations. The standard deviation was then found for each area and compared.

Reasonable error bars for the focal point location of the parabolic reflector were determined by comparing twenty-five Monte Carlo simulation measurement sets. For each set, the measured data had a random Gaussian distribution number, with the mean equal to zero and variance equal to the reasonable measurement error, added (Couto, Damasceno and Pinheiro de Oliveira 2012). These simulated measurements were then fitted the same way as the actual data and then compared using standard deviation and mean (see Figure).

Results

The raw data for the ruler measurements are given in the Appendix (Table 1). Reasonable error for these measurements was determined to be ± __ inches; this lead to a focal point variance of ___ from the Monte Carlo simulations.

Using the nonlinear regression technique, the best fit surfaces were generated for both measurement techniques. The reflector was cut ___ inches away from the paraboloid’s vertex, with its displacement in the x direction equal to __ inches and the displacement in the y direction equal to ___ inches. The focal length of the reflector was found to be ____ inches.

The raw data for the ultrasonic measurements are given in the Appendix (Table 2). Reasonable error for these measurements was determined to be ± __ inches; this lead to a standard deviation of ___ from the Monte Carlo Simulations and a mean of ___.

Discussion and Conclusions

The focal length of the reflector was found to be ___ ± ___ inches. The best fit measurement differences between the two techniques is shown in Table 3 .The ruler has much more accurate results, which can be attributed to both the large number of points measured and the level of accuracy. However, the time required for measurements is much greater. The parabolic reflector has to be calibrated for the tilt of both the feed and the reflector itself, as well as minute adjustments with aim (Aubin and Bates 2002). For this reason, someone wanting to repeat the work done may want to measure fewer points with a quicker method, such as the ultrasonic tape measure and make corrections in real-time.

Talking with experts in the field, the expected focal length for the parabolic reflector was between six and seven feet, which is in general agreement with the results here. The computer generated parabolic reflector that was made from the best fit software in MATLAB is similar in shape and dimension to the actual shape as well, which gives further reason to conclude that the focal point of the reflector is at (__,___,__).

Acknowledgements

The author would like to thank Dr. Roland Lawrence for his numerous contributions to the project, as well as for advice and mentoring. In addition, the author would like to thank Jefferson Lab’s Applied Research Center for use of their facilities.

Literature Cited

Aubin, John F. and Mark Bates. 2002. "Compact Range Performance Effects in Interferometer Testing and Related Statistical Analysis of Field Probe Measurements." ORBIT/FR. http://www.microwavevision.com/sites/www.microwavevision.com/files/files/ORBIT-FR-CompactRangePerformanceEffects-03-04-05-Aubin.pdf.

Brown, Robert G. 2007. Polarization of Plane Waves. December 28. Accessed 2 17, 2014. http://www.phy.duke.edu/~rgb/Class/phy319/phy319/node37.html.

Burnside, Walter D. and Inder J Gupta. 2005. Compact and Spherical Range Design, Application and Evaluation. Tuscon: Ohio State University ElectroScience Laboratory. ftp://esl.eng.ohio-state.edu/pub/people/ljs/raytheon%20short%20course/Raytheon-Part%202.ppt.

Chia, Tse-Tong, Nikolai Balabukha, Yeow-Beng Gan, Wee-Jin Koh, and Tat-Soon Yeo. 1995. "A Compact Range for RCS & Antenna Measurements: System Description." Department of Electrical Engineering, University of Singapore 419-423. http://www.ece.nus.edu.sg/asl/welcome/Db/Publication/Printed%20Papers/A%20Compact%20Range%20for%20RCS%20Measurement.pdf.

Couto, Paulo Roberto Guimaraes, Jailton Carreteiro Damasceno, and Sergio Pinheiro de Oliveira. 2012. "Monte Carlo Simulations Applied to Uncertainty in Measurement." In Theory and Applications of Monte Carlo Simulations, by Victor Chan. Accessed April 20, 2014. doi:10.5772/53014.

Fitzpatrick, Richard. 2007. Spherical Mirrors. July 14. http://farside.ph.utexas.edu/teaching/316/lectures/node136.html.

Shields, Michael W. and Alan J. Fenn. 2007. "A New Compact Range Facility for Antenna and Radar Target Measurements." Lincoln Laboratory Journal 381-391. http://www.ll.mit.edu/publications/journal/pdf/vol16_no2/16_2_08Shields.pdf.

Way, Jeff, Bill Griffin, Mark Bellman, and Randy Smith. 2000. TRW's New Compact Antenna Test Range. M.I. Technologies. http://www.mitechnologies.com/papers/00/TRW's%20New%20Compact%20Antenna%20Test%20Range.pdf.

Appendix

Tables

Table 1. Raw data of ruler measurements.

X (inches) / Y (inches) / Z (inches)
0.00 / 0.00
2.58 / 2.58
5.16 / 5.16
7.74 / 7.74
10.32 / 10.32
12.90 / 12.90
15.48 / 15.48
18.06 / 18.06
20.64 / 20.64
23.22 / 23.22
25.80 / 25.80
28.38 / 28.38
30.96 / 30.96
33.54 / 33.54
36.12 / 36.12
38.70 / 38.70
41.28 / 41.28
43.86 / 43.86
46.44 / 46.44
49.02 / 49.02
51.60 / 51.60
54.18 / 54.18
56.76 / 56.76
59.34 / 59.34
61.92 / 61.92
64.50 / 64.50

Table 2. Raw data of ultrasonic tape measure measurements.

X (inches) / Y (inches) / Z (inches)

Table 3. A Comparison of the various results found by both methods of measurement.

Measurement / Standard Deviation
of Focal Lengths / Reasonable Measurement Error / Mean / Δx / Δy / Focal Length
Ruler
Ultrasonic

Figures

Figure 1. A wave interfering with itself after passing through two holes due to a phase difference.

Figure 2. A spherical wave propagated a long distance becomes a plane wave.

Figure 3. A far-field range being used to test electromagnetic properties of a jet. Source: http://voprosik.net/wp-content/uploads/2012/01/286.jpg

Figure 4. The parabolic reflector being measured.

Figure 5. The jig made for accurate and precise separation on the grid.

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