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Differential Near Field Holography for Small Antenna Arrays by PDF

152 Pages·2011·3.28 MB·English
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Differential Near Field Holography for Small Antenna Arrays by Brian A. Janice A Thesis Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE in partial fulfillment of the requirements for the Degree of Master of Science in Electrical and Computer Engineering by ____________________________________ August 2011 APPROVED: ____________________________________________________ Dr. Jeffrey Herd (MIT Lincoln Laboratory) ____________________________________________________ Professor Reinhold Ludwig ____________________________________________________ Professor Sergey Makarov, Major Advisor Abstract Near-field diagnosis of antenna arrays is often done using microwave holography; however, the technique of near-field to near-field back-propagation quickly loses its accuracy with measurements taken farther than one wavelength from the aperture. The loss of accuracy is partially due to windowing, but may also be attributed to the decay of evanescent modes responsible for the fine distribution of the fields close to the array. In an effort to achieve better resolution, the difference between these two phase-synchronized near-field measurements is used and propagated back. The performance of such a method is established for different conditions; the extension of this technique to the calibration of small antenna arrays is also discussed. The method is based on the idea of differential backpropagation using the measured/simulated/analytical data in the near field. After completing the corresponding literature search authors have found that the same idea was first proposed by P. L. Ransom and R. Mittra in 1971, at that point with the Univ. of Illinois [1],[2]. This method is basically the same, but it includes a few distinct features: 1. The near field of a (faulty) array under test is measured at 1.52.5 via a near field antenna range. 2. The template (non-faulty) near field of an array is simulated numerically (full-wave FDTD solver or FEM Ansoft/ANSYS HFSS solver) at the same distance – an alternative is to use measurements for a non-faulty array. 3. Both fields are assumed (or made) to be coherent (synchronized in phase). 4. A difference between two fields is formed and is then propagated back to array surface using the angular spectrum method (inverse Fourier propagator). The corresponding result is the surface (aperture) error field, F . This approach is more precise than the Z inverse Rayleigh formula used in [1] since the evanescent spectrum may be included into consideration. 5. The error field magnitude, F , peaks at faulty elements (both amplitude and phase Z excitation fault). 6. The method inherently includes all mutual coupling effects since both the template field and the measured field are full-wave results. ii Acknowledgements I would like to thank the Electrical & Computer Engineering and Mathematical Sciences Departments of Worcester Polytechnic Institute for the knowledge they have endowed me during my graduate and undergraduate studies; without their contributions, I would have not been able to complete this work. In particular, Professor Sergey Makarov has provided me with more knowledge, motivation, and confidence in my abilities than any student could ask for. I would also like to thank Dr. Francesca Sciré-Scappuzzo for her belief in my abilities, support, and insistence on my completion of a graduate degree. To the staff of MIT Lincoln Laboratory, in particular Drs. Jeffrey Herd and Sean Duffy: I have never been so pleased to learn so much in what felt like such little time. Your patience and manner is grand. Also, if not for Stephen Targonski and Hans Kvinlaug allowing me to use their lab and near-field range, this work would be incomplete. To my committee: Professor Reinhold Ludwig, Dr. Jeffrey Herd, and Professor Sergey Makarov. Thank you for your time and patience – your belief in my work is an honor. This work is a tribute to the sacrifices my mother has made in recognition of my potential. She has put me in the position I am in today. iii Abstract ........................................................................................................................................... ii  Acknowledgements ........................................................................................................................ iii  Table of Figures ............................................................................................................................ vii  Table of Tables ............................................................................................................................... x  Chapter 1: Introduction ................................................................................................................... 1  Fundamental Antenna Quantities ................................................................................................ 2  Antenna Pattern ....................................................................................................................... 2  Radiation Lobes and Beamwidth ............................................................................................ 4  Polarization ............................................................................................................................. 6  Antenna Arrays ........................................................................................................................... 9  Chapter 2: Algorithm Specification and Numerical Modeling ..................................................... 13  The FDTD Algorithm ............................................................................................................... 13  Source Modeling ....................................................................................................................... 18  Boundary Conditions ................................................................................................................ 