Principles of Planar Near-Field Antenna Measurements [2 ed.] 1839536993, 9781839536991

Table of contents :
Cover
Contents
About the authors
List of abbreviations
Foreword
Preface
1 Introduction
1.1 The phenomena of antenna coupling
1.2 Characterisation via the measurement process
1.2.1 Free space radiation pattern
1.2.2 Polarisation
1.2.3 Bandwidth
1.3 Assumed (suppressed) time dependency
1.4 The organisation of the book
References
2 Maxwell’s equations and electromagnetic wave propagation
2.1 Electric charge
2.2 The electromagnetic field
2.3 Accelerated charges
2.4 Maxwell’s equations
2.5 The electric and magnetic potentials
2.5.1 Static potentials
2.5.2 Retarded potentials
2.6 The inapplicability of source excitation as a measurement methodology
2.7 Field equivalence principle
2.8 Characterising vector electromagnetic fields
2.9 Summary
References
3 Introduction to near-field antenna measurements
3.1 Introduction
3.2 Antenna measurements
3.3 Forms of near-field antenna measurements
3.4 Plane rectilinear near-field antenna measurements
3.5 Chambers, screening and absorber
3.6 RF subsystem
3.7 Robotics positioner subsystem
3.8 Near-field probe
3.9 Generic antenna measurement process
3.10 Summary
References
4 Plane-wave spectrum representation of electromagnetic waves
4.1 Introduction
4.2 Overview of the derivation of the plane-wave spectrum
4.3 Solution of the scalar Helmholtz equation in Cartesian co-ordinates
4.3.1 Introduction to integral transforms
4.3.2 Fourier transform solution of the scalar Helmholtz equation
4.4 On the choice of boundary conditions
4.5 Operator substitution (derivative of a Fourier transform)
4.6 Solution of the vector Helmholtz equation in Cartesian co-ordinates
4.7 Solution of the vector magnetic wave equation in Cartesian co-ordinates
4.8 The relationship between electric and magnetic spectral components
4.9 The free space propagation vector
4.10 Plane-wave impedance
4.11 Interpretation as an angular spectrum of plane waves
4.12 Far-field antenna radiation patterns: approximated by the angular spectrum
4.13 Stationary phase evaluation of a double integral
4.14 Co-ordinate free form of the near-field to angular spectrum transform
4.15 Reduction of the co-ordinate free form of the nearfield to far-field transform to Huygens’ principle
4.16 Far-fields from non-planar apertures
4.17 Microwave holographic metrology (plane-to-plane transform)
4.18 Far-field to near-field transform
4.19 Radiated power and the angular spectrum
4.20 Summary of conventional near-field to far-field transform
References
5 Measurements – practicalities of planar near-field antenna measurements
5.1 Introduction
5.2 Sampling (interpolation theory)
5.3 Truncation, spectral leakage, and finite area scan errors
5.4 Antenna-to-antenna coupling (transmission) formula
5.4.1 Behaviour of evanescent plane wave mode coefficients
5.4.2 Simple scattering model of a near-field probe during a planar measurement
5.5 Effect of acquiring near-field data using an electric dipole probe
5.6 Rotationally symmetric,
polarised, near-field probe
5.7 Evaluation of the conventional near-field to far-field transform
5.7.1 Standard techniques for the evaluation of a double Fourier integral
5.8 General antenna coupling formula: arbitrarily orientated antennas
5.9 Plane-polar and plane bi-polar near-field to far-field transform
5.9.1 Boundary values known in plane polar co-ordinates
5.9.2 Boundary values known in plane bi-polar co-ordinates
5.10 Regular azimuth over elevation & elevation over azimuth co-ordinate systems
5.11 Polarisation basis and antenna measurements
5.11.1 Cartesian polarisation basis – Ludwig I
5.11.2 Polar spherical polarisation basis
5.11.3 Azimuth over elevation basis – Ludwig II
5.11.4 Copolar and cross-polar polarisation basis – Ludwig III
5.11.5 Circular polarisation basis – RHCP and LHCP
5.12 Linear and circular polarisation bases – complex vector representations
5.13 Overview of antenna alignment corrections
5.13.1 Scalar rotation of far-field antenna patterns
5.13.2 Vector rotation of far-field antenna patterns
5.13.3 Rotation of copolar polarisation basis – generalised Ludwig III
5.13.4 Generalised compound vector rotation of far-field antenna patterns
5.14 Brief description of near-field co-ordinate systems
5.14.1 Range fixed system (RFS)
5.14.2 Antenna mechanical system
5.14.3 Antenna electrical system
5.14.4 Far-field azimuth and elevation co-ordinates
5.14.5 Ludwig III co-polar and cross-polar definition
5.14.6 Probe alignment definition (single port probe)
5.14.7 General vector rotation of antenna radiation patterns
5.15 Directivity & gain
5.15.1 Directivity
5.15.2 Calculating the power radiated in a direction-cosine coordinate system
5.15.3 Direct evaluation of directivity for a uniformly illuminated square aperture
5.15.4 Gain
5.15.5 Gain-transfer (gain-comparison) method
5.15.6 Approximation of the gain of a rectangular pyramidal horn
5.16 Calculating the peak of a pattern
5.16.1 Peak by series solution
5.16.2 Peak by polynomial fit
5.16.3 Peak by centroid
5.17 Estimating the position of a phase centre from far-field data
5.18 Summary
References
6 Probe pattern characterisation
6.1 Introduction
6.2 Effect of the probe pattern on far-field data
6.3 Desirable characteristics of a near-field probe
6.3.1 Open-ended rectangular waveguide probes
6.3.2 Dual-polarised waveguide probes
6.3.3 Broadband probes
6.3.4 Other less commonly encountered types of near-field probes – dipoles
6.4 Acquisition of quasi far-field probe pattern
6.4.1 Sampling scheme
6.4.2 Electronic system drift (Tie-scan correction)
6.4.3 Channel balance correction
6.4.4 Assessment of chamber multiple reflections
6.4.5 Correction for rotary errors
Box 6.1. AAPC – circumscribing three non-linear points on a plane
6.4.6 Remote source antenna tilt-angle correction
6.4.7 Re-tabulation of probe vector pattern function
6.4.8 Alternate interpolation formula
6.4.9 Approximate unwrapping of two-dimensional phase functions
6.4.10 True far-field probe pattern
6.5 Finite element model of open ended rectangular waveguide probe
6.6 Probe displacement correction
6.7 Channel balance correction
References
7 Computational electromagnetic model of a planar near-field measurement process
7.1 Introduction
7.2 Linear superposition of electric dipoles
7.3 Method of sub-apertures
7.4 Aperture set in an infinite perfectly conducting ground plane
7.4.1 Plane wave spectrum antenna–antenna coupling formula
7.5 Vector Huygens method
7.6 Kirchhoff–Huygens method
7.7 Current elements method
7.8 Equivalent currents method – near-field to far-field transform, antenna diagnostics, and range reflection suppression
7.9 Generalised technique for the simulation of near-field antenna measurements
7.9.1 Mutual coupling and the reaction theorem
7.10 Near-field measurement simulation
7.11 Reaction theorem
7.11.1 Lorentz reciprocity theorem (field reciprocity theorem)
7.11.2 Generalised Reaction theorem
7.11.3 Mutual impedance and the Reaction theorem
7.12 Full wave simulation of a planar near-field antenna measurement
7.13 Summary
References
8 Antenna measurement analysis and assessment
8.1 Introduction
8.2 The establishment of the measure from the measurement results
8.2.1 Measurement errors
8.2.2 The sources of measurement ambiguity and error
8.2.3 The examination of measurement result data to establish the measure
8.3 Measurement error budgets
8.3.1 Applicability of modelling error sources
8.3.2 The empirical approach to error budgets
8.3.3 Applicability of the digital twin to assessing error budgets
8.3.4 Truncation
8.3.5 Numerical truncation and rounding error
8.3.6 Probe x,y (in-plane) position error
8.3.7 Aliasing (data point spacing)
8.3.8 Systematic phase, e.g. drift
8.3.9 Dynamic range
8.3.10 Summary
8.4 Illustration of the compilation of range assessment budgets
8.5 Quantitative measures of correspondence between data sets
8.5.1 The requirement for measures of correspondence
8.6 Comparison techniques
8.6.1 Examples of conventional data set comparison techniques
8.6.2 Novel data comparison techniques
8.7 Summary
References
9 Advanced planar near-field antenna measurements
9.1 Introduction
9.2 Active alignment correction
9.2.1 Acquisition of alignment data in a planar near-field facility
9.2.2 Acquisition of mechanical alignment data in a planar near-field facility
9.2.3 Example of the application of active alignment correction
9.3 Amplitude only planar near-field measurements
9.3.1 Plane-to-plane phase retrieval algorithm
9.3.2 Plane-to-plane phase retrieval algorithm – with aperture constraint
9.4 Non-iterative phase retrieval technique
9.4.1 Current phase retrieval techniques
9.4.2 AUT phase and probe position reconstruction from four or more reference antenna phase measurements
9.4.3 Simulation of the measurement system
9.5 Traditional position correction algorithms, in plane
and z plane corrections
9.5.1 Taylor-series expansion
9.5.2 K-Correction method
9.6 Non-canonical transforms, plane wave spectrumbased treatment
9.6.1 Matrix inversion method – solution of a system of equations
9.6.2 Non-uniform FFT-based algorithms
9.6.3 Non-canonical transform
9.7 Compressive sensing
9.7.1 Introduction to compressive sensing
9.7.2 Defective element detection using compressive sensing
9.7.3 Compressive sensing applied to a 2D array
9.7.4 Practical implementation
9.7.5 Compressive sensing using near-field scanning
9.7.6 Summary of compressing sensing technique
9.8 Three antenna extrapolated gain measurements
9.9 Partial scan techniques
9.9.1 Auxiliary translation
9.9.2 Rotations of the AUT about the z-axis
9.9.3 Auxiliary rotation – bi-planar near-field antenna measurements
9.9.4 Near-field to far-field transformation of probe-corrected data
9.9.5 Applicability of the poly-planar technique
9.9.6 Complete poly-planar rotational technique
9.10 Concluding remarks
References
Appendices
A.1 Appendix A: Other theories of interaction
A.1.1 Examples of postulated mechanisms of interaction
A.2 Appendix B: Measurement definitions as used in the text
A.3 Appendix C: An overview of co-ordinate systems
A.3.1 Antenna mechanical system
A.3.2 Antenna electrical system
A.3.3 Far-field plotting systems
A.3.4 Direction cosine
A.3.5 Azimuth over elevation
A.3.6 Elevation over azimuth
A.3.7 Polar spherical
A.3.8 Azimuth and elevation (true-view)
A.3.9 Range of spherical angles
A.3.10 Transformation between co-ordinate systems
A.3.11 Co-ordinate systems and elemental solid angles
A.3.12 Relationship between co-ordinate systems
A.3.13 Azimuth, elevation and roll angles
A.3.14 Euler angles
A.3.15 Quaternion
A.3.16 Elemental solid angle for a true-view co-ordinate system
A.4 Appendix D: Trapezoidal discrete Fourier transform
A.5 Appendix E: Calculating the semi-major axis, semiminor axis and tilt angle of a rotated ellipse
A.6 Appendix F: Fast Fourier transform algorithm
A.7 Appendix G: Asymptotic far-field form of the Kirchhoff–Huygens formula
References
Index
Back Cover

Citation preview

IET ENERGY ENGINEERING 95

Principles of Planar Near-Field Antenna Measurements

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Principles of Planar Near-Field Antenna Measurements 2nd Edition Stuart Gregson, John McCormick and Clive Parini

The Institution of Engineering and Technology

Published by SciTech Publishing, an imprint of The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). † The Institution of Engineering and Technology 2023 First published 2023 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Futures Place Kings Way, Stevenage Hertfordshire SG1 2UA, United Kingdom www.theiet.org While the author and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the author nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the author to be identified as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library

ISBN 978-1-83953-699-1 (hardback) ISBN 978-1-83953-700-4 (PDF)

Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon Cover Image - Close up of a Dual Robotic Antenna Measurement System. (Courtesy of Boeing)

Contents

About the authors List of abbreviations Foreword Preface

xiii xv xix xxi

1 Introduction 1.1 The phenomena of antenna coupling 1.2 Characterisation via the measurement process 1.2.1 Free space radiation pattern 1.2.2 Polarisation 1.2.3 Bandwidth 1.3 Assumed (suppressed) time dependency 1.4 The organisation of the book References

1 1 4 6 8 8 12 12 13

2 Maxwell’s equations and electromagnetic wave propagation 2.1 Electric charge 2.2 The electromagnetic field 2.3 Accelerated charges 2.4 Maxwell’s equations 2.5 The electric and magnetic potentials 2.5.1 Static potentials 2.5.2 Retarded potentials 2.6 The inapplicability of source excitation as a measurement methodology 2.7 Field equivalence principle 2.8 Characterising vector electromagnetic fields 2.9 Summary References

15 15 16 17 20 27 27 27

3 Introduction to near-field antenna measurements 3.1 Introduction 3.2 Antenna measurements 3.3 Forms of near-field antenna measurements 3.4 Plane rectilinear near-field antenna measurements 3.5 Chambers, screening and absorber 3.6 RF subsystem 3.7 Robotics positioner subsystem

39 39 39 46 49 51 53 58

31 32 34 36 37

viii

Principles of planar near-field antenna measurements 3.8 Near-field probe 3.9 Generic antenna measurement process 3.10 Summary References

63 66 68 68

4

Plane-wave spectrum representation of electromagnetic waves 69 4.1 Introduction 69 4.2 Overview of the derivation of the plane-wave spectrum 70 4.3 Solution of the scalar Helmholtz equation in Cartesian co-ordinates 71 4.3.1 Introduction to integral transforms 71 4.3.2 Fourier transform solution of the scalar Helmholtz equation 72 4.4 On the choice of boundary conditions 84 4.5 Operator substitution (derivative of a Fourier transform) 85 4.6 Solution of the vector Helmholtz equation in Cartesian co-ordinates 86 4.7 Solution of the vector magnetic wave equation in Cartesian co-ordinates 89 4.8 The relationship between electric and magnetic spectral components 89 92 4.9 The free space propagation vector k 4.10 Plane-wave impedance 93 4.11 Interpretation as an angular spectrum of plane waves 95 4.12 Far-field antenna radiation patterns: approximated by the angular spectrum 97 4.13 Stationary phase evaluation of a double integral 100 4.14 Co-ordinate free form of the near-field to angular spectrum transform 106 4.15 Reduction of the co-ordinate free form of the near-field to far-field transform to Huygens’ principle 109 4.16 Far-fields from non-planar apertures 111 4.17 Microwave holographic metrology (plane-to-plane transform) 112 4.18 Far-field to near-field transform 114 4.19 Radiated power and the angular spectrum 118 4.20 Summary of conventional near-field to far-field transform 121 References 123

5

Measurements – practicalities of planar near-field antenna measurements 5.1 Introduction 5.2 Sampling (interpolation theory) 5.3 Truncation, spectral leakage, and finite area scan errors 5.4 Antenna-to-antenna coupling (transmission) formula 5.4.1 Behaviour of evanescent plane wave mode coefficients 5.4.2 Simple scattering model of a near-field probe during a planar measurement 5.5 Effect of acquiring near-field data using an electric dipole probe 5.6 Rotationally symmetric, x-polarised, near-field probe

125 125 125 127 133 141 144 146 150

Contents 5.7

5.8 5.9

5.10 5.11

5.12 5.13

5.14

5.15

5.16

Evaluation of the conventional near-field to far-field transform 5.7.1 Standard techniques for the evaluation of a double Fourier integral General antenna coupling formula: arbitrarily orientated antennas Plane-polar and plane bi-polar near-field to far-field transform 5.9.1 Boundary values known in plane polar co-ordinates 5.9.2 Boundary values known in plane bi-polar co-ordinates Regular azimuth over elevation & elevation over azimuth co-ordinate systems Polarisation basis and antenna measurements 5.11.1 Cartesian polarisation basis – Ludwig I 5.11.2 Polar spherical polarisation basis 5.11.3 Azimuth over elevation basis – Ludwig II 5.11.4 Copolar and cross-polar polarisation basis – Ludwig III 5.11.5 Circular polarisation basis – RHCP and LHCP Linear and circular polarisation bases – complex vector representations Overview of antenna alignment corrections 5.13.1 Scalar rotation of far-field antenna patterns 5.13.2 Vector rotation of far-field antenna patterns 5.13.3 Rotation of copolar polarisation basis – generalised Ludwig III 5.13.4 Generalised compound vector rotation of far-field antenna patterns Brief description of near-field co-ordinate systems 5.14.1 Range fixed system (RFS) 5.14.2 Antenna mechanical system 5.14.3 Antenna electrical system 5.14.4 Far-field azimuth and elevation co-ordinates 5.14.5 Ludwig III co-polar and cross-polar definition 5.14.6 Probe alignment definition (single port probe) 5.14.7 General vector rotation of antenna radiation patterns Directivity & gain 5.15.1 Directivity 5.15.2 Calculating the power radiated in a direction-cosine coordinate system 5.15.3 Direct evaluation of directivity for a uniformly illuminated square aperture 5.15.4 Gain 5.15.5 Gain-transfer (gain-comparison) method 5.15.6 Approximation of the gain of a rectangular pyramidal horn Calculating the peak of a pattern 5.16.1 Peak by series solution 5.16.2 Peak by polynomial fit 5.16.3 Peak by centroid

ix 151 152 157 162 164 177 187 190 190 191 193 195 198 204 208 208 211 213 214 217 217 218 218 219 219 219 219 220 220 222 224 228 229 231 235 235 237 239

x

6

7

Principles of planar near-field antenna measurements 5.17 Estimating the position of a phase centre from far-field data 5.18 Summary References

240 243 243

Probe pattern characterisation 6.1 Introduction 6.2 Effect of the probe pattern on far-field data 6.3 Desirable characteristics of a near-field probe 6.3.1 Open-ended rectangular waveguide probes 6.3.2 Dual-polarised waveguide probes 6.3.3 Broadband probes 6.3.4 Other less commonly encountered types of near-field probes – dipoles 6.4 Acquisition of quasi far-field probe pattern 6.4.1 Sampling scheme 6.4.2 Electronic system drift (Tie-scan correction) 6.4.3 Channel balance correction 6.4.4 Assessment of chamber multiple reflections 6.4.5 Correction for rotary errors 6.4.6 Remote source antenna tilt-angle correction 6.4.7 Re-tabulation of probe vector pattern function 6.4.8 Alternate interpolation formula 6.4.9 Approximate unwrapping of two-dimensional phase functions 6.4.10 True far-field probe pattern 6.5 Finite element model of open ended rectangular waveguide probe 6.6 Probe displacement correction 6.7 Channel balance correction References

245 245 246 248 250 253 254

Computational electromagnetic model of a planar near-field measurement process 7.1 Introduction 7.2 Linear superposition of electric dipoles 7.3 Method of sub-apertures 7.4 Aperture set in an infinite perfectly conducting ground plane 7.4.1 Plane wave spectrum antenna–antenna coupling formula 7.5 Vector Huygens method 7.6 Kirchhoff–Huygens method 7.7 Current elements method 7.8 Equivalent currents method – near-field to far-field transform, antenna diagnostics, and range reflection suppression 7.9 Generalised technique for the simulation of near-field antenna measurements 7.9.1 Mutual coupling and the reaction theorem

257 257 259 264 265 268 270 273 273 277 280 283 285 289 289 290

293 293 294 296 299 304 306 308 314 324 338 339

Contents

xi

7.10 Near-field measurement simulation 7.11 Reaction theorem 7.11.1 Lorentz reciprocity theorem (field reciprocity theorem) 7.11.2 Generalised Reaction theorem 7.11.3 Mutual impedance and the Reaction theorem 7.12 Full wave simulation of a planar near-field antenna measurement 7.13 Summary References

342 345 345 350 352 353 357 357

8 Antenna measurement analysis and assessment 8.1 Introduction 8.2 The establishment of the measure from the measurement results 8.2.1 Measurement errors 8.2.2 The sources of measurement ambiguity and error 8.2.3 The examination of measurement result data to establish the measure 8.3 Measurement error budgets 8.3.1 Applicability of modelling error sources 8.3.2 The empirical approach to error budgets 8.3.3 Applicability of the digital twin to assessing error budgets 8.3.4 Truncation 8.3.5 Numerical truncation and rounding error 8.3.6 Probe x,y (in-plane) position error 8.3.7 Aliasing (data point spacing) 8.3.8 Systematic phase, e.g. drift 8.3.9 Dynamic range 8.3.10 Summary 8.4 Illustration of the compilation of range assessment budgets 8.5 Quantitative measures of correspondence between data sets 8.5.1 The requirement for measures of correspondence 8.6 Comparison techniques 8.6.1 Examples of conventional data set comparison techniques 8.6.2 Novel data comparison techniques 8.7 Summary References

359 359 359 360 363

9 Advanced planar near-field antenna measurements 9.1 Introduction 9.2 Active alignment correction 9.2.1 Acquisition of alignment data in a planar near-field facility 9.2.2 Acquisition of mechanical alignment data in a planar near-field facility 9.2.3 Example of the application of active alignment correction 9.3 Amplitude only planar near-field measurements 9.3.1 Plane-to-plane phase retrieval algorithm

413 413 413 416

365 368 369 370 371 373 376 376 379 381 382 382 383 388 388 389 389 393 408 410

417 420 426 427

xii

Principles of planar near-field antenna measurements 9.3.2 Plane-to-plane phase retrieval algorithm – with aperture constraint 9.4 Non-iterative phase retrieval technique 9.4.1 Current phase retrieval techniques 9.4.2 AUT phase and probe position reconstruction from four or more reference antenna phase measurements 9.4.3 Simulation of the measurement system 9.5 Traditional position correction algorithms, in plane and z plane corrections 9.5.1 Taylor-series expansion 9.5.2 K-Correction method 9.6 Non-canonical transforms, plane wave spectrum-based treatment 9.6.1 Matrix inversion method – solution of a system of equations 9.6.2 Non-uniform FFT-based algorithms 9.6.3 Non-canonical transform 9.7 Compressive sensing 9.7.1 Introduction to compressive sensing 9.7.2 Defective element detection using compressive sensing 9.7.3 Compressive sensing applied to a 2D array 9.7.4 Practical implementation 9.7.5 Compressive sensing using near-field scanning 9.7.6 Summary of compressing sensing technique 9.8 Three antenna extrapolated gain measurements 9.9 Partial scan techniques 9.9.1 Auxiliary translation 9.9.2 Rotations of the AUT about the z-axis 9.9.3 Auxiliary rotation – bi-planar near-field antenna measurements 9.9.4 Near-field to far-field transformation of probe-corrected data 9.9.5 Applicability of the poly-planar technique 9.9.6 Complete poly-planar rotational technique 9.10 Concluding remarks References

Appendices Index

431 434 435 438 441 451 452 459 464 464 472 475 483 483 486 488 495 502 507 507 517 517 520 523 531 537 541 545 545 551 595

About the authors

Professor Stuart Gregson is an honorary visiting professor at Queen Mary University of London and the director of Operations & Research at Next Phase Measurements. He received his BSc degree in Physics in 1994 and his MSc degree in Microwave Solid State Physics in 1995 both from the University of Portsmouth. He received his PhD degree in 2003 from Queen Mary University of London. From his time with: Airbus, Leonardo, NSI-MI and the National Physical Laboratory; Prof. Gregson has developed special experience with near-field antenna measurements, finite array mutual coupling, computational electromagnetics, installed antenna and radome performance prediction and design, compact antenna test range design & simulation, electromagnetic scattering, 5G OTA measurements and has published numerous peer-reviewed research papers on these topics regularly contributing to and organizing industrial courses on these subject areas. He is a fellow of the Antenna Measurement Techniques Association, a fellow of the Institution of Engineering and Technology, a fellow of the Institute of Physics and is a chartered engineer and physicist. Dr John McCormick has extensive experience in many areas of metrology ranging from wet chemistry laboratories to free-space electromagnetic measurement facilities. His expertise relevant to these volumes relates to his involvement over a number of decades in research and development related to naval and airborne radar systems along with RCS and EW. This experience has been gained in the course of his working relationships with DERA, BAE Systems SELEX ES and latterly Finmeccanica, where he has been engineering lead on a range of advanced programmes that required the development and implementation of diverse and novel measurement techniques. He holds degrees at BA, BSc, MSc and PhD levels, is a fellow of the Institute of Physics, a fellow of the Institution of Engineering Technology and is a chartered physicist and chartered engineer. Additionally, he takes a strong and active interest in the encouragement of public awareness of all areas of science and engineering especially within the school environment where he acts as a schools STEM ambassador. Professor Clive Parini is a professor of antenna engineering at Queen Mary University of London and until his part retirement in October 2021 headed the Antenna & Electromagnetics Research Group. He received his BSc(Eng) degree in Electronic Engineering in 1973 and PhD in 1976 both from Queen Mary University of London. After a short period with ERA Technology Ltd., he joined Queen Mary

xiv

Principles of planar near-field antenna measurements

University of London as Lecturer (1977), moving to reader (1990) and then professor (1999). He has published over 400 papers on research topics including array mutual coupling, array beam forming, antenna metrology, antennas for mobile and on-body communications, millimetrewave compact antenna test ranges, millimetrewave integrated antennas, quasi-optical systems and antenna applications for metamaterials. In 1990, he was one of three co-workers to receive the IEE Measurements Prize for work on near-field reflector metrology. He is a fellow of the IET and a past member and chairman of the IET Antennas & Propagation Professional Network Executive Team. He is a past member of the editorial board and past honorary editor for the IET Journal Microwaves, Antennas & Propagation. In 2009, he was elected a fellow of the Royal Academy of Engineering.

List of abbreviations

Abbreviation Definition 3D AAPC AC AES AESA AMS AMTA APC AR AUT Az CAE CATR CDF CEM CP CW dB DC DFT DUT EFT EHF EIRP El EM EMI EMPL ESA ET FCC FDTD FF FF-MPAC FFT GBM GPIB GSM GTD

three dimensional advanced antenna pattern correction alternating current antenna electrical system active electronically scanned array antenna mechanical system antenna measurement techniques association antenna pattern comparison axial ratio antenna under test azimuth computer-aided engineering compact antenna test range cumulative distribution function computational electromagnetic circularly polarised continuous wave decibel direct current discrete Fourier transform device under test electromagnetic field theory extremely high frequency effective isotropic radiated power elevation electromagnetic electromagnetic interference equivalent multipath level European Space Agency edge taper federal communications commission finite difference time domain far-field far-field multi-probe anechoic chamber fast Fourier transform Gaussian beam mode general purpose interface bus Global System for Mobile Communications, originally Groupe Spe´cial Mobile geometric theory of diffraction

xvi

Principles of planar near-field antenna measurements

HF HVAC I IDFT IEEE IET IF IFFT ITU LF LHCP LI LII LIII LO LP LPDA MARS MC MHM MoM NATO NBS NF NIST OEFS OEWG OMT OTA PCU PDF PIN PNF PO PTD PTFE PTP PWS PWSC Q QMUL QZ RA RADAR RAL RAM RCS RF RFS RHCP RI RMS RSA

high frequency heating ventilation and air conditioning in-phase receiver channel inverse discrete Fourier transform Institute of Electrical and Electronics Engineers Institution of Engineering and Technology intermediate frequency inverse fast Fourier transform International Telecommunication Union low frequency left hand circular polarisation Ludwig 1st definition of cross-polarisation Ludwig 2nd definition of cross-polarisation Ludwig 3rd definition of cross-polarisation local oscillator linearly polarised log periodic dipole array mathematical absorber reflection suppression mirror cube microwave holographic metrology method of moments North Atlantic Treaty Organization National Bureau of Standards near-field National Institute of Standards and Technology opto-electric field sensor open ended waveguide probe orthogonal mode transducer over the air power control unit probability density function p-type intrinsic n-type construction for diode junction planar near-field physical optics physical theory of diffraction polytetrafluoro-ethylene plane-to-plane plane wave spectrum plane wave spectrum components quadrature receiver channel Queen Mary, University of London quiet zone range assessment radio detection and ranging Rutherford Appleton Laboratory radar absorbent material radar cross-section radio frequency range fixed system right hand circular polarisation range illuminator root mean square remote source antenna

List of abbreviations RSS Rx SAR SD SFD SGA SGH SHF SMA SNR SRD SWR TE TEM TM TRP Tx UHF UTD UWB VHF VNA VSWR WG

root sum square receive synthetic aperture radar standard deviation saturating flux density standard gain antenna standard gain horn super high frequency subminiature A signal-to-noise ratio step recovery diode standing wave ratio transverse electric transverse electric and magnetic transverse magnetic total radiated power transmit ultra high frequency universal theory of diffraction ultra wideband very high frequency vector network analyser voltage standing wave ratio waveguide

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Foreword

Second edition of Principles of Planar Near-field Antenna Measurements Ed Joy, Professor Emeritus, Georgia Institute of Technology 2 February 2023, Boulder, Colorado Significant advancements have been made in Planar Near-field Measurements in the 16 years since the publication of the first edition of Principles of Planar Nearfield Antenna Measurements. The second edition of Principles of Planar Near-field Antenna Measurements updates the first addition with these advances. The authors present these advances in a clear and understandable manner, including their practical implementation, recommended measurement procedures and necessary mathematics. The authors, Prof. Stuart Gregson, Dr John McCormick, and Prof. Clive Panini, are leading practitioners, researchers and teachers of planar near-field antenna measurement techniques. Many of the advancements presented were developed by them. A highly visible update is the use of an abundance of color photos, figures, and diagrams, notably, of modern near-field measurement systems and equipment including robots and drones. New topics are spread throughout the book to document the advances in near-field measurement practice and theory which take advantage of the modern equipment. First and foremost, the second edition sticks to its roots as a book all about planar near-field measurements. Chapters on Maxwell–Heaviside equations, near-field antenna measurement techniques, plane wave spectra, practical implementation of planar near-field systems, near-field probe characterization, the computational model of the planar near-field measurement process, near-field measurement error analysis, and advanced measurement techniques are still there, with additions. The second edition retains the first edition’s collection of useful appendices with the addition of a Fast Fourier Transform Appendix. Major additions found in Chapter 5 are the plane-polar near-field transform and the plane-bipolar transform, including transforms with acceleration. A detailed discussion of zero padding and associated phase compensation, insights into nearfield evanescent waves and their properties, closer examination of the dipole as a near-field probe, the advantages of a rotationally symmetric near-field probe, the use of parabolic and Taylor-series approximations of an antenna’s main beam to determine main beam direction and phase center, derivation of the directivity of a

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Principles of planar near-field antenna measurements

uniformly illuminated square aperture, and the NRL formula for the approximate gain of a pyramidal horn also have been added to Chapter 5. Addition of examples of common near-field probes in Chapter 6 include a rectangular open-ended wave guide, a dielectrically loaded open-ended waveguide, a dual polarized circular open-ended waveguide, an open boundary wideband dualridged horn, a linearly polarized log periodic array, and a miniature electric nearfield probe. The discussion on the merits of the various near-field probes has been expanded. The increase in computer modeling of planar near-field antenna measurement systems over the last 16 years has led to the increased size of Chapter 7. Six methods for the simulation of an AUT near-field are presented using: an array of electric dipoles; an array of sub-apertures; a plane-to-plane transform; Kirchhoff– Huygens’ method; current elements method; and equivalent currents method. These methods are fully explained and recommendations are given for the best use of each. Chapter 7 concludes with a full wave simulation of a planar near-field antenna measurement. A commercially available method of moments tool was used to model a measurement of a standard gain horn with an open-ended rectangular probe on a planar near-field measurement system, illustrating the possibilities and challenges of full wave simulations. The computer modeling tools developed in Chapter 7 are put to use in Chapter 8 to help assess planar near-field measurement system errors and form a range error budget. Error assessments presented include: planar measurement truncation, numerical truncation and rounding, in plane position error, out of plane position error, aliasing, RF system amplitude and phase drift with time and receiver dynamic range. The chapter concludes with a planar near-field range assessment example of the measurement of a standard gain horn with a rectangular open-ended waveguide. The example includes 18 sources of measurement uncertainty. Chapter 9, Advanced planar near-field measurements, is a fun chapter, a peek into the future of planar near-field measurements. It is interesting to see how many of the advanced topics presented in First Edition of Principles of Planar Near-Field Measurements have become standard practice. We are treated to a new list of advances, including: drones and robotic test systems and non-iterative phase recovery methods; non-conical near-field to far-field transforms – matrix inversion method, with examples including use for plane-polar transforms; non-conical nearfield to far-field transforms – non-uniform FFT for irregular sampling; examples of using non-conical transforms for near-field measurements using drones; comprehensive sensing for array diagnostics using plane wave spectrum processing; and a three-antenna extrapolated gain method also using plane wave spectrum processing. This second edition is a brilliant collection of the body-of-knowledge of planar near-field measurements with extensions to the future. I highly recommend this book for antenna measurement practitioners, researchers, and teachers, and for those who need a thorough introduction to planar near-field antenna measurements.

Preface

So often, it is the very everyday nature of the physical phenomena around us that blind us to their universality and their importance, both in how we understand and use them in our environment. The list of technological advances over the ages engineered by exploiting these so often ignored or unappreciated phenomena would run to a work of thousands of pages crammed with ingenuity, inventiveness and insight. Our entire technological society is riddled with examples of devices, tools and mechanisms that are based on the existence of these physical phenomena, designed and manufactured by engineering techniques based on, and exploiting the fundamental physical laws that govern these phenomena. As with so many other of the technological wonders of the present day that are taken for granted, countless generations must have dreamt of gazing down on the dark side of the moon. Only a few decades separate us from that day when the crew of Apollo 8 were the first humans to see that sight so permanently hidden from other humans by a manifestation one of the most universal of all observed physical phenomena, the Coupling of Harmonic Systems. Every schoolboy and girl know that despite their physical isolation, the harmonic oscillation of the earth rotating on its axis is coupled to the periodic orbit of the moon so that the same side of the moon always points towards the earth. Of course, what the crew of Apollo 8 saw was conveyed to us back here on the earth by making use of the same universal phenomena of coupled harmonic systems, except that in this case they were coupled electronic, as opposed to massive gravitational, systems. No one who has studied electronic engineering, to any appreciable level, has escaped from hours spent in the pursuit of the solution to problems concerning the arrangement of resistances, capacitances, and inductances in circuits, to produce harmonic systems that have in turn their associated resonant frequencies, bode plots, and Q factors. However, much of what is involved in the modern electronic technologies is based on the existence of harmonic circuits and the universally observed phenomena that these circuits couple together. By way of illustration, in essence, the entire field of electromagnetic compatibility (EMC) is an attempt to minimise the extent to which systems couple. Conversely, the fields of communications and radar in turn both involve attempts to maximise this coupling. So, the extent to which electronic systems interact, as a result of this coupling, is fundamental to large swathes of electronic engineering and therefore also to our modern technological society. This interaction can be minimised by using a variety of strategies, physical isolation, screening, the judicious choice of systems components to separate

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Principles of planar near-field antenna measurements

resonant frequencies are all viable, but this coupling can never be completely removed. However, for many systems, the existence of this coupling and its exploitation for the transfer of information in the form of a signal is imperative to the successful operation of the technologies. This of course means that for these technologies, techniques and components must be developed that maximise the coupling between harmonic electronic systems. Many strategies have been employed to maximise this coupling and the subsequent transfer of information between the systems. However, if the systems are physically isolated from each other and no fixed or transmission line can be established between them, as with the earth and the moon, the free space between them must be exploited as a medium to facilitate the interaction between these apparently isolated systems. The most commonly used strategy for enhancing the interaction between such isolated electronic systems is the inclusion of circuit elements within the electronic systems that enhance this interaction, these individual circuit components are usually referred to as Antennas. It is not the purpose of this book to hypothesis or examine in great detail the mechanisms by which the interaction facilitated by the antennas between the electronic systems occurs, although a variety of such mechanisms and the basis of their associated mathematical algorithms are briefly discussed in Appendix A. In this text, the interaction will primarily be described in terms of the propagation of an angular spectrum of transverse, to take account of polarisation, waves propagating in a non-dispersive medium, these waves being consistent with solutions to Maxwell’s equations. In a large variety of circumstances, this is a particularly successful algorithm for the description of the interaction in question, but the treatment, in this volume, will be such that other hypothesised interaction mechanisms and their attendant mathematical algorithms will not be precluded by the explanations introduced. One of the most common techniques adopted to characterise, predict, and quantify this coupling between electronic circuits is to attempt to reduce the problem of circuit coupling to that of antenna performance. Thus, by characterising antennas in a known circuit configuration, the extent to which they enhance coupling in other situations can be predicted. This is the fundamental procedure adopted in antenna test ranges, where the inclusion of antennas in a configuration of two coupled circuits, usually referred to as the transmit, (Tx), and the receive, (Rx), circuits allows this measurement process to be performed. This means that the characterisation of the antennas in this circuit configuration can be used to predict the response of other circuit configurations which include the same antennas. The accurate characterisation of how the presence of antennas will affect the coupling of electronic circuits can be accomplished using a number of different range configurations, one of the most accurate being the antenna near-field range. This technique allows the characterisation of antennas where measurements are made in close physical proximity to the antennas and thus these measurements can be performed in small highly controlled environments where extraneous noise and interference, mechanical, environmental and electromagnetic, can be eliminated or effectively suppressed. This means that highly stable, repeatable measurements

Preface

xxiii

from which the antenna characteristics can be extracted are possible. All measurement techniques have their limitations and ranges of applicability, not least near-field antenna measurements. However, the necessary information required to inform and influence the design of systems in which antennas are used to enhance the coupling between electronic circuits can be obtained by the skilful and expert use of such antenna test ranges. Therefore what is intended, in the following chapters, is an initial examination of the properties of antennas that allow them to enhance the free-space interaction of electronic systems. This will then be followed by the description of the theory of an effective, efficient, and accurate methodology for characterising these properties using the antenna measurement technique of planar near-field scanning. This will be followed by a review of the practical implications of making such measurements in terms of, techniques, instrumentation, processing, and analysis of data. The utility of the planar methodology is then illustrated with example measurement campaigns. These include a discussion of the characterisation of high gain instruments, electrically large reflector assemblies, planar array antennas along with the ability to transform back to array elements in the aperture plane, to confirm element excitations, and to optimise the overall antenna performance. The determination and compilation of measurement uncertainty budgets is included together with an illustration of the way in which computational electromagnetic simulation, and the concept of the digital twin, may be exploited in their development. Some of the latest advances in such methodologies will be examined particularly with respect to the introduction of statistical image classification techniques which aim to assess the accuracy, sensitivity, and repeatability of given data. These techniques are applicable to all types of antenna pattern, both measured and theoretical, and so are of interest to a wide range of readers who undertake, or are required to interpret antenna radiation pattern data. Finally, many of the most recent advances in the technique which deal with measurement correction, the introduction of noncanonical transforms, and partial scan techniques based on auxiliary translations and rotations to produce poly-planar near-field data sets will be described. This will involve an explanation of the measurement techniques, the assessment of the additional terms introduced in the error budget associated with the technique, and the theoretical basis of the transforms developed to allow their deployment. A large number of facilities exist world-wide, and these techniques will be of interest to current and new planar near-field users alike, as they enable the maximum size of the antenna that can be measured within a given facility to be significantly increased. Furthermore, many of the techniques developed within this technique will be of great utility to those practitioners working with positioning systems comprising uninhabited air vehicles, i.e. drones, or multi-axis industrial robots where the non-canonical scanning data transformation and post-processing techniques will be of enormous utility. In summary, the updated and expanded volume will provide a comprehensive introduction and explanation of both the theory and practice of planar near-field measurements, from its basic postulates and assumptions, to the intricacies of its deployment in complex and demanding measurement scenarios.

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Principles of planar near-field antenna measurements

The International System of Units (SI) is used exclusively. This text uses the approximation m0 = 4p10–7 NA–2. Following the redefinition of SI base units, the kilogram, ampere, kelvin and mole, on 20 May 2019, the difference between this value of m0 and the SI (experimental) value of m0 is less than 110 9 in relative value which is negligible in the context of the uncertainty budgets discussed herein. However, this assumption should be noted and re-examined periodically as it does subtlety affect the permittivity of vacuum, impedance of vacuum and admittance of vacuum. Numbers in parenthesis () denote equations whilst numbers in brackets [] denote references. Our thanks to the many individuals who generously gave assistance, advice, and support. We gratefully acknowledge the invaluable suggestions, corrections and constructive criticisms of the many people who gave freely of their time to review the manuscript at various stages throughout its preparation. However, any errors or lack of clarity must remain the responsibility of the authors, who would welcome any and all such mistakes being brought to their attention. The authors are grateful to their wives, (Catherine Gregson, Imelda McCormick and Claire Parini) and children, (Elizabeth Gregson and Robert Parini), whose unwavering understanding, support, encouragement and good humour, were necessary factors in the completion of this work. A special vote of thanks must be devoted to Catherine for her tireless work on the manuscript. To the companies and individuals who generously provided copyright consent. The authors are also extremely grateful to Prof. Ed Joy for his many valuable comments and suggestions as well as for his generous encouragement. There are many useful and varied sources of information that have been tapped in the preparation of this text; however, mention must be made of four books which have been of particular help to the authors, and will be referred to throughout. In no particular order. H.P. Hsu, Applied Fourier Analysis, Harcourt Brace College Publishing. H. Anton, Calculus with Analytic Geometry, John Wiley & Sons, Inc. M.R. Spiegel, Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis, Schaum Publishing Company. R.H. Clarke and J. Brown, Diffraction Theory and Antennas, Ellis Horwood Ltd. Although the nomenclature and development of the theory of planar near-field measurements as presented within this text has not followed that of the National Institute of Standards and Technology (NIST) format, the technical publications originating from that organisation have also been a rich source of valuable information. In particular. D.M. Kerns, Plane-Wave Scattering-Matrix Theory of Antennas and AntennaAntenna Interactions, Nat. Bur. Stand. Monograph 162. A.C. Newell, Planar Near-Field Antenna Measurements, Electromagnetic Fields Division, National Institute of Standards and Technology, Boulder, CO. Stuart Gregson, John McCormick, Clive Parini London & Edinburgh January 2023

Chapter 1

Introduction

1.1 The phenomena of antenna coupling This chapter describes the phenomenological basis of antenna measurements and attempts to set out the processes and techniques that are developed in this text against the background of what can actually be observed about the action of antennas as circuit elements. As is abundantly clear from the title, this volume has been penned for a very specific purpose, to explain clearly, concisely and in an understandable form the theory and practice of Planar Near-Field Antenna Measurements. Again, as already stated in the preface, to do this the volume will confine itself to considering the radiative coupling between electronic systems in free space for a number of very sound reasons. First, in almost every practical engineering circumstance, this is the mode in which antennas are utilised. If coupling between systems that are not physically separated by large distances is required, various forms of transmission lines can be utilised; however, large separation distances almost invariably require the use of antennas. Communication systems contain transmit, (Tx) and receive, (Rx) subsystems, which use at least two antennas, broadcast systems, and considerably more. Radars may, or may not, use the same antenna for their Tx and Rx subsystems, and the coupling may well be profoundly affected by the scattering from some target, but in essence, we are still considering coupling between electronic sub-systems. Careful consideration of all electronic systems that utilise antennas as a component reveal that it is the extent of this coupling that is fundamental to their operation. Replace Rx antenna with another transducer, e.g. bolometer for power detection and rectifier for rectenna (power transmission by microwave signal), and you have summed up almost every possible engineering circumstances except for those systems designed to detect transmissions from naturally occurring radiation sources, e.g. radiometers. Second, the fundamental electromagnetic (EM) properties of antennas are very limited. If you place a passive antenna in an incident field, it will scatter energy, and depending on the power flux incident, it may get a little hot, that is, about all. It is a common fallacy to assign properties to antennas that actually belong to the systems within which they are embedded. By way of example, often antennas are

2

Principles of planar near-field antenna measurements

described as being reciprocal devices, in fact by definition reciprocity is a property that can only be applied to objects that are either sources or sinks for energy [1]. Thus, it is correct to say that electronic systems that are reciprocal, e.g. that include generators and loads and satisfy the usual requirements of isotropy, will not have this property affected by the inclusion of passive antennas within them. However, to talk of antennas in the absence of well-defined terminals, where load impedance’s and source voltages are attached, being reciprocal is to misinterpret the concept of reciprocity. In fact, almost all of the properties usually assigned to antennas, and of engineering importance, are in fact properties of systems containing antennas. Antennas do not by themselves transmit or receive energy across free space they are simply bi-directional transducers that can be included as circuit components in electronic systems. It is systems that include antennas as components that have the emergent property that they can radiatively couple power across large ranges in free space. Thus, the characterisation of antennas via measurements to quantify their performance is really the characterisation of highly specified electronic systems containing Tx and Rx antennas that can be used to predict the response of other systems in which the same or similar antennas will be utilised. Finally, as has already been stated in the preface, this volume will not attempt to hypothesise or examine the variety of postulated physical mechanisms by which the interaction, which is optimised by the antennas, between the electronic systems occurs. However, as it will attempt to develop a thorough theoretical explanation of near-field antenna measurements, it must, at least, be consistent with physical law. Very little that is encountered within the discipline of electronic engineering is of a more relativistic nature than the operation of antennas. By definition, the interaction facilitated by the presence of antennas occurs at the speed of light thus relativity cannot be completely ignored when considering the action of antennas. In this text, only Tx and Rx antennas that are in translational equilibrium will be considered and their velocities relative to each other will be specified to be zero. Additionally, the EM interactions will be observed from an inertial reference frame coincident with the fiducial mechanical datum of the Tx antenna. These conditions make it possible to consider the antenna characterisation without reference to any relativistic effects associated with a multiplicity of reference frames or to any known inertial effects. This simplifies the explanations, without invalidating them in more complex situations, and makes it convenient to consider the measurement process in terms of classical electromagnetic field theory (EFT) based on the work of the famous Scottish Physicist, James Clerk-Maxwell. Maxwell’s equations, whose mathematical form as we know them today, have much to do with simplification provided by the renowned German experimentalist Heinrich Hertz and are themselves relativistically invariant. Therefore, the solution of the Helmholtz wave equation that will be developed within the text is automatically compliant with this facet of physical law. Thus, it is largely possible to develop a consistent theoretical explanation of near-field antenna measurements based on classical EFT. However, classical EFT has limitations built into its basic structure and, as with all other postulated mechanisms for the interaction of antennas across

Introduction

3

space and time, will produce inconsistent answers when used outside of its range of applicability. One of the most striking limitations in the theory becomes manifest when dealing with any single antenna. Calculations of the power required to accelerate the electrons within antennas, to produce currents, that provide values for the energy and momentum transferred to these distributions of electrons can be achieved fairly easily. However, if the amount of energy and momentum transferred to or from the EM field as a result of these accelerations is calculated, it is found that the values calculated from circuit theory and the values calculated from field theory do not agree. Thus, the radiation resistance of the antenna will define an equivalent resistance required to dissipate a given amount of power, which is not equal to the power transferred to the EM field around the antenna, meaning that for an individual antenna, energy and momentum are not conserved locally. Globally for the level of energy to fall at one point in a space, it must increase elsewhere, for energy to be conserved there must therefore be a change in the energy distribution and density in the space. This rate of change of energy with respect to time, dE/dt = power will, as a result of the requirement for local conservation, be manifest as a flux between the regions in the space where the energy density is changing. Thus, the existence of an energy density at a point in space, in the absence of a flux, does not imply the development of power, an important factor when consideration is given to the near-fields of any antenna. This local aspect of conservation requires that the power dissipated in the antenna via its radiation resistance be equal to that developed in the flux around it, and this is not the case for calculations based on simple classical EFT. Therefore, the energy and momentum dissipated from the antenna do not equate to that in the flux propagating away from the antenna. Nothing could be more at odds with all of classical physical science than the concepts that the law of energy conservation in time, and the law of momentum conservation in space can be violated. Many theoretical mechanisms have been postulated to account for this apparent anomaly, some of which invoke the concept of non-point-like charged particles being involved, e.g. Poincare stresses [2]. Others involve the retention of the advanced wave solution to the Helmholtz wave equation, most famously the Wheeler Feynman absorber theory of radiation [3]. The absorber theory is particularly attractive as it is by definition relativistically invariant, as will be expanded upon in Chapter 4, and work extending and eliminating some of its limitations by John Cramer at Washington State University in Seattle [4] has made it also applicable within the realms of quantum as well as classical mechanics. However, making use of the reciprocal nature of systems including antennas allows the consideration of systems where the power launched into the EM field is coupled out by the presence of the same or other antennas. Therefore, the local power imbalance identified in a system containing one antenna will be eliminated when considering any set of systems where energy is coupled into then out of free space. Thus, any physical system containing a Tx antenna and at least one

4

Principles of planar near-field antenna measurements

Rx antenna will satisfy the local conservation laws of energy and momentum for the systems containing the antennas. In summary: Almost every imaginable engineering situation will involve coupling between Tx and Rx antennas (this coupling is what engineers are actually interested in). Almost every possible measurement scenario will involve both Tx and Rx antennas (it is what can actually be measured). Finally, classical electromagnetism is applicable in situations that involve both Tx and Rx antennas (it is a situation that can actually be described rigorously and accurately using theory). For all these reasons, this volume will confine itself to antenna measurements designed to characterise the radiative coupling between electronic systems in free space, a homogeneous linear isotropic dielectric medium, facilitated by the presence of antennas within the systems, but what exactly do we mean by measurements?

1.2 Characterisation via the measurement process Measurement is often defined as ‘The quantitative determination of a physical magnitude adopted as a standard, or by means of a calibrated instrument. The result of a measurement is thus a numerical value expressing the ratio between the magnitude under examination and a standard magnitude regarded as a unit’ [5]. A definition originally made popular by Lord Kelvin in the nineteenth century. However, the passage of years resulting in an increased level of understanding of the physical process of measurement, along with the introduction of system-centred concepts and the development of information theory, has led to a more generalised view of the measurement process. This gives an information conversion definition of measurement as ‘Consisting of information transfer with accompanying energy transfer. Energy cannot be drawn from a system without altering its behaviour, hence all measurements affect the quantity being measured. Measurements therefore are a carefully balanced combination of physics, (energy transfer) and applied mathematics, (information transfer)’ [6]. This information extraction concept is particularly applicable to antenna measurements were the test range holds many similarities to a communication system. Although the procedure involves a Tx signal, which initially contains no information, as it is completely predictable, information can be extracted from the input Tx signal via its comparison with the output Rx signal. Whatever conceptual model of the measurement process is adopted, as will be highlighted in this volume, the raw data from any near-field antenna measurement must be processed to provide predictions of the parameters of the antenna that are required. Thus, the measurement process must be consistent with the formal

Introduction

5

mathematical definition of measurement [7], which is included for completeness as shown in Box 1.1.

Box 1.1 For a well-defined, non-empty class of extra-mathematical entities Q, let there exist upon that class a set of empirical relationships R = {R1, R2, . . . , Rn,}. Let us further consider a set of numbers N (in general a subset of the set of real numbers R), and let there be defined on that set a set of numerical relationships P = { P1, P2, . . . , Pn,}. Let there exist a mapping M with domain Q and a range in N, M: Q !N which is a homomorphism of the empirical relationship system and the numerical relationship . The triplet P = constitutes a scale of measurement of Q. It is required that M be a well-defined operational procedure, it is called the fundamental measurement procedure of Q. ni [ N the image of qi [ Q under M will be denoted by ni = M(qi), ni will be called the measure of qi on the scale P, qi the measurand and Q the measured class. There will in general be other procedures of mapping Q onto N denoted by M0 : Q ! N such that M(qi) = M0 (qi) either for all qi [ Q or qi [ Q0 where Q0 ⊂ Q any such procedure is a measurement procedure Q or Q0 on the scale P. The statistical nature of establishing the measure in any practical circumstance from the measurements (measure [Box 1.1] empirically being defined as the limit in the behaviour of the measurement procedure as the number of measurement trials tends to infinity) is discussed further in Chapter 5. So, whether the measurement process is thought of in purely physical terms or also as the extraction of knowledge from a system containing information, it is clear that there must be a very well-defined and controlled measurement procedure. Additionally, the process must have a clearly defined measurand and a recognisable scale or standard against which the response or behaviour of the system during the measurement procedure can be assessed. For near-field antenna measurements, the measurand is the power coupled into the Rx system, the source of both the energy and the information bearing signal transferred in the process. The scale is relative to a reference signal, which can be calibrated against a standard and the measurement procedure is the stated subject of this volume, near-field antenna measurements. Having defined exactly what the measurand in the measurement system is, how can measurements of the power coupled between such systems be used to characterise the antennas embedded within them and give a measure of the antennas performance in another circumstance. In electronic engineering, the interaction between such systems is usually described in terms of a number of useful design parameters that are ascribed to the Tx and Rx antennas. The most important free space parameters being the following.

6

Principles of planar near-field antenna measurements

1.2.1

Free space radiation pattern

Antennas do not radiate equally in all directions, and the concept of an isotropic radiator is useful as a standard relative to which any other antenna’s performance can be quantified but is in theory and practice impossible to construct. Therefore, the variation in the ratio of the radiated power, as a function of angle relative to the fixed mechanical datum of any antenna, is an important parameter. Figure 1.1 illustrates a co-ordinate system against which this variation can be judged with the z- or f-axis conforming to the mechanical datum, often referred to as boresight, of the Tx antenna. Figure 1.2 illustrates the relative angular position of the Tx and Rx systems as Rx moves around a circular path at a fixed value of f with q varying along the circular path and with Tx at its centre. For any sufficiently large fixed value of R, the extent to which power is transferred between the two antennas relative to the value at q = 0 would vary as a function of the angle q. Clearly this variation in relative power would also be a function of the angle f so, the so-called radiation pattern of the antenna would actually be a function of q and f the angles which define the direction of the displacement R, the path between Tx and Rx antennas. The pattern function is an important parameter of any antenna and, assuming the magnitude of R, the distance between the two antennas is electrically large, for two such antennas the variation along a segment of the circular path shown in Figure 1.2 could be of the Cartesian form shown in Figure 1.3. Here, the maximum-recorded signal is normalised to unity, i.e. zero on a dB scale. An alternative to the Cartesian form is the polar plot in which the pattern is plotted with the amplitude in the radial axis and the angle in the azimuthal. This is a generally Theta axis

yAMS θ^ –

ϕ^ – Chi axis

kAMS

ϕ

θ zAMS Phi axis

xAMS

Figure 1.1 Illustrating the antenna co-ordinate system, where the square represents the antenna aperture

Introduction

7

YAMS ZL Rx R Zg

θ

~

Tx

ZAMS

Figure 1.2 Orientation of Tx and Rx antenna, including all circuit components 0

Amplitude [dB]

–10 –20 –30 –40 –50 –60 –60

–40

–20

0 θ [deg]

20

40

60

Figure 1.3 Recorded power normalised to zero as Rx is moved around the circle typically less commonly encountered form of presentation, however, it has benefits when representing antennas with very broad pattern functions. The measurement of absolute levels of power coupled or the levels relative to calibration standards are also possible but again detailed discussion of this will be delayed until a simple model of radiating structures is developed in Chapter 2.

8

Principles of planar near-field antenna measurements

There are also, those concepts relevant to the characterisation of antennas as circuit elements that cannot by definition be measured by, conventional near-field scanners, but that are still relevant to them. These include scattering parameters and their relevance to definitions of gain in terms of accepted as opposed to delivered power in any circuit and their possible use in scattering matrix descriptions of Tx and Rx antennas. A logarithmic scale is generally used to present antenna pattern plots as it tends to focus attention on the regions where the largest values are encountered thereby compressing the scale in a way that renders the graphical interpretation perhaps easier to interpret, i.e. it tends to enhance the main-beam region of the plot. In Figure 1.3, the peak of the pattern has been normalised to 0 dB with the rest of the pattern plotted relative to this (arbitrary) value. To establish an absolute y-axis datum, we need to introduce the concept of gain (or directive gain), which enables the antenna engineer to compare different antennas directly with one another. The concept of directivity and gain is discussed in detail in Chapter 5.

1.2.2

Polarisation

Having measured this pattern function, you might be forgiven for assuming that you now know everything about the angular variations in coupled power between the Tx and Rx systems. However, for any position of Tx and Rx, where either of the antennas to be rotated about their mechanical datum, i.e. about phi axis for the Tx antenna as per Figure 1.1, a variation in the amount of power coupled as a function of f would be observed. This variation is ascribed to the polarisation of the antenna and various polarisation bases that can be used to describe this polarisation will be developed over the course of the text. Figure 1.4 being a typical measured response of what would be termed a linearly polarised Rx antenna’s response if Tx had the same so-called polarisation. This concept of polarisation will be further examined in the light of the development of the Helmholtz wave equation developed in succeeding chapters.

1.2.3

Bandwidth

This is the range of frequencies, f, in Hz over which the antenna is effective in facilitating the EM interaction. Figure 1.5 illustrates a plot of the measured power in a receiver as a function of frequency for boresight Tx and Rx, where a 3 dB and 10 dB bandwidth for an antenna is marked. For a full characterisation, such a bandwidth response would of course be required for each combination of pattern and polarisation. By systematically varying the position, orientation, and frequency of excitation of the Tx antenna relative to the Rx antenna in terms of the parameters q, f, f, and R, it is possible to characterise the interaction between the Tx and Rx sub-systems in terms of, pattern, polarisation and bandwidth. However, one other additional class of measurement can be included that is directly related to the action of antennas as initiators of harmonic coupling between electronic circuits. The basis of all free space antenna measurement techniques, and indeed much of Electromagnetism, is the assumption that the antennas under tests (AUTs) and the systems used in any test procedures behave in a linear fashion, in fact lack of

Introduction 0 –5 –10 –15

Amplitude (dB)

–20 –25 –30 –35 –40 –45 –50 0

50

100

150

200

250

300

350

Phi (deg)

Figure 1.4 The variation in coupled power as a function of f 0 –3 –6 –9 dB –12 –15 –18 –21 8

8.5

9

9.5 10 10.5 Frequency/GHz

11

11.5

12

Figure 1.5 Showing normalised pattern at q = f = zero from 8 to 12 GHz

9

10

Principles of planar near-field antenna measurements

linearity will be a source of uncertainty in the measurement process that will be referred to in Chapter 8. Such linear systems can be described by linear differential equations like (1.1) shown: cn :

d n f2 ðtÞ d n1 f2 ðtÞ df2 ðtÞ þ c0 :f2 ðtÞ ¼ f1 ðtÞ þ c : þ :::::::c1 : n1 dtn dtn1 dt

(1.1)

For any simple system that can support harmonic oscillations, this can be truncated to a second-order equation of the form: c2 :

d 2 f2 ðtÞ df2 ðtÞ þ c0 f2 ðtÞ ¼ f1 ðtÞ þ c1 : dt2 dt

(1.2)

where the constants and functions in the equation can be related to the usual circuit parameters of capacitance (C), inductance (L), voltage (V), charge (q) and resistance (R), to give L:

d 2 qðtÞ dqðtÞ 1 þ :qðtÞ ¼ V ðtÞ þ R: 2 dt dt C

(1.3)

This is an equation relating to the circuit parameters, with a harmonic solution that will be familiar to any student of alternating current (AC) theory. However, such an equation is inadequate to describe the harmonic solutions present in a circuit at radio or microwave frequencies. If a voltage is applied to such a circuit, this voltage will be propagated through the circuit at approximately the speed of light, 0.3 billion metres per second. Thus, at 50 Hz, this will produce a spatial harmonic variation in the circuit voltage that will be cyclic over some 6 million metres. Therefore for any circuit harmonically oscillating in time at 50 Hz, it is reasonable to assume that the voltage and currents are constant at all points in the circuit at any specified time. However, at the microwave frequency of 10 GHz, the associated cyclic spatial variation of the currents and voltages in any circuit will be repetitive over a distance of the order of 3 cm. At this frequency, assuming the circuit itself is at least of the order of a few centimetres in length, the currents and voltages in that circuit will vary harmonically both as a function of when and where they are observed. A harmonic system in which such oscillations are a function of space and time will be described by a partial differential equation. Such an equation that linearly relates the rate of change of the variables with respect to time and space is a wave equation. This means that measurements of the instantaneous power made at different points in the circuit at the same time will give different results, and these results can be related to provide a measure of the relative phase of the harmonic oscillation at the different points in the circuit. These measurements that can be used to assign a phase to the harmonic coupling are the other additional class of measurements that can be made on antennas. As will be discussed in Chapter 3, these measurements are the source of the in phase and at quadrature data that will be fundamental to the near-field measurement process.

Introduction

11

All of the above measurements can be made in an effort to characterise the coupling between antennas; however, one point that has been briefly mentioned will need further explanation. Figure 1.2 shows an Rx system including antenna placed at a position along the circular path, and Figure 1.3 shows the variation in the antenna pattern with angle q. The text then goes on to state that provided R is large enough the pattern will just be a function of q and f, but how electrically large must R be for this to be true. When R is small, of the order of a few wavelengths in free space, the extent of coupling between the circuits is profoundly affected by the instantaneous distribution of charge on the surfaces of the antennas. The ratio of the power coupled is strongly dependent on R. This is the so-called reactive region around an antenna where reactive coupling dominates radiative coupling. As the distance R increases, the power coupled between the circuits is no longer dominated by this charge distribution, Figure 1.6 illustrates such a situation. In Figure 1.6, the displacement from Tx to Rx is again labelled R but many displacements, e.g. R’ are also paths between the antennas, meaning that in the situation illustrated, there is no unique path between the two antennas. Only when R is infinitely large will the displacements R and R’ be effectively the same. Therefore, only when R is infinitely large can we define a unique path with a definite length where all parts of the two antennas are effectively at equal distances apart, when this is so, the two antennas are in each other’s far-field. In practice, since in the vast majority of engineering situations, we are concerned with antennas that are at large but finite distances apart, the far-field is defined as being when all parts of the Tx antenna are effectively at the same distance from the Rx antenna. At this distance, the angular field distribution is essentially, but not strictly, independent of R. The distances at which R can be considered to be large enough to define a

YAMS Z Rx

R᾿ R θ

Zg ~

Tx

ZAMS

Figure 1.6 Two of the possible paths between the Tx and Rx antenna

12

Principles of planar near-field antenna measurements

far-field region will be examined in the following chapters when a mechanism of interaction based on classical EFT will be expounded. Finally, that region in which the coupling is dominated by radiation but the distance R is not sufficiently large to uniquely define a single path is called the radiating near-field, this is the region surrounding the antennas that will be, in many senses, the practical focus of this text. In engineering situations, as a result of how they are employed, almost invariably it is the far-field performance of antennas that is of interest. However, for practical reasons that will be examined in this text, the ability to make near-field measurements that can be used to predict far-field parameters is extremely important in antenna engineering. This short description of the observables and hence measurable parameters associated with antennas concludes the initial phenomenological description of the action and characterisation of antennas. In the rest of the following chapters, a particular model explaining the interaction between Tx and Rx systems via the application of classical EFT will be developed and used to explain and examine antenna measurements, procedures and analysis [8].

1.3 Assumed (suppressed) time dependency The time factor of EM field data can be specified either as being exp(jwt) or as exp (iwt). In RF and antenna measurements, most commercial test equipment provides the output with an assumed exp(jwt) time dependency. In practice, this convention can be confirmed by increasing the measurement distance slightly where if the time dependence is positive, i.e. exp(jwt) the phase dependence will be given by exp(–jkr) and, the measured phase will therefore decrease when the distance increases. This text assumes a positive, suppressed, time dependency of exp(jwt).

