tool for MIMO antenna analysis and design. CMT allows us to examine each physical mode of the antenna aperture, and to access its many physical parameters ...
ABSTRACT YANG, BINBIN. A Modal Approach to Compact MIMO Antenna Design. (Under the direction of Dr. Jacob J. Adams.) MIMO (Multiple-Input Multiple-Output) technology offers new possibilities for wireless communication through transmission over multiple spatial channels, and enables linear increases in spectral efficiency as the number of the transmitting and receiving antennas increases. However, the physical implementation of such systems in compact devices encounters many physical constraints mainly from the design of multi-antennas. First, an antenna’s bandwidth decreases dramatically as its electrical size reduces, a fact known as antenna Q limit; secondly, multiple antennas closely spaced tend to couple with each other, undermining MIMO performance. Though different MIMO antenna designs have been proposed in the literature, there is still a lack of a systematic design methodology and knowledge of performance limits. In this dissertation, we employ characteristic mode theory (CMT) as a powerful tool for MIMO antenna analysis and design. CMT allows us to examine each physical mode of the antenna aperture, and to access its many physical parameters without even exciting the antenna. For the first time, we propose efficient circuit models for MIMO antennas of arbitrary geometry using this modal decomposition technique. Those circuit models demonstrate the powerful physical insight of CMT for MIMO antenna modeling, and simplify MIMO antenna design problem to just the design of specific antenna structural modes and a modal feed network, making possible the separate design of antenna aperture and feeds. We therefore develop a feed-independent shape synthesis technique for optimization of broadband multi-mode apertures. Combining the shape synthesis and circuit modeling techniques for MIMO antennas, we propose a shape-first feed-next design methodology for MIMO antennas, and designed and fabricated two planar MIMO antennas, each occupying an aperture much smaller than the regular size of λ/2 × λ/2. Facilitated by the newly developed source formulation for antenna stored energy and recently reported work on antenna Q factor minimization, we extend the minimum Q limit to antennas of arbitrary geometry, and show that given an antenna aperture, any antenna design based on its substructure will result into minimum Q factors larger than or equal to that of the complete structure. This limit is much tighter than Chu’s limit
based on spherical modes, and applies to antennas of arbitrary geometry. Finally, considering the almost inevitable presence of mutual coupling effects within compact multiport antennas, we develop new decoupling networks (DN) and decoupling network synthesis techniques. An information-theoretic metric, information mismatch loss (Γinfo ), is defined for DN characterization. Based on this metric, the optimization of decoupling networks for broadband system performance is conducted, which demonstrates the limitation of the single-frequency decoupling techniques and room for improvement.
© Copyright 2017 by Binbin Yang All Rights Reserved
A Modal Approach to Compact MIMO Antenna Design
by Binbin Yang
A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy
Electrical Engineering Raleigh, North Carolina 2017
APPROVED BY:
Dr. Brian A. Floyd
Dr. Brian L. Hughes
Dr. Michael B. Steer
Dr. Ruian Ke
Dr. Jacob J. Adams Chair of Advisory Committee
DEDICATION To my father, Ming-Yu Yang (杨明玉), whose endless love, encouragement and extraordinary confidence in me makes me who I am today, yet to whom I owe the most in my life.
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BIOGRAPHY Binbin Yang was born in Nanyang, Henan Province, China in 1988. He received his Bachelor’s degree in Electrical Engineering from Hunan University, Changsha, China, in 2010, and his Master’s degree in Electrical Engineering from the University of Chinese Academy of Sciences, Beijing, China, in 2013. From 2013 to 2017, he pursued his doctoral degree at North Carolina State University, Raleigh, NC, USA, and worked as a research assistant at Antennas and Electromagnetics Lab under Dr. Jacob J. Adams. His research interests include MIMO antennas, characteristic mode theory, RF and microwave systems, wireless communication and computational electromagnetics.
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ACKNOWLEDGEMENTS First and foremost, I would like to thank my mentor and advisor in my doctoral education, Dr. Jacob J. Adams. I am really grateful to him for offering me the opportunity and assistantship to pursue my Ph.D. degree here at North Carolina State University. He is not only an experienced researcher, but also a great mentor offering valuable suggestions and guidance. I benefit a lot from his insight, perseverance and professional attitude, and really appreciate his trust and support. He has allowed great freedom and flexibility in managing my time and choosing interested research topics. I feel very lucky to have him as my advisor while I am learning to be an independent researcher. I sincerely express my gratitude to my other committee members, Dr. Brian A. Floyd, Dr. Brian L. Hughes, Dr. Michael B. Steer and Dr. Ruian Ke, for their professional suggestions and helpful discussions, which significantly improved the quality of my research work. I am also grateful to all the professors who has taught me at NC State, for imparting their knowledge that prepares me for future careers. I would like to thank all the collaborators in EARS project: Dr. Jacob J. Adams, Dr. Brian A. Floyd, Dr. Brian L. Hughes, Shaohan Wu, Wuyuan Li, Charley Wilson and Dr. Lopamudra Kundu. Their many valuable suggestions and inspirational discussions throughout the past four years really help shape my research work. I also want to thank Futurewei Technologies at Bridgewater, New Jersy, and my mentors Sean Ma and Leonard Piazzi for offering me the internship opportunity and sponsoring part of my research work on DN synthesis during the summer of 2016. To all my colleagues at Antennas and Electromagnetics Lab (AEL) at NCSU, Meng Wang, Kurt Schab, Shruti Srivastava, Clifford Muchler, Danyang Huang, Vivek Bharambe and Munirah Boufarsan, thank you all for the discussions in office, feedback in our group meetings and lending a hand on lab fabrication and measurement. I would like to say thank you to many friends in China, Ying-Yong Zhang, Dr. FengMan Liu, Dr. Zhi-Hua Li, Dr. Da-Quan Yu and Dr. Hai-Fei Xiang. They have supported me in one way or another during my Master education. I also owe special thanks to Dr. Daniel Guidotti, whose friendship and supportive advices have been with me throughout my Master and Ph.D. career. I owe eternally to my beloved father, Ming-Yu and my elder sister, Yan. My debt to them is immeasurable.
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At last, I would like to thank my wife, Xiao, and our precious two kids she has been raising most of the time all by herself. My life in the past four years has been so joyfully fulfilled because of their love, encouragement, support and sometimes distraction.
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TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1 Introduction . . . . . . . . . . . 1.1 MIMO System . . . . . . . . . . . . . . 1.2 Challenges in Compact MIMO System 1.2.1 Antenna Physical Limits . . . . 1.2.2 Mutual Coupling . . . . . . . . 1.3 MIMO Antenna . . . . . . . . . . . . . 1.4 Decoupling Network . . . . . . . . . . 1.5 Overview of the Dissertation . . . . . .
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Chapter 2 Characteristic Mode Theory . . . 2.1 Numerical Modeling of Radiating Objects 2.1.1 PEC Objects . . . . . . . . . . . . 2.1.2 Dielectric Objects . . . . . . . . . . 2.1.3 Method of Moment . . . . . . . . . 2.2 Characteristic Mode Theory . . . . . . . . 2.2.1 Orthogonality . . . . . . . . . . . . 2.2.2 Modal Expansion . . . . . . . . . . 2.2.3 Eigenvalues . . . . . . . . . . . . . 2.2.4 Characteristic Modal Q Factor . . .
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Chapter 3 Broadband Circuit Models for MIMO Antennas 3.1 Circuit Model for Arbitrary Wire MIMO Antennas . . . . 3.1.1 Formulation of Multi-port Admittances Using CMT 3.1.2 Proposed Multi-port Circuit Model . . . . . . . . . 3.1.3 Demonstration Examples . . . . . . . . . . . . . . . 3.2 Circuit Model for Probe-fed Planar MIMO Antennas . . . 3.2.1 Formulation of Multi-port Impedances Using CMT 3.2.2 Proposed Multi-port Circuit Model . . . . . . . . . 3.2.3 Calculation of the Circuit Elements . . . . . . . . . 3.2.4 Demonstration Examples . . . . . . . . . . . . . . . 3.3 Application to Feed Specification . . . . . . . . . . . . . . 3.3.1 Visualizing Input Parameters Using Heat Maps . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4 Feed-independent Shape Synthesis of MIMO Antennas . . . 4.1 Challenges in MIMO Antenna Shape Synthesis . . . . . . . . . . . . . . .
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A Simple Feed Network to Extract Characteristic Modes of Symmetric Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Framework of Antenna Shape Synthesis . . . . . . . . . . . . . . . . . . . 4.3.1 Binary Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Optimization Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Minimize the Sum of the Modal Qs . . . . . . . . . . . . . . . . . 4.4.2 Minimize the Sum of the Normalized Modal Qs and Occupied Area 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5 Systematic Design of Planar MIMO Antennas . . . . . . . . . 5.1 Shape-first Feed-next Design Methodology . . . . . . . . . . . . . . . . . 5.2 Design Example 1: Two-Port Planar MIMO Antenna on Air Substrate . 5.2.1 Shape Synthesis of Self-Resonant MIMO Antennas . . . . . . . . 5.2.2 Feed Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Fabrication and Measurement . . . . . . . . . . . . . . . . . . . . 5.3 Modeling of Antennas on Dielectric Substrates . . . . . . . . . . . . . . . 5.3.1 Spectral Domain DGF of Microstrip Substrate . . . . . . . . . . . 5.3.2 Spatial Domain DGF Using DCIM . . . . . . . . . . . . . . . . . 5.3.3 Numerical Examples of Dyadic Green’s Functions . . . . . . . . . 5.3.4 Extension of Planar Antenna Circuit Model to Microstrip Antennas 5.4 Design Example 2: Two-Port Microstrip MIMO Antenna . . . . . . . . . 5.4.1 Shape Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Feed Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Fabrication and Measurement . . . . . . . . . . . . . . . . . . . . 5.4.4 Antenna Pattern and Correlation . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 6 Physical Limits of Antennas of Arbitrary Geometry . . . . . 96 6.1 Problem of Q Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2 Relation Between Impedance Matrices of A Structure and Its Substructures 98 6.3 Substructure Eigenvalue Bound . . . . . . . . . . . . . . . . . . . . . . . 100 6.4 Limit on Minimum Tuned Q Factor . . . . . . . . . . . . . . . . . . . . . 102 6.4.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.5 Limit on Untuned Modal Q Factors . . . . . . . . . . . . . . . . . . . . . 104 6.5.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter 7 Decoupling Networks for MIMO Antennas . . . . . . . . . . . 108 7.1 A Decoupling Network Based on Characteristic Port Modes . . . . . . . 109 7.1.1 Mathematics and Topology of the DN . . . . . . . . . . . . . . . 109
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7.1.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . Decoupling Network Synthesis Using TLs . . . . . . . . . . . . . 7.2.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 7.2.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . Broadband Mulitport Matching Network Optimization . . . . . 7.3.1 An Information-theoretic Metric for DN Characterization 7.3.2 Broadband MN Optimization . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 8 Conclusion and Future 8.1 Summary of Dissertation . . . 8.1.1 Contributions . . . . . 8.2 Future Directions . . . . . . .
Directions . . . . . . . . . . . . . . . . . . . . .
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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A Source Formulation of Antenna Stored Energy and Appendix B Multi-port Transformer Network . . . . . . . . . . Appendix C Dyadic Green’s Function of Microstrip Substrate . C.1 Field Green’s Function . . . . . . . . . . . . . . . . . . . C.2 Potential Green’s Function . . . . . . . . . . . . . . . . . Appendix D Eigenvalue Equivalence . . . . . . . . . . . . . . . Appendix E Code on N-port Network Synthesis Using TLs . .
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. . . . . . Q Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF TABLES Table 3.1 Modal parameters of the first three characteristic modes of the 0.5m strip dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 3.2 Circuit Elements of mode 1 . . . . . . . . . . . . . . . . . . . . . . Table 3.3 Circuit Elements of mode 2 . . . . . . . . . . . . . . . . . . . . . . Table 3.4 Circuit Elements of mode 3 . . . . . . . . . . . . . . . . . . . . . . Table 3.5 The DC Capacitance Calculated Based on MoM. . . . . . . . . . . Table 3.6 The Circuit Parameters Calculated for the Slotted Patch With The Reference Point r 0 = (3 cm, −2 cm, z). . . . . . . . . . . . . . . . . Table 3.7 The Transformation Ratios of The Four Feed Schemes for the Slotted Patch With The Reference Point r 0 = (3 cm, −2 cm, z). . . . . . . Table 4.1 The Modal Qs of the Optimized Two-Port and Four-Port Substructures at 500 MHz Using Cost Function 2. . . . . . . . . . . . . . . Table 5.1 Dyadic Green’s Functions of Single Layer Microstrip Substrate . . Table 5.2 Comparison of the Realized Gains (dBi) at the Center Frequencies From Simulation (2.435 GHz) and Measurement (2.485 GHz) . . .