19  Array Geometry Under Study and Near Field Structure .......................................................... 22  Chapter 3: Near to Near/Far Field Transformations ..................................................................... 26  Near to Far Field Transformations ............................................................................................ 28  Fraunhofer Diffraction .......................................................................................................... 28  Equivalent Magnetic Currents and MoM ............................................................................. 29  Rayleigh Diffraction Integral ................................................................................................ 35  Near to Nearer Field Transformations ...................................................................................... 37  Chapter 4: Idea of Differential Backpropagation ......................................................................... 43  Algorithm .................................................................................................................................. 46  Simulated Backpropagation Results ......................................................................................... 49  iv Extensions ................................................................................................................................. 51  Chapter 5: Measured Results ........................................................................................................ 53  Antenna Measurements ............................................................................................................. 53  Hologram Error Sources ........................................................................................................... 55  Array Under Study .................................................................................................................... 59  Numerical vs. experimental backpropagation ........................................................................... 61  Hybrid (Numerical – Experimental) Backpropagation ............................................................ 71  Chapter 6: Potential Extension to Array Calibration & Conclusions ........................................... 74  Conclusions ............................................................................................................................... 76  References ..................................................................................................................................... 78  Appendix A: Backpropagation for all array elements. ................................................................. 81  Appendix B: Backpropagation for partially attenuated element. ................................................. 82  Appendix C: Backpropagation for element partially out of phase elements ................................ 83  Appendix D: Backpropagation for all array elements with 0.32 spacing. .................................. 85  Appendix E: FDTD MATLAB Codes .......................................................................................... 86  main.m ...................................................................................................................................... 86  array_4x4_cw.m ........................................................................................................................ 89  constructor.m ............................................................................................................................ 93  viewer.m.................................................................................................................................. 101  fdtd.m ...................................................................................................................................... 103  abc_murfirst.m ........................................................................................................................ 129  abc_super.m ............................................................................................................................ 130  fc.m ......................................................................................................................................... 131  plot_nearfield.m ...................................................................................................................... 131  v Appendix F: Backpropagation MATLAB Code ......................................................................... 138  vi Table of Figures Figure 1 Antenna pattern representations of the same antenna array: Top left: the field pattern in a linear scale represents the magnitude of the electric or magnetic field as a function of angular space. Top right: the power pattern in a linear scale represents a plot of the square of the magnitude of the electric or magnetic field as a function of the angular space. Bottom: the power pattern in the dB scale represents the magnitude of the electric or magnetic field (in decibels) as a function of angular space. ............................................................................................................ 3  Figure 2 Half-Power Beamwidth (HPBW) (top) First Null Beamwidth (FNBW) (bottom) Example for the same antenna array as shown in Fig. 1. The HPBW is approximately 26 while the FNBW is approximately 61 ..................................................................................................... 6  Figure 3 Examples of linear (left) and circular (right) polarization. The polarization is the curve traced by the end point of the vector representing the instantaneous electric field. ....................... 7  Figure 4 Frequency dependent transmission coefficient (dB) of two dipoles (transmit/receive network). The red curve pertains to the two dipoles with the same polarization plane (top left). The light blue curve pertains to the network when one dipole is rotated 30 (top right). The violet curve pertains to the network when one dipole is rotated 60 (bottom left). The dark blue curve pertains to the network when one dipole is rotated 90 (bottom right). ......................................... 