1.4 The organisation of the book The mathematical nature of the predictive algorithms associated with theory will be rigorously examined and included in the text. However, if the reader wishes to take as read certain of the key assumptions and results they will find that full derivations are only included within numbered boxes as per Box 1.1 which mathematically defines measurement. The arguments put forward in the text can be followed without recourse to these numbered boxes on an initial or subsequent examination, but they are included to provide a full, complete and rigorous explanation within the body of the text. This separation of much of the fundamental theory behind the plane wave spectrum and its use in near-field scanning also has other advantages. It allows these derivations to be utilised out with the main thrust of the text and makes them readily available to the reader less interested in the measurement process and more concerned with plane wave techniques and its application in other engineering areas. Throughout the text as the arguments developed move from the theoretical nature of interaction of antennas through the reasons behind the choice of

Introduction

13

parameters chosen to characterise antennas along with the implications and practicalities of antenna measurement procedures, the text will continue to be rigorously illustrated, described and explained. Along with the complete mathematical development of the theory of near-field measurements, which the reader can choose to follow or take as assumed from the contents of the boxes, the text will attempt to inform and advise on the practical implications of the use of near-field antenna measurements. This will extend to the assessment of near-field measurement data as an input to engineering tools and the development of practical methodologies for the analysis of the results of such measurements. Chapter 2 mainly concerns itself with the fundamental relationship between field and charge, and this forms the basis for Chapter 3 that introduces the near-field scanning technique. Chapter 4 examines the theory of the plane wave spectrum, the theoretical basis of the near-field to far-field processing concept, and Chapter 5 deals with the practicalities of near-field measurements. Chapter 6 explains the nature and requirement for probe characterisation in near-field scanner measurements, while Chapter 7 develops effective modelling concepts that can be used to assess any nearfield scanning procedure. Chapter 8 describes the representative theory of measurements and how the impact of this theory on the accuracy of measured data sets can be assessed together with the compilation of range uncertainty budgets. Finally, while throughout attempting to develop consistent logical explanations of all the relevant aspects for near-field scanning, the very latest developments in near-field scanning are discussed in Chapter 9, including the poly-planer technique, drone and industrial robotic arm-based measurements, extrapolation range measurements and array antenna diagnostics using modern compressive sensing. However, the explanations of these methods and techniques will have to be delayed until a model of radiating structures, and how the near-field scanning measurement procedure relates to it, based on EFT is developed and explained. This is the subject of the following chapters in this volume.

References [1] Olver A.D., The Handbook of Antenna Design, vol. 1, Peter Peregrinis Ltd, UK, 1986, pp. 11–12. [2] Feynman R.P., The Feynman Lectures on Physics, vol. 2, Addison Wesley Publishing, Boston, MA, 1964, pp. 28.1–28.10. [3] Wheeler J.A. and Feynman R.P., “Interaction with the absorber as the mechanism of radiation”, Reviews of Modern Physics, vol. 17 nos. 2 and 3, 1945, pp. 157–181. [4] Cramer J.G., “The transactional interpretation of quantum mechanics”, Reviews of Modern Physics, vol. 58, 1986, pp. 647–688. [5] Scrivenor P. (Ed.), “New Caxton Encyclopaedia”, Caxton Publishing Company, London, 1964. [6] Stein P.K., “Measurement Engineering”, 1st ed., Stein Engineering Services, Phoenix, AZ, USA, 1964.

14

Principles of planar near-field antenna measurements

[7] Sydenham P.H., “Measuring Instruments Tools of Knowledge and Control”, Peter Peregrinus Ltd in Association with Science Museum London, 1979, Steleaus UK and New York, NY. [8] IEEE Standard 145-1993 (Revision of IEEE Standard 145-1983), IEEE Standard Definition of Terms for Antennas. Sponsor Antenna Standards Committee IEEE of the Antennas and Propagation Society. Approved March 18, 1993. IEEE Standards Board.

Chapter 2

Maxwell’s equations and electromagnetic wave propagation

2.1 Electric charge It is an empirical fact that under investigation, electric charge appears to exist in two forms, usually but not exclusively referred to as positive and negative. This chapter will attempt to explain the action of antennas in terms of an explanation of the nature of the interaction of these different types of charge when they are in motion within the structure of an antenna. This explanation of this interaction will be developed via the concept of the electromagnetic field. Although the text in the main attempts to confine itself to the classical representations of field concepts, as antenna theory concerns the propagation of electromagnetic energy between physically remote antennas at the speed of light it is not possible to completely ignore the relativistic aspects of antenna theory. This does not invalidate the approach that will be adopted as the principle equations used, based on the seminal work of James Clerk–Maxwell, are as will be seen in the text themselves relativistically invariant in form. However, as the concept of relative motion is fundamental to any understanding of the concept of magnetism it can be helpful to bear in mind during any descriptions of a priori principles of special relativity, these being: ●

●

The principle of the constancy of the speed of light: which states that the speed of light in vacuum will be measured to be c
3  108 ms1 in all inertial frames of reference, irrespective of the state of motion of the frame. The principle of relativity: which states that the laws of physics can be expressed in the same form in all inertial frames of reference. Therefore, any description or explanation of any physically observable phenomena that is not invariant between inertial reference frames is not consistent with physical law.

Only very occasionally within the arguments constructed within this text will it be necessary to return to these principles but without them being explicitly stated it is unclear that they do in fact underpin the entire theory of classical electromagnetism within the framework of physical law. Another empirically established fact is the conservation of electric charge. This means that electric charge can be neither created nor destroyed thus any change in its distribution within space must involve the motion of charged particles. This can

16

Principles of planar near-field antenna measurements

be summed up by a continuity equation, rj þ

@r ¼0 @t

(2.1)

where r is the charge density and j is the current density It can be seen from (2.1) that any change in charge density within a volume constitutes a current density flowing out of or into that volume.

2.2 The electromagnetic field An electromagnetic field can be thought of as constituting that ‘state of excitement’ induced in space by the presence of a possibly time dependent, distribution of electric charge that has the potential to act on other charges if they are present within the field. The action on any test charges [1] present will be such that it alters, or tends to alter, their state of motion. Although there is only one field, the electromagnetic field associated with charge distributions, historically it has been split into the concept of an electric and a magnetic field due to the different circumstances under which both are most easily observed. The electric field E can be expressed in terms of the Coulomb force law as F ¼

q1  q2 R 4pR2

(2.2)

where F is the force that acts on q1, by definition equal and opposite to that which acts on q2; R is the distance between charges q1 and q2; R is the unit displacement vector in the direction defining the displacement between q1 and q2; q1 and q2 are two distinct point-like distributions of charge. Thus, F ¼ q1 E

(2.3)

where E is defined as E ¼

q2 R 4pR2

(2.4)

Thus, in this case, we are defining E as the field producing the force acting on q1, i.e. the field produced by the presence of q2. These simple formulas quantify the forces acting on any stationary point charge, a test charge, at a point in space where the field E is present. Where force is itself defined as that which alters or tends to alter the motion of bodies. Of course, as a direct result of the principle of relativity and the equivalence of inertial reference frames, all states of transitional equilibrium are equivalent. Therefore, in a similar fashion it must be possible to define the force that acts on the test charge even if it is initially in a state of uniform motion, as opposed to being stationary. However, although a distribution of separate charges may be individually in translational equilibrium they may be in motion relative to each other and

Maxwell’s equations and electromagnetic wave propagation

17

therefore there will be no inertial frame relative to which all of the charges constituting the distribution are stationary. To take account of this relative motion, a second vector B is defined that relates the force that acts on the test charge in the presence of fields at the points in space the charge instantaneously occupies when it is in motion. From the experiment, it is found that these fields apply a force F on the charge q moving with velocity v which is given by the Lorentz force law as F ¼ qðE þ v  B Þ

(2.5)

Clearly as v ! 0 the Lorentz Force law tends to the Coulombic formulation. In this formulation, relativistic effects can be taken into account by modifying the mass of the particle so that it becomes a function of the relative velocity and the rest mass. Additionally, the vectors E and B will vary as a function of the inertial state of any observer. However, a range of Lorentz invariant parameters, e.g. E, H, [2], can be defined which allow transformations of the E and B fields between inertial reference frames. This allows the Lorentz force law to be a considered a fundamental law of physics and to act as the definition of the vector quantities E and B. Another inevitable consequence of special relativity, with respect to the principle of the constancy of the speed of light, has to be taken into account in any theory that attempts to describe a mechanism for the interaction of physically remote antennas. At any point in time, the field produced at a point in space remote from the charge distribution, that is its source, is not equal to the field that would be produced by the charge distribution at that instant in time. Since time elapsed is equal to distance divided by speed, it is in fact the field that would be created by the charge distribution a period of time equal to the magnitude of the displacement of the test charge from the source divided by the speed of light. Thus, at any point in time, the field at a point in space mirrors the charge distribution that was present at a point in the past equal to the magnitude of the displacement from the source divided by the speed of light. This in turn means that the effect of any change in the charge distribution will take a finite amount of time to act on the test charge. Therefore, any change in the field will be retarded by a period of time directly proportional to the magnitude of the displacement of the test charge from the source. The concept that the finite velocity of propagation retards the effects of the variation of any field source across space is crucial to the development of classical electromagnetic field theory. This is especially important when electromagnetism is framed in terms of a theory of potentials that will also be retarded by the constant, large, but finite speed of light.

2.3 Accelerated charges Figure 2.1 illustrates the well-known tool for the representation of electric fields in free space, field lines [3]. It shows a positive charge situated at point A at time t1. As illustrated in Figure 2.2, the charge is accelerated to point B at a time t 1 + t/2.

18

Principles of planar near-field antenna measurements

A

B

Figure 2.1 Fields lines around an isolated +ve charge

A

B

Figure 2.2 Curvature of field lines associated with accelerating charge

It is then moved back to point A at a time t1 + t, and Figure 2.3 illustrates the pattern of field lines around the point charge after it has arrived back at A. If a further period of time equal to t is allowed to elapse with the charge held stationary a situation similar to Figure 2.4 would be observed. Clearly if the point charge was subject to alternating, sinusoidal, timeharmonic displacements between points A and B, an arrangement of field lines similar to Figure 2.5 could be expected.

Maxwell’s equations and electromagnetic wave propagation

A

19

B

Figure 2.3 Curvature of field lines associated zero total displacement at t1+t

A

B

Figure 2.4 Curvature of field lines associated with propagation after t1+2t From Figures 2.1 to 2.5, it can be seen that accelerated motion of a charge will result in curvature in the field lines (those areas of the figures that are shaded in grey). As a result of the finite speed of propagation, the retardation of the transverse disturbance of the field can be seen to radiate outwards with a speed of c. It will be shown in the next section where we examine the relationships between E and B as described by Maxwell’s equations that this retarded transverse disturbance is in fact the basis of an electromagnetic wave. Where the changing electric field and its

20

Principles of planar near-field antenna measurements

A

B

Figure 2.5 Field lines associated with propagation of harmonic displacement

associated changing magnetic field form a propagating harmonic disturbance through space.

2.4 Maxwell’s equations Classically, the relationships between the components of any electromagnetic field are described by Maxwell’s field equations and by the equations representing the properties of the medium in which that field exists. Maxwell’s equations can be written in differential form as, e.g. [4–7], rD ¼r

(2.6)

rB ¼0

(2.7)

rE ¼

@B @t

rH ¼J þ

@D @t

(2.8) (2.9)

The definitions and units of these quantities are: E is the electric field intensity in volts per meter, H is the magnetic field intensity in amperes per meter, J is the current density composed from the impressed, or source electric current, and the conduction electric current density all of which are in amperes per square meter, D is the electric flux density in coulombs per square meter, B is the magnetic flux

Maxwell’s equations and electromagnetic wave propagation

21

density in Weber’s per square meter, and r is the charge density in coulombs per cubic meter.

Box 2.1 In this text, we are taking Maxwell’s equations to be the fundamental postulates, or axioms from which we will be developing classical electromagnetic (EM) field theory. However, it should be born in mind that they themselves can be derived as theorems for other sets of postulates, e.g. coulombs law and the already stated postulates of special relativity can be used as a starting point to derive Maxwell’s equations [2]. For these differential equations to hold, the field vectors must be well behaved, i.e. continuous functions of position and time with continuous derivatives, that are both single-valued and bounded. By classically, we imply that the problems discussed only require consideration of lengths that are large compared to atomic dimensions and charge magnitudes so that recourse to quantum field theories, such as quantum electrodynamics (QED), can be avoided. More exactly, classical concepts can be characterised by assuming that: 1. 2. 3.

The world is divisible into distinct continuous elements. The state of each element can be described in terms of dynamic variables that can be specified with infinite precision. Exact laws that define the change of the system in terms of the dynamic variables can describe the interdependency between parts of a system.

Since only alternating, sinusoidal, time-harmonic quantities are to be considered, the time dependency of the complex representations of the electromagnetic jwt field vectors can be taken pffiffiffiffiffiffiffi to be of the form e where to six decimal places e = 2.718282 and j ¼ 1 is the imaginary unit. Here, w ¼ 2pf is the angular frequency, and f represents the temporal frequency measured in Hz. This complex exponential form of spatial and time variation of the fields is used for convenience where it is understood that the actual field quantities are obtained by taking the real part of the complex quantity, i.e., n o E ¼ E0 cos ðwtÞ ¼ Re E0 ejwt (2.10) n o H ¼ H0 cos ðwtÞ ¼ Re H0 ejwt (2.11) Although in principal it is equally valid to take the imaginary part of these complex quantities throughout, it offers no obvious advantage and is not adopted. Essentially then, and for mathematical convenience, we are simply utilising a complex representation of a real wave. When using this notation the time factor is usually suppressed, i.e. the complex exponentials are cancelled on either side of the relevant expressions, and this convention is adopted throughout. Although it is

22

Principles of planar near-field antenna measurements

conventional in electromagnetism and optics to adopt a positive time dependence,* in the study of quantum mechanics and solid-state physics, the opposite time dependence is more often adopted. Here, the term frequency domain is used to denote complex time-harmonic quantities with the ejwt factor suppressed. Thus, the plane-wave spectrum method as developed herein can be considered to be a frequency domain method where the formulas developed consider waveforms comprising a single frequency. Practically, when taking electrical measurements using a vector network analyser or receiver this means that to be able to use the formula that are developed herein we must use ratioed measurements. That is to say, we use the ratio of the measurements taken from two receivers with our equations. Here, one receiver measures the so-called “test” signal which is assumed to vary during the course of the acquisition, e.g. as a function of position; the other being the “reference” signal which is assumed to be independent and fixed for the duration of the acquisition. This “ratioed” measurement has the effect of dividing out the ejwt factor which is common to each of the two measurements and means that we obtain data that may be used directly with the frequency domain formula without the need for further alteration. We are of course free to reintroduce the time dependency at any point if we wish to visualise how the field behaves as a function of time which can be instructive. Returning to our development, clearly, as the electromagnetic field vectors are of the form ejwt , the following operator substitution can be utilised, Dn ¼

@n  ðjwÞn @tn

(2.12)

This simply states that differentiating electromagnetic field vectors with respect to time is equivalent to multiplying the field vectors by the imaginary unit and the angular velocity of the field which is assumed to be fluctuating in a sinusoidal fashion. Crucially, the simplification afforded by restricting ourselves to considering purely monochromatic waveforms in no way restricts the analysis since any angular frequency may be considered to be a component of a Fourier series, or in the limit a Fourier integral, thus enabling this analysis to be applied to arbitrary waveforms. Some of the field components contained within Maxwell’s equations can be related to one another through the properties of the medium in which the fields exist: B ¼ mH

(2.13)

D ¼ eE

(2.14)

J ¼ sE

(2.15)

* This is also the time dependency that is adopted by most commercially available vector network analyzers (VNA) and receivers.

Maxwell’s equations and electromagnetic wave propagation

23

Here, m is the magnetic permeability of the medium, e is the permittivity of the medium, i.e. the dielectric constant, and s is the specific conductivity. In general, e and m are complex tensors that are functions of field strength, however, for the case of free space antenna problems they can usually be approximated by real constants. A vacuum, in classical electromagnetic field theory, can be taken to consist of a source and sink free, simple linear homogeneous and isotropic free-space region of space in which harmonic time varying fields are measured. In such an environment, no charges are present, the current density will necessarily be zero and the resistance of the medium is infinite, i.e. zero conductivity, thus r ¼ 0, J ¼ 0 and s ¼ 0. In this case, Maxwell’s simultaneous differential equations reduce to two homogeneous, i.e. equated to zero, and two non-homogeneous expressions namely, rE ¼0

(2.16)

rH ¼0

(2.17)

r  E ¼ jwmH

(2.18)

r  H ¼ jweE

(2.19)

Eliminating the magnetic field intensity from these equations yields, r  r  E ¼ jwmðr  H Þ ¼ w2 meE

(2.20)

Thus, the most general solution of Maxwell’s equations in terms of the material constants and the angular frequency of the electromagnetic radiation is r  r  E  w2 meE ¼ 0

(2.21)

Similarly, eliminating the electric field intensity, the magnetic field can be expressed as, r  r  H  w2 meH ¼ 0

(2.22)

These expressions are often referred to as complex vector wave equations, which constitute the most general forms of the wave equation. These wave equations are usually expressed in a simpler form that is particularly convenient for problems involving Cartesian coordinate systems. Using the vector identity, r 2 A ¼ rð r  A Þ  r  r  A

(2.23)

and recalling that r  E ¼ 0 then, rðr  E Þ ¼ rð0Þ ¼ 0

(2.24)

Thus, the complex vector wave equation can be rewritten as r2 E þ w2 meE ¼ 0

(2.25)

24

Principles of planar near-field antenna measurements

This is known as the vector Helmholtz equation. Similarly, as r  H ¼ 0 the magnetic field can be expressed as r2 H þ w2 meH ¼ 0

(2.26)

Crucially, and a point that is easily (and often) overlooked, is that the vector operator substitution used to obtain the Helmholtz equation from the general wave equation is only valid in Cartesian coordinates. Thus, r2 E ðr ; tÞ and r2 H ðr ; tÞ must be calculated in terms of x, y and z. If instead the problem is cast in another coordinate system, e.g. spherical, the general wave equation must be solved instead. In general, the one-dimensional transverse wave equation can be expressed as @ 2 uðx; tÞ 1 @ 2 uðx; tÞ  2 ¼0 @x2 c @t2

(2.27)

where c is taken to denote the velocity of the wave. Assuming again that the wave is sinusoidal in form this can be expressed as d 2 uðxÞ w2 þ 2 u ðx Þ ¼ 0 c dx2

(2.28)

Through a comparison with the Helmholtz equation, we find that the phase velocity c of the electromagnetic wave can be expressed in terms of the properties of the medium through which the wave is propagating as 1 c ¼ pffiffiffiffiffi ¼ f l me

(2.29)

where f is the frequency in hertz, l is the wavelength in meters and c is the velocity in meters per second.† Here the radical, or root, is assumed positive. For convenience, a positive constant k is defined as pffiffiffiffiffi k ¼ w me (2.30) The exact reason for adopting this definition of k is presented in Chapter 4, however, for the time being the adoption of this definition of k can be justified on the basis of notational convenience. Clearly then the constant k, often termed the

† It was Heinrich Hertz (1857–1894) who provided the first experimental evidence for the existence of the electromagnetic wave as hypothesized by Maxwell and who arguably was the first antenna engineer. He achieved this by discharging an induction coil and by observing a large spark at the spark-gap transmitter, and a small spark at the receiver. By ca. 1888, he was also able to demonstrate that these EM waves: travelled in straight lines, were polarized, could be reflected by a zinc screen, were refracted by a prism made of pitch, could produce standing waves (where he was able to measure the wavelength) and verified the finite speed of light by multiplying together the wavelength and the frequency. However, he did not immediately recognize the practical applications of his work, and so it was instead Guglielmo Marconi, the inventor of commercial radio, who is credited for that innovation.

Maxwell’s equations and electromagnetic wave propagation

25

wave number or propagation constant, is simply related to the wavelength as k ¼ 2pf

pffiffiffiffiffi 2p me ¼ l

(2.31)

If the medium in which the field exists is a vacuum then, pffiffiffiffiffiffiffiffiffi k ¼ w m0 e 0

(2.32)

Here, m0 and e0 are the permeability and the permittivity of free space, respectively, and k0 is thence used to denote the free space propagation constant. Thus, the Maxwell’s equations can be transformed into the following vector Helmholtz, or wave, equations: r2 E þ k02 E ¼ 0

(2.33)

r2 H þ k02 H ¼ 0

(2.34)

The velocity of an electromagnetic wave is unambiguous when considering simple solutions, i.e. plane waves. However, as the wave equation also admits solutions representing standing waves, the concept of a velocity of an electromagnetic wave can become a little ambiguous. In 1975, the Fifteenth Confe´rence Ge´ne´rale des Poids et Mesures (CGPM), Resolution 2 (CR, 103 and Metrologia, 1975, 11, 179–180): entitled recommended value for the speed of light, adopted the speed of propagation of electromagnetic waves in vacuum as c ¼ 299792458m=s

(2.35) 9

where the estimated uncertainty is 410 . As this uncertainty principally corresponded to the uncertainty in the characterisation of the meter, in 1975, the Seventeenth CGPM decided that the meter should be defined to be the length of the path travelled by light in a vacuum during the time interval of 1/299,792,458 s. Thus, by redefining the unit of length, the velocity of light in a vacuum could be defined to be exactly 299,792,458 m/s. The choice of e or m will define a system of units. Following the redefinition of SI base units in 2019, the value of the vacuum permeability in SI units is proportional to the dimensionless fine-structure constant, m0 ¼ 1:25663706212ð19Þ  106 Henry=m

(2.36)

The classical value was m0 ¼ 4p  107 ¼ 1:2566370614:::  106 Henry=m

(2.37)

Thus, following the redefinition of SI base units, the kilogram, ampere, kelvin and mole, on the 20th of May 2019, the difference between this value of m0 and the SI (experimental) value of m0 is less than 1109 in relative value which is negligible in the context of the uncertainty budgets discussed herein. As this is determined experimentally, this needs to be reassessed periodically. Similarly, following

26

Principles of planar near-field antenna measurements

the redefinition, for the meter, kilogram and second (MKS) base system of units, the permittivity of free-space which is measured in units of farad per meter is e0 ¼ 8:8541878128ð13Þ:::  1012 Farad=m

(2.38)

The classical value was e0 ¼

1 ¼ 8:854187817:::  1012 Farad=m m0 c 2

(2.39)

So again, the difference is much smaller than the uncertainties we are likely to encounter. Returning to our development, if we assume that we are working with a Cartesian co-ordinate system, and noting that if we are to utilise the Helmholtz equation we have no choice in this matter, the electric field may be expressed as E ðx; y; zÞ ¼ b e x Ex ðx; y; zÞ þ be y Ey ðx; y; zÞ þ b e z Ez ðx; y; zÞ

(2.40)

The Laplacian operator r2 when expressed in Cartesian coordinates can be obtained from r2 ¼ r  r     @ @ @ @ @ @ b b ex þ b ey þ b ey  ex þ b ey þ b ey ¼ @x @y @y @x @y @y ¼

(2.41)

@ @ @ þ þ @x2 @y2 @z2 2

2

2

Hence, we may separate the field components and write the vector Helmholtz equation as three equivalent uncoupled scalar Helmholtz equations,  2  @ Ex ðx; y; zÞ @ 2 Ex ðx; y; zÞ @ 2 Ex ðx; y; zÞ 2 þ þ þ k0 Ex ðx; y; zÞ b (2.42) ex ¼ 0 @x2 @y2 @z2  2  @ Ey ðx; y; zÞ @ 2 Ey ðx; y; zÞ @ 2 Ey ðx; y; zÞ 2 b þ þ þ k E ð x; y; z Þ (2.43) ey ¼ 0 0 y @x2 @y2 @z2  2  @ Ez ðx; y; zÞ @ 2 Ez ðx; y; zÞ @ 2 Ez ðx; y; zÞ 2 b þ þ þ k E ð x; y; z Þ (2.44) ez ¼ 0 0 z @x2 @y2 @z2 Similar expressions also hold for Hx ðx; y; zÞ, Hy ðx; y; zÞ and Hz ðx; y; zÞ. Therefore, all of the components of the electromagnetic field obey the scalar differential wave (Helmholtz) equation, @ 2 uðx; y; zÞ @ 2 uðx; y; zÞ @ 2 uðx; y; zÞ þ þ þ k02 uðx; y; zÞ ¼ 0 @x2 @y2 @z2

(2.45)

Using vector notation, this can readily be expressed in a more compact form as r2 u þ k02 u ¼ 0

(2.46)

Maxwell’s equations and electromagnetic wave propagation

27

This differential equation can be solved by direct integration using Green’s theorem to yield the Kirchhoff integral theorem. Additionally, however, Chapter 4 will set out an alternative methodology for utilising this equation that is more convenient when considering most, but not all, problems encountered in the study of planar near-field antenna metrology. However, a methodology that will be instructive and gives insight into the nature of the electromagnetic interaction that is often successfully adopted to evaluate the electromagnetic fields produced and subsequently radiated by moving charges is to consider the fields E and B to be the resultants of potentials.

2.5 The electric and magnetic potentials 2.5.1 Static potentials The electric potential, f, is a scalar potential that is a function of position and time defined by ð 1 rðr0 Þ dt (2.47) fðr ; tÞ ¼ 4pe0 jr  r0 j where r is a point at which the potential is being evaluated and r0 is the location of the charge density element, and the static electric field is given by E ¼ rf

(2.48)

Alternatively, the magnetic potential, A, is defined as being a vector given by ð j ðr0 Þ m A ðr ; tÞ ¼ dt (2.49) 4p jr  r0 j And the static magnetic field is given by B ¼rA

(2.50)

Equations (2.47) and (2.49) for the potentials give the static electric and magnetic fields, however, as explained in Section 2.2, at any point that is spatially separated from the source of the potential a finite amount of time must pass before the influence of the source can affect the potential.

2.5.2 Retarded potentials Figure 2.6 illustrates that the potential at point P at time t is determined by the position of the potential source at position A at the earlier time of tt where the distance s = ct = [rr0 (tt)], and B is the position of the source at time t. Thus, for a distributed charge of density r the potential at position r and time t due to the charge in the vicinity of r0 depends on the value of r at the previous time

28

Principles of planar near-field antenna measurements cτ A

B

P

s r'(t-τ)

r'(t)

r O

Figure 2.6 Figure showing potential at time t from point charge moving from A to B

t-[r-r 0 ]/c; therefore, the potential of the entire charge is ð 1 rðr0 ; t  jr  r0 j=cÞ 0 dt fðr ; tÞ ¼ 4pe0 jr  r 0 j

(2.51)

This formula for the potential that is calculated to take account of the finite speed of light is referred to as the retarded scalar potential, and a similar argument can be followed to establish a retarded vector potential, ð 0 0 m j ðr ; t  jr  r j=cÞ 0 A ðr ; tÞ ¼ (2.52) dt 0 4p jr  r j Therefore, (2.48) does not apply for time-dependent systems and must be modified to take account of the finite but constant speed of light to be: E ¼ rf 

@A @t

(2.53)

Since the curl of any grad  0 this satisfies Maxwell’s third equation as @ ðr  A Þ @t @ @B r  E ¼  ðr  A Þ ¼  @t @t

r  E ¼ r  ðrfÞ 

(2.54) (2.55)

which is in agreement with (2.8). Additionally, (2.50) is still correct for timedependent systems as the div of any curl  0, r  ðr  A Þ ¼ r  B ¼ 0

(2.56)

From (2.6), e1 r  D ¼ r2 f 

@ ðr  A Þ ¼ e1 r @t

(2.57)

Maxwell’s equations and electromagnetic wave propagation

29

If 1/v2@ 2f/@t2 is inserted into (2.57), where v is the propagation velocity, r2 f 

@ 1 @2f r 1 @2f ðr  A Þ þ 2 2 ¼ þ 2 2 @t v @t e v @t

(2.58)

Rearranged to give r2 f 

1 @2f r 1 @2f @  ¼   ðr  A Þ v2 @t2 e v2 @t2 @t

(2.59)

This can be simplified to give r2 f 

1 @2f r @F ¼  v2 @t2 e @t

(2.60)

1 @f v2 @t

(2.61)

where F ¼rA þ

By substitution into (2.9) in a similar fashion, it can be shown that r2 A 

1 @2A ¼ mj þ r  F v2 @t2

(2.62)

Both of these equations can be simplified if we make F ¼rA þ

1 @f ¼0 v2 @t

(2.63)

Equations (2.60) and (2.62) then become r2 f 

1 @2f r ¼ 2 2 v @t e

(2.64)

r2 A 

1 @2A ¼ mj v2 @t2

(2.65)

Clearly if the condition set out in (2.63) where F = 0 is met, then (2.64) and (2.65) are decoupled in that (2.64) now defines f in terms of charge density without reference to current density and (2.65) does likewise for A and current density. The condition that F = 0, referred to as the Lorentz condition, can in fact always be satisfied due to the nature of the definitions of f and A. If A is transformed to A0 A ! A 0 ¼ A þ rc and f ! f0 ¼ f þ

@c @t

where c is a function of position and time.

(2.66)

(2.67)

30

Principles of planar near-field antenna measurements Since E0 ¼ E þr

@c @  rc ¼ E @t @t

(2.68)

and B 0 ¼ B þ r  ðrcÞ ¼ B

(2.69)

The use of either f or f0 or A or A0 is arbitrary as E and B will remain unchanged. Transformations such as (2.66) and (2.67) are referred to as gauge transformations and potentials that satisfy the Lorentz condition are said to belong to the Lorentz gauge. The retarded potential (2.51) and (2.52) are solutions of the decoupled (2.64) and (2.65) in combination with the Lorentz condition therefore they provide a consistent method for the solution of Maxwell’s equations. They allow the sources r and j to be the inputs that can be used to calculate f and A, which in turn allow the calculation of E and B. This may appear to be a long and convoluted methodology for the calculation of fields from their charge and current density sources, but it is usually a much easier process than attempting to evaluate E and B directly from Maxwell’s equations.

Box 2.2 In fact this procedure of moving from sources to potentials of electric and magnetic fields can be avoided if EM theory is expressed in terms of vector quantities that are themselves Lorentz invariant. And by definition, the scalar product of four vectors and four vectors with the d’Lambertian vector operator are Lorentz invariant. This requires that the quantities to be used to describe the EM interaction are all four vectors, these being: 1.

2. 3.

4.

The Interval between any two events in spacetime: R1,2 = [r1,2, c(t1-t2)]. Its four components being the ordinary three-dimensional distance vector r and the time difference. The Four-Potential: A = [A, f/c], which contains the vector potential A and the scalar electrostatic potential. The Current Density: J =[J, cr], which contain the current density and the charge density. In fact this vector describes J as a current in the three spatial dimensions and r as a current in the temporal dimension. The Propagation Vector: k = [k, w/c], composed of the ordinary propagation vector k and the frequency w of a relativistic wave.

The use of these terms means that the entire EM interaction can be summarised as   1 @2 2 1.1 r  2 2 A ¼ m0 J ¼ r2 A c @t

Maxwell’s equations and electromagnetic wave propagation

31

In terms of a four-vector A and J. This expression allows the calculation of all EM phenomena without recourse to either E or B, thus confirming the conclusions of the Aharonov Bohm thesis as to the reality of the A field and the Aharonov–Bohm effect that has been subsequently experimentally confirmed [8]. Additionally this formulation of EM theory is particularly effective in Lagrangian formulations of EM theory where the Schwarzschild invariant S = (J A) = (J.Arf) can be interpreted when integrated with respect to all four coordinates as the Action of the EM system. In fact in the seminal work [9], Feynman, Morinigo and Wagner make the comment, ‘The guts of all electromagnetism is contained in the specification of the interaction of current and field as JA0 . In Chapter 1, the action of antennas was described in terms of the observation that spatially separated physical systems that include antennas will form coupled harmonic systems. These can be used to transfer power in the form of induced voltages and currents and thus develop a signal in the Rx sub-system that contains information as to the nature of the harmonic excitation of the Tx sub-system. Thus, although the action of the antennas is described in terms of voltages and currents, the physical manifestations of f and j, developing power in spatially remote antennas as shown above, a model of the propagation between them can be constructed in terms of abstract fields and potentials.

2.6 The inapplicability of source excitation as a measurement methodology The application of this methodology that provides an algorithm for the calculation of the free-space electromagnetic field propagation away from or to an antenna structure in terms of currents and charge distributions excited on a mechanical structure is widely used in antenna design. As basic circuit theory and methods designed to cope with guided wave paths in systems are based on the concepts of current and voltage, the interface of these techniques with antenna design methods is extremely fortuitous. This is particularly so in the areas of antenna feeds where a guided wave structure must be interfaced with an antenna designed to radiate the power delivered by the feed into or out of free space. However, in terms of metrology, the concept of excitation currents on a structure has considerable limitations. The direct probing of the surface of any radiating structure to determine charge and current distributions, as a result of the constraints associated with the transfer of signal along a conducting path from any single or array of sensors, is an extremely intrusive measurement methodology. Although to some extent all measurement procedures are intrusive, in general the end result of such a RF surface probing procedure is a very considerable alteration in the existing state of the system in the course of the measurement procedure.

32

Principles of planar near-field antenna measurements

This alteration can in theory be compensated for via the use of theoretical models. However, if the measurement procedures results are essentially the results of a modelling procedure that is no more accurate than the original modelling procedure included as part of the design process, then measurements cannot be used to confirm the accuracy or effectiveness of any design. The accuracy of any compensating mathematical processing can also be brought into question due its possible inconsistency with theory. In that, as a result of space contraction, and therefore the non-invariance of volume under the Lorentz transformation, current and charge density are not separately relativistically invariant quantities. Therefore, any extrapolation from raw measurement data from only one of these to processed measured data may be done without a firm basis in the physical processes involved. However, an alternative strategy can be used to determine the radiated fields from an electromagnetic source that can be more amenable to the constraints of metrology.

2.7 Field equivalence principle The field equivalence principle is the process of replacing the actual sources that create an electromagnetic field over some closed surface, S, with equivalent sources located on that same surface. It is in fact a theoretical statement of Huygen’s principle that any wavefront can be viewed as being made up of secondary sources of spherical waves. Each point on a primary wavefront can be considered to be a new source of a secondary spherical wave and that a secondary wavefront can be constructed as the envelope of these secondary spherical waves, at the same frequency as illustrated in Figure 2.7. An illustrative computer animation of Huygen’s principle can be found at [10]. Figure 2.8 defines the field equivalence principle, where a set of electric and magnetic current sources create the radiated electric, E, and magnetic, H, fields over an arbitrary closed surface S. The wavefronts that create the radiated field E1, H1 at point P in the left-hand diagram can be alternatively created by equivalent electric and magnetic current sources Js and Jm on the surface S of the right-hand diagram, so creating the same Wave front at time t+∆t

Wave front at time t

Figure 2.7 Huygen’s principle

Maxwell’s equations and electromagnetic wave propagation

E, H

Sources

No sources •P(E1,H1)

33

E, H

Js

•P(E1,H1) Jm

Surface S

Figure 2.8 The field equivalence principle

radiated field at point P and indeed everywhere outside the enclosing surface S which now contains no sources. In order that the total field throughout the whole of the volume space (both internal and external to S) is a valid solution to Maxwell’s equations, the equivalent sources must conform to the proper boundary conditions at S between the internal and external E and H fields at the surface S as well as the radiation condition at infinity. By postulating a null field inside S, the equivalent surface currents are given by (2.70) and (2.71), J s ¼ a n  H ðs Þ

(2.70)

J m ¼ a n  E ðs Þ

(2.71)

where the unit vector an represents the surface normal and E(s) and H(s) represent the tangential electric and magnetic fields at the surface S. A particularly valuable modification to the field equivalence comes about when we note that the zero field with S cannot be disturbed by changing the material properties within S, for example that of a perfect electric conductor. In this case, at the moment the electric conductor is introduced, the electric current on the surface S, Js, is short-circuited. This leaves just the magnetic current, Jm, over the surface S and it radiates in the presence of the perfect electric conductor to give the correct fields E1 and H1 at point P in Figure 2.8. Similarly the dual of this process can be enacted such that the material properties are replaced by a perfect magnetic conductor, so short-circuiting the magnetic current, Jm, thus leading to a purely electric current Js radiating in the presence of a perfect magnetic conductor. The real utility of the field equivalence approach comes when the surface S becomes an infinity flat plane, since here the problems reduce to determining how a magnetic surface current, Jm, radiates in the presence of a flat perfect electric conductor of infinite extent. From image theory, this problem is reduced to that of Figure 2.9, where in (a) the presence of the electric conductor short-circuits Js and the removal of the infinite electric conductor by image theory (b) lead to the doubled magnetic surface current in an unbounded medium (c), from which the radiating fields can be determined to the right of the conducting plane. By duality the use of a perfect magnetic conductor reduces the problem to that of a doubled electric surface current radiating in an unbounded medium.

34

Principles of planar near-field antenna measurements Infinite electric conductor

Images

Js

Js

Js

Js=0

Jm

Jm

Jm

2Jm

S (a)

S (b)

S (c)

Figure 2.9 A application of image theory to the radiating magnetic current in presence of a perfect infinite electric conductor The above process of equivalent fields thus leads to a convenient way forward in that we can measure the electric field on an infinite plane close the antennas physical structure, from which we can derive the generating magnetic current on this scanned plane, which can then be used to determine the far-field characteristics of the antenna. This process of planar near-field measurement will be developed in the next chapter.

2.8 Characterising vector electromagnetic fields We have seen in Chapter 1 (Figure 1.1) that it is convenient to represent the radiating plane aperture of an antenna in rectangular coordinates and the radiated far-field in spherical coordinates. Although the choice of spherical coordinates for the far-field is near universal, the choice of coordinate system to represent the antenna in order to calculate its radiation pattern very much depends on the structure of the antenna. Considerable mathematical simplifications in calculating the radiation pattern of a given antenna can be achieved by the choice of a matching coordinating system. For example, the use of cylindrical coordinates to calculate the radiation of a circular open-ended waveguide offers considerable simplification over the use of a rectangular aperture coordinate frame. In this section, we will illustrate the process of determining the radiated field by considering the z directed Hertzian dipole, and this will also serve to define the concept of the plane wave. The fundamental solution for the wave equation in vector potential A (2.65) is the retarded vector potential of (2.52) at a single source point with current js is A ðr ; tÞ ¼

0 0 m j s ðr ; t  jr  r j=cÞ 4p jr  r 0 j

(2.72)

This is clearly a function of both source point and field point and defines the ‘action at a distance’ property of the electromagnetic wave. Equation (2.72) is often called the Green’s Function because by definition a Green’s function is the solution to a differential equation for a unit source. We will now consider the radiation from this

Maxwell’s equations and electromagnetic wave propagation

35

z

Ar Aϕ θ

r Aθ

ϕ

y

x

Figure 2.10 Coordinate system for infinitesimal dipole over its length l infinitesimal small (with respect to the radiation wavelength) current element, often termed the Hertzian dipole, Figure 2.10. For this dipole, we have a constant current. j s ðr0 ; t  jr  r0 j=cÞ ¼ Job a z ejkr

(2.73)

Thus, the vector potential can be written as A ¼b az

mJo l jkr e 4pr

(2.74)

Since the far-field is to be expressed in spherical coordinates, the following vector identity, see Chapter 5, 32 3 2 3 2 Ar Ax sin q cos f sin q sin f cos q 4 Aq 5 ¼ 4 cos q cos f cos q sin f  sin q 54 Ay 5 (2.75) Af  sin f cos f 0 Az yields, A ¼ ðb a r cos q  b a q sin qÞ

mJo l jkr e ¼ A r ðq; rÞ þ A q ðq; rÞ 4pr

(2.76)

As js is located in the z direction, it is independent of the far-field angle f and so d/df = 0 thus, H ¼rA

(2.77)

gives the magnetic field at the observation point r. Expressing the cross product in spherical coordinates (e.g., see reference [11]) with d/df = 0 gives:   1 @ðrAq ðq; rÞÞ @Ar ðq; rÞ b  H ¼ (2.78) af r @r @q

36

Principles of planar near-field antenna measurements Yielding only an Hf component in the far-field. For the electric field, we have E ¼

1 ðr  H Þ jwe

with Hr = Hq = d/df = 0, which yields:   1 1 @ðHf sin qÞ Er ¼ jwe r sin q @q   1 1 @ðrHf Þ Eq ¼ jwe r @r Evaluating the three non-zero field components gives:   jkJo l sin q 1 1 Hf ¼ þ 2 ejkr 4p r jkr   Zo Jo l 1 1 cos qejkr þ Er ¼ 2p r2 jkr3   jkZo Jo l sin q 1 1 1 Eq ¼ þ  ejkr 4p r jkr2 k 2 r3

(2.79)

(2.80) (2.81)

(2.82) (2.83) (2.84)

Near to the dipole 1/r2 and 1/r3 terms dominate, whereas in the far-field only 1/r terms are significant. Thus, the far-fields are given by jkJo l sin q jkr e 4pr jkZo Jo l sin q jkr e Eq ¼ 4pr Hf ¼

(2.85) (2.86)

and we note that Eq/Hf = Zo the free space wave impedance. Thus, the infinitesimal dipole radiates a locally plane wave in the radial direction r. It should be noted that computing the average power flow (Poynting vector) in the case of the near-field (1/r2 and 1/r3 terms only) results in zero power flow indicating that the field is reactive. For the far-field case, real power flow is achieved.

2.9 Summary Thus, this chapter has described a theory that describes propagation through free space as a process of the propagation of electromagnetic waves, these waves being directly related to the acceleration of charged particles. A description of these waves is then provided based on Maxwell’s equations and the derivation of the scalar Helmholtz equations. Then a description of the sources of these waves as being retarded potentials that then produce fields was provided. Although the bases of this explanation are charge and current densities on the mechanical structure that constitutes an antenna, direct measurement methods to

Maxwell’s equations and electromagnetic wave propagation

37

assess these sources are not viable, therefore an alternative equivalent fields model was developed which in turn suggested other measurement methodologies. The next chapter will go on to investigate and describe this alternative nearfield measurement technique.

References [1] J. Manners and S. Ross, Discovering Physics Block B Unit 1, Open University, 1994, pp. 9–12. [2] L. Corsor, Electromagnetism, Freeman and Co., San Francisco, CA, 1978, pp. 261–262. [3] R. Schmitt, Electromagnetics Explained, Newnes, Amsterdam, 2002, pp. 25–26. [4] R.E. Collin and F.J. Zucker, Antenna Theory, McGraw Hill, New York, NY, 1969. [5] S. Silver, Microwave Antenna Theory and Design, McGraw Hill, New York, NY, 1949. [6] J.A. Stratton, Electromagnetic Theory, McGraw Hill, New York, NY, 1941. [7] J.D. Krauss, Antennas, McGraw Hill, New York, NY, 1950. [8] A. Tonomura, The Quantum World Revealed by Electron Waves, World Scientific, Singapore, 1998. [9] R.P. Feynman, F.B. Morinigo, and W.G. Wagner, Feynman Lecturers on Physics, Addison-Wesley, Boston, MA, vol. 3, p. 36, 1995. [10] Demonstration of Huygen’s Principle, http://www.sciencejoywagon.com/ physicszone/lesson/otherpub/wfendt/huygens.htm [11] C. Balanis, Antenna Theory and Design, 2nd ed., John Wiley & Sons, New York, NY, 1997p. 920.

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Chapter 3

Introduction to near-field antenna measurements

3.1 Introduction In the previous chapter, we found that if we can determine the near-field on an infinite plane close to a radiating antenna, we can subsequently determine the radiated far-field. We shall see in subsequent chapters the mathematical process by which this transformation can be undertaken in the most efficient way and consider the limitations imposed by a finite planar scan and how such limitations can be mitigated. In this chapter, we consider the practicalities of measuring the near electric field to provide a data set that can be processed.

3.2 Antenna measurements By way of introduction, we will start by considering conventional far-field antenna measurement. The far-field radiation pattern is characterised by ● ● ●

Spatial amplitude variation Spatial phase variation Spatial polarisation variation

Ideally, these can be determined by placing the antenna under test (AUT) in a perfect plane-wave field and mechanically rotating the antenna about the relevant coordinate whilst measuring the received amplitude and/or phase. Using the coordinate frame shown in Figure 3.1 and with the antenna aperture electric field polarised in the y direction, the measurement configuration takes the form of Figure 3.1. To acquire the E-plane all that is required is to rotate the source polarisation by 90
and to rotate the AUT about the f-axis by 90
, this being shown in Figure 3.2. These two f cuts are often termed the principle plane cuts. Figure 3.3 shows a cross-polar cardinal cut which is achieved, in this case, by rotating the AUT by 90
in f about its boresight direction. Figure 3.4 contains the 45
inter-cardinal cut. Here, as the polarisation of the AUT and the incoming plane wave are no longer aligned (or orthogonal) hence the reduction in the peak level in the inset radiation pattern. Note, the general shape of the pattern is also changed as we are now no longer observing one of the principal plane cuts.

40

Principles of planar near-field antenna measurements

Antenna aperture field (E1, H1)

AUT at ϕ = 0º

In-coming vertically polarised perfect planewave (E2, H2), ψ = 90º

VNA/receiver

Position controller

0

18.000 GHz

–5 –10

Amplitude (dB)

–15

Received amplitude Pr =

{∫ 1 2

–25 –30 –35 –40

}

H-plane cut

–45

2

–50

(E1 × H2 – E2 × H1) . n^ ds

S2

–20

–55 –60

–80

–60

–40

–20

0 Theta

20

40

60

80

Angular position θ

Figure 3.1 H-plane radiation pattern acquisition

Antenna aperture field (E1, H1)

AUT at ϕ = –90º

VNA/receiver

In-coming vertically polarised perfect planewave (E2, H2), ψ = 0º

Position controller

0

18.000 GHz

–5 –10

Amp (dB)

–15

{∫ 1 2

S2

–25

–35

Received amplitude Pr =

–20

–30

E-plane cut

–40

}

(E1 × H2 – E2 × H1) . n^ ds

2

–45 –50 –55 –60

–80

–60

–40

–20

0 Theta

20

40

60

80

Angular position θ

Figure 3.2 E-plane radiation pattern acquisition By using two axes of movement, such as shown in Figure 3.5, the incoming plane wave can be fixed in polarisation in which case: ● ●

Azimuth pattern is then E-plane pattern Elevation pattern is the H-plane pattern

Introduction to near-field antenna measurements

Antenna aperture field (E1, H1)

AUT at ϕ = –90º

VNA/ receiver

41

In-coming vertically polarised perfect planewave (E2, H2), ψ = 90º

Position controller

18.000 GHz 0 –5 –10

Amp (dB)

–15 –20 –25 –30

Received amplitude Pr =

{∫ 1 2

^

–35

}

2

(E1 × H2 – E2 × H1) . n ds

S2

Cross-polar cut

–40 –45 –50 –55 –60

–80

–60

–40

–20

0 Theta

20

40

60

80

Angular position θ

Figure 3.3 Cross-polar radiation pattern acquisition

Antenna aperture field (E1, H1)

AUT at ϕ = –45º

VNA/receiver

In-coming vertically polarised perfect planewave (E2, H2), ψ = 90º

Position controller

0

18.000 GHz

–5 –10

Amp (dB)

–15 –20 –25 –30 –35

Received amplitude Pr =

{∫ 1 2

S2

–40

}

(E1 × H2 – E2 × H1) . n^ ds

2

45º inter-cardinal cut

–45 –50 –55 –60

–80

–60

–40

–20

0 Theta

20

40

60

80

Angular position θ

Figure 3.4 45
inter-cardinal radiation pattern acquisition Although the E-plane and H-plane terminology is widely used and is helpful in this context, it is, however, ambiguous when describing circularly polarised antennas. By taking azimuth cuts for many different values of elevation, one can build up a set of radiation pattern cuts and using interpolation a three-dimensional

42

Principles of planar near-field antenna measurements AZIMUTH

Co AUT -po lar E-f iel d

Incoming planewave

EL ELEVATION

AZ

Sphere surrounding the AUT

Figure 3.5 Principle plane cut acquisition using azimuth over elevation turntable

Power (dB)

–5 0

–10

–10

–15

–20

–20

–30

–25 –30

–40

–35

–50 1

–40 0.5

1 0.5

0

0

–0.5 v

–45

–0.5 –1

–1

u

Figure 3.6 Isometric view of three-dimensional radiation pattern view of the radiation pattern can be achieved. This can then be viewed as a contour or isometric plot, the latter being shown in Figure 3.6. By way of a comparison, Figure 3.7 contains a three-dimensional polar plot of the same antenna radiation pattern illustrating another style of presentation. Here, the red circle is on the xz-plane, the blue circle is on the yz-plane and the green circle is on the xy-plane. Although there are a great many ways in which the plane-wave illumination of the AUT can be achieved in practice, their mechanisms can be considered to divide into two categories, direct and indirect collimation. Those that rely upon direct collimation include free space ranges, reflection ranges, i.e. compact antenna test

Introduction to near-field antenna measurements

43

Y –5 –10 –15 –20 –25 –30 X –35 Z –40 –45

Figure 3.7 Three-dimensional polar plot of radiation pattern

Wave-fronts Source antenna Nearly plane-wave

R = range length

Figure 3.8 Plane wave created from far-field source ranges (CATR) and refraction, i.e. dielectric lens ranges. Indirect techniques include all forms of near-field ranges, i.e. planar, cylindrical and spherical. The most basic direct method is to generate the plane wave from a portion of a spherical wave front. This can be achieved by having a source antenna at a long distance from the AUT and so the AUT aperture sees a nearly plane wave when R is large, as shown in Figure 3.8. To ensure near-plane wave conditions at the AUT aperture, the phase taper across the AUT aperture is controlled to be a maximum variation of 22.5
. Referring to Figure 3.9 this can be expressed as 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2 2p @ D R2 þ  RA (3.1) Df ¼ l 2

44

Principles of planar near-field antenna measurements 'I

D

R

Figure 3.9 Far-field phase taper geometry

0 1  2 !12 2pR @ D 1þ Df ¼  1A l 2R

(3.2)

Taking the first term of the Taylor series yields, Df 

pD2 4lR

(3.3)

If Df < p8 or 22.5
, then we get the far-field distance. The value of 22.5 stems from the optical analogue having been proposed by Lord Rayleigh [1] and is empirically derived. Broadly, this equates to the minimum distance at which a null can be observed between the main beam and the first side-lobe. Thus, in practice, this leads to an optimistic assessment of the far-field distance: R

2D2 l

(3.4)

This value of R is approximately equivalent to the displacement where only one unique path exists between the AUT and the test antenna, as per Chapter 1, in the limit if only one path existed then Df = zero. However, this book is concerned with the indirect approach of illuminating the AUT via a near-field probe. To illustrate the behaviour of the fields radiating from the antenna Figure 3.10 shows how the fields gradually change as a function of distance from the reactive near-field through the intermediate region to the far-field. The right hand side of Figure 3.10 shows how the amplitude varies from the reactive near-field to the asymptotic far-field.

AUT-probe multiple reflections

R < 1.6λ

3λ

Far-field (Fraunhoffer) 2D2 4N 2 ≈ R> λ k0

Reactive near-field

r

N = k0rMRE+n1

S21 Freq = 8 (GHz)

–20

Amplitude (dB)

0

Radiating near-field (Fresnel)

Reactive near-field

–25

–30

–35 0

5

15

20

25

λ

lim (|S21|) = R→∞

Figure 3.10 Illustration of fields from a radiating antenna.

10

PR λ 2 G G = PT 4R T R

46

Principles of planar near-field antenna measurements

3.3 Forms of near-field antenna measurements Three commonly employed co-ordinate systems that are utilised for taking nearfield antenna measurement are spherical, cylindrical and planar. These systems are generally considered preferable since not only is the vector Helmholtz equation separable in each of these systems, in practice, positioner subsystems that employ them can be conveniently constructed. These systems are illustrated schematically in Figure 3.11. Here, the circles denote the positions where the near-field data would be taken when using standard canonical sampling schemes. In principle, spherical, cylindrical and planar techniques are endeavouring to derive a complex vector field function at a large distance from the antenna, from the sampling of similar complex data over a well-understood surface at a much smaller distance. This facilitates the testing of electrically large antennas in a controlled, indoor, environment. In all cases, the acquisition of the near vector field is accomplished by placing a probe at a particular position pointing in a particular direction and allowing the electric field which surrounds the probe to generate an observable excitation current. The difference in potential between the probe and a reference is sampled in phase and at quadrature. Provided that two such orthogonal complex voltages are sampled over a well-defined surface at regular intervals, the principal of modal expansion can be utilised to determine the amplitudes and phases of an angular spectrum of plane, cylindrical or spherical waves. This enables the computation of the electric and magnetic fields at any distance from the AUT, and hence the computation of the fields when infinitely far removed from the radiator, which results in a true far-field vector pattern. Despite the obvious similarities between the theoretical descriptions at the generic level, the differing geometries result in a significant divergence in the specific implementation of each. In order for the spherical near-field range (SNFR) to characterise the propagating near-field component, a test probe is held at rest, whilst the AUT is nodded in f and rotated in q, where q and f are conventional spherical co-ordinates. This results in the path of travel of the probe describing a spherical surface that is attached to the AUT. This experimental set-up is in stark contrast to the planar

Spherical near-field acquisition geometry

Cylindrical near-field acquisition geometry

Planar near-field acquisition geometry

Figure 3.11 Near-field acquisition geometries

Introduction to near-field antenna measurements

Plane bi-polar

Plane polar

47

Plane rectilinear

Figure 3.12 Planar acquisition geometries near-field scanner (PNFS) where it is the AUT, which remains at rest, whilst a small light probe is scanned across the aperture of the AUT. The cylindrical near-field scanning (CNFS) technique utilises a hybrid measurement configuration in which the AUT is rotated in azimuth (xy-plane), whilst a scanning probe is moved linearly in z. Planar measurement configuration can be subdivided into three mechanically convenient acquisition systems, which are illustrated schematically in Figure 3.12. Again, the circles denote the positions where the near-field data would be taken when using standard canonical sampling schemes. The sampled, usually electric, field components are typically chosen to coincide with the tangent to the measurement surface and are orientated such that samples are taken along paths that are parallel to lines of increasing ordinate. As stated in Chapter 2, Helmholtz equation from the general wave equation is only valid in Cartesian co-ordinates. So, strictly, the vector Helmholtz equation is not necessarily solved in either plane-polar or plane-bi-polar co-ordinate systems, rather the sampled field components are resolved onto an equivalent Cartesian polarisation basis, whereupon the transformation to the angular spectrum can be performed either directly or by use of approximation. It is possible to solve the wave equation in plane-polar coordinates [2] and bi-polar coordinates [3], but this approach offers limited advantage over the ‘cost’ of the more mathematically complicated transformation process. The plane bi-polar configuration is similar to plane-polar, except the probe sweeps out an arc in two dimensions rather than just one. Mechanically, this is the simplest positioning system to construct, as only two rotary joints are required to connect the RF output from the probe to the input of the network analyser, with no variation in phase over the scan plane. Thus, this design is cost effective; however, the transform is more complex, and the technique has remained comparatively unpopular. Probe pattern correction is complicated by the fact that the probe is rotated by a different amount for every measurement point. This can be addressed in three ways: mechanically by counter rotating the probe; algorithmically by utilising the generalised transmission formula developed below; by utilising a rotationally symmetrical probe and deploying the probe pattern correction in the usual way. We will consider this process in more detail in Chapter 5. The plane-polar scanning technique offers a similar simplification in mechanical complexity, as only a single linear probe trajectory is required. Probe

48

Principles of planar near-field antenna measurements

pattern correction is handled using one of the techniques described above for plane bi-polar geometry. Again, an increased computational complexity results from utilising a curvilinear co-ordinate system; however, the angular spectrum can be obtained from a FFT/Bessel or a Jacobi-Bessel series representation of the sampled electric field. The key benefit of this approach is that if the linear translation stage is used to acquire the near-field along a radius, then as the diameter is twice the radius, then we are able, in principle, to acquire data across disks that are larger than the test system, and potentially are larger than the enclosing test chamber. Mechanically, the plane rectilinear configuration is the most demanding, as the accuracy of the far-field pattern depends crucially upon the accurate positioning and orientation of the probe whilst sampling near-field data. The measured phase is the relative phase between the signal channel and the reference channel. However, as the electrical path of the test channel must pass through moving parts of the robotic positioner, it must not be allowed to change in electrical length for risk of disturbing the value of the measured phase. Although this is in common with all forms of near-field measurement, it is particularly challenging for planar measurements as the acquisition planes tend to be larger and the probe is displaced linearly. Rotary joints can be used to form a pantograph arrangement; however, this requires the use of a large number of rotary joints. This can be undesirable as each joint introduces loss, discontinuities, will not yield perfect phase stability, and can limit the upper frequency limit of operation. Alternatively, a more popular moving discontinuity arrangement, i.e. rolling bend, can be employed. However, in practice, maintaining the form of the discontinuity is extremely difficult, although improvements in both RF cables and rotary joints have eased this problem. On the positive side of the balance sheet, the transformation and probe pattern correction processes are straightforward and highly efficient. In conclusion, although measurements can be made utilising mechanically convenient plane-polar or plane-bi-polar geometries, these systems attain a considerable degree of mechanical simplicity at the expense of either computational efficiency or the loss of rigour, i.e. the introduction of numerical approximations, e.g. polynomial interpolation. Alternatively, measurements can be made using the mechanically inconvenient plane rectilinear geometry that yields considerable advantages in terms of computational efficiency and simplicity. Despite the mechanical difficulties inherent within the plane rectilinear geometry, its mathematical and computational simplicity has made it by far the most popular of the near-field techniques used in industry today. As there are a great many different methods for acquiring antenna pattern data, as is expounded above, a degree of care is required to insure that the most appropriate method is selected for a particular antenna. If real-time testing is required, or the antenna transmits or received a complex waveform, i.e. not a CW wave, or the phase cannot be measured, or if only pattern cuts or boresight gain sweeps are required, then a very likely direct far-field measurement technique will be appropriate, e.g. a far-field range or compact antenna test range. If the far-field distance is larger than the available space, then a CATR is probably needed. Alternatively, a near-field measurement may be viable. The choice here will primarily depend on the antenna. If the test antenna had gain larger than circa 15 dBi and if the pattern is only required over the forward half-space, e.g. out to no more than circa 80
, then

Introduction to near-field antenna measurements

49

START Overhead scanning arm SNF YES FF/QFF range NO 2D2 >r available λ

NO

Direct measurement techniques

YES

NO

YES

Can we measure phase?

Max FF angle < 80°?

Principal plane cuts, only?

YES

Planar NF system

YES

YES CATR

Real-time testing?

Directivity > ~15 dBi?

ϕ/θ Spherical NF system

NO YES Broad pattern in Az and El?

YES NO

NO

Gravitationally Sensitive AUT

NO

Cylindrical NF system

NO Indirect measurement techniques

Figure 3.13 Antenna measurement system selection decision maze one of the planar near-field methods may be appropriate. If the antenna has a fanbeam pattern, i.e. it is very broad in one axis then; a cylindrical near-field measurement will likely be required. If, however, the antenna is low gain with fullsphere pattern data required, then a spherical measurement will be needed. Here, the choice of spherical system will be governed by whether it is possible to rotate the AUT with respect to the local gravity vector. If the antenna can be rotated, then a conventional ‘f over q’ type positioning system may be used. If, however, the antenna is gravitationally sensitive, then some form of articulated spherical nearfield system will likely be needed. This decision tree, or maze, can be seen illustrated in Figure 3.13. More information on near-field antenna measurements, and antenna measurements in general, is available in the open literature, e.g. [4–7].

3.4 Plane rectilinear near-field antenna measurements Conventionally, planar near-field measurement systems operate by sampling the amplitude and phase of the propagating near-field at regular intervals, on a plaid monotonic grid over a planar surface, which is tangential to that of the antenna aperture plane, and located a few wavelengths in front of it. This arrangement is illustrated in Figure 3.14. The probe must be in the propagating near-field region, not the reactive nearfield, as evanescent coupling is omitted from the antenna–antenna coupling formulae (Chapter 5). An electrically small, i.e. a low gain, low scattering crosssection antenna, commonly referred to as a near-field probe is used to radiate at each of the pre-selected points within the planar surface. Typically, the measurements are made on a lattice that corresponds to a regular rectangular Cartesian grid along paths that are parallel with the x- and y-axes. Acting as a transmitter the field from the probe produces a quasi-spherical wave within the free-space port of the AUT. Amplitude, phase, and polarisation of the radiation from the near-field probe are held constant for all positions of the probe, and a plane wave is synthesised at the surface by the superposition of these

50

Principles of planar near-field antenna measurements

y

x Ey Ex

r'

0

Z=Zt

z

Measurement point

Antenna under test Sampling plane

Figure 3.14 Co-ordinate system for planar scanning

quasi-spherical waves. This process is repeated, but with the probe antenna rotated through 90
about the normal to the plane to form an orthogonally polarised plane wave since, as shown in Chapter 4, Section 4.6, two orthogonal tangential field components are required to determine the complete polarisation properties of the AUT. Just as the free-space port plane wave is formed through the superposition of the large number of quasi-spherical waves, the response of the AUT to the synthesised plane wave is formed by the superposition of the responses of the AUT to the quasi-spherical waves. In essence, and outside of the reactive region, any change in the spatial distribution of the field in free-space on the plane over which the probe is scanned cannot occur at a rate greater than that determined by the free-space wavelength. Therefore, at a specific frequency, spatial rates of change on the surface will all be the same if the propagation is only in one direction through the surface. Given that the only variable that can change the spatial rate of change on the plane, if the frequency is constant, is the angle at which the propagation passes through the plane and hence its projection onto the plane, each different rate of change represents propagation at a different angle and is the source of the concept of a spectrum of monochromatic plane waves all propagating through the plane at different angles. The synthesised plane wave can then be steered to other directions by linear phase shifting. This process of phase shifting and summing can be recognised as the Fourier transform and is carried out using either the discrete Fourier transform (DFT) algorithm or when appropriate, the efficient fast Fourier transform (FFT) algorithm.

Introduction to near-field antenna measurements

51

3.5 Chambers, screening and absorber The prospect of testing an antenna indoors and conducting antenna experiments in an environmentally controlled anechoic chamber is attractive, since it eliminates some of the severe problems associated with direct far-field testing. These include the cost of antenna transport, test configuration and the cost of real estate. Additional expenses are introduced through schedule disturbance caused by precipitation, temperature instability, high winds causing Tx and Rx tower movement and the more subtle climactic effects on antenna performance induced, for example, by variations in direction and level of solar radiation. These difficulties often result in the poor repeatability between measurements and, consequently, result in an inability to calibrate and correct for these uncertainties. However, using conventional far-field measurements means that the 2D2/l range length limitation restricts this to only electrically small antenna apertures. The compact antenna test range (CATR) overcomes this length limitation by using a collimating reflector to convert the spherical wave from a feed horn to a pseudo plane wave (the so-called quiet zone) that illuminates the AUT [4]. However, the need to control reflector edge diffraction and the cost of the reflector surface with rms surface accuracy of order l/100 make this a costly alternative. Alternatively, and for cases where the phase may be measured, the nearfield approach offers a controlled physical environment (such as temperature, humidity and cleanliness for space-craft applications) as well as full electromagnetic screening to avoid outside interference for the experimental arrangement, within a relatively small anechoic chamber. This full electromagnetic screening may also have beneficial effects as certain antennas, e.g. active electronically scanned arrays, may be required to be tested in transmit. Thus, the use of a screened environment may have health and safety and/or security advantages. An anechoic chamber is designed to simulate a reflection less, free-space environment for electromagnetic testing. This permits the test environment to be carefully controlled in terms of scattering (RF multipath), electromagnetic compatibility (RF noise), temperature, seismic stability, humidity and cleanliness. However, this tends to limit the maximum size of antennas that can be accommodated within the test environment. In practice, the anechoic environment is not perfect, as typical high-quality wide band radar absorbing material (RAM) has a normal incidence monostatic reflection coefficient of ca. –40 dB. The bistatic reflection coefficient of most RAM will degrade as the angle of incidence becomes larger, whilst the monostatic backscatter is always present. A measure of RAM performance is its reflectivity, which is ratio of received signal with absorber to that of signal received when absorber replaced by metal plate (Figure 3.15). Typical values of reflectivity range from: 20 dB to 50 dB for normal incidence depending on cone size relative to operational wavelength.