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Table 7.1 Characteristic Impedance and Electrical Length of All the TL Branches121 Table 7.2 Component values of the optimized 2nd order LC π matching network128
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LIST OF FIGURES Figure 1.1 Illustrative diagram of a MIMO system. . . . . . . . . . . . . . . Figure 1.2 A Compact MIMO Receiver Architecture. . . . . . . . . . . . . . Figure 2.1 Illustration of the first four characteristic modes (currents and far field patterns) on a rectangular patch. . . . . . . . . . . . . . . . . Figure 2.2 Input admittance of a center-fed 0.5m dipole by superposition of its modal responses (2nd mode not excited by the center feed). . . Figure 2.3 Characteristic modal Q factors of a rectangular plate (120 mm × 76 mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.1 Equivalent circuit model for wire MIMO antennas [1]. . . . . . . . Figure 3.2 Modal circuit templates: (a) series RLC circuit, and (b) high-pass circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.3 Configuration of the two-port dipole. . . . . . . . . . . . . . . . . Figure 3.4 Comparison of the results from our circuit model (solid lines) and full wave FEKO simulation (dashed lines): (a) S parameters from high-pass circuit model, and (b) Q factors of the self impedances. Figure 3.5 Y parameters of high-pass circuit model (solid lines), in comparison with the results from full wave simulation (dashed lines). . . . . . Figure 3.6 Z parameters of high-pass circuit model (solid lines), in comparison with the results from full wave simulation (dashed lines). . . . . . Figure 3.7 Comparison of the results from our circuit model (solid lines) and full wave FEKO simulation (dashed lines): (a) S parameters from series RLC circuit model; (b) Q factors of the self impedances. . . Figure 3.8 The variation of the eigenvalue spectrum of a rectangular patch antenna (12 cm × 7.6 cm, h = 5 mm) at two different feed positions: r 1 = (3 cm, 0, z) for solid lines, and r 1 = (4 cm, 2 cm, z) for dashed lines [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.9 An illustration of the virtual feed concept for an arbitrary planar antenna. Virtual feeds are introduced at r i and r j with a reference port located at r 0 [2]. . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.10 The equivalent circuit model for planar MIMO antennas [2]. . . . Figure 3.11 The process to calculate the parameters in the circuit model in Figure 3.10 [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.12 The geometry of the slotted patch [2]. . . . . . . . . . . . . . . . . Figure 3.13 The comparison of S parameters from port simulation (marker) and circuit model (solid lines) of the slotted patch for (a) feed scheme 1, (b) feed scheme 2, (c) feed scheme 3 and (d) feed scheme 4 [2]. . .
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Figure 3.14 The comparison of Z parameters from port simulation (dashed lines) and circuit model (solid lines) of the slotted patch for feed scheme 1. Figure 3.15 The comparison of Y parameters from port simulation (dashed lines) and circuit model (solid lines) of the slotted patch for feed scheme 1. Figure 3.16 (a) the S11 (dB) map of the slotted patch at the first modal resonance frequency (0.8565 GHz), with the optimal feed position marked as P (−1.022 cm, −1.052 cm, z) and the worst-case feed position marked as W ((4.107 cm, 1.31 cm, z), (b) the S11 map at the second modal resonance frequency (1.6033 GHz), (c) the broadband predicted S11 (solid line) for the optimal feed compared with that from the port simulation (dash line) in FEKO, and (d) the broadband predicted S11 (solid line) for the worst-case feed compared with that from the port simulation (dash line) in FEKO [2]. . . . . . . . . . . . . . . Figure 3.17 (a) the S11 (dB) map of the triangular patch at the first modal resonance frequency (1.1669GHz), with the optimal feed position marked as P (−1.286 cm, 0.8033 cm, z), and (b) the broadband predicted S11 (solid line) compared with that from the port simulation (dash line) in FEKO [2]. . . . . . . . . . . . . . . . . . . . . . . . Figure 4.1 Illustrative examples for (a) the one plane of symmetry for the two-port case and (b) the two planes of symmetry for the four-port case [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 4.2 The definition of even and odd modes for the two-port case based on the type of symmetry of the current distribution [3]. . . . . . . Figure 4.3 The definition of even and odd modes for the four-port case based on the type of symmetry of the current distribution [3]. . . . . . . Figure 4.4 Qs of the two eigen port modes (markers), compared with the characteristic modal Qs (solid lines) of the two-port strip dipole antenna [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 4.5 Qs of the four eigen port modes (markers), compared with the characteristic modal Qs (solid lines) of the crossed strip dipole antenna. The even-odd and odd-even modes are degenerate [3]. . Figure 4.6 MIMO System framework [3]. . . . . . . . . . . . . . . . . . . . . Figure 4.7 An illustrative diagram of genetic algorithm. . . . . . . . . . . . . Figure 4.8 Mechanism of extracting MoM Z matrix for substructures. . . . . Figure 4.9 The entire conducting plate used for shape optimization, with triangular mesh and two planes of symmetry enforced [3]. . . . .
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Figure 4.10 Shape optimization results for cost function 1 with four modes considered: (a) The best fit antenna at the 3rd, 15th, 100th, and 800th generations, (b) characteristic modal Qs of an early stage (3rd generation) structure, compared with those of the whole plate, and (c) characteristic modal Qs of an intermediate stage structure (100th generation), compared with those of the whole plate [3]. . . Figure 4.11 The optimization results based on cost function 2 for (a) two-port case and (b) four-port case; and their characteristic modal Qs compared with the modal Qs of the whole plate: (c) two-port case, (d) four-port case [3]. . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.1 The concept of shape-first feed-next design methodology. . . . . . Figure 5.2 The evolution of the shape optimization with three iterative mesh and 80 generations in each iteration [4]. . . . . . . . . . . . . . . Figure 5.3 The modal significance of the optimized two port patch antenna. Figure 5.4 (a) the S11 map at 1.2GHz, (b) the S21 map at 1.2GHz with one fixed feed at P1(0.01125m,0.007125m,z), and (c) the S parameter curve based on the chosen fixed feed P1 and the chosen 2nd feed at P2(0.0075m, 0.007125m, z) in (b), compared with the simulation result based on FEKO (markers) [4]. . . . . . . . . . . . . . . . . Figure 5.5 (a) The prototype of the 2-port MIMO antenna over air substrate, and (b) the measurement result (S12 overlaps with S21 due to reciprocity). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.6 A single layer microstrip substrate. . . . . . . . . . . . . . . . . . Figure 5.7 The example spatial DGF for Gxx A from (a) literature [5], and (b) our DGF implementation (normalized in the figure for easy comparison). Figure 5.8 The example spatial DGF for Gq from (a) literature [5], and (b) our DGF implementation (normalized in the figure for easy comparison). Figure 5.9 The broadband response of a bowtie antenna calculated using our customized MoM code (dashed line) and the commercial MoM solver FEKO (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.10 Modeling of an arbitrary two-port rectangular microstrip antenna using our circuit model (solid lines), in comparison with the result from full wave simulation in FEKO (dashed lines). . . . . . . . . Figure 5.11 S11 heat map of the microstrip rectangular patch at its first resonant frequency generated using our circuit model. . . . . . . . . . . . . Figure 5.12 Iterative shape optimization of two-port patch antenna over microstrip substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.13 Modal significance of the final optimized antenna aperture. . . . Figure 5.14 (a) S11 heat map of the optimized antenna at 2.4 GHz, and (b) S21 heat map of the optimized antenna at 2.4 GHz with the first port being fixed at P1 (4.76 mm, 0). . . . . . . . . . . . . . . . . . . .
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Figure 5.15 Comparison of the antenna input S parameters from our circuit model (solid lines) and full wave simulation in FEKO (marked lines). Figure 5.16 (a) The prototype of the 2-port MIMO antenna over microstrip substrate, and (b) the simulation result in HFSS (dashed lines) and measurement result (solid lines). . . . . . . . . . . . . . . . . . . . Figure 5.17 Normalized radiation patterns of port 1: (a) E plane (φ = 0), and (b) H plane (φ = 90); the solid lines are the measurement results at center frequency (2.485 GHz) and the dashed lines are the simulation results at center frequency (2.435 GHz). . . . . . . . . . . . . . . Figure 5.18 Normalized radiation patterns of port 2: (a) H plane (φ = 0), and (b) E plane (φ = 90); the solid lines are the measurement results at center frequency (2.485 GHz) and the dashed lines are the simulation results at center frequency (2.435 GHz). . . . . . . . . . . . . . . Figure 5.19 The envelope correlation coefficient calculated from the measured 3D radiation pattern and the S parameters. . . . . . . . . . . . .
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93 94
Figure 6.1 An illustrative example of a structure Ω and its one arbitrary ¯ ⊆ Ω. . . . . . . . . . . . . . . . . . . . . . . . . . 99 substructure Ω Figure 6.2 An example of bound on minimum tuned Q factors: all the considered substructures have minimum Q factors higher than the complete structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Figure 6.3 Convergence plot of the five independent shape optimization, each targeting at the minimization of one individual untuned modal Q factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Figure 6.4 Relation between the tuned Q factor and the untuned Q factor of an antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Figure 7.1 The concept of decoupling network. . . . . . . . . . . . . . . . . . Figure 7.2 The multi-port transformer network. . . . . . . . . . . . . . . . . Figure 7.3 (a) Illustration of the antenna decoupling network based on characteristic port mode using ABCD parameters; (b) details within the DN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 7.4 (a) two parallel λ/4 monopoles, spaced by λ/8; (b) S parameters of the parallel monopoles (S11 overlaps with S22 due to symmetry and S12 overlaps with S21 due to reciprocity). . . . . . . . . . . . . . . Figure 7.5 (a) the topology of the DN based on LC π networks; (b) corresponding decoupled S parameters of the parallel λ/4 monopoles. . . . . Figure 7.6 (a) the topology of the DN based on CM port modes; (b) corresponding decoupled S parameters of the parallel λ/4 monopoles (S21 is very low, thus not shown in this scale). . . . . . . . . . . . Figure 7.7 The topology of the generalized TL π network (two-port case) . .
xiii
109 110
112
115 115
116 117
Figure 7.8 (a) the geometry of the two-port square patch antenna, (b) the original S parameter behavior of the two-port patch (S12 overlaps with S21 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 7.9 Simulation of the synthesized TL-DN in Keysight ADS. . . . . . . Figure 7.10 The decoupled S parameters of the coupled two-port patch antenna using the TL-based DN (S12 overlaps with S21 ). . . . . . . . . . . Figure 7.11 A simple MIMO system in S parameters. . . . . . . . . . . . . . . Figure 7.12 The performance of a DN based on generalized π network, designed at a single frequency: (a) decoupled S parameters, (b) mismatch loss in linear scale and (c) resulting capacity of the MIMO system with this DN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 7.13 The performance of the optimized broadband DN based on 2ndorder LC-π network: (a) decoupled S parameters, (b) mismatch loss in linear scale and (c) resulting capacity of the MIMO system with this DN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119 120 120 122
126
127
Figure B.1 The multi-port transformer network. . . . . . . . . . . . . . . . .
151
Figure C.1 A single layer microstrip substrate with an x-oriented unit dipole.
153
xiv
Chapter 1 Introduction This past decade has seen a significant boost in the number of wireless devices, and it is estimated that by 2030, 75% of the world’s population will have mobile connectivity and 60% should have broadband access [6]. However, as is well known, the capacity of conventional wireless communication is fundamentally limited by the available frequency bandwidth and transmission power [7], both of which are scarce resources. How to serve the ever-increasing number of users with even better performance given these limited resources is nowadays one of the key topics in wireless communication industry. While some have sought to increase the available spectrum by operating at millimeter frequency range, such as proposed for future 5G [8] or by dynamically allocating the available spectrum, so-called Spectrum Collaboration [9], another option that comes at almost no cost is to seek technologies that make efficient use of the available frequency spectrum. Some example technologies of this kind are Full-duplex Radio [10] and Multiple-Input Multiple-Output (MIMO) Systems [11]. MIMO system is a more popular choice not only because of its easier implementation, but also because of its scalable increase in spectral efficiency.
1.1
MIMO System
The Multiple-Input Multiple-Output (MIMO) concept as a technique to increase wireless system capacity was first investigated in 1990s [11, 12]. Since then, it has been under
1
n1
y1
x1
n2
x2
y2
TX
RX nN
xM
H
yN
Channel Matrix
Figure 1.1 Illustrative diagram of a MIMO system.
research worldwide [13, 14, 15, 16, 17, 18, 19]. Figure 1.1 gives an illustrative diagram of a generic MIMO system. Different from a conventional single antenna system, a MIMO system has multiple transmitting (M ) and multiple receiving antennas (N ). Each receiving antenna can receive signals from all the transmitting antennas, creating M × N signal transmission paths. MIMO system is not the first time when multiple antennas have been used in wireless communication. Early on, multiple antennas (e.g. antenna array) are used to enhance system signal-to-noise ratio (SNR) through beamforming by concentrating antenna radiation energy on a specific direction and beamwidth. Later on, multi-antennas are used to provide spatial diversity, countering the multipath fading effects. Basically, with multiple receiving antennas at certain distance in between, signals from different propagation paths can be picked up independently by the multiple antennas, and then added together constructively. Since the chance of all the signal paths suffering from strong multipath fading is much lower, the system bit error rate (BER) is significantly reduced in this way. However, the spatial diversity from multiple antennas has limited benefits. What makes multiple antenna system powerful and appealing is the extra spatial multiplexing it offers. With multiple independent signal propagation path between the multiple transmitting and multiple receiving antennas, we can transmit multiple independent data streams through
2
those independent paths. Take the example in Figure 1.1. If assuming no signal and noise correlation at both transmitter and receiver sides, the capacity of the MIMO system increases linearly with respect to the minimum number of transmitting/receiving antennas (min {M, N }) [11, 12], at no more cost on either power or spectrum resources. Due to this significant advantage in spectral efficiency over conventional single-antenna systems, MIMO has become an essential element in many wireless communication standards, such as 4G LTE, WIMAX and 5G.