9  Figure 5 uv-space plot of the array factor for a 10x10 array of /2 spaced elements. Top left shows a uniformly excited array; Top right shows an array with a progressive 60 phase shift on one axis; Bottom shows an array with a progressive 60 phase shift on both axes. ..................... 11  Figure 6 Array factors of a 10x10 array of /2 spaced elements with no progressive scan angle, but an amplitude taper. The graph on the left shows a uniformly excited array with the -13dB side lobes. The graph on the right shows an array with a Dolph-Tschebyscheff taper applied for - 26dB side lobes. ............................................................................................................................ 12  Figure 7 1D visual example of the FDTD “leap frog” algorithm ................................................. 15  Figure 8 3D visual example of the FDTD “leap frog” algorithm ................................................. 16  Figure 9 4x4 array geometry under study. .................................................................................... 22  Figure 10 Geometry of Fraunhofer diffraction problem. The blue section is the aperture. .......... 28  vii Figure 11 Example of non-uniform amplitude pattern of individual elements in a 4x4 array of patch antennas. The top left is a corner element, the top right is an inner element, and the bottom is an edge element. ........................................................................................................................ 38  Figure 12 Three numerically computed spatial Fourier spectra (spectrum magnitudes) in k-space for a 4×4 array of /2 spaced patch antennas over a larger ground plane/reflector. All magnitude spectra are obtained for the co-polar electric field at the distance of /8 from the aperture plane. The observation plane is approximately twice as large as the array itself . The first plot (top left) is the spectrum for the non faulty array with all radiators driven by identical generators; the second plot (top right) is the spectrum for a faulty array where the generator for radiator 22 is shorted out; the third plot (bottom) is the difference spectrum between first two, which shall be used for the identification of a faulty element. ............................................................................. 45  Figure 13 Lattice representation of surface with points stored within a matrix. Each square represents the differential area. ..................................................................................................... 47  Figure 14 The differential area of matrix that should not be included in the surface area calculation is shown in red. Overlapping red regions must be subtracted twice. ......................... 48  Figure 15 Example of how the placement of the transmitting antenna along the grid changes the orientation relative to the test antenna. This directive property, along with the polarization, must be taken into account when observing measurements. The red circles portray the different part of the horn pattern seen by the array when the horn is at different locations. .................................. 54  Figure 16 Example of smoothed hologram due to windowing in k-space, an effect of a finite measurement plane [8] .................................................................................................................. 56  Figure 17 Hologram of 4x4 patch array modeled in MATLAB with different windows in k- space. Top left pertains to k2 k2 0.55k with a 42.5% error; Top right pertains to x y k2 k2 0.65k with a 42.0% error; Middle left pertains to k2 k2 0.75k with a 39.9% error; x y x y Middle right pertains to k2 k2 0.85k with a 39.1% error (fake); Bottom pertains to x y k2 k2 0.95k with a 405% error. ............................................................................................. 58  x y Figure 18 Top – A 4x4 array of patches with a corporate feed and posterior-fastened aluminum ground plane; bottom – the same array in the near-field range. ................................................... 60  viii Figure 19 Simulated (Ansoft/ANSYS HFSS) current distribution on the ground plane when one of the array elements is detuned. ................................................................................................... 61  Figure 20 Squared differential hologram of non-faulty measured and simulated (HFSS) data with the simulated data multiplied by a variable phase. From top left, the phases are -40, -30, - 25, -20, -14 (optimal). It can be seen that the measured array may not have been completely parallel with the plane of elevation. .............................................................................................. 72  Figure 21 Top - Array factor of uniformly excited 4x4 array with /2 spacing (black) and array with corner element attenuated by 3dB (red). Bottom - Array factor of uniformly excited 4x4 array with /2 spacing (black) and array with inner element attenuated by 3dB (red). ............ 75  ix Table of Tables Table 1 Electric field distributions in different observation planes (/2 spacing). ....................... 23  Table 2 Backpropagation parameters corresponding to a minimum restoration error of the co- polar E-field at the distance of /8 from the top of the antenna array in the near field. ............... 49  Table 3 Backpropagated fields in three cases (/2 spacing). Element 22 is shorted out for a faulty array. ............................................................................................................................................. 50  Table 4 Backpropagation results (num. to num. and experiment to experiment) – last row. ...... 63  Table 5 Hybrid backpropagation results (numerical to experiment) – last row. .......................... 73  x

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Abstract. Near-field diagnosis of antenna arrays is often done using microwave .. Chapter 4: Idea of Differential Backpropagation . antenna is by using a parabolic reflector. kd nj. I kd mj. I. AF. 1. 1 sin sin. 1 exp cos sin. 1 exp β φ θ β φ θ.
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