52

Principles of planar near-field antenna measurements T

R

θ

Figure 3.15 Measurement of RAM reflectivity

Figure 3.16 Pyramid absorbing material

Box 3.1 Pyramidal absorber, as illustrated in Figure 3.16 provides a smooth transition from free-space 377 W impedance to the 0 W impedance of the metal plate that backs the absorber. The taller the absorber the more smooth the transition and the better the reflectivity. Chambers are often constructed inside fully shielded enclosures, such as shown in Figure 3.17. They are used where a high level of magnetic, electric and microwave shielding is required. Typical shielding values when tested to IEEE299 are ● ●

100 dB @ 1–500 MHz in the electric field 100 dB @ 1–18 GHz in microwave (far-field)

Introduction to near-field antenna measurements

(a)

(b)

53

(c)

(d)

Figure 3.17 (a) Fully screened chamber, (b) details of door, (c) details of corner construction and cable entry and (d) door seal detail

3.6 RF subsystem A typical RF subsystem for a planar near-field application is shown in Figure 3.18 and is essentially a two-arm microwave interferometer where a probe and AUT are inserted into the test arm. In practice, this is realised with a standard vector network analyser (VNA) measurement system controlled via GPIB or Ethernet via a central computer (Figure 3.18). This configuration is based around a VNA operating in remote mixing mode. For

Principles of planar near-field antenna measurements Y

50 GHz PNA

AUT OEWG probe

1 2 3 4 5 6 7 8 COM

54

RF ( k02 so that the waves decay exponentially as they propagate outward in the positive z-direction and remain finite as z ! 1. If instead the opposite sign had been chosen, then this would represent an exponentially increasing wave that would become infinitely large as r became infinitely large. This can be seen to be true because if kz contained a positive imaginary component namely kz ¼ kzr þ jkzi where kzr and kzi are both real then, ejðkx xþky yþfkzr þkzi gzÞ ¼ ejðkx xþky yþkzr zÞ ekzi z

(4.43)

which clearly contains an exponential increasing amplitude factor. If, however, this solution was finite in this limit, then this solution could not be rejected. In the case of z < 0, j should be changed to +j. This topic is considered in further detail below in Chapter 5 where we consider the physical implications. Thus, we may write that the solution of the scalar Helmholtz equation is derived from the Fourier transform of the boundary conditions in a more compact form as ð ð  1 1 1
F kx ; ky ejðkx xþky yþkz zÞ dkx dky (4.44) uðx; y; zÞ ¼ 2 4p 1 1 This is clearly a physically satisfying result. By way of illustration let us consider the behaviour of the free space Greens function, as in the limit of the farfield it behaves in a manner which is similar to that of a general far-field antenna pattern function. The free space Greens function, which is a radial spherical mode and experiences a soft singularity when r ¼ 0, can be expressed as y¼

ejk0 r r

(4.45)

Clearly, as the amplitude of the free space Greens function reduces by an amount that is inversely proportional to distance, this implies that the rate of change of amplitude with distance will tend to zero in the far-field as the distance tends to

80

Principles of planar near-field antenna measurements Y-axis

P Far field point

Δz θ Z-axis Z0 1st measurement plane

2nd measurement plane

Figure 4.1 Schematic representation of plane-to-plane translation infinity. Conversely, the rate of change of the phase of the free space Greens function is obviously independent of distance as it is a constant. Consequently, the corresponding angular spectra of two coplanar field distributions, i.e. two antenna apertures, spaced apart by a few wavelengths in z will differ only by a phase factor. Displacing one plane by a distance z0 in the z-axis will result in a change in path length of z0 cos q where q is the polar angle, i.e. that angle measured away from the positive z-axis, cf. Figure 4.1. The free-space electrical length, i.e. the phase, is related to the physical length l by, f ¼ k0 l

(4.46)

where f denotes the electrical length. Thus, the difference in phase between the two angular spectra will be characterised by f ¼ k0 z0 cos q ¼ kz z0

(4.47)

which is in agreement with our rigorous analysis presented above. By way of a check, we may substitute the integral solution into the left-hand side of the scalar Helmholtz equation and check to ensure that the sum of the terms is zero. This yields,

ð ð  jðkx xþky yþkz zÞ @2 1 1 1
F kx ; ky e S¼ 2 dkx dky @x 4p2 1 1

ð ð  jðkx xþky yþkz zÞ @2 1 1 1
F kx ; ky e dkx dky þ 2 @y 4p2 1 1 (4.48)

ð ð  jðkx xþky yþkz zÞ @2 1 1 1
F kx ; ky e dkx dky þ 2 @z 4p2 1 1

ð ð  jðkx xþky yþkz zÞ 1 1 1
e F k ; k dk dk þk02 x y x y 4p2 1 1

Plane-wave spectrum representation of electromagnetic waves

81

where if this solution satisfied the wave equation S, which represents the sum of these terms, will be identically zero. Utilising the operator substitution for the derivative of a Fourier transform obtains, S¼

kx2 ky2

1 4p2 1 4p2

1 kz2 2 4p þk02

1 4p2

ð1 ð1 1 1

ð1 ð1

1 1

ð1 ð1

1 1 ð1 ð1 1 1


 F kx ; ky ejðkx xþky yþkz zÞ dkx dky
 F kx ; ky ejðkx xþky yþkz zÞ dkx dky




F kx ; k y e

(4.49) jðkx xþky yþkz zÞ

dkx dky


 F kx ; ky ejðkx xþky yþkz zÞ dkx dky

Whereupon, dividing by the Fourier transform of the solution of the scalar Helmholtz equation yields, S ¼ k02  kx2  ky2  kz2

(4.50)

Clearly then, S ¼ 0 and hence these solutions satisfy the scalar Helmholtz equation. This can be seen to follow from the fact that each component wave satisfies Maxwell’s equations, consequently any resultant obtained by superposition will likewise satisfy the field equations. Although, as illustrated earlier, the Helmholtz equation is a direct consequence of Maxwell’s equations, the converse is not true. Consequently, care must be taken to ensure that any solution of the wave equation also constitutes a solution of Maxwell’s equations. An alternative, more conventional, but ultimately less universally applicable technique for solving the scalar Helmholtz equation is presented in Box 4.2.

Box 4.2 As shown earlier, the scalar Helmholtz equation in rectangular co-ordinates is @ 2 uðx; y; zÞ @ 2 uðx; y; zÞ @ 2 uðx; y; zÞ þ þ þ k 2 uðx; y; zÞ ¼ 0 @x2 @y2 @z2

(4.51)

The method of separation of variables seeks to find a solution of the form of, uðx; y; zÞ ¼ X ðxÞY ðyÞZ ðzÞ

(4.52)

82

Principles of planar near-field antenna measurements

Substituting this expression into the Helmholtz equation yields, @ 2 ðX ðxÞY ðyÞZ ðzÞÞ @x2 2 @ ðX ðxÞY ðyÞZ ðzÞÞ þ @y2 2 @ ðX ðxÞY ðyÞZ ðzÞÞ þ @z2

0¼

(4.53)

þ k 2 X ðxÞY ðyÞZ ðzÞ Or, 0 ¼ Y ðyÞZ ðzÞ

@ 2 X ðx Þ @x2

þ X ðxÞZ ðzÞ

@ 2 Y ðy Þ @y2

þ X ðxÞY ðyÞ

@ Z ðzÞ @z2

(4.54)

2

þ k 2 X ðxÞY ðyÞZ ðzÞ Dividing by the solution yields, 0¼

1 @ 2 X ðx Þ X ðxÞ @x2 1 @ 2 Y ðy Þ þ Y ðyÞ @y2 1 @ 2 Z ðzÞ þ Z ðzÞ @z2 þ k2

(4.55)

Consider rearranging the above expression to obtain, 1 @ 2 X ðx Þ 1 @ 2 Y ðy Þ 1 @ 2 Z ðzÞ 2 ¼ k   X ðxÞ @x2 Y ðyÞ @y2 Z ðzÞ @z2

(4.56)

The left-hand side is a function of x alone, whilst the right hand depends upon y and z only. So a function of x is equated to a function of y and z; however x, y and z are all independent variables. This can only be true if each side is equal to a constant, a constant of separation. Thus let, 1 @ 2 X ðx Þ ¼ kx2 X ðxÞ @x2

(4.57)

Plane-wave spectrum representation of electromagnetic waves 1 @ 2 Y ðy Þ ¼ ky2 Y ðyÞ @y2

(4.58)

1 @ 2 Z ðzÞ ¼ kz2 Z ðzÞ @z2

(4.59)

where kx2 ; ky2 ; kz2 are referred to as separation constants and for convenience they are chosen to be the square of another constant. By substituting these expressions into the expression, we obtain the relationship between the separation constants, kx2  ky2  kz2 þ k 2 ¼ 0

(4.60)

k 2 ¼ kx2 þ ky2 þ kz2

(4.61)

The separated ordinary differential equation may be written in a more convenient form as @ 2 X ðx Þ @ 2 Y ðy Þ þ kx2 X ðxÞ ¼ þ ky2 Y ðyÞ 2 @x @y2 @ 2 Z ðzÞ þ kz2 Z ðzÞ ¼ @z2 ¼0

(4.62)

As no derivatives with respect to x or y appears, we have succeeded in reducing a partial differential equation to an ordinary differential equation. These expressions are all of the same form and are referred to as harmonic equations. Any solution of the harmonic equation we shall call a harmonic function: @ 2 U ðu Þ þ ku2 U ðuÞ ¼ 0 @u2

(4.63)

One possible solution of this differential equation is ejku u . As any linear combination of harmonic functions is also a harmonic function, more general wave functions may be obtained by summing over all possible choices for the two independent separation parameters, i.e. uðx; y; zÞ ¼

1 X

1 X


 F kx ; ky ejkx x ejky y ejkz z

(4.64)

kx ¼1 ky ¼1

Finally, the most general wave functions can be constructed by integrating over all values of the separation parameters, i.e. ð1 ð1
 F kx ; ky ejkx x ejky y ejkz z dkx dky (4.65) uðx; y; zÞ ¼ 1 1

83

84

Principles of planar near-field antenna measurements

4.4 On the choice of boundary conditions Within the above derivation, it has been assumed that the aperture illumination function f ðx; yÞ is known over the entire z = 0 plane, and is small when x and y are large. In practice, all real sources, i.e. sources that occupy a finite region of space and radiate finite power, must be of finite extent so that their illumination function can be expressed mathematically with no loss of accuracy,  f ðx; yÞ When; x1  x  x2 and y1  y  y2 (4.66) f ðx; yÞ ¼ 0 Elsewhere Clearly then, the limits of integration can be collapsed from infinite to finite values as ð y 2 ð x2
 f ðx; yÞejðkx xþky yÞ dxdy F kx ; ky ¼ =ff ðx; yÞg ¼ (4.67) y2

x1

Now, depending upon which of the field components f is taken to represent can result in the imposition of various characteristics on the resulting angular spectrum. The most commonly used choice is to assume that f is taken in turn to denote each of the two orthogonal electric field components that are tangential to the antenna aperture plane so that the following Fourier transform pairs are formed,
 (4.68) Fx kx ; ky z ¼ 0 , fx ðx; y; z ¼ 0Þ
 Fy kx ; ky z ¼ 0 , fy ðx; y; z ¼ 0Þ

(4.69)

In this case, the boundary conditions are akin to specifying that the aperture illumination function is set in an infinite, perfectly conducting ground plane, i.e. a perfect electrical conductor (PEC). Thus, outside the aperture, i.e. on the infinitely conducting plane the following relationship holds: b n  E ðx; y; z ¼ 0Þ ¼ 0

(4.70)

Here b n is taken to denote the surface unit normal, which in this example has a component purely directed in the positive z-axis. Thus, it is more convenient to determine the total electric and magnetic field vectors from the tangential components of the electric field, rather than from the magnetic field. However, outside the aperture on the surface of the infinitely conducting material the normal electric field component will not necessarily be identically zero, i.e. b n  E ðx; y; z ¼ 0Þ 6¼ 0

(4.71)

Plane-wave spectrum representation of electromagnetic waves

85

Thus, if the normal component of the angular spectrum is obtained directly from the Fourier transform of the normal electric field component the limits of integration will not collapse to the area of the aperture and instead must be obtained from, ð1 ð1
 (4.72) fn ðx; yÞejðkx xþky yÞ dxdy F n kx ; ky ¼ 1 1

If the limits of integration are reduced to a finite region of the x–y plane, then this is necessarily equivalent to setting the aperture in a perfect magnetic conductor (PMC) where b n  H ðx; y; z ¼ 0Þ ¼ 0

(4.73)

b n  E ðx; y; z ¼ 0Þ ¼ 0

(4.74)

and,

The introduction of this additional, and often unnecessary, assumption can cause confusion when comparing values for the normal field component obtained by applying the plane-wave condition and from direct integration. These differences are generally more pronounced for wide out pattern angles.

4.5 Operator substitution (derivative of a Fourier transform) Within the development of the PWS representation, a general operator substitution was utilised within the integral transform method of solution of the scalar Helmholtz equation which is a second-order differential equation, i.e. a hyperbolic equation where n = 2. The derivation of this operator substitution introduces some additional restrictions that are not generally noted within the literature, and that have an importance for non-coplanar analysis. Provided that the reader is prepared to accept the operator substitution,  n @ uðx; tÞ ¼ ðjsÞn U ðs; tÞ (4.75) = @xn ðx;tÞ , and all lower derivatives, and only utilise this relationship in cases where @ @xun1 exist and tend to zero as jxj ! 1, Box 4.3 may be omitted. n1

Box 4.3 Let the Fourier transform of the solution uðx; tÞ with respect to x be, ð1 U ðs; tÞ ¼ =fuðx; tÞg ¼ uðx; tÞejsx dx (4.76) 1

86

Principles of planar near-field antenna measurements We shall be required to assume that the solutions uðx; tÞ and its derivaðx;tÞ are small for large values of jxj and approach zero as x ! 1. Let: tive @u@x @uðx; tÞ ux ðx; tÞ ¼ (4.77) @x Consider, =fux ðx; tÞg ¼

ð1 1

ux ðx; tÞejsx dx

(4.78)

By using the method of definite integration by parts when u ¼ ejsx then Ð jsx du ¼ jse dx and dv ¼ ux ðx; tÞdx then v ¼ ux ðx; tÞdx ¼ uðx; tÞ we obtain that, ð1 ð1 udv ¼ ejsx uðx; tÞj1  uðx; tÞjsejsx dx (4.79) =fux ðx; tÞg ¼ 1 1

1

As uðx; tÞ ! 0 when x ! 1 then, ð1 uðx; tÞejsx dx ¼ jsU ðs; tÞ =fux ðx; tÞg ¼ js

(4.80)

1

Hence,  @uðx; tÞ ¼ jsU ðs; tÞ = @x Now, a repeated application of this formula will yield,  2 @ uðx; tÞ = ¼ s2 U ðs; tÞ @x2

(4.81)

(4.82)

where we have been forced to assume that ux ðx; tÞ ! 0 when x ! 1. Through the repeated application of the formulae above the general operator substitution can be obtained and written as  n @ uðx; tÞ ¼ ðjsÞn U ðs; tÞ: (4.83) = @xn ðx;tÞ and all lower derivatives exist and are small The assumption that @ @xun1 for large values of jxj and approach zero as x ! 1 is strictly adhered. This can readily be extended to two or more dimensions to obtain the general operator substitution quoted in Section 4.3.2. n1

4.6 Solution of the vector Helmholtz equation in Cartesian co-ordinates As both Maxwell’s equations and the Helmholtz equation must be satisfied, some additional restrictions are necessarily placed upon any possible solution. The

Plane-wave spectrum representation of electromagnetic waves

87

implications of this can be investigated further by substituting the above Fourier integral solution into r  E ¼ 0. Thus, the divergence of the electric field can be expressed in terms of the general solution as

ð ð
 jðkx xþky yþkz zÞ @ 1 1 1 F kx ; ky e ex 0¼ dkx dky  b @x 4p2 1 1

ð ð
 jðkx xþky yþkz zÞ @ 1 1 1 (4.84) e þ F k ; k dk ey dk x y x y b @y 4p2 1 1

ð ð
 jðkx xþky yþkz zÞ @ 1 1 1 þ F kx ; ky e dkx dky  be z @z 4p2 1 1 Essentially then we are attempting to solve Maxwell’s equations by trying to obtain a solution of the wave equation (which is the Fourier transform part of the above equation) whose divergence is zero. The divergence of the field is zero because we are attempting to find a solution in free space thus no ‘sources’ or ‘sinks’ are present. So, utilising the operator substitution for the derivative of a Fourier transform yields, ð1 ð1
 j Fx kx ; ky ejðkx xþky yþkz zÞ dkx dky 0 ¼ 2 kx 4p 1 1 ð1 ð1
 Fy kx ; ky ejðkx xþky yþkz zÞ dkx dky þky 1 1

þ kz

ð1 ð1

1 1


 Fz kx ; ky ejðkx xþky yþkz zÞ dkx dky Þ

(4.85)

Differentiating both sides with respect to kx ; ky and dividing by common factors obtains kx Fx ðkx ; ky Þ þ ky Fy ðkx ; ky Þ þ kz Fz ðkx ; ky Þ ¼ 0

(4.86)

When rewritten using vector notation this can be expressed succinctly as k F ¼0

(4.87)

Alternatively, this can be expressed in an equivalent form in terms of the radial spherical unit vector, b er  F ¼ 0

(4.88)

Hence, only two field components may be specified independently, the third being fixed since, Fz ðkx ; ky Þ ¼ 

kT  FT ðkx ; ky Þ kx Fx ðkx ; ky Þ þ ky Fy ðkx ; ky Þ ¼ kz kz

(4.89)

Thus, the longitudinal spectral component is derived from the two-dimensional aperture field by enforcing this plane-wave condition. This requires special

88

Principles of planar near-field antenna measurements

treatment when the propagation vector lies within the x–y-plane since the denominator tends to zero (kz =0). Thus at first sight it would appear that the longitudinal electrical field component would become infinite. By examining the rate of change of the numerator and the denominator in this limit, the value of the longitudinal field component is found to be finite over all space. This can be shown from L’Hoˆpital’s rule which states that in the limit the ratio of the numerator and the denominator is equal to the ratio of the derivatives of the numerator and the denominator,  0  f ðx Þ f ðx Þ (4.90) ¼ lim 0 lim x!a g ðxÞ x!a g ðxÞ Thus when expressed in terms of spherical co-ordinates L’Hoˆpital’s rule, when applied to the longitudinal spectral component, can be seen to be,  lim fF g ¼ lim  sin q cos jEx þ sin q sin jEy z q!p=2 cos q q!p=2 8 9 d
> = < sin q cos jEx þ sin q sin jEy > (4.91) ¼ lim  dq d > q!p=2 > ; : ðcos qÞ dq Hence, the angular spectrum remains finite over all space as  cos q cos jEx þ cos q sin jEy lim fFz g ¼ lim ¼0 sin q q!p=2 q!p=2

(4.92)

By using the plane-wave condition, it is possible to write the total, i.e. Ex, Ey, Ez, electric field in free space purely in terms of the tangential spectral components as
# ð ð "
 k T  FT k x ; k y 1 1 1 ejðkx xþky yþkz zÞ dkx dky E ðx; y; zÞ ¼ 2 F T kx ; k y  b ez 4p 1 1 kz (4.93) where
 F T kx ; ky ; z ¼ 0 ¼

ð1 ð1 1 1

ET ðx; y; z ¼ 0Þejðkx xþky yÞ dxdy

(4.94)

Thus, it is crucial that all spectral components remain finite for all values that kx and ky can take. If this were not the case, due to the antireductionist Fourier relationship between the spatial and spectral domains, the longitudinal component of the electric field would become infinite everywhere in space which is obviously unsatisfactory. As will be demonstrated below, the finiteness of the PWS can be illustrated when expressing these relationships in spherical coordinates where the one over cosine of theta term vanishes. This also provides a computationally stable platform to integrating across the unit-circle which would otherwise cause a singularity within the integral. This is of utility when computing fields that contain

Plane-wave spectrum representation of electromagnetic waves

89

non-homogeneous plane wave spectra, e.g. in some close coupling applications such as when evaluating the mutual coupling between radiating elements within finite arrays.

4.7 Solution of the vector magnetic wave equation in Cartesian co-ordinates It is important to note that as the scalar wave equation, r2 u þ k02 u ¼ 0, is equally valid for both electric and magnetic fields, then exactly the same integral transform procedure that was utilised to solve the electric wave equation can be used to solve the magnetic wave equation. Hence, by an exchange of the relevant variables, it can be shown that it is possible to write the total, i.e. Hx, Hy, Hz, magnetic field in free space purely in terms of the tangential magnetic spectral components as
# ð ð "
 kT  GT kx ; ky 1 1 1 ejðkx xþky yþkz zÞ dkx dky H ðx; y; zÞ ¼ 2 GT kx ; ky  ^e z 4p 1 1 kz (4.95) where
 GT kx ; ky ; z ¼ 0 ¼

ð1 ð1 1 1

HT ðx; y; z ¼ 0Þejðkx xþky yÞ dxdy

(4.96)

and are the magnetic spectral components which are the direct analogue of the electric spectral components F. Clearly then all observations made regarding the nature of the integral transform solution of the electric field are equally valid for the integral transform solution of the magnetic field. It will be shown below that a simple relationship exists between the electric spectral components and the magnetic field thus the magnetic spectra are not often considered. This is used below, however, when computing power.

4.8 The relationship between electric and magnetic spectral components The magnetic field anywhere in the forward half-space can be obtained from the angular spectra by using Maxwell’s equation, H ðx; y; zÞ ¼ 

1 r  E ðx; y; zÞ jwm

(4.97)

to determine the magnetic field from the electric field. Hence, the magnetic field can also be determined from the spectral components. Correspondingly, the specification of two orthogonal tangential electric field components over the surface of a plane uniquely determines the magnetic field and thus the entire electromagnetic

90

Principles of planar near-field antenna measurements

six-vector is uniquely specified, i.e. ð ð
 1 1 1 E ðx; y; zÞ ¼ 2 F kx ; ky ejk  r dkx dky 4p 1 1 ð ð
 1 1 1 1 r 2 H ðx; y; zÞ ¼  F kx ; ky ejk  r dkx dky jwm 4p 1 1

(4.98) (4.99)

Expressing the magnetic field in terms of the magnetic spectra yields, ð ð
 1 1 1 1 r G kx ; ky ejk  r dkx dky ¼  2 4p 1 1 jwm ð ð
 1 1 1  2 F kx ; ky ejk  r dkx dky 4p 1 1 (4.100) Exchanging the order of integration and differentiation and cancelling like terms yields, ð1 ð1 ð ð
 jk  r 1 1 1 G kx ; ky e r dkx dky ¼  jwm 1 1 1 1
  F kx ; ky ejk  r dkx dky

(4.101)

In order that this can be converted into a more convenient form, let us first consider the divergence of the electric spectra namely, 2 3 b ex b ey b ez 6@ @ @ 7 7 rA ¼6 4 @x @y @z 5 Ax A y A z





@Ay @Ax @Az @Ay @Az @Ax  b  þb  (4.102) ¼b ex ey ez @y @z @x @z @x @y where for convenience the new vector A has been defined to be,
 A ¼ F kx ; ky ejk  r (4.103)
 Here, F kx ; ky is a constant with respect to the variables x, y and z. Hence,

 rA ¼b e x jky Az þ jkz Ay  b e y ðjkx Az þ jkz Ax Þ þ b e z jkx Ay þ jky Ax h

i r  A ¼ jk0 b e x bAz  gAy  b e y ðaAz  gAx Þ þ b e z aAy  bAx

(4.104)

(4.105)

Plane-wave spectrum representation of electromagnetic waves 2

be x r  A ¼ jk0 4 a Ax

b ey b Ay

3 b ez g 5 Az

91

(4.106)

Since the unit vector in the radial direction is b e r ¼ ab e x þ bb e y þ gb ez

(4.107)

the curl of the electric field is related to the electric field through a 90 phase change, a linear scaling of amplitude, and by taking a cross product with the propagation vector as r  A ¼ jk0b e r  A ¼ jk  A

(4.108)

Or


 
 r  F kx ; ky ejk  r ¼ jejk  r k  F kx ; ky

Thus, ð1 ð1 1


 1 G kx ; ky ejk  r dkx dky ¼ wm 1

(4.109)

ð1 ð1 k 1 1


  F kx ; ky ejk  r dkx dky Differentiating both sides with respect to kx and ky yields,

 1 G kx ; ky ejk  r ¼ k  F kx ; ky ejk  r wm

(4.110)

(4.111)

or

 1 G kx ; ky ¼ k  F kx ; ky wm

(4.112)

Thus, the electric and magnetic spectra are related to one another through a cross product with the propagation vector and a simple linear scaling. Taking the inverse Fourier transform of this will yield an expression for the magnetic field in terms of the electric angular spectra hence, ð1 ð1
 1 H ðx; y; zÞ ¼ 2 k  F kx ; ky ejk  r dkx dky (4.113) 4p wm 1 1 Or when expressed in terms of the tangential components this becomes, "
# ð1 ð1
 kT  F T kx ; ky 1 ez H ðx; y; zÞ ¼ 2 k  F T kx ; ky  b 4p wm 1 1 kz jðkx xþky yþkz zÞ dk dk e x

y

(4.114)

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Principles of planar near-field antenna measurements

These expressions thus predict the field everywhere in space once the field has been specified over a plane. Within the formulation of this transform method, an assumption is made that the region under consideration is a source or sink free, simple homogeneous and isotropic free-space region where a source is taken to represent a current in time. Hence, although it is possible to derive the fields for the whole space using spectral techniques, those results obtained for the region z < 0 will be in error, as this is conventionally taken to be behind the antenna aperture plane. That region of space directly behind the antenna aperture plane contains the radiating structure and will necessarily contain both sinks and sources. Thus in practice, these expressions are limited to specifying the field distribution within only the forward half space, i.e. where z > 0. Clearly the electric spectra can be obtained from the magnetic spectra. The transverse nature of the spectral components can be readily established by substituting the respective Fourier solutions into the two remaining Maxwell equations. Thus, since we have previously shown that r  E ðx; y; zÞ ¼ 0 implies that,
 (4.115) k  F kx ; ky ¼ 0 and similarly, r  H ðx; y; zÞ ¼ 0 will imply that,
 k  G kx ; ky ¼ 0

(4.116)

Thus both the electric and magnetic spectral components are transverse, i.e. no component of F or G lies in the direction of propagation k.

4.9 The free space propagation vector k The free-space propagation constant k was introduced in Chapter 2 as a convenient way of bundling together the angular frequency and the material properties of the medium through which the field is propagating. In this section, we derive the length of the propagation vector and show that it has a magnitude equal to the free space propagation constant. In other words, for a wave to propagate, the electric and magnetic fields must be related in a way that is dependent upon the properties of the material through which the fields exist. Motivated by this, let us first consider the relationship between the electric and magnetic angular spectra namely,

 1 G kx ; ky ¼ k  F kx ; k y wm

(4.117)

Taking the vector cross product of another vector, k, and this term yields, 

 (4.118) k  k  F kx ; ky ¼ wmk  G kx ; ky Using the vector identity, A  ðB  C Þ ¼ B ðA  C Þ  C ðA  B Þ

(4.119)

Plane-wave spectrum representation of electromagnetic waves

93

yields,





 k  k  F k x ; k y ¼ k k  F k x ; k y  F k x ; k y ðk  k Þ

(4.120)

However as k  k ¼ jk j2 ¼ k02 and as shown previously, k  F ¼ 0 then,

 F kx ; ky k02 ¼ wmk  G kx ; ky (4.121) Recalling that,

 1 F kx ; ky ¼  k  G kx ; ky we

(4.122)

Then clearly,

 F kx ; ky k02 ¼ w2 meF kx ; ky Factorising yields,
2 
 w em  k02 F kx ; ky ¼ 0

(4.123)

(4.124)




Hence, for nontrivial solutions, i.e. solutions for which F kx ; ky 6¼ 0, this is satisfied when, k02 ¼ w2 em

(4.125)

Hence, the vector k has length |k|, pffiffiffiffiffi 2p ¼ jk j k0 ¼ w em ¼ l

(4.126)

Where again the positive radical is chosen.

4.10 Plane-wave impedance The expressions derived above can be conveniently simplified if it is noted that the ratio of E and H for a transverse electromagnetic (TEM), i.e. a plane, wave is a constant with units of ohms namely, rffiffiffiffiffi E m0 ¼ 376:7 W (4.127) Z0 ¼ H e0 Thus, the magnetic field can be expressed more conveniently in terms of the electric field and the propagation vector as ð1 ð1
 1 H ðx; y; zÞ ¼ 2 k  F kx ; ky ejk  r dkx dky (4.128) 4p k0 Z0 1 1 or simply, H ðx; y; zÞ ¼

1 4p2 Z0

ð1 ð1 1 1


 be r  F kx ; ky ejk  r dkx dky

(4.129)

94

Principles of planar near-field antenna measurements

Box 4.4 A useful scaling factor can be obtained by examining the ratio of the planewave solutions of the electric and magnetic fields. Consider a purely x-polarised plane-wave propagating in the direction of the positive z-axis thus, E ¼ jE jb e x ¼ E0 ejkz zb ex : Thus, the corresponding magnetic field would be,  1 k0

k0 b E0 ejkz zb H ¼ ðk  E Þ ¼ e z  E0 ejkz zb ex ¼ ey : wm wm wm

(4.130)

(4.131)

Now consider the ratio of the electric and magnetic field vectors for a TEM wave namely, C¼

E k E E ¼ ¼ T: H k H HT

(4.132)

Here, E is the electric field intensity in units of volts per meter and H is the magnetic field intensity in units of amps per meter. ET and HT are the tangential components of the electric and magnetic fields, respectively. Thus, as C is the ratio of transverse (TEM waves have no electric or magnetic field component in the direction of propagation) orthogonal components of E and H it has units of volts per amp which, by definition, is an ohm, i.e. a measure of resistance or more generally, impedance. Explicitly then this ratio, which has units of ohms, is the impedance that a plane wave would experience when propagating in a perfect vacuum when the electric and magnetic field vectors are oscillating in phase, i.e. TEM, is usually designated the symbol Z0 where rffiffiffi E E0 ejkz z wm 2pf lm m m Z0 ¼ ¼ k0 ¼ cm ¼ pffiffiffiffiffi ¼ : (4.133) ¼ ¼ jk z z H k0 2p em e wm E0 e Thus, Z0 is the impedance of free space, has units of ohms, and in general is a function of the properties of the material in which the field exists. Classically, the impedance of free space can be defined exactly in terms of p and the speed of light in a vacuum, c. When quoted to nine decimal places Z0 is Z0 ¼ 376:730313461 W: However, this is often quoted approximately as rffiffiffi m pffiffiffiffiffiffiffiffiffi ¼ m2 c2 120p: Z0 ¼ e

(4.134)

(4.135)

Plane-wave spectrum representation of electromagnetic waves

95

Following the SI redefinition, the presently accepted value for the impedance of free space is Z0 ¼ 376:730313668ð57Þ W:

(4.136)

where the impedance of free space is subject to experimental measurement uncertainty as only the speed of light in vacuum c has an exact defined value. Conversely, if the electric and magnetic fields are not orthogonal and are out of phase with one another, the wave impedance will contain a non-zero reactive component thus this quantity represents an impedance, not a resistance. The inverse of impedance is admittance. Thus, the plane-wave admittance of free space Y0 is Y0 ¼

1 Z0

(4.137)

and is measured in units of ohm1 (W1) which are more often termed siemens (S).

4.11 Interpretation as an angular spectrum of plane waves Classically, a wave can be thought of as being a complex function of two (or more) co-ordinates of space and time. It is possible to classify a wave by noting the nature of the surface over which the phase remains constant. Waves are referred to as being plane if their equal phase surfaces are planar. Each of the infinite numbers of perpendiculars to these surfaces is called a wave normal and defines the direction in which the wave propagates. If Fðx; y; zÞ ¼ constant describes an equal phase surface, then rFðx; y; zÞ will be the direction in which the phase varies most rapidly. This quantity is referred to as the vector phase constant t . By convention, this will be the direction in which the phase decreases most rapidly, i.e. t ¼ rF

(4.138)

Applying this to the term F ejk  r within the integrand above where k is real yields, t ¼ rðk  r Þ ¼ k

(4.139)

Here the quantity F ejk  r can be thought of as representing a uniform homogeneous plane wave of amplitude F , propagating in the direction k, and as shown above, constitutes an exact solution of Maxwell’s fundamental field equa  tions. The function F kT can therefore be thought of as a superposition of plane waves travelling in different directions with different amplitudes, i.e. an ‘angular spectrum’ of plane waves or, a ‘PWS’. When k is complex, however, i.e.

96

Principles of planar near-field antenna measurements

k ¼ b  ja , then the phase constant will also become complex as t ¼ rðk  r Þ ¼ k ¼ b  ja

(4.140)

Hence,

F ejk  r ¼ F e

j

b  ja

 

r

¼Fe

jb

r

ea  r

(4.141)

This represents equal-phase surfaces that are perpendicular to b and equal amplitude surfaces that are perpendicular to a i.e. in-homogeneous plane waves where the wave propagates in the direction b and are attenuated in the direction a . Here, the position vector is described by r ¼ x^x þ y^y þ z^z

(4.142)

and the direction of propagation is

 k ¼ kx^x þ ky^y þ kz^z ¼ k0 u^x þ v^y þ w^z

(4.143)

Where in spherical co-ordinates, u; v; w are the direction cosines sin q cos f, sin q sin f and cos q respectively, where q is the polar, or zenith angle, and f is the azimuthal angle. This is represented schematically in Figure 4.2. This, therefore, is the basis for the principle that ‘the electric or magnetic field at any point in space can be constructed from an infinite number of plane-waves of different amplitudes propagating in different directions’. It is of interest to note that although the interpretation of the particular solution in transform space as a PWS is of utility, this interpretation was not relied upon within the above derivation. Consequently, the question as to the existence of the PWS and the physical reality

z

Equiphase surface

(x,y,z) r θ

(kx,ky,kz )

k y

x

ϕ

Figure 4.2 Co-ordinate system for plane-wave propagation

Plane-wave spectrum representation of electromagnetic waves

97

of unbounded plane waves in general bears little relevance to this discussion. This is crucial, as the infinite plane wave is physically impossible since the total energy transported across an equiphase surface would be infinite. The attraction of the PWS method stems from the relative simplicity with which plane waves can be expressed and manipulated mathematically as building blocks to synthesise real arbitrary field distributions. The relation between the angular spectrum and the aperture distribution involves no approximations or aperture size restrictions. Consequently, the PWS method is inherently more accurate than other older theories, as the approximations that must inevitably be introduced occur at a later stage of the analysis, i.e. during the asymptotic derivation of far-field parameters. The field produced by a bounded current distribution is in any direction outgoing when infinitely far removed from the source. Thus, a single analytic representation as a superposition of plane waves travelling in diverse directions cannot be valid over more than a half-space. A plane wave that propagates outward in a given direction in one half-space will necessarily travel inwards in the opposite direction in the other half-space. Similarly, unbounded inhomogeneous plane waves that decay exponentially in one direction are matched by an exponential growth in the opposite direction that eventually becomes infinite. It is a violation of physical reality, and therefore unacceptable, to allow a solution to increase without limit. As the general solution is formed from the superposition of homogeneous and inhomogeneous plane waves, it is necessary to restrict the PWS representation to the field radiated into a half space by a surface current density flowing on a bounding infinite plane.

4.12 Far-field antenna radiation patterns: approximated by the angular spectrum The beauty of the angular spectrum approach lies within the simplicity with which far-field parameters can be obtained from plane-wave mode coefficients. The method of stationary phase is used to simplify the two-dimensional inverse Fourier transform that is required to obtain spatial quantities from the angular spectra. This results in the reduction of the inverse transform to a multiplication by a trigonometric scalar obliquity factor. For the case of conventional near-field antenna measurements, the obliquity factor is centred about the range boresight which happens to be coincident with the antenna boresight. For the case where the AUT is not aligned to the axes of the range, care must be taken to insure that the cosine term applied is in the correct co-ordinate system. In order that this issue can be resolved, the necessary integrals are evaluated whilst paying particular attention to the co-ordinate systems. As illustrated above and assuming that the integrals involved can be evaluated, the specification of two tangential electric field components, or two tangential magnetic fields, can be used to determine the electric and magnetic fields anywhere in the forward half space. Consequently, no assumption is introduced concerning the relationship between the aperture electric and aperture magnetic fields as either

98

Principles of planar near-field antenna measurements

can be used to determine the fields over an entire half space. This provides a convenient mechanism for the derivation of radiation patterns in the far-field, by recognising that the far-field can be represented by a spherical surface of large radius, centred about the radiator. Here two assumptions are required: first that the dimensions of the aperture are of finite extent, i.e. the field sources lying in the half-space behind the aperture plane are finite and occupy a finite volume. Consequently, the equivalent current distribution over the aperture is of finite area. Second, the radius must be large compared to the wavelength and the greatest dimension of the aperture. In the majority of cases, these requirements are satisfied and the double integral, ð ð  1 1 1 h
E ðx ¼ ru; y ¼ rv; z ¼ rwÞ ¼ 2 F T kx ; ky ; z ¼ 0 4p 1 1
 kT  F T kx ; ky ; z ¼ 0  ejrðkx uþky vþkz wÞ dkx dky b ez kz (4.144) can be evaluated asymptotically using a two-dimensional form of the stationary phase algorithm and, ð1 ð1
 F T kx ; ky ; z ¼ 0 ¼ ET ðx; y; z ¼ 0Þejðkx xþky yÞ dxdy (4.145) 1 1

Obtaining asymptotic far-field parameters equates to evaluating the electric and magnetic fields over the surface of a sphere, or hemisphere in the case of a half space, described by x ¼ r sin q cos f ¼ ru

(4.146)

y ¼ r sin q sin f ¼ rv

(4.147)

z ¼ r cos q ¼ rw

(4.148)

Here r is fixed and the spherical angles q and f are varied such that, 0  f  2p p 0q 2

(4.149) (4.150)

The polar spherical grid can be thought of as being that grid that is most closely related to a positioner that consists of an upper roll rotator, i.e. f, to which the AUT is attached, and a lower rotator, i.e. q, upon which the upper f rotator is attached. As the AUT is attached to the roll positioner, the AUT will rotate about the roll axis that is therefore the polar axis, cf. Chapter 5. The field point is obtained by rotating the horizontal theta positioner and vertical roll positioner through the angles q and f where the order is unimportant. For antenna measurement, this arrangement has the advantage that it moves the AUT through only a small portion of the test zone, and it places the blockage that results from the AUT mount entirely in the back hemisphere. Moving the AUT by only a small amount minimises errors

Plane-wave spectrum representation of electromagnetic waves

99

associated with imperfections in the illumination of the test zone and can render probe pattern correction unnecessary, ðX ; Y Þ ¼ ðq; fÞ

(4.151)

Here, q and f define the direction to the field point through, br ¼ sin q cos fbe x þ sin q sin fb e y þ cos qbe z

(4.152)

Hence, q ¼ arccos ðwÞ

v f ¼ arctan u

(4.153) (4.154)

The definition of an equatorial spherical co-ordinate system is identical to the polar spherical case. The difference purely results from the application of a 90 rotation in q, in order that the main beam of the radiator points along the positive x-axis (through the equator), rather than along the positive z-axis (through the pole). Although this system is commonly used for acquiring antenna radiation patterns in spherical near-field facilities, it is not used within this text and is only included here for the sake of completion. The far-field is defined such that, E far ¼ lim fE g

(4.155)

H far ¼ lim fH g

(4.156)

r!1

and, r!1

The double integral above, with the spectrum function known, is in general difficult to evaluate. However, the approximate asymptotic method of stationary phase algorithm may be used to evaluate this integral, of which a detailed derivation is presented below, so that we may write the solution of the vector wave equation in the far-field as derived in Section 4.11 and Box 4.5, E ðru; rv; rwÞ j


 ejkr cos qF kx ; ky lr

(4.157)

Here, the polar angle q is measured away from the normal vector (the zenith or positive z-axis). Crucially in the limit of the far-field, the two-dimensional inverse Fourier transform is essentially reduced to a multiplication with a simple trigonometric function. However, this expression is only valid for certain stationary points which can be expressed as kx ¼ k0 sin q cos f

(4.158)

ky ¼ k0 sin q sin f

(4.159)

kz ¼ k0 cos q

(4.160)

100

Principles of planar near-field antenna measurements

Alternatively, this can be expressed in terms of the x-, y- and z-axis direction cosines as kx ¼ k0 a

(4.161)

ky ¼ k0 b

(4.162)

kz ¼ k0 g

(4.163)

Conventionally, the 1r term and the unimportant phase factor ejkr are divided out of far-field antenna patterns. From the above expression and recalling that k  F ¼ 0 then clearly, k E ¼0

(4.164)

Hence, in the far-field, the electric and magnetic fields are locally planar and tangential to the direction of wave propagation. Furthermore, the magnetic field may be obtained from the electric field as H ¼

1 b er  E Z0

(4.165)

This functional relationship between the angular spectrum and the radiation pattern is exact only when the observation point is infinitely far removed from the aperture. Importantly, no assumption is introduced concerning the relationship between the electric and magnetic fields in the aperture, as only the electric field or the magnetic field is required to completely specify the radiation pattern. Here, the principal variation of the far-field pattern is governed by the Fourier transform of the aperture illumination function. The slowly varying geometric terms are conventionally interpreted as ‘obliquity factors’ and are associated with the unit vectors that arise from the aperture polarisation. This formulation implies that there is no one-to-one correspondence between any field point P and the field at any point on the aperture plane; rather, the field at P is an integrated effect of the contributions from every point on the aperture plane. Crucially, the angular spectrum was obtained directly from the sampled fields. The inverse transform was thence obtained from these angular spectra implying that the cosinusoidal term is referred to the normal of the sampling surface, i.e. the obliquity factor should be applied in the sampling co-ordinate system, irrespective of the orientation of the radiator.

4.13 Stationary phase evaluation of a double integral Lord Kelvin initially formulated this principal for use with one-dimensional integrals, Math. Phys. Papers IV, 303–306, 1910. The discussion herein constitutes an expanded version of that developed by Colin and Zucker [6]. The derivation of this asymptotic solution is lengthy and rather involved so merely the result is quoted

Plane-wave spectrum representation of electromagnetic waves

101

here and the detailed workings are instead consigned to the text box. Eðx ¼ ru; y ¼ rv; z ¼ rwÞ j


 ejkr cos qF kx ; ky lr

(4.166)

Essentially then, in the far-field at the stationary points the two-dimensional inverse Fourier transform reduces to a simple functional geometric term and a normalisation constant.

Box 4.5 The method of stationary phase is in effect an algorithm for asymptotically evaluating the following integral, ðð
 1 Eðx; y; zÞ ¼ 2 F kx ; ky ejk  r dkx dky : (4.167) 4p D Here, F ðx; yÞ and k  r are real and continuous on D and jk jjr j is a large positive number. The remainder of this section is concerned with the evaluation of this sole integral. Consequently, placing the final result aside, this section adds little to the understanding of the underlying physics and can thus be ignored on first acquaintance. Assuming that the integrand oscillates rapidly along the integration path except at the points where k  r is stationary at the single point ðk1 ; k2 Þ, then the exact value of the integral may only be given by the contributions arising from the stationary points. A point is considered stationary when,      @k  r  @k  r  ¼ ¼ 0: (4.168)   @kx  kx ¼ k1 @ky  kx ¼ k1 k ¼ k k ¼ k y

2

y

2

Strictly, such a point within the domain of the integration can be referred to as a critical point of the first kind. Thus, the stationary points can be found by applying these conditions to the following relationship: k  r ¼ k x x þ ky y þ kz z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ¼ r kx sin q cos f þ ky sin q sin f þ k02  kx2  ky2 cos q : (4.169) Thus, d ðk  r Þ kx cos q ¼ sin q cos f  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0: dkx k2  k2  k2 0

x

y

(4.170)

102

Principles of planar near-field antenna measurements

Hence, the stationary points can be expressed as sin q cos f ¼ k1 kx ¼ kz cos q and, sin q sin f ¼ k2 : ky ¼ kz cos q Using, k02 ¼ kx2 þ ky2 þ kz2 ¼ kz2

(4.171)

(4.172)

2

2 2 sin2 q cos2 f 2 sin q sin f 2 2 sin q þ k þ k þ 1 ¼ k z z z cos2 q cos2 q cos2 q (4.173)

Thus by choosing the positive radical, kz ¼ k0 cos q:

(4.174)

Expanding, k  r in a Taylor’s series in two variables about these stationary points yields, k  r ¼ k  rjk ¼ k x 1 ky ¼ k2

   1 2@ k r  þ ðk x  k 1 Þ  2 @kx2  kx ¼ k1 k ¼ k y 2   2 @ 2 k  r  1
þ ky  k2  2 @ky2  kx ¼ k1 k ¼ k y 2  
 @ 2 k  r  þ þ ðk x  k 1 Þ k y  k 2  @kx @ky  kx ¼ k1 k ¼ k y 2 2

Hence, the integral becomes,

(4.175)

ðð
 1 Eðx; y; zÞ ¼ 2 F kx ; k y 4p D    0       2 2
 1 @ k  r 1 @ k  r 2    B 2 k k j@k  r  þ ð  k Þ þ  k   x 1 y 2  kx ¼ k1 2 @kx2  kx ¼ k1 2 @ky2  kx ¼ k1 k ¼ k k ¼ k k ¼ k y 2 y 2 y 2 e  1 
 @ 2 k  r  C þðkx  k1 Þ ky  k2 þ   Adkx dky (4.176)  @kx @ky  kx ¼ k1 k ¼ k y 2

Plane-wave spectrum representation of electromagnetic waves

103

Let us define the following substitutions: A¼

1 @2k  r ; 2 @kx2

(4.177)

B¼

1 @2k  r ; 2 @ky2

(4.178)

C¼

@2k  r ; @kx @ky

(4.179)

and u ¼ kx  k1 ;

(4.180)

v ¼ ky  k2 :

(4.181)

Recalling that kz ¼ k0 cos q then at the stationary points,    sin q cos f sin q sin f  x þ kz y k  r  k ¼ k ¼ kz 1  x cos q cos q k ¼ k y 2
 þ kz z ¼ k0 r sin2 qcos2 f þ sin2 qcos2 f þ cos2 q ¼ k0 r:

(4.182)

Taking the first three terms in each dimension the integral becomes, ðð
 2 2 1 Eðx; y; zÞ 2 F kx ; ky ejðkrþu Aþv BþuvC Þ dkx dky : (4.183) 4p D du dv Exchanging the variable using, dk ¼ 1, dk ¼ 1 and bringing the index y pendent variables outside the integral yields, ðð  2 2 1
Eðx; y; zÞ 2 F kx ; ky ejkr ejðu Aþv BþuvCÞ dudv: (4.184) 4p D

Let pffiffiffi x ¼ u A; pffiffiffi y ¼ v B:

(4.185) (4.186)

Then,
 1 pffiffiffiffiffiffi F kx ; ky ejkr Eðx; y; zÞ 2 4p AB



ðð e D



ffiffiffi j x2 þy2 þpCxy AB

dxdy

(4.187)

104

Principles of planar near-field antenna measurements

By completing the square for the x variable in the exponential, we may write 

 2 ðð j xþ pCyffiffiffi þy2
1 C2 
 4AB 2 AB 1 pffiffiffiffiffiffi F kx ; ky ejkr e Eðx; y; zÞ dxdy 2 4p AB D (4.188) Here we may extend the domain of the integral to infinity, as the contribution outside the domain is negligible,

2 ð ð
2
 jkr 1 1 j xþ2pCyffiffiffi 2 C 1 AB pffiffiffiffiffiffi F kx ; ky e e ejy 14AB dxdy Eðx; y; zÞ 2 4p AB 1 1 (4.189) Performing the integration on the x variable first using the exchange of ffi where dx ¼ dx then using the standard integral, variable x ¼ x þ 2pCyffiffiffiffi AB rffiffiffi ð1 p jp jaðxx0 Þ2 e 4: e dx ¼ (4.190) a 1 where a ¼ 1 and x0 ¼ 0 we may express the electric field as ð
 jkr pffiffiffi jp 1 jy2
1 C2  1 4AB dy 4 pffiffiffiffiffiffi F kx ; ky e Eðx; y; zÞ pe e 4p2 AB 1

(4.191)

2

C and x0 ¼ 0 hence, Using the standard integral again where a ¼ 1  4AB sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 pffiffiffi p 1 p p pffiffiffiffiffiffi F kx ; ky ejkr pej 4 (4.192) Eðx; y; zÞ ej 4 : C2 1  4AB 4p2 AB

Thus with no further approximation,
 1 1 p pffiffiffiffiffiffi F kx ; ky ejkr pej2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Eðx; y; zÞ 2 4p2 AB 1 C

(4.193)

4AB

Hence, Eðx; y; zÞ

 1
2pj F kx ; ky ejkr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 4p 4AB  C 2

(4.194)

The constants A, B and C may be evaluated as follows: 1 @2k  r 2 @kx2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  r @2

¼ kx sin q cos f þ ky sin q sin f þ k02  kx2  ky2 cos q 2 2 @kx (4.195)

A¼

Plane-wave spectrum representation of electromagnetic waves 0

105

1

r @ B kx cos q C @sin q cos f  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 @kx k02  kx2  ky2 0 1

A¼

rB ¼ @

2

(4.196)

kx2 cos q

cos q C 3=2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 k0  kx2  ky2 k02  kx2  ky2

Hence, A¼



r sin2 q cos2 f 1þ 2k0 cos2 q

(4.197)

Similarly,

r sin2 q sin2 f 1þ B¼ 2k0 cos2 q

(4.198)

Furthermore, @2k  r @kx @ky qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  @2

¼ r kx sin q cos f þ ky sin q sin f þ k02  kx2  ky2 cos q @kx @ky (4.199) 0 1

C¼

C ¼ r

@ B kx cos q C @sin q cos f  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA @ky k2  k2  k2 x

0

0 B ¼ rkx cos q@

C¼

y

1 ky k02



kx2



ky2

C 3=2 A

rsin2 q cos f sin f k0 cos2 q

(4.200)

(4.201)

So, 4AB  C 2 ¼ 4





r sin2 q cos2 f r sin2 q sin2 f 1þ 1 þ 2k0 cos2 q 2k0 cos2 q



r2 sin4 qcos2 jsin2 f r2 1 ¼ 2 k0 cos4 q k0 cos2 q

(4.202)

106

Principles of planar near-field antenna measurements

Hence, 2p 2pk0 cos q j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ j 2 r 4AB  C

(4.203)

Finally, the asymptotic evaluation of the electric field, E far ¼ lim fE g, r!1 can be expressed as Eðx ¼ ru; y ¼ rv; z ¼ rwÞ j


 ejkr k0 cos qF kx ; ky : 2pr

(4.204)


 ejkr cos qF kx ; ky : lr

(4.205)

Or, Eðx ¼ ru; y ¼ rv; z ¼ rwÞ j

Within the above derivation, certain assumptions have been made, the most significant of these being: ●

●

●

Truncation of the Taylor series to only the first three terms in each dimension. Assumption that k : r is a smoothly varying function about the stationary points (without a strict definition of smooth). An implicit assumption is made that no singularities are present.

4.14 Co-ordinate free form of the near-field to angular spectrum transform Thus far, the analysis has been restricted to the consideration of problems where the boundary conditions f ðx; yÞ are known in the same co-ordinate system, as that required by the solution uðx; y; zÞ. Techniques for the removal of this restriction constitute the topic for this section. If the field can be represented as an area of a planar surface located at the origin with a radiating field distribution across it, we may express two tangential and mutually orthogonal field components as





 ET rT ¼ b (4.206) u x Ex rT þ b u y Ey rT Here, the tangential component of the position vector is expressed as rT ¼ b ux x þ b uy y

(4.207)

Thus, the equivalent tangential spectral components may be evaluated using



 ð1 ð1

 j k  r T T dxdy (4.208) F T kT ¼ ET rT e 1 1

Plane-wave spectrum representation of electromagnetic waves

107

where the tangential components of the propagation may be expressed as kT ¼ ^u x kx þ ^u y ky

(4.209)

Hence, the Cartesian far-field electric components can be written in terms of the tangential spectral components as



 3 2



 jkr aF k þ bF x T y kT e ^u z 5 g4Fx kT ^u x þ Fy kT ^u y  E ðru; rv; rwÞ j lr g (4.210)





 i ejkr h  Fx kT g^u x þ Fy kT g^u y  aFx kT þ bFy kT ^u z E ðru; rv; rwÞ j lr (4.211) or, E ðru; rv; rwÞ j



i ejkr h gFT kT  ^u z ^u r  FT kT lr

(4.212)

Here, the tangential spectral components have been written in vector form as FT ¼ Fx ^u x þ Fy ^u y þ 0^u z

(4.213)

The unit vector from the sampling plane to the observation point is ^ u r ¼ a^u x þ b^u y þ g^u z

(4.214)

To place this in a co-ordinate free form, this expression must be modified to account for any displacement and rotation of the plane with respect to the origin of the co-ordinate system. Here primed co-ordinates are used to denote those coordinates associated with the source, whilst those associated with the observation point are not primed, see Figure 4.3. Angular spectra are obtained by integrating the complex tangential electric field components over the sampling plane. As in general, the source and observation frames of reference will not be coincident and synonymous account must therefore be made for this when determining equivalent angular spectra. As can be seen from Figure 4.3, the relationship between the position vectors can be expressed as r0 þ r00 ¼ r

(4.215)

A displacement between the origin of the co-ordinate system and the sampling plane can be accommodated through the introduction of a differential phase change 0 of the form ejk ^u r  r . The introduction of this phase factor can be seen to follow directly from the shifting property of the Fourier transform [2]. The ‘illumination’ factor, g ¼ cos q where q is measured from the outward-facing unit normal to the sampling plane normal, can be expressed generally as ^u r00  ^n . Hence,

108

Principles of planar near-field antenna measurements y

Sampling plane

Ey' r' nˆ

Ex'

θ

O x

r r''

z

Ey (P)

Ex (P) P

Figure 4.3 Generalised co-ordinate system for solution of vector Helmholtz equation

E ðru; rv; rwÞ j

00



i ejkr h ^ ^ ^ ^  k  F  ð u n ÞF n u 00 00 T T T kT r r lr00

(4.216)

Using, A  ðB  C Þ ¼ ðA  C ÞB  ðA  B ÞC

(4.217)

then, 00

  ejkr

E ðru; rv; rwÞ j 00 u^ r00  FT kT  ^n lr

(4.218)

Hence, E ðru; rv; rwÞ j

ejkr lr00

00

 ð

 0 ^u r00  ET 0 rT0 ejk0 ^u r00 r da0  ^n S

(4.219) Here, the electric field is correctly resolved onto a Cartesian polarisation basis in the sampling co-ordinate system. These unit vectors can be readily rotated so that the polarisation basis can be rewritten in terms of any preferred observation frame of reference. This expression is only valid for the positive half space with respect to the sampling plane, or when ^n  ^u r00 0. Near-field

Plane-wave spectrum representation of electromagnetic waves

109

Translated measurement plane

AUT

Reconstructed aperture plane

Measurement plane

Figure 4.4 Schematic representation of plane to plane transform performed with and without alignment correction

data can be recovered directly from the rotated spectral components using, ð ð  1 1 1 
E ðx; y; zÞ ¼ 2 F kx ; ky ejðkx xþky yþkz zÞ dkx dky (4.220) 4p 1 1 Here, there is no requirement for any further isometric transformations and all usual numerical techniques for improving the efficiency of the transformation can be utilised. As illustrated in Figure 4.4, the recovered plane will not be coplanar with sampling surface. Furthermore, even if those issues associated with polarisation are ignored then the differences between the illumination functions are not characterised with a simple linear phase taper.

4.15 Reduction of the co-ordinate free form of the nearfield to far-field transform to Huygens’ principle As was described in Chapter 3 in some detail, Huygens’ principle [7–9] can be stated as ‘each point on a primary wave-front can be considered to be a new source of a secondary spherical wave, of the same frequency, and that a secondary wavefront can be constructed as the superposition of these secondary spherical waves

110

Principles of planar near-field antenna measurements

with due regard to their phase differences’. When expressed mathematically, the electric field, at a point P, radiated by a Huygens surface S, is [10]: ð i ejkr00 j h ^u r00  ðE  ^n Þ e ð^ da (4.221) uÞ¼ r00 l S Here, r00 is the distance between element and field point, l is the wavelength, k0 is the wave-number, ur00 is the unit vector from element to field point, n is the unit normal vector at the element and E is the total, or tangential, aperture electric field at the element. This statement of Huygens’ formula is rigorous, provided that the observation point is more than a few wavelengths from the source and that the scalar product of n and ur00 is positive. The geometry for this statement of the Huygens’ formula can be found illustrated in its conventional form in Figure 4.5. Importantly, the derivation of this relationship requires the observation point to be placed in the far-field of the infinitesimal Huygens element. Thus, this expression cannot be used to model the behaviour of the fields in the reactive near-field where evanescent fields become significant. This is unimportant in the simulation of near-field measurements as these measurements should be acquired outside of this region of space. As such, this form of Huygens’ principle is utilised in Chapter 6 within the near-field measurement simulation software where this formulation can be utilised in cases where the field is known over a non-planar surface. Huygens’ principle can be obtained directly from the co-ordinate free form of the near-field to far-field transformation by reducing the area of the aperture plane until in the limit, it constitutes an elemental, i.e. infinitesimal, Huygens source. From Section 4.14, it can be established that the near-field to far-field transform can be expressed as  ð

00

 0 ejkr E ðru; rv; rwÞ j 00 ^u r00  ET 0 rT0 ejk0 ^u r00 r da0  ^n lr S (4.222) P

r S

O’ O

r’’

ur’’

r’ n

Figure 4.5 Co-ordinate system for Huygens’ principle

Plane-wave spectrum representation of electromagnetic waves

111

Thus, in the limit when the aperture is reduced to a Huygens element, i.e. when da ¼ lim fS g, and the observer is in the far-field so that r00 and r are parallel, S!0

dE ðru; rv; rwÞ j

  0 ejkr

^u r  ET 0 rT0 ejk0 ^u r00 r da0  ^n lr

(4.223)

Hence, integrating over the complete surface yields, E ðru; rv; rwÞ j

ejkr lr

ð h

i 0 ^u r  ET 0  ^n ejk0 ^u r r da0

(4.224)

S

Here, it is important to stress that the co-ordinate free form of the angular spectrum representation of an electromagnetic field is the most general representation, as it contains information pertaining to both propagating and evanescent fields. Furthermore, as Huygens’ principle essentially constitutes a special case of the angular spectrum approach, the same assumptions and limitations bind it, i.e. any deficiency within the Fourier approach will also represent a deficiency with this Huygens method.

4.16 Far-fields from non-planar apertures As shown earlier, the far electric field can be obtained from the near electric fields using the following expression:  ð1 ð1 ejk0 r

^u r  ^n E ðra; rb; rgÞ ¼ j f ðx; yÞejk0 ðaxþbyþgzÞ dxdy (4.225) lr 1 1 Often, it is convenient to express the surface profile on which the near-field is defined as a function of two co-ordinates, x and y, where the co-ordinates are plaid, monotonic and equally spaced. For a smooth surface defined by the function, g ðx; y; zÞ ¼ z  f ðx; yÞ ¼ 0

(4.226)

Thus, z ¼ f ðx; yÞ then the outward facing, positive, normal is given by



 @g @g @x @y 1 ^e x  ^e y þ ^e z n ¼ nx^e x þ ny^e y þ nz^e z ¼  (4.227) rg rg rg where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 @g @g þ þ1 rg ¼ @x @y

(4.228)

Multiplying by –1 produces the negative orientated, i.e. inward facing, normal vector. Generally, the partial derivatives are obtained by taking finite differences.

112

Principles of planar near-field antenna measurements

The elemental surface area da is given by,   dxdy   da ¼  n 

(4.229)

z

Hence,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 @g @g þ þ1 da ¼ dxdy @x @y

(4.230)

Thus, the far electric field can be obtained from, ð ð

 ejk0 r 1 1 E ðra; rb; rgÞ ¼ j f ðx; yÞejk0 ðaxþbyþgzÞ ^u r  b n lr 1 1 !12 2 2 @g @g þ þ 1 dxdy: (4.231) @x @y

 Crucially, the obliquity factor b ur  b n has been brought inside the integral, as n will vary over the surface of integration and, the unit surface normal b b u r ¼ ab e x þ bbe y þ gb ez

(4.232)

This formula can be used in cases where the field is known over a non-planar surface, provided that the surface is smooth. Specifically, the function describing the surface must be continuous, as must all of the first partial derivatives. Here, the arbitrary but smooth surface is expressed as a function of two co-ordinates x and y where the co-ordinates are plaid, monotonic and equally spaced. Finally, within visible space, the angular spectrum can be obtained from, !12 ð ð

 @g 2 @g 2 1 1 1 jk0 ðaxþbyþgzÞ b F ðk0 a; k0 bÞ ¼ f ðx; yÞe þ þ 1 dxdy ur  b n g 1 1 @x @y (4.233) Although useful, these expressions are presented here more for the sake of completion as they are not automatically in a form that is amenable for use with probe uncorrected data, unless the orientation of the probe remains fixed with respect to the test antenna across the acquisition interval, e.g. when acquiring data using uninhabited air vehicles, i.e. drones, when the probe orientation is fixed, e.g. vertically; or when using multi-axis industrial robotic based systems. This topic is examined in greater detail below in the following chapters, e.g. Chapter 9.

4.17 Microwave holographic metrology (plane-to-plane transform) The theoretical framework for the plane-to-plane transform has already been developed within Chapter 4. It is given special attention here as in part because it

Plane-wave spectrum representation of electromagnetic waves

113

is of practical utility and in part because it has received such a great deal of attention in the open literature. As shown earlier, knowledge of the tangential components of the electric field enables the entire electromagnetic six-vector to be determined everywhere within a half space. This, of course, enables the fields over one plane in space to be used to determine the field over another plane in space. The reconstruction of near-field data over a plane in space, other than the measurement plane, is accomplished by the application of a differential phase change. This can be seen to be analogous to a defocusing of the far-field image. Thus in summary, ð1 ð1
 f ðx; y; z ¼ 0Þejðkx xþky yÞ dxdy (4.234) F k x ; ky ; z ¼ 0 ¼ 1 1

And, uðx; y; zÞ ¼

1 4p2

ð1 ð1 1 1


 F kx ; ky ; z ¼ 0 ejðkx xþky yþkz zÞ dkx dky

(4.235)

Hence, the field over one plane can be used to calculate the field over the surface of another, parallel plane displaced by an amount z in the z-axis. When expressed compactly using operator notation this reduces to,   (4.236) uðx; y; zÞ ¼ =1 =ff ðx; y; z ¼ 0Þgejkz z This reconstructed plane can be located at any of an infinite number of planes that are in the region of space at, or in front of the instruments aperture. It is when the fields are reconstructed at a plane that is coincident with the AUT’s ‘aperture plane’ that this process is of most utility. The antenna aperture can be conveniently thought of as that surface in space, which represents the transition between the majority conduction current and displacement current regions defined by the presence of a charge distribution. A near-field measurement is typically constructed so that the field produced by the antenna is sampled over a region of space in which there is an absence of divergence contained within that field. Therefore, the planeto-plane transform process results in knowledge of only the radiating components and provides no knowledge as to the stored energy component unless the probe is in very close proximity to the AUT in which case evanescent fields can couple. However, providing there is a few wavelengths separation, cf. Chapter 5, and providing the RF subsystem has a finite dynamic range then practically these fields will be sufficiently small that they will be absent. This is also examined within the range assessment and uncertainty budget section below. Thus, the field is reconstructed with a resolution of at best half a wavelength only. Although the field can be reconstructed with an infinitely fine sample spacing, the amount of information contained within the resulting pattern is still limited due to the absence of reactive, i.e. evanescent, fields. The technique of microwave holographic diagnosis, i.e. the recovery of the antenna aperture illumination function, is now a well-established method of non-intrusive, non-destructive characterisation of antenna assemblies as it can clearly show faulty radiating elements or incorrectly adjusted transmit and

114

Principles of planar near-field antenna measurements

receive modules within a phased array antenna, or an incorrectly aligned feed or reflector within a reflector antenna assembly. Sometimes the aperture illumination function is referred to as being a hologram. This is perhaps a little confusing as a hologram is usually taken to mean an image that contains both amplitude and phase information. As such, the measurement plane itself, or even the far-field, could equally well be referred to as being a hologram thus this terminology is depreciated.