1.2
Challenges in Compact MIMO System
Though it is mentioned in Section 1.1 that MIMO system promises a linear increase in spectral efficiency with respect to the minimum number of transmitting/receiving antennas, as we will see, many factors in physical implementation could undermine the system performance. This is especially true in compact devices such as mobile terminals, because of many constraints imposed on MIMO antenna design due to the limited physical space.
1.2.1
Antenna Physical Limits
One important design constraint comes from the physical limits on antenna bandwidth. As was shown in Chu’s pioneering work [20], an antenna’s Q factor increases dramatically as its electrical size reduces. Based on [21], an antenna’s matched fractional bandwidth is inversely proportional to its Q factor. Therefore, the smaller the electrical size of the antenna is, the higher its Q factor is and the narrower the antenna bandwidth is after matching. This is a significant constraint for antenna design in compact devices, which usually occupy a relatively small form factor. Aside from antenna bandwidth, there is a physical limit on antenna radiation efficiency as well. As is shown in [22, 23], the radiation efficiency of small antennas also decreases as their electrical size decreases, making designing small antennas difficult. It is worth mentioning that in [22], the author proved that there are a limited number of radiationefficient modes within a given antenna aperture, and any antenna design by removing certain antenna areas within the aperture will only reduce the radiation efficiency of all
3
the modes. This observation is instructional for small multimode antenna designs. Therefore, we conclude that we cannot design antennas smaller without sacrificing its performance, because of the physical limits on antenna bandwidth and efficiency. Unfortunately, a lot of product development is moving towards miniaturization, as manifested by the mobile terminals and small wearable devices. This requires us to make best use of the available space for antenna design.
1.2.2
Mutual Coupling
Another challenge in MIMO system implementation is the mutual coupling effect between multiple antennas, an effect that is well known in antenna array design [24]. When multiple antennas are closely spaced (such as in compact devices), the electromagnetic waves radiated from one antenna could induce current and voltages at its neighboring antennas. This mutual coupling effect will distort antenna radiation patterns, change the antenna impedance matching, and cause correlation in the transmitted or received signals, which directly corresponds to the reduction in MIMO system capacity [14]. This mutual coupling effect poses a significant challenge to MIMO system implementation in compact devices. On the antenna side, due to mutual coupling, all the antenna elements have to be considered together as one aperture, making MIMO antenna design a single multi-port antenna design problem. The traditional array design concept that a single antenna element can be designed and then assembled into an array topology, does not apply here for compact devices. On the system level, in order to combat the mutual coupling effect, a multiport decoupling/matching network is usually connected between the multiport antenna and the RF loads [25]. As will be explained in detail later on, the design of multi-port matching network is still an area under study, and not many design techniques are available. To facilitate the following discussion, Figure 1.2 offers a simple diagram of a compact RF MIMO receiver front end. The first block represents a MIMO antenna with multiple ports, receiving signals from free space, and the second block represents a multiport matching network that matches the coupled antenna system to the multiple RF loads (LNA, Mixer, etc.). In this dissertation, we will focus on MIMO antenna and multiport matching network design.
4
MIMO Antenna
Matching Network
Multiport Matching Network
ZA
RX
RX 1
RX N
Figure 1.2 A Compact MIMO Receiver Architecture.
1.3
MIMO Antenna
From what we have described above, it is obvious that MIMO antenna should have multiple ports, and it is desirable to have those ports being isolated and well matched (if no matching network is connected). Though MIMO antennas share many properties, its design could be drastically different if used in different applications. In applications where antenna is not constraint by physical aperture size and spacing, such as in base station, MIMO antenna design can be simplified to the design of antenna arrays, where antenna pattern and gain can be well controlled by proper design of the single element. This is the most simple MIMO antenna design method, but is not practical for many spacelimited applications. In order to reduce the occupied space, many researchers propose MIMO antennas with multiple closely spaced elements, and rely on the manipulation of antenna element and ground plane geometry (such as cutting slots or rotation) to reduce mutual coupling [26, 27, 28]. This type of design lacks a systematic design methodology, therefore does not provide much insight for other antenna designers. In this dissertation, we will focus on MIMO antenna design for compact devices, typically with a size smaller than λ/2 × λ/2. Due to the strong mutual coupling effects, all the antennas have to be considered altogether and designed as a single-aperture multi-port antenna, whether physically connected or not. Not many systematic design methods exist for such kind of compact MIMO antennas. In [29], a systematic design method for rectangular multimode
5
patch antennas is proposed based on cavity models, where analytical eigenfunctions exist and are employed for antenna response expansion and prediction. A three-port MIMO antenna is demonstrated therein. However, this approach can only be applied to antennas of canonical geometry, and it is also worth noting that the designed antenna aperture has the electrical size of 1.43λ × 1.43λ, which is large enough to apply the antenna array design method. Several interesting design has been reported based on phone chassis and excitation of the structural characteristic modes [30, 31, 32] using coupling elements. This idea of using the entire phone chassis for antenna design is optimal in terms of bandwidth. However, the way to shape the phone chassis or adding coupling elements is arbitrary, and relies on quite amount of cut-and-try before reaching an optimal design. In this dissertation, we will look at MIMO antenna design from a modal perspective, develop a systematic design methodology for compact MIMO antennas and demonstrate the design methodology with antenna prototypes.
1.4
Decoupling Network
When MIMO antenna design does not meet the requirement of isolation, or external environment contributes to mutual coupling, a multiport decoupling/matching network is usually employed to combat the mutual coupling effects and to provide a multiport matching between the antenna and the load, as shown in Figure 1.2. Though singleport matching has been extensively studied since Fano’s development of broadband matching limitations [33, 34] and many network synthesis techniques have been developed for matching network synthesis [35], not many theoretical work has been developed for multiport systems. In general, there is no available theory regarding the physical realizability of a multi-port network achieving a given broadband frequency behavior. Therefore, most of the available decoupling techniques in the literature deal with single frequency decoupling. Frequency independent decoupling has only been observed when the load has certain symmetry (e.g. circular symmetry) [36, 37], in which case the real and imaginary parts of the system impedance share the same frequency independent decoupling vector. In most of the literature, the reported decoupling techniques are either
6
for specific applications [38, 39], or with certain symmetry in network [37]. Very few works have been reported on general decoupling networks. An early one is developed by Geren [40], who ingeniously separates the decoupling of the imaginary and real parts of the load impedance into two steps. However, the proposed implementation using directional couplers is complicated. A most recent one is the work of Ding [41], who proposed a generalized π network as the decoupling network. However, there are an infinite number of implementations for the single frequency design, and it is still unclear which decoupling network offers broader bandwidth. In this dissertation, we will look at new decoupling network design techniques, define metrics for matching network characterization, and conduct MN optimization for broadband system performance.
1.5
Overview of the Dissertation
The organization of this dissertation is as follows. Chapter 2 gives a brief background review on characteristic mode theory, establishing a theoretical background and notation convention for later chapters. In Chapter 3, we present efficient broadband circuit models for wire and planar multiport antennas using characteristic mode theory. Besides the usefulness for MIMO antenna feed specification, the circuit models also demonstrate the possibility to model the antenna aperture behavior using only structural modes and without including physical feeds, making possible feed-independent antenna shape synthesis. Chapter 4 presents a feed-independent shape synthesis technique for MIMO antennas using characteristic mode theory and genetic algorithms, where physical limit on modal antenna Q factors is empirically observed. In Chapter 5, we develop a shape-first feed-next design methodology for MIMO antennas based on the research work in Chapter 3 and 4. Two MIMO antennas are fabricated and measured as the verification of the proposed design methodology. In Chapter 6, employing a new source formulation for antenna Q factor and stored energy which is applicable to antennas of arbitrary geometry, we present a new physical limit on the minimum Q factor of antennas of arbitrary geometry, which serves as a
7
tighter bound than Chu’s spherical Q limit. In Chapter 7, we present some new techniques for decoupling network design, and optimization of broadband multi-port matching networks. Chapter 8 summarizes the main contributions of the dissertation and suggests scopes for future work.
8
Chapter 2 Characteristic Mode Theory Characteristic mode theory (CMT) is a modal analysis technique for antennas of arbitrary shape, originally developed by Garbacz in 1968 [42] and then reformulated by Harrington in 1971 [43]. With characteristic mode analysis (CMA), antenna designers can obtain the eigen responses of an antenna, a powerful set of information that was previously only available to closed waveguides and resonant cavities. The access to these eigen responses could potentially offer new physical insights on antenna design and analysis. Especially in the most recent decade, CMT has drawn a lot of attention from antenna researchers and designers, and lots of interesting research work [44, 45, 46, 47, 48, 49, 50, 51, 31, 52] has been conducted based on CMT. All these research work on CMT not only demonstrated the power of CMT, but also significantly advanced its capability and applications. Because of the orthogonality between different characteristic modal currents and far fields as we will introduce, they are potentially useful for MIMO antenna design. If these different characteristic modes can be excited independently, each mode can be used as a MIMO channel, resulting into very low mutual coupling (signal correlation) between different ports (channels). This motivates our employment of CMT for MIMO antenna design. In order to set the background and notations for future chapters, we here give a brief review of CMT, its formulation, solution and interpretation of many modal parameters.
9
2.1
Numerical Modeling of Radiating Objects
The physics of all macroscopic electromagnetic problems are the same, namely Maxwell’s equations. However, different ways to formulate the problem allows us to solve the problem using different numerical techniques. Antenna problems are typically solved using magnetic vector potential and electric scalar potential equations, where the unknown parameters are the current and charge distributions on the antenna structure. Using the Lorenz gauge (∇ · A = −jωµΦ), the two potential equations can be written as ∇2 A + k 2 A = −µ0 J
∇2 Φ + k 2 Φ = −
ρ 0
(2.1)
(2.2)
where A is the magnetic vector potential, Φ is the electric scalar potential, and J and ρ are respectively the source current and charge densities. µ0 and 0 are the free space permeability and permittivity respectively. The solutions of the above two equations are well known as ZZZ ¯ (r, r0 ) · J(r0 )dV 0 A(r) = µ G (2.3) A V0
Φ(r) = −
1 jω
ZZZ V
0
∇0 · J(r0 )Gq (r, r0 )dV 0
(2.4)
In free space,
where
¯ (r, r0 ) = ¯IG (r, r0 ) G A 0
(2.5)
Gq (r, r0 ) = G0 (r, r0 )
(2.6)
0
e−jk|r−r | G0 (r, r ) = 4π|r − r0 | 0
(2.7)
In equation (2.4), the charge density is represented in terms of current density by invoking the current continuity equation. The scattered electric field from the antenna is related to those potentials as
10
Es (r) = −jωA(r) − ∇Φ(r)
2.1.1
(2.8)
PEC Objects
For a PEC radiating object, the boundary condition requires that total tangential electric field on the surface is zero, which means, n ˆ × (Es + Ei ) = 0
(2.9)
or using Harrington’s operator representation [43, 53], n ˆ × (L(J) − Ei ) = 0
(2.10)
L(J) = jωA(r) + ∇Φ(r)
(2.11)
where
2.1.2
Dielectric Objects
For a dielectric object, the total E field inside the object is proportional to the total polarization current. Therefore, we can write the equation as Es + Ei =
J(r) jω(ˆ(r) − 0 )
(2.12)
or in operator form L(J) = jωA(r) + ∇Φ(r) +
J(r) = Ei jω(ˆ(r) − 0 )
(2.13)
where ˆ(r) = (r) − jσ(r)/ω is the complex material permittivity including loss effect, and σ(r) is the conductivity of the dielectric material.
11
2.1.3
Method of Moment
Using method of moments (MoM) [53] and proper basis functions (RWG, SWG) [54, 55], the operator equations in (2.10) and (2.13) can be readily converted into matrix equations. For example, if using RWG basis function [54] and expanding the current density on PEC P structure as J = N n=1 In ψn and use ψm (m = 1, 2, ...N ) as test function (ψn and In are respectively the n-th RWG basis function and the expansion coefficient), then the moment method matrix equation for (2.10) can be expressed as: ZI = V
(2.14)
where Zmn = jωµ0
Z Z S
S
ψm (r) · ψn (r0 )G0 (r, r0 )dS 0 dS− j Z Z (∇ · ψm (r))(∇ · ψn (r0 ))G0 (r, r0 )dS 0 dS (2.15) ω0 S S
and Vm =
Z S
ψm (r) · Ei dS
(2.16)
The details on how to implement MoM for antenna modeling will not be covered here, and interested readers can refer to [53, 56] for further reading. In this dissertation, the MoM code we use and develop for PEC antennas is mainly based on that in [57]. SWG basis could be used to model dielectric objects and we have some development on that as well [58]. These MoM codes turn out to be very useful for customized CM analysis, antenna shape synthesis and circuit modeling, as will be obvious to readers later on.