4.18 Far-field to near-field transform It is customary to present far-field antenna pattern functions in terms of spherical angles. Often, far-field pattern functions are presented tabulated as a function of the spherical angles azimuth and elevation, rather than the polar spherical angles q and f. The azimuth over elevation grid can be thought of as being that grid that is most closely related to a positioner that consists of an upper azimuth rotator, to which the AUT is attached, and a lower elevation positioner upon which the azimuth rotator is mounted. As the AUT is attached to the azimuth positioner, the AUT will rotate about the azimuth axis that is therefore the polar axis. The field point is obtained by rotating the horizontal azimuth positioner and vertical elevation positioner through the angle Az and El where the order is unimportant. A detailed overview of co-ordinate systems, including the azimuth over elevation system can be found in Appendix C. Here, it is sufficient to recall that Az and El define the direction to the field point through, br ¼  sin ðAzÞcos ðElÞb e x þ sin ðElÞb e y þ cos ðAzÞcos ðElÞb ez

(4.237)

So that, Az ¼ arctan

u w

El ¼ arcsin ðvÞ

(4.238) (4.239)

The inclusion of the minus sign in the x co-ordinate is useful, as it signifies that the observer is positioned behind the antenna looking out at the far-field pattern. Other choices are possible; however, this is in agreement with that which is usually adopted in space, and airborne RADAR applications. It is important to keep in mind the difference between the range co-ordinate system and the antenna coordinate system. These azimuth and elevation angles are measured with reference to the AUT and are fixed to it. Thus, one should try and imagine being bolted to the antenna and seeing the chamber rotate as the positioners move. This is not the usual way we conceptualise these things as we are used to standing stationary on the floor of the chamber and watching the AUT rotate within the fixed chamber, and not the converse. This is discussed in greater detail in Chapter 5 and in [11]. Let us assume that the far-field pattern is tabulated on a plaid monotonic equally spaced azimuth over elevation grid. The integration required to obtain

Plane-wave spectrum representation of electromagnetic waves

115

near-field parameters from far-field pattern functions must therefore be modified accordingly as ð ð  1 1 1
E ðx; y; zÞ ¼ 2 A kx ; ky ejðkx xþky yþkz zÞ dkx dky (4.240) 4p 1 1 Since kx ¼ k0 a and ky ¼ k0 b then by differentiation we obtain the following relationships, 2p da l 2p db dky ¼ l dkx ¼

(4.241) (4.242)

Then, 1 E ðx; y; zÞ ¼ 2 l

ð1 ð1 1 1

A ða; bÞejk0 ðaxþbyþgzÞ dadb

(4.243)

This is a very convenient form of the integral as the Fourier variables a and b are direction cosines that are not scaled by the free space propagation constant k0. Thus, the unit circle is a2 + b2 = 1, which represents the range that the integration limits may be collapse to as we are considering homogeneous PWS only. The application of the change of variable formula for double integrals can be applied to the Fourier transform of the boundary conditions so that all quantities can be rewritten in terms of the angular spectrum. Using the initial boundary condition specified in a regular azimuth and elevation co-ordinate system where the transformation between direction cosine and azimuth and elevation angles could be expressed as follows, cf. Figure 5.38: a ¼  sin ðAzÞcos ðElÞ

(4.244)

b ¼ sin ðElÞ

(4.245)

g ¼ cos ðAzÞcos ðElÞ

(4.246)

Thus, 1 E ðx; y; z ¼ 0Þ ¼ 2 l

ð1 ð1

A ð sin ðAzÞcos ðElÞ; sin ðElÞÞ     jk ð sinðAzÞcosðElÞxþsinðElÞyÞ  @ ða; bÞ  e @ ðAz; ElÞdAzdEl 1 1

(4.247)

where      @a  @ ða; bÞ   @Az    @ ðAz; ElÞ ¼  @b   @Az

 @a  @El  ¼ @a  @b  @a  @b @b  @Az @El @El @Az  @El

(4.248)

116

Principles of planar near-field antenna measurements Thus, @a @ ¼ ð sin ðAzÞcos ðElÞÞ ¼  cos ðAzÞcos ðElÞ @Az @Az @b @ ¼ ðsin ðElÞÞ ¼ cos ðElÞ @El @El @a @ ¼ ð sin ðAzÞcos ðElÞÞ ¼ sin ðAzÞsin ðElÞ @El @El @b @ ¼ ðsin ðElÞÞ ¼ 0 @Az @Az Hence,    @ ða; bÞ    @ ðAz; ElÞ ¼  cos ðAzÞcos ðElÞcos ðElÞ  0

(4.249) (4.250) (4.251) (4.252)

(4.253)

So that, E ðx; y; zÞ ¼

1 l2

ð1 ð1 1 1

A ðAz; ElÞ  expðjk ð sin ðAzÞcos ðElÞx

þ sin ðElÞy þ cos ðAzÞcos ðElÞzÞÞ  cos ðAzÞcos2 ðElÞdAzdEl

(4.254)

This expression will allow the determination of the propagating near-field, on a planar surface from far-field data that has been tabulated on the surface of a sphere, using an azimuth over elevation positioner system, i.e. the basis of spherical microwave holographic metrology. When implemented numerically, the integration is separable, if the azimuth integral is evaluated first, as alpha is a function of azimuth and elevation whereas beta is a function of elevation only. When the range of azimuth and elevation angles represents a full sphere, this formula is required to be applied separately for the forward and back half spaces. A similar procedure can be used to derive similar expressions for the case where the far-field pattern has been tabulated on a regular spherical grid. Here the direction cosines are related to the polar spherical angles through, a ¼ sin q cos f (4.255) b ¼ sin q sin f

(4.256)

g ¼ cos q

(4.257)

So that, @a @ ¼ ðsin q cos fÞ ¼ cos q cos f @q @q @a @ ¼ ðsin q cos fÞ ¼  sin q sin f @j @j

(4.258)

(4.259)

Plane-wave spectrum representation of electromagnetic waves @b @ ¼ ðsin q sin fÞ ¼ cos q sin f @q @q @b @ ¼ ðsin q sin fÞ ¼ sin q cos f @j @j

117

(4.260) (4.261)

Hence,   @ ða; bÞ @a @b @a @b   @ ðq; jÞ ¼ @q  @j  @j  @q ¼ cos q cos f sin q cos f þ sin q sin f cos q sin f Or simplifying with trigonometric identities yields,   @ ða; bÞ   @ ðq; jÞ ¼ sin q cos q

(4.262)

(4.263)

Hence, 1 E ðx; y; zÞ ¼ 2 l

ð p ð p=2þj1 p 0

A ðq; jÞejk0 ðaxþbyþgzÞ sin q cos qdqdf

(4.264)

For the case where we wish to recover propagating fields only, the infinite range of integration in q reduces to the real finite limit, ð ð 1 p p=2 E ðx; y; zÞ ¼ 2 A ðq; jÞejk0 ðaxþbyþgzÞ sin q cos qdqdf (4.265) l p 0 This expression is very useful. In addition to enabling microwave holography to be applied to spherical measurement, whether they are taken using a far-field range, a compact antenna test range or a spherical range, they show how to relate the plane wave and spherical-wave expansions to one another, they also show how to circumnavigate the soft singularity in the normal field component that is encountered on the kz = 0 (q = 90 ) circle. To illustrate this final point, the planewave condition can be expressed as k A ¼0

(4.266)

Thus, the normal plane-wave component can be expressed in terms of the tangential plane-wave components as Az ¼ 

Ax sin q cos f þ Ay sin q sin f cos q

(4.267)

Thus, the normal near electric field component can be obtained from, Ez ðx; y; zÞ ¼ 

1 l2

ð p ð p=2 p 0

Ax ðq; fÞsin q cos f þ Ay ðq; fÞsin q sin f jk0 ðaxþbyþgzÞ e sin q cos qdqdf cos q

(4.268)

118

Principles of planar near-field antenna measurements Simplifying yields, Ez ðx; y; zÞ ¼ 

1 l2

ð p ð p=2 p 0




 Ax ðq; fÞcos f þ Ay ðq; fÞsin f ejk0 ðaxþbyþgzÞ sin2 qdqdf

(4.269) Here, the soft singularity at the unit circle has been removed and the normal near-field component can be recovered without encountering an inconvenient divide by zero error. As noted earlier, this expression enables fields to be produced that remain finite for all values of q. Although here we are assuming that there are no fields outside of the unit circle, i.e. there are no non-homogeneous plane wave spectra, if there were, then the limit of the q integration may be extended to include complex angles. This is discussed further in Chapter 5. This would allow fields to be computed that include reactive fields which become important when evaluating the coupling between radiating elements within a closely coupled array.

4.19 Radiated power and the angular spectrum The time averaged power flux that is transmitted per unit area across an interface, in this case an aperture plane, radiated by an arbitrary current density can be expressed as [12], ð

 1  ET  HT  b (4.270) e z dxdy Pr ¼ Re 2 aperture where ET ¼ Exbe x þ Eyb ey

(4.271)

HT ¼ Hxb e x þ Hyb ey

(4.272)

and,

This is equivalent to considering the power radiated by an arbitrary current density flowing in the plane. Here, E and H are assumed to be peak values of the field. If instead E and H had denoted root mean square (RMS) values of the field, then the factor of one half must be omitted. Taking the cross product of the tangential electric field and the complex conjugate of the tangential magnetic field yields,   b e x be y b e z 



  ET  HT  b e z ¼  Ex Ey 0   b e z ¼ Ex Hy  Ey Hx b ez  b ez  Hx Hy 0  ¼ Ex Hy  Ey Hx :

(4.273)

Plane-wave spectrum representation of electromagnetic waves

119

Hence, ð

 1   Ex Hy  Ey Hx dxdy Pr ¼ Re 2  aperture ð ð 1 1 Ex Hy dxdy  Re Ey Hx dxdy ¼ Re 2 2 aperture aperture

(4.274)

Using Parseval’s theorem in two dimensions [2], ð1 ð1 ð ð 
 1 1 1
f ðx; yÞg  ðx; yÞdxdy ¼ 2 F kx ; ky G kx ; ky dkx dky 4p 1 1 1 1 (4.275) Obtains, 1 Pr ¼ 2 Re 8p

ð 1 ð 1 1 1

Fx Gy dkx dky



1  2 Re 8p

ð 1 ð 1 1 1

Fy Gx dkx dky



(4.276) Now, 1 k F jk jZ0 
o 1 n
b e x kk F z  kz F y þ b e y ðk z F x  k x F z Þ þ b e z k x Fy  k y F x : ¼ jk jZ0 (4.277)

G ¼

Now as k E ¼0

(4.278)

then, Fz ¼ 

k x Fx þ k y F y kz

(4.279)

So,
 

k y k x F x þ k y Fy 1 1 b G ¼ k F ¼  k z Fy ex  kz jk jZ0 jk jZ0
 k x k x Fx þ k y F y þbe y kz Fx þ kz
 þbe z kx Fy  ky Fx

(4.280)

120

Principles of planar near-field antenna measurements So,

1 9   = k k F þ k F x x y x y 1 Adkx dky Re Fx @kz Fx þ Pr ¼ 2  ; 8p jk jZ0 : 1 1 kz  9 8 0

1 = kx2 þ ky2 , for the spectral field function. If the expressions for the plane-to-plane transform that was developed in the preceding sections are used to propagate the tangential components of the electric field, and provided that the angular spectrum of the sampling near-field probe is known then the antenna-to-antenna coupling formula, " # PBy ða; bÞ j PBx ða; bÞ Sx ¼ Sy l PCy ðb; aÞ PCx ðb; aÞ 

1 g



1  b2 ab





ejk ðax0 þby0 þgz0 Þ Ax ða; bÞ ab   2 Ay ða; bÞ  ð1  a Þ g (7.31)

can be used to incorporate probe pattern effects. Here the nomenclature of Chapter 5 has been maintained. When this is incorporated into the plane-to-plane transform expressions, the x-polarised and y-polarised measured field can be obtained from, ( " #

PBy ða; bÞ PBx ða; bÞ sx ðx; y; z ¼ d Þ 1 j ¼= sy ðx; y; z ¼ d Þ l PCy ðb; aÞ PCx ðb; aÞ 1  g



1  b2 ab



ejkz d =fEx ðx; y; z ¼ 0Þ g ab   = Ey ðx; y; z ¼ 0Þ ð1  a2 Þ g (7.32)

Here, sx, sy are the measured, i.e. probe coupled, near-field data. Unfortunately, whilst this method is efficient, as all of the transforms can be accomplished by means of the fast Fourier transform, as expressed here, it is both limited to planar geometry and is somewhat incestuous using as it does an inversion of the microwave holographic metrology and probe-correction formula. It is possible however to extend this to other acquisition geometries, cf. [4] by rotating the plane-wave spectrum of the AUT prior to evaluating the coupling. By performing this in a

306

Principles of planar near-field antenna measurements

point-wise fashion, this approach can be used to simulate cylindrical, spherical, etc. acquisition geometries. A general-purpose method for computing simulations of near-field measurements that include the effects of the measuring probe is presented below in the final section of this chapter. The utility of this more advanced, and significantly more complex, simulation methodology stems from noting that it does not rely upon either the angular spectrum method or the probe pattern correction expressions used until this point in the text.

7.5 Vector Huygens method The angular spectrum method that is employed above to produce an initial very useful, but somewhat limited, near-field simulation can be deployed in a very similar fashion to produce a far more effective near-field simulation tool. As is shown in Chapter 4, the vector Huygens method is essentially an alternative deployment of the angular spectrum method that, for the purposes of simulating near-field measurements, is perhaps a more useful one. The far-field form of Huygens’ principle can be obtained directly from the co-ordinate free form of the near-field to far-field transformation of Chapter 4 by reducing the area of the aperture plane until in the limit, it constitutes a single elemental, i.e. infinitesimal, Huygens source. From Chapter 4, the general coordinate free near-field to far-field transform can be expressed as,


ð  00
  ejkr u r  r0 da0  b E ðx; y; zÞ  j 00 b ET 0 rT 0 ejk0b u r00  n (7.33) lr S Now, if the size of the radiating aperture is made infinitesimally small, two convenient things happen. First, the observer will necessarily be situated in the far-field of the infinitesimal radiating aperture provided only that they are a displaced by any finite amount, i.e. by a couple of wavelengths or so; and second, as the aperture is an infinitesimally small distance across, it is reasonable to assume that the electric field is constant over the surface of the aperture. Thus, in the limit when the aperture is reduced to a Huygens element, i.e. when da ¼ lim fS g, and the observer is in the farS!0

field, i.e. more than a few wavelengths away from the element, 00     ejkr  u r00  ET 0 rT 0 da0  b n E ðx; y; zÞ  j 00 b lr

(7.34)

Now, this logic can be easily extended to consider apertures of finite extent. If the finite aperture is divided into infinitesimal Huygens sources then the total field can be obtained by integrating the contribution over each of these individual radiators. Specifically the field at a point in space resulting from an infinitesimal Huygens elemental can be expressed as 00     ejkr  u r00  ET 0 rT 0 da0  b n (7.35) dE ðx; y; zÞ  j 00 b lr

Computational electromagnetic model

307

So that the total field at a point in space can be expressed as, E ðx; y; zÞ 

j l

ð

    ejkr00 b da0 u r00  ET 0 rT 0  b n r00 S

(7.36)

Where S denotes the finitely large aperture. The magnetic field can be obtained from, H ðx; y; zÞ 

j lZ0

ð

h    i ejkr00 b da0 u r00  b u r00  ET 0 rT 0  b n r00 S

(7.37)

Here, r00 is the distance between element and field point, l is the wavelength, k0 is the wave-number, ur00 is the unit vector from element to field point, n is the unit normal vector at the element and E is the total, or tangential, aperture electric field at the element. This statement of Huygens’ formula is rigorous, provided that the observation point is more than a few wavelengths from the elemental source (i.e. in the far-field of the elemental source) and that the scalar product of n and ur00 is positive. The geometry for this statement of the Huygens’ formula can be found illustrated in its conventional form in Chapter 4. Usefully, as the surface unit normal varies over the aperture S with aperture electric and aperture magnetic fields then these expressions can be used to calculate the electric and magnetic fields at the observation point ðx; y; zÞ from smooth nonplanar apertures. Furthermore, provided that the observation point is displaced by more than a few wavelengths from the radiating aperture, the field points can be distributed arbitrarily throughout space. Thus, this field propagation algorithm can be used to produce near-field simulations over any desired sampling surface, i.e. plane rectilinear, plane-polar, plane-bi-polar, cylindrical, polar spherical, equatorial spherical, non-canonical, etc. Practically, the elemental far-field requirement manifests itself within these expressions with the inverse r term that will become large as r becomes small and in the limit will tend to infinity as r tends to zero. This aside, provided r is of the order of a couple of wavelengths, these expressions can be used to produce reliable pattern predictions. As the derivation of these relationships requires the observation point to be placed in the far-field of the Huygens element, neither of these expressions can be used to model the behaviour of the fields in the reactive near-field where evanescent fields become significant. Again, this is unimportant in this area of application as near-field antenna measurements are only made outside this region of space. Here, it is important to stress that the co-ordinate free form of the angular spectrum representation of an electromagnetic field is the most general representation, as it contains information pertaining to both propagating and evanescent fields. Furthermore, as the Huygens’ principle essentially constitutes a special case of the angular spectrum approach, the same assumptions and limitations bind it; i.e. any deficiency within the Fourier approach will also represent a deficiency with this implementation of Huygens method.

308

Principles of planar near-field antenna measurements

7.6 Kirchhoff–Huygens method The Kirchhoff–Huygens’ principle is a powerful technique for determining the field in a source-free region outside a surface from knowledge of the field distributed over that surface and is in effect a direct integration of Maxwell’s equations. It is applicable to arbitrary-shaped apertures over which both the electric and magnetic fields are prescribed. The Kirchhoff–Huygens’ principle has the important benefit that the field point will contain both the propagating and reactive fields. Unfortunately, this added rigour is sought at the expense of an increase in the requisite computational effort and, significantly, the added requirement that the magnetic field must be known over the radiating surface in addition to the electric field as was the case previously. When expressed mathematically the electric field, at a point P, radiated by a closed Huygens surface S [5], ð 1 Ep ¼ ½jwmðn  H Þy þ ðn  E Þ  ry þ ðn  E Þryda (7.38) 4p S Here, y represents the following spherical function: y¼

ejk0 r r0

0

(7.39)

Now,



r0 ¼ r  r0

(7.40)

Thus, r0 y ¼

@y @y @y be x0 þ b b e y0 þ e @x0 @y0 @z0 z0

(7.41)

By multiplying the numerator and denominator by @r’ yields, r0 y ¼

@y @r0 @y @r0 @y @r0 b b b þ þ e e e @r0 @x0 x0 @r0 @y0 y0 @r0 @z0 z0

(7.42)

Now,




0 0 0 @y @ ejk0 r jk0 ejk0 r ejk0 r 1 ¼  y ¼ jk ¼   0 r0 r0 r0 2 @r0 @r0 r0

(7.43)

Also,
12  @r0 @  x  x0 ¼ ¼ ðx  x0 Þ2 þ ðy  y0 Þ2 þ ðz  z0 Þ2 @x0 @x0 r0

(7.44)

Similarly, @r0 y  y0 ¼ @y0 r0

(7.45)

Computational electromagnetic model @r0 z  z0 ¼ 0 @z0 r

309

(7.46)

Combining these results yields, r0 y ¼ 







y  y0 1 z  z0 1 x  x0 1 yb yb e jk   jk   jk  yb e e z0 0 0 0 x0 y0 r0 r0 r0 r0 r0 r0 (7.47)

Thus,

  1 x  x0 y  y0 z  z0 1 r0 b b b jk0 þ 0 þ þ þ y y ¼ jk e e e 0 x y z 0 0 0 r r0 r 0 r0 r0 r0

r0 y ¼

(7.48) Or,
r0 y ¼

jk0 þ

 1 0 br y r0

(7.49)

Hence, the general vector Kirchhoff–Huygens’ formula becomes, Ep ¼

1 4p

ð S





1 1 jwmðn  H Þy þ ðn  E Þ  jk0 þ 0 br 0 y þ ðn  E Þ jk0 þ 0 br 0 y da r r

(7.50) Upon factorising, we obtain the required result, Ep ¼

1 4p

ð S


 0   1 ejk0 r jk0 þ 0 jwmðn  H Þ þ ðn  E Þ  br 0 þ ðn  E Þbr 0 da r r0

(7.51) Since, Hp¼

1 4p

ð ½jweðn  E Þy þ ðn  H Þ  ry þ ðn  H Þryda

(7.52)

S

Then similarly, Hp¼

1 4p

ð S





1 1 jweðn  E Þy þ ðn  H Þ  jk0 þ 0 br 0 y þ ðn  H Þ jk0 þ 0 br 0 y da r r

(7.53) As before, factorising yields the required result,
 ð 0  1 1 ejk0 r 0 0 jk0 þ 0 Hp¼ jweðn  E Þ þ ðn  H Þ  br þ ðn  H Þbr da r0 4p S r (7.54)

310

Principles of planar near-field antenna measurements P

r S

r’

u O

r0 n

Figure 7.7 Co-ordinate system for Kirchhoff–Huygens formula This expression yields the field at a point in space outside the radiating Huygens surface from an integral of the electric and magnetic fields over the closed surface S. Here, da is an elemental area of S. The geometry for this statement of the Kirchhoff–Huygens’ formula can be found illustrated in its conventional form in Figure 7.7. These expressions can be converted into an equivalent but more compact form that provides the asymptotic far electric and magnetic fields. The details of this can be found presented below in Appendix G. The Kirchhoff–Huygens theory is exact, provided that the field is known exactly over an entire closed surface. The closed surface can take the form of an infinite plane together with an infinite radius hemisphere. If the source is finite then, from the radiation condition, it can be seen that no contribution to the total field arises from any part of the hemispherical portion of the surface. For these equations to be implemented numerically, we need to have an effective way of calculating the unit normal and elemental area for a general surface, Box 7.2 contains a simple method for evaluating these quantities from knowledge of the surface profile.

Box 7.2: Calculation of surface normal and elemental area Let the surface over which the electric and magnetic fields are known be expressed as, g ðx; y; zÞ ¼ 0

(7.55)

The outward pointing surface normal can be formed from the cross product of two non-parallel tangent vectors. Thus, provided that the Cartesian co-ordinates are tabulated such that, x ¼ f ðu; vÞ

(7.56)

y ¼ f ðu; vÞ

(7.57)

z ¼ f ðu; vÞ

(7.58)

Computational electromagnetic model

311

then two tangential vectors a and b can be formed from, dx b dy b dz b i þ j þ k du du du dx dy dz b ¼ bi þ bj þ b k dv dv dv a ¼

(7.59) (7.60)

Hence, the surface unit normal can be obtained from, b n ¼

a b ja  b j

(7.61)

The inward-pointing unit surface normal will be anti-parallel with this vector. Finally, the elemental area can be found by evaluating, da ¼ ja  b j

(7.62)

Again, as with the vector Huygens formula, this field propagation algorithm can be used to produce near-field simulations over any desired sampling surface. Usefully, the requirement for the field point to be displaced from the Huygens surface by a few wavelengths is removed. In contrast then, these expressions yield reliable predictions within the reactive near-field region. However, the inverse r term is still present and although these expressions yield reliable fields for smaller ranges than is the case for the vector Huygens formula, it will still cause these formulas to become unstable for very small values of r. This is an unfortunate consequence of the fact that electromagnetic field theory is essentially a classical theory and as such ignores quantum effects. If r becomes very small, then the field point and the source are essentially separated by distances that approach the atomic scale and instead a quantum–mechanical formulation should be adopted. Thus, whilst there exist algorithms that seek to resolve the singularity encountered when r is exactly zero, [6] no physical theory is known to yield valid unambiguous results on such sub-nuclear scale. The utility of these expressions when it comes to the simulation of near-field measurements is that they permit the field everywhere in space to be computed from knowledge of the fields over a finite closed surface. This enables full-wave three-dimensional computational electromagnetic simulation tools to be used to solve for the fields around some, comparatively small tractable, radiating structure whereupon the Kirchhoff–Huygens method can be used to calculate the fields resulting from this radiator over a somewhat larger surface. Such simulation techniques can include the finite difference time domain method, the finite element method or the method of moments. In this way, measurement simulations of great accuracy can be produced comparatively simply and easily using, essentially rigorous, but computationally intensive solvers.

312

Principles of planar near-field antenna measurements

To illustrate the applicability of the Kirchhoff–Huygens formula, an openended rectangular WR90 field probe radiating at 10 GHz was modelled. A waveguide was chosen, as it is a low gain instrument with a front to back ratio of only ca. 10 dB. As such, it is not an instrument generally considered as being amenable for characterisation in a planar facility. Importantly, in order that the technique could be verified, the waveguide section was placed in free space, i.e. the aperture plane was not set in a perfectly conducting ground plane. The Cartesian components of the electric and magnetic fields were then obtained from the finite element-modelling package over the surface of a closed ‘Huygens’ box. The y-polarised component of the electric field is shown in Figures 7.8 and 7.9. Here, the field within the waveguide section has been set to zero. If the field within the waveguide section had not been set to zero, the total far-field pattern would be zero as the source would be located outside the surface of integration. This would imply that the same amount of field would have been flowing into the

60 Top view

z (mm)

50

40

45 40 35 50

30 50 0

0

y (mm)

20

x (mm)

–50 –50

10

Figure 7.8 Near-field power sampled over cubic surface viewed from above

60 Bottom view 50

40 z (mm)

45 40 35 –50

–50 0

0 x (mm)

30

20 50

50

y (mm) 10

Figure 7.9 Near-field power sampled over cubic surface viewed from below

Computational electromagnetic model

313

Power (dB)

volume as was flowing out. These fields were transformed to the far-field and compared with the far-field patterns predicted by the full-wave simulation. The resulting patterns can be found presented in Figures 7.10 and 7.11. The agreement is clearly encouraging with an equivalent multipath level (EMPL) less than –50 dB everywhere and an average value of approximately –70 dB. Here, as with the Fourier implementation, corner and edge element averaging has been employed in order that the intersection between adjacent planes could be rigorously handled. In the following plots, the red trace represents the far-field pattern function as predicted by a proprietary full-wave three-dimensional CEM solver, the cyan trace is the far-field as predicted by only the top plane, the blue trace represents the far-field as Ey freq 10 (GHz) 5 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 –55 –60 –65 –70 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 El (deg)

CEM solver Top plane Top & side planes All sides HFSS-All sides EMPL

30

40

50

60

70

80

90

Power (dB)

Figure 7.10 Far-field full sphere elevation pattern cut Ey Freq 10 (GHz) 5 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 –55 –60 –65 –70 –180 –160 –140 –120 –100 –80 –60 –40 –20 0 20 40 Az (deg)

CEM solver Top plane Top & side planes All sides HFSS-All sides EMPL

60

80

100 120 140 160 180

Figure 7.11 Far-field full sphere azimuth pattern cut

314

Principles of planar near-field antenna measurements

predicted by the top and four side planes, the black trace represents the far-field pattern as predicted by all of the sides and the magenta trace is the EMPL between the far-field pattern from all sides and that produced by the simulation. Clearly, the far-field pattern function is free from spurious ripple, and full sphere far-field results have been obtained. Importantly, as illustrated in Figure 7.11, the correct far-field pattern function was obtained only when the bottom plane was included. The structure that is evident within the EMPL trace is probably a result of the fact that the simulation produces only a quasi-far-field pattern, with a radius of 20 m, rather than a true asymptotic far-field pattern. The requirement for the inclusion of the bottom plane follows from the requirement to perform the pattern integration over a closed surface. This also illustrates how a full-wave solver can be used to initialise the simulation of a near-field measurement that is much larger than that which could be reliably placed within the solution space of the simulation tool, or that requires a greater degree of flexibility than that which is available from the general purpose tool.

7.7 Current elements method The current elements method is an alternative field propagation method to those developed above. The current element method replaces the fields with an equivalent surface current density Js which is used as an equivalent source to the original fields. This is an approach that has found great utility in the design and simulation of reflector antennas, including compact antenna test ranges. The equivalent surface current density can be obtained from the magnetic fields and the surface unit normal using, cf. Chapter 2, Js¼b n H

(7.63)

Here, b n denotes the outward pointing unit surface normal and H the magnetic fields at the surface s where the source current, Idl = Jsds. As is shown in Box 7.3, the fields radiated by an electric current element can be expressed as,  1  ry  J s ds 4p

dH ðPÞ ¼

(7.64)

Here, we are computing the elemental magnetic field from the vector potential where, y¼

ejk0 r r0

0

So that, cf. previous section,
 1 ry ¼ jk0 þ 0 yb er r

(7.65)

(7.66)

Computational electromagnetic model

315

Thus,
 0  1 1 ejk0 r  b dH ðPÞ ¼ jk0 þ 0 e r  J s ds r0 4p r

(7.67)

This is an exact expression. Integrating the elemental magnetic field results in the total magnetic field, H ðP Þ ¼

1 4p

ð

  1 þ jk0 r0 jk0 r0 b e ds er Js r0 2 S

(7.68)

The corresponding elemental electric fields can be obtained, approximately, from the elemental magnetic fields using the far-field TEM condition. That is to say, approximately, dE ¼ Z0 ðdH  b e rÞ

(7.69)

This is a very good approximation providing that the field point is more than a few wavelengths from the elemental source. As before, the total electric and magnetic fields can be obtained from the elemental fields by integrating across the surface of the impressed sources. Conversely, if the field point is close to the surface, then the electric field can be obtained using, cf. Box 7.3,

 jwm0 1 J s y þ 2 J s  r ry ds dE ðPÞ ¼  4p k0

(7.70)

Or, E ðP Þ ¼ 

Box 7.3:

jwm0 4p

ð s

J sy þ


  1 1 yb J  r jk þ u ds 0 r0 k02 s

(7.71)

Derivation of the current element method

The time-harmonic Maxwell electromagnetic field equations with source volume current density Js and source charge density r can be expressed as, cf. Chapter 2, r  E ¼ jwmH

(7.72)

r  H ¼ jweE þ J s

(7.73)

er  E ¼ r

(7.74)

mr  H ¼ 0

(7.75)

316

Principles of planar near-field antenna measurements

Since mr  H ¼ 0, H can be represented by the curl of another vector as it obeys the vector identity, r  ðr  A Þ ¼ 0

(7.76)

This is discussed in greater detail below at the bottom of this box. Thus, we may define the magnetic vector potential A as, rA ¼H

(7.77)

Substituting into (7.72) yields, r  E ¼ jwmðr  A Þ

(7.78)

Or, r  ðE þ jwmA Þ ¼ 0

(7.79)

The term in brackets can be considered an E field and since an electric field E is given by a scalar potential, we can define the scalar electric potential F such that, E þ jwmA ¼ rF

(7.80)

Or, E ¼ jwmA  rF

(7.81)

Where the vector identity, r  rF ¼ 0

(7.82)

holds. Substituting H ¼ r  A into (7.73) yields, r  r  A ¼ jweE þ J s

(7.83)

Using the vector identity, r  r  A ¼ r ð r  A Þ  r2 A

(7.84)

Then, rðr  A Þ  r2 A ¼ jweE þ J s

(7.85)

Or from the expression of the E field in (7.81) above, rðr  A Þ  r2 A ¼ J s  jweðjwmA þ rFÞ

(7.86)

Hence, rðr  A Þ  r2 A ¼ J s  jwerF þ w2 meA

(7.87)

Computational electromagnetic model

317

Rearranging yields, J s ¼ r2 A  rðr  A Þ  jwerF þ w2 meA

(7.88)

Collecting terms, r2 A þ w2 meA  rðjweF þ r  A Þ ¼ J s

(7.89)

However, A has not been fully specified as no divergence relationship has been defined. Thus, let us define the divergence of A as being, r  A ¼ jweF

(7.90)

This is the so-called Lorentz condition, jweF þ r  A ¼ 0

(7.91)

We chose this definition so that we could reduce our expression (7.92). Thus, r2 A þ w2 meA ¼ J s

(7.92)

So that A can be found for a given Js. A Green’s function solution to this equation for a single current element Js is, 1 ejk0 r 1 ¼ Js J y r 4p 4p s

A ¼

(7.93)

which gives the radiated magnetic vector potential due to the single current element, where we have defined the Green’s function to be, y¼

ejk0 r r

(7.94)

Thus, for a surface s, we can sum up each current element on s and write that, ð 1 A ¼ J yds (7.95) 4p s s which is often termed the radiation integral. As we defined A such that, H ¼rA Then, H ¼

ð   1 r  J s y ds 4p s

(7.96)

(7.97)

Using the standard differential vector identity, r  ðgF Þ ¼ rg  F þ g ðr  F Þ

(7.98)

318

Principles of planar near-field antenna measurements

Then,     r  yJ s ¼ ry  J s þ y r  J s

(7.99)

However, as r operates in the far-field (which need only be a wavelength or two away from the current element) and Js only on the surface then, rJs ¼0 So that,   r  yJ s ¼ ry  J s

(7.100)

(7.101)

Hence, ð 1 H ¼ ry  J s ds 4p s

(7.102)

As r operates only on the coordinates of the source point then,
 1 (7.103) ry ¼ jk0 þ ybe r r Where, y¼

ejk0 r r

And b e r is the unit vector in the direction of r. Hence,
 ð  1  1 b H ¼ e r  J s jk0 þ yds 4p s r

(7.104)

(7.105)

The corresponding elemental electric fields can be obtained, approximately, from the elemental magnetic fields using the far-field TEM condition. However, it can also be obtained rigorously as follows. Let us recall the definition of scalar potential. We may therefore write that, E þ jwmA ¼ rF

(7.106)

Then, E ¼ jwmA  rF

(7.107)

As, r  A ¼ jweF

(7.108)

rA ¼ F jwe

(7.109)

Computational electromagnetic model

319

Then, rð r  A Þ ¼ rF jwe

(7.110)

Thus, E ¼ jwmA þ

r ðr  A Þ jwe

Recalling that for a surface, ð 1 A ¼ J yds 4p s s

(7.111)

(7.112)

Then, E ¼

ð ð   jwm 1 1 J s yds þ r r  J s y ds 4p s jwe 4p s

Factorising yields,

ð   jwm 1 r r  J s y ds E ¼ J sy  4p s ðjweÞðjwmÞ

(7.113)

(7.114)

Now, 1 c2 c2 l2 1 ¼ ¼ ¼ ¼ 2 2 2 2 2 2 4p f 4p w em w k0

(7.115)

Thus, jwm E ¼ 4p

ð s

 1 J s y þ 2 J s  r ry ds k0

(7.116)

Here, r operates only on the coordinates of the source point so that as noted above,
 1 (7.117) er ry ¼ jk0 þ yb r In summary, we needed to find the electric and magnetic fields radiated by a current distribution Js on a surface enclosing the antenna. Using Maxwell’s equations directly involves solving two coupled differential equations. So, to simplify the problem, we introduce a MAGNETIC VECTOR POTENTIAL A which provides a convenient intermediate step leading to a single equation for the radiation field of the antenna due to Js.

320

Principles of planar near-field antenna measurements

Let us now examine the mathematical procedure we used above in a little more detail. As noted, we defined the magnetic vector potential, A, such that the curl of A equalled the magnetic field H, i.e., rA ¼H

(7.118)

As we shall now show, we can have different vector potentials that have the same H. This is because H is determined from A by differentiation, cf. the Cartesian case where,



bi b bj k







 @Ax @Az H ¼

@ @ @

@Az @Ay b  þ  j

@x @y @z @y @z @z @x

A A A

x y z ¼ bi
 @Ay @Ax þb  (7.119) k @x @y Thus, adding a constant to A will not change the magnetic field H. However, we can show that there is more flexibility here than this. If we have a vector potential A that gives the correct magnetic field H for a real situation, we can consider what other different A0 would also give rise to the same magnetic field. That is to say both A and A0 must have the same curl, i.e. r A ¼ rA0 ¼ H

(7.120)

or equivalently, r  A 0  r  A ¼ r  ðA 0  A Þ ¼ 0

(7.121)

Now, if the curl of a vector is zero, it must be the gradient of some scalar field, say F, since, r  ðrFÞ ¼ 0

(7.122)

Thus, r  ðA 0  A Þ ¼ r  ðrFÞ

(7.123)

So that if A is a valid vector potential, then so too is A0 for any scalar field F as, A 0 ¼ A þ rF

(7.124)

with both A and A0 leading to exactly the same magnetic field H. As we have so much flexibility in the choice of A, it is preferable if we select an A which makes the mathematics more convenient. We do that as follows. Now, although A and

Computational electromagnetic model

321

A0 must have the same curl, they do not need to have the same divergence as, r  A 0 ¼ r  A þ r2 F

(7.125)

0

Thus, we can make rA equal to anything at all depending upon the choice of F. In our case, we have chosen the divergence of vector A to be proportional to the negative of the scalar field (where we have omitted the prime on A), r  A ¼ jweF

(7.126)

We did this purely for mathematical convenience so that we could make a term vanish and enable us to more easily relate the surface current Js to the vector potential A. Specifically, so that we could write that, jweF þ r  A ¼ 0

(7.127)

which we called, the Lorentz condition. In truth, the right choice will depend upon the particular problem that we are considering but here, this was the appropriate one. Thus, this gauge condition is merely a mathematical procedure for coping with redundant degrees of freedom. The utility of this approach can be illustrated with the example of a 28 GHz, 5G New Radio (NR) massive-MIMO antenna. The near electric and magnetic fields were exported from a three-dimensional full-wave computational electromagnetic solver over a plane that was 0.1 wavelengths from the antenna, cf. Figures 7.12 and 7.13. This relatively small distance was selected so as to minimise truncation effects and was transformed to the far-field using the plane-wave-spectrum method and the current elements method.

0 60

–5 –10

40

60

150

40

100

20

50

0

0

0

–25 –30

–20

–20

–50

–40

–100

–60

–150

Phase (deg)

y (m)

–20

y (m)

20

Amplitude (dB)

–15

–35 –40

–40 –45

–60 –50

0 x (mm)

50

–50

Figure 7.12 Ey near-field amplitude obtained from simulation

–50

0 x (mm)

50

Figure 7.13 Ey near-field phase obtained from simulation

322

Principles of planar near-field antenna measurements

The respective far-electric-fields can be found presented in Figures 7.14–7.25. Here, from the inspection of these figures, some differences are apparent, as is expected based upon the differences between the transformations; however, the overall degree of agreement is very encouraging. Although this approach is useful for CEM simulation, when used as a basis for a near-field to far-field transformation of measured data, the issue of deriving the magnetic fields arises. If a near-field probe is used that samples the magnetic nearfields, then this may present a solution. However, typically the probes that are used sample the near-electric fields in which case we would need to obtain the magnetic fields from those measured electric fields. This can be resolved in a number of

1

1

0

0.8

0

0.8 –10

–10 0.6

–20 –30

0 –40

–0.2 –0.4

–20

0.4 0.2 v

–30

0 –40

–0.2 –0.4

–50

–0.6

–50

–0.6 –60

–60

–0.8 –1 –1 –0.8 –0.6 –0.4 –0.2

–0.8 –1 –1 –0.8 –0.6 –0.4 –0.2

–70 0 u

0.2 0.4 0.6 0.8

1

Figure 7.14 Ex far-field amplitude obtained using the PWS method

1

1 150

0.6

100

0.4

0.8

150

0.6

100

0.4

0

–0.2

–50

–0.4

50

0.2 v

0

Phase (deg)

50

0.2 v

0.2 0.4 0.6 0.8

Figure 7.15 Ex far-field amplitude obtained using the current elements method

1 0.8

0

0

–0.2

–50

–0.4 –100

–0.6 –0.8 –1 –1 –0.8 –0.6 –0.4 –0.2

–70 0 u

Phase (deg)

v

0.2

Amplitude (dB)

0.4

Amplitude (dB)

0.6

–150 0 u

0.2 0.4 0.6 0.8

1

–100

–0.6 –0.8 –1 –1 –0.8 –0.6 –0.4 –0.2

–150 0 u

0.2 0.4 0.6 0.8

1

Figure 7.16 Ex far-field phase Figure 7.17 Ex far-field phase obtained obtained using the PWS using the current elements method method

Computational electromagnetic model 1

323

1

0

0

0.8

0.8

–10

–10

0.6 –20 –30

0 –40

–0.2 –0.4

–20

0.4 0.2 v

v

0.2

Amplitude (dB)

0.4

–30

0 –40

–0.2 –0.4

–50

Amplitude (dB)

0.6

–50

–0.6

–0.6

–60

–60

–0.8

–0.8 –1 –1 –0.8 –0.6 –0.4 –0.2

0 u

0.2 0.4 0.6 0.8

–1 –1 –0.8 –0.6 –0.4 –0.2

–70

1

0 u

0.2 0.4 0.6 0.8

–70

1

Figure 7.19 Ey far-field amplitude Figure 7.18 Ey far-field amplitude obtained using the current obtained using the PWS elements method method

PWS method Ey

1

0.8

0.6

0.6

0.4

0 –50

–0.4 –0.6

–100

–0.8

–150

50

0.2 v

0

–0.2

Phase (deg)

50

0.2 v

100

100

0.4

–1 –1 –0.8 –0.6 –0.4 –0.2

150

0.8

150

0

0

–0.2

–50

–0.4 –100

–0.6 –0.8

0 u

0.2 0.4 0.6 0.8

1

Figure 7.20 Ey far-field phase obtained using the PWS method

Phase (deg)

1

–1 –1 –0.8 –0.6 –0.4 –0.2

–150 0 u

0.2 0.4 0.6 0.8

1

Figure 7.21 Ey far-field phase obtained using the current elements method

ways however most commonly this is recast in terms of an electric field integral equation to relate the near-fields to the equivalent magnetic currents [7]. This can be converted to a matrix form through the method of moments which can be solved using the conjugate gradient method, or a least square solution can be sought if the matrix is rectangular. This method is developed in the next section. In summary, the current elements method provides a very accurate method for the propagation of fields providing the magnetic field is available to initiate the process. Fortunately, modern full-wave CEM solvers provide this information and as such this method finds considerable utility in simulating larger near-field measurements of more representative antennas.

324

Principles of planar near-field antenna measurements 0

0 PWS method Current elements

–10

–10

–15

–15

–20 –25 –30

–25 –30 –35

–40

–40

–45

–45 –0.8

–0.6

–0.4

–0.2

0 u

0.2

0.4

0.6

0.8

–50 –1

1

Figure 7.22 H-cut copolar amplitude comparison of PWS and current elements method

–0.6

–0.4

–0.2

0 v

0.2

0.4

50

50 Phase (deg)

100

0

0.6

0.8

1

PWS method Current elements

150

100

0

–50

–50

–100

–100 –150

–150 –1

–0.8

Figure 7.23 V-cut copolar amplitude comparison of PWS and current elements method

PWS method Current elements

150

Phase (deg)

–20

–35

–50 –1

PWS method Current elements

–5

Amplitude (dB)

Amplitude (dB)

–5

–0.8

–0.6

–0.4

–0.2

0 u

0.2

0.4

0.6

0.8

1

Figure 7.24 H-cut copolar phase comparison of PWS and current elements method

–1

–0.8

–0.6

–0.4

–0.2

0 v

0.2

0.4

0.6

0.8

1

Figure 7.25 V-cut copolar phase comparison of PWS and current elements method

7.8 Equivalent currents method – near-field to far-field transform, antenna diagnostics, and range reflection suppression As was shown in the preceding section, it is possible to transform near-fields to the far-field using a transform that utilised an equivalent surface current. This is a very powerful an accurate method; however, it has the inherent disadvantage of utilising the near-magnetic-fields, rather than the near-electric-fields that are usually the starting point when taking near-field measurements. This section presents a related

Computational electromagnetic model

325

technique that is predicated on similar ideas, although here we shall assume that we have observables that are proportional to the near-electric-field. As we shall show, in the equivalent magnetic current approach, we use conventional near-field data to obtain an equivalent magnetic current over a convenient surface that encloses the AUT which can then be used to obtain the desired electric fields elsewhere in space including the far-field. An electric field integral equation is derived which relates the measured near-field to the equivalent magnetic currents which is solved using an efficient, all be it resource intensive, moment method procedure [8] with point matching which converts the integral equation into an equivalent matrix-equation which can be solved in one of a number of ways, e.g. by using the least squares conjugate gradient (LSQR) method [9]. The LSQR algorithm is a particular implementation of the conjugate gradient method. A discussion of this algorithm is beyond the scope of this text and is instead left to the open literature. This method can be used effectively to recover the currents in the near vicinity of an arbitraryshaped AUT as well as to suppress range reflections. Results are presented below and used to illustrate each of these features. First, as we have assumed above, during the development of the plane-wave spectrum method, let us assume that the antenna is placed in one half-space and is radiating into the other, forward half-space, with an infinite xy-plane dividing the two regions. If an infinite electric conducting sheet is postulated on the one side of the surface, then in this case, only the tangential components of the electric fields need be specified on the surface. The magnetic current is then provide by, (   2 E a ðr 0 Þ  b n over the aperture 0 J m ðr Þ ¼ (7.128) 0 elsewhere Now, the radiated fields must be determined in the presence of the conducting sheet which results in the solutions only being available for the forward half-space. However, as before, we are only concerned with the source and sink free, linear, homogeneous and isotropic region of space which we may assume corresponds to z  0 so this is not too much of an impediment. The electric vector potential F may be defined as [10] (cf. the previous section where analogously we defined a magnetic vector potential, A), F ðr Þ ¼

e 4p

ð

Jm ðr0 Þ jk0 R 0 e ds R S

(7.129)

Here, the free-space propagation constant is denoted by k0, primed variables are associated with the source point, and un-primed variables are associated with the field point thus the distance R can be expressed as, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (7.130) R ¼ jr  r0 j ¼ ðx  x0 Þ2 þ ðy  y0 Þ2 þ ðz  z0 Þ2

326

Principles of planar near-field antenna measurements We may obtain the electric fields from the electric vector potential F as [10], 1 E ð r Þ ¼  ð r0  F ð r 0 Þ Þ e

(7.131)

So that, E ðr Þ ¼ 


 ð Jm ðr0 Þ jk0 R 0 1 e r0  ds R 4p S

(7.132)

Or equivalently when exchanging the order of integration and differentiation and recalling that AB = BA then,  ð
1 1 E ðr Þ ¼ Jm ðr0 Þ  r0 ejk0 R ds0 (7.133) 4p S R The three-dimensional Green’s function can be expressed as,



0 jk0 r  r

jk0 R e e ¼ g ðr ; r 0 Þ ¼ R jr  r 0 j Thus, the crucial expression we require can be written as, ð 1 E ðr Þ ¼ Jm ðr0 Þ  r0 gðr ; r0 Þds0 4p S

(7.134)

(7.135)

Here E(r) is the field that is measured over the planar acquisition surface S located at a distance of a few wavelengths from the AUT, Jm denotes the surface magnetic current sheet that we seek to calculate that gives rise to the electric fields E(r) that we have measured in Cartesian coordinates and resolved onto a Cartesian polarisation basis and where the gradient operator can be expressed as, r0 ¼

@ @ @ be x0 þ 0 b e y0 þ 0 b e z0 0 @x @y @z

(7.136)

Thus we may write that where the reader is referred to Section 7.6 above for the detailed derivation of this, r0 y ¼


jk0 þ

1 R




 x  x0 y  y0 z  z0 b b b e x0 þ e y0 þ e z0 y R R R

(7.137)

Or,

 i
ejk0 R h 1 0 0 0 ðx  x Þbe x0 þ ðy  y Þb ry¼ e y0 þ ðz  z Þbe z0 jk0 þ R2 R 0

(7.138)

We have now derived the key integral equation that the equivalent currents method is predicated upon. Equivalently in the matrix form, we can write that,

Computational electromagnetic model

b be y

ex 0

J ð r Þ J ðr 0 Þ mx my Jm ðr0 Þ  r0 g ðr ; r0 Þ ¼

0 0

@gðr ; r Þ @g ðr ; r Þ

@x0 @y0



be z

0 J mz ðr Þ

@g ðr ; r0 Þ

@z0

327

(7.139)

Expanding yields,


0  @gðr ; r0 Þ 0 @g ðr ; r Þ J m ðr 0 Þ  r 0 g ð r ; r 0 Þ ¼ b e x J my ðr0 Þ  J ð r Þ mz @z0 @y0
0 0  0 @g ðr ; r Þ 0 @g ðr ; r Þ  J mx ðr Þ þ be y J mz ðr Þ @x0 @z0
0  @g ðr ; r0 Þ 0 @g ðr ; r Þ  J ð r Þ þb e z J mx ðr0 Þ my @y0 @x0 (7.140)

Hence, 2

@g ðr ; r0 Þ @z0

0 6 3 ð 6 E x ðr Þ 0 6 4 Ey ðr Þ 5 ¼ 1 6  @gðr ; r Þ 0 4p S 6 @z0 6 E z ðr Þ 0 4 @g ðr ; r Þ @g ðr ; r0 Þ @y0 @x0 2

3 @g ðr ; r0 Þ  2 3 @y0 7 7 Jmx ðr0 Þ @g ðr ; r0 Þ 7 74 Jmy ðr0 Þ 5ds0 7 @x0 7 Jmz ðr0 Þ 5 0 (7.141)

Typically, for planar scanning, only the x- and y-polarised near-electric field components are sampled, also recalling the comments above regarding the assumed boundary conditions where the z-polarised surface current component is zero, i.e. Jmz = 0 thus, 2 3 @g ðr ; r0 Þ



ð 0 1 6 E x ðr Þ 0 7 Jmx ðr0 Þ @z ds0 ¼ (7.142) 4 5 E y ðr Þ @g ðr ; r0 Þ Jmy ðr0 Þ 4p S  0 @z0 Where,
 @g ðr ; r0 Þ ejk0 R 1 0 b b ðz  z Þ jk0 þ e z0 ¼ e 0 R2 @z0 R z Hence we may write that,
 ð jk0 R 1 e 1 0 E x ðr Þ ¼ Jmy ðr0 Þds0 ðz  z Þ jk0 þ 4p S R2 R
 ð jk0 R 1 e 1 0 Jmx ðr0 Þds0 ðz  z Þ jk0 þ E y ðr Þ ¼  4p S R2 R

(7.143)

(7.144) (7.145)

328

Principles of planar near-field antenna measurements

Having now derived an expression for the measured x- and y-polarised fields Ex and Ey in terms of an integral involving the unknown equivalent surface currents Jmy and Jmx, a Method of Moments (MoM) approach can be employed to accomplish this. Here, we are only considering the propagating fields. We may now use the sampling theorem to replace the continuous electric fields with a set of samples spaced at a maximum of half-wavelength apart, and the current sheet with an array of fictitious magnetic dipoles that are also spaced at a maximum of half-wavelength apart. This latter assumption is equivalent to using a delta function as the expansion function for the current source within the integrand and enables us to replace the integration with a summation which in turn permits us to express this in an equivalent matrix form yielding, 2 3 3 2 32 Jmy ðr0 1 Þ G1;1 G1;2    G1;l Ex ðr1 Þ 6 Ex ðr2 Þ 7 6 G2;1 G2;2    G2;1 76 Jmy ðr0 2 Þ 7 6 7 7 6 76 (7.146) 6 .. 7 ¼ 6 .. 7 .. 76 .. .. .. 4 4 . 5 4 . 5 . . . .  5 Gk;1 Gk;2    Gk;l Ex ðrk Þ Jmy r0 l A more extensive consideration of this treatment can be found in [11,12]. Thus, when written compactly in matrix notation, we obtain,   ½Ex  ¼ ½G Jmy (7.147)   Ey ¼ ½G½Jmx  (7.148) Here, [E] and [Jm] are column vectors, while the elements in the matrix [G] can be obtained by evaluating,
 ð  1 ejk0 R  1 0 Gk;l ¼ jk z  z þ (7.149) ds0 k 0 l 4p Sk;l R2k;l Rk;l Where Sk,l is the area of the patch k,l. As commented above, if we assume that the measured fields are outside the reactive near-field region, which is a standard assumption that is necessitated if probe compensation is to be used and practically, if AUT-to-probe multiple reflections are to be kept to acceptable levels then we may recover the propagating near-fields at any point in the forward half-space, and in the true far-field without loss of rigour as we have an infinitesimal current element using,
  1 ejk0 R  1 0 (7.150) zk  zl jk0 þ Gk;l ¼ 4p R2k;l Rk;l Here it must be remembered that the reconstructed currents must be more than a wavelength away from the measurement surface for reliable results to be obtained. We may thus obtain the excitation coefficients for the fictitious magnetic dipole array if we can solve the system of equations,   (7.151) ½G1 ½Ex  ¼ Jmy   ½G1 Ey ¼ ½Jmx  (7.152)

Computational electromagnetic model

329

If the number of measured near-field points equals the number of fictitious magnetic dipoles, then this system of equations may be solved uniquely. If the number of measured points is greater than the number of dipoles, i.e. current elements, then this may be solved in a least squares sense, e.g. by using the LSQR method [9]. The LSQR algorithm is a particular implementation of the conjugate gradient method that has proven to be particularly effective in dealing with these sorts of problems. Once the coefficients for the dipole array have been determined then we may compute a new [G] matrix that will produce the electric fields at whichever points in space we require, including the far-field using,   (7.153) ½Ex  ¼ ½G Jmy   Ey ¼ ½G½Jmx  (7.154) A degree of care should be exercised in selecting the position and number of current elements as it is relatively easy to obtain an unreliable solution. For example, if the reconstructed array is too large, then unreliable far-field can be produced. Very quickly, the [G] matrix can become excessively large as it has as many columns as there are fictitious magnetic dipoles, and as many rows as there are measured near-field points. The true far-field can be obtained when using the following approximation for the far-field Green’s function where we have assumed that R > r0 , g ðr ; r 0 Þ ¼

ejk0 R ejk0 r jk0 ðux0 þ vy0 þ wz0 Þ  e R r

(7.155)

Here, typically, the unimportant spherical phase factor and inverse r term would be suppressed, and u, v, and w are direction cosines to the point in the far-field. To illustrate the use of this method, we have simulated the orthogonal, tangential, planar near electric field amplitude and phase data which can be found in Figures 7.26–7.29. Near-field Ex

Near-field Ex

0

1.5

1.5 150

–5 1

1

–10

100

0

–25 –30

–0.5 –35

0.5

0

0 –50

–0.5

–40

–1

50

–100

–1

–45 –1.5 –1.5

(a)

–1

–0.5

0 x (m)

0.5

1

1.5

–50

Phase (deg)

y (m)

–20

y (m)

0.5

Amplitude (dB)

–15

–150 –1.5 –1.5

(b)

–1

–0.5

0 x (m)

0.5

1

1.5

Figure 7.26 Near-field horizontally polarised simulated planar near-fields of a 5 GHz offset reflector antenna. (a) PNF Ex amplitude [dB] and (b) PNF Ex phase [deg].

330

Principles of planar near-field antenna measurements Near-field Ey

Near-field Ey

0

1.5

1.5 150

–5 1

1

–10

100

–25 –30

–0.5

50

0

0

Phase (deg)

0

0.5 y (m)

y (m)

–20

Amplitude (dB)

–15 0.5

–50

–0.5 –35 –40

–1

–100

–1

–45 –1.5 –1.5

–1

(a)

–0.5

0 x (m)

0.5

1

1.5

–150 –1.5 –1.5

–50

–1

–0.5

(b)

0.5

0 x (m)

1

1.5

Figure 7.27 Near-field vertically polarised simulated planar near-fields of a 5 GHz offset reflector antenna. (a) PNF Ey amplitude [dB] and (b) PNF Ey Phase [deg].

Reconstructed Mx

0.5

0

Reconstructed Mx

0.5

150

–5 0.3

–10

0.3

100

–25 –0.1

–30

50

0.1

0 –0.1

Phase (deg)

y (m)

–20

y (m)

0.1

Amplitude (dB)

–15

–50

–35 –0.3

–40

–100

–0.3

–45 –0.5 –0.5

(a)

–0.3

–0.1

x (m)

0.1

0.3

0.5

–50

–150 –0.5 –0.5

(b)

–0.3

–0.1

x (m)

0.1

0.3

0.5

Figure 7.28 x-polarised reconstructed equivalent currents over the source plane which here was chosen to contain the vertex of the offset parabolic reflector. (a) Mx amplitude [dB] and (b) Mx phase [deg].

Here, the near-fields are tabulated on a plaid, monotonic, and equally spaced plain rectilinear grid. This then can be considered to be a simulation of a horizontally co-polarised antenna measurement that is truncated, as it is necessarily finitely large, but is in all other respects represents our truth model. Although only the orthogonal tangential electric fields are presented, this method may be used to compute the complete near electromagnetic six-vector. Additionally, the same simulation technique can be used to compute the ideal, i.e. reference, truncation-free, far-field pattern against which we can compare all of our transformed data.

Computational electromagnetic model

331

First-order truncation effect set in at 70 . Here, the measurement plane was 0.5 m (ca. eight wavelengths) in front of the antenna. Here, the fictitious magnetic dipoles were spaced half wavelength apart in the xy-plane. The equivalent far-field was then computed from the reconstructed array of fictitious current dipoles. The far-fields can be seen presented and compared against reference far-fields that were obtained directly from the antenna simulation software and can be seen presented in Figures 7.30–7.37 which compare respective copolar and cross-polar amplitude and phase patterns that are presented in the form

Reconstructed My

0.5

0

Reconstructed My

0.5

150

–5 0.3

–10

0.3

100

–25 –0.1

–30

50

0.1

0 –0.1

Phase (deg)

y (m)

–20

y (m)

0.1

Amplitude (dB)

–15

–50

–35 –0.3

–100

–0.3

–40 –45

–0.5 –0.5

–0.3

–0.1

(a)

x (m)

0.1

0.3

0.5

–150 –0.5 –0.5

–50

–0.3

–0.1

(b)

x (m)

0.1

0.3

0.5

Figure 7.29 y-polarised reconstructed equivalent currents over the source plane which here was chosen to contain the vertex of the offset parabolic reflector. (a) My amplitude [dB] and (b) My phase [deg].

–10

60

Far-field Ex

90

0

0 –10

60

–20

–20 30

–40

–30

El (deg)

El (deg)

–30 0

Amplitude (dB)

30

0 –40 –30

–30

–50

–50 –60

–90 –90

(a)

Amplitude (dB)

Far-field Ex

90

–60 –70 –60

–30

0 Az (deg)

30

60

90

–60

–90 –90

(b)

–60 –70 –60

–30

0 Az (deg)

30

60

90

Figure 7.30 Comparison between copolar far-field amplitude patterns comparing reference and equivalent currents transformed data. (a) Reference far-field Ex amplitude [dB] and (b) far-field Ex amplitude [dB] from equivalent current method.

Principles of planar near-field antenna measurements Far-field Ex

90

Far-field Ex

90

150

150 60

60 100

100 30

0

0

El (deg)

50 Phase (deg)

El (deg)

30

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50 0

0

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Phase (deg)

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–60

–30

(a)

0 Az (deg)

30

60

–90 –90

90

–60

–30

(b)

0 Az (deg)

30

60

90

Figure 7.31 Comparison between copolar far-field phase patterns comparing reference and equivalent currents transformed data. (a) Reference far-field Ex phase [deg] and (b) far-field Ex phase [deg] from equivalent current method.

Far-field Ey

90

60

0

90

–10

60

Far-field Ey

0 –10 –20

–20

–30

El (deg)

–40

Amplitude (dB)

El (deg)

–30 0

0 –40 –30

–30

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–90 –90

(a)

Amplitude (dB)

30

30

–60 –70 –60

–30

0 Az (deg)

30

60

90

–60

–90 –90

(b)

–60 –70 –60

–30

0 Az (deg)

30

60

90

Figure 7.32 Comparison between cross-polar far-field amplitude patterns comparing reference and equivalent currents transformed data. (a) Reference far-field Ey amplitude [dB] and (b) far-field Ey amplitude [dB] from equivalent current method.

of two-dimensional false colour checkerboard plots, or one-dimensional cardinal cuts. From inspection of these results, it is clear that they are in very encouraging agreement. Furthermore, it can be seen that the reference, i.e. truncation free, far-field pattern and the transformed measurement, starts to diverge at circa

70 in azimuth and elevation, which is where we would expect the first-order truncation effect set in.

Computational electromagnetic model Far-field Ey

90

Far-field Ey

90 150

150

60

60 100

100 30

0 –50

–30

El (deg)

0

Phase (deg)

50

50 0

0

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Phase (deg)

30 El (deg)

333

–100 –60

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–60

–30

(a)

0 Az (deg)

30

60

–150 –90 –90

90

–60

–30

(b)

0 Az (deg)

30

60

90

Figure 7.33 Comparison between cross-polar far-field phase patterns comparing reference and equivalent currents transformed data. (a) Reference far-field Ey phase [deg] and (b) far-field Ex phase [deg] from equivalent current method.

Far-field Ex, RMS dB diff = –64.2 (dB)

0

Far-field Ex 150

–10 100 Phase (deg)

Amplitude (dB)

–20 –30 –40

50 0 –50

–50 –100 –60 –70

Reference Transform equivalent currents dB diff

–80

–60

–40

–20

0 20 Az (deg)

40

60

80

Reference Transform equivalent currents

–150 –80

–60

–40

–20

0 20 Az (deg)

40

60

80

Figure 7.34 Comparison of copolar far-field azimuth cuts comparing reference and equivalent currents transformed amplitude (left) and phase (right) patterns. (a) Amplitude comparison and (b) phase comparison.

Next, we examine the PNF measurement of a NRL compliant x-band SGH at 8.2 GHz. In this measurement, the near-field was acquired so that the far-field pattern would be considered to be valid out to ca. 60 in azimuth and elevation. The measurements were taken in an open laboratory environment where the test site and positioning equipment were devoid of radar-absorbent material. This particular point is expounded upon below. The measured near-field amplitude and phase horizontally polarised near electric-field component can be seen presented in

334

Principles of planar near-field antenna measurements Far-field Ex, RMS dB diff = –51.7 (dB)

0

Far-field Ex 150

–10 100 Phase (deg)

Amplitude (dB)

–20 –30 –40

50 0 –50

–50 –100 –60 –70

Reference Transform equivalent currents dB diff

–80

–60

–40

–20

0 20 El (deg)

40

60

Reference Transform equivalent currents

–150 –80

80

–60

–40

–20

0 20 El (deg)

40

60

80

Figure 7.35 Comparison of copolar far-field elevation cuts comparing reference and equivalent currents transformed amplitude (left) and phase (right) patterns. (a) Amplitude comparison and (b) phase comparison. Far-field Ey, RMS dB diff = –67.7 (dB)

0

Far-field Ey 150

–10 100 Phase (deg)

Amplitude (dB)

–20 –30 –40

50 0 –50

–50 –100 –60 –70

Reference Transform equivalent currents dB diff

–80

–60

–40

–20

0 20 Az (deg)

40

60

80

Reference Transform equivalent currents

–150 –80

–60

–40

–20

0 20 Az (deg)

40

60

80

Figure 7.36 Comparison of cross-polar far-field azimuth cuts comparing reference and equivalent currents transformed amplitude (left) and phase (right) patterns. (a) Amplitude comparison and (b) phase comparison.

Figure 7.38. Conversely, the reconstructed fictitious magnetic dipole array amplitude and phase values can be seen presented in Figure 7.39. This was used to compute the equivalent far-field which can be seen compared with equivalent pattern data as obtained using the conventional plane-wave spectrum-based nearfield to far-field transform, cf. Figures 7.40 to 7.43 inclusive. Here, the far-field patterns can be seen presented without probe compensated being applied as probe compensation is omitted from the standard equivalent currents method. The equivalent current-derived far-fields have been effectively

Computational electromagnetic model Far-field Ey

Far-field Ey, RMS dB diff = –139.2 (dB)

0

335

Reference Transform equivalent currents dB diff

150 100 50

Phase (deg)

Amplitude (dB)

–50

0 –50

–100

–100 Reference Transform equivalent currents

–150 –150

–80

–60

–40

–20

0 20 El (deg)

40

60

–80

80

–60

–40

–20

0 20 El (deg)

40

60

80

Figure 7.37 Comparison of cross-polar far-field elevation cuts comparing reference and equivalent currents transformed amplitude (left) and phase (right) patterns. (a) Amplitude comparison and (b) Phase comparison.

Near-field Ex

0.5

0

Near-field Ex

0.5

150

–5 0.3

–10

0.3

100

–30

–25

0

–0.1

Phase (deg)

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50

0.1 y (m)

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y (m)

0.1

Amplitude (dB)

–15

–50

–35 –0.3

–40

–100

–0.3

–45 –0.5 –0.5

(a)

–0.3

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0.1 x (m)

0.3

0.5

–50

–150 –0.5 –0.5

(b)

–0.3

–0.1

0.1

0.3

0.5

x (m)

Figure 7.38 Near-field horizontally polarised planar near electric-field of an x-band NRL compliant x-band SGH. (a) PNF Ex amplitude [dB] and (b) PNF Ex phase [deg].

smoothed by the process of distilling the measured data down to an array of magnetic dipoles that comprise far fewer elements than the original near-field measurement contained samples and that was distributed over a much smaller region of space. Dipoles may be placed with half-wavelength separation, or more finely spaced if desired for the purposes of visualizing the field, although the actual spatial resolution is, by virtue of the absence of reactive fields contained within the near-field measurement, still limited to a half-wavelength. Again, the array of magnetic dipoles must be displaced away from the measurement plane by an amount that is larger than a wavelength. In this instance, the distance between the

Principles of planar near-field antenna measurements Reconstructed My

0.15

0

Reconstructed My

0.15

150

–5 0.1

100

–25 –30

–0.05

–35

0.05 y (m)

–20 0

Amplitude (dB)

–15

0.05 y (m)

0.1

–10

0 –50

–0.05

–100

–40

–0.1

50

0

–0.1

–45 –0.15 –0.15

–0.1

(a)

–0.05

0 x (m)

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0.1

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0.15

Phase (deg)

336

–150 –0.15 –0.15

–0.1

(b)

–0.05

0 x (m)

0.05

0.1

0.15

Figure 7.39 y-polarised reconstructed equivalent currents over the source plane which here was chosen to intersect with the antenna aperture plane. (a) My amplitude [dB] and (b) My phase [deg].

0

Far-field Ex

60

0

–10 30

–40 –50

–30

–20 –30

El (deg)

0

Amplitude (dB)

–20 –30

El (deg)

–10 30

0 –40 –50

–30

–60 –60 –60

(a)

–30

0 Az (deg)

30

60

–70

Amplitude (dB)

Far-field Ex

60

–60 –60 –60

(b)

–30

0 Az (deg)

30

60

–70

Figure 7.40 x-polarised far-field amplitude pattern. (a) Far-field Ex amplitude [dB] from plane-wave spectrum method and (b) far-field Ex amplitude [dB] from equivalent current method showing effective scattering suppression. SGH and the near-field probe was 0.27 m, or just over seven wavelengths, which easily satisfied this requirement. The advantage of selecting the antenna aperture plane for the position of the fictitious magnetic dipole array is that the position in which the currents are most tightly constrained, thereby requiring the smallest number of dipoles reducing the computational effort, memory requirements, and aiding in the numerical stability of the method. For this reason, this is a recognised method that can be utilised for the purposes of attenuating range reflections, bias-leakage error, etc. This is a consequence of the ability of the method to restrict the allowed equivalent antenna sources to a tightly confined region with only those antenna fields which are consistent with the source region being reconstructed and included within the transform. Conversely, spurious echo contributions deriving

Computational electromagnetic model Far-field Ex

Far-field Ex

60 150

150

100

30

100

30

0

El (deg)

50 Phase (deg)

El (deg)

50 0

0

0

–50 –30

–100

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–100

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–30

0 Az (deg)

(a)

30

–150 –60 –60

60

Phase (deg)

60

337

–30

(b)

0 Az (deg)

30

60

Figure 7.41 x-polarised far-field phase pattern. (a) Far-field Ex phase [deg] from plane-wave spectrum method and (b) far-field Ex phase [deg] from equivalent current method showing effective scattering suppression.

Far-field Ex

–10

–10

–20

–20

–30 –40 –50

–30 –40 –50

–60

Plane wave spectrum Equivalent currents method

–70

(a)

Far-field Ex

0

Amplitude (dB)

Amplitude (dB)

0

–60

Plane wave spectrum Equivalent currents method

–70 –80

–60

–40

–20

0 20 Az (deg)

40

60

80

(b)

–80

–60

–40

–20

0 El (deg)

20

40

60

80

Figure 7.42 Comparison of x-polarised far-field amplitude pattern. (a) Far-field Ex amplitude [dB] from plane-wave spectrum method and (b) farfield Ex amplitude [dB] from equivalent current method.

from radiators situated outside of the source domain are suppressed [13,14] as is the case here where the measurements were taken in an open laboratory environment using a robotic positioning sub-system that was devoid of RF absorber. Thus, this method offers an alternative, single frequency, spatial filtering scattering suppression technique that does not require any special dedicated hardware or broad-band measurements. This method is, in many respects, similar in capability to the more commonly utilised, more widely adopted, mode filtering-based alternative, cf. [15,16], and for the planar case specifically, cf. [17] with a thorough development of both the theory and the practice being presented in [18]. As probe compensation is not naturally

338

Principles of planar near-field antenna measurements Far-field Ex 150

100

100

50

50 Phase (deg)

Phase (deg)

Far-field Ex 150

0

0

–50

–50

–100

–100 Plane wave spectrum Equivalent currents method

–150

(a)

–80

–60

–40

–20

0 20 Az (deg)

40

60

80

Plane wave spectrum Equivalent currents method

–150

(b)

–80

–60

–40

–20

0 20 Az (deg)

40

60

80

Figure 7.43 Comparison of x-polarised far-field phase pattern. (a) Far-field Ex amplitude [dB] from plane-wave spectrum method and (b) far-field Ex amplitude [dB] from equivalent current method.

incorporated into this method in its standard form, in the period since its first inception, it has perhaps found greatest utility taking probe compensated far-field data as its starting point and using it for non-destructive, non-intrusive antenna diagnostics, and for post-processing to suppress range reflections, etc. For a more general transform that includes full probe compensation, see for example Volume 2 of Ref. [13].

7.9 Generalised technique for the simulation of near-field antenna measurements The following simulation technique essentially entails a method of moment solution of a two antenna-coupling problem. Essentially, this only differs from a standard method of moments solution by choice of large, complex sources and the necessary use of generalised field propagation formula. The advantage of a method of moments type solution over, say FDTD or FE methods, is that with these other techniques, the space between the antennas, even if it is a vacuum, must be considered (i.e. meshed). Clearly, this can become prohibitive for the case where the radiators are separated by any finite distance. In contrast, a method of moments approach only considers the radiators so the separation between the antennas is unconnected to the amount of computational effort required in obtaining the solution. Provided that the antennas in conjunction with the circuits in which they are placed, including the source and load, are reciprocal then the mutual coupling between two antennas can be found from knowledge of the fields radiated by these antennas in isolation and from the reaction theorem [19]. If the fields radiated by an antenna are known over a convenient, arbitrary, closed surface that surrounds the antenna, then the field radiated by this antenna at any point in the region of space outside this surface can be obtained from the Kirchhoff–Huygens principal. As

Computational electromagnetic model Fields known over this surface (1)

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Fields to be evaluated over this surface (2)

Tx origin (x0, y0, z0)

origin (x1, y1, z1)

Figure 7.44 Geometry of Kirchhoff–Huygens reaction theorem illustrated schematically in Figure 7.44, if the Kirchhoff–Huygens formula is used to obtain the fields radiated by antenna 1 over a closed surface that surrounds antenna 2, provided that the fields radiated by antenna 2 are also known over this surface, then the surface integral form of the reaction theorem can be used to calculate the mutual impedance between these antennas. The mutual admittance between the antennas can then be found from the mutual impedance. These admittances can be used to populate an admittance matrix from which the equivalent normalised scattering matrix can be easily obtained. The transmission scattering coefficient, when evaluated with the two antennas suitably displaced, can be recognised as constituting a single sampling node within a near-field measurement. By repeating this calculation for every point in the near-field measurement, a full acquisition can be constructed. As the displacement and orientation of the coupled antenna system can be chosen arbitrarily, provided only that the enclosing surfaces do not intersect, any near-field (or quasi far-field) measurement system can be simulated. Hence, simulations produced from this procedure can be used to rigorously verify the corrections made for the modal receiving coefficients (e.g. plane-wave) of the scanning probe during the near-field to far-field transformation process. The Kirchhoff–Huygens method has been described in detail in Chapter 4 and an introduction to the reaction theorem and how it can be used to obtain the mutual coupling coefficient between two antennas is presented in the following section.

7.9.1 Mutual coupling and the reaction theorem Provided that the electric and magnetic field vectors (E1, H1) and (E2, H2) are of the same, i.e. monochromatic, frequency then the mutual impedance, Z21, between antenna 1 and 2 in the environment described by e, m can be expressed, from the reaction theorem, in terms of a surface integration as [19], refer to Section 7.11 for derivation: ð V21 1 Z21 ¼ ¼ ðE  H 1  E 1  H 2 Þ  b n ds (7.156) I11 I11 I22 S2 2

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Principles of planar near-field antenna measurements

Here, n is taken to denote the outward pointing surface unit normal. The subscript 1 denotes parameters associated with antenna 1 while the subscript 2 denotes quantities associated with antenna 2, i.e. S2 is a surface that encloses antenna 2, but not antenna 1. Here, I11 is the terminal current of antenna 1 when it transmits and similarly, I22 is the terminal current of antenna 2 when it transmits. From reciprocity, the mutual impedance, Z12 = Z21, and is related to the coupling between two antennas. Clearly then the mutual impedance will also be a function of the displacement between the antennas, their relative orientations, and their respective polarisation properties. The terminal current of these transmitting antennas, I11 and I22, can be obtained from the knowledge of the power injected at the port, P1 and P2 which is typically taken to be unity, and is specified within the modelling tool, and the port impedance Z1 or Z2 using, rffiffiffiffiffi P1 (7.157) I11 ¼ Z1 The power injected into the space that lies outside the Huygens surface can be readily obtained by integrating the complex Pointing vector over the Huygens surface S using, ð

ðE  H Þ  b n ds

Prad ¼ Re

(7.158)

S

Here, and as in common with many commercially available electromagnetic modelling tools, E and H are assumed to be RMS values of the field. If instead E and H denote peak values of the field, then a factor of one half must be included. The self-impedance Z11 and Z22 can be obtained in many ways. However, they are perhaps most easily obtained from whichever three-dimensional fullwave electromagnetic solver which was used to obtain the radiated fields from the isolated antennas. As an admittance is merely the reciprocal of an impedance, an admittance matrix [Y] representing these two port-coupled systems can be readily populated so that, ½Y  ¼

Y11 Y21

Y12 Y22

(7.159)

It is well known that the re-normalised scattering matrix, [SW], can be calculated from this admittance matrix and is used to describe what fraction of the signal is transmitted, or reflected at each port, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ½SW  ¼ ½YW ð½Z   ½ZW Þð½Z  þ ½ZW Þ1 ½ZW  (7.160) Here, [YW] = ([ZW])1 and is a diagonal matrix with the desired normalising admittance as the diagonal entries, i.e. the admittance of the attached transmission

Computational electromagnetic model

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line which in this case will be equal to the port impedance Z1 = Z2 = ZTE. This can be expressed mathematically as, ½YW  ¼ YW dij

(7.161)

With di,j denoting the Kronecker delta where i and j are positive integers,  dij ¼

1ði ¼ jÞ 0ði 6¼ jÞ

(7.162)

The elements S1,2 = S2,1 of [SW] are the complex transmission coefficients for the coupled antenna system which are taken to represent a single point in the nearfield measurement. This then completes what is essentially a method of moments analysis of this system. To summarise, the near-field measurement simulation algorithm consists of the following steps: 1. 2. 3. 4. 5.

Use (7.51) and (7.54) to translate the fields of antenna 1 to antenna 2. Use (7.156) to evaluate the mutual impedance and thus the mutual admittance. Populate the admittance matrix for the two-antenna system. Use (7.160) to determine the complex mutual coupling coefficient Repeat steps 1–4 inclusive for each sample point in the near-field scan.

Rotate the probe to simulate the sampling of a second orthogonal near-field component and repeat steps 1–5 inclusive to generate a second scan and thus complete the near-field acquisition. The magnitude of the mutual coupling coefficient between a pair of polarisationmatched loss-less dipoles that are perfectly matched to their respective source and load, and that are in the far-field of one another can be obtained from the Friis [20] transmission formula: PR ¼ lim ðjS21 jÞ ¼ R!1 PT




l 4pR

2 GT GR

(7.163)

Conversely, the mutual coupling coefficient can be obtained by taking the E and H-fields from a half-wavelength dipole from a full wave-solver, and using the physical optics reaction theorem algorithm set out above. As the antennas that are transmitting and receiving are of exactly the same design, then GR = GT. The gain of one of these dipoles at 10 GHz along the x-axis was approximately 2.16 dB. Figure 7.45 contains a comparison of the mutual coupling obtained using these two contrasting methods. From inspection, it is clear that the agreement is encouraging with differences only becoming more pronounced as the separation becomes smaller, i.e. in the region where the far-field approximation within the Friis transmission equation is most unreliable. To illustrate the utility of this generalised, but rather involved and computationally intensive, simulation method the following section presents a comparison between a near-field measurement and a near-field simulation.

342

Principles of planar near-field antenna measurements –10 PO-RT algorithm Friis –15

S21 (dB)

–20

–25

–30

–35

–40

0

0.05

0.1

0.15 Distance (m)

0.2

0.25

0.3

Figure 7.45 Mutual coupling between adjacent dipoles

7.10 Near-field measurement simulation Figure 7.46 contains a schematic representation of the near-field probe and a standard gain horn (SGH) that was used as an AUT. An electrically small AUT was chosen as its properties could be obtained directly from a full-wave computational electromagnetic (CEM) solver. The faint grey ellipsoidal surface that can be seen to enclose these instruments represents the Huygens surface that was used with the physical optics reaction theorem CEM simulation. The scanning probe consisted of an WG16 chamfered open-ended rectangular waveguide probe combined with a surface wave absorbing (SWAM) cone that was designed to minimise reflections from the mechanical interface located towards the rear of the probe. A detailed description of the modelling and verification thereof can be found within the literature [21]. Again similarly good agreement was attained between the CEM model of the SGH and measurements taken using a compact antenna test range (CATR). A near-field measurement of the SGH was taken using a planar near-field antenna test range. The acquisition window was chosen to be 0.8 m in the x–y plane whilst the distance between the SGH aperture and the near-field probe aperture in the z-direction was set at 10.0 cm, i.e. approximately 3 wavelengths (3l) at 10.0 GHz. This separation insured that the probe was outside the reactive nearfield, whilst making the first-order truncation angle as large as possible ( 83 in azimuth and elevation) and attempted to minimise the detrimental effects arising from multiple reflections that can be set-up between the AUT and the scanning near-field probe. Although phenomena arising from the first two of these effects

Computational electromagnetic model

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will be included within the measurement simulation, multiple reflections between the AUT and the probe are ignored. The separation between adjacent measurement points was one half wavelength as this satisfied the Nyquist sampling criteria, and in the absence of errors in the position at which measurements are taken, will guarantee alias free far-field patterns over the entire forward half-space. The SGH was installed in the range so that it was principally “y-polarised” with respect to the axes of the near-field range. Figures 7.47 and 7.48 contain a comparison of horizontal and vertical cuts through the simulated near-field measurement and the actual near-field measurement.

Figure 7.46 Near-field probe (left) and AUT (SGH) (right) 0 Measured Simulated –10

Magnitude (dB)

–20

–30

–40

–50

–60 –0.8

–0.6

–0.4

–0.2

0 x-axis (m)

0.2

0.4

Figure 7.47 Near-field cut in x-axis

0.6

0.8

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Principles of planar near-field antenna measurements 0 Measured Simulated

–10 –20

Magnitude (dB)

–30 –40 –50 –60 –70 –80 –90 –0.8

–0.6

–0.4

–0.2

0 y-axis (m)

0.2

0.4

0.6

0.8

Figure 7.48 Near-field cut in y-axis

The agreement between the simulation and the measurement is encouraging in both planes and it suggests that the simulation technique is capable of creating a simulation of a measurement taken on a near-field range. In particular, the cut in the y-axis, Figure 7.48, where there is less energy as you move away from the centre shows very good agreement. It is only when the magnitude of the coupling is at lower levels, i.e. < 40 dB, that the characteristics of the two lines begin to diverge. Inevitably, the measurement contains uncertainties arising from imperfections in alignment (both in translation, x, y, z and rotation in azimuth, elevation, roll), multi-path (scattering from the chamber walls and the frame of the robotic positioner), multiple reflections between the antenna and the probe, and imperfections in the manufacture of both the probe and AUT. None of these error terms are included within the near-field simulation. Unfortunately, the evaluation of the transmission coefficient S21 requires the computationally expensive evaluation of a sextuple integration to obtain each nearfield sample point. However, the simulation of an entire near-field measurement constitutes a coarsely granular problem. Specifically, each sampling node can be evaluated independently of every other sampling node and in this way, provided only that a sufficiently large array, i.e. cluster, of computers is available, the total processing time is in principle equal to the time taken to evaluate a single measurement point. Finally, it is important to highlight that this technique neither relies on the modal expansion method for the representation of electromagnetic waves nor does it utilise inversions of conventional probe pattern correction algorithms. Thus, it

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provides an independent approach to the verification of other near-field to far-field transformation algorithms.

7.11 Reaction theorem The generalised method of moments solution outlined in the preceding section made extensive use of the Reaction theorem and used several expressions, as is, without explanation or proof. The following sections are unnecessary if all that is required is to implement the techniques developed above; however, they are included here to aid and add detail to the technique whilst not disturbing the development of the near-field modelling technique developed above. First, the reciprocity theorem is developed. Then, this is used as the basis for the Reaction theorem that is used to relate circuit parameters to field parameters which is the form of the reaction integral which is central to the near-field modelling technique set out above.

7.11.1 Lorentz reciprocity theorem (field reciprocity theorem) Consider two elemental current sources, I1 at r ¼ r1 and I2 at r ¼ r1 where both sources oscillate with the same angular frequency w. Here, I1 produces E1 ; H1 and I2 produces E2 ; H2 . For I1, r  E1 ¼ M1  jwmH1

(7.164)

r  H1 ¼ J1 þ jweE1

(7.165)

For I2, r  E2 ¼ M2  jwmH2

(7.166)

r  H2 ¼ J2 þ jweE2

(7.167)

Let us first consider the divergence. r  E1 ¼ M1  jwmH1 to obtain,   H2  r  E1 ¼ H2  M1  jwmH2  H1 Now dot E1 with r  H2 ¼ J2 þ jweE2 to obtain,   E1  r  H2 ¼ E1  J2 þ jweE1  E2

First,

dot

H2

with

(7.168)

(7.169)

Subtracting these expressions yields,     E1  r  H2  H2  r  E1 ¼ E1  J2 þ H2  M1 þ jweE1  E2 þ jwmH2  H1 (7.170)

346

Principles of planar near-field antenna measurements Now since, (7.171) r  ðA  B Þ ¼ B  ðr  A Þ  A  ðr  B Þ Hence,         E1  r  H2  H2  r  E1 ¼ r  H2  E1 ¼ r  E1  H2 (7.172) Thus,

  r  E1  H2 ¼ E1  J2 þ H2  M1 þ jweE1  E2 þ jwmH2  H1 (7.173)

Now dot E2 with r  H1 ¼ J1 þ jweE1 to obtain,   E2  r  H1 ¼ E2  J1 þ jweE2  E1

(7.174)

Now dot H1 with r  E2 ¼ M2  jwmH2 to obtain,   H1  r  E2 ¼ H1  M2  jwmH1  H2

(7.175)

Subtracting these expressions yields,     E2  r  H1  H1  r  E2 ¼ E2  J1 þ H1  M2 þ jweE2  E1 þ jwmH1  H2 (7.176) Now since, r  ðA  B Þ ¼ B  ðr  A Þ  A  ðr  B Þ

(7.177)

Hence,         E2  r  H1  H1  r  E2 ¼ r  H1  E2 ¼ r  E2  H1 (7.178) Thus,

  r  E2  H1 ¼ E2  J1 þ H1  M2 þ jweE2  E1 þ jwmH1  H2 (7.179)

Subtracting these equations yields,

    r  E1  H2 þ r  E2  H1 ¼ E1  J2 þ H2  M1 þ jweE1  E2 þ jwmH2  H1 E2  J1  H1  M2  jweE2  E1  jwmH1  H2

(7.180) Upon cancellation this becomes,     r  E1  H2 þ r  E2  H1 ¼ E1  J2 þ H2  M1  E2  J1  H1  M2 (7.181)

Computational electromagnetic model

347

Or,   r  E1  H2  E2  H1 ¼ E1  J2 þ H2  M1  E2  J1  H1  M2 (7.182) This is the Lorentz reciprocity theorem in differential form. Taking the volume integral of both sides of this expression yields,     ∭ V r  E1  H2  E2  H1 dv ¼ ∭ V E1  J2 þ H2  M1  E2  J1  H1  M2 dv

(7.183) Now recalling the divergence theorem, n ds ∭ V r  D dv ¼ ∯ D  b

(7.184)

S

So that,     n ds ¼ ∭ V r  E1  H2  E2  H1 dv ∯ E1  H2  E2  H1  b S

(7.185) Thus, 

 ∯ E1  H2  E2  H1 S







n ds ¼ ∭ V E1  J2 þ H2  M1  E2  J1  H1  M2 dv b

(7.186) This is the Lorentz reciprocity theorem in the integral form. So summarising yields,     ∯ E1  H2  E2  H1  b n ds ¼ ∭ r  E1  H2  E2  H1 dv V

S

  ¼ ∭ V E2  J1 þ H1  M2  E1  J2  H2  M1 dv   ¼ ∯ E2  H1  E1  H2  b n ds S

(7.187) Here, the closed surface S encloses the volume V, and the surface normal points out of the volume. Suppose that each of the sources is contained within non-intersecting finite, closed volumes V1 and V2, respectively. Thus, V1 contains only J1, M1 while V2 contains only J2, M2 hence,     ∯ E1  H2  E2  H1  b n ds ¼ ∭ V1 E2  J1  H2  M1 dv (7.188) S1

348

Principles of planar near-field antenna measurements Similarly,     n ds ¼ ∭ V2 H1  M2  J2  E1 dv ∯ E1  H2  E2  H1  b

(7.189)

S2

This is a commonly used form of reciprocity and it is derived by assuming that J1 and J2 are the only sources of E1, H1, E2, and H2 i.e. there are no assumed magnetic sources. Alternatively, by assuming that both sources are outside the volume of integration yields,   n ds ¼ 0 ∯ E1  H2  E2  H1  b

(7.190)

S1

Within Box 7.4 further attention is given to the physical interpretation of this relationship.

Box 7.4 If this assumption is not made and the sources are in each other’s far-field since:     ∯ E1  H2  E2  H1  b n ds ¼ ∭ V1 E2  J1  J2  E1 dv S1

(7.191) In the far-field, r E ¼0¼E r

(7.192)

H ¼ r  E =z

(7.193)

and,

Here, z is used to denote the impedance of the medium in which the field is propagating. From, A  ðB  C Þ ¼ ðA  C ÞB  ðA  B ÞC (7.194)       E1  H2 ¼ E1  r  E2 =z ¼ E1  E2 =z r  E1  r E2 =z (7.195) Since E1  r ¼ 0     E1  H2 ¼ E1  E2 =z r  0 ¼ E1  E2 =z r

(7.196)

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349

And       E2  H1 ¼ E2  r  E1 =z ¼ E2  E1 =z r  E2  r E1 =z (7.197) Since E2  r ¼ 0     E2  H1 ¼ E2  E1 =z r  0 ¼ E2  E1 =z r

(7.198)

Therefore, in this case:     E1  H2  E2  H1 ¼ E1  E2 =z r  E2  E1 =z r ¼ 0

(7.199)

Therefore, if J1 and J2 are localised and, as the cross product terms are zero, i.e. there are no incoming waves from infinitely far away. ∭ V1 ðE2  J1  E1  J2 Þdv ¼ ∯ 0  nds

(7.200)

Therefore ∰ ðE2  J1 Þdv ¼ ∰ ðE1  J2 Þdv

(7.201)

This is the usually quoted form of the Rayleigh–Carson Reciprocity theorem. However, if the theorem is considered in the simplest of physical situations i.e. for dipoles as described in Section 7.4.1, then the integrals will disappear and only the products of the fields and dipoles need to be considered, i.e. E2  P 1 ¼ E1  P2

(7.202)

Further insight can be gained by performing a dimensional analysis on these products in terms of units where,
 volt  ðcoulomb  metreÞ E 1  P2 metres ¼ ð
volt  coulomb  Þ (7.203) joule  coulomb ¼ coulomb ¼ joule Thus, the Reaction theorem can be viewed as an energy conservation equation where the dot product terms relate to the partition of energy between the physical structures carrying the currents, i.e. Newton’s third law relating actions to equal and opposite reactions, and the cross-product terms represent a flux across the boundary of a surface containing these structures.

350

Principles of planar near-field antenna measurements

7.11.2 Generalised Reaction theorem Maxwell’s equations for the fields within S2 but outside the source region can be expressed as, r  E 1 ¼ jwmH 1

(7.204)

r  H 1 ¼ jweE 1

(7.205)

r  E 2 ¼ jwmH 2

(7.206)

r  H 2 ¼ jweE 2

(7.207)

Now, the “reaction” of antenna 1 on antenna 2 is defined to be, ð h2; 1i ¼ ðE 2  H 1  E 1  H 2 Þ  b n ds

(7.208)

S2

Here, S2 is a surface that encloses antenna 2, but not antenna 1. For the sake of completion, the reaction of antenna 2 on antenna 1 can similarly be defined as, ð n ds (7.209) h1; 2i ¼ ðE 1  H 2  E 2  H 1 Þ  b S1

Similarly, S1 is a surface that purely encloses antenna 1 but not antenna 2. For simplicity, the reaction formula can be expressed as, ð (7.210) h2; 1i ¼ ðE 2  H 1  E 1  H 2 Þ  ds S2

Where, ds ¼ b n ds

(7.211)

Now, let the surface of integration be changed from S2 to Sm þ ST thus, ð ð h2; 1i ¼ ðE 2  H 1  E 1  H 2 Þ  ds þ ðE 2  H 1  E 1  H 2 Þ  ds Sm

ST

(7.212) If the metal surface Sm is assumed to be perfectly conducting, then E1 and E2 will only have normal components thus the term E  H will only have tangential components that are orthogonal to the surface normal thus the integral over the metal surface will vanish. Hence, ð ðE 2  H 1  E 1  H 2 Þ  ds (7.213) h2; 1i ¼ ST

Now let, E 2 ¼ V22 e

(7.214)

H 2 ¼ I22 h

(7.215)

Computational electromagnetic model Then,

351

ð

h2; 1i ¼ ST

ðV22 e  H 1  E 1  I22 h Þ  ds

(7.216)

Now, the voltage and current induced at antenna 2 when antenna 1 transmits are defined in a similar way by setting, E 1 ¼ V21 e

(7.217)

H 1 ¼ I21 h

(7.218)

Thus, ð h2; 1i ¼

ðV22 e  I21 h  V21 e  I22 h Þ  ds

(7.219)

ST

As the vector mode functions are related by, h ¼b n e

(7.220)

Where they are normalised to unity through, ð e  e ds ¼ 1

(7.221)

ST

Thus, ð ðV22 I21 e  ðb n  e Þ  V21 I22 e  ðb n  e ÞÞ  ds

h2; 1i ¼ ST

(7.222)

ð ðe  ðb n  e ÞÞ  ds

h2; 1i ¼ ðV22 I21  V21 I22 Þ

(7.223)

ST

Now as, A  ðB  C Þ ¼ ðA  C ÞB  ðA  B ÞC Then,

ð ½ðe  e Þb n  ðe  b n Þe   ds

h2; 1i ¼ ðV22 I21  V21 I22 Þ ST

ð h2; 1i ¼ ðV22 I21  V21 I22 Þ Now since,

(7.224)

Ð

ST ST e

ð ðe  b n Þe  ds

ðe  e Þb n  ds 

(7.225) (7.226)

ST

 e ds ¼ 1 and as ds ¼ b n ds then,

b n  ds ¼ b n b n ds ¼ ds

(7.227)

Thus,



ð h2; 1i ¼ ðV22 I21  V21 I22 Þ 1  ðe  b n Þe  ds ST

(7.228)

352

Principles of planar near-field antenna measurements Now, as the fields are tangential then ð ðe  b n Þe  ds ¼ 0

(7.229)

ST

Thus, h2; 1i ¼ V22 I21  V21 I22

(7.230)

Now, as the received current I21 is assumed to be zero as the terminals are assumed to be open circuited when receiving then, this becomes, h2; 1i ¼ V21 I22

(7.231)

Hence, ð V21 I22 ¼  S2

ðE 2  H 1  E 1  H 2 Þ  b n ds

(7.232)

This is of importance as it constitutes a “mixed type” reciprocity theorem. Finally then, ð 1 V21 ¼  ðE  H 1  E 1  H 2 Þ  b n ds (7.233) I22 S2 2

7.11.3 Mutual impedance and the Reaction theorem The mutual impedance Z21 between antenna 1 and antenna 2 in the environment ðe; mÞ can be expressed in terms of the electric and magnetic fields as [21], Z21 ¼

V21 I11

(7.234)

Thus, Z21 ¼

1 I11 I22

ð S2

ðE 2  H 1  E 1  H 2 Þ  b n ds

(7.235)

Here, I11 is the terminal current of antenna 1 when it transmits and b n is the outward pointing unit normal. Similarly, I22 is the terminal current of antenna 2 when it transmits. The mutual impedance Z12 ¼ Z21 represents the coupling between two antennas and will therefore be a function of the distance separating the two antennas, the relative antenna orientations, and their respective polarisation properties. Similarly, Z12 ¼

1 I11 I22

ð S1

ðE 1  H 2  E 2  H 1 Þ  b n ds

(7.236)

Computational electromagnetic model

353

7.12 Full wave simulation of a planar near-field antenna measurement If the frequency is sufficiently low and the AUT and probe sufficiently simple, then it may be possible to simulate a planar near-field antenna measurement using a fullwave three-dimensional computational electromagnetic solver. In this section, a method of moments based commercially available tool was used to simulate the measurement of an x-band standard gain horn by an x-band open-ended rectangular waveguide probe. Here, to simplify the simulation no absorber collar was used on the probe or the SGH however that can in principle be included if more accurate simulations are required. The absorber was omitted so as to be able to keep the simulation times to reasonable limits. The difficulty associated with simulating a near-field measurement involving acquisitions taken over two-dimensional surfaces is that a new simulation must be run for every position of the probe. This means that an nm point measurement would take 2nm individual simulations for a single frequency if dual polarised acquisitions were needed, which is generally the case. Here, the results of a conventional xy-plane rectilinear near-field acquisition are considered and are illustrated schematically in Figure 7.49 to simulate the measurement of an x-band (WR90) pyramidal horn by an x-band WR90 rectangular waveguide probe testing at 8.2 GHz. One complete full-wave

Z/N

Y/V X/U

Figure 7.49 Planar-near-field measurement model shown with a 10 cm separation in the z-axis between the probe and the SGH with an offset in the xy position

354

Principles of planar near-field antenna measurements

EM simulation is required per point within the simulated plane rectilinear acquisition meaning that in this case, assuming half wavelength sample spacing, required 3131 = 961 individual S21 simulations, and this assumed the use of symmetry to minimise the computational effort. This is a very accurate simulation as it includes both multiple reflections between the AUT and probe and evanescent coupling. The results of this simulation can be seen presented in Figures 7.50 and 7.51 which respectively show the amplitude and phase of the transmission coefficient S21 in the form of a false-colour checkerboard plot. Here, the largest field intensity can be seen in the centre of the simulated “measurement” where the probe is passing over the aperture of the rectangular pyramidal horn. The field outside of this projected aperture is far lower, with diffraction effects also being clearly visible which agrees with what one would generally expect to see in practice as the measurement is taken across a plane that is parallel to but displaced from the aperture plane of the AUT. The phase plot shows a relatively flat phase function across the aperture of the AUT with the phase changing outside of this region. By way of a comparison, Figures 7.52 and 7.53 contain results that are equivalent to those shown in Figures 7.50 and 7.51; however, here, an infinitesimal Hertzian dipole probe was used as opposed to a finitely large rectangular OEWG probe. Although the respective results are similar some differences do exist and it is important recognise them and as such plots containing horizontal and vertical cuts through the amplitude patterns can be seen presented in Figures 7.54 and 7.55, respectively. Here, the red trace denotes the field sampled using a dipole probe, i.e. that used by theory, with the OEWG probe results being represented by the blue trace. From inspection of these plots, it is evident that the simulated OEWG measurement contains a larger amount of ripple outside of the geometric projection of the aperture of the AUT. This is a result of multiple reflections, i.e. between the AUT and the probe. As observed above, this is a real effect and is something that is seen in

0 0.25

0.2

–10

0.15

100

0.15

–15

0.05

–20

0

–25

–0.05

–30

–0.1

–35

–0.15 –40

–0.2

–45

–0.25 –0.2

–0.1

0 x (m)

0.1

0.2

0.3

–50

Figure 7.50 Simulated Ex polarised amplitude pattern measurement including OEWG probe effects

0.1

50

0.05 y (m)

0.1 y (m)

150

0.25

–5

0.2

0

0

–0.05 –50

–0.1 –0.15

–100

–0.2 –0.25

–150 –0.2

–0.1

0 x (m)

0.1

0.2

0.3

Figure 7.51 Simulated Ex polarised phase pattern measurement including OEWG probe effects

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355

0 0.25

0.2

–10 –15

0.1

0.05

–20

0.05

0

–25

–0.05

–30

y (m)

y (m)

100

0.15

0.15 0.1

150

0.25

–5

0.2

50

0

0

–0.05 –50

–0.1

–0.1

–35

–0.15 –0.2 –0.25 –0.2

–0.1

0 x (m)

0.1

0.2

0.3

–0.15

–40

–0.2

–45

–0.25

–50

Figure 7.52 Simulated Ex polarised amplitude pattern measurement assuming ideal Hertzian dipole probe

–100 –150 –0.2

–0.1

0 x (m)

0.1

0.2

0.3

Figure 7.53 Simulated Ex polarised phase pattern measurement assuming ideal Hertzian dipole probe

0 Dipole OEWG –10

Amplitude (dB)

–20

–30

–40

–50

–60

–0.25 –0.2 –0.15 –0.1 –0.05

0 0.05 x (m)

0.1

0.15

0.2

0.25

Figure 7.54 Comparison of horizontal cut through simulated Ex polarised amplitude pattern using OEWG and dipole probes

356

Principles of planar near-field antenna measurements 0 Dipole OEWG –10

Amplitude (dB)

–20

–30

–40

–50

–60

–0.25 –0.2 –0.15 –0.1 –0.05

0 0.05 y (m)

0.1

0.15

0.2

0.25

Figure 7.55 Comparison of vertical cut through simulated Ex polarised amplitude pattern using OEWG and dipole probes

practical measurements. However, due to the extended processing time required, it is something that is usually absent from most measurement simulations. From inspection of the respective phase plots, it is evident that this coupling has upset the measured phase although this is far less obvious. The second effect that is clear is the change in the shape of the pattern cuts. This is a consequence of the convolution between the fields in the AUT and the fields in the probe. In essence, the dipole probe performs an average of the field incident upon it. As the probe is infinitesimally small, this averaging is also infinitesimal resulting in the dipole probe sampling the field at a single point in space, with a single direction of polarisation. However, the OEWG has a finitely large aperture meaning that this averaging is performed over a finitely large region of space. The coupling of the field into the aperture of the OEWG probe results in the measured signal being a combination of the field associated with the AUT and the field associated with the probe. The central task of many near-field to far-field transformation algorithms, as developed within the preceding chapters, concerns compensating for this probe pattern effect. Thus, one of the great utilities of this measurement simulation approach is that probe effects are included enabling the data provided to be used for the purposes of accurately and precisely verifying complete transformation algorithms. Unfortunately, the simulation times involved can be prohibitive thereby providing motivation for the development and use of the measurement simulation techniques developed above.

Computational electromagnetic model

357

7.13 Summary Within this chapter, a number of simulation techniques have been developed, each offering a different balance of sophistication and effort. Ultimately, it would perhaps be preferable to utilise a full wave three-dimensional electromagnetic solver to tackle these problems and with the passage of time this is becoming an ever more feasible option. However, until these methods can be deployed on a sufficiently large scale and can provide results in a sufficiently compact time scale, other alternative solutions will remain attractive. For many applications, the comparatively simple vector Huygens method is sufficient and indeed it was used in the preparation of many of the simulated data sets that are utilised within Chapter 9 to illustrate and verify some of the more advanced transformation and correction techniques. The current elements method is a very accurate approach for the simulation of reflector antennas with this offering a very convenient AUT model that can be used to construct a range of test nearfield measurement configurations efficiently and flexibly [20]. Unfortunately, it is not possible to use these techniques to model every phenomenon that can be observed in a near-field measurement system, e.g. multiple reflections between the AUT and the probe, but there is perhaps sufficient choice detailed above that the vast majority of situations that are likely to be encountered in practice, can be accommodated.

References [1] J.A. Estefan, “Survey of model-based systems engineering (MBSE) methodologies”, Incose MBSE Focus Group, 25 (2007): 8. [2] M. Grieves, “Digital twin: manufacturing excellence through virtual factory replication”, White Paper 1 (2015): 1–7. [3] H.C. Chen, “Theory of electromagnetic waves, a co-ordinate-free approach”, in: International Student Editions in Related Fields, McGraw-Hill, New York, NY, 1985, p. 350. [4] A.C. Newell and S.F. Gregson, “Computational electromagnetic modelling of near-field antenna test systems using plane wave spectrum scatting matrix approach”, in: AMTA, October, 2014. [5] S. Silver, Microwave Antenna Theory and Design, 1st ed., McGraw Book Company Inc., New York, NY, 1949, p. 83 [6] R. Mills, Space Time and Quanta an Introduction to Contemporary Physics, W.H. Freeman, New York, NY, 1994, pp. 327–336. [7] P. Petre and T.K. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach”, IEEE Transactions on Antennas and Propagation, 40(11) (1992): 1348–1356. [8] R.F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, NY, 1961.

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[9] C.C. Paige and M.A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares”, ACM Transactions on Mathematical Software, 8(1) (1982): 43–71. [10] A.W. Rudge, K. Milne, A.D. Olver, and P. Knight, “The handbook of antenna design, Vol. 1”, in: IEE Electromagnetic Waves Series 15, Peter Peregrinus, London, 1982, p. 383, ISBN 0-906048-82-6. [11] T.K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM”, IEEE Transactions on Antennas and Propagation, 47(3) (1999): 566–573. [12] R.F. Harrington, Field computation by moment methods, Krieger, Melbourne, FL, 1987. [13] J.L.A. Quijano, L. Scialacqua, J. Zackrisson, L.J. Foged, M. Sabbadini, and G. Vecchi, “Suppression of undesired radiated fields based on equivalent currents reconstruction from measured data”, IEEE Antennas and Wireless Propagation Letters, 10 (2011): 314–317. [14] F. Cano-Facilla, S. Burgos, F. Martin, and M. Sierra-Castan˜er, “New reflection suppression method in antenna measurement systems based on diagnostic techniques”, IEEE Transactions on Antennas and Propagation, 59(3) (2011): 941–949. [15] G.E. Hindman and A.C. Newell, “Reflection suppression in a large spherical near-field range”, in: AMTA 27th Annual Meeting & Symposium, Newport, RI, October 2005. [16] G.E. Hindman and A.C. Newell, “Reflection suppression to improve anechoic chamber performance”, in: AMTA Europe 2006, Munich, Germany, March 2006. [17] S.F. Gregson, A.C. Newell, G.E. Hindman, and M.J. Carey, “Extension of the mathematical reflection suppression technique to the planar near-field geometry,” in: Antenna Measurement Techniques Association Annual Symposium Proceedings, Denver, CO, October 2010, pp. 94–100. [18] C. Parini, S. Gregson, J. McCormick, D. Janse van Rensburg, and T. Eibert, Theory and Practice of Modern Antenna Range Measurements, 2nd expanded ed., vols. 1 and 2, IET Press, London, 2020. [19] J.H. Richmond, “A reaction theorem and its application to antenna impedance calculations”, IRE Transactions on Antenna and Propagation, 9 (1961): 515–320. [20] C.A. Balanis, Antenna Theory Analysis and Design, 2nd ed., John Wiley and Sons Inc., New York, NY, 1997, p. 86. [21] R.W. Lyon, S.F. Gregson, C. Mitchelson, and J. McCormick, “Computational electromagnetic modelling of a probe employed in planar near field antenna measurements”, in: ICAP 2003, Exeter, UK.

Chapter 8

Antenna measurement analysis and assessment

8.1 Introduction Up until this point, this volume has described and explained the techniques and concepts involved in performing near-field antenna measurements in a systematic form. This has been done by: ● ●

●

●

●

addressing the fundamental phenomenological action of antennas, developing a conceptual model of the interaction of antennas based on classical electromagnetic field theory, deriving a methodology for the solution of the differential equations of classical electromagnetic field theory based on the plane wave spectrum approximation, describing the details of a measurement technique (near-field scanning), which can be used to make measurements of the coupling between two antennas, the AUT and a probe antenna, and finally making predictions of the far-field radiative performance of the AUT based on the plane wave spectrum approximation.

However, the problem of deriving the measure, i.e. the true value of the magnitude of a physical quantity, such as the power flux density transmitted in a given direction relative to an antenna, from a measurement process has not yet been addressed. Note, strictly this is the primary measure of the measurand in question, other secondary measures can be derived from the measurement results but unless specifically stated, in this chapter, measure will be taken to mean primary measure.

8.2 The establishment of the measure from the measurement results The concept of measurement is interpreted differently in different sciences and therefore by definition in different areas of engineering and technology. As described in Chapter 1, the information extraction model is a particularly applicable concept for the cognitive evaluation of antenna measurements. In near-field scanning, we are primarily concerned with microwave frequencies and as millions of years of human evolution have left our species without sense organs that respond to

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Principles of planar near-field antenna measurements

such frequencies sensory cognition is largely irrelevant, leaving only rational cognition as a tool for the evaluation of microwave phenomena. Rational cognition involves analysis, synthesis translation and inference of and from images or representations of reality that are not sensory and is thus performed in the domain of abstraction. By definition observation, experimentation and evaluation are used in rational cognition and when the observations of physical phenomena are designed to extract quantitative information these observations are of course termed measurements. However, in order to meet the requirement that the results of measurements are quantitative representations in the abstract domain of physical phenomena, severe restrictions as to the form of the measurement procedure must be imposed and adhered to, to reduce ambiguity. The minimisation of the effects of error sources in measurements will result in the reduction of ambiguity in the mapping that transforms the magnitude of the real physical quantity to its representational state in the abstract measurement domain. As well as the usual empirical definition of a scale, such a mapping is also often referred to, ‘as the scale of the measurement’, see Appendix B. The possibility/practicality of establishing a non-ambiguous empirical scale as a ratio, relative to a standard in multiples of the resolution of the measurement system, will be addressed in this chapter. Before it is possible to devise strategies that identify, quantify and/or eliminate the error sources that introduce ambiguity into the measurements, it is necessary to examine the nature and mode of their occurrence.

8.2.1

Measurement errors

Figure 8.1 illustrates the relationship between the measured physical quantity and the measurement representation that constitutes the result of the measurement. Clearly the integrity of the mapping M which defines the scale over which the measurements are made is fundamental to the accuracy of the measurement process and the accuracy with which the measurements can be used to define the measure of the measurand. Where again we would define the measurand, (q), the physical Physical State Set

Measurement Process M: Q N

Representational Symbol Set

q1

n1

q2

n2

Q

N

Figure 8.1 Pictorial representation of the set theoretic model of measurement

Antenna measurement analysis and assessment

361

quantity to be measured, the measurement result as the image of the measurand in the representational set and the measure (n) as the true value of the physical quantity being measured using the scale, cf. Appendix B. The operation including the mapping, (M), where the elements of set Q are mapped to set N, can be viewed as a transfer function where an input nonmathematical physical quantity is mapped to an output signal, which is a representation of the physical state on a scale, i.e. N ¼ MðQÞ

(8.1)

Where Q ¼ ðq1 ; q2 ; q3 ; :::qi Þ is the set of physically realisable values of the measurand, the domain set of the transfer function and N ¼ ðn1 ; n2 ; n3 ; :::ni Þ is the image of Q under the mapping M, the image set of the transfer function.

Box 8.1 Note: This representation of the measurand in a symbol set can take many forms, e.g.: ● ● ● ●

A number in a table or array. A point on a graph. A colour on a graphic, or a pseudo colour plot. An arrangement of raised indentations on a Braille script.

These are all examples of representative symbols that could be used where the guiding principle of effective representation in measurement theory is that the relationship between the symbols representing the measured values and the actual true values of the measurand should be of the same logical form [1]. In fact, this procedure of moving from sources to potentials to electric and magnetic fields can be expressed in terms of the individual elements of the sets: n ¼ mðqÞ

(8.2)

Where an equivalence exists of the form n  q under the mapping m. Remembering that ni ¼ mðqi Þ will only be true if m is a one to one mapping. If, as is usually the case for antenna measurements, the representation is numerical the level of the measurement of any variable describes how much information the numbers associated with the variables contains, whether these are represented on, nominal/categorical, ordinal/ranked, interval/scaled or ratio/metric, scales [2]. The advantages to be gained by considering the different types of information in measurement data will be illustrated in Section 8.3 when data assessment techniques based on the different measurement levels will be discussed.

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Principles of planar near-field antenna measurements

Box 8.2 ●

●

●

●

●

●

Note: For antenna measurements, an interval, as opposed to categorical or ordinal scale, calibrated to make it metric, is usually used where these different types of scale can best be described in terms of level of measurement by the following example. Conventionally two antennas AUT1 and AUT2 may have their boresight gains measured and the results for AUT1 be quoted as 9 dBi and for AUT2 21 dBi. These results could have been mapped onto a categorical scale where the representation of AUT 1 might have been categorised as low gain and AUT2 as high gain. In this case, a nominal level of measurement would have been made defining a categorical variable. These might have been appropriate categories on the nominal scale or not e.g. AUT1 may in fact have higher gain than AUT2 just not on boresight. Measurements of AUT1 and AUT2 could have been mapped into an ordinal scale where under the scale mapping AUT1 and AUT2 would have been ranked. In this case, an ordinal level of measurement would have been made defining a ranked variable, with only the ordinal level of information represented. In fact as already stated in near-field antenna measurements, the AUT gain is usually measured relative to an arbitrary reference signal so AUT1 and AUT2 gains are mapped onto an interval scale. Where AUT2s gain is represented by not only a larger value, but by a value that falls 16 equal intervals above the unitary value assigned to AUT1. In this case, an interval level of measurement would have been made defining the scaled variable, with only the interval level of information represented. Note: although in practice AUT2s gain might be thought of as 16 times greater than AUT1s since the zero point on any interval scale is arbitrary the ratios between numbers on that scale are not meaningful. Thus strictly, the operations of multiplication and division cannot be defined at this level of measurement, which is more accurately described by the complete Presburger as opposed to finitely specified, consistent, but incomplete Peano arithmetic [3]. Finally if the measurements of AUT1 and AUT2 are compared directly with a defined standard, in this case the theoretical gain of an isotropic radiator these measurements can be considered to consist of a mapping onto a representative metric scale. Where, as already stated, AUT1 might have a gain of 9 dBi and AUT2 of 21 dBi relative to the isotropic standard. In this case, a ratio level of measurement would have been made defining a metric variable, with the ratio relative to the isotropic standard, level of information represented.

Antenna measurement analysis and assessment q

n = m(q)

363

n

Figure 8.2 Simple functional block diagram of a measurement

●

Clearly the information required to represent the boresight gain of AUTs 1 and 2 on these categorical, ordinal or interval scales is contained within the metric data. However if there is a requirement to analyse the data, the different aspects and levels of measurement represented on the different scales are amenable to very different statistical methodologies. As a result, considerable insight into the nature of any measurement procedure can be obtained by an approach that does not limit itself to just examining the interval or metric levels of information represented.

Figure 8.2 again shows how a theoretical measurement procedure can be viewed as the operation of a transfer function on an input member of a physical state set, the measurand, to map it onto an output representative abstract symbol set, the measurement result [4]. In this theoretical case, the measurement result = measure of the measurand. Thus, such a measurement system would provide a measurement result that would unambiguously and accurately define the measure of the measurand. However, under no practical circumstances can the result of a single measurement on a scale give the absolute magnitude of a measure as in fact all measurement procedures have inherent measurement errors associated with them irrespective of the extent of control exercised within them. This means that the result of any measurement cannot be taken as representative of the measure of the measurand unless the extent and nature of the ambiguities introduced by error sources into the measurement procedure are analysed, assessed and quantified.

8.2.2 The sources of measurement ambiguity and error The introduction of errors into any measurement process can be considered to be the result of two main factors. First, the transfer function can introduce an error into the measurement result where the mapping itself is only approximately equivalent to the true scale mapping, both as a result of systemic errors in the analogue mapping process and as a result of external possibly random error sources, i.e. me ffi m

(8.3)

where me is the actual as opposed to the theoretical scale mapping of the measurement system. For example, the response of the receiver in a near-field antenna

364

Principles of planar near-field antenna measurements q

ne = me(q)

ne

em1 em2

Figure 8.3 Measurement ambiguity introduced by mapping errors sources

qe

ne = m(qe)

ne

eq1 eq2

Figure 8.4 Measurement ambiguity introduced by input error sources measurement systems is assumed to be linear, in practice it will only be approximately linear over a given range of values or under a given range of external environmental conditions. This will cause the scale mapping of any practical measurement system to be, at least to some extent, in error. Thus as illustrated in Figure 8.3 possibly as a result of external influences, error sources em1 and em2, the mapping is the cause of ambiguity. Second, the input member of the physical state set, the measurand, may not be the only input that the system is mapping onto the output as a signal. Again for example, all antenna ranges attempt to measure the antenna pattern as the direct path relative power flux density propagating in a given direction with the antenna in free space. In practice, the antenna will be mounted within a test range that will be the source of scattering, this scattered field will produce multi-path within the range, which also acts as an error source inducing an ambiguous input to the measurement system. Thus in the example, shown in Figure 8.4, qe 6¼ q

(8.4)

where qe is the actual ambiguous input value and q is the value if no other error sources were present. This means that the model of a measurement system represented by the transfer function equation (8.1) may actually be represented by: ne ¼ me ðqÞ

(8.5)

ne ¼ mðqe Þ

(8.6)

or

or as illustrated in Figure 8.5: as result of both input and mapping error sources most likely, ne ¼ me ðqe Þ

(8.7)

Antenna measurement analysis and assessment qe

eq1 eq2

ne = me(qe)

365

ne

eq1 eq2

Figure 8.5 Measurement ambiguity introduced by mapping and input errors Where the output the measurement result, ne, has an error associated with either or both of the input and scale mapping being inaccurate. Whereas for an unambiguous measurement procedure, an equivalence exists of the form: n  q under the mapping m

(8.8)

For a practically realisable measurement system, ne ffi qe under the mapping me

(8.9)

The end result is that unless the errors associated with the scale mapping of the measurand input can be systemically quantified and removed by post processing of the output signal, in any practical measurement system there will be ambiguity in the measurement results. This means that the measurement result cannot accurately and unambiguously represent the measure of the measurand. How then can a measurement process be used to establish the true value of any physical quantity? This will be considered further in subsequent sections.

8.2.3 The examination of measurement result data to establish the measure In cases where the ambiguity is introduced by the mapping me the result will be that the element in the domain is mapped to a possible set of values, i.e. q ) Nq  fng

(8.10)

where Nq  N . Formally, the mapping transforms q into a random variable and Nq is the set of values of this variable. This can be illustrated by the functional block diagram shown in Figure 8.6 where the transfer function maps the input member of the physical state set onto a distribution of possible values each separated by a multiple of the scales resolution, e. The random variable n, which is an image of the element q in the domain Nq, will have a distribution, pq(n), where the assignment of a number of elements in Nq is possible but one state in Nq will be preferred. If there is a preferred value it can be supposed that the probability of the mapping, n ¼ me ðqÞ

(8.11)

366

Principles of planar near-field antenna measurements qe

ne = me(qe)

n–2ε n–ε n n+ε

eq1 eq2

eq1 eq2

n+2ε

Figure 8.6 Measurement ambiguity introduced by mapping onto a distribution is either greater or smaller than the mapping, n þ e ¼ me ðqÞ

(8.12)

Where e > 0. Therefore, the selection of the elements of the set Nq, which maps state q, is described by a measure, p 2 h0; 1i For which, X pq ðnÞ ¼ 1

(8.13)

(8.14)

n2Nq

or, ð ðNq Þ

pq ðnÞdn ¼ 1

(8.15)

dependant on the discrete or continuous nature of the  domain.  Thus, the mapping takes the form of the assignment relation q ) n; pq ðnÞ as opposed to the expression q ) n, where n [ N. This illustrates that, for all theoretical and practical measurement systems, measures are in fact probabilities [5], calculated from mappings of random variables onto a distribution which defines a probability density.

Box 8.3 Of particular interest in low-resolution measurement systems is the definition of the Lebesgue measure of a set where the determination of the measure is based on the creation of closed intervals and thus can be applied to a nondense distribution within an image set. Theoretically this methodology of integrating the content or measure of a non-dense set is very important as any practically realisable measurement scale will be composed of rational multiples based on the resolution of the scale and thus will by definition will not be a dense set. Thus strictly any probability calculus based on the PeanoJordon content derived from a Rieman integral is inaccurate [6]. However, under most practical circumstances, the resolution of the scale will be such as

Antenna measurement analysis and assessment

367

to render the scale effectively continuous and not discrete over the rational intervals of the resolution. Therefore in practice and in particular for nearfield antenna measurement systems this theoretical limitation of the probability calculus is not significant The assignment relation describes a complete set of possible events that will be assigned to q with a probability of 1, as per (8.14). To define a measurement scale only one possible element n*, being the best possible mapping must be selected from Nq to define the measure of the measurand. This selection will be based on the decision D where: n ¼ Dðpq Þ

(8.16)

Many different decision rules can be followed dependant on the type of measurement and nature of the measurand. However, by following the measurement procedure to produce measurement results, that can be analysed statistically in terms of their probability, a maximum-likelihood value and an accompanying confidence interval can be assigned to a given value and distribution in the measurement domain. Thus n* represents a maximum-likelihood measure and can be taken as the true value of the measurand. As already stated, in this text, the primary measure defines the maximumlikelihood value but other secondary measures that refer to tendencies of spread, kurtosis and skewness are all calculable. For any useful measurement system, therefore, the spread of possible values defined by a confidence interval at a given level of probability is a vital secondary measure of the data set. For cases where the ambiguity is introduced by input errors, a similar argument can be followed to describe the errors introduced by the measurement process. In turn, this distribution relating to the ambiguity in the input allows the inclusion of this and all other error sources in an overall total distribution, pt(n) which can be used in the assignment of a maximum-likelihood value and any other measure that is required. This is not the only methodology for deriving the total error or combined uncertainty for a measurement process. In practice for the ease of evaluation, the errors can be viewed in terms of orthogonal Type A, (statistically derived) and Type B, (deterministically derived) error sources as opposed to input and mapping sources where the random and deterministic sources are evaluated to produce orthogonal distributions. Other methodologies can be adopted to calculate the combined uncertainty dependant on how the correlation between the input and output are evaluated and compared to the theoretical assignment relations between them, however in practice all depend on a statistical analysis of measurement data. From the above discussion, it is clear that the measurement process tentatively illustrated in Figure 8.2 consisting primarily of a mapping of the physical state onto a representative symbolic set is over simplistic. The mapping is in fact an

368

Principles of planar near-field antenna measurements

assignment relationship that assigns the measure of the measurand to the maximum-likelihood value within the symbolic set according to a distribution pt(n) where the mapping and input are subject to measurement errors as per Figure 8.6. Therefore, in order to use a measurement system accurately, a knowledge of the distribution pt(n) will be necessary to assign the maximum-likelihood value and any confidence interval associated with measure to the measurement results. However, in practice, a priori knowledge of any distributions is very rare, this means that in any practical measurement system, the distribution must be established using the measurement procedure. Since the measurement procedure consists of mapping the element q into a random variable n with a distribution pt (n) the only possible practical methodology for establishing pt(n) is to repeat the measurement process until the measurement results display the nature of the distribution. This, theoretical, necessity of characterising the probability distribution is the basis of the requirement that all measurement procedures need repetition to assign a measure with any degree of confidence. Only measurements repeated infinitely many times can assign a measure with 100% confidence, hence the requirement to express measurement results with a  confidence level at a given level of significance, see Appendix B. If there is no assignment of uncertainty, i.e. the establishment of secondary measures of central tendency or spread of the results of a measurement, then it is not possible to define any useful level of confidence in the accuracy of the measure, i.e. the maximum-likelihood value of the measurand. Thus, without repetition of the measurement process, the ambiguities introduced into the measurement process either by lack of fidelity in the transfer function or as additional unwanted inputs to the system will not allow the derivation of the measure. However, in practice, only a limited number of measurements will be possible, perhaps only one, so some method of assigning a measured value to the measurement results must be put in place. The most usual method of doing this is to try and establish the nature and extent of the distributions that will be produced by the measurement process by characterising the measurement system a priori to the measurements being performed.

8.3 Measurement error budgets The usual pragmatic approach adopted is to establish an error budget for any measurement facility that is indicative of the error levels that can be expected and to quote the uncertainty and thus accuracy of the measurements performed within the facility in terms of this indicative error budget. This error budget consists of a quantitative assessment of the impact of the combination of individual error sources on the measurement result along with an estimation of the likely impact on the accuracy to which the measure of the measurand can be predicted. Thus, an accurate and complete examination of the measurement errors associated with a given antenna or set of antennas can be used to assign an expected uncertainty, via extrapolation, to other antennas measured in the same facility.

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8.3.1 Applicability of modelling error sources The assessment of measurement and test facilities to establish the diversity and magnitude of possible error sources that can introduce ambiguity into measurement results is a well-developed subject. It does not depend entirely on empirical evidence, and the use of simulations and other assessment techniques in error analysis can be valuable. However, it should be recalled that all simulation techniques are based on models and their associated algorithms all of which, at least under particular circumstances, are more or less inaccurate. In this text, a model of the electromagnetic interaction based on Maxwell’s equations has been developed, as it is particularly applicable to the action of antennas and antenna measurements. However, it is important to realise that accurate models, and measurements of the interaction of electromagnetic radiation with solid objects, e.g. antennas must at least to some extent acknowledge the discrete particulate nature of the solid state. For macroscopic structures like antennas and antenna test ranges, the errors produced by the uncertainty relations associated with momentum/position, energy/time could be assumed to be negligible and within the noise of the system. For example in his classic text ‘Quantum Theory’, David Bohm calculates the inaccuracies associated with the standard macroscopic voltmeter pointing needle to be of the order of 1026 m, i.e. 1012 times smaller than a nucleus, a very small error indeed. However, although the quantum mechanical aspects of the discrete nature of matter may only produce small error terms the discrete nature of matter does contribute larger random fluctuations that produce errors in the system. These fluctuations referred to as noise have a variety of sources. The parameter that describes the distribution of energy amongst the discrete particles, i.e. the temperature gives rise to noise via a number of different thermal mechanisms. The most important of these within electronic measurement systems being Johnson noise resulting from the thermal motion of the conduction electrons in the solid. Flicker noise due to quality and stability of components may also be present and noise solely due to the discrete nature of the electrons carrying any current can at small signal levels also produce noise. Although cooling and quality control of components can reduce these effects, in any practical measurement system, they cannot be completely removed. Thus, the error of the resolution of the scale being the limiting error only applies above a certain noise floor level, and, for any electronic measurement system, this system noise floor is a fundamental limitation in the accuracy of any measurement. These noise error sources can to an extent be modelled statistically and as such can be accommodated with classical EM theory. However, the addition of thermodynamic aspects into the modelling of a measurement system’s response can only be approximate as at root classical EM theory is based on differential equations and the continuous charge distributions associated with extended objects. The actual behaviour of discrete charged matter, in the circumstances we are primarily concerned with, is characterised by statistical concepts based on Fermionic half integer spin particulate distributions and these are concepts that have no classical interpretation.

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Principles of planar near-field antenna measurements

Thus accurate modelling of such effects requires either the abandonment of classical concepts or an ad hoc marriage of incompatible concepts to produce a hybrid model of the system response. If very high degrees of confidence in the results are required, this is a process that only has limited applicability. The same limitations implied the application of individually robust but collectively incompatible concepts could also be seen to be present for a range of other environment aspects of any practical measurement system.

8.3.2

The empirical approach to error budgets

This suggests that in the absence of some overriding definitive standard or infallible model, although modelling can be used to identify error sources, the only practical methodology for assessing the ability of any test facility to make measurements is by way of repetition of the measurement procedure. This repetition can be accomplished without alteration in the measurement configuration, to simply address repeatability and precision, or with the inclusion of parametric variations to assess sensitivity. The parametric variations can also be used to assess the accuracy of the measurement if enough thought is devoted to the nature and extent of the parametric variations to be used, along with the types of analysis that are to be employed in the assessment process. In essence, the desirability of such empirically based schemes is based on the same three practicalities highlighted as the basic reasons for attempting to characterise antenna performance in Chapter 1: With careful control and design of the measurement procedures, it will produce results that reflect what it is that we are actually interested in. Since the assessment scheme is based on utilising the measurement procedure, it reflects what we can actually measure. Since the data sets produced will have the same structure as the measurement data, it is particularly amenable to a range of rigorous mathematical/statistical examination techniques that can be justified rigorously and accurately using theory.

●

●

●

The ingenious use of measurement repetition methodologies within a nearfield range to examine precision, sensitivity and accuracy can be used to assess the impact of the main possible error sources for a near-field range. These being: 1. 2. 3. 4. 5. 6. 7. 8.

Probe relative pattern Probe polarisation ratio Probe gain measurement Probe alignment error (angular errors) Normalisation constant (for specific types of gain measurement) Impedance mismatch factor AUT alignment Data point spacing (aliasing)

Antenna measurement analysis and assessment 9. 10. 11. 12. 13. 14. 15. 16. 17.