2.2
Characteristic Mode Theory
Using the notation in Harrington’s paper [43], characteristic modes are the solution of the following generalized eigenvalue problem over an arbitrary antenna aperture XJn = λn RJn
12
(2.17)
where X and R are the imaginary and real parts of the MoM Z matrix of the antenna, and λn and Jn are the eigenvalue and eigencurrent of the n-th mode. This generalized eigenvalue equation can be readily solved using Matlab at any frequency to get a series of characteristic modes, but it takes some effort to correlate the eigen solutions at different frequencies, a problem called broadband mode tracking [59]. In this work, the CM solver in [60] is used for modal solution and mode tracking.
2.2.1
Orthogonality
As explained in [43], all the characteristic modes have orthogonal far field patterns and orthogonal current distributions on the antenna aperture. Mathematically, these orthogonality properties can be represented as: JH m RJn = δmn
(2.18)
JH m XJn = λn δmn
(2.19)
JH m ZJn = (1 + jλn )δmn
(2.20)
1 ZZ Em · E∗n dS = δmn η S∞
(2.21)
where δmn is equal to 1 when m = n, and 0 otherwise. As an example, Figure 2.1 shows the first four characteristic modes of a rectangular patch placed 5 mm above an infinite ground plane. From the definition of characteristic modes, these modes have orthogonal current distribution and orthogonal far field patterns. The orthogonal properties between different characteristic modes make CMT a useful tool for MIMO antenna design, as there is no coupling between different orthogonal modes. In future Chapters, we will explain how CMT can be used for MIMO antenna design.
13
Figure 2.1 Illustration of the first four characteristic modes (currents and far field patterns) on a rectangular patch.
2.2.2
Modal Expansion
The orthogonality property of characteristic modes makes them a useful set of basis for antenna response expansion. Given the characteristic modal responses, we can expand the antenna response for an arbitrary excitation in terms of the modal responses. For example, for a certain excitation field Ei , the resulted response of a given antenna can be expanded in terms of its characteristic modal responses as, J=
∞ X
αn Jn
(2.22)
αn En
(2.23)
n=1
E=
∞ X n=1
hEi ,Jn i where αn = 1+jλn is the weighted excitation coefficient of the n-th mode and represents RRR i how strongly a mode is excited, and hEi , Jn i = E · Jn dV is the symmetrical product of the excitation field and the modal current.
14
Besides the current and field pattern, many other secondary parameters, such as admittance Y, impedance Z, Q factor and radiation efficiency have also been studied in terms of characteristic modes, and found decomposable into these orthogonal CM modes in one way or another. For example, in [47], the characteristic modal Q factors are studied, and found useful for input port Q factor expansion. In [23, 61], characteristic modal efficiency is calculated by introducing loss to the characteristic modal solution of the PEC (perfect electric conductor) antennas, and total efficiency at the input port can also be expanded in terms of the modal efficiency. In [2], we have explained in detail the modal expansion of impedance Z for planar antennas. We here illustrate the modal expansion for admittance Y. As discussed in [62], if assuming a 1V gap voltage source at certain feed position (node i), the total admittance at the feed can be expressed as Yii =
X hEi , Jn i X Jn (i) Jtot (i) X = αn Jn (i) = Jn (i) = Jn (i) 1V n n 1 + jλn n 1 + jλn
(2.24)
Similarly, the mutual admittance between port j and excitation port i can be expressed as Yji =
X hEi , Jn i X Jn (i) Jtot (j) X αn Jn (j) = = Jn (j) = Jn (j) 1V n n 1 + jλn n 1 + jλn
(2.25)
As an example, Figure 2.2 shows how the input admittance of a center-fed 0.5 m dipole can be decomposed into a series of modal admittances. The green and red lines are the modal responses of the first and the third characteristic modes respectively, while the black dash lines represent the total input response. The 2nd mode is not excited by the center feed. These modal expansions of antenna input parameters provides great insight for our antenna circuit modeling work, which will be presented in detail in Chapter 3.
15
Gin (mS)
15 Re(Y 1 )
10
Re(Y 3 ) Re(Y in )
5 0 2
4
6
8
10 # 108
6
8
10 # 108
f 10
Im(Y )
B in (mS)
1
Im(Y3)
5
Im(Yin )
0 -5 -10 2
4
f
Figure 2.2 Input admittance of a center-fed 0.5m dipole by superposition of its modal responses (2nd mode not excited by the center feed).
2.2.3
Eigenvalues
Besides the modal current and modal field, there is valuable information we can extract from the eigenvalues as well. As shown in [43], the characteristic eigenvalue is related to the physical field as λn = ω
ZZZ
(µ|Hn |2 − |En |2 )dV
(2.26)
Therefore, depending on the sign of λn , we can classify mode n as either capacitive (λn negative) or inductive (λn positive). When λn = 0, the total reactive energy cancels each other, and we say that the mode is self-resonant. The frequency at which λn = 0 is denoted as the self-resonant frequency of that mode. Note that, we can access the resonant frequency of each characteristic mode without even exciting the antenna with physical feeds, and it is also independent of antenna feed positions in general. This is an
16
important observation for feed-independent antenna shape synthesis, as will be discussed in Chapter 4. Another important parameter that we can derive from the eigenvalues is modal significance, which is defined as M Sn =
1 |1 + jλn |
(2.27)
Modal significance shows clearly how each mode is close to resonance at each frequency. It achieves the maximum value of 1 at its resonant frequency (when λn = 0) and falls off when eigenvalues are large.
2.2.4
Characteristic Modal Q Factor
One of the important parameters that we will use extensively in MIMO antenna design is the characteristic modal Q factor, and we here provide the details on its definition and calculation. The calculation of antenna Q factor is not an easy task, because of the difficulty in calculating stored energy in the antenna system, and we will review the Q factor problem in detail in Chapter 6. Here we will just highlight the recent source formulation [63] that we employ for characteristic modal Q factor calculation. As explained in detail in Appendix A, based on the source formulation [63, 47], antenna tuned Q factor can be calculated as Q(I) =
2ω max{W e , W m } max{IH Xe I, IH Xm I} = Prad IH RI
(2.28)
The calculation of the radiation power (Prad ) and the stored electric and magnetic energy (W e ,W m ) are simplified as quadratic operations on antenna current as e Wvac =
1 H I Xe I 4ω
(2.29)
m = Wvac
1 H I Xm I 4ω
(2.30)
17
Figure 2.3 Characteristic modal Q factors of a rectangular plate (120 mm × 76 mm).
1 (2.31) Prad = IH RI 2 where Xe and Xm are respectively the operators for the stored electric and magnetic energy, and their definition can be found in Appendix A. R is the real part of the MoM Z matrix. After solving the CMT eigenvalue equation (2.17), we have access to all the characteristic modal currents. By applying the modal current into (2.28), we can easily calculate the Q factor for each characteristic mode. As an example, Figure 2.3 shows the characteristic modal Q factors of a rectangular plate with a dimension of 120 mm × 76 mm. Since the antenna matched fractional bandwidth is inversely proportional to the antenna Q factor [21, Eq. 87], these modal Q factors tell us the achievable bandwidth each mode can offer if individually excited and matched. As can be observed from Figure 2.3, there are a limited number of broadband modes in this aperture. For example, at 500 MHz, only 3 modes have Q factors less than 200, implying that it is impossible to extract a 4th mode with a reasonable bandwidth. In future Chapters, we will need to calculate characteristic modal Q factors for MIMO antenna circuit modeling and antenna shape synthesis.
18
Chapter 3 Broadband Circuit Models for MIMO Antennas As introduced in Chapter 2, characteristic modes serves as a complete basis for current, field and admittance expansion. This not only offers great physical insight in terms of antenna operation, but also could be used for antenna circuit modeling. In this Chapter, we propose efficient analytical circuit models for MIMO antennas, that could not only be useful for MIMO antenna feed specification, but also for other purposes, such as antenna time-domain analysis, broadband matching/decoupling network development, and re-configurable antenna design. In order for the reader to appreciate the proposed MIMO antenna circuit model, I will first point out the inefficiency of conventional circuit modeling techniques. The conventional brute force method for circuit modeling is based on observation of antenna response and approximating the system as parallel resonances in series [64], series resonances in parallel [46], or even arbitrary lumped network blocks with enough degree of freedom through rational function approximation [65]. The direct application of these methods to the modeling of MIMO antenna encounters significant challenges because of the increasing 2 degrees of freedom ( N 2+N entries in the impedance matrix for an N port antenna). Though several researchers have explored the multi-port system modeling using vector fitting [66] and other rational interpolation techniques [67], no attempts have been made on synthesizing a matrix of rational functions using simple circuit elements. Moreover, these
19
brute-force methods do not provide any physical insight on antenna operation and can only generate analytical models when the system input responses are known a priori. In this Chapter, based on CMT, we propose a physics-based technique for multi-port antenna modeling. This method is straightforward and general because it models the antenna’s eigen response instead of port responses, while specific port responses can be expanded in terms of these eigen responses. Therefore, with this technique, we do not generate a specific circuit model for a specific number of ports at fixed positions, but generate a general circuit model for an arbitrary number of ports at arbitrary positions. In the following several sections, circuit modeling techniques for wire antenna and planar antennas will be explained in details respectively, demonstrating the simple mathematical formulation, physical intuition, and high accuracy this modal approach provides.
3.1
Circuit Model for Arbitrary Wire MIMO Antennas
The first type of antenna we looked at is wire antennas, though this specific circuit model could also be applied to antennas with wire feeds or even strip feeds. As opposed to the planar antenna circuit model to be introduced in Section 3.2, the circuit model in this section applies to antennas where feed is implicitly modeled as part of the antenna geometry, and the current at the feeds can be accessed from characteristic modal analysis.
3.1.1
Formulation of Multi-port Admittances Using CMT
Referring back to Chapter 2, the input self and mutual admittance of an antenna with delta gap voltage source can be expressed in terms of characteristic modes as Yii = and Yji =
M X
Jm (i) Jm (i) m=1 1 + jλm M X
Jm (i) Jm (j) m=1 1 + jλm
20
(3.1)
(3.2)
where λm and Jm are the eigenvalue and eigencurrent of the m-th mode. If we choose an arbitrary node p as reference port, and denote the modal admittance at the reference port as J2 (p) Ym = m (3.3) 1 + jλm then the self and mutual admittance in Equation 3.1 and 3.2 can be written as Yii = and Yji = Denoting αim = expression as
Jm (i) Jm (p)
M X
J2m (i) Ym 2 m=1 Jm (p)
(3.4)
M X
Jm (i)Jm (j) Ym J2m (p) m=1
(3.5)
(J is real for characteristic modes), we can simplify the above
Yii =
M X
2 αim Ym
(3.6)
αim αjm Ym
(3.7)
m=1
and Yji =
M X m=1
For an N -port antenna system, the input admittance matrix can be written as Y = AYm AT where
α 11 α21 A= α 31
αN 1
α12 α13 . . . α22 α23 . . . α32 α33 . . . ............. αN 2 αN 3 . . .
21
(3.8)
α1M α2M α3M αN M
(3.9)
and
Y 0 0 1 0 Y2 0 Y m = 0 0 Y3 .. .. .. . . . 0 0 0
... ... ... .. .
0 0 0 .. .
. . . YM
(3.10)
It is also worth noting that the reference ports for all the modes do not need to be the same, in which case, both the modal admittance and the corresponding column in Equation (3.9) need to be defined with respect to its corresponding reference port.
3.1.2
Proposed Multi-port Circuit Model
The mathematical representation in (3.8) might be a little bit abstract, but the physical meaning it represents is very neat once understood. Namely, any port response, self or mutual, can be decomposed into some weighted sum of the modal response (Ym ). After some investigation, it is found that Equation (3.8) can be physically represented by a generalized multi-port transformer network loaded with the modal admittances. Interested readers can refer to Appendix B for detailed mathematical derivations. Based on this observation, we here propose the circuit model for (3.8) as the topology shown in Figure 3.1. The transformation ratio between port i and mode m is 1 : αim , where αim is defined as Jm (i) . (3.11) αim = Jm (p) Because characteristic mode is relatively frequency independent (current distribution does not change very much vs frequency), we can assume the transformation ratios as constants (usually the value at the modal resonant frequency is chosen). Therefore, the transformation ratios are straightforward to calculate, once the reference port and the input ports are specified. For wire-feed antennas with 1D triangle basis, Jm (i) is the same as the current coefficient Im (i) obtained from solving MoM matrix equation, but for strip-feed antennas, namely antennas with a gap voltage source at the edge of an RWG basis function, Jm (i) = Im (i)l(i), where l(i) is the edge length of the i-th RWG basis.