371

Measurement area truncation Probe x, y, z positional errors (Note: sometimes transverse and longitudinal positional errors are treated separately) AUT/probe multiple reflections Receiver nonlinearly Receiver dynamic range System phase error (flexing cable, rotary joints, etc.) Room scattering Leakage and cross talk Unspecified random errors in amplitude and phase

Some of these errors being inherently random, for example random errors #17, and some are systemic, e.g. truncation error #9. In this volume, it is not proposed that these individual errors be examined along with methodologies for there quantitative acquisition and classification as type A or type B. Ample literature exits as to the identification, e.g. [7] and quantitative acquisition, e.g. [8] of error sources and a through, extensive presentation is provided in Chapter 10 of [9]. However, the implications of these error sources on the integrity of data sets recorded during antenna measurements and therefore their applicability as sources of information to antenna performance must be addressed. To do this, the output data sets generated in repeated trials, either with or without deterministic parameter variation, must be quantitatively analysed, compared and assessed to establish the level of correspondence between them.

8.3.3 Applicability of the digital twin to assessing error budgets As noted above, the development of general-purpose tools for the simulation of antenna measurement systems and the construction of a digital twin is of interest for several reasons, one of which being the assessment of terms within the range assessment budget and verification of data correction algorithms, e.g. probe pattern correction, position correction, etc. which constitute a crucial constituent of the planar data transformation and post-processing. This method can be seen illustrated in Figure 8.7 where the example of the verification of various components within the data transformation and post processing chain is illustrated. A number of strategies for the simulation of planar near-field antenna measurements were presented and illustrated in the previous chapter. Here, we will illustrate a few examples that highlight the utility and convenience of the concept when constructing range assessments, and validating data transformation and post processing algorithms. Of the items noted above that are typically considered to comprise the range uncertainty budget, the following can be assessed, at least in part, through the use of a digital twin, and in a convenient and straight forward manner: ● ●

Probe x, y position error. Probe z position error (planarity).

372

Principles of planar near-field antenna measurements

Aperture illumination function

• • • • •

Simulated data: Plane rectilinear Rotated plane Cylindrical Auxiliary translation Auxiliary rotation

Transform to far-field using trial algorithm.

• • • •

Comparison of data in far-field co-ordinate system: Overlay contours Overlay cuts Calculate EMPL Calculate correlation statistics

Figure 8.7 Illustration of software verification data processing chain

● ● ● ●

Aliasing. Truncation. Systematic phase, e.g. drift. Dynamic range.

These are now illustrated in the following sections where the example of a simulated planar near-field measurement of a 5 GHz, C-band, 30 cm by 40 cm elliptical offset parabolic reflector antenna with a focal length of 50 cm will be examined with an antenna to probe separation of 70 cm. The 70 cm distance places the measurement plane a little over three wavelengths away from the feed which is a position that would be reasonable in practice from both a mechanical interference point of view, and electrically as it places the probe outside of the reactive nearfield region. The simulated near-field measurement had a data point spacing of a half wavelength in both x- and y-axes and an estimated maximum far-field pattern angle validity of 60 in both azimuth and elevation, based on a scan plane size of 1.4 m in x and y (see Section 5.3). The measurement was simulated using the

Antenna measurement analysis and assessment

373

current element method that was developed in Chapter 7, with extensive details of how this method can be used to simulate an offset reflector antenna available in [9]. Although in principle this can also be used to example plane-polar, plane-bi-polar etc. measurement systems, here we will limit ourselves to the plane-rectilinear case for the sake of brevity as in principle, these alternative geometries may be treated in a very similar fashion.

8.3.4 Truncation Simulated orthogonal, tangential, planar near electric field amplitude and phase data can be found presented in Figures 8.8–8.11. Here, the near-fields are tabulated on a plaid, monotonic and equally spaced plain rectilinear grid. This then, can be considered to be an unperturbed near-field simulation of a horizontally co-polarised antenna measurement that is truncated, as it is necessarily finitely large, but is in all

–10

1

–0.5

0 x [m]

0.5

1

–30 0 –40 –50

–1

–60

–1

–20

–0.5

–50

–1

–10

0.5 y [m]

–40 –0.5

Amplitude [dB]

–30 0

0

1

–20

0.5 y [m]

Ey freq 5 [GHz]

0

–70

Amplitude [dB]

Ex freq 5 [GHz]

–60

–1

–0.5

0 x [m]

0.5

1

–70

Figure 8.8 Ex pol simulated near-field Figure 8.9 Ey pol simulated near-field amplitude pattern amplitude pattern Ex freq 5 [GHz]

Ey freq 5 [GHz] 150

150

1

1 100

100 0.5

–50

–0.5

50 0

0

–50

–0.5

–100 –1

Phase [deg]

0

0

y [m]

50 Phase [deg]

y [m]

0.5

–100 –1

–150 –1

–0.5

0 x [m]

0.5

1

–150 –1

–0.5

0 x [m]

0.5

1

Figure 8.10 Ex pol simulated near-field Figure 8.11 Ey pol simulated near-field phase pattern phase pattern

374

Principles of planar near-field antenna measurements

other respects represents our truth model. Although only the orthogonal tangential electric fields are presented, this method may be used to compute the complete near electromagnetic six-vector, in both the propagating (as is used here) and reactive near-fields, which can be useful on occasion. Additionally, the same simulation technique can be used to compute the ideal truncation-free, far-field pattern against which we can compare all of our transformed data, cf. Figure 8.7. Although in this case the antenna is linearly polarised, in general circularly polarised and slant linear antennas, etc. may also be simulated providing a very flexible and powerful measurement simulation. This data was transformed to the far-field and was compared with equivalent far-field data that was obtained directly from the simulator. Figures 8.12 and 8.13 show the far-field copolar and cross-polar cardinal cuts obtained from the near-field to far-field transform compared with the equivalent cut obtained directly from the simulation together with the dB difference level. Similarly, Figures 8.14 and 8.15 show the equivalent far-field phase cuts.

0

Ex RMS dB diff –61.37 [dB]

0 Direct FF PNF-FF dB diff

–10

–20 Amplitude [dB]

Amplitude [dB]

Direct FF PNF-FF dB diff

–10

–20 –30 –40

–30 –40

–50

–50

–60

–60

–70 –1

Ey RMS dB diff –72.08 [dB]

–0.5

0 u

0.5

–70 –1

1

Figure 8.12 Ex far-field amplitude pattern comparison

–0.5

0.5

1

Figure 8.13 Ey far-field amplitude pattern comparison

Ex

Ex Direct FF PNF-FF

150

Direct FF PNF-FF

150

100

100

50

50

Phase [deg]

Phase [deg]

0 u

0 –50

0 –50 –100

–100

–150

–150 –0.8 –0.6 –0.4 –0.2

0 u

0.2

0.4

0.6

Figure 8.14 Ex far-field phase pattern comparison

0.8

1

–1

–0.5

0 u

0.5

Figure 8.15 Ey far-field phase pattern comparison

1

Antenna measurement analysis and assessment

375

The RMS dB difference level can be used as a measure of adjacency between the respective amplitude cuts, see below in this chapter for a more expansive discussion on pattern comparison. Usefully, these RMS dB difference levels can be used as input to the range assessment and are a clear illustration of phenomena associated with the first and second order truncation effects. Here, although it can be argued that the difference level also contains a contribution arising from numerical truncation and rounding from the different numerical algorithms, by far the largest contribution comes from the truncation of the planar near-field data set that was used to compute the far-field pattern. This comparison is encouraging as the simulation was predicated upon the current-element method, whereas the planar near-field to far-field transform utilises the plane-wave spectrum method that has been developed within this text. As such, these represent two very different methods for accomplishing the same ends. A further illustration of the usefulness of this approach is to further truncate the planar near-field data set and to re-compute the far-field patterns. Here, the nearfield data was truncated to a 2 2 m square acquisition with the resulting far-field amplitude results being presented in Figures 8.16 and 8.17 where we see the RMS dB difference level has increased by ca. 3 dB on the copolar and ca. 5 dB on the cross-polar. In this way, it is possible to estimate the impact that truncation has on a given measurement and when considering a new facility, to decide with some degree of certainty how large the system must be in order to achieve some specified uncertainty. Although truncation was not the first component within the range uncertainty budget, we have chosen to start with this as it is necessarily present in all planar near-field measurements and simulated measurements, unless that is the measurement plane is chosen to coincide with the antenna aperture plane, which is clearly an unrealistic choice in reality.

0

Ex RMS dB diff –58.50 [dB]

0 Direct FF PNF-FF dB diff

–10

–20 Amplitude [dB]

Amplitude [dB]

Direct FF PNF-FF dB diff

–10

–20 –30 –40

–30 –40

–50

–50

–60

–60

–70 –1

Ey RMS dB diff –67.33 [dB]

–0.5

0 u

0.5

1

Figure 8.16 Ex far-field amplitude pattern comparison for a truncated 2 2 m square acquisition plane

–70 –1

–0.5

0 u

0.5

1

Figure 8.17 Ey far-field amplitude pattern comparison for a truncated 2 2 m square acquisition plane

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Principles of planar near-field antenna measurements

8.3.5

Numerical truncation and rounding error

The effects of truncation and rounding can be illustrated easily through simulation. The simulated near-field measurement can be perturbed and the resulting far-field compared with the unperturbed result. As an example, the near-field simulation used in the previous section was retransformed only here, holding the (x,y) probe position to mm, i.e. rounded to three decimal places. The resulting far-field patterns can be found presented in Figures 8.18 and 8.19. Here, the RMS dB difference level was 62 dB. This is a fairly insidious issue that can become significant when working at higher frequencies where this can manifest itself with the appearance of spurious side-lobes and increasingly unreliable phase functions. In contrast, rounding the amplitude to three decimal places in dB form and the phase to three decimal places in decimal degree form yields an RMS dB difference level of ca. 160 dB, which is well below the other terms in the facility range uncertainty budget for this example. Conversely, rounding to only two decimal places increased the RMS dB difference level to ca. 94 dB, which is also smaller than the other uncertainty components in the budget. However, the low cost of modern digital mass storage suggests that the cost of adopting one additional decimal place for each of the amplitudes and phases when recording the electric fields is a very worthwhile and inexpensive investment. In general, as soon as difference is small compared to the wavelength (ca. 1% is a good measure) this issue tends to become unimportant. This analysis also serves to highlight the criticality of positional accuracy in near-field measurements which is addressed in the next section. 0

Ex RMS dB diff –62.16 [dB]

0 Direct FF PNF-FF dB diff

–10

–20 Amplitude [dB]

Amplitude [dB]

Direct FF PNF-FF dB diff

–10

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Figure 8.18 Ex far-field amplitude Figure 8.19 Ey far-field amplitude pattern comparison for (x,y) pattern comparison for (x,y) probe position rounded to probe position rounded to three decimal places three decimal places

8.3.6

Probe x,y (in-plane) position error

As suggested by the results presented within the previous section, positional accuracy of the near-field probe is an important issue. A commonly encountered issue when acquiring planar near-field data with continuous motion of the probe,

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i.e. when taking data on the fly, in bidirectional modes which is generally preferred as it avoids the time taken for the near-field probe to ‘fly back’ to the start prior to taking each cut in order that the data is taken with the probe always travelling in the same direction, is data registration. This means that any backlash in the positioning system or dwells between the time when the receiver is triggered to take a measurement and the VNA actually taking the measurement can results in some degree of bidirectional position error. This form of x,y positioning error can be readily incorporated into the simulation with an example of the effect that this has on the simulated near-field data being presenting in Figures 8.20 and 8.21. Here a positioning error of 1 cm, i.e. a sixth of a wavelength, was introduced into every other cut to illustrate the nature of the effect that this can have on the corresponding farfield patterns. The simulation assumed that the measurement was taken while scanning in the y-axis so the error in the registration of the near-field data can be seen by comparing adjacent vertical cuts. This can be seen in both the amplitude and phase patterns compared to Figures 8.8 and 8.10. The near-field data in Figures 8.20 and 8.21 was then transformed to the farfield, without compensation for the error in position registration, with the far-field cardinal cuts presented in Figures 8.22 and 8.23. Here, we can see the RMS dB difference level is 68 dB on the horizontal cut and 61 dB for the vertical cut. However, it is also evident that while the greatest effect can be seen for the vertical cut, at wide angles in the horizontal cut, the difference level increases noticeably. If we were to increase the registration error to a half wavelength, then a spurious grating lobe would appear centred at u = 1. Generally, in modern test systems, these errors are sufficiently small as to be effectively negligible. However, they can be encountered on occasion and when they are, the resulting phenomena are deleterious. Whilst this is important to note, perhaps a more commonly encountered positional error results from axis linearity. The precise nature of the positional error will depend to a large extent upon the Ex freq 5 [GHz]

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Figure 8.20 Ex pol simulated near-field Figure 8.21 Ex pol simulated nearfield phase pattern for x,y amplitude pattern for x,y positioning error positioning error

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Figure 8.22 Ex pol u-cut far-field amplitude pattern for x,y positioning error

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Figure 8.23 Ex pol v-cut far-field amplitude pattern for x,y positioning error

design of planar near-field positioning system concerned and a range of mathematical models can be constructed to approximate this. An alternative approach would be to measure the position of the probe across the acquisition interval by means of a laser tracker. To obtain the accuracy required this would generally be accomplished in a stop-motion mode so as to be able to integrate out the positional uncertainty, this can yield useful information because timing errors would be omitted. However, these are generally small and can be determined using alternative means. The grid with which this mechanical data is tabulated is usually far coarser than that which is used to acquire near-field data as a consequence of the stop-motion measurement mode. Thus, practically, the positional errors are typically interpolated onto a finer abscissa by means of spine interpolation which is appropriate as we may reasonably assume these positional errors are, on the macroscopic scale, slowly varying functions of position. In the following illustration, a positional error in the x-axis was assumed to be 3.5 mm peak-to-peak with an RMS error of 0.8 mm, in the y-axis a peak-to-peak error of 0.9 mm with an RMS error of 0.3 mm, and in the z-axis a peak-to-peak error of 0.2 mm with an RMS error of 0.1 mm were assumed. Again, 5 GHz near-fields were simulated and transformed to the far-field before being compared to the equivalent ideal patterns. These can be seen presented in Figures 8.24 and 8.25 where we see the RMS dB difference levels were 69 dB and 77 dB for the horizontal and vertical cuts respectively. Here, we have illustrated the use of this for analysing in-plane position errors. It is possible to compute equivalent results for planarity errors. It is also possible to use analytical formula to obtain a reasonable, very similar as it turns out, estimate which requires far less effort. The Ruze equation is an expression that relates the gain of an antenna to the RMS random surface errors for a reflector antenna [10]. The equation was initially developed for use with parabolic reflector antennas, however it was later adapted for use with phased arrays which is a case that is very closely aligned to the PNF measurement case that we are concerned with here. For

Antenna measurement analysis and assessment 0

Ex RMS dB diff –68.72 [dB]

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Figure 8.24 Ex pol u-cut far-field amplitude pattern for x, y,z positioning errors

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Figure 8.25 Ex pol v-cut far-field amplitude pattern for x,y, z positioning errors

the phase array case, a factor of 2 is divided out from the exponential which follows from the fact that for a reflector antenna (or CATR), the effect of a surface deformation is twice as important as it has an effect on both the incoming and reflected outgoing waves. For the phased arrays case, and the PNF case, this is not so. Thus, the Ruze equation may be stated as, 2

GðeÞ ¼ G0 eð2pe=lÞ

(8.17)

Here G(e) is the modified gain, G0 is the gain of the antenna and e is the RMS z-directed positional error. Thus, for a frequency of 19 GHz and an RMS planarity of 0.086 mm the boresight gain error is 0.005 dB which yields and equivalent error/signal level of 64.4 dB. Repeating this via simulation obtained a value of 65 dB, which was almost the same. However, the simulation reveals that this is dominated by the error in the main-beam region, and as such this is a pessimistic estimation. Repeating this and excluding the main beam region yielded a value that was significantly smaller, i.e. 84 dB. Usefully the Ruze equation can be adapted so that the effect that this has in other directions can be examined, e.g. 60 as, 2

GðeÞ ¼ G0 eð2pe sinðqÞ=lÞ

(8.18)

Thus, at 60 , the gain error reduces to 0.001 dB, with an equivalent E/S level of 76.7 dB. The importance of this simulation is really that we may use the simulator to examine the upper bound uncertainty that a given positional error would have on a particular measurement. This will be revisited in the next chapter where we use a similar approach to verify the effectiveness of a compensation technique.

8.3.7 Aliasing (data point spacing) Aliasing is an error that we should not encounter providing we adhere to the rules that are prescribed by the sampling theorem. In practice, we would wish to acquire

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data at a little finer than a half wavelength so as to provide some margin. However, when testing electrically large, high gain antennas where wide out pattern information is not required, we may take data at less than this and speed up our acquisition times. A common example would be when testing high gain satellite communications antennas, e.g. deployable Gregorian reflector assemblies, which at most require pattern data out to 8.7 which is the angle the Earth subtends from geostationary altitude. The results of sampling at 0.75l in both x- and y-axes can be seen presented in Figures 8.268.29 respectively. Whilst clearly these patterns are grossly incorrect for wide out pattern angles, for regions closer to the main beam direction the aliasing error may be sufficiently small as to be acceptable. Especially, when consider that for the case of our satellite communications example, the edge of the Earth is at u = 0.15 where the dB difference level is ca. 70 dB below the peak of the pattern. Ex RMS dB diff –39.78 [dB]

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Figure 8.26 Ex pol u-cut far-field amplitude pattern for 0.75 wavelength sampling

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Figure 8.27 Ey pol u-cut far-field amplitude pattern for 0.75 wavelength sampling

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Figure 8.28 Ex pol u-cut far-field phase pattern for 0.75 wavelength sampling

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Figure 8.29 Ey pol u-cut far-field phase pattern for 0.75 wavelength sampling

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8.3.8 Systematic phase, e.g. drift RF system phase drift is a commonly encountered issue in near-field measurements. Generally this results from variations in the temperature of the test system with mechanical changes in the guided wave path contributing towards further variations in the measured phase. To illustrate the effect of this, a very simple case of a linear 10 phase change as a function of the x-axis position was incorporated into the near-field measurement. This broadly corresponds to the idea that the acquisition is taken by scanning the y-axis (in a short period of time) and stepping the x-axis. Conceptually, this corresponds to electrically scanning the beam of the AUT horizontally which can be seen in Figures 8.30 and 8.31. The lobed form of the dB difference level for the horizontal cut corresponds to the effect that one has when two patterns are misaligned which is what we would expect to see here. The RMS dB difference level has increased significantly with significant consequences for the ability of the measurement to correctly determine the pointing of the beam. This was a very simple, perhaps unrepresentative, example. However, we may readily adapt the form of the phase drift to include other behaviours, e.g. a sinusoidal variation of phase, or perhaps a triangular wave form which can be observed resulting from the behaviour of many heating ventilation and air-conditioning systems (HVAC) which tend to turn on periodically and then allow the chamber’s temperature, and therefore phase, to drift gradually between times. The harmonic nature of this behaviour tends to result in errors adding up coherently in certain directions such that the phenomena are more pronounced on certain side lobes only, with the angle being largely dependent upon the corresponding spatial frequency of the phase drift. This can be seen illustrated in Figure 8.32 where a phase variation of 2 peak-to-peak was superimposed on the near-field data that had 25 full cycles applied across the x-axis scan. From inspection of this figure, it is clear that the side lobes in the region of u = 0.5 were mainly affected with higher spatial frequencies

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Figure 8.30 Ex pol u-cut far-field amplitude pattern for x position phase change

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Figure 8.31 Ex pol v-cut far-field amplitude pattern for x position phase change

1

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Principles of planar near-field antenna measurements 0

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Figure 8.32 Ex pol u-cut far-field amplitude pattern for cyclic phase error in x

corresponding to the wider out sidelobes being corrupted. This also highlights the quite significant effect that comparatively small errors have providing they contain a strong spectral component.

8.3.9

Dynamic range

The impacts of dynamic range errors on transformed far-field data can be established by superimposing a random signal on the simulated near-field data. Typically the noise would have the form of a normally distributed amplitude error and a uniformly distributed phase error. This stems from the way that phase is determined within the receiver or VNA. The effect of adding a random signal at a 70 dB level on the simulated planar near-field measurement can be shown in Figures 8.33 and 8.36. This 5 GHz data was transformed to the far-field and compared with the ideal far-field pattern. In this simulation, the noise was added across the entire near-field acquisition interval as a constant signal level rather than being incorporated as a noise source that is dependent upon the signal level itself. For the case of a planar near-field simulation, this provides a pessimistic estimation.

8.3.10 Summary In this section, we have attempted to illustrate the way in which simulation and the concept of a digital twin can be used to design antenna measurement, evaluate certain terms within the facility level uncertainty budget. With greater computational effort other components of the uncertainty budget may be evaluated by means of

Antenna measurement analysis and assessment Ex freq 5 [GHz]

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Figure 8.33 Ex pol simulated near-field Figure 8.34 Ex pol simulated nearamplitude pattern with field phase pattern with 70 dB noise added 70 dB noise added

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Figure 8.35 Ex pol u-cut far-field amplitude pattern with 70 dB noise added

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Figure 8.36 Ey pol u-cut far-field amplitude pattern with 70 dB noise added

computational electromagnetic simulation (CEM), many of which were developed within the preceding chapter; however, the purpose of this section was to highlight the simplicity with which the behaviour of the measurement system can be explored and the ease with which useful upper bound values can be determined that may be used to guide the design of a given antenna measurement or test facility.

8.4 Illustration of the compilation of range assessment budgets As noted above, a detailed description of the range uncertainty budget can be found in the open literature with a lengthy and very detailed treatment being found in [9].

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Principles of planar near-field antenna measurements

It is worthwhile nevertheless to summarise the steps in compiling range assessment uncertainty budget with the focus being on tailoring this to the planar implementation. An example, fictitious, range uncertainty budget is shown in Figure 8.38. Here, dummy numbers have been entered into the table and do not intend to indicate representative values for the items concerned. As this budget is intended to illustrate the planar case, probe positioning errors have been broken out into in-plane, i.e. (x,y) errors and planarity, i.e. z, position errors. In compiling this table, it is assumed that those numbers denoted in red represent values that are entered into the table, whilst those in black are computed. The choice here is unimportant and is instead used to illustrate the arithmetic steps required. Values are determined experimentally, analytically, or by virtue of CEM simulation. Once determined, these values would be used to populate the table shown in Figure 8.38 where the error sources have been divided amongst six categories (Column 5) which are: 1. 2. 3. 4. 5. 6.

Probe/illuminator related errors. Mechanical/positioner related errors. Absolute power level related errors. Processing related errors. RF sub-system related errors. Environmental errors.

These loose categories are also denoted by the shading within the table. Next we shall turn our attention to the arithmetic of compiling the range uncertainty estimate. If a measurement, or simulation is used to determine an error to signal level (E/S) in dB then this would be entered into the table directly. This is the case for the majority of the values shown in this example, which are highlighted in red in the column labelled ‘Estimated E/S level (dB)’. The E/S level in dB itself may be determined, for example, from the RMS dB difference level between two measurements or simulations where a single parametric change has been introduced. First, each data set is converted to linear form from dB, i.e. using, I ðnÞ ¼ 10ðI ðnÞdB =20Þ

(8.19)

The dB difference level z(n)dB can be computed using, zðnÞdB ¼ 20log10 ðjI1 ðnÞ  I2 ðnÞjÞ

(8.20)

where I1(n) and I2(n) denote the respective data sets. This measure of adjacency can be plotted against the respective data sets to illustrate the level of agreement attained. However here, we are interested in the root mean square (RMS) dB difference level which can be calculated using, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0v u N u1 X (8.21) ðI1 ðnÞ  I2 ðnÞÞ2 A zRMSdB ¼ 20log10 @t N n¼1

Antenna measurement analysis and assessment

385

Here, N denotes the number of points in each cut or pattern and is the measure utilised in the previous section. This can be computed using the full twodimensional pattern or most often for a pair of cuts where the patterns have, usually, been normalised. For high gain antennas, it is more common to use cuts as the proportion of data points containing large amounts of power is larger than if a two-dimensional pattern data set were used. This is therefore likely to result in a pessimistic assessment for the uncertainty. However, as the goal of this exercise is typically to compile an upper-bound estimate of the expected measurement uncertainty, this generally represents a sensible, pragmatic, conservative, choice. The relationship between this and other measures is illustrated in Figure 8.37. Here, xi is the individual sample, x = mean(xi) is the mean value, D = xi – x is the difference, RMSD ffi s (functions in question will all have a zero mean value, in which case the RMS value reduces to s), s is the standard deviation (68.3%), 2s = s + 6.02 dB (95.5%), 3s = s + 9.54 dB (99.7%). By way of an illustration, Figure 8.38 contains an example, fictitious range assessment for a typical planar near-field system. Here, the 18 individual terms (18 since we have split transverse and longitudinal position error out) are listed together with their associated estimated uncertainty (dB) and an estimated error over signal level, i.e. E/S level, (dB). The results that were noted in the previous section, for example, would be entered into the assessment table under the E/S (dB) column. Then,

0 Data Mean σ 2σ 3σ

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2,000

4,000

6,000

8,000 10,000 Sample

12,000

14,000

16,000

Figure 8.37 Relationship between statistical measures

18,000

Estimated 18 Term Uncertainty Budget Term # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Item Probe relative pattern Probe polarization purity Probe alignment AUT alignment Probe (x,y), position error Probe z position error Gain reference uncertainty Normalization constant Impedance mismatch Aliasing (data point spacing) Truncation Receiver linearity Systematic phase error (inc. cable flex & temp effects) Leakage (and cross talk) Receiver dynamic range Multiple reflection (between AUT and probe) Chamber reflection (room scattering) Random amplitude and phase errors Gain / Directivity Total (RSS) Uncertainty at 1σ

Red values are entered values Estimated Uncertainty (dB) 0.24 0.22 0.19 0.17 0.15 0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.05 0.04 0.04 0.03 0.52

Red values are entered values Estimated E/S Level (dB) –31.00 –32.00 –33.00 –34.00 –35.00 –36.00 –37.00 –38.00 –39.00 –40.00 –41.00 –42.00 –43.00 –44.00 –45.00 –46.00 –47.00 –48.00 –24.20

Level Description Total (RSS) Uncertainty @ 1σ @ – 30 dB SLL Total (RSS) Uncertainty @ 1σ @ – 35 dB SLL Total (RSS) Uncertainty @ 1σ @ – 40 dB SLL Total (RSS) Uncertainty @ 1σ @ – 45 dB SLL

SLL Uncertainty (dB) 9.40 13.00 17.10 21.56

SLL relative to peak of pattern –30.00 –35.00 –40.00 –45.00

dB Increase in RMS for 1σ 0.0

Level Description Total (RSS) Uncertainty @ 2σ @ – 30 dB SLL Total (RSS) Uncertainty @ 2σ @ – 35 dB SLL Total (RSS) Uncertainty @ 2σ @ – 40 dB SLL Total (RSS) Uncertainty @ 2σ @ – 45 dB SLL

SLL Uncertainty (dB) 13.80 17.99 22.50 27.21

SLL relative to peak of pattern –30.00 –35.00 –40.00 –45.00

dB Increase in RMS for 2σ 6.0

Level Description Total (RSS) Uncertainty @ 3σ @ – 30 dB SLL Total (RSS) Uncertainty @ 3σ @ – 35 dB SLL Total (RSS) Uncertainty @ 3σ @ – 40 dB SLL Total (RSS) Uncertainty @ 3σ @ – 45 dB SLL

SLL Uncertainty (dB) 16.71 21.14 25.80 30.60

SLL relative to peak of pattern –30.00 –35.00 –40.00 –45.00

dB Increase in RMS for 3σ 9.5

Class Probe/Illuminator related errors Probe/Illuminator related errors Probe/Illuminator related errors Mechanical/Positioner related errors Mechanical/Positioner related errors Mechanical/Positioner related errors Absolute power level related errors Absolute power level related errors Absolute power level related errors Processing related errors Processing related errors RF sub-system related errors RF sub-system related errors RF sub-system related errors RF sub-system related errors Environmental errors Environmental errors Environmental errors

Error Type Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Systematic Random

E/S Level (linear) 0.028 0.025 0.022 0.020 0.018 0.016 0.014 0.013 0.011 0.010 0.009 0.008 0.007 0.006 0.006 measurement at multiple z distances 0.005 multiple displacement measurements / MARS processing 0.004 comparison of repeat measurement data sets 0.004 0.062 Comment simulation / probe pattern uncertainty simulation / probe polarization uncertainty simulation / alignment uncertainty simulation / alignment uncertainty - assumed zero simulation simulation SGH gain uncertainty uncertainty in AUT and SGH pattern peaks reflection coefficient uncertainty repeat measurement at reduced data point spacing simulation / measurement truncation pre-measured linearity data used to distort NF data stability measurement (e.g. overnight) repeat measurements with AUT and probe terminated

Figure 8.38 Example (containing fictitious values) of range assessment budget

Antenna measurement analysis and assessment the upper bound uncertainty can be obtained from the E/S level using,    E=SjdB  Upper Bound UncertaintydB ¼ 20log10 1 þ 10 20

387

(8.22)

If instead the estimated uncertainty were known, i.e. the upper bound uncertainty then the equivalent E/S level may be obtained using,   Upper Bound UncertaintyjdB   20 1 (8.23) E=S dB ¼ 20log10 10 When combining the individual terms into a single combined total E/S level, then the E/S level must be converted first to linear form before the RSS total can be computed. The linear E/S level is obtained using, E=SjdB 20

E=S ¼ 10

Where the RSS total of the 18 individual terms is obtained from, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 18 uX ðE=S ðnÞÞ2 E=S ¼ t RSS

(8.24)

(8.25)

n¼1

This can be expressed in the dB form as, E=SRSS jdB ¼ 20log10 ðE=SRSS Þ

(8.26)

Finally, we may compute the total upper bound uncertainty for the total 18 terms using the usual formula,    E=SRSS jdB  (8.27) Total RSS Upper Bound UncertaintydB ¼ 20log10 1 þ 10 20 The power of this technique is that we may compute the effect that the total uncertainty would have on another side lobe at a different level, or, as was illustrated above, we may compute the level for two standard deviations or three standard deviations. Thus, to illustrate this, the total RSS upper bound uncertainty at 1s at a given side lobe level SLL can be computed using,    E=SRSS jdB SLL  20 (8.28) LeveldB ¼ 20log10 1 þ 10 Here, SLL = 30 dB if we wish to estimate the upper bound uncertainty on side lobe that is 30 dB below the peak of the pattern at 1s. To compute the uncertainty at 2s, 95.5%, then we would use,    E=SRSS jdB þ6:02SLL  20 (8.29) LeveldB ¼ 20log10 1 þ 10 And lastly, for 3s, 99.7% we would write,    E=SRSS jdB þ9:54SLL  20 LeveldB ¼ 20log10 1 þ 10

(8.30)

The compilation of an example fictitious range assessment table is illustrated in Figure 8.38.

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Principles of planar near-field antenna measurements

8.5 Quantitative measures of correspondence between data sets 8.5.1

The requirement for measures of correspondence

In this text, as with other measures, the measure of correspondence is the true value of some quantity that characterises the similarities or differences between data sets. Attempts to produce objective quantitative measures of correspondence between data sets that can be used to assess the accuracy, sensitivity and repeatability associated with the production of such antenna data have been widely reported [11–13]. The utility of such comparisons or, measures of adjacency between data sets, lies not only in their ability to determine the degree of similarity between various data sets but also in their ability to categorise the way in which these sets differ. Without the ability to produce such metrics of similarity, any assessment as to the integrity of a data set is necessarily reduced to subjective value judgements. The data sets produced by near-field scanners comprise near-field measurements and post processed far-field predictions. Almost invariably it is the accuracy of the far-field predictions that are of importance so in this section we will confine ourselves to assessment of these data sets, although all the techniques could be deployed for the assessment of the raw near-field data. Classically the antenna pattern is considered as defining the relative power flux density propagating to or from an antenna in a specific direction, usually confined to power associated with a field of a given polarisation. However, the limitations imposed by this scheme, particularly with regard to the antinomy of the electron mass/energy and its implications for radiation resistance [14], mean that this interpretation of the action of antenna-to-antenna coupling can have limited applicability and can also in certain circumstances only provide limited insight. Although this concept has demonstrated its advantages in this text, alternative interpretations consistent with physical law can provide additional insight as to how an antenna pattern might be interpreted and thus assessed and compared with other data sets representing antenna patterns. Descriptions of antenna patterns based on more fundamental physical interpretations, which concentrate on the irreversible macroscopic process of measurement can be useful in assessing the process of radiative emission/absorption and are compatible with classical EM theory [15]. The Schrodinger wave equation, the Dirac equation, or Quantum Electrodynamics (QED) all can be useful as conceptual models when attempting to assess data sets produced as a result of measuring antenna-to-antenna coupling. This is because they all are empirically based interpretations that concentrate on the process of measurement prediction as opposed to the mechanism of electromagnetic interaction. Here, the antenna pattern is described by electron–photon–electron interactions that can only be specified by the probability of interaction, where this resultant probability is formed by the superposition of complex probability amplitudes. Thus, the antenna pattern, that is classically considered as defining the relative power flux density propagating to or from an antenna, is more correctly described

Antenna measurement analysis and assessment

389

as the probability of discrete electron–photon–electron interactions. Here, the probability of interaction is given over known solid angles, relative to the AUT placed at the centre of the inertial frame of reference. Consequently, the AUT pattern can be legitimately interpreted as a frequency distribution for these interactions that, when normalised to unity, can be recognised as an angular probability density function describing the process of electromagnetic interaction.

8.6 Comparison techniques Previously, the comparison of such large data sets that can be recognised as probability density distributions has been significantly simplified by the techniques of statistical pattern recognition [12]. The application of statistical techniques is particularly appropriate to antenna patterns as stated above when the nature of the pattern is not constrained to the conventional classical interpretation, i.e. is not restricted to being considered as an angular spectrum of electromagnetic waves propagating in diverse directions. Furthermore, the statistical approach has the inherent advantage that it can be used to consider the global, i.e. non-local, features of the data set and distils the complexity of the pattern into an alternative, dimensionally reduced, set of virtually unique features that can be utilised to describe the data. This extraction of global features is of particular relevance for antenna patterns as it takes account of the inherently anti-reductionist and holistic nature of the integral transforms that relate the aperture excitation to the angular far-field pattern. The holistic nature of the respective domains can be readily expounded as a change in any part of the spatial domain will result in a corresponding change to every part of the spectral domain and vice versa.

8.6.1 Examples of conventional data set comparison techniques An often-adopted technique for the quantitative comparison of antenna pattern data sets is the calculation of an equivalent multi-path level (EMPL). This can be thought of as the amplitude necessary to force the different pattern values to be equal. If no account is to be taken of the phase of the patterns, as is often the case when assessing far-field data, then the EMPL can be expressed in terms of the amplitude of the samples as,    jjI1 ði; jÞj  jI2 ði; jÞjj  EMPLdB ¼ 20log10 (8.31) 2 Here, the factor of a half has been included as it is assumed that the ‘correct’ value lies between the two measured samples. Typically these will be normalised so that the peak of one or other data set is 1 V, i.e. 0 dB. Figure 8.39 illustrates the calculated EMPL comparison of cuts through the pattern for two repeat scans of a high gain antenna.

390

Principles of planar near-field antenna measurements 10 0

Normalised gain/dB

–10 –20 –30

Scan 1 Scan 2 EMPL

–40 –50 –60 –70 –80 –90 –60 –50 –40 –30 –20 –10 0 10 20 Azimuth angle/degrees

30

40

50

60

Figure 8.39 Cuts through two scans and their calculated EMPL Scans 1 and 2 in Figure 8.39 agree very closely; however, the calculated EMPL shown on the same plot clearly shows the extent and the nature of the differences between them. The differences can be seen to be a function of the signal strength as the EMPL closely mirrors the variation in the pattern. Also a left to right EMPL trend can be seen implying an asymmetry in the measurements. The EMPL level is an easily evaluated metric that is conceptually simple as; broadly, it represents the size of signal required to make the two different signals the same. Thus, this technique is highly sensitive to the presence of constant or varying amplitude displacements between the comparison data sets. The techniques can also be extended to take account of phase information by taking the difference between the respective complex in-phase and quadrature voltages separately. It is often used to represent the uncertainty associated with a given data set, i.e. analogous to an error bar. The principal limitation of this technique is that it is a local feature comparison and as such it fails to produce a single, or small number, of coefficients that can be used to describe the data set. Instead, it produces a value for each element in the comparison data sets. This not only results in the EMPL having to be presented graphically but also requires that the comparison data sets should contain an equal number of elements, although this difficulty can often be resolved with the use of interpolation. However, the difficulties of displaying the EMPL of a two-dimensional data set like a full antenna pattern means that graphical representation can be problematic. It is for this reason only a cut through a full data set is shown in this text. Additionally as this is a local interval assessment technique, the results are often sensitive and discontinuous obscuring subtler underlying features. Smoothing, i.e. by taking a ‘boxcar average’ can mitigate such effects, although this is undesirable as the fidelity of the response is compromised.

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Other conventional measures of correspondence, not so closely associated with antenna measurements, can also be used to assess the alignment of the data sets. The peak signal to noise ratio (PSNR) is used to measure the difference between two data sets where the elements have values that lie in the range, 0  jI ðiÞj  1. The PSNR is often given in decibel units (dB), which can be used to measure the ratio of the peak signal, 1 V, and the difference between two data sets I1 ði; jÞ and I2 ði; jÞ, using the formula: 0 1 PSNRjdB

B C B C 1 B ¼ 20log10 BsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC C N @ 1P 2A ð I ð i; j Þ  I ð i; j Þ Þ 1 2 N

(8.32)

i¼1

Clearly when I1 ði; jÞ ¼ I2 ði; jÞ for all values of (i,j), the two data sets are identical thus, the peak signal to noise ratio in this case will be infinite. Although there are several different definitions for the signal-to-noise ratio, this choice is commonly employed for the purposes of digital image processing. For the data displayed in Figure 8.39, the PSNR is 54.5 dB a single numerical result that can be used to describe the correspondence of the two displayed data sets. Thus the entire data sets illustrated in Figure 8.39 can be reduced to a single value quantitatively defining the correspondence between the sets. This has the advantage of brevity and accuracy; however, much useful information is lost in reducing the dimension of the data. PSNR is the ratio of the largest signal to the arithmetic root mean square of the differences between the respective data sets, and as such the presence of a constant offset between data sets will dominate the value of the PSNR. For the case of near-field antenna measurements where the system is essentially acting as an interferometer between a reference and test path, an accurate absolute reference can only be obtained by way of a gain calibration, which is difficult and often inaccurate. Although the PSNR approach yields a single coefficient, it has the complication of having an infinite range, i.e. 0  k  ?, when expressed in dB. In practice, this metric is found to be enormously sensitive with patterns that are essentially very similar yielding very large differences. Although this technique is global, i.e. it takes account of differences between every part of each data set, it fails to take account of phase information and as it is a purely interval technique, it is sensitive to the influence of outlying points. Finally, the evaluation of the PSNR requires that the respective data sets contain the same number of elements. If two signals, such as antenna patterns, vary similarly point to point, then a measure of their similarity may also be obtained by taking the sum of the products of the corresponding pairs of points. If the two sequences of numbers are independent and random, the sum of the products will tend to zero as the number of pairs of points is increased to infinity, as all numbers positive and negative are equally likely. If, however, the sum is finite and non-zero, this will indicate a degree of correlation. A negative result will occur if one sequence increases as the

392

Principles of planar near-field antenna measurements

other decreases. Thus, the cross-correlation coefficient r between two data sequences I1 and I2 of equal length can be expressed as, r¼

N 1X I1 ðnÞI2 ðnÞ N n¼1

(8.33)

The 1/N term is included in the definition of the cross-correlation to insure that the result is independent of the number of sampled points. Unfortunately, however, the value of the correlation coefficient will greatly depend on the absolute values of the respective data sets. This can be overcome by normalising the coefficient to the range 1  r  1. This in turn can be accomplished by normalising the crosscorrelation coefficient by the factor, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! ! ! u u N N N N X X u 1X u X 1 1 t I 2 ðnÞ I 2 ðn Þ ¼ t I12 ðnÞ I22 ðnÞ N n¼1 1 N n¼1 2 N n¼1 n¼1 (8.34) Thus, the normalised correlation coefficient can be expressed as, N P

I1 ðnÞI2 ðnÞ n¼1 r ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N N ffi P 2 P 2 I 1 ðn Þ I2 ðnÞ n¼1

(8.35)

n¼1

This is usually known as a cross-correlation coefficient and, as shown above, it is normalised so that its value always lies in the range, 1  r  1 where +1 implies perfect correlation, 0 signifies no correlation and –1 represents opposite signals, i.e. signals out of phase by p. For the data sets shown in Figure 8.39, the cross-correlation coefficient is 0.999941 highlighting the very close correlation between the two data sets. The cross-correlation coefficient is a computationally expensive, generalpurpose technique for obtaining a single quantitative correctly normalised measure of adjacency, a technique that is often used to calibrate time delays or offsets between theoretically identical signals. For the case of antennas, this would equate to determining the pointing error of a known antenna pattern function. It is a holistic metric that compares data over the entire extent of the two data sets although, unlike the previous techniques, zero padding the smaller data set can accommodate data sets of differing sizes. The cross-correlation coefficient can take account of amplitude and phase data provided that the data sets are represented in the rectilinear form; however, as it is a purely interval technique that essentially relies upon a summation process, it is both potentially numerically unstable and sensitive to the presence of outlying points. In practice, minor differences between otherwise similar patterns are not well discriminated. For the case of the data sets presented in Figure 8.39 where a cross-correlation coefficient of 0.999941 was calculated, as with

Antenna measurement analysis and assessment

393

many observed levels of uncertainty for near-field antenna facilities, the differences were mainly reported in the third or the fourth decimal place.

8.6.2 Novel data comparison techniques The measurement data sets that are produced in near-field antenna test ranges are complex and the integral transforms that are used on them are holistic. This suggests that assessment methods that are based on extracting features from the patterns that are global to the entire pattern, as opposed to specific to localised areas, would be useful in the assessment process. Therefore, the identification of pattern features that are a function of the entire pattern that can then be analysed to calculate a measure of comparison or adjacency in the feature space are desirable. A statistical interval measurement of correspondence based on calculating the moments of the antenna pattern when it is treated as a probability distribution has already been reported [12]. Moments of a probability density function describing area, centroid, variance, kurtosis and skewness, as shown below, yield 15 numerical values that characterise the data. These calculated numerical values, effectively dimensionally reduce the data set to 15 numbers derived from the first 5 moments of the data which represent 15 global or universal features of the entire twodimensional D (q,f) pattern. A similar techniques being possible with a smaller number of moments for data sets that only comprise cuts, e.g. (q). For data points i,j with signal value E(i,j): 0th Moment (area) A¼

I X J X

Eði; jÞ

(8.36)

i¼0 j¼0

1st Moment (centroid) Centroid in the x direction: mx ¼

I X J 1X ðkx ði; jÞ Eði; jÞÞ A i¼0 j¼0

(8.37)

Centroid in the y direction. my ¼

I X J

 1X ky ði; jÞ Eði; jÞ A i¼0 j¼0

(8.38)

Where kx (i,j), ky (i,j) represent two co-ordinate points for the data point (i,j) 2nd Moment (variance) Variance is a measure of the width or variation of a distribution. The variance of a distribution in the x direction is, I X J h i X mx2 ¼ ðkx ði; jÞ  mxÞ2 Eði; jÞ (8.39) i¼0 j¼0

394

Principles of planar near-field antenna measurements The covariance is, mxy ¼

I X J X



ðkx ði; jÞ  mxÞ ky ði; jÞ  my Eði; jÞ

(8.40)

i¼0 j¼0

The variance in the y dimension is, I X J h

i X 2 ky ði; jÞ  my Eði; jÞ my2 ¼

(8.41)

i¼0 j¼0

3rd Moment (skewness) Skewness is a measure of asymmetry of a distribution around the sample mean. If the skewness is negative, the data are spread out more to the left of the mean than to the right of the mean. If the distribution is perfectly symmetrical, then the skewness is zero. The skewness in the x dimension is, I X J h i X ðkx ði; jÞ  mxÞ3 Eði; jÞ (8.42) mx3 ¼ i¼0 j¼0

The x dimension co-skewness is, i

 ðkx ði; jÞ  mxÞ2 ky ði; jÞ  my Eði; jÞ

(8.43)

The y dimension co-skewness is, I X J h i X

2 ðkx ði; jÞ  mxÞ ky ði; jÞ  my Eði; jÞ mxy2 ¼

(8.44)

mx2 y ¼

I X J h X i¼0 j¼0

i¼0 j¼0

The skewness in the y dimension is, I X J h

i X 3 ky ði; jÞ  my Eði; jÞ my3 ¼

(8.45)

i¼0 j¼0

4th Moment (kurtosis) Kurtosis is a measure of the peakness or flatness of a distribution relative typically to a normal distribution, i.e. how outlier-prone a distribution is. A distribution with positive kurtosis is termed leptokurtic (clear peak to the distribution) whilst a distribution with negative kurtosis is termed platykurtic (flat top to the distribution) and intermediate values are termed mesokurtic: mx4 ¼

I X J h X

ðkx ði; jÞ  mxÞ4 Eði; jÞ

i (8.46)

i¼0 j¼0

mx3 y ¼

I X J h X i¼0 j¼0

i

 ðkx ði; jÞ  mxÞ3 ky ði; jÞ  my Eði; jÞ

(8.47)

Antenna measurement analysis and assessment mx2 y2 ¼

I X J h i X

2 ðkx ði; jÞ  mxÞ2 ky ði; jÞ  my Eði; jÞ

395 (8.48)

i¼0 j¼0

mxy3 ¼

I X J h X

i

3 ðkx ði; jÞ  mxÞ ky ði; jÞ  my Eði; jÞ

(8.49)

i¼0 j¼0

my4 ¼

I X J h

X

ky ði; jÞ  my

4

Eði; jÞ

i (8.50)

i¼0 j¼0

This set of 15 separate calculations can therefore be used to condense and reduce the dimensionality of the data in the far-field predicted files from arrays containing thousands of numerical entries to column vectors with 15 elements. Although the column vectors are far more difficult to interpret from the point of view of visual observation, since they are virtually unique to any given pattern they contain sufficient condensed information to define and compare data sets numerically. This 15-dimensional column vector virtually uniquely describes pattern data sets; however, some method of condensed data set comparison must be attempted. Any vector in a space can have another vector defined for which the inner product of the two is zero, i.e. the two vectors are orthogonal. Therefore, the product of a vector a with another vector b is zero if a and b are orthogonal to each other. This can be illustrated in the three-dimensional case for a and b. If two vectors in three-dimensional space are orthogonal to each other, ax bx þ ay by þ az bz ¼ 0

(8.51)

where the subscripts refer to the three spatial dimensions. If we are required to find b such that it is orthogonal this is simply accomplished since by and bz can be chosen arbitrarily, and then bx will be given by: bx ¼

ay by  az bz ax

(8.52)

Therefore, if a is known, it is a simple task to calculate another vector, b, that is orthogonal to it. As already stated the product of any vector with another that is orthogonal to it is zero so this operation of taking the inner product can be used to compare vectors. For the vectors above and another vector c, if b c = 0 then a = c. However, if b c 6¼ 0, then a 6¼ c. Additionally the extent to which the inner product of b and c varies from zero can form the basis of a metric that can be used to compare the differences between vectors, and a vector correlation error can be assigned to the patterns. The angles the vectors present to each other can be used as a measure of similarity, thus the magnitude and the argument are, mag ¼ bx cx þ by cy þ bz cz

(8.53)

396

Principles of planar near-field antenna measurements From the definition of the dot product, cos q ¼

b c jb jjc j

(8.54)

Thus, bx cx þ by cy þ bz cz cos q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2x þ b2y þ b2z c2x þ c2y þ c2z

(8.55)

Here, the extension to the 15, or in principle to an arbitrary, dimensional case is obvious. These two numbers can be treated as the abscissa and ordinate for a point on a plane [12]. The rectangular representation of the feature plane possesses an ambiguity, i.e. soft singularity, as when the magnitude is zero the argument can take on any value. If, however, the plot is plotted in its polar form, this can be avoided as, x ¼ ðb c Þcos q

(8.56)

y ¼ ðb c Þsin q

(8.57)

In practice, this technique has been found to yield a reliable, sensitive measure of the degree of similarity between two patterns. Although there are an infinite number of choices for the b, that are all at a normal to a, an ambiguity is not introduced as the inner product between these b vectors and the c feature vector will be the same. The method is similar to a simple correlation based on normalising two vectors to unity and taking their product, where a product of unity means that the two vectors are identical. Both methods can be expanded to take account of the angles that the feature vectors present to each other and basically each method defines the extent of similarity between the data sets. In one case, identical data sets being defined by unity and in the other by zero. Figure 8.40 illustrates the side lobe levels of four measurements made of the same antenna. The plots represent cuts along the azimuth axis, scaled in wave vector space and dB below maximum, of the predicted far-field patterns from measurements made over 3 3 m and 4 4 m scan planes. The smaller scans 16 and 19 represent repeat measurements separated by a period of hours and the scans 17 and 18 are each the product of scans made over a larger scan plane, to reduce truncation, but separated in time by a number of days. By inspection, it can be seen that the scans appear to be separated into two groups, suggesting that truncation effects are involved, but the extent of the variation between scans for such small differences is difficult to extract. Thus, by inspection, the variation in the patterns between measurements is difficult to judge; it could, however, be estimated by using the statistical image classification technique based on moments. For the entire patterns from which the four cuts shown above are taken, 15 component feature vectors were constructed from their moments. A feature vector

Antenna measurement analysis and assessment

397

–35

Gain/dB

–40 –45 –50 –55 –60 –0.7

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

Azimuth angle/radians Ba_16

Ba_17

Ba_18

Ba_19

Figure 8.40 Cuts through the four data sets

Angle vs modulas data sets (16) and (19) 0.28

Angle

0.279 0.278 0.277 0.276 0.275 309

309.1

309.2

309.3

309.4

309.5

Modulas

Figure 8.41 Points for patterns (16) and (19), a 3 3 m scan separated by hours for an equivalent isotropic pattern can be calculated and used as a metric to scale the measured pattern vectors. Since the individual moments of any pattern were not linearly related to each other, being summations of higher powers of the measured signal, they therefore vary greatly in size. Knowing that the product of two orthogonal vectors is zero, the inner product of the scaled pattern vectors was taken with a test vector which was orthogonal to the isotropic feature vector. The product of the isotropic vector and the test vector would be zero and the product of the pattern vectors with the test vector would define an interval data level of correlation between them. The angle that the vectors present to each other and the modulus of each of the four resulting vectors were calculated. Their values were plotted against one another as a single point, respectively, for each pattern. Figures 8.41 and 8.42 show the points for

398

Principles of planar near-field antenna measurements Angle vs modulas data sets (17) and (18) 0.275

Angle

0.2748 0.2746 0.2744 0.2742 0.274 410

415

420

425

430

Modulas

Figure 8.42 Points for patterns (17) and (18), a 4 4 m scan separated by days

Angle vs modulas data sets (16), (17), (18) and (19) 0.277 0.2765 Angle

0.276 0.2755 0.275 0.2745 0.274 300

320

340

360 380 Modulas

400

420

Figure 8.43 Clustering of the four points

measurements (16) and (19), the 3 3 m data sets, and measurements (17) and (18), the 4 4 m data sets, respectively. When plotted on the same graph, Figure 8.43, the four points are clearly consistent with the nature of the measured patterns, as e.g. (16) and (19) are the two closest, in fact indistinguishable on this scale, as they were derived from the data of two identical subsequent measurements. Measurements 17 and 18 represent data sets acquired over identical larger scan planes but also they show the possible error effects of drift and reproducibility, see Appendix B, as data sets 17 and 18 do not cluster as closely as 16 and 19. Therefore, very similar measured patterns should be characterised by points that cluster closely together and alternatively, distant points should be features of very different patterns. This means that by means of the method described above, the complexity of this classification problem can be reduced down to a single point per pattern and the quantitative comparison of data sets can be accomplished using the well-developed statistical technique of cluster analysis [16].

Antenna measurement analysis and assessment

399

Whether conventional or novel all the data assessment so far examined have addressed the interval nature of the data sets. However, there are two specific aspects of the near-field antenna measurement methodology that handicaps any interval pattern assessment of antenna patterns produced by near-field scanning: ● ●

The very high dynamic range of the measurement system. The interferometric nature of the measurement and the lack of uniformity of the reference source.

Both of these mean that interval assessment of the data sets can lead to misleading results as such an interval methodology depends on absolute signal levels, while the measurement technique is based on relative interferometric test and reference signal level measurements. An ordinal measure of association that overcomes this limitation can be derived if the interval nature of the data is ignored. If ranked, in terms of the amplitude, all antenna data sets sampled over the same intervals and containing the same number of elements are bijections between the set and itself, i.e. permutations of the same elements [17]. The only possible variation is in where these elements are to be found in the data sets, therefore all data sets containing the same number of elements represent different permutations of the same data. Thus, it is the similarity of the permutations that are assessed and by inference also the data from which the permutations are constructed. This provides the opportunity to construct a measure of association based on the inverse permutation of data sets with respect to each other. This will produce a metric of correspondence that is immune to many of the pathological inconsistencies of such large interval data sets that affect interval assessment techniques. Any proposed objective measure of correlation, or association, between data sets based on this methodology would be desired to be: ●

● ● ●

a single coefficient, independent of scaling or shift due to the differences in reference levels, insensitive to the large dynamic range of the data, normalised i.e. give correlation value ranging between –1 and 1, and finally, symmetrical or commutative to the operation of correspondence.

If we assume a suitable methodology of defining a single coefficient and normalising it, a value between –1 and 1 can be found. As the range of values in the permutation is limited to the number of elements in the set, the dynamic range is also limited but not restrictive. Additionally permutations mirror Abelian symmetry under a group operation [17] and, therefore, are by definition symmetrical and commutative. Thus such a measure of association based on the correlation of the permutations derived from the data sets is possible. Within the image processing community, such a measure has already been devised and implemented [18] and it can be applied to the assessment of antenna patterns [13]. Following the development of [18], this measure is expressed in terms of a rank permutation which is obtained by sorting the data in ascending

400

Principles of planar near-field antenna measurements

order and then labelling each element with integers accordingly, i.e. ½1; 2; 3; ; n where n is the number of elements in the set. The correlation between two rankings can be considered to constitute a measure of closeness, or distance. For a set of amplitude values I1 and I2 , let p1 be the rank of element I1i among I1 and p2 be the rank of element I2i among I2 . If the ranks are not unique, i.e. two elements have the same value then the elements are ranked so that the relative spatial ordering between elements is preserved. A composition permutation s is defined such that si is the rank of the element in I2 that corresponds to the element with rank pi1 in I1 . Hence, for the case of a perfect positive correlation, s ¼ ð1; 2; 3; nÞ, where n is the number of elements in the set. The definition of a distance metric to assess the distance between s and the identity permutation u ¼ ð1; 2; 3; nÞ will result in a measure of the distance between p1 and p2 . The distance vector dmi at each si is defined as the number of sj where j ¼ 1; 2; 3; i which are greater than i. This can be expressed as, dmi ¼

i X

 J sj > i

(8.58)

j¼1

where J ðBÞ in an indicator function which is defined as,  1 when B true J ðB Þ ¼ 0 when B false

(8.59)

Here, dmi can be thought of as a measure of the number and the extent to which the elements are out of order. If I1 and I2 were perfectly correlated, then the distance measure will become a vector of zeros, i.e. dm ðs; uÞ ¼ ð0; 0; 0; ; 0Þ

(8.60)

The maximum value that any component of this distance vector can take is n=2, which occurs for the case of a perfect negative correlation. Finally, a coefficient of correlation can be obtained from the vector of distance measures as, k ðI 1 ; I 2 Þ ¼ 1 

2maxni¼1 dmi n=2

(8.61)

Here, if I1 and I2 are perfectly correlated, then ðs ¼ uÞ and k ¼ 1. When I1 and I2 are perfectly negatively correlated then k ¼ 1. A flow chart that schematically represents the procedure involved in computing either k is presented as in Figure 8.44 along with examples of the use of the techniques on a few small simple data sets. Consider the two small data sets I1 and I2 where, I1 ¼ ½10; 20; 30; 40; 60; 50

(8.62)

I2 ¼ ½10; 20; 30; 40; 50; 60

(8.63)

and,

Antenna measurement analysis and assessment First antenna pattern

Second antenna pattern

Obtain ranking

Obtain ranking

401

Compute permutation

Compute distance vector

Evaluate ordinal correlation coefficient

Figure 8.44 Flow chart for ordinal assessment technique Let the rank of I1 and I2 be p1 and p2 respectively. Clearly, and,

p1 ¼ ½1; 2; 3; 4; 6; 5

(8.64)

p2 ¼ ½1; 2; 3; 4; 5; 6

(8.65)

Now, s is a composition permutation. To find the first element of s, search through the elements of p1 to find the element containing the value 1 and make a note of its index. We use the value of the element of p2 with the index corresponding to the index of the element already found in p1 as the value of the first element of s. This is then repeated for each element in s. So consider finding the first element of s, here, the first element in p1 is equal to 1. So, take the first element in p2 and place it in the first element of s. Now consider trying to find the fifth element of s for

402

Principles of planar near-field antenna measurements

example. So, here the sixth element of p1 is equal to 5. Now, the sixth element of p2 contains the value 6 thus the fifth element of s contains the value 6. Repeating this procedure for each of the elements of s in turn yields, s ¼ ½1; 2; 3; 4; 6; 5

(8.66)

dmi is a distance vector where the value of the ith element of dmi depends upon the sum of a function that counts the number of out of order elements. This function contains the number of concurrent out of order elements, i.e. if one element is out of order the function takes the value 1, if two elements are next to one another and out of order the function takes the value 2, etc. Thus as s contains only one out of order element then, dmi ¼ ½0; 0; 0; 0; 1; 0

(8.67)

Since the maximum value of dmi is 1, then the coefficient of correlation is, k ðI 1 ; I 2 Þ ¼ 1 

2 1 1 ¼ 6=2 3

(8.68)

Several more examples are presented below with just the results shown. Example 1. Perfect negative correlation Let, p1 ¼ ½6; 5; 4; 3; 2; 1 and p2 ¼ ½1; 2; 3; 4; 5; 6 . Then, s ¼ ½6; 5; 4; 3; 2; 1

(8.69)

Hence, dmi ¼ ½1; 2; 3; 2; 1; 0

(8.70)

Thus, k ð I1 ; I 2 Þ ¼ 1 

2 3 ¼ 1 6=2

(8.71)

Example 2. Partial negative correlation Let, p1 ¼ ½1; 2; 6; 5; 4; 3 and p2 ¼ ½1; 2; 3; 4; 5; 6 . Then, s ¼ ½1; 2; 6; 5; 4; 3

(8.72)

Hence, dmi ¼ ½0; 0; 1; 2; 1; 0

(8.73)

Thus, k ð I1 ; I 2 Þ ¼ 1 

2 2 1 ¼ 6=2 3

(8.74)

Antenna measurement analysis and assessment

403

Example 3. Insensitivity to texture Let, p1 ¼ ½1; 3; 2; 4; 6; 5 and p2 ¼ ½1; 2; 3; 4; 5; 6 . Then, s ¼ ½1; 3; 2; 4; 6; 5

(8.75)

Hence, dmi ¼ ½0; 1; 0; 0; 1; 0

(8.76)

Thus, k ðI 1 ; I 2 Þ ¼ 1 

2 1 1 ¼ 6=2 3

(8.77)

Here we obtain the same coefficient of correlation as we did for the first example when p1 ¼ ½1; 2; 3; 4; 6; 5 and p2 ¼ ½1; 2; 3; 4; 5; 6 , although clearly there are more out of order elements, i.e. two, in this case. Such an occurrence is an example of insensitivity to ‘texture’ (or peak magnitude of d) within a data set. Although this is clearly a disadvantage, it is not thought to be of primary importance when considering antenna radiation patterns. The ordinal process of ranking the data to produce permutations takes no account of the absolute amplitude, the interval nature of the data, or spatial angles at which the data is found. Thus, every region of the pattern is judged to be equally important in the calculation of k irrespective of the amplitude of the measurement result. However, the ordinal measure of association can be readily modified to take account of different regions of interest by re-tabulating the data in such a way as to attribute more samples to regions of greatest interest prior to ranking the data. This approach minimises the impact of numerical instabilities as observed when using a purely interval assessment technique, whilst also minimising the impact of lowlevel spurious signals as discussed above. It will in fact produce a permutation that is weighted to take more account of the specific property of the patterns that is judged to be important, e.g. higher signal levels. Assuming that the patterns are sufficiently well sampled, this re-tabulation, which can readily be determined for the case of antenna radiation patterns, can be accomplished rigorously through the application of the sampling theorem i.e. Whittaker interpolation. Alternatively, this can be performed efficiently albeit with approximation, using piecewise polynomial functions, i.e. cubic spline or cubic convolution interpolation. This will produce data sets that are biased towards the characteristics of the areas of interest by having more data points within these areas. An example of this technique would be to produce a hybrid interval/ordinal assessment technique based on placing more data points in the set in areas where the signal strength is higher. The idea of choosing to place the most samples where the field intensity is greatest is equivalent to choosing a sampling increment that is, at least to some extent, inversely proportional to the intensity of the field at that point. Figure 8.45 illustrates cuts through two data sets and the calculated EMPL

404

Principles of planar near-field antenna measurements 0

Normalised gain [dB]

–10 –20 Ant 1 Ant 2 EMPL

–30 –40 –50 –60 –70 –50 –40 –30 –20 –10 0 10 20 Azimuth angle [deg]

30

40

50

Figure 8.45 Cuts through antenna patterns and EMPL for Ant 1 and Ant 2

between them. The measurements results Ant 1 and Ant 2 are of the same antenna with the measurement setup altered so that the noise level in Ant 2 is greater at angles off boresight than for Ant 1. Clearly from the figure, the EMPL reflects the presence of the noise in the measurements as the EMPL does not appear to be related to the signal level and in fact is smaller in regions where the recorded signal level is greatest. The use of the ordinal assessment technique illustrated above defines a k value of 0.8800 for the comparison between the two data sets. If, however, we are more interested in the regions of the pattern where the signal level is high, the data could be re-tabulated prior to the ordinal assessment in terms of its absolute interval values. Figure 8.46 illustrates, what were initially the same data sets and their EMPL, re-tabulated such that the data sampling interval is a function of the signal level and thus there are far more data points in the regions of high signal level where we wish to concentrate our analysis. Here, the x-axis is still angular; we have merely deformed the pattern within that coordinate system. Clearly the main beam of the antenna where the signal level is the highest now has far more data points than was previously the case. Thus, ordinal assessment will be biased to reflect the relationship between the two data sets in the regions of high signal level. In this case, this results in a new k value of 0.9158 implying a closer correspondence between the two data sets in the regions in which we have most interest. Thus, a hybrid interval/ordinal technique has been demonstrated that can be used to assess the data if a bias based on signal level is required. Clearly the introduction of bias need not be based on signal amplitude, but angle or rate of change of signal level are examples of parameters that can be used to bias the data sets prior to assessment. The choice

Antenna measurement analysis and assessment

405

0

Normalised gain [dB]

–10 –20 –30

Ant 1

–40

Ant 2 EMPL

–50 –60 –70 –50 –40 –30 –20 –10 0 10 20 Azimuth angle [deg]

30

40

50

Figure 8.46 Re-tabulated cuts through antenna patterns and EMPL for antennas 1 and 2

of variation in sample rate as a function of parameter of course depends on the nature of the correspondences that are being sought. The ordinal and hybrid interval ordinal methodologies both overcome many of the disadvantages displayed in traditional and novel interval assessment strategies but they do place constraints on the types of data sets that can be compared. The comparison of permutations requires the two data sets to be, either identical in terms of sampling interval and extent and number of data points, or for it to be possible to interpolate the sets to arrive at a situation where these conditions hold. For complex multidimensional data sets containing many different angles and frequencies, this is often impossible. Some of these difficulties can be overcome and different data set structures can be compared if prior to the ordinal assessment, the data sets are categorised and then the relative frequencies associated with the categorisation are the subject of the ordinal measure of correspondence. Although there are a great many ways of categorising a given data set, one of the simplest is to divide the interval data set into a number of amplitude bins and to count how many elements fall within each bin, i.e. a categorical interval methodology. Each data set that is compared will provide a single histogram that can be normalised before subsequently being processed to provide the measure of correspondence. Normalisation would usually be accomplished by ensuring the total summation of the frequencies of the two sets to be compared that was equal while the relative frequencies for the bins in each data set remained constant. Figure 8.47 illustrates two simple histograms constructed from the data sets illustrated in Figure 8.45.

Principles of planar near-field antenna measurements Numbers of elements in each bin

406

Histograms of Ant and Ant 2 30 25 20 15 10 5 0

Ant 1 Ant 2

1

2

3

4

5

6

7

8

9

10

Bins

Figure 8.47 Histograms of data sets for antennas 1 and 2

Levels

Sorted data and required bin levels. 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 –55 0

10

20

30

40 50 60 70 Numbers of data points

80

90

100

Figure 8.48 Sorted amplitudes and required bin levels for antenna 1

Ten bins placed linearly between –50 dB and 0 dB were used to construct the histograms and clearly the two histograms are similar therefore the deployment of the range of assessment techniques already illustrated in the text that would lead to objective measures of correspondence between these data set histograms. However, if the comparison of different data sets can be reduced to a comparison of their amplitude histograms, a range of highly developed techniques primarily developed in the areas of image processing can be deployed. Particularly accurate and effective are techniques based on the concept of histogram equalisation [19]. Histogram equalisation is concerned with producing a histogram where there are equal numbers of entries in each bin. Usually, this is done by varying the levels and sizes of the bins until an equal number of points are to be found in each. For example, if the data for antenna 1 is sorted in terms of the number of data points at given levels, Figure 8.48 will be produced where there are required to be 10 points in each bin. From this figure, the levels that will be required to equalise a histogram of the data can be calculated and are shown in Figure 8.48.

Antenna measurement analysis and assessment

407

Numbers of elements in each bin

Equalised histogram for Ant 1 18 16 14 12 10 8 6 4 2 0

Ant 1 Ant 2

1

2

3

4

5

6

7

8

9

10

Number of elements in total bins

Figure 8.49 Antenna 1 histogram equalised using levels as shown in Figure 8.47

Cumulative best fit for Ant 1 100 90 80 70 60 50 40 30 20 10 0

Ant 1 Ant 2

0

1

2

3

4 5 6 Number of bins

7

8

9

10

Figure 8.50 Best fit line for Ant 1 cumulative data and Ant 2 scatter around it

If a histogram of antenna 2 were constructed using the same bins, then unless the patterns of antennas 1 and 2 were identical, the histogram for antenna 2 would not be equalised. Figure 8.49 shows the equalised histogram for antenna 1 using the values as shown in Figure 8.48 to define the bins and the resulting unequalled histogram of antenna 2. From Figure 8.49, it is clear that the bin levels calculated to equalise antenna 1 have not equalised antenna 2, and thus again there are clear quantifiable differences identified in the two patterns. If the above data is plotted, as a cumulative frequency as per Figure 8.50, it is clear that many standard regressive techniques can be deployed on the data to calculate measures of correspondence. The expression of the difference between two antenna patterns as the regression of a linear function allows a great deal of freedom as to which data assessment methodologies should be used; however, care should be taken when deciding on which assessment techniques to use. Figure 8.51 illustrates two very different theoretically possible antenna patterns that will have the same histograms so care must be exercised at an early stage in the choice of assessment methodology.

Principles of planar near-field antenna measurements 5

5

0

0

–5

–5

–10

–10

–15

–15

–20

–20

Level

Level

408

–25

–25

–30

–30

–35

–35

–40

–40

–45

–45

–50

50

100

150

Element

200

250

300

–50

50

100

150

200

250

300

Element

Figure 8.51 Different patterns with identical histograms, Set 1 (left) and Set 2 (right)

8.7 Summary The theory of measurement that is described but not rigorously illustrated in the early parts of this chapter highlights the requirement for obtaining quantitative, holistic and or local, measures of similarity between data sets that have been acquired within any measurement system. In addition to more conventional interval techniques usually used in antenna pattern assessment, i.e. PSNR, EMPL, and cross-correlation coefficient, other newer and more sophisticated techniques have been presented. All of the conventional and novel comparison techniques have particular areas of applicability where their specific characteristics are suited to the abstraction of the large data sets to distil their important or relevant features so that these can be quantitatively assessed. However, of the conventional assessment techniques, the EMPL has been found to be particularly useful for graphically illustrating the differences between two one-dimensional pattern cuts, whilst the ordinal measure of association has been found to be particularly adept at describing differences between two-dimensional pattern functions. In principal, the hybrid ordinal-interval technique should offer significant advantages over the ordinal technique as it takes account of the interval nature of the data set, an aspect that is primary to the statistical moments methodology. However, it is clear that the new and novel antenna measurement techniques being pioneered at present offer an assessment challenge if the large volumes of data these techniques generated are to be quantitatively, effectively and concisely analysed and summarised. All the data assessment techniques at root depend on reducing the dimensionally of the data sets to make them more easily accessible. Antenna patterns acquired in test ranges may contain tens or hundreds of thousands of individual data points and the quantitative assessment of such large data sets is close to impossible without distilling the data down to manageable levels. However, it should be

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409

remembered that all data reduction techniques will involve the loss of some information from the data sets. Thus a combination of the techniques is recommended so that inaccurate conclusions are not drawn, e.g. Figure 8.51 shows two plots that would yield identical histograms but if the interval methodology based on moments was also used on the data that exact nature of their differences could be extracted. Although these techniques, for the ease of explanation, have been illustrated on small data sets or cuts through much larger data sets as will be illustrated in Chapter 9 they are particularly effective when the nature and size of data sets make conventional assessment techniques problematic. All of the above arguments confirm the applicability of such assessment methodologies to the measurement process. They also show how they can be used to enhance the understanding and interpretation of measurement data and thus lead to the extraction of information from measurement data. However, the use of such objective quantitative methods of assessment additionally allows the interpretation of the results by a whole new variety of users. Essentially a near-field antenna range, equipped with all of its shielding, absorbing, positioning, RF and processing equipment, can: ●

●

●

●

Sense its electromagnetic environment, the measurement process is essentially a quantified sensing process in which the system records a signal level as a function of the position/orientation of the probe. Quantitatively assess the data it produces using the types of procedures discussed above. Deploy a range of decision-making artificial intelligence algorithms based on the output of the processing that allow the test range. Without outside intervention, pursue certain courses of action, i.e. a low level of correlation between repeat scans that can be highlighted by objective quantified assessment techniques could initiate a self-calibration procedure.

Although not at present comprehensively implemented in any range, known by the authors, this could provide the antenna test range itself with the ability to instigate actions in response to the decisions it takes as a result of examining the processed data. By definition [20], a machine that can sense its environment processes information from its sensors and other internal information, decides what to do next and then executes that response is a robot. Thus, the new variety of users mentioned above could include AI machines, as the decisions may need to be made in the presence of levels of uncertainty, autonomous robotic systems. These systems, as well as running the measurement procedure, can use objective measures of correspondence, examine data for inconsistencies and decide to run diagnostics on the instrumentation, on the measurements procedure or the AUT in response to decisions arrived at as a result of data processing and examination. Such intelligent autonomous measurement systems, acting on decisions made on the processed data, by removing the requirement for human involvement at every stage of course offer many prospective advantages with respect to control, reliability and cost.

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Principles of planar near-field antenna measurements

References [1] Langer S.K., An Introduction to Symbolic Logic, 3rd ed., Dover Publications, Mineola, NY, 1967, pp. 42–43. [2] Stevens S.S., Handbook of Experimental Psychology, G. Wiley, New York, NY, 1951. [3] Barrow J.D., Impossibility, the Limits of Science and the Science of Limits, Comments on the Doctoral Thesis Godel K, Oxford University Press, Oxford, 1998, pp. 222–223. [4] Boros A., Measurement Evaluation, English Translation Gabor 1989, Elsevier Amsterdam, 1989, pp. 34–42. [5] Andrews P.P. and Grewal M.S., Kalman Filters Theory and Practice Using Matlab, 2nd ed., John Wiley and Sons, New York, NY, 2001. pp. 102–103. [6] Geilert W., Gottwald S., Hellwich M., Kastner H., and Kustner H., VNR Concise Encyclopaedia of Mathematics, Van Nostrand Reinhold, New York, NY, 1989, pp. 686–687. [7] Newell A.C., “Error analysis techniques for planar near-field measurements”, IEEE Transactions on Antennas and Propagation, vol. 36, no. 6, pp. 754–768, 1988. [8] McCormick J., The Use of Secondary Spatial Transforms in Near-Field Antenna Measurements, Ph.D. Thesis, The Open University, April 1999. [9] Parini C., Gregson S., McCormick J., van Rensburg D.J., and Eibert T., Theory and Practice of Modern Antenna Range Measurements, 2nd Expanded Edition ed., vol. 1, IET Press, London, 2020. [10] Ruze J., “Antenna tolerance theory – a review”, Proceedings of the IEEE, vol. 54, no. 4, pp. 633–640, 1966. [11] McCormick J. and Da Silva E., “The use of an auxiliary translation system in near-field antenna measurements”, In Proceedings of the International Conference on Antennas and Propagation, April 1997, Edinburgh, vol. 1, p. 1.90. [12] Gregson S.F. and McCormick J., “Image classification as applied to the holographic analysis of mis-aligned antennas”, Presented at ESA ESTEC, 1999. [13] Gregson S.F., McCormick J. and Parini C.G., “Measuring wide angle antenna performance using small planar scanners”, In IEE ICAP 2001, UMIST. [14] Feynman R.P., The Feynman Lectures on Physics, vol. II, Chapter 28, pp. 28.1–28.10, Addison Wesley Publishing Company, San Diego, CA, 1964. [15] Field J. H., “Relationship of quantum mechanics to classical electromagnetism and classical relativistic mechanics”, European Journal of Physics, 25, pp. 385–397, 2004. [16] Day F., Hand D.J., Jones M.C., Cunn A.D., and McConway K.J., The Elements of Statistics, Open University/Prentice Hall, Hoboken, NJ, 1995, Section 14.2, pp. 528–547.

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[17] Stewart I.S., Concepts of Modern Mathematics, Dover Publications Ltd., Mineola, NY, 1995, pp. 100–109. [18] Bhat D.N. and Nayar S.K., “Ordinal measures for visual correspondence”, Technical Report, CUCS-009-96, Columbia University Centre for Research in Intelligent Systems, 1996. [19] Russ J.C., The Image Processing Handbook, 3rd ed., IEEE Press, Piscataway, NJ, 1999, pp. 233–239. [20] Course Team, Robotics and the Meaning of Life, The Open University, 2004, Section 1.2, pp. 3–4.

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Chapter 9

Advanced planar near-field antenna measurements

9.1 Introduction This chapter presents a brief introduction to a number of more advanced and more recently developed topics associated with planar near-field antenna measurements. These include active alignment correction which seeks to improve the accuracy with which the boresight direction of an antenna is known; position correction techniques for improving the flatness and straightness of the sampling grid; noncanonical transforms, phase-recovery which enables measurements to be made where obtaining a direct phase reference would be impractical including testing using drones and industrial robotic systems; microwave holographic metrology which is used for performing non-destructive non-invasive aperture diagnostics; compressive sensing which is used for efficient industrial testing; three-antenna extrapolated gain and polarisation measurements, auxiliary translation and auxiliary rotation which are used to minimise truncation; finishing with the poly-planar technique which is used to mitigate measurement truncation.

9.2 Active alignment correction The importance of accurately determining the electrical boresight direction of an antenna with respect to a mechanical datum can be illustrated by considering the implications of an error in pointing a high gain space telecommunications antenna aboard a geostationary satellite. By the way of illustration, Figure 9.1 contains a schematic representation of Europe and North Africa as seen from an orbit radius of 42,164.14 km, i.e. an altitude of 35,786 km and a mean Earth radius of 6,378.14 km, with a sub-satellite latitude of 0
and a sub-satellite longitude of 13
. Clearly, an error in the pointing of the geostationary antenna of as little as one degree will displace the pattern by many kilometres over the surface of the earth. For example, consider the distance between London (51.500
, 0.083
) and Paris (48.833
, 2.333
) as shown in Figure 9.1. The distance between London and Paris over the surface of the Earth can be calculated from Napier’s spherical trigonometry cosine rule for sides [1] and is found to be 343.2 km. This corresponds to an angle of less than 0.4
as seen from the geostationary orbit.

414

Principles of planar near-field antenna measurements 10 9 8 El (deg)

7 6 5 4 3 2 –6

–5

–4

–3

–2

–1

0 1 Az (deg)

2

3

4

5

6

7

Figure 9.1 Plot of Europe and North Africa as viewed from geostationary orbit As edge-of-cover (EOC) gain slopes of 10 or 20 dB per degree are becoming common place, and as coverage regions are typically defined with reference to geographical features, i.e. coastlines, rivers, political borders, etc., fractions of a degree error in antenna pointing between design and in-service performance can have significant implications on mission effectiveness. This is further complicated as the spacecraft mission lifetime is dictated by the usage of fuel, that in turn, depends on the tolerances set on spacecraft attitude control, and by the fact that the antenna alignment cannot typically be adjusted in orbit. Similarly, the polarisation purity of linear antennas, required for polarisation reuse schemes, dictates that antenna alignment in test and installation is known to fractions of a degree. Another example that requires accurate antenna to range alignment is the configuration and pattern measurement of large active antennas. A notable example was the test and measurement campaign of the advanced synthetic aperture RADAR (ASAR) antenna that had a dimension of 10 m  1.3 m and utilised an array of 320 active transmit/receive (TR) modules. The antenna weighed approximately 750 kg and was required to be a deployed structure designed for a zero gravity environment. Figure 9.2 shows the ASAR antenna deployed in a planar near-field test facility in Portsmouth together with its crucial zero-g mechanical support structure. Testing such an array therefore poses challenges for the measurement facilities that are not encountered in typical applications. Specifically, obtaining the electrical boresight relative to a fixed mechanical datum is crucial as the null-to-null beam width is just a little more than half a degree. The combined mass of the instrument and its zero gravity support is such that accurately aligning the antenna to the axis of the range was found to be quite impossible. Thus, existing alignment techniques for planar near-field testing are unable to deliver the required degree of accuracy and are generally inappropriate for use with such large, heavy gravitationally sensitive antenna assemblies.

Advanced planar near-field antenna measurements

415

Figure 9.2 The Astrium Limited ASAR antenna developed for the European Space Agency Envisat Program shown under test in the Large PNFS. Picture courtesy of Astrium. The normal aim of a range measurement process is to characterise the radiation pattern of the AUT at a very great, or infinite, distance with reference to an angular or other co-ordinate system defined with respect to the mechanical interface. This data can then be utilised to establish the extent to which the instrument fulfils its requirements. The angular accuracy required is usually in the order of 0.02
, particularly where the antenna is to be mounted on a spacecraft intended for low Earth, or especially geo-stationary orbit. If an instrument is characterised whilst its axes are not perfectly aligned to those associated with the range, a non-rectilinear correction must be applied to the data set so that range independent predictions can be made. The co-ordinate free form of the near-field to far-field transform detailed in Chapter 4 together with the probe pattern correction algorithm that was set out in Chapter 5, automatically enable far-field parameters to be determined relative to a fixed mechanical interface that is not necessarily aligned to the axes of the range. Thus, if the orientation of this mechanical interface is known, the sampled nearfield data can be used to provide alignment corrected far-field patterns. In the

416

Principles of planar near-field antenna measurements

sections that follow three techniques are set out that can each be used to acquire this data. Essentially, for the case of the planar near-field to far-field transformation, the application of alignment correction data is handled rigorously by expanding the plane wave spectrum (PWS) on an irregular grid in the range system. This irregular space corresponds to a regular angular domain in the antenna mechanical system. With the transformation of the measured Cartesian field components from the range polarisation basis into the antenna polarisation basis, the required isometric rotation is completed. The scanning probe can be thought of as a device that spatially filters the fields received from different parts of the AUT. In a planar range, the effects include something similar to a direct multiplication of the far-field probe pattern with the far-field antenna pattern. This can be seen to be a direct result of the nature of the convolution theorem [2] and can be visualised directly from the mechanical operation of the scanner. It is not usually possible to neglect these effects in the planar range because of the large angles of validity required and the short measurement distance employed.

9.2.1

Acquisition of alignment data in a planar near-field facility

The antenna-to-range alignment can be acquired by optical means through the use of optical co-ordinate measuring devices such as a laser tracker, via theodolites, mirrors and optical cubes, or by utilising a precision mechanical contact probe. A mobile optical three co-ordinate measuring machine determines points in space relative to the tracker by means of reflecting a laser beam off an optical target that usually consists of a corner reflector, i.e. a trihedral, of various sizes depending on the accuracy required. The larger the size of the trihedral, the greater the accuracy obtained. The device is aligned with the local gravity vector before the polar and azimuth angles are obtained from high precision angular encoders within the tracking head of the device. The distance |r|, however, is measured with an interferometer by counting interference fringes between the reflected and transmitted waves. In this way, the Cartesian co-ordinates of a point in space can be determined relative to the co-ordinate measuring machine directly from the measured spherical co-ordinates. Such a system is typically capable of acquiring a stationary target over an angular range of 235
in f and 45
q with an angular accuracy of 0.14 sec, and a distance of between 25 mm and 35 m. As the distance ordinate is acquired by counting wavelengths, the uniformity of optical density of the medium in which the laser is propagating is crucial in order that reliable data can be obtained. Typically, such laser interferometers can suffer from ca. 50 mm of noise per 3 m distance and this can restrict the maximum dimension of the object to be measured. Additionally, these accuracies are usually obtained by integrating over time and as such result in the inability to acquire positional information when the probe is in motion necessitating a stop motion approach.

Advanced planar near-field antenna measurements

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By acquiring a number of points, typically at least three across the measurement plane and a similar number of points across the aperture of the AUT, the alignment of the AUT with respect to the range can be readily determined. Although, by acquiring a large number of points, this technique can be seen to be flexible, accurate, and repeatable, the cost of such devices at the current time can be prohibitive. However, when available, alignment information acquired with a laser tracker can be used with the alignment compensation techniques presented below in Section 9.5. Alternatively, an antenna can be aligned optically to the axes of a planar facility with the use of a theodolite. Although a theodolite can be used to acquire the spherical angles to a target, i.e. a tooling ball, unlike the laser tracker, the distance to the target is not measured. However, if a precision length bar with targets mounted at each end is used with suitable software, this method can provide a full three-dimensional coordinate measurement of a point with reference to three fixed targets forming a Cartesian co-ordinate system. A theodolite can also be used to acquire the normal of a mirror, or other optically flat object i.e. optical cube, a practice commonly termed auto collimating. In this case, an antenna can be aligned to the axes of the range if the AUT has attached a mirror whose normal is parallel with the mechanical boresight and the range has a similarly well aligned reflector. Thus, the object of the alignment process is to make the reference and AUT mirrors parallel. The roll can then be determined relative to the local gravity vector by means of an inclinometer. In practice, however, this can become both impractical and inaccurate, as it often proves impossible to view simultaneously both the reference and antenna reflectors. In this eventuality, an auxiliary mirror is placed on the probe carriage and aligned to the reference mirror that can subsequently be translated in the xy-plane to a position that is convenient. However, without the provision of a mechanism for applying fine mechanical adjustments, such alignments are impossible to perform with the required degree of accuracy. A detailed description of such techniques can be found in [3]. Alternatively, if the scanning probe is replaced with a precision mechanical contacting probe an alternative accurate, cost effective alignment process can be affected. However, as the interface for alignment information is by way of correctly normalised orthogonal direction cosine matrices, the method of derivation of alignment information is unimportant and any of these techniques can be accommodated.

9.2.2 Acquisition of mechanical alignment data in a planar near-field facility The following acquisition of alignment data in a planar facility is based upon the premise of being able to acquire four points on the antenna mechanical interface plane in the range co-ordinate system. In practice, this is achieved by replacing the near-field probe with a precision mechanical contacting probe or laser distance measuring instrument. Such mechanical contacting probes are commonly used for

418

Principles of planar near-field antenna measurements YRFS

YAMS

XAMS

AUT aperture plane ZAMS XRFS

ZRFS

Figure 9.3 Co-ordinate system of AUT installed in planar facility surface profiling and co-ordinate measuring. From these four points, we can construct four normal vectors where the average angle between each can be used to calculate a root mean squared (RMS) angle that can be taken as an indication of the measurement error. The correct projection of each Cartesian components of the antenna system onto each Cartesian component of the range system determines the antenna to range mechanical alignment direction cosine matrix. For the case where there is a suitable datum available on the antenna, a roll angle can be deduced from any of the edge vectors. In Figure 9.3, the antenna mechanical system is denoted by the abbreviation AMS whilst the range fixed system is denoted the abbreviation RFS. Consider a planar-aperture antenna as shown in Figure 9.3. We measure the range co-ordinates of four points on the antenna aperture plane. From these four points, we construct four normal vectors, one from each of the permutations of the three points. The angles between each normal and the others are calculated and the RMS of these angles taken as an indication of measurement error. We can thus obtain a single mean normal from the average of the four individual normal vectors. This vector is za ¼ ZAMS and hence, we enter this as the bottom row of the ½B matrix: 2 3 2 3 2 3 xfrs B11 B12 B13 xa 4 ya 5 ¼ 4 B21 B22 B23 5  4 yfrs 5 (9.1) zfrs za B31 B32 B33 i.e. B31 ¼ Za  Xrfs ¼ nx

(9.2)

B32 ¼ Za  Yfrs ¼ ny

(9.3)

B33 ¼ Za  Zrfs ¼ nz

(9.4)

Advanced planar near-field antenna measurements

419

Now, we assume initially that the antenna is not rolled around the Za axis. Hence, B12 ¼ Xa  Yfrs ¼ 0

(9.5)

From this, we obtain, B211 þ B213 ¼ 1

(9.6)

Xa  Za ¼ 0

(9.7)

and,

So that, B11 B31 þ B13 B33 ¼ 0

(9.8)

Hence, we may deduce the following and therefore populate the top row of [B] from the bottom row of [B], 1 B11 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffi B2 1 þ B312

(9.9)

33

B12 ¼ 0 B13 ¼

B11 B31 B33

(9.10) (9.11)

The middle row can then be obtained from the cross product of the bottom and top rows of [B] thus specifically, B21 ¼ B32 B13  B33 B12

(9.12)

B22 ¼ B33 B11  B31 B13

(9.13)

B23 ¼ B31 B12  B32 B11

(9.14)

Where the determinant of [B] = 1. If as illustrated in Figure 9.4, we now obtain a measure of the AUT roll around Za, we may incorporate this by pre-multiplying ½B by the Zroll matrix, 2 3 cos f sin f 0 4  sin f cos f 0 5 (9.15) 0 0 1 Here, f is the angle of roll around the Za axis. For the case where there is a suitable datum available on the antenna, this angle may be deduced from the information we already have. For example, if the four points acquired are the corners of the rectangular aperture of a horn, then we may deduce the AUT roll from the top edge vector. We may also use the bottom edge vector and take an average to improve accuracy.

420

Principles of planar near-field antenna measurements YRFS

XAMS

YAMS vt AUT aperture plane

q ZAMS

nˆ

XRFS

P

ZRFS

Figure 9.4 Co-ordinate system of AUT installed in planar facility including roll Here, b n is a unit normal to the aperture plane, P is orthogonal to b n and yRFS, and n and vT is the top therefore resides within the RFS xz-plane, q is orthogonal to P and b edge vector. Clearly, from Figure 9.4, we may write that (where x, y, z are in the RFS),   b y bz  x b p ¼ n b (9.16) y ¼  nx ny nz  ¼ nzb x þ nxbz 0 1 0 n is a unit normal to Here, b   b b y  x  n q ¼p n ¼ z 0  nx ny

the aperture, hence,  bz    nx  ¼ nx nyb x þ n2z þ n2x b y  nz nybz nz 

(9.17)

Thus, 0

1 q  v t C B roll ¼ arcsin @  A  q  v t 

(9.18)

These expressions that are used to derive the antenna to range direction cosine matrices can be used within an active alignment verification programme.

9.2.3

Example of the application of active alignment correction

In order that the validity of the alignment techniques discussed above can be thoroughly tested, a low gain instrument, standard gain horn (SGH), can be

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acquired at a variety of different orientations with respect to the range axes. A low gain instrument is preferable here, as the signal levels are relatively high at large angles so that errors in the isometric rotation are clearly observable. The antenna to range alignment is measured, as described above, in each case and then the data transformed. The four antenna to range alignments used were: Set 1. AUT nominally aligned to the range. Set 2. AUT nominally aligned to the range but the scanning probe rotated around the range z-axis. Set 3. AUT miss-aligned in azimuth to the range. Set 4. AUT grossly miss-aligned in azimuth, elevation and roll to the range. The co-ordinates of the four corner points of the aperture of the SGH are acquired whilst the AUT was orientated in each of the positions described above. For Sets 1 and 2, see Table 9.1. Direction Cosine matrix for Sets 1 and 2: 2 3 9:99994536E  01 9:62894467E  04 3:16256120E  03 ½A ¼ 4 9:64654645E  04 9:99999381E  01 5:55088973E  04 5 3:16202475E  03 5:58136719E  04 9:99994845E  01 For Set 3, see Table 9.2. Direction cosine matrix for set 3. 2 9:97777254E  01 1:05650315E  02 ½A ¼ 4 1:72130108E  02 9:94705988E  01 6:43759556E  02 1:02217306E  01

3 6:57946161E  02 1:01309969E  01 5 9:92676865E  01

Table 9.1 Acquired co-ordinates of corners of SGH for set 1 and 2

Bottom left Top left Top right Bottom right

x

y

z

268.985 268.967 280.447 280.453

0.265 8.2470 8.2510 0.291

3.2612021 3.2622638 3.2297205 3.2211745

Table 9.2 Acquired co-ordinates of corners of SGH for set 3

Bottom left Top left Top right Bottom right

x

y

z

266.840 266.770 278.236 278.294

3.403 5.104 5.062 3.420

1.7299882 2.5996785 3.3406798 2.4692953

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Principles of planar near-field antenna measurements For Set 4, see Table 9.3. Direction cosine matrix for set 2 8:69528605E  01 ½A ¼ 4 4:93672376E  01 1:44079562E  02

4. 4:91830703E  01 8:62891045E  01 1:16282431E  01

3 4:49729277E  02 1:08197175E  01 5 9:93111679E  01

The corrected far-field data can be found in terms of pseudo colour (checkerboard) plots as shown in Figures 9.5–9.8. The far-field plots consists of Ludwig III vertical co-polarisation and crosspolarisation data tabulated on a regular 81-element by 81-element grid in an azimuth over elevation co-ordinate system. The data has only been plotted out to 40
in azimuth and elevation in order that the entire far-field data set should be free from first-order truncation effects. Some differences are evident in the cross-polar patterns which are primarily a result of differences in the antenna-to-probe multiple reflections. Microwave holographic metrology (MHM) or aperture diagnostics is a powerful technique that can also be usefully extended with the incorporation of active alignment correction. Figures 9.9 and 9.10 contain plots of reconstructed near-field planar illuminations derived from the same near-field data set which was acquired whilst the antenna was grossly mis-aligned within the range. Table 9.3 Acquired co-ordinates of corners of SGH for set 4

Bottom left Top left Top right Bottom right

y

z

271.346 267.075 277.059 281.300

1.175 8.546 14.210 6.868

3.0288871 2.1037310 1.5844619 2.5065814

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Advanced planar near-field antenna measurements Copolar power (dB)

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Figure 9.7 Far-field copolar and cross-polar pattern of set 3

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Figure 9.8 Far-field copolar and cross-polar pattern of set 4

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Figure 9.10 Reconstructed amplitude and phase function with active alignment correction applied

Figure 9.9 illustrates the aperture illumination function of the circular array antenna in the absence of active alignment correction. Conversely, Figure 9.10 shows the reconstructed aperture illumination function once the near-field data has been corrected. The differences between the two sets of results are not characterised by a simple linear phase taper. The uncorrected results are focused when x = 0, where the aperture plane and the translated measurement plane intersect, and become progressively diffracted as the magnitude of x increases and the two planes diverge in space. However, when corrected, the reconstructed image is focused, i.e. free from diffraction effects, as only then is the reconstructed plane coincident and synonymous with the antenna aperture plane and shows a clear resemblance with the physical aperture as shown in Figure 9.11. The small vertical phase taper evident on the corrected aperture phase function results from the small elevation pointing error which was not compensated for.

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Figure 9.11 Photograph of circular slotted waveguide array antenna (courtesy of SELEX)

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Figure 9.12 Ex polarised horizontal cut of measured and translated near-field data, amplitude (left) and phase (right) Obtaining a quantitative verification of an aperture diagnostics algorithm is difficult in the absence of detailed design information. However, as the aperture plane is merely one of an infinite number of planes in space, the verification of a plane-to-plane translation would yield confidence in the procedure.

426

Principles of planar near-field antenna measurements Ex power (dB)

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Figure 9.13 Power propagating in the near zone of an antenna The differences between the translated and measured data sets are small with an EMPL of typically –50 dB in both planes as illustrated in Figure 9.12 which shows a comparison of the amplitude and phase horizontal cuts obtained from the measured and translated near-field data sets. The phase functions are very similar with a small constant phase offset observable that results from small amounts of drift in the system that is an artefact of a gradual change in temperature within the facility between the two acquisitions. The agreement shown would be less impressive if the measurements were made at the larger AUT-to-probe separation that is a direct consequence of increased measurement truncation. By way of an illustration, as the plane-wave spectrum method essentially enables the entire electromagnetic six-vector to be determined over an entire half-space, it is not only possible to reconstruct the fields over a plane but also over a sphere or any other surface of interest. This is illustrated in Figure 9.13 where the near-field has been computed over the xz- and yx-planes and which show the field as it radiates away from the antenna.

9.3 Amplitude only planar near-field measurements The phase retrieval problem arises in applications of electromagnetic theory in which wave phase is apparently lost or is impractical to measure and only intensity data are available. The planar near-field methodology as treated herein requires holographic measurements to be made. In other words, in order that the angular

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spectrum can be obtained, knowledge of both the amplitude and the phase of the field must be available. However, direct measurement of phase becomes progressively more difficult as the frequencies concerned become higher. Matters are further compounded since even antennas that are comparatively modest in physical size when operating at high frequencies will constitute electrically large instruments. Electrically large scan planes pose significant difficulties for the planar methodology which requires that a probe be moved across a correspondingly electrically large (many hundreds of wavelengths across) scan plane whilst maintaining the same positional tolerances and phase stability of the moving guided wave path, both as a function of position and time. Thus, an alternative method that removes the requirement to measure phase is often thought desirable in circumstances such as these. Also, losses inherent within the guided wave path limit the dynamic range of the receiving system increasing the noise within the measured signal and degrading the accuracy of the measured phase. Many alternatives are available for recovering the phase from amplitude only measurements; however, in the following sections, only the most applicable and only those that are most readily implemented are discussed. Specifically, the use of reference beam addition is not considered, instead the use of multiple intensity distributions which permit the use of iterative computational procedures are developed as this experimental arrangement is perhaps more convenient for the majority of experimentalists.

9.3.1 Plane-to-plane phase retrieval algorithm The plane-to-plane (PTP) algorithm essentially entails taking two amplitude only measurements over parallel planes in the near-field of the antenna that are separated by a known distance as illustrated in Figure 9.14. Essentially, we are using the plane-to-plane transform as used for microwave holographic metrology to calculate the field over one plane from knowledge of that field over another. Now as was shown above in Chapter 4, the angular spectrum and the boundary conditions, i.e. the measurement, are related to one another through the Fourier relationship, ð1 ð1   uðx; y; z ¼ 0Þejðkx xþky yÞ dxdy F kx ; ky ¼ =fuðx; y; z ¼ 0Þg ¼ 1 1

(9.19) and,

    uðx; y; zÞ ¼ =1 F kx ; ky ejkz z ð ð  1 1 1  ¼ 2 F kx ; ky ejðkx xþky yþkz zÞ dkx dky 4p 1 1

(9.20)

Thus, if the field is known over one plane, which can be defined to be at z = 0, then the field over another parallel plane, i.e. the plane to plane transform, can be expressed as,   (9.21) uðx; y; zÞ ¼ =1 =fuðx; y; z ¼ 0Þgejkz z

428

Principles of planar near-field antenna measurements 2nd Measurement plane 1st Measurement plane AUT aperture plane

AUT (Rx) RSA (Tx)

Figure 9.14 Measurement configuration of the PTP algorithm

Here, as only knowledge of the propagating field is known, the limits of integration in the spectral domain are truncated to visible space. This equates to imposing a filter function on the spectral field components that remove all evanescent components. As these components decay exponentially away from the aperture, then as the field is propagated towards the aperture, they will exponentiate and could cause the algorithm to become unstable. In practice as the measurements are made outside the reactive near-field region, fields in invisible space will only enter into the algorithm through numerical noise. Using this plane-to-plane transform that can be implemented very efficiently using the fast Fourier transform, the plane-to-plane phase retrieval algorithm can be described as follows: 1. 2. 3. 4.

Measure the amplitude of the field over plane 1. Measure the amplitude of the field over plane 2. Use PTP transform to propagate the AUT aperture fields to plane 2 from plane 1 which yields an amplitude and phase at plane 2. Retain the phase but replace the amplitude estimation at plane 2 with the measured amplitude at plane 2.

Advanced planar near-field antenna measurements 5.

429

Use PTP transform to propagate the fields back to plane 1, yielding an amplitude and phase at plane 1. Retain the phase but replace the amplitude estimation at plane 1 with the measured amplitude at plane 1. Repeat steps 3–6 until amplitude on plane 1 (or plane 2) has converged to within a prescribed tolerance. Transform the fields to the far-field using standard algorithm.

6. 7. 8.

Figure 9.15 contains example plots of the measured amplitude taken across two parallel planar surfaces with a separation between the AUT and the probe of z = 0.105 m and z = 0.235 m respectively taken at mm wave frequencies. These amplitude patterns can be used with the plane-to-plane phase retrieval algorithm as described above to reconstruct the associated phase patterns. When the phase retrieval algorithm has converged sufficiently, the measured amplitude and reconstructed phase functions can be transformed to the angular spectrum. Figure 9.16 contains a comparison of the angular spectra obtained from a direct Ex (dB)

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Figure 9.16 Comparison of angular spectrum obtained through direct measurement (left) and phase retrieval (right)

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Principles of planar near-field antenna measurements

holographic measurement and from the plane-to-plane phase retrieval algorithm. The figure on the right was derived using retrieved phase function. Clearly, the angular spectra as recovered by conventional amplitude and phase, i.e. coherent, measurements agree with those obtained from phase recovery. However, the phase retrieval patterns clearly contain a greater amount of speckle noise. This is perhaps more apparent in Figure 9.17 which contains cardinal cut of the respective patterns together with the equivalent multipath level. Although the cuts agree the impact of the noise is clearly illustrated in the EMPL that is perhaps as little as 20 dB below the pattern functions out to 25
and then less beyond this angular region. Figure 9.18 contains the reconstructed aperture illumination function of the antenna as recovered from the coherent, amplitude and phase, measured data. Conversely, Figure 9.19 contains the reconstructed aperture illumination function as recovered from the amplitude only measurement and an application of the iterative plane-to-plane algorithm. Although encouraging, to obtain results with this limited degree of agreement took ca. 20,000 iterations of the plane-to-plane phase BT antenna

BT antenna Plane 1 Plane 2 Phase retrieval EMPL

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Figure 9.17 Measured planes 1 and 2 and resulting phase retrieved angular spectra along with the EMPL compared to the true spectra Ex (dB)

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Figure 9.18 Aperture illumination function derived from coherent measurement

Advanced planar near-field antenna measurements Ex (dB) phase recovery

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Figure 9.19 Aperture illumination function derived from amplitude only measurement recovery algorithm. In general as the separation between the two measurement planes is increased, the convergence rate increases, as the difference between the respective patterns is more significant; however, the additional separation also increases truncation within the measurement which will inevitably degrade the quality of the measurements. Clearly maintaining good alignment between the measurements is crucial to the success of the technique. The convergence rate of this algorithm is clearly very slow and an alternative algorithm, with a faster rate of convergence would be highly desirable. This therefore is the motivation for the inclusion of an aperture constraint that is currently the most widely used form of the phase recovery technique that is discussed in the following section.

9.3.2 Plane-to-plane phase retrieval algorithm – with aperture constraint The plane-to-plane (PTP) algorithm essentially entails taking two amplitude only measurements over parallel planes in the near-field of the antenna and separated by a known distance. However, if some additional information concerning the construction of the antenna is available, then the convergence rate of the algorithm can be significantly improved. One simple modification of the plane-to-plane algorithm is to incorporate knowledge of the aperture illumination function of the antenna, providing of course that the antenna has a well-defined aperture. However, many higher gain antennas, such as reflector type antennas or phased array antennas, do have well-constrained apertures and as such filtering out the fields outside of the physical aperture is both possible and convenient. The modified plane-to-plane phase retrieval algorithm can be described as follows: 1. 2. 3. 4.

Measure the amplitude of the field over plane 1. Measure the amplitude of the field over plane 2. Guess the amplitude and phase of the antenna aperture illumination function. Truncate the fields to the physical extent of the antenna aperture.

432 5. 6. 7. 8. 9. 10. 11. 12. 13.

Principles of planar near-field antenna measurements Use PTP transform to propagate the AUT aperture fields to plane 1. Replace the amplitude estimation at plane 1 with the measured amplitude at plane 1. Use PTP transform to propagate the fields back to the AUT aperture plane. Truncate the fields to the physical extent of the antenna aperture. Use PTP transform to propagate the AUT aperture fields to plane 2. Replace the amplitude estimation at plane 2 with the measured amplitude at plane 2. Use PTP transform to propagate the fields back to the AUT aperture plane. Repeat steps 4–11 until amplitude on plane 1 (or plane 2) has converged to within a prescribed tolerance. Transform the fields to the far-field using standard algorithm.

Figure 9.20 contains cardinal cuts of the conventional and phase recovered patterns together with the equivalent multipath level. Here, the degree of agreement attained between the two sets of data can be seen to be significantly better than was achieved previously which is further demonstrated by the reduction in the EMPL. Figure 9.21 contains a comparison of the angular spectra obtained from a direct holographic measurement and from the plane-to-plane phase retrieval algorithm with the aperture constraint imposed. Here, the figure on the right was derived using the retrieved phase function. The reconstructed aperture illumination function shown in Figure 9.22 further illustrates the success of the technique as the phase recovered results are clearly in good agreement with the phase measured results. However, the effect of the application of the aperture plane spatial filter can clearly be seen with the field outside the physical aperture having been suppressed which is most clearly seen in the horizontal plane. The importance of this can perhaps be best illustrated by comparing the number of iterations required in order to obtain these results. Previously, we needed

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Figure 9.20 Measured planes 1 and 2 and resulting phase retrieved angular spectra along with the EMPL level compared to the true spectra

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Advanced planar near-field antenna measurements 1

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Figure 9.21 Comparison of angular spectra

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Figure 9.22 Reconstructed aperture illumination function amplitude (left) and phase (right) 20,000 iterations whilst with the modified algorithm only 200 iterations were required. This algorithm is very similar to the previous one (same building blocks merely put together in a slightly different way) only here we apply an additional aperture constraint. In this case, this merely corresponds to truncating fields that are outside the antenna aperture which is assumed to be a disk of diameter 0.254 m. Essentially, these algorithms are the microwave analogue of attempting to reconstruct a hologram from two related, but different photographs. Although impressive, the improvement in efficiency has been gained at the cost of a loss in generality. For both this and the previous phase recovery algorithm, the image comparison techniques of Chapter 8 can be implemented to aid in the assessment of the convergence. One notable issue is in the recovery of the phase of an electronically scanned array antenna. Here, these algorithms struggle to converge to the correct result unless the initial guess for the phase function includes a linear taper that approximates, to some degree, the phase distribution of the actual scanned antenna.

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Phase retrieval algorithms do find application in a number of specialist applications but currently are not found widely deployed in industry. Even at millimetre wave frequencies and above, it is common practice to use specialised RF subsystems and optimised guided wave paths in order that phase information can be obtained. The recent interest in the use of uninhabited air vehicles (UAV), i.e. drones, for near-field measurements has somewhat reinvigorated the interest in phase recovery. The next section provides an introduction to an alternative method for accomplishing this task.

9.4 Non-iterative phase retrieval technique A great many different techniques can be deployed to address the problem of recovering the phase of a measurement with non-iterative approaches having clear advantages in terms of computational effort, real-time phase recovery, and the potential for greater accuracy. There are a number of instances in NF/FF measurements when measuring the phase directly with the probe connected to a VNA is problematic. The use of drones is becoming increasing popular for making in situ antenna measurements. The work of [4] offers a highly comprehensive study of the errors and uncertainty sources of far-field (FF) drone-based measurements, in this case undertaken at 14.5 GHz with an offset reflector AUT with gain of order 30 dBi. Outdoor slant range antenna pattern measurements taken at both 35
and 45
AUT elevation angles with drone transmitting ranges of 350 m and 700 m are reported and compared with reference patterns obtained from spherical near-field measurements made at the ESA-ESTEC test facility. Both copolar and cross-polar antenna pattern measurements were performed and levels of equivalent error signal (difference between drone measured amplitude pattern and reference pattern) of 56 dB (co-polar azimuth) and 48 dB (x-polar azimuth) were achieved. There are clear limitations to the current drone measurement technologies however. For electrically large objects or antennas operating at VHF, the height limitations of typical drones (120 m in the UK) are such they will normally be in the near-field of these antennas. Also, in the case of large antenna farms, it is often not possible to tilt an antennas elevation down sufficiently low for unobstructed slant range FF drone-based measurements. Nearfield (NF) drone measurements are thus needed, but have been limited to date by the need to measure both amplitude and phase. The ultimate limit to the accuracy in a NF measurement system is dominated by both the positional and RF phase accuracy. Currently used NF techniques include flying an optical fibre cable tether to provide the phase reference [5–7], as well as using phase retrieval techniques with multiple amplitude scans at different NF distances as described above [8,9], which have both measurement time and accuracy penalties for physically large structures. Tethered systems offer good performance and [10] describes a 2.45 GHz measurement of an 8 dB gain horn AUT using a laser tracker to determine the drone-based probe location and a tethered optical fibre for phase reference. NF phase accuracy was better than 9
and the transformed FF

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pattern exhibited pattern errors of 0.6 with 40 dB signal to noise (SNR), we shall see below that this MSE can be achieved via conventional convex optimisation based CS with (M/N) of than the length of time required to perform the measurement. True value: Value consistent with a definition of a given particular quantity.

Appendices

559

Measure: True value of a measurand evaluated via a measurement procedure, i.e. after correction, the limit in the measurement results as the number of measurement trials tends to infinity. In the absence of infinity many trials, it can only be defined as the maximumlikelihood value with a confidence interval at a given level of significance. Confidence interval: An interval in which one can be confident, with a given level of probability that a parameter lies. In measurement theory, the parameter is the measure of the measurand. Level of significance: The quantified probability that defines the level of confidence that a value falls within an interval, e.g. 5% significance means that in 95 out of 100 trials; the measurement result can be expected to fall within the interval. Specifically the level of significance of the outcome of any trial is the probability, on the null hypothesis of any result being out with a specified interval. n.b. A more detailed definition can be found in [2] Error source: Any variable that can affect the highly controlled set of operations, i.e. the measurement procedure, and thus produce an error in the measurement result. Error (of measurement): Result of measurement minus a true value of the measurand, used to establish a level of confidence in the dispersion of measurement results around the measure. Measurement – true value = error: Error/true value = relative error. Uncertainty in measurement: Parameter associated with the result of any measurement trial that characterises the dispersion of the values that could reasonably be attributed to the measurand, i.e. the result of the evaluation aimed at characterising the range within which the true value of a measurand is estimated to fall, with a given level of confidence. Standard uncertainty: Uncertainty in the measurement derived from the error associated with the functional relationship between a single error source and the uncertainty parameter. For Type A errors, a statistical correlation will have to be used in the absence of a functional relationship. Combined uncertainty: Uncertainty in the measurement derived from the errors associated with the functional relationship between all the error sources and the uncertainty parameter. For

560

Principles of planar near-field antenna measurements

Type A errors, a statistical correlation will have to be used in the absence of a functional relationship. Expanded uncertainty the definition of quoted uncertainty in the measure: The total uncertainty in the measure derived from all the errors associated with the functional relationships between all the error sources and the uncertainty parameter taking into account the distributions of the individual uncertainties and a coverage factor to define a confidence interval at a given level of significance. Coverage factor: A factor included in the calculation of the expanded uncertainty in any measurement, assuming normally distributed errors it = 2, to approximate a 95% confidence interval in which the measurand could reasonably be attributed to occur. Accuracy of measurement: Closeness of the agreement between a measurement result derived from following the measurement procedure to determine the true value of the measurand. In the absence of an infinity of trials and therefore a definite measure, the probability of proximity to and dispersion around the true value defined via the expanded uncertainty.

A.3 Appendix C: An overview of co-ordinate systems The following section presents a concise summary of perhaps the most commonly encountered coordinate systems before progressing to discuss the various methods for representing the relationships between them. A very extensive treatment of coordinate systems and polarisation bases can be found in [3] and the purpose of this section is not to supplant that. However, within this text, reference is made to many of these co-ordinate systems and the transformations between them, so a brief summary is appropriate and the following sections aim to provide that.

A.3.1

Antenna mechanical system

The antenna mechanical system (AMS) co-ordinate axes form a right-handed set nominally orientated coincident and synonymous with the RFS axes. Thus, looking in the +Zams direction, they are orientated as follows: +Xams axis is horizontal and increases towards the left, +Yams axis is vertically and increases upwards. This system is used for plotting the far-field patterns.

A.3.2

Antenna electrical system

The AES co-ordinate axes form a right-handed set nominally orientated coincident and synonymous with the AMS axes as follows: +Xaes parallel to +Xams, +Yaes parallel to +Yams, +Zams parallel to +Zaps. Thus, looking in the +Zaes direction, the nominal orientation is: +Xaes axis: horizontally orientated and increases towards the left, +Yaes axis is vertical and increases upwards and +Zaes axis increases towards the far-field. The +Zaes axis defines the electrical boresight of the antenna,

Appendices

561

and the copolar and cross-polar patterns are often resolved onto this system. The AES is attached to the AMS and moves with it. The AES may also be rotated about any or all of its axes.

A.3.3

Far-field plotting systems

It is often assumed that the grids upon which the near- or far-field antenna data as tabulated are plaid, monotonic and equally spaced. Whilst not necessary from a theoretical stand point, these conditions greatly simplify the recording process for a robotic positioner as well as simplifying the tasks of numerical integration, differentiation and interpolation. Mathematically, this implies that the output grids can be reconstructed from the expressions, X ¼ X0 þ DX ðm  1Þ

(A.13)

Y ¼ Y0 þ DY ðn  1Þ

(A.14)

Where m and n are positive integers, n ¼ 1; 2; 3;   N

(A.15)

m ¼ 1; 2; 3;   M

(A.16)

Here, X0 ; Y0 are the starting values of the grid in the x- and y-plotting axes respectively, DX ; DY are the incrementing values in the x- and y- plotting axes respectively. Although a great many, essentially equivalent, far-field plotting systems can be utilised, perhaps the most commonly adopted ones are: 1. 2. 3. 4. 5.

Direction cosine (u,v-space). Azimuth over elevation. Elevation over azimuth. Polar spherical. True-view (azimuth and elevation).

By way of illustration, Figure A.1 illustrates schematically the relationship between the polar spherical and azimuth over elevation angles. These systems are introduced and discussed in the following sections. Furthermore, relationships between the various co-ordinate systems can be obtained by equating the respective direction cosine components.

A.3.4

Direction cosine

The direction cosine grid can be conceptualised as a projection of a sphere on a plane that is orthogonal to the z-axis. ðX ; Y Þ ¼ ðu; vÞ

(A.17)

Here, u and v are the first two co-ordinates to the field point hence, br ¼ ðub e x ; vb e y ; wb e zÞ

(A.18)

562

Principles of planar near-field antenna measurements Theta axis

yAMS R θˆ

ϕˆ

–xAMS

Chi axis

kAMS

P θ

ϕ

El

O

zAMS xAMS

Az AUT

Phi axis

Q

Figure A.1 Comparison of Az/El and polar spherical angles

Where u and v are related to the spherical co-ordinates by, u ¼ sin q cos f

(A.19)

v ¼ sin q sin f

(A.20)

For the forward half space, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w ¼ cos q ¼ 1  u2  v2

(A.21)

For the back half space, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w ¼ cos q ¼  1  u2  v2

(A.22)

In the true far-field, the electric and magnetic fields will be identically zero for the reactive, i.e. non-visible, regions of uv-space. Thus, E ðu; vÞ ¼ 0 when u2 þ v2 > 1

(A.23)

H ðu; vÞ ¼ 0 when u2 þ v2 > 1

(A.24)

Figure A.2 illustrates a far-field pattern when tabulated in a regular direction cosine co-ordinate system. Here, the main beam has been steered to azimuth = 25 , elevation = 20 to help illustrate the impact that a particular choice of orientation has on the plotted pattern. The ‘K-space’ co-ordinate system is related to the direction cosine coordinate system through the linear scaling of the free-space

Appendices 1

0

0.75

–10

0.5

–30 0 –40 –0.25

Amplitude [dB]

–20

0.25 v

563

–50

–0.5

–60

–0.75 –1 –1 –0.75 –0.5 –0.25

0 u

0.25 0.5 0.75

1

–70

Figure A.2 Pattern plotted in direction cosine co-ordinate system propagation constant, k0, thus, kx ¼ k 0 u

(A.25)

ky ¼ k 0 v

(A.26)

kz ¼ k0 w

(A.27)

A.3.5

Azimuth over elevation

The azimuth over elevation grid can be thought of being that grid that is most closely related to a positioner that consists of an upper azimuth rotator, to which the AUT is attached, and a lower elevation positioner upon which the azimuth rotator is attached. As the AUT is attached to the azimuth positioner, the AUT will rotate about the azimuth axis that is therefore the polar axis. The field point is obtained by rotating the horizontal azimuth positioner and vertical elevation positioner through the angle Az and El where the order is unimportant: ðX ; Y Þ ¼ ðAz; ElÞ

(A.28)

Where Az and El define the direction to the field point through, br ¼ ð sin ðAzÞcos ðElÞb e x þ sin ðElÞb e y þ cos ðAzÞcos ðElÞb e zÞ

(A.29)

Hence, Az ¼ arctan

u w

El ¼ arcsin ðvÞ

(A.30) (A.31)

564

Principles of planar near-field antenna measurements 90

0

60

–10

El [deg]

–30 0 –40

Amplitude [dB]

–20 30

–30 –50 –60

–90 –90

–60

–60

–30

0 Az [deg]

30

60

90

–70

Figure A.3 Pattern plotted in azimuth over elevation co-ordinate system

The minus sign included with the x-axis co-ordinate is used to denote that the observer is standing behind the antenna looking in the positive z direction. An alternative, and equally valid, choice would be to associate the minus sign with the y-axis. Figure A.3 illustrates a far-field pattern when tabulated in a regular azimuth over elevation co-ordinate system. Although in the region around boresight, the pattern appears to be unchanged, towards the north, El = 90 , and south, El = 90 , poles the pattern is distorted to the extent that round objects will appear square, cf. q = 90 contour.

A.3.6

Elevation over azimuth

The elevation over azimuth grid can be thought of being that grid that is most closely related to a positioner that consists of an upper elevation rotator, to which the AUT is attached, and a lower azimuth positioner upon which the elevation rotator is attached. As the AUT is attached to the elevation positioner, the AUT will rotate about the elevation axis that is therefore the azimuthal axis. The field point is obtained by rotating the horizontal azimuth positioner and vertical elevation positioner through the angle Az and El where the order is unimportant: ðX ; Y Þ ¼ ðAz; ElÞ

(A.32)

Where Az and El define the direction to the field point through, br ¼ ð sin ðAzÞb e x þ cos ðAzÞsin ðElÞb e y þ cos ðAzÞcos ðElÞb e zÞ

(A.33)

Appendices 90

0

60

–10

565

El [deg]

–30 0 –40

Amplitude [dB]

–20 30

–30 –50 –60

–90 –90

–60

–60

–30

0 Az [deg]

30

60

90

–70

Figure A.4 Pattern plotted in elevation over azimuth co-ordinate system

Hence, Az ¼ arcsin ðuÞ v El ¼ arctan w

(A.34) (A.35)

As before, the minus sign included with the x-axis co-ordinate is used to denote that the observer is standing behind the antenna looking in the positive z direction. Figure A.4 illustrates a far-field pattern when tabulated in a regular elevation over azimuth co-ordinate system.

A.3.7

Polar spherical

The polar spherical grid can be thought of being that grid that is most closely related to a positioner that consists of an upper roll rotator, i.e. f, to which the AUT is attached, and a lower rotator, i.e. q, upon which the upper f rotator is attached. This is sometimes referred to as the ‘model tower’ arrangement. As the AUT is attached to the roll positioner, the AUT will rotate about the roll axis that is therefore the polar axis. The field point is obtained by rotating the horizontal theta positioner and vertical roll positioner through the angles q and f where the order is unimportant. For antenna measurement, this arrangement has the advantage that it moves the AUT through only a small portion of the test zone, and it places the blockage that results from the AUT mount entirely in the back hemisphere. Moving the AUT by only a small amount minimises errors associated with imperfections in the illumination of the test zone and can render probe pattern correction

566

Principles of planar near-field antenna measurements

unnecessary: ðX ; Y Þ ¼ ðq; fÞ

(A.36)

Where q and f define the direction to the field point through, br ¼ ðsin q cos fbe x þ sin q sin fbe x þ cos qb e xÞ

(A.37)

Hence, q ¼ arccos ðwÞ v f ¼ arctan u

(A.38) (A.39)

The definition of an equatorial spherical co-ordinate system is identical to the polar spherical case. The difference purely results from the application of a 90 rotation in q, in order that the main beam of the radiator points along the positive x-axis (through the equator), rather than along the positive z-axis (through the pole). Figure A.5 illustrates a far-field pattern when tabulated in a regular polar spherical co-ordinate system.

A.3.8

Azimuth and elevation (true-view)

ðX ; Y Þ ¼ ðAz; ElÞ

(A.40)

Where Az and El define the direction to the field point through, Az ¼ q cos f

(A.41)

0

180 150

–10

120 90

ϕ [deg]

–30

30 0

–40

–30 –60

Amplitude [dB]

–20

60

–50

–90 –120

–60

–150 –180 90

0 30 60

–70 θ [deg]

Figure A.5 Pattern plotted in polar spherical co-ordinate system

Appendices 90

0

60

–10

567

El [deg]

–30 0 –40

–30

Amplitude [dB]

–20 30

–50 –60 –90 –90

–60

–60

–30

0 Az [deg]

30

60

90

–70

Figure A.6 Pattern plotted in azimuth and elevation (true-view) co-ordinate system El ¼ q sin f Hence, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ Az2 þ El2
 El f ¼ arctan Az

(A.42)

(A.43) (A.44)

The polar spherical angles q, f are related to the direction cosines through the normal expressions. The minus sign included with the x-axis co-ordinate is used to denote that the observer is standing behind the antenna looking in the positive z direction. Figure A.6 illustrates a far-field pattern when tabulated in a regular azimuth and elevation (true-view) co-ordinate system.

A.3.9

Range of spherical angles

For any given direction in space, as referenced to a given frame of reference, it is possible to use an infinite number of different, but equivalent, spherical angles to describe the relationship. However, in most areas of application, the spherical angles are limited to modulo 360 , or equivalently modulo 2p radians. However, even if the range of the spherical angles is limited, it is still possible to describe a given direction in more than one way. An implicit assumption has been made within this text concerning the range of the angles (variables) q and f. These are, 0  q  180

(A.45)

180  f  180

(A.46)

568

Principles of planar near-field antenna measurements y

–θ

ϕ +180 x θ

ϕ

Figure A.7 Conventional and alternate spherical angles

An alternative but equally valid choice is, 180  q  180

(A.47)

0  f  180

(A.48)

Typically, this is convenient for displaying cuts, as only one value of f is required to specify an entire great circle cut. Conversion between the two systems is facilitated through, If q < 0 then q ¼ q; f ¼ f þ 180

(A.49)

The inverse mapping is, if f < 0 then q ¼ q; f ¼ f þ 180

(A.50)

These relationships can be justified from Figure A.7. Finally, as an example, the point ðq ¼ 20; f ¼ 30Þ is the same point in space as ðq ¼ 20; f ¼ 150Þ for a fixed length of r. Figure A.8 illustrates a far-field pattern when tabulated in a regular polar spherical co-ordinate system.

A.3.10

Transformation between co-ordinate systems

Table A.1 comprises a summary of the various co-ordinate systems discussed above and illustrates how the various parameters can be related to one another. For example, the spherical angles can be related to the azimuth over elevation angles as, q ¼ arccos ðwÞ ¼ arccos ðcos ðAzÞcos ðElÞÞ

(A.51)

Appendices 0

180 150

–10

120 90

–30

30 0

–40

–30 –60

Amplitude [dB]

–20

60 ϕ [deg]

569

–50

–90 –120

–60

–150 –180 –9 0 –6 0 –3 0 0

–70 θ [deg]

Figure A.8 Pattern plotted in polar spherical co-ordinate system – alternate sphere

f ¼ arctan

v u


¼ arctan

tanðElÞ  sin ðAzÞ

 (A.52)

Or conversely the azimuth and elevation angles can be related to the spherical angles as, u Az ¼ arctan ¼ arctanðtan q cos fÞ (A.53) w El ¼ arcsin ðvÞ ¼ arcsin ðsin q sin fÞ

(A.54)

Indeed, by transforming via the direction cosines, it is possible to convert from any one set of co-ordinates to any other set of co-ordinates. As a note of caution, however, when calculating the inverse tangent, it is important that the four quadrant inverse tangent is used. This function will return angles over a full 180 angular range rather than over the more limited 90 range that is returned by the conventional inverse tangent function.

A.3.11 Co-ordinate systems and elemental solid angles The expression for the elemental solid angle for each of the systems described above is presented in Table A.2. A proof of the true-view expression and direction cosine expression can be found presented in the sections below.

A.3.12 Relationship between co-ordinate systems Passive transformation matrices are matrices that post-multiply a point vector to produce a new point vector and are merely a change in the coordinate system. The

Table A.1 Transformation between different coordinate parameters Co-ordinate system

x-axis

y-axis

z-axis

Direction cosine k-Space Azimuth over elevation Elevation over azimuth Polar spherical True-view (azimuth & elevation)

u

v

w

 sin ðAzÞcos ðElÞ  sin ðAzÞ sin qpcos f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi El

sin Az2 þ El 2 cos tan1 Az

sin ðElÞ cos ðAzÞsin ðEl Þ sin qpsin f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi El

sin Az2 þ El2 sin tan1 Az

cos ðAzÞcos ðElÞ cos ðAzÞcos ðElÞ cos qpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos Az2 þ El 2

kx k0

ky k0

kz k0

Appendices

571

Table A.2 Expressions for the elemental solid angle Co-ordinate system

Co-ordinates

dW

Direction cosine Azimuth over elevation Elevation over azimuth Polar spherical True-view

ðu; vÞ ðAz; ElÞ ðAz; ElÞ ðq; fÞ ðAz; ElÞ

1 cos q dudv cos ðElÞdAzdEl cos ðAzÞdAzdEl sin ðqÞdqdf sinc ðqÞdAzdEl

relationship between two co-ordinate systems can be defined with the use of a fourby-four homogeneous transformation matrix namely, 2 03 2 3 2 3 A1;1 A1;2 A1;3 A1;4 x x 6 y0 7 6 A2;1 A2;2 A2;3 A2;4 7 6 y 7 6 07¼6 7 6 7 (A.55) 4 z 5 4 A3;1 A3;2 A3;3 A3;4 5  4 z 5 1 0 0 0 1 1 Or, 2 3 2 03 x x 6y7 6 y0 7 6 0 7 ¼ ½ A  6 7 4z 5 4z5 1 1

(A.56)

Here, the elements A1,4, A2,4, and A3,4 represent a translation between the origins of the respective frames of reference. The three-by-three sub matrix, 2 3 2 3 ^e x0  ^e x ^e x0  ^e y ^e x0  ^e z A1;1 A1;2 A1;3 4 A2;1 A2;2 A2;3 5 ¼ 4 ^e y0  ^e x ^e y0  ^e y ^e y0  ^e z 5 (A.57) ^e z0  ^e x ^e z0  ^e y ^e z0  ^e z A3;1 A3;2 A3;3 contains the rotational information relating these frames of reference. This can also be expressed in terms of the cosine of the angles between the various combinations of unit vectors. This is also why these are termed direction cosine matrices. Specifically then, 2 3 2 3 cos q1;1 cos q1;2 cos q1;3 A1;1 A1;2 A1;3 4 A2;1 A2;2 A2;3 5 ¼ 4 cos q2;1 cos q2;2 cos q2;3 5 (A.58) A3;1 A3;2 A3;3 cos q3;1 cos q3;2 cos q3;3 Now, in essence, we are merely projecting each of the unit vectors of one coordinate system onto each of the unit vectors of the other. Therefore, each row can be considered to represent a vector describing the orientation of the unit vector of the primed co-ordinate system in terms of the un-primed co-ordinate system. Similarly, each of the columns can be considered to represent a vector describing the orientation of the unit vector of the un-primed co-ordinate system in terms of

572

Principles of planar near-field antenna measurements

the primed co-ordinate system. As the rotations that we are considering are isometric, the distance of a point from the origin in one system will be exactly the same in each co-ordinate system, i.e. it is invariant under the transformation. This can be expressed mathematically as, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A.59) l ¼ x2 þ y2 þ z2 ¼ x0 2 þ y0 2 þ z0 2 Similarly, the length of a vector will also remain invariant under these transformations. Clearly then, a unit vector will have unit length in every system. Thus, the magnitude of each of the row vectors will be one. Similarly, the magnitude of the column vectors will also be one. This can be expressed conveniently as, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A.60) 1 ¼ A2i;1 þ A2i;2 þ A2i;3 where i ¼ 1; 2; 3 and, 1¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A21;i þ A22;i þ A23;i where i ¼ 1; 2; 3

(A.61)

Also, as the un-primed unit vectors will be mutually orthogonal, which will also be the case for the primed unit vectors then knowledge of any two row vectors will enable the third to be obtained by taking the cross product of the other two. Furthermore, as the unit vectors in each co-ordinate system are orthogonal, only two of any three vectors can be chosen arbitrarily, the third being recoverable from the cross product of the other two. Here, this implies that knowledge of any two rows will enable the third to be determined and similarly, knowledge of any two columns will enable the third to be deduced. For example, Að3; 1Þ ¼ Að1; 2ÞAð2; 3Þ  Að1; 3ÞAð2; 2Þ

(A.62)

Að3; 2Þ ¼ Að1; 3ÞAð2; 1Þ  Að1; 1ÞAð2; 3Þ

(A.63)

Að3; 3Þ ¼ Að1; 1ÞAð2; 2Þ  Að1; 2ÞAð2; 1Þ

(A.64)

Conversely, the inverse transformation can be accomplished with, 2 03 2 3 x x 6 y0 7 6y7 1 6 0 7 ¼ ½A   6 7 4z 5 4z5 1 1

(A.65)

The adoption of a four-by-four matrix, with its inherent redundancy, is preferable as the matrix inverse, and thus the inverse transformation, only exists for square matrices. An added advantage of this definition is that the four-by-four alignment matrices can be obtained directly from most engineering computer-aided design (CAD) packages. By way of illustration, rotations about the x-, y-, and

Appendices z-axes are represented respectively 2 1 0 0 6 0 cos qx sin qx Rx ¼ 6 4 0  sin qx cos qx 0 0 0 2

cos qy 6 0 Ry ¼ 6 4 sin qy 0 2

cos qz 6  sin qz 6 Rz ¼ 4 0 0

0 1 0 0

 sin qy 0 cos qy 0 sin qz cos qz 0 0

0 0 1 0

by the three matrices: 3 0 07 7 05 1

573

(A.66)

3 0 07 7 05 1

(A.67)

3 0 07 7 05 1

(A.68)

The derivation of the rotation matrix can either be obtained from the use of trigonometric identities or from geometry. To illustrate this for the case of a positive rotation about the positive z-axis, it is clear from the diagram below and from trigonometry that, x0 ¼ x cos qz þ y sin qz

(A.69)

y0 ¼ x sin qz þ y cos qz

(A.70)

0

z ¼z

(A.71)

A similar construction can be used for rotations about the x- and y-axes respectively. Translation of Tx, Ty, Tz in the x-, y- and z-axes respectively can be y y’

θz x’

θz

θz x

z, z’

Figure A.9 Illustration of a positive rotation about the positive z-axis

574

Principles of planar near-field antenna measurements

implemented using, 2 3 1 0 0 Tx 6 0 1 0 Ty 7 7 T ¼6 4 0 0 1 Tz 5 0 0 0 1

(A.72)

A series of transformation matrices may be concatenated into a single matrix by multiplication. If A1, A2, and A3 are transformation matrices to be applied in order, and the matrix A is the product of the three matrices. Thus, ððP  A 1 Þ  A 2 Þ  A 3 P  ððA 1  A 2 Þ  A 3 Þ ¼ P  A

(A.73)

Where the multiplication is non-commutative and, A ¼ ðA 1  A 2 Þ  A 3

(A.74)

In this way, any sequence of rotations can be constructed by sequentially multiplying out the necessary rotations and translations. Often only the rotational relationship between two systems is considered. This is often the case when considering far-field patterns. In this case, the three-by-three element sub-matrix can be considered alone and is used instead of the homogeneous four by four transformation matrix. Utilisation of the three-by-three direction cosine matrix has an important benefit. As the direction cosine matrix is orthogonal and normalised the matrix whose elements are all real, the inverse is identically equal to the matrix transpose. This means that obtaining the inverse transformation is essentially reduced to a matter of reordering of the elements within the matrix, which is both trivial and numerically robust. When direction cosine matrices are used, the determinant of the matrix should be calculated and any significant deviation from unity can be treated as being indicative of a bad direction cosine matrix, as the matrix should be normalised to unity. Occasionally a good direction cosine matrix can be reported as faulty, if the number of significant figures used to represent the matrix is insufficient. Typically, all direction cosine matrices should be treated as being of type double precision in order that truncation and rounding errors remain acceptably small. This follows from noting that typically the smallest angular increment that is usually of interest at microwave frequencies from a rotary position encoder is 0.01 , or when expressed in terms of a direction cosine this deviates from unity in the eighth decimal place. Furthermore, the act of multiplying out one or more direction cosine matrices can further compromise the data, as the cumulative rounding error can increase appreciably.