22
1:αN1
1:αN2
1:αNM
1:α21
1:α22
1:α2M
1:α11
1:α12
1:α1M
Port N
Port 2
Port 1
6E 6
Y1
Y2
YM
Fig.Figure 1. A3.1 general broadband circuit model for a MIMO Equivalent circuit model for wire MIMO antennas [1]. antenna. The externally accessible ports are shown at left while circuits representing the frequency response and Each modal admittance a lumped coupling Ybetween ports areas shown atresonant right. circuit. We could m can be modeled
1E 1
1E 1
use series resonant circuit, high-pass circuit or even higher order circuits if needed. We here only consider 2nd order systems as theythe demonstrate in all the antenna impedance is desired, can enough be fit accuracy to simple models. Thesystem, modalthree admittances typically cases template we studied.circuit To model a 2nd order parameters are are needed: R, L and well-modeled by simple pass [3], parameters though others C. From the modal admittance Ym ,high we can alsocircuits extract three out: modal Fig. 2. Th couldfrequency be usedfres depending on theresistance desired level of accuracy. In factor this Qres . strip dipole resonance , modal radiation Rrad and modal quality case, onthe all mutual impedances are fully Depending the expressions circuit topology,for different RLC values can be derived. analytical, theminpotentially useful for system For series RLCmaking circuit shown Figure 3.2 (a), the relation betweenmodeling the values are:
and calculation of matching and capacity bounds.
rad An additional advantageRof= R this physics-based approach is (3.12) We have accurate ana that the accuracy and applicable bandwidth of the resulting method is sca model are scalable. If only a narrowband model is required, one and gives ph or two blocks representing significant characteristic modes couple thro may suffice. To extend the bandwidth, more modes can be 23 analytical mo added in a systematic way at frequencies where their decoupling n contribution to the overall admittance becomes significant. calculations Similar to [3], the effects of many higher order modes can be Additional e lumped into a single shunt capacitance to ground.
L
C
C L
R
(a)
R
(b)
Figure 3.2 Modal circuit templates: (a) series RLC circuit, and (b) high-pass circuit.
Qres Rrad ωres
(3.13)
1 ωres Qres Rrad
(3.14)
L= C=
For high-pass circuit shown in Figure 3.2 (b), the relation between the values are: q 1 R = (2 + Qres (Qres + 4 + Q2res )Rrad ) 2 r
L=
Rrad 3 + Q2res +
q
1
√
2ωres
r
2/(Qres (Qres + C=
q
4 + Q2res + Qres 4 + Q2res
Qres
(3.15)
(3.16)
q
Rrad ωres
4 + Q2res ))
(3.17)
Both series RLC and high-pass circuit will be used in following examples. As was shown in [46], high-pass circuit tend to offer better performance over a broadband frequency range than series RLC circuit. Here are the steps to follow to get the circuit model: • First, conduct broadband characteristic mode analysis by solving the eigenvalue equation; • Second, model each modal admittance at the reference port as resonant circuits, by first calculating Rrad , Qres and ωres , and then converting them to RLC values based on (3.12)-(3.17) depending on the circuit template we use;
24
• Third, calculate the transformation ratios between the observation and reference ports using (3.11); • At last, refer to equation (3.8) for analytical expression of input port parameters and figure 3.1 for circuit configuration. Theoretically, we can include as many modes as possible, but in many simple small antennas, only the first few modes are needed to model the antenna response, while the effect of the remaining higher order modes can be lumped together as a capacitance in parallel as they all contribute a small amount of capacitive reactance. The self and mutual admittance can therefore be simplified as Yii =
K X
2 αim Ym + jωCHOM
(3.18)
m=1
and Yji =
K X
αim αjm Ym
(3.19)
m=1
where K is the number of modes used to model the antenna (K M ), and CHOM is the contribution from the remaining higher order modes. In this simple treatment, we assumed that there is negligible amount of coupling between the ports contributed by the higher order modes.
3.1.3
Demonstration Examples
To demonstrate the validity of the proposed circuit model in Figure 3.1, we here apply it to an example problem: a two-port dipole. Two-port systems are considered here for simplicity, though an arbitrary number of ports can be included. 3.1.3.1
Two-Port Dipole
In this example, a two port strip dipole with asymmetric feeds will be considered. The strip dipole is 0.5m long and 2mm wide, with one port at the center, and the other port at one-quarter length from the end, as is shown in Figure 3.3.
25
L/4 Port 2 Port 1
Figure 3.3 Configuration of the two-port dipole.
For small antennas, only a finite number of modes will be needed in order to model the system. In this case, we only modeled three modes, and the higher order modes all add a small amount of capacitive reactance, and are thus lumped as a capacitor (neglecting the mutual terms contributed by the higher order modes). Following the analysis in Section 3.1.2, the radiation resistance of the three modes are calculated from (3.3). The resonant frequency can be obtained from the CMA (frequency at which λn = 0). The Q factor of each mode at its resonant frequency can be calculated based on the source formulation in (A.10). The modal parameters of the first three modes are calculated and listed in Table 3.1, where mode 1 and 3 use the port at the center as the reference port, while mode 2 uses the port at quarter dipole length as the reference port (considering the fact that the 2nd modal current has a null at the center, might be sensitive to numerical errors). The corresponding RLC circuit elements of the three modes are then calculated from the modal parameters based on Equation (3.12)-(3.17), and are given in Table 3.2, 3.3 and 3.4 respectively. The corresponding transformation coefficients between the observation ports and the reference port are
α11 α12 α13 1 1.4195e − 11 1 A= = α21 α22 α23 0.7353 1 0.6839
26
(3.20)
Table 3.1 Modal parameters of the first three characteristic modes of the 0.5m strip dipole
mode
fres (MHz)
Qres
Rrad (Ω)
1
283.58
6.524
70.96 (center)
2
580.42
9.284
91.76 (quarter)
3
877.5
11.51
108.8 (center)
Table 3.2 Circuit Elements of mode 1
mode 1
Series RLC
High Pass
R (Ω)
70.922
3090.1
L (nH)
259.79
268.8
C (pF)
1.2131
1.1987
Table 3.3 Circuit Elements of mode 2
mode 2
Series RLC
High Pass
R (Ω)
91.6590
8000.8
L (nH)
228.14
237.63
C (pF)
0.3344
0.32005
Table 3.4 Circuit Elements of mode 3
mode 3
Series RLC
High Pass
R (Ω)
109.661
14523
L (nH)
226.78
229.69
C (pF)
0.1446
0.14429
The effect of higher order modes are lumped into a capacitor (0.24 pF in this case) connected in parallel at the two input ports respectively. With the information of the modal circuit elements, the transformation ratios and the higher order mode capacitance CHOM , the input port parameters can be obtained from (3.8) (or specifically, (3.18) and (3.19) with higher orde modes truncation).
27
S
S
Q Q
S
(a)
(b)
Figure 3.4 Comparison of the results from our circuit model (solid lines) and full wave FEKO simulation (dashed lines): (a) S parameters from high-pass circuit model, and (b) Q factors of the self impedances.
The circuit model is then constructed and simulated in Keysight ADS. Figure 3.4 (a) shows the two-port S parameters from our circuit model (solid lines), compared with the result from full wave simulation in FEKO (dash lines), and Figure 3.4 (b) shows the Q factors calculated from the self impedances. Very good agreement is observed between our model and full wave simulation. Figure 3.5 and 3.6 shows broadband Y and Z parameters from our circuit model (solid lines), compared with the full wave simulation results (dash lines). Very good agreement between our circuit model and the full wave simulation result is observed as well. In order for the reader to appreciate the accuracy of the high-pass circuit model, Figure 3.7 shows the S parameter and Q factor result from the circuit model based on series RLC circuit templates. Though S parameters do not show much difference from the high-pass circuit model in Figure 3.4, the Q factors deviate significantly from the full wave simulation, demonstrating that the high-pass circuit template models the broadband antenna response more accurately, a phenomenon noted in [46, 20]. We attribute the deviation between our circuit model and the full wave simulation results to several factors:
28
0.015 0.010 0.005
Re
Dash line: full wave simulation Solid line: model Re
0.000 Im
-0.005
Im
-0.010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 freq, GHz
(a)
(b)
0.015 0.010 0.005
Dash line: full wave simulation Solid line: model
Re
Re
0.000 -0.005
Im
Im
-0.010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 freq, GHz
(c)
(d)
Figure 3.5 Y parameters of high-pass circuit model (solid lines), in comparison with the results from full wave simulation (dashed lines).
29
Re
Im Im
Re
(a)
(b)
1000 800 600 400
Re
Dash line: full wave simulation Solid line: model Im Im
200 Re
0
-200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 freq, GHz
(c)
(d)
Figure 3.6 Z parameters of high-pass circuit model (solid lines), in comparison with the results from full wave simulation (dashed lines).
30
0 -5
S
S
-10
Q
-15
Dash line: full wave simulation Solid line: model
-25 -30 0.1
Q
S
-20
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
freq, GHz
(a)
(b)
Figure 3.7 Comparison of the results from our circuit model (solid lines) and full wave FEKO simulation (dashed lines): (a) S parameters from series RLC circuit model; (b) Q factors of the self impedances.
• The modal admittance Ym is modeled using 2nd order RLC circuit templates (could be improved if higher order high-pass circuits are used); • The transformation coefficients are assumed constant for simplicity (though not strictly necessary); • The coupling between ports contributed by the higher order modes are neglected, and only their contribution to the self terms are considered as lumped CHOM . Overall, the proposed circuit model using high-pass circuits and transformer networks provides a simple and physically intuitive circuit model for wire and wire-fed MIMO antennas with arbitrary geometry. To the best of the author’s knowledge, this is the first time a general multiport antenna can be systematically modeled using simple circuits. Though rational fitting techniques for multiport systems [66, 67] exist, it remains difficult to synthesize a matrix of rational functions. Moreover, our antenna circuit model based on the modal approach allows quick access to input parameters for arbitrary feed combinations (by simply recalculating the transformation ratios).
31
3.2
Circuit Model for Probe-fed Planar MIMO Antennas
In Section 3.1.2, we have proposed a circuit model for arbitrary antennas with wire structure or wire probe feeds. Therefore, we can model a planar antenna with probe feeds using the circuit model in Figure 3.1. However, as will be explained in one simple example, this is not an efficient way to do that. Consider the example of a probe-fed rectangular patch over an air substrate (12 cm × 7.6 cm, h = 5 mm). To obtain the input parameters (S, Z, or Y parameters), the patch is typically excited with a delta-gap source placed on a wire probe between the ground plane and patch radiator. Thus, the physical probe must be included in the 3D model when computing the impedance matrix and the characteristic modes. However, as evident from Figure 3.8, the modal significance (and hence the eigenvalues) of the patch change when the feed is placed in different positions. The eigencurrents will also vary with feed position. In other words, if a physical feed probe is placed into the model, the set of basis functions generated will differ for each feed position. A new eigenvalue problem must then be solved for each position, making for an extremely inefficient process that is little more useful than a conventional port-based simulation. Instead, we propose a modeling approach in which a single eigendecomposition of the feed-free planar radiator can be used to compute the self or mutual input parameters for any number of probes placed at any position on the structure. The central concept is the introduction of virtual feeds between the radiator and ground plane as illustrated in Figure 3.9. These feeds are not placed in the physical 3D model and thus do not have associated rows and columns in the MoM impedance matrix. Consequently, the eigenvalues and eigenvectors are independent of the location of the feed probes. This approach can be viewed as an extension of the approximate analytical eigen-expansions found in classical theory [68, 69, 70]. However, these are restricted to simple patch geometries (rectangular, circular, etc.), whereas our method could be applied to arbitrary shapes by employing the exact numerical eigenmodes computed from characteristic mode theory.
32
1
12 cm
r2
y
7.6 cm
0.8
0
x
r1
MS
Solid lines: feed at r
0.6 Dash lines: feed at r 1 2 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6
0.4 0.2 0 0
0.5
1
1.5 f (GHz)
2
2.5
3
Figure 3.8 The variation of the eigenvalue spectrum of a rectangular patch antenna (12 cm × 7.6 cm, h = 5 mm) at two different feed positions: r 1 = (3 cm, 0, z) for solid lines, and r 1 = (4 cm, 2 cm, z) for dashed lines [2]. 𝑧 𝑦 𝑥
𝒓𝑗
𝒓𝑖 𝒓0
ℎ
𝜖𝑟
𝜇𝑟
Figure 3.9 An illustration of the virtual feed concept for an arbitrary planar antenna. Virtual feeds are introduced at r i and r j with a reference port located at r 0 [2].
3.2.1
Formulation of Multi-port Impedances Using CMT
Consider the example planar antenna over a ground plane shown in Figure 3.9. The antenna consists of a planar radiator of arbitrary shape over a substrate with height h, backed by an infinite ground plane. No physical feed probes are included in the model, but virtual feeds at r i (xi , yi , z) and r j (xj , yj , z) represent the two potential feed positions. A reference feed that will be used to simplify our model is located at r 0 (x0 , y0 , z). In this case, the self and mutual port impedances between the two potential feeds can be written
33
as [71] hi, ii − Zii = − 2 = Ii
RRR
hi, ji − Zji = − = Ii Ij
RRR
E ii · J i dv Ii2
(3.21)
E ji · J j dv Ii Ij
(3.22)
where E ii and E ji are the total fields at position r i and r j when the excitation probe is RR at r i , and J i is the excitation current density at r i , and Ii = J i · zˆ dS. Based on characteristic mode theory (CMT) [43], the total field at position r i and r j can be expanded in terms of the characteristic eigen-field as E ii =
X
αm E m (r i ) =
m
E ji =
X
X m
αm E m (r j ) =
X m
m
hE m , J i i E m (r i ) 1 + jλm
(3.23)
hE m , J i i E m (r j ) 1 + jλm
(3.24)
m ,J i i where E m and λm are the eigen-field and eigenvalue of the mth mode, αm = hE is 1+jλm RRR the expansion coefficient, and hE m , J i i = E m · J i dv. The substrate height h is often small in terms of wavelength. Therefore, to compute the RRR inner product of the mth mode with the ith excitation current, hE m , J i i = E m · J i dv, we assume the modal E field is uniform in the vertical direction. The feed probe at position r i is modeled as a uniform current filament,
J i = zˆδ(x − xi , y − yi )
0 ≤ z ≤ h.