A.3.13

Azimuth, elevation and roll angles

Any number of angular definitions for describing the relationship between the two co-ordinate systems are available. However, if the angles azimuth, elevation and roll are used, where the rotations are applied in this order, we may write the

Appendices

575

equivalent direction cosine matrix as, ½ A ¼ ½ A 1   ½ A 2   ½ A 3  Specifically, 2 3 cos ðrollÞ sin ðrollÞ 0 ½A1  ¼ 4  sin ðrollÞ cos ðrollÞ 0 5 0 0 1 2 3 1 0 0 ½A2  ¼ 4 0 cos ðelÞ  sin ðelÞ 5 0 sin ðelÞ cos ðelÞ 2 3 cos ðazÞ 0  sin ðazÞ 5 ½ A3  ¼ 4 0 1 0 sin ðazÞ 0 cos ðazÞ

(A.75)

(A.76)

(A.77)

(A.78)

These transformation matrices can be easily derived either from geometry, or from trigonometric identities. Here, in accordance with the rules of linear algebra, the first rotation matrix is written to the right. When multiplied out ½A can be explicitly expressed as Arow,column, A1;1 ¼ cos ðrollÞcos ðazÞ þ sin ðrollÞsin ðelÞsin ðazÞ

(A.79)

A1;2 ¼ sin ðrollÞcos ðelÞ

(A.80)

A1;3 ¼ cos ðrollÞsin ðazÞ  sin ðrollÞsin ðelÞcos ðazÞ

(A.81)

A2;1 ¼  sin ðrollÞcos ðazÞ þ cos ðrollÞsin ðelÞsin ðazÞ

(A.82)

A2;2 ¼ cos ðrollÞcos ðelÞ

(A.83)

A2;3 ¼ ðsin ðrollÞsin ðazÞ þ cos ðrollÞsin ðel Þcos ðazÞÞ

(A.84)

A3;1 ¼  cos ðelÞsin ðazÞ

(A.85)

A3;2 ¼ sin ðelÞ

(A.86)

A3;3 ¼ cos ðelÞcos ðazÞ

(A.87)

Where the rotations are understood to have been performed in the following order: 1. 2. 3.

Rotate about the negative y-axis through an angle azimuth. Rotate about the negative x-axis through an angle elevation. Rotate about the z-axis through an angle roll.

When az = 0, el = 0, roll = 0, the direction cosine matrix will be the identity matrix and specifies that no rotations are to be applied, i.e., 2 3 1 0 0 ½A ¼ 4 0 1 0 5 ¼ ½I  (A.88) 0 0 1

576

Principles of planar near-field antenna measurements

Clearly, from this matrix, it can be seen that these angles can be obtained from the matrix ½A as,
 A31 (A.89) az ¼ arctan A33 el ¼ arcsin ðA32 Þ
 A12 roll ¼ arctan A22

(A.90) (A.91)

Many other definitions for rotating frames of reference exist. These include the triad of Euler angles (see below), or the yaw pitch, and roll angles. However, the azimuth, elevation and roll definition is most convenient when presenting data tabulated in a regular azimuth over elevation co-ordinate system.

A.3.14

Euler angles

As an alternative to the azimuth, elevation and roll rotations described above the triad of Euler angles are often utilised to represent the relationship between two frames of reference. 1. 2. 3.

Rotate about the z-axis through an angle f. Rotate about the new y-axis through an angle q. Rotate about the new z-axis through an angle c.

Specifically, if the angles f, q, and c are used, where the rotations are applied in this order, we may write the equivalent direction cosine matrix as, ½A  ¼ ½A 1   ½A 2   ½A 3 

(A.92)

Where, 2

cos ðcÞ ½A1  ¼ 4  sin ðcÞ 0 2 cos ðqÞ 0 ½A 2  ¼ 4 0 1 sin ðqÞ 0 2 cos ðfÞ ½A3  ¼ 4  sin ðfÞ 0

3 0 05 1 3  sin ðqÞ 5 0 cos ðqÞ 3 sin ðfÞ 0 cos ðfÞ 0 5 0 1

sin ðcÞ cos ðcÞ 0

(A.93)

(A.94)

(A.95)

When multiplied out ½A, can be explicitly expressed as Arow,column, A1;1 ¼ cos c cos f cos q  sin c sin f

(A.96)

A1;2 ¼ cos f sin c þ cos c cos q sin f A1;3 ¼  cos c sin q

(A.97) (A.98)

Appendices A2;1 ¼  cos f cos q sin c  cos c sin f

577 (A.99)

A2;2 ¼ cos c cos f  cos q sin c sin f

(A.100)

A2;3 ¼ sin c sin q

(A.101)

A3;1 ¼ cos f sin q

(A.102)

A3;2 ¼ sin f sin q

(A.103)

A3;3 ¼ cos q

(A.104)

Here, the three angles are referred to as Euler angles. Conversely, the three Euler angles can be determined from the direction cosine matrix as, q ¼ arccos ðA33 Þ If q 6¼ 0 then,
 A23 c ¼ arctan A13
 A32 f ¼ arctan A31

(A.105)

(A.106)

(A.107)

However, if q ¼ 0, then a zero divide by zero ambiguity is introduced. In this case, we must use, c¼0 f ¼ arctan




A12 A22



(A.108) (A.109)

Clearly, when the q rotation is zero, the f and c rotations are equivalent and thus either rotation may be used. A conversion between the azimuth, elevation and roll angles and the three Euler angles can be accomplished readily by equating the elements of their respective direction cosine matrices.

A.3.15 Quaternion As shown above, and further illustrated in Section 9.2.2, only four elements of the nine elements within the direction cosine matrix can be chosen arbitrarily. As such, it is sometimes desirable to record the relationship between two coordinate systems in a more compact form. It is not the purpose of this section to give a through presentation of quaternions as this is beyond the scope of this text, instead the discussion will be limited to the use of quaternions in implementing coordinate transforms. Mathematically, quaternions can be considered to be a non-commutative extension of complex numbers. By way of an analogy, complex numbers are represented as a sum of real and imaginary parts. Similarly, a quaternion can also

578

Principles of planar near-field antenna measurements

be written as a linear combination of real and hyper-complex parts, Q ¼ q0 þ q1 i þ q2 j þ q3 k

(A.110)

Where, i2 ¼ j2 ¼ k 2 ¼ ijk ¼ 1

(A.111)

Thus, as a complex number can be represented as a point on a two-dimensional plane, a quaternion can be considered to be a point in a four-dimensional space. A quaternion can be computed from a direction cosine matrix using, 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A11 þ A22 þ A33 þ 1 2 1 q1 ¼ ðA23  A32 Þ 4q0 1 q2 ¼ ðA31  A13 Þ 4q0

q0 ¼

q3 ¼

1 ðA12  A21 Þ 4q0

(A.112) (A.113) (A.114) (A.115)

Conversely, a direction cosine matrix can be constructed from a quaternion using, A11 ¼ 2q20  1 þ 2q1 q2

(A.116)

A12 ¼ 2q1 q2 þ 2q0 q3

(A.117)

A13 ¼ 2q1 q3  2q0 q2

(A.118)

A21 ¼ 2q1 q2  2q0 q3

(A.119)

A22 ¼ 2q20  1 þ 2q22

(A.120)

A23 ¼ 2q2 q3 þ 2q0 q1

(A.121)

A31 ¼ 2q1 q3 þ 2q0 q2

(A.122)

A32 ¼ 2q2 q3  2q0 q1

(A.123)

A33 ¼ 2q20  1 þ 2q23

(A.124)

As is the case for vectors, the length, or norm, of a quaternion is of utility. This can be calculated from, pffiffiffiffiffiffiffiffiffi (A.125) jQj ¼ Q Q Here, the superscript star is used to denote the complex conjugate of Q so that Q þ Q ¼ 2q0 . Thus, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A.126) jQj ¼ q20 þ q21 þ q22 þ q23

Appendices

579

If Pq and Qq are quaternions and are expressed using the vector form of a quaternion then, P q ¼ p0 þ P

(A.127)

Qq ¼ q0 þ Q

(A.128)

When expressed in this form, the quaternions can be multiplied together using, Rq ¼ Pq Qq ¼ p0 q0  P  Q þ p0 Q þ q0 P þ P  Q

(A.129)

Here, a dot denotes the scalar dot product and the cross denotes the vector cross product. When expanded out and expressed in the element form this equates to, r0 ¼ p0 q0  p1 q1  p2 q2  p3 q3

(A.130)

r1 ¼ p0 q1 þ p1 q0 þ p2 q3  p3 q2

(A.131)

r2 ¼ p0 q2 þ p2 q0 þ p3 q1  p1 q3

(A.132)

r3 ¼ p0 q3 þ p3 q0 þ p1 q2  p2 q1

(A.133)

As was the case for the multiplication of direction cosine matrices, quaternion multiplication is equivalent to the concatenation of several sequential rotations. As was the case for direction cosine matrices, multiplications are non-commutative, i.e., Pq Qq 6¼ Qq Pq

(A.134)

Inverting a quaternion rotation produces the inverse rotation and the inverse of a quaternion is equal to the complex conjugate of that quaternion thus, 1 ¼ q0  q1  q2  q3 Q1 q ¼ ðq0 þ q1 þ q2 þ q3 Þ

(A.135)

Hence, computing an inverse rotation using the quaternion representation requires less effort than accomplishing the same task using direction cosine matrices. All rotations can be represented by a single rotation about an axis in space. The axis and angle of that rotation can be calculated from the quaternion using, f ¼ 2arccos ðq0 Þ v ¼ v1b e x þ v 2b e y þ v3b ez ¼

(A.136) h

1 q1b e x þ q2b e y þ q3b ez sin ðf=2Þ

i (A.137)

Here, f is used to denote the angle of rotation and v represents the axis of the rotation. Conversely, the quaternion can be computed from the angle and axis of rotation using,
 f (A.138) q0 ¼ cos 2

580

Principles of planar near-field antenna measurements
 f 2
 f q2 ¼ v2 sin 2
 f q3 ¼ v3 sin 2 q1 ¼ v1 sin

(A.139) (A.140) (A.141)

In addition to being a more efficient recording method, quaternions have the advantage that computing inverse rotations is made significantly easier than is the case for direction cosine matrices. Also, whilst the multiplication of two direction cosine matrices can, in the presence of truncation and rounding errors, produce a direction cosine matrix that is not a rotation, i.e. the components naturally drift which violates the orthonormality constraints. Quaternions have no such problem, i.e. the multiplication of two quaternions will always yield a rotation, all be it, perhaps the wrong rotation. It is also a comparatively simple matter to adjust for numerical drift, one merely needs to compute the norm of the quaternion and then divide each component by it. This takes far fewer operations than matrix orthonormalisation which would be required in order to attempt to correct a direction cosine matrix.

A.3.16

Elemental solid angle for a true-view co-ordinate system

Consider evaluating the following integral where the pattern is tabulated on a regular polar spherical grid, ð ðp 1 p 2 (A.142) P¼ jET ðq; fÞj2 sin q dqdf Z0 p 0 Where Et is the RMS value. Now suppose that we have the field tabulated on a regular true-view grid, i.e. ET ðX ; Y Þ where, X ¼ q cos f

(A.143)

Y ¼ q sin f

(A.144)

The problem of evaluating the total radiated power of an antenna when the pattern has been tabulated using a true-view coordinate system could easily be encountered. Rather than recalculating the pattern on a more convenient plotting system, it would be convenient if it were possible to evaluate this integral directly in this system. Motivated by this, the original integral can be expressed as, ðð 1 @ ðq; fÞ dXdY (A.145) jET ðX ; Y Þj2 sin q P¼ Z0 X 2 þY 2 p42 @ ðX ; Y Þ

Appendices

581

Where,    @q @q   @ ðq; fÞ  @X @Y  ¼  @ ðX ; Y Þ  @f @f    @X @Y Clearly, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ X2 þ Y2
 Y f ¼ arctan X Evaluating the elements of the matrix yields, ffi @q @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ¼ X 2 þ Y 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @X @X X2 þ Y2 ffi @q @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y ¼ X 2 þ Y 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 @Y @Y X þ Y2 @f @ ¼ @X @X @f @ ¼ @Y @Y





 Y Y arctan ¼ 2 X X þ Y2


arctan


 Y X ¼ 2 X X þ Y2

(A.146)

(A.147) (A.148)

(A.149)

(A.150)

(A.151)

(A.152)

Thus, @ ðq; fÞ X X Y Y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 2 2 2 2 þ Y þ Y2 @ ðX ; Y Þ X X X þY X þY X2 þ Y2 ¼ 3 ðX 2 þ Y 2 Þ2 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 X þ Y2

(A.153)

Recalling the relationship between the polar angle and the plotting axes yields, @ ðq; fÞ 1 ¼ @ ðX ; Y Þ j q j Thus the integral can be expressed as,   ðð   1 2 sin q P¼ dXdY j E T ðX ; Y Þj  Z0 X 2 þY 2 p42 q 

(A.154)

(A.155)

582

Principles of planar near-field antenna measurements Or, P¼

1 Z0

ðð 2 X 2 þY 2 p4

jET ðX ; Y Þj2 jsinc qjdXdY

(A.156)

Where in the limit as q tends to zero sincq tends to one. Alternatively, when expressed purely in the plotting coordinates, this becomes, ðð  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2 (A.157) E ð X ; Y Þ sinc X2 þ Y2 dXdY P¼ j j  T Z0 X 2 þY 2 p42

A.4 Appendix D: Trapezoidal discrete Fourier transform Following [5], consider the following testing integral, ð2 x2 ejwx dx F ðwÞ ¼

(A.158)

2

Integrating this yields,  

2 jejwx 2 2 F ðwÞ ¼  3 w x þ 2jwx  2 w 2

(A.159)

Thus the exact function can be expressed as, F ðwÞ ¼ 





j2 2 2w þ 2jw  1 e2jw þ 2w2 þ 2jw þ 1 e2jw w3

(A.160)

This was evaluated numerically using the conventional DFT, FFT and then with the trapezoidal transforms where, x ¼ 2 þ nDx

(A.161)

w ¼ mDw

(A.162)

n ¼ 0; 1; 2; . . . ; N  1

(A.163)

m ¼ 0; 1; 2; . . . ; M  1

(A.164)

Dx ¼ 0:05

(A.165)

Dw ¼

2p MDx

Here, N1 = 80 and M = 256. The DFT was evaluated as, XN 1 f ðmDxÞejwx F ðwÞ  FDFT ff ðmDxÞg ¼ ejwDx=2 Dx n¼0

(A.166)

(A.167)

Appendices

583

The FFT was evaluated as, F ðwÞ  FFFT ff ðmDxÞg ¼ ejwð2þDx=2Þ Dx

XN 1 n¼0

f ðmDxÞej2pmn=M

(A.168)

Note, some FFT’s adopt the opposite sign convention. The trapezoidal DFT was evaluated as, F ðwÞ  FTDFT ff ðmDxÞg ¼ sincðqÞFDFT ff ðmDxÞg

1 jq e  sincðqÞ FDFT fDf ðmDxÞg þ j2q

(A.169)

Either the DFT or FFT algorithm can be employed herein. Furthermore, q¼

wDx 2

(A.170)

Df ðnDxÞ ¼ f ððn þ 1ÞDxÞ  f ðnDxÞ

(A.171)

In practice, the slope of the function is obtained by central differencing, whilst a right difference is taken at the left-hand side and a left difference is taken at the right-hand side, this can be expressed explicitly as follows: 1 Df ðnDxÞ ¼ ðf ððn þ 1ÞDxÞ  f ððn  1ÞDxÞÞ 2

(A.172)

Df ðnDxÞ ¼ f ððn þ 1ÞDxÞ  f ðnDxÞ

(A.173)

Df ðnDxÞ ¼ f ðnDxÞ  f ððn  1ÞDxÞ

(A.174)

Results obtained from the DFT and the trapezoidal transforms can be found compared with the exact result in Figures A.10 and A.11. Clearly, the first-order trapezoidal DFT offers a significant improvement over the more commonly employed zero-order rectangular DFT. A further comparison is made in Table A.3. These values differ from those presented in the original reference. This discrepancy is most likely a result of typing errors within that paper and 0 Exact DFT EMPL

–10

100

–30

Phase [deg]

Mag [dB]

–20

–40 –50 –60

50 0 –50

–100

–70 –80

Exact DFT

150

–150 50

100

m

150

200

250

5

10

m

15

20

Figure A.10 Comparison of exact and discrete rectangular transforms

25

584

Principles of planar near-field antenna measurements 0 Exact Trapezoidal DFT EMPL

–20

50 Phase [deg]

–40 Mag [dB]

Exact Trapezoidal DFT

100

–60 –80

0

–50

–100 –100 –120

50

100

m

150

200

250

5

10

m

15

20

25

Figure A.11 Comparison of exact and discrete trapezoidal transform

Table A.3 Comparison of exact result, trapezoidal, FFT and DFT results m

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

Exact

DFT

FFT

Trapezoidal

Mag (dB)

Arg (deg)

Mag (dB)

Arg (deg)

Mag (dB)

Arg (deg)

Mag (dB)

Arg (deg)

0.00 15.28 18.52 20.30 22.34 25.11 29.17 36.17 60.23 37.26 33.13 31.74 31.87 33.31 36.35 42.44 72.27 42.99 38.31 36.51 36.31 37.46 40.23 46.06 79.32 46.42

90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00 90.00

0.00 15.28 18.45 20.12 22.00 24.55 28.32 34.90 57.36 35.61 30.93 28.98 28.48 29.20 31.40 36.39 59.19 35.94 29.79 26.50 24.57 23.64 23.71 25.35 32.17 31.27

90.00 87.75 91.24 89.47 89.97 90.63 88.51 93.66 33.75 93.34 88.59 90.59 90.00 89.41 91.42 86.61 157.50 86.41 91.47 89.38 90.02 90.56 88.67 92.88 78.75 117.45

0.00 15.28 18.45 20.12 22.00 24.55 28.32 34.90 57.36 35.61 30.93 28.98 28.48 29.20 31.40 36.39 59.19 35.94 29.79 26.50 24.57 23.64 23.71 25.35 32.17 31.27

90.00 87.75 91.24 89.47 89.97 90.63 88.51 93.66 33.75 93.34 88.59 90.59 90.00 89.41 91.42 86.61 157.50 86.41 91.47 89.38 90.02 90.56 88.67 92.88 78.75 117.45

0.00 15.28 18.52 20.30 22.35 25.12 29.17 36.17 60.29 37.26 33.13 31.74 31.88 33.32 36.35 42.44 72.61 43.01 38.32 36.52 36.32 37.46 40.24 46.06 79.45 46.42

90.00 90.00 89.99 90.01 89.98 90.03 89.96 90.08 89.02 90.04 89.99 90.00 90.01 89.97 90.04 89.91 92.50 89.94 90.02 90.00 89.99 90.02 89.96 90.06 89.84 89.76

Appendices

585

the use of single precision arithmetic within the original calculation. The results shown here were generated using double precision arithmetic throughout. This is a first-order integration technique so it is exact for functions that are of first or zero order, i.e. piecewise linear or piecewise constant. Thus, for a ‘top hat’ function, the results obtained numerically with this formula were found to be in agreement with the exact result to machine precision, i.e. approximately –300 dB EMPL. For this reason, this integration method has been verified with a quadratic testing function.

A.5 Appendix E: Calculating the semi-major axis, semiminor axis and tilt angle of a rotated ellipse From Chapter 5, Ef ¼ E2 cos ðwt þ DfÞ

(A.175)

This can be written as, Ef ðr ; tÞ ¼ E2 ðr Þ½cos wt cos g þ sin wt sin g ¼ pf ðr Þcos wt þ qf ðr Þsin wt

(A.176)

Where, pf ðr Þ ¼ E2 ðr Þcos g

(A.177)

qf ðr Þ ¼ E2 ðr Þsin g

(A.178)

In general, it is possible to write, E ðr ; tÞ ¼ p ðr Þcos wt þ q ðr Þsin wt

(A.179)

So pq ðr Þ ¼ E1 ðr Þ; qq ðr Þ ¼ 0; yields Eq ðr ; tÞ

(A.180)

Thus, g = 0 gives linear polarisation and g = 90 gives circular polarisation (for E1 = E2) and elliptical polarisation otherwise. We can write,

(A.181) E ðr ; tÞ ¼ Re Uðr Þejwt U ðr Þ ¼ p ðr Þ þ jq ðr Þ

(A.182)

Now looking at E point r = r0 as time varies, end point of E describes an ellipse defined by p and q. Now we can write,   p þ jq ¼ ða þ jb Þejf (A.183)

586

Principles of planar near-field antenna measurements Here, f is any scalar. Thus we can write, a ¼ p cos f þ q sin f

(A.184)

b ¼ p sin f þ q cos f

(A.185)

If we now choose f so that a and b are perpendicular to each other then, by orthogonality, a  b ¼ ðp cos f þ q sin fÞ  ðp sin f þ q cos fÞ ¼ 0

(A.186)

We can now write, E ðr ; tÞ ¼ ða þ jb ÞejðwtfÞ ¼ a cos ðwt  fÞ þ b sin ðwt  fÞ

(A.187)

Taking Cartesian axes with origin at r0 and with x and y directions along a and b yields, Ex ¼ a cos ðwt  fÞ

(A.188)

Ey ¼ a sin ðwt  fÞ

(A.189)

Which is the parametric equation of an ellipse with semi major axis = a, semi minor axis = b, and tilt angle = f. Thus, solving for these three parameters will fully specify the ellipse. By simple geometry, it can be shown that p and q are thus semi radi of the ellipse measured in the q and f directions, see Figure 5.48 in Chapter 5. Simplifying the dot product of a and b yields,  

p 2 þ q 2 cos f sin f þ p  q cos2 f  sin2 f ¼ 0 (A.190)  1 2 p þ q 2 sin 2f þ p  q cos 2f ¼ 0 2

(A.191)

Hence we obtain the desired result, tan 2f ¼

2p  q p2  q2

(A.192)

If g is used to denote the angle between the vectors p and q, then this can be expressed as, tan 2f ¼

2pq cos g p2  q2

Now consider,  2 ða Þ2 ¼ p cosf þ q sinf

(A.193)

(A.194)

Appendices

587

Expanding yields, a2 ¼ p2 cos2 f þ q2 sin2 f þ 2p  q cos f sin f

(A.195)

Or, a2 ¼ p2 cos2 f þ q2 sin2 f þ p  q sin 2f

(A.196)

Expanding yields, a2 ¼

p2 q2 ðcos 2f þ 1Þ þ ð1  cos 2fÞ þ p  q sin 2f 2 2

(A.197)

Simplifying obtains, a2 ¼

1

1 2 p þ q2 þ p2  q2 cos 2f þ p  q sin 2f 2 2

(A.198)

Returning to the expression for the rotation and considering the tangent of an angle yields a useful trigonometric identity namely, tan2 f ¼

sin2 f 1  sin2 f

(A.199)

Thus, tan2 2f ¼

4ðp  q Þ2

(A.200)

ðp2  q2 Þ2

Hence, 4ðp  q Þ2 sin2 2f ¼ 1  sin2 2f ðp2  q2 Þ2



(A.201)

2 p2  q2 sin2 2f ¼ 4ðp  q Þ2  4sin2 2fðp  q Þ2

p2  q2

2

þ 4ðp  q Þ2 ¼

4ðp  q Þ2 sin2 2f

(A.202)

(A.203)

Thus we obtain the first of our two necessary substitutions: 2ðp  q Þ sin 2f ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp2  q2 Þ2 þ 4ðp  q Þ2

(A.204)

588

Principles of planar near-field antenna measurements The second can be obtained from the first using, 2p  q p2



q2

¼

2ðp  q Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos 2f ðp2  q2 Þ2 þ 4ðp  q Þ2

(A.205)

Hence, we find the second of our two substitutions, p2  q2 cos 2f ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp2  q2 Þ2 þ 4ðp  q Þ2

(A.206)

Thus, if these substitutions are used to simplify the expression of a2, we obtain, a2

1 2 p þ q2 2 2 1 ðp2  q2 Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp2  q2 Þ2 þ 4ðp  q Þ2  2 2 p q þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp2  q2 Þ2 þ 4ðp  q Þ2 ¼

(A.207)

Or, 2

3 2 2 2 2 ð p  q Þ þ 4ðp  q Þ

16 7 a2 ¼ 4 p2 þ q2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 2 2 2 ðp2  q2 Þ þ 4ðp  q Þ

(A.208)

Hence, a2 ¼

 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 p þ q2 þ ðp2  q2 Þ2 þ 4ðp  q Þ2 2

(A.209)

This can be expressed in terms of the angle between the vectors p and q as,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 h 2 p þ q2 þ p4 þ q4  2p2 q2 þ 4p2 q2 cos2 g (A.210) a2 ¼ 2 a2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 h 2 p þ q2 þ p4 þ q4 þ 2p2 q2 ð1 þ 2cos2 gÞ 2

Thus the final result is, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 h 2 p þ q2 þ p4 þ q4 þ 2p2 q2 cos 2g a¼ 2

(A.211)

(A.212)

Appendices

589

Following a similar procedure, we can obtain a similar result for b. Thus, b2 ¼ p2 sin2 f þ q2 cos2 f  p  q sin 2f

(A.213)

So that, b2 ¼

1

1 2 p þ q2  p2  q2 cos 2f  p  q sin 2f 2 2

(A.214)

Hence, " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2

1 2 2 2 2 2 b ¼ p þ q  ðp  q Þ þ 4 p  q 2 2

Thus as required, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 h 2 p þ q2  p4 þ q4 þ 2p2 q2 cos 2g b¼ 2

(A.215)

(A.216)

A.6 Appendix F: Fast Fourier transform algorithm Figure A.12 presents an example of a one-dimensional, radix-2, decimation in frequency (DIF) fast Fourier transform (FFT) algorithm, which is written in the form of a MATLAB script. Here, the complex vector f contains the input data. This is an in-place FFT, so the transformed data is computed and stored in the vector f. The vector f must have a length that is equal to a positive integer power of two, e.g. 128, elements long. When the variable isign = 1, this algorithm will perform a FFT. Alternatively, when isign =1, this will compute an inverse FFT. For more information about the discrete Fourier transform, see for example Ref. [5, specifically p. 53]. Here, the DFT can be expressed as equating to an efficient computation of, F ð1  k  N Þ ¼

N X

f ðnÞej2pðk1Þðn1Þ=N

(A.217)

n¼1

When using this FFT algorithm, it is important to recognise that this has the opposite time dependency of that used by planar near-field theory as developed within this text. Thus, to use this algorithm, we must conjugate the fields on input to this subroutine and then conjugate the transformed fields that are returned. Thus, the near-field data would first be zero-padded to make the array sizes a power of two in each dimension. This routine would then be called to transform the data along the rows and then down the columns, or vice versa, to compute the twodimensional FFT of the data. The resulting data would need to be FFT shifted to place the DC component at the centre of the array and the phase conjugated as noted above. The FFT algorithm presented below agreed with the built-in FFT that is provided with MATLAB version R2022a to ca. 1e15 in the real and imaginary

590

Principles of planar near-field antenna measurements

Figure A.12 Example power of 2 fast Fourier transform algorithm

Appendices

591

parts, although it ran some 130 times slower than the internal function when used to process the same data.

A.7 Appendix G: Asymptotic far-field form of the Kirchhoff–Huygens formula As is shown in Chapter 7, the general vector Kirchhoff–Huygens formula can be expressed as
 ð  0   1 1 ejk0 r jk0 þ 0 Ep ¼ jwmðn  H Þ þ ðn  E Þ  br 0 þ ðn  E Þbr 0 da r0 4p S r (A.218) Importantly, these expressions can be significantly simplified for the asymptotic far-field case. In this section, we will tackle this problem. Since, cf. Chapter 7, r ¼ r0  r0

(A.219)

So,   jr0 j ¼ r0 ¼ r  r0 

(A.220)

Clearly, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 r0 ¼ ðrx  r0x Þ2 þ ry  r0y þ ðrz  r0z Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ r2 þ r2 þ r2 þ r2  2r r  2r r  2r r r0 ¼ rx2 þ r0x x 0x y 0y z 0z y z 0y 0z

(A.221) (A.222)

Thus, r0 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ r02  2r  r0

(A.223)

Using the first two terms of the binomial expansion for r0 yields, r0 ¼ r 

r  r0 ¼ r  br  r0 r

(A.224)

Thus, e

jk0 r0

¼e

jk0

  br  r0  r

¼ ejk0 r ejk0br  r0

Now, as is shown in Chapter 7,
 0 1 ejk0 r 0 br r0 y ¼ jk0 þ 0 r0 r

(A.225)

(A.226)

Thus, neglecting   products of small quantities and thus introducing an error of the order of O r10 2 then,

592

Principles of planar near-field antenna measurements 0

r0 y ¼

jk0 ejk0 r 0 jk0 e br ¼ r0

jk0

  br  r0  r r0

br 0

(A.227)

Since in the far field r and r0 are parallel, then for the purposes of evaluating amplitude, the magnitude of r can be considered to be the same as the magnitude of r0 with no loss of precision thus,   jk0 b r  r0  r jk0 r0 jk0 e jk0 e br 0 ¼ br (A.228) r0 y ¼ 0 r r These expressions can be substituted into the general formula for the electric field thus, ð 1 ejk0 r Ep ¼ ½jwmðn  H Þ þ jk0 fðn  E Þ  br þ ðn  E Þbr gejk0br r0 da 4p r S (A.229) Since in the far-field, the electric field will be normal to the direction of propagation, ð 1 ejk0 r Ep ¼ (A.230) ½jwmðn  H Þ þ jk0 ðn  E Þ  br ejk0br r0 da 4p r S  ð  1 jk0 ejk0 r wm br  ðn  E Þ þ Ep ¼  ðn  H Þ ejk0br r0 da (A.231) r 4p k0 S Now, 

1 jk0 ejk0 r k0 ejk0 r ejk0 r ejk0 r k0 ejk0 r p ¼ ¼ ¼ ¼ r jr4p 2jrl k0 r 2jl k0 r jl2 4p

(A.232)

And, wm 2pf m 2pf lm m ¼ cm ¼ pffiffiffiffiffi ¼ ¼ ¼ k0 k0 2p em

rffiffiffi m ¼Z e

(A.233)

Thus, Ep ¼

ejk0 r p k0 r jl2

ð

½br  ðn  E Þ þ Z ðn  H Þejk0br r0 da

(A.234)

S

Now the term n  H will contain a radial component that must be zero in the far field. Consider the following term: br  fðn  H Þ  br g

(A.235)

Appendices

593

Using, A  ðB  A Þ ¼ ðA  A ÞB  ðA  B ÞA ¼ jA j2  ðA  B ÞA

(A.236)

Then br  fðn  H Þ  br g ¼ jbr jðb n  H Þ  ðbr  ðb n  H ÞÞbr

(A.237)

Since jbr j ¼ 1 and in the far field ðbr  ðb n  H ÞÞbr ¼ 0 then, br  fðn  H Þ  br g ¼ b n H

(A.238)

Thus, the final expression for the far-field becomes, ð ejk0 r p ½br  ðn  E Þ þ Zbr  fðn  H Þ  br gejk0br r0 da Ep ¼ k0 r jl2 S (A.239) Where the magnetic field can be obtained from the electric field using, 1 Hp ¼ br  Ep Z

(A.240)

Here, the unimportant spherical phase factor and inverse radius term would be suppressed in the formula for the far-electric and magnetic fields.

References [1] J.A. Wheeler and R.P. Feynnan, ‘Interaction with the absorber as a mechanism of radiation’, Reviews of Modern Physics, 1945;17(267). [2] ‘Encyclopaedia of Statistical Sciences’, Vol. 8, Camuel, Kotz and Norman, John Wiley and Son, New York, NY, pp. 466–467. [3] C. Parini, S. Gregson, J. McCormick, D.J. van Rensburg, and T. Eibert, Theory and Practice of Modern Antenna Range Measurements, 2nd expanded ed., vols. 1 and 2, IET Electromagnetic Waves Series 55, The Institution of Engineering and Technology, UK. [4] M.S. Narasimhan and M. Karthikeyan, ‘Evaluation of Fourier transform integrals using FFT with improved accuracy and its applications’, IEEE Transactions on Antennas and Propagation, 1984;AP-32(4):404–408. [5] A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, Hoboken, NJ, 1989.

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Index

absorber 51, 59, 62, 252, 353 absorber theory 3, 555–7 accelerated charges 17–20 AC theory: see alternating current (AC) theory active alignment correction 127, 413 application of 420–6 planar near-field facility acquisition of alignment data in 416–17 acquisition of mechanical alignment data in 417–20 additive white Gaussian noise (AWGN) 487 advanced planar near-field antenna measurements 413 active alignment correction 413 acquisition of alignment data 416–17 acquisition of mechanical alignment data 417–20 example of the application of 420–6 compressive sensing technique 483–6, 507 applied to 2D array 488–95 compressive sensing using near-field scanning 502–7 defective element detection using 486–8 practical implementation 495–502 non-canonical transforms, plane wave spectrum-based treatment 464 matrix inversion method 464–72

non-canonical transform 475–83 non-uniform fast Fourier transform (NUFFT)-based algorithms 472–5 non-iterative phase retrieval technique 434 AUT phase and probe position reconstruction 438–43 current phase retrieval techniques 435–7 millimetre wave robotic arm-based NF measurement system 448–50 near-field to far-field antenna measurement performance 443–8 partial scan techniques 517 auxiliary rotation 523–31 auxiliary translation 517–20 bi-scan auxiliary rotation technique 537–8 complete poly-planar rotational technique 541–5 experimental procedure 542–5 experimental set up 541–2 Kirchhoff–Huygens principle 533–7 poly-planar technique, applicability of 537–41 rotations of the AUT about the z-axis 520–3 tri-scan auxiliary rotation technique 538–41 plane-to-plane phase retrieval algorithm 427–31 with aperture constraint 431–4

596

Principles of planar near-field antenna measurements

three antenna extrapolated gain measurements 507–17 traditional position correction algorithms 451 K-correction method 459–64 Taylor-series expansion 452–9 aliasing 379–80 alternating current (AC) theory 10 amplitude only planar near-field measurements 426 plane-to-plane phase retrieval algorithm 427–31 with aperture constraint 431–4 AMS: see antenna mechanical system (AMS) angular spectra 101, 107 antenna alignment corrections 208 copolar polarisation basis, rotation of 213–14 far-field antenna patterns generalised compound vector rotation of 214–17 scalar rotation of 208–11 vector rotation of 211–12 antenna coupling, phenomena of 1–4 antenna coupling formula 157–62 antenna electrical system 218–19, 560–1 antenna measurement analysis and assessment 359 comparison techniques 389 conventional data set 389–93 novel data 393–408 data sets, quantitative measures of correspondence between 388–9 establishment of measure from measurement results 359 measurement errors 360–3 measurement result data, examination of 365–8

sources of measurement ambiguity and error 363–5 measurement error budgets 368 aliasing 379–80 applicability of modelling error sources 369–70 applicability of the digital twin to assessing 371–3 dynamic range 382 empirical approach to 370–1 numerical truncation and rounding error 376 probe x, y (in-plane) position error 376–9 systematic phase 381–2 truncation 373–5 range assessment budgets, illustration of the compilation of 383–7 antenna measurements 39–44 antenna mechanical system (AMS) 218, 560 antenna plotting system (APS) 219 antenna radiation patterns, general vector rotation of 219–20 antenna-to-antenna coupling formula 133, 304–6 evanescent plane wave mode coefficients, behaviour of 141–4 simple scattering model of near-field probe during planar measurement 144–6 antenna under test (AUT) 8, 39, 42–3, 46, 49–50, 294, 438–43 APS: see antenna plotting system (APS) arbitrarily orientated antennas 157–62 assumed (suppressed) time dependency 12 asymptotic far-field form of Kirchhoff–Huygens formula 591–3

Index AUT: see antenna under test (AUT) auxiliary rotation 523 continuity of field function at intersection 525–6 modified near-field to far-field transform 531 modified probe pattern correction 528–30 normal field component, reconstruction of 527–8 simple transform 524–5 simple transform, deficiency of 526–7 spectral leakage and suitability of windowing functions 530–1 auxiliary translation 517–20 AWGN: see additive white Gaussian noise (AWGN) azimuth, elevation and roll angles 574–6 azimuth and elevation (true-view) co-ordinate system 566–7 azimuth over elevation grid 193–5, 563–4 and elevation over azimuth co-ordinate systems 187–90 bandwidth 8–12 base transceiver station (BTS) system 483 Bayesian Compressive Sensing (BCS) framework 485 BCS framework: see Bayesian Compressive Sensing (BCS) framework Bessel function 168–9, 185 bi-scan auxiliary rotation technique 537–8 boundary conditions 84–5 broadband probes 254–7 BTS system: see base transceiver station (BTS) system

597

Cartesian co-ordinates 23–4, 26, 186, 326 scalar Helmholtz equation in 71 Fourier transform solution 72–83 integral transforms 71 solution of vector Helmholtz equation in 86–9 solution of vector magnetic wave equation in 89 Cartesian far-field electric components 107 Cartesian field components 147–8 Cartesian polarisation basis 190 CATR: see compact antenna test range (CATR) CDF: see cumulative distribution function (CDF) CEM model: see computational electromagnetic (CEM) model centroid, calculation peak of a pattern by 239–40 CGPM: see Confe´rence Ge´ne´rale des Poids et Mesures (CGPM) chamber multiple reflections, assessment of 268–70 chambers 51–3 channel balance correction 265–8, 289–90 circular polarisation basis 198–204 classical electromagnetism 551–2 classical quantum mechanics 552–3 CNFS technique: see cylindrical near-field scanning (CNFS) technique collective electrodynamics 554–5 communication systems 1 compact antenna test range (CATR) 42–4, 48, 51, 130, 342 complete poly-planar rotational technique 541 experimental procedure 542–5 experimental set up 541–2

598

Principles of planar near-field antenna measurements

complex vector wave equations 23 compressive sensing (CS) technique 483–6, 507 defective element detection using 486–8 practical implementation 495–502 2D array, compressive sensing applied to 488 refinements to basic CS approach 490–5 using near-field scanning 502–7 computational electromagnetic (CEM) model 293, 342 current elements method 314–24 electric dipoles, linear superposition of 294–6 equivalent currents method 324–38 full wave simulation of a planar near-field antenna measurement 353–6 infinite perfectly conducting ground plane, aperture set in 299 antenna-to-antenna coupling formula 304–6 Kirchhoff–Huygens method 308–14 modelling tool 250 mutual coupling and reaction theorem 339–42 near-field measurement simulation 342–5 reaction theorem 345 generalised 350–2 Lorentz 345–9 mutual impedance and 352 sub-apertures, method of 296–8 vector Huygens method 306–7 Confe´rence Ge´ne´rale des Poids et Mesures (CGPM) 25 conventional data set comparison techniques, examples of 389–93 co-ordinate free form 106–9

co-ordinate systems 560 antenna electrical system 560–1 antenna mechanical system (AMS) 560 azimuth, elevation and roll angles 574–6 azimuth and elevation (true-view) co-ordinate system 566–7 azimuth over elevation grid 563–4 co-ordinate systems and elemental solid angles 569 direction cosine 561–3 and elemental solid angles 569, 580–2 elevation over azimuth 564–5 Euler angles 576–7 far-field plotting systems 561 polar spherical 565–6 quaternion 577–80 range of spherical angles 567–8 relationship between 569–74 transformation between 568–9 copolar and cross-polar polarisation basis 195–8 copolar polarisation basis, rotation of 213–14 Coulomb force law 16 cross-polar radiation pattern acquisition 41 CS technique: see compressive sensing (CS) technique cumulative distribution function (CDF) 443, 492 current elements method 314–24 cylindrical near-field scanning (CNFS) technique 47 data point spacing 379–80 data sets, quantitative measures of correspondence between 388–9 defective element detection using compressive sensing 486–8

Index DFT algorithm: see discrete Fourier transform (DFT) algorithm DGPS: see differential GPS (DGPS) differential GPS (DGPS) 436 Digital Twin (DT) 293 dipole antennas 257 Dirac delta function 167 Dirac equation 388, 553 direction cosine coordinate system 561–3 calculation of radiated power in 222 directivity 220–2 Dirichlet 73 discrete Fourier transform (DFT) algorithm 50, 152, 156, 589 double Fourier integral, standard techniques for evaluation of 152 discrete Fourier transform (DFT) 156 fast Fourier transform (FFT) 152–5 Fourier transforms with improved accuracy 157 double integral, stationary phase evaluation of 100–6 DT: see Digital Twin (DT) dual-polarised waveguide probes 253–4 dynamic range 382 electric and magnetic potentials retarded potentials 27–31 static potentials 27 electric and magnetic spectral components, relationship between 89–92 electric charge 15–16 electric dipole probe, effect of acquiring near-field data using 146–50 electric dipoles, linear superposition of 294–6

599

electromagnetic (EM) properties of antennas 1 electromagnetic field 16–17, 159 electromagnetic model, computational: see computational electromagnetic model electromagnetic screening 51 electromagnetism 8 electronic system drift 264–5 elemental solid angle for co-ordinate systems 569, 580–2 elevation over azimuth 564–5 EMPL: see equivalent multipath level (EMPL) E-plane radiation pattern acquisition 40 equivalent currents method 324–38 equivalent multipath level (EMPL) 285, 389, 404, 436, 444–50, 470 error budgets: see measurement error budgets Euler angles 576–7 evanescent plane wave mode coefficients, behaviour of 141–4 far-field antenna patterns and elevation co-ordinates 219 generalised compound vector rotation of 214–17 radiation patterns 97–100 scalar rotation of 208–11 vector rotation of 211–12 far-field data, effect of probe pattern on 246–7 far-field phase taper geometry 44 far-field plotting systems 561 far-fields from non-planar apertures 111–12 far-field transform near-field transform to 121–3 to near-field transform 114–18

600

Principles of planar near-field antenna measurements

fast Fourier transform (FFT) algorithm 50, 152–5, 167, 301, 589–91 FFT algorithm: see fast Fourier transform (FFT) algorithm field equivalence principle 32–4 field reciprocity theorem 345–9 finite area scan errors 128–9 finite element model of open ended rectangular waveguide probe 285–9 Fourier approach 307 Fourier integral transform 129 Fourier transform 71–4, 77, 81, 85, 87, 100, 115, 167, 181, 185, 452, 470 with improved accuracy 157 solution of scalar Helmholtz equation 72–83 four-point bi-linear formula 278 free space propagation vector 92–3 free space radiation pattern 6–8 full wave simulation of planar near-field antenna measurement 353–6 gain 228–9 gain-transfer (gain-comparison) method 229–31 generalised reaction theorem 350–2 general-purpose interface bus (GPIB) card 62 generic antenna measurement process 66–7 GPIB card: see general-purpose interface bus (GPIB) card Green’s function 27, 34, 79–80 heating ventilation and air-conditioning (HVAC) systems 265, 381 Helmholtz equation 2–3, 8, 24, 26, 46–7, 72, 77, 81, 555 Hertzian dipole 160

histogram equalisation 406 H-plane radiation pattern acquisition 40 Huygens element 306–7 Huygens’ principle 32, 109–11, 307, 531 HVAC systems: see heating ventilation and air-conditioning (HVAC) systems image theory 33–4 infinite perfectly conducting ground plane, aperture set in 299–306 integral transforms 71 interaction, postulated mechanisms of 551 absorber theory 555–7 classical electromagnetism 551–2 classical quantum mechanics 552–3 collective electrodynamics 554–5 relativistic quantum mechanics 553–4 interpolation formula 277 four-point bi-linear formula 278 nearest neighbour 277 phase interpolation 279–80 six point formula 279 three point linear formula 278 interpolation theory 125–7 K-correction method 459–64 Kirchhoff–Huygens principal 308–14, 338–9, 531, 533–7 asymptotic far-field form of 591–3 Kirchhoff integral theorem 27, 71 Laplacian operator 26 least squares conjugate gradient (LSQR) method 325, 329, 468, 471 left hand circularly polarised (LHCP) field 198–204

Index leptokurtic 394 LHCP field: see left hand circularly polarised (LHCP) field L’Hoˆpital’s rule 88 linear and circular polarisation bases 204–8 local oscillator (LO) signal 54 log-periodic dipole array antenna (LPDA) 254, 256 Lorentz condition 29 Lorentz reciprocity theorem 345–9 LO signal: see local oscillator (LO) signal LPDA: see log-periodic dipole array antenna (LPDA) LSQR method: see least squares conjugate gradient (LSQR) method Ludwig I 190 Ludwig II 193–5 Ludwig III 195–8, 213–14, 219 matrix inversion method 464–72 maximum directivity 220 maximum radial extent (MRE) 170, 488 Maxwell’s equations 2, 20–7, 66–7, 69, 71–2, 81, 87, 89, 92, 554–5 Maxwell’s equations and electromagnetic wave propagation 15 accelerated charges 17–20 electric and magnetic potentials retarded potentials 27–31 static potentials 27 electric charge 15–16 electromagnetic field 16–17 field equivalence principle 32–4 measurement methodology 31–2 vector electromagnetic fields, characterising 34–6

601

MBSE/MBD: see Model-Based Systems Engineering and Development (MBSE/MBD) measurement, defined 4, 557–60 measurement error budgets 368 aliasing 379–80 applicability of digital twin to assessing 371–3 applicability of modelling error sources 369–70 dynamic range 382 empirical approach to 370–1 numerical truncation 376 probe x, y (in-plane) position error 376–9 rounding error 376 systematic phase 381–2 truncation 373–5 measurement methodology 31–2 measurement process, characterisation via 4 bandwidth 8–12 free space radiation pattern 6–8 polarisation 8 measurement results, establishment of measure from 359 measurement errors 360–3 measurement result data, examination of 365–8 sources of measurement ambiguity and error 363–5 measurements 125 antenna alignment corrections 208 copolar polarisation basis, rotation of 213–14 generalised compound vector rotation of far-field antenna patterns 214–17 scalar rotation of far-field antenna patterns 208–11

602

Principles of planar near-field antenna measurements

vector rotation of far-field antenna patterns 211–12 antenna coupling formula 157–62 antenna-to-antenna coupling formula 133 evanescent plane wave mode coefficients, behaviour of 141–4 simple scattering model of near-field probe 144–6 arbitrarily orientated antennas 157–62 azimuth over elevation and elevation over azimuth co-ordinate systems 187–90 directivity 220–2 electric dipole probe, effect of acquiring near-field data using 146–50 far-field data, estimating the position of phase centre from 240–2 finite area scan errors 128–9 gain 228–9 gain-transfer (gain-comparison) method 229–31 linear and circular polarisation bases 204–8 near-field co-ordinate systems 217 antenna electrical system 218–19 antenna mechanical system 218 antenna radiation patterns, general vector rotation of 219–20 far-field azimuth and elevation co-ordinates 219 Ludwig III co-polar and cross-polar definition 219 probe alignment definition 219 range fixed system (RFS) 217 near-field measurements 150–1 near-field to far-field transform 151 discrete Fourier transform (DFT) 156 fast Fourier transform (FFT) 152–5

Fourier transforms with improved accuracy 157 peak of a pattern, calculation of 235 by centroid 239–40 by polynomial fit 237–9 by series solution 235–7 plane bi-polar co-ordinates, boundary values known in 177–87 plane polar co-ordinates, boundary values known in 164–77 polarisation basis and antenna measurements 190 azimuth over elevation basis 193–5 Cartesian polarisation basis 190 circular polarisation basis 198–204 copolar and cross-polar polarisation basis 195–8 polar spherical polarisation basis 191–3 radiated power calculation in direction-cosine coordinate system 222 rectangular pyramidal horn, approximation of gain of 231–5 rotationally symmetric probes 150–1 sampling (interpolation theory) 125–7 spectral leakage 129 truncation 127–30 uniformly illuminated square aperture, direct evaluation of directivity for 224–8 x-polarised antenna 150 measurement system, simulation of 441, 450–1 AUT phase and probe position reconstruction 442–3 millimetre wave robotic arm-based NF measurement system 448–50 near-field to far-field antenna measurement performance 443–8

Index mechanical ground support equipment (MGSE) 58 mesokurtic 394 Method of Moments (MoM) approach 234, 328 MGSE: see mechanical ground support equipment (MGSE) MHM: see microwave holographic metrology (MHM) microwave holographic metrology 112–14 microwave holographic metrology (MHM) 299, 422 Model-Based Systems Engineering and Development (MBSE/MBD) 293 MoM approach: see Method of Moments (MoM) approach MRE: see maximum radial extent (MRE) mutual coupling and reaction theorem 339–42 mutual impedance and reaction theorem 352 Naval Research Laboratory (NRL) 231, 235 nearest neighbour interpolation 277 near-field acquisition geometries 46 near-field antenna measurements 4, 39 absorber 51 antenna measurements 39–44 chambers 51–3 electromagnetic screening 51 forms of 46–9 generic antenna measurement process 66–7 near-field probe 63–6 plane rectilinear 49–50 radio frequency (RF) subsystem 53–7 robotics positioner subsystem 58–63

603

near-field co-ordinate systems 217 antenna electrical system 218–19 antenna mechanical system 218 antenna radiation patterns, general vector rotation of 219–20 far-field azimuth and elevation co-ordinates 219 Ludwig III co-polar and cross-polar definition 219 probe alignment definition 219 range fixed system (RFS) 217 near-field data 108–9 near-field measurement system 54, 66, 150–1, 342–5 near-field probe 63–6 desirable characteristics of 248 broadband probes 254–7 dipole antennas 257 dual-polarised waveguide probes 253–4 open-ended rectangular waveguide probes 250–3 near-field scanner (NFS) 519 near-field scanning, compressive sensing using 502–7 near-field to far-field transformation 151 double Fourier integral, standard techniques for evaluation of 152 discrete Fourier transform (DFT) 156 fast Fourier transform (FFT) 152–5 Fourier transforms with improved accuracy 157 of probe-corrected data 531 Kirchhoff–Huygens principle 533–7 near-field transform far-field transform to 114–18 to far-field transform 121–3 NFS: see near-field scanner (NFS) non-canonical transform 475–83

604

Principles of planar near-field antenna measurements

plane wave spectrum-based treatment 464 matrix inversion method 464–72 non-canonical transform 475–83 non-uniform fast Fourier transform (NUFFT)-based algorithms 472–5 non-iterative phase retrieval technique 434 antenna under test (AUT) phase and probe position reconstruction 438–41 current phase retrieval techniques 435–7 measurement system, simulation of 441, 450–1 AUT phase and probe position reconstruction 442–3 millimetre wave robotic arm-based NF measurement system 448–50 near-field to far-field antenna measurement performance 443–8 non-planar apertures, far-fields from 111–12 non-uniform fast Fourier transform (NUFFT)-based algorithms 472–5 novel data comparison techniques 393–408 NRL: see Naval Research Laboratory (NRL) NUFFT-based algorithms: see nonuniform fast Fourier transform (NUFFT)-based algorithms numerical truncation 376 Nyquist criteria 283 Nyquist frequency 167–8, 171 OEWG probe: see open ended rectangular waveguide (OEWG) probe OMT: see orthogonal mode transducer (OMT)

open ended rectangular waveguide (OEWG) probe 64, 250–3, 253, 263, 285–9, 354, 356 operator substitution 85–6 orthogonal mode transducer (OMT) 254, 289 partial scan techniques 517 auxiliary rotation 523 continuity of field function at intersection 525–6 modified near-field to far-field transform 531 modified probe pattern correction 528–30 normal field component, reconstruction of 527–8 simple transform 524–5 simple transform, deficiency of 526–7 spectral leakage and suitability of windowing functions 530–1 auxiliary translation 517–20 complete poly-planar rotational technique 541 experimental procedure 542–5 experimental set up 541–2 near-field to far-field transformation of probe-corrected data 531 Kirchhoff–Huygens principle 533–7 poly-planar technique, applicability of bi-scan auxiliary rotation technique 537–8 tri-scan auxiliary rotation technique 538–41 rotations of AUT about z-axis 520–3 PCU: see power control unit (PCU) peak of a pattern, calculation of 235 by centroid 239–40 by polynomial fit 237–9

Index by series solution 235–7 peak signal to noise ratio (PSNR) 391 PEC: see perfect electrical conductor (PEC) perfect electrical conductor (PEC) 84–5 phase interpolation 279–80 phase with respect to space (PWS) 524 planar near-field facility acquisition of alignment data in 416–17 acquisition of mechanical alignment data in 417–20 planar near-field scanner (PNFS) 46–7 plane bi-polar co-ordinates, boundary values known in 177–87 plane bi-polar measurement 187 plane-bi-polar near-field data 183 plane polar co-ordinates, boundary values known in 164–77 plane-polar scanning technique 47, 164 plane rectilinear near-field antenna measurements 49–50 plane-to-plane (PTP) phase retrieval algorithm 427–31 with aperture constraint 431–4 plane-to-plane translation formula 302 plane-wave impedance 93–5 plane waves, interpretation as angular spectrum of 95–7 plane wave spectrum (PWS) method 69–70, 96–7, 304–6, 416 derivation of 70–1 plane-wave spectrum representation of electromagnetic waves 69 boundary conditions 84–5 Cartesian co-ordinates solution of vector Helmholtz equation in 86–9 solution of vector magnetic wave equation in 89 co-ordinate free form of near-field

605

to angular spectrum transform 106–9 to far-field transform to Huygens’ principle 109–11 electric and magnetic spectral components, relationship between 89–92 far-field antenna radiation patterns 97–100 far-fields from non-planar apertures 111–12 far-field to near-field transform 114–18 free space propagation vector 92–3 interpretation as angular spectrum of plane waves 95–7 microwave holographic metrology 112–14 near-field to far-field transform 121–3 operator substitution 85–6 plane-wave impedance 93–5 plane-wave spectrum, derivation of 70–1 radiated power and angular spectrum 118–21 scalar Helmholtz equation 71 Fourier transform solution of 72–83 integral transforms 71 stationary phase evaluation of double integral 100–6 platykurtic 394 PNFS: see planar near-field scanner (PNFS) Poincare´ sphere 203 Poisson’s equation 281–2 polarisation 8 polarisation basis and antenna measurements 190 azimuth over elevation basis 193–5 Cartesian polarisation basis 190 circular polarisation basis 198–204

606

Principles of planar near-field antenna measurements

copolar and cross-polar polarisation basis 195–8 polar spherical polarisation basis 191–3 polar spherical 565–6 polar spherical polarisation basis 191–3 polynomial fit, calculation peak of a pattern by 237–9 poly-planar technique, applicability of bi-scan auxiliary rotation technique 537–8 tri-scan auxiliary rotation technique 538–41 power control unit (PCU) 62 Poynting vector 36 probe alignment definition 219 probe displacement correction 289 probe pattern 245 channel balance correction 289–90 on far-field data 246–7 near-field probe, desirable characteristics of 248 broadband probes 254–7 dipole antennas 257 dual-polarised waveguide probes 253–4 open-ended rectangular waveguide probes 250–3 open ended rectangular waveguide (OEWG) probe 285–9 probe displacement correction 289 quasi far-field probe pattern, acquisition of 257 assessment of chamber multiple reflections 268–70 channel balance correction 265–8 correction for rotary errors 270–3 electronic system drift 264–5 four-point bi-linear formula 278 interpolation formula 277–80

nearest neighbour 277 phase interpolation 279–80 probe vector pattern function, re-tabulation of 273–7 remote source antenna tilt-angle correction 273 sampling scheme 259–64 six point formula 279 three point linear formula 278 true far-field probe pattern 283–4 two-dimensional phase unwrapping 280–3 probe vector pattern function, re-tabulation of 273–7 probe x, y (in-plane) position error 376–9 PSNR: see peak signal to noise ratio (PSNR) PTP phase retrieval algorithm: see plane-to-plane (PTP) phase retrieval algorithm PWS: see phase with respect to space (PWS); plane wave spectrum (PWS) method QED: see quantum electrodynamics (QED) quantum electrodynamics (QED) 21, 388, 553 quasi far-field probe pattern, acquisition of 257 alternate interpolation formula 277 four-point formula (bi-linear) 278 nearest neighbour 277 phase interpolation 279–80 six point formula 279 three point formula (linear) 278 approximate unwrapping of twodimensional phase functions 280–3 assessment of chamber multiple reflections 268–70

Index channel balance correction 265–8 electronic system drift (Tie-scan correction) 264–5 probe vector pattern function, re-tabulation of 273–7 remote source antenna tilt-angle correction 273 rotary errors, correction for 270–3 sampling scheme 259–64 true far-field probe pattern 283–4 quaternion 577–80 radars 1 radiated power and angular spectrum 118–21 radiated power calculation in directioncosine coordinate system 222 radio frequency (RF) subsystem 53–7 range assessment budgets, illustration of compilation of 383–7 range fixed system (RFS) 217 ratioed measurements 22 reaction theorem 345 generalised 350–2 Lorentz 345–9 mutual impedance and 352 rectangular pyramidal horn, approximation of gain of 231–5 “reference” signal 22 relativistic quantum mechanics 553–4 remote source antenna (RSA) tilt-angle correction 273 restricted isometry property (RIP) 485 retarded potentials 27–31 RFS: see range fixed system (RFS) RF subsystem: see radio frequency (RF) subsystem RHCP field: see right hand circularly polarised (RHCP) field

607

right hand circularly polarised (RHCP) field 198–204 RIP: see restricted isometry property (RIP) RMS values: see root mean square (RMS) values robotics positioner subsystem 58–63 root mean square (RMS) values 118 rotary errors, correction for 270–3 rotationally symmetric probes 150–1 rounding error 376 RSA tilt-angle correction: see remote source antenna (RSA) tilt-angle correction Ruze equation 379 Rx antenna 1–2, 4–8, 11–12 scalar Helmholtz equation 71 Fourier transform solution of 72–83 integral transforms 71 scalar rotation of far-field antenna patterns 208–11 Schrodinger wave equation 388 second-order truncation effect 128 semi-major axis calculation of rotated ellipse 585–9 series solution, calculation peak of a pattern by 235–7 SGH: see standard gain horn (SGH) simple scattering model of near-field probe during planar measurement 144–6 single port probe (SPP) 141 SMCs: see spherical mode coefficients (SMCs) SNF measurement: see spherical near-field (SNF) measurement SNFR: see spherical near-field range (SNFR) spectral leakage 129

608

Principles of planar near-field antenna measurements

spherical angles, range of 567–8 spherical mode coefficients (SMCs) 484 spherical near-field (SNF) measurement 484 spherical near-field range (SNFR) 46 SPP: see single port probe (SPP) standard gain horn (SGH) 342, 420 standing wave ratio (SWR) 251 static potentials 27 stationary phase evaluation of double integral 100–6 sub-apertures, method of 296–8 suppressed time dependency 12 SWR: see standing wave ratio (SWR) systematic phase 381–2

true far-field probe pattern 283–4 truncation 127–30, 373–5 2D array, compressive sensing applied to 488 refinements to basic CS approach 490–5 two-dimensional phase unwrapping 280–3 Tx antenna 1–7, 11–12

tangential magnetic spectral components 89 Taylor expansion 452 Taylor-series correction formula 456 Taylor-series expansion 452–9 TEM: see transverse electromagnetic (TEM) three antenna extrapolated gain measurements 507–17 three point linear formula 278 tie-scan correction 264–5 tilt angle calculation of rotated ellipse 585–9 traditional position correction algorithms, in plane and z plane corrections 451 K-correction method 459–64 Taylor-series expansion 452–9 transverse electromagnetic (TEM) 93 trapezoidal discrete Fourier transform 582–5 tri-scan auxiliary rotation technique 538–41

vector electromagnetic fields, characterising 34–6 vector Helmholtz equation 24, 46–7 in Cartesian co-ordinates 86–9 vector Huygens method 306–7 vector magnetic wave equation in Cartesian coordinates 89 vector network analyzers (VNA) 22 vector rotation of far-field antenna patterns 211–12 VNA: see vector network analyzers (VNA) voltage standing wave ratio (VSWR) 132 VSWR: see voltage standing wave ratio (VSWR)

UAV: see uninhabited air vehicles (UAV) uniformly illuminated square aperture, direct evaluation of directivity for 224–8 uninhabited air vehicles (UAV) 434, 479

wave normal 95 Wheeler Feynman absorber theory of radiation 555 x-polarised antenna 150