(3.25)
Applying these simplifications into (3.21)-(3.24) and substituting (3.23)-(3.24) into (3.21)-(3.22), the self and mutual impedance can be simplified as Zii = −
X m
Zji = −
X m
z (r i )h z Em E (r i )h 1 + jλm m
(3.26)
z Em (r i )h z E (r j )h. 1 + jλm m
(3.27)
34
If we write the modal impedance at a reference point r 0 as Zm = −
z (Em (r 0 )h)2 , 1 + jλm
(3.28)
the self and mutual impedance can be written with reference to Zm as Zii =
2 Zm γi,m
(3.29)
γj,m γi,m Zm
(3.30)
z (r i ) Em . z Em (r 0 )
(3.31)
X m
Zji =
X m
where γi,m =
For an N -port antenna system, the input impedance matrix can be written as Z = ΓZm ΓT where
γ 11 γ21 Γ= γ 31
γN 1
and
γ12 γ13 . . . γ22 γ23 . . . γ32 γ33 . . . ............ γN 2 γN 3 . . .
Z 0 0 1 0 Z2 0 0 0 Z3 Zm = .. .. .. . . . 0 0 0
... ... ... .. .
(3.32)
γ1M γ2M γ3M γN M 0 0 0 .. .
. . . ZM
(3.33)
(3.34)
Therefore, it is obvious from (3.32) that the self and mutual impedance terms in (3.29) and (3.30) are simply weighted sums of the modal impedance at the reference point in (3.28). Since all of these quantities in (3.33) and (7.14) depend only on the eigenvalues
35
and eigen-fields generated by the eigen analysis of the feed-free structure, we need only one eigendecomposition to compute the self and mutual impedances of any number of ports located at any position.
3.2.2
Proposed Multi-port Circuit Model
In principle, the modal and port impedances in (3.28)-(3.30) could be evaluated numerically. However, our observations indicate that using direct numerical values for the eigen-fields generates non-physical responses at low frequencies. To resolve this problem, we develop an accurate analytical representation of the impedances via a circuit model, as we have done in Section 3.1.2. The self impedance in (3.29) is a number of modal reference impedances connected in 2 series and scaled by a factor γi,m . From (3.31), the factor γi,m is a ratio of the eigen-field at the port location relative to the field at the reference point. Though each eigen-field is a frequency-dependent complex value, they are dominated by a large imaginary part and vary slowly over a wide frequency range around their resonance. Therefore, we approximate the ratio γi,m as a constant real number. Though this is not strictly necessary, it allows us to conceive of the sum in (3.29) as many modal impedances connected in series through different transformers with turns ratios γi,m . Two types of modes are observed for the feed-free patches. One type includes nonresonant loop modes (λm > 0 for all frequencies), which are not strongly excited by the vertical current filament, and the effect of which are neglected here. The other type includes those that are strongly excited by the current filament, and here are denoted as resonant modes (λm = 0 at resonance frequencies). All the resonant modes excited by the current filament have negative λm below resonance frequency and positive λm above resonance frequency. As a result, the modal impedance in (3.28) will be inductive below resonance and capacitive above resonance, which corresponds to the behavior of a parallel resonance, i.e., 1 Zm ≈ 1 (3.35) 1 + jωCm + jωL Rm m When we compute the self impedance in (3.29), we find that a single-feed microstrip patch antenna looks like a short circuit at DC. This is evidently a non-physical result
36
Port N
Port 2
Port 1
M1N M2N
γN,1:1
γ2,1:1
Lprobe1
Lprobe2
M12
LprobeN
Cdc
γ1,1:1
Z1
γN,2:1
γ2,2:1
γ1,2:1
Z2
γN,M:1
γ2,M:1
γ1,M:1
ZM
Figure 3.10 The equivalent circuit model for planar MIMO antennas [2].
since it is well-known that the probe-fed patch should behave like an open circuit at low frequencies. We remedy this problem by adding the DC capacitance of the antenna aperture to our model. Though this does not derive directly from the modal theory, a similar approach is taken in other studies [72, 73] when feed probe is absent in the model. Furthermore, to account for the inductances contributed by the physical feed, the self and mutual inductances of the probes are computed analytically. Taking into account of the DC capacitance, the probe inductance and contribution
37
from the resonant modes, the circuit level representation of the self and mutual impedances can be written as X 1 2 + jωLpi + γi,m Zm jωCdc m
(3.36)
X 1 + jωMji + γj,m γi,m Zm jωCdc m
(3.37)
Zii = Zji =
where the Zm are modeled as a parallel resonances from (3.35). Then based on the expressions in (3.36)-(3.37), a general circuit model for planar antennas is shown in Figure 3.10. Interestingly, it is similar to the power plane circuit model in [72], but takes into account the radiation effect at the open boundaries of the antenna. Each modal impedance Zm is modeled as a parallel resonant circuit, and is coupled to each port i through a transformer with the transformation ratio of γi,m . Negative transformation ratios are allowable and result in merely a change of transformer polarity.
3.2.3
Calculation of the Circuit Elements
The process to calculate the circuit parameters in Figure 3.10 is illustrated in Figure 3.11. It is critical to note that in Figure 3.11 the DC parameters and Steps 1-4 in the right-hand process flow need to be completed only once for a fixed antenna aperture, and are independent of the location and number of feeds. In fact, only the transformer ratios in the circuit model are a function of feed position. Thus, once the eigendecomposition is completed and the other elements are calculated, a change in location of the virtual feeds is simply a change in the transformer ratios which can be computed with little computational effort. In this way, the input parameters everywhere on the aperture can be quickly computed with a high degree of accuracy, as we will demonstrate in the following sections. The modal circuit parameters (Rm , Lm , Cm and γi,m ) are obtained from the CM analysis of the feed-free structure, while the DC parameters (Cdc , Lpi and Mij ) are determined separately.
38
DC parameters (Cdc, Lprobe, Mij)
Step 1:
CM analysis (λm, Jm)
Step 2:
Select the dominant modes
Step 3:
Modal parameters (fres (λm=0)) (Q factor at fres) (E field at r0 at fres)
Step 4:
Modal circuit parameters (Rm, Lm, Cm)
Step 5:
Port related parameters (E field at ri at fres) (transformation ratio γi,m)
Repeat step 5 for a new feed position, or an additional port
Complete circuit model
Figure 3.11 The process to calculate the parameters in the circuit model in Figure 3.10 [2].
3.2.3.1
Determination of DC Parameters (Cdc , Lpi and Mij )
The DC capacitance of the planar antenna is calculated in Matlab using MoM [53] based on the same triangular mesh as used for the high frequency simulation. Rather than Gaussian Quadrature for the numerical integration, a 9-element sub-division [57] is used for simplicity. Table 3.5 verifies our calculated DC capacitances with results calculated from other sources [74] using adaptive meshing. Less than ≤ 3% error is observed. The self and mutual inductances of the probes are calculated analytically based on the formulation in [75] as
Lself
v u
µ0 l u l = l ln + t1 + 2π ρ ρ
39
!2
−
s
2
ρ 1+ l
ρ 1 + + l 4
(3.38)
M=
µ0 l l ln + 2π D
v u u t
1+
l D
!2
−
s
1+
D l
2
+
D l
(3.39)
where l is the length, ρ is the radius of the probe and D is the distance between two wires. The mutual inductance is usually negligibly small when D l. 3.2.3.2
Determination of Zm (Rm , Lm , Cm ) and γi,m
The determination of the modal circuit elements relies on the CM analysis of the feed-free antenna structure, as illustrated in Figure 3.11 in step 1-5. In step 1, the eigenvalues and eigenvectors of the feed-free antenna structure are first calculated using a Matlab-based MoM [57] and CM solver [46]. In step 2, the dominant modes within the band of interest are selected for circuit modeling by choosing only modes with a large modal significance. In step 3, the modal parameters (fres ,Qrad ,E m ) are obtained through postprocessing. The resonant frequency is first determined directly from the eigenvalue spectrum (the resonance occurs when λm = 0). Then, the modal Q factors at the resonant frequencies are computed from the eigen current distribution on the antenna [63] (or can be estimated as Qrad = f2 dλ [76]). At last, the modal E fields at the reference point r 0 at the resonant df frequencies are calculated from the eigen current distribution. In step 4, the modal circuit parameters (Rm , Lm , Cm ) are calculated from three parameters: modal resonance frequency fres , Q factor at resonance frequency Qrad and radiation resistance Rrad . The radiation resistance is calculated as, z Rrad = Re{Zm }|f =fres = Re{−(Em (r 0 )h)2 }
(3.40)
Table 3.5 The DC Capacitance Calculated Based on MoM.
Shape
Cdc (MoM)
Cdc (from [74])
Parallel plate(1m×1m, h=0.1m)
119.03 pF
115.50 pF
Circular disk (radius=1m)
70.92 pF
70.81 pF
40
The modal circuit parameters in (3.35) can then be calculated based on these three parameters as Rm = Rrad (3.41) Cm =
Qrad 2πfres Rrad
(3.42)
Lm =
Rrad 2πfres Qrad
(3.43)
In step 5, the modal E fields at the resonant frequencies at each port position are calculated similarly from the eigen current distribution, and then the transformation ratio γi,m are obtained based on (3.31). At last, by substituting the DC parameters, the modal circuit parameters and the transformation ratios into (3.36)-(3.37), the circuit model is completed. The benefit of having a reference point r 0 is that the modal impedances Zm only need to be determined once, while all the other feed positions are coupled to the reference modal impedance through a transformer with the transformation ratio γi,m , defined in (3.31). This approach is very efficient in that only step 5 in Figure 3.11 needs to be repeated for a new feed position or for an added port. The reference point is arbitrary as long as it is not at the exact null of the E field of the desired modes. 3.2.3.3
Truncation of Higher Order Modes
In principle, all the modes can be included in the circuit model. However, for simplicity in real applications, only several dominant modes need to be considered, while the effect of all the higher order modes can be modeled as a lumped inductance, LHOM , which is directly in series with the self impedance, and resulting to a modified form for (3.36) and (3.37) as M X 1 2 Zii ≈ + jωLpi + γi,m Zm + jωLHOM (3.44) jωCdc m=1 Zji ≈
M X 1 + jωMji + γj,m γi,m Zm . jωCdc m=1
(3.45)
The higher order mode inductance, LHOM , can be obtained by subtracting the reactance at the reference port based on circuit model from the reactance at the reference port
41
based on port simulation at a single frequency. Though the effect of higher order modes is slightly different at different feed positions, we assume it is the same for all the feed positions and that inter-port coupling due to the higher order modes is negligible. These assumptions have negligible effect on the accuracy in the cases we have examined.
3.2.4
Demonstration Examples
To validate the proposed circuit model, several examples are given in this section. Air dielectrics are considered in all the models for simplicity (the case for microstrip substrates is considered in Chapter 5). All the metals are assumed PEC (perfect electric conductor) with zero thickness. All the circuit models are simulated in Keysight ADS (Advanced Design System) according to the schematic in Figure 3.10, while the port simulations (with a physical feed probe in the model) are used as benchmarks and conducted in FEKO. 3.2.4.1
A Slotted Rectangular Patch
In order to demonstrate the general validity of the proposed circuit model, we select an example that does not have analytical solutions. Consider the slotted rectangular patch (10.5 cm × 6.5 cm, h = 5 mm) shown in Figure 3.12. Though this is a common type of design, the slot is cut arbitrarily and without any design intent.
Figure 3.12 The geometry of the slotted patch [2].
42
Table 3.6 The Circuit Parameters Calculated for the Slotted Patch With The Reference Point r 0 = (3 cm, −2 cm, z).
Mode NO. m 1 2 3 4 5
Rm (Ω) 3288.3 1.505 193.20 55.483 7.2365
Lm (nH) 1.7033 0.0047 0.7229 0.1538 0.0228
Cm (pF) 20.271 2110.2 8.2637 26.659 135.4
Cdc (pF)
Lprobe (nH)
LHOM (nH)
16.424
2.236
1.254
Table 3.7 The Transformation Ratios of The Four Feed Schemes for the Slotted Patch With The Reference Point r 0 = (3 cm, −2 cm, z).
Mode NO. m 1 2 3 4 5
Feed scheme 1 γ1,m γ2,m 1.4764 1.1701 -9.7558 -1.3589 -0.8260 1.1557 -0.4877 1.9267 0.2973 5.0298
Feed scheme 2 γ1,m γ2,m 1.4764 0.2414 -9.7558 10.8259 -0.8260 0.7701 -0.4877 -0.9159 0.2973 -4.3617
Feed scheme 3 γ1,m γ2,m 0.9902 -1.2451 1.7462 -0.4755 1.1659 -0.7669 1.1794 3.1444 1.3473 -2.6778
Feed scheme 4 γ1,m γ2,m 1.0985 -1.5177 -0.9216 -23.948 0.3918 0.639 0.1536 -1.7966 -1.3209 1.2876
In order to further demonstrate the generality of the circuit model, four arbitrary feed schemes are chosen: feed scheme 1 (r 1 (4 cm, 1.5 cm, z), r 2 (4.5 cm, −2.5 cm, z)), feed scheme 2 (r 1 (4 cm, 1.5 cm, z), r 2 (0, −2.5 cm, z)), feed scheme 3 (r 1 (3 cm, −2.7 cm, z), r 2 (−4.7 cm, 2.7 cm, z)), and feed scheme 4 (r 1 (2.4 cm, −0.5 cm, z), r 2 (0.6 cm, 2.7 cm, z)). The reference point is arbitrarily chosen as r 0 (3 cm, −2 cm, z). The modal impedances at the reference point are modeled as parallel RLC circuits based on the steps 1-4 in Figure 3.11. The DC parameters and modal reference impedances are given in Table 3.6. These parameters are independent of the feeding scheme. To complete the analysis, the impedance at all the considered feed positions are obtained by determining the transformation ratios between the port and the modal impedance. In contrast, with a port simulation (using a physical feed probe), an indepen-
43
dent simulation must be conducted for each of the four feed schemes. Table 3.7 shows the transformation ratios from the reference modal impedance to the port impedance for the four feed schemes of the slotted patch. The predicted S parameters for the four feed schemes are compared with those from four separate port simulations (one for each feeding scheme) and shown in Figure 3.13 (a)-(d). Figure 3.14-3.15 shows the comparison results on Z and Y parameters. Despite the highly complex frequency responses, good agreement is observed over the wide frequency range (0.1 GHz-3 GHz) for all the four feed schemes. Several factors may contribute to the deviation between the circuit model and the full wave simulation: • This circuit model based on the virtual probe concept is essentially an approximate approach; • The modal impedance is modeled as simple parallel RLC circuits (Note that we did not notice improvement when alternative TE high-pass circuit templates are used); • The transformation ratios are assumed constant for simplicity (though not strictly necessary); • The truncation of higher order modes neglects mutual coupling between the observation ports caused by higher order modes, and also simplifies its effect on self input impedances. (If this is the dominating factor, including more modes will improve the accuracy.) Overall, this circuit model based on virtual probe concept provides an efficient and simple circuit model for planar MIMO antennas of arbitrary geometry. To the best of the author’s knowledge, this is the first time a general multiport planar antenna can be systematically and efficiently modeled using simple circuits. Though rational fitting techniques for multiport systems [66, 67] exist, it remains difficult to synthesize a matrix of rational functions. Moreover, our antenna circuit model based on the modal approach allows quick access to input parameters for arbitrary feed combinations, whereas the traditional fitting/interpolation techniques require a priori knowledge of the input parameters and therefore can only provide model for given fixed feeds.
44
r1
r1
r2
r2
(a)
(b)
r2 r1
r2
r1
(c)
(d)
Figure 3.13 The comparison of S parameters from port simulation (marker) and circuit model (solid lines) of the slotted patch for (a) feed scheme 1, (b) feed scheme 2, (c) feed scheme 3 and (d) feed scheme 4 [2].
45
300 200 Re
100 0 Im
-100 -200 -300 0.0
0.5
1.0
1.5
2.0
2.5
Re
Im
Dash line: full wave simulation Solid line: model
3.0
freq, GHz
(a)
(b) 300
Re
200 100 0 Im
Im
-100
Re
Dash line: full wave simulation Solid line: model
-200 -300 0.0
0.5
1.0
1.5
2.0
2.5
3.0
freq, GHz
(c)
(d)
Figure 3.14 The comparison of Z parameters from port simulation (dashed lines) and circuit model (solid lines) of the slotted patch for feed scheme 1.
46
Re
Re
Im
Im
(a)
(b)
Re
Re
Im
Im
(c)
(d)
Figure 3.15 The comparison of Y parameters from port simulation (dashed lines) and circuit model (solid lines) of the slotted patch for feed scheme 1.
47
3.3
Application to Feed Specification
The circuit models proposed in previous sections has the ability to quickly evaluate the input parameters at very large numbers of feed locations. With one CM analysis of a feed-free structure, it can predict the impedances at any arbitrary feed position. This enables us to efficiently visualize the input parameters, such as S, Z or Y (self and mutual), on the antenna structure, and select the optimal feed position.
3.3.1
Visualizing Input Parameters Using Heat Maps
Characteristic mode theory allows us to consider the antenna design as two independent steps: 1) finding the optimal radiator geometry, 2) selecting the optimal feed location. Here, we assume that the design of a suitable radiating element has been completed and only the second step of feed specification remains. In order to select the optimal feed location, we would need to interrogate many possible locations using a port-based simulation, but using the virtual feed approach described here, we can simply observe the values of S11 at all potential feed points with a single CM simulation. As an example, the slotted patch in Figure 3.12 and a triangular patch are considered. Based on the process described in Figure 3.11, the modal circuit parameters need only be determined once at the reference point. Then the calculation of Z11 at any other feed position only requires the transformation ratios γi,m at the resonance frequencies (step 5 in Figure 3.11), which can be generated much more quickly than a new port simulation. Our goal is to evaluate the input parameters at every node in the triangular mesh using the proposed approach. Then, the S11 (relative to 50 Ω) is plotted at every point on the aperture as a heat map. All the port simulations in this Section are performed in FEKO, while the circuit model and S11 heat maps are generated directly in Matlab. Figure 3.16 (a) shows the S11 (dB) map of slotted patch at its first modal resonance frequency (0.8565 GHz) as calculated from the expansion of the characteristic modes using virtual feeds. Over 600 feed positions are evaluated which would require an immense amount of time to compute using conventional probe-based modeling. The red regions represent the feed positions that will give a high reflection coefficient, while the blue regions correspond to low reflection coefficient. Figure 3.16 (b) shows the S11 map at
48
the second modal resonance frequency, 1.6033 GHz. The heat maps at both resonant frequencies are quite different, indicating that these maps could be combined to selectively excite one mode but not the other or both modes simultaneously. By visualizing the distribution of S11 at the first modal resonance frequency (Figure 3.16 (a)), the optimal feed position is chosen at P (−1.022 cm, −1.052 cm, z), while one worst-case feed position is also selected at W (4.107 cm, 1.31 cm, z). Figure 3.16 (c) compares the broadband S11 predicted from the circuit model with the S11 obtained from port simulation in FEKO at position P. Figure 3.16 (d) shows the comparison of S11 from the circuit model and FEKO simulation at position W. Good agreement in the responses is observed for both cases over the wide frequency range, 100 MHz ∼ 3 GHz. It is also clear that, as desired, point P excites the fundamental mode at 0.8565 GHz while point W is very poorly matched at this frequency. Similar to the process for generating the S11 map for the slotted patch, Figure 3.17 (a) show the S11 map of the triangular patch, at its first modal resonance frequencies (1.1669 GHz). Through observation of the S11 heat maps, the optimal feed positions are selected and marked as P in Figure 3.17 (a). Figure 3.17 (b) show the broadband S11 predicted from the circuit model, compared with the S11 obtained from port simulation in FEKO. There is clearly a band of positions that will result in a match to the 50 Ω system. Although these maps provide a useful intuitive design tool, they have not been used previously in antenna analysis because the computational overhead in generating hundreds of port-based simulations is significant. However, this new method allows a much faster, if slightly approximate, evaluation of large distributions of feed points. Interested readers can refer to [2] for information on the time efficiency of the proposed circuit model and more numerical examples. This visual map is a unique way of evaluating planar antennas and could generate research on new topics such as optimization methods for feed selection. Furthermore, this can easily be extended to mutual coupling analysis of multi-port antennas on compact apertures and various other applications where additional ports may be loaded with tunable elements for reconfiguration.
49
Find the answer for yourself if you really want to know, instead of casually asking for an explanation. Think by and for yourself.
Fi n d th e a ns wer fo ry ou rs el f i f y ou re al ly wa nt to kno w, i ns te ad of ca su al l y ask in g for an e xp la na tio n. Th in k by a nd fo ry oursel f.
S11 at 2nd resonance
Dipole approximation
0.04
0.04 -1
-2
-2
0.02
W
-4
-4 -5
0
-6
P
y (m)
y (m)
0.02
-3
-6
0 -8
-7 -10
-8
-0.02
-0.02
-9 -12
-10
-0.04 -0.06
-11
-0.04
-0.02
0
0.04
0.02
-0.04 -0.06
0.06
-14
-0.04
-0.02
0 x (m)
x (m)
(a) 0
-5 -5
FEKO sim Model
-10
S Parameter (dB)
S Parameter (dB)
0.04
0.06
(b)
0
-15 -20
66
56
FEKO sim Model
-10
-15
-25 -30 0
0.02
0.5
1
1.5 f (GHz)
2
2.5
3
-20 0
0.5
1
1.5
2
2.5
3
Freq (GHz)
(c)
(d)
Figure 3.16 (a) the S11 (dB) map of the slotted patch at the first modal resonance frequency (0.8565 GHz), with the optimal feed position marked as P (−1.022 cm, −1.052 cm, z) and the worst-case feed position marked as W ((4.107 cm, 1.31 cm, z), (b) the S11 map at the second modal resonance frequency (1.6033 GHz), (c) the broadband predicted S11 (solid line) for the optimal feed compared with that from the port simulation (dash line) in FEKO, and (d) the broadband predicted S11 (solid line) for the worst-case feed compared with that from the port simulation (dash line) in FEKO [2].
50
Fi n d th e a ns wer fo ry ou rs el f i f y ou re al ly wa nt to kno w, i ns te ad of ca su al l y ask in g for an e xp la na tio n. Th in k by a nd fo ry oursel f.
0
Averaging over abs(S11)
FEKO sim Model
0.04 -1 -2
y (m)
0.02
-3
P
0
-4 -5 -6 -7
-0.02
S Parameter (dB)
-5
-8 -9
-0.04
-10 -15 -20 -25
-10 -11
-0.06 -0.08 -0.06 -0.04 -0.02
0 x (m)
0.02
0.04
0.06
0.08
-30 0
0.5
(a)
1
1.5 f (GHz)
2
2.5
3
(b) 46
Figure 3.17 (a) the S11 (dB) map of the triangular patch at the first modal resonance frequency (1.1669GHz), with the optimal feed position marked as P (−1.286 cm, 0.8033 cm, z), and (b) the broadband predicted S11 (solid line) compared with that from the port simulation (dash line) in FEKO [2].
3.4
Summary
In this chapter, we developed circuit models for wire and planar MIMO antennas. Both theoretical formulation and numerical results are provided to demonstrate the proposed models. The limiting factors on the accuracy and ways to improve them are discussed as well. These circuit models are very efficient in terms of time consumption in that only with one characteristic mode analysis, the input parameters at arbitrary feed positions for arbitrary number of ports can be obtained immediately. These circuit models enables us to visually locate the optimal feed positions on the antenna aperture, a useful feature for antenna feed design. The validity of the circuit models further implies that we can consider an antenna’s shape and feed design as independent steps. In general, we can seek an optimal antenna aperture first without considering feed, and once the optimal antenna aperture is obtained, the optimal feed positions can be specified using these circuit modeling and visualization techniques.
51
Chapter 4 Feed-independent Shape Synthesis of MIMO Antennas As we have observed from the circuit models in Chapter 3, an antenna’s port response is fundamentally determined by the characteristic modes of the antenna aperture, whereas different feed positions only couple to the modes differently. This indicates that the antenna aperture is of fundamental importance to optimal antenna design, and should be considered first in antenna design. It also indicates that the antenna aperture could be designed without even considering feeds, as all the port responses can be derived from the modal responses of the antenna aperture. In this chapter, we will look at feed-independent antenna shape synthesis of MIMO antennas. Throughout this dissertation, we will mainly focus on MIMO antenna design within a limited space, usually smaller than λ/2 × λ/2. In such limited space, antenna aperture needs to be optimized to make use of any available bandwidth it can offer, and we don’t have the luxury to design the MIMO antenna as multiple independent antenna elements. Because of mutual coupling effects, all the antennas need to be considered as a single-aperture, whether they are physically connected or not.
52
4.1
Challenges in MIMO Antenna Shape Synthesis
Antenna shape synthesis has been studied by many researchers using different optimization techniques [77, 78, 79]. However, most of them are for single-port antennas, and in most cases, antenna shape synthesis is conducted with a specific feed position. The prior specification of feed position unavoidably enforces certain constraint on antenna shape optimization. This constraint becomes even more significant for MIMO antennas, because in this case not only input matching, but also high isolation and commensurate bandwidth are demanded between the ports. With prior specified feeds, the optimized antenna is only optimal for the specific feed positions, and could be far from the ultimate optimal solution that is available when all possible feed positions are subject to optimization. In order to find the optimal solution, it might be tempting to include all the possible feed positions into the optimization problem. This, though technically possible, could significantly increase the complexity of the problem, especially for MIMO antennas where multiple feeds are required. This also raises the problem of redundant full wave simulation of the same antenna aperture for many different feed combinations. To avoid the constraint set by the feed positions, several researchers has developed techniques for antenna shape synthesis without prior feed specification [49] using characteristic mode theory (CMT) [43]. As introduced in Chapter 2, CMT offers access to many antenna response parameters without actually exciting the antenna. Most of the reported work has employed CMT for shape synthesis of single port antennas [49], none for MIMO antennas. The way the cost function and Q factors are calculated in those reported works cannot be easily applied to multi-port antennas either. In this chapter, we will show how this feed-independent shape synthesis technique can be extended to MIMO antennas.
4.2
A Simple Feed Network to Extract Characteristic Modes of Symmetric Antennas
Before discussing antenna shape synthesis, we here demonstrate first that the characteristic modes can be accessed physically for certain type of antennas with a simple feed network. Consider antennas with one plane and two planes of symmetry, as shown in Figure 4.1.
53
a
a
b
d
c
b
(a)
(b)
Figure 4.1 Illustrative examples for (a) the one plane of symmetry for the two-port case and (b) the two planes of symmetry for the four-port case [3].
The driving point admittance matrix of the two port system is
Y11 Y12 YA = Y12 Y11
(4.1)
which can be diagonalized respectively by a set of simple frequency invariant decoupling vectors [52] 1 1 1 (4.2) V2 = √ 2 1 −1 The diagonalized admittance matrix looking into this decoupling network is
Λ=
Λ 11
Λ22
=
Y + Y12 11
Y11 − Y12
(4.3)
where Λ11 and Λ22 are the eigenvalues of the Y matrix in (4.1). For a four-port antenna with two planes of symmetry, as shown in Figure 4.1 (b), where a, b, c and d correspond to the four symmetric feeds for Ports 1-4, there exists a similar property. The driving point admittance matrix of a four port symmetric antenna
54
is
Y 11 Y12 YA = Y13 Y14
Y12 Y11 Y14 Y13
Y13 Y14 Y11 Y12
Y14 Y13 Y12 Y11
(4.4)
which can be diagonalized using the frequency invariant decoupling vector [52]
1 1 1 1 1 1 −1 −1 1 V4 = 2 1 −1 1 −1 1 −1 −1 1
(4.5)
The decoupled admittance matrix is
Λ 11 Λ=
Λ22 Λ33 Λ44
(4.6)
where, Λ11 = Y11 + Y12 + Y13 + Y14 Λ22 = Y11 + Y12 − Y13 − Y14 Λ33 = Y11 − Y12 + Y13 − Y14 Λ44 = Y11 − Y12 − Y13 + Y14
(4.7)
are the eigenvalues of the Y matrix in (4.4). Invoking the expansion of the self and mutual admittances in terms of CM modes in Chapter 2, the decoupled port admittances for the two-port case can be expanded in
55
even (Λ11)
odd (Λ22)
Figure 4.2 The definition of even and odd modes for the two-port case based on the type of symmetry of the current distribution [3].
terms of characteristic modes as Λ11 = Y11 + Y12 =
Λ22 = Y11 − Y12 =
(Jn (a) + Jn (b)) Jn (a) 1 + jλn n X Jn (a)Jn (a) =2 even modes 1 + jλn
X
(Jn (a) − Jn (b)) Jn (a) 1 + jλn n X Jn (a)Jn (a) =2 odd modes 1 + jλn
(4.8)
X
(4.9)
In (4.8) and (4.9) we define the modes with symmetric current at the two positions as even modes and the modes with opposite symmetric current as odd modes, as indicated in Figure 4.2. Because of the even and odd current symmetry, only the even characteristic modes will contribute to Λ11 in (4.8), and only the odd characteristic modes will contribute to Λ22 in (4.9). For electrically small antennas, the eigenvalues of the higher order modes are much larger than that of the lower order modes, resulting into small excitation coefficient and negligible contribution of the higher order modes to the driving point admittance. The eigen port admittances in (4.8) and (4.9) then become Λ11 ≈ 2
Jeven 1 (a) Jeven 1 (a) = 2Yeven 1 1 + jλeven 1
56
(4.10)
Λ22 ≈ 2
Jodd 1 (a) Jodd1 (a) = 2Yodd 1 1 + jλodd 1
(4.11)
Because the Q is related to the normalized derivative of the antenna impedance (or admittance) as [21] v u
ω0 u t ∂R QZ ≈ 2R ∂ω0
!2
∂X |X| + + ∂ω0 ω0
!2
(4.12)
the Qs of the two eigen port modes in (4.10) and (4.11) are identical to the Qs of the fundamental even and the odd characteristic modes, i.e., QΛ11 ≈ QCM even1 QΛ22 ≈ QCM odd1
(4.13)
This direct relation in (4.13) now offers two simple performance metrics, QCM even1 and QCM odd1 , for compact symmetric two-port antennas and reduces the analysis of a two-port antenna into that of two independent characteristic modes. Moreover, the Q of each mode is only dependent on the antenna’s structure, irrespective of any feed. Similarly, the decoupled eigen admittances for four-port symmetric MIMO antennas in Figure 4.1 (b) with four symmetric feeds a, b, c and d can be expressed in terms of the characteristic modal eigenvalues and eigenvectors as (Jn (a) + Jn (b) + Jn (c) + Jn (d)) Jn (a) 1 + jλn n X (Jn (a) + Jn (b) − Jn (c) − Jn (d)) = Jn (a) 1 + jλn n X (Jn (a) − Jn (b) + Jn (c) − Jn (d)) = Jn (a) 1 + jλn n X (Jn (a) − Jn (b) − Jn (c) + Jn (d)) = Jn (a) 1 + jλn n
Λ11 = Λ22 Λ33 Λ44
X
(4.14)
If the effect of higher order modes can be ignored, each of the four combinations in (4.14) is equivalent to a single characteristic mode (even-even, even-odd, odd-odd or odd-even mode), as indicated in Figure 4.3, and the Q factors of the four eigen port modes are also equivalent to that of the four corresponding characteristic modes.
57
even-even (Λ11)
even-odd (Λ22)
odd-even (Λ44)
odd-odd (Λ33)
Figure 4.3 The definition of even and odd modes for the four-port case based on the type of symmetry of the current distribution [3].
These results show that characteristic modes of those symmetric antennas can be potentially accessed using simple decoupling networks. As an example, we consider a 0.5 m strip dipole, with the two feeds symmetrically spaced about the center at a distance of 0.125 m from the ends. Figure 4.4 shows a comparison between the Qs of the eigen port modes calculated from (4.12) and the characteristic modal Qs calculated from source formulation in (A.10). The Qs of the eigen port modes agree very well with the Qs of the first even and odd characteristic modes, confirming our analysis. Above 600 MHz, the second odd mode becomes the dominant contributor to the odd port mode quality factor, QΛ22 , which begins to align with that of the second odd mode. As another example, Figure 4.5 shows the Qs of the eigen port modes of a crossed strip dipole antenna with four ports spaced symmetrically about the center at a distance of 0.125m from the ends, compared with the characteristic modal Qs. Again, the Qs of the four eigen port modes (Λ11 , Λ22 , Λ33 and Λ44 ) agree very well with the Qs of the corresponding characteristic modes (even-even, even-odd, odd-odd and odd-even), verifying our analysis. Note that the Qs of Λ22 and Λ44 overlap as degenerate modes. Throughout this Chapter, the Q of eigen port modes are named as QΛ11 , QΛ22 , QΛ33 and
58
200 QΛ11
2mm
QΛ22
150
Q
0.5m
Qodd1 Qeven1
100
Qodd2 Qeven2
50
0
600 400 f (MHz)
200
800
1000
Figure 4.4 Qs of the two eigen port modes (markers), compared with the characteristic modal Qs (solid lines) of the two-port strip dipole antenna [3].
200
QΛ11 QΛ22 QΛ33
150
QΛ44 QEO
Q
QOE QOO
100 even-odd2 odd-even2 even-even
50
odd-odd2
200
QEO2 QOE2
odd-odd
QOO2
even-odd odd-even
0
QEE
600
400
800
f (MHz)
Figure 4.5 Qs of the four eigen port modes (markers), compared with the characteristic modal Qs (solid lines) of the crossed strip dipole antenna. The even-odd and odd-even modes are degenerate [3].
59
Antennas
Matching Network
Load
Λ11
ZA
Multiple Single-port Matching Network
Decoupling Network
ZL
ZL
Λ22
Figure 4.6 MIMO System framework [3].
QΛ44 , while the Q of characteristic modes are named as QEE , QEO (Qeven ), QOO and QOE (Qodd ).
4.3
Framework of Antenna Shape Synthesis
The analysis in previous section demonstrates that characteristic modes of symmetric MIMO antennas can be accessed using simple decoupling networks, reducing MIMO antenna design to the design of multiple characteristic modes. Referring to Figure 4.6 for the MIMO system we are looking at, if considering ideal decoupling and matching networks, the only limiting factor in MIMO system performance is the antenna bandwidth (or Q factor). Therefore in the antenna shape optimization, we can only seek to minimize the antenna Q factors.
4.3.1
Binary Genetic Algorithm
We here employ genetic algorithm (GA) for antenna shape optimization, because of its popularity in antenna shape optimization history [78, 80]. The mechanism of GA can be generally explained using the diagram in Figure 4.7. In each generation, a certain number of populations are generated from previous generation. Each population corresponds to a chromosome with all the to-be-optimized parameters as its genes. The fitness of all the populations will be evaluated against the cost function, and if the cost function criterion is not satisfied, the populations will be fed into the GA operator to generate the off-springs
60
Populations
Mutation
Crossover
Reproduction
Evaluation
Fitness Satisfied?
No
Yes End
Figure 4.7 An illustrative diagram of genetic algorithm.
(the next generation). The algorithm is evolutionary because in each generation, only the populations with better fitness values are selected for reproduction of next generation, and the best chromosome in each generation is always passed along to the next generations. By controlling the number of generations (Ngen ), the number of populations in each generation (Npop ) and the mutation rate, the effectiveness of the entire algorithm can be controlled. Interested readers can refer to [78] for further details. Specifically, the GA we use for antenna optimization is targeted at optimizing the meshed antenna geometry so that the desired antenna behavior or response can be obtained after enough number of searches. The gene in each chromosome corresponds to the presence or absence of a mesh element (triangle). MoM is a very suitable tool for this analysis, because the entire MoM Z matrix is just the reaction matrix between all the basis elements in MoM meshing. Therefore, the removal of a mesh element corresponds just to the removal of the rows and columns in the MoM Z matrix that is contributed by
61
Figure 4.8 Mechanism of extracting MoM Z matrix for substructures.
that mesh element, as illustrated in Figure 4.8.
4.3.2
Optimization Setup
Figure 4.9 shows the complete aperture for an example optimization, the size of which is 120 mm × 76 mm, similar to the size of a cellular handset. The optimization frequency here is chosen as 500 MHz, and the total aperture size is 0.2λ × 0.127λ. For all optimizations, the number of generations is set as 800 and the population is 50.
Figure 4.9 The entire conducting plate used for shape optimization, with triangular mesh and two planes of symmetry enforced [3].
62
After each generation, the chromosomes with costs below the mean cost are kept, while the rest are regenerated through mating. Each pair of mating parents for an offspring are obtained through tournament selection from 6 randomly chosen chromosomes out of the total retained chromosomes. Then the rebuilt 50 chromosomes are subject to 2-8% mutation, with three best chromosomes kept unmutated.
4.4 4.4.1
Numerical Results Minimize the Sum of the Modal Qs
In the MIMO system framework shown in Figure 4.6, the multi-port antenna problem is reduced to multiple independent port modes using frequency invariant decoupling networks. Assuming ideal matching networks for each decoupled port, the Q factor of each eigen port solely determines the matching difficulty [33]. Thus our optimization parameter focuses only on the Q of each eigen port mode, which was demonstrated in Section 4.2 to be approximately the same as the lowest order even or odd characteristic modes. Maximizing the bandwidth of the fundamental even and odd modes is equivalent to minimizing their Qs. Thus, the first cost function is intuitively chosen as cost1 = (Qodd1 + Qeven1 )/2
(4.15)
for two-port MIMO antennas, and cost1 = (QOE + QEO + QOO + QEE )/4
(4.16)
for four-port MIMO antennas, where Qodd1 and Qeven1 is the Q factor of the first odd and even characteristic modes defined in Figure 4.2, and QOE , QEO , QOO and QEE are the Q factor of the four fundamental eigen modes (odd-even, even-odd, odd-odd and even-even) defined in Figure 4.3. Equal weights are chosen with no assumption of any differences between the desired modes or any knowledge of fundamental performance limits. Though some non-trivial shapes can be obtained using (4.15) for the two-mode case, the result soon converges to the whole plate when more modes are considered. Figure
63
3rd generation
15th generation
800th generation
100th generation (a)
200
200 QOE(Plate) QEO(Plate)
150
150
QOO(Plate)
QOE(Plate)
QEE(Plate)
100
Q
Q
QEO(Plate) QOO(Plate)
QOE(GA)
100
QEO(GA)
Q (Plate)
QOO(GA)
EE
Q (GA) OE
50
QEE(GA)
50
QEO(GA) QOO(GA)
0
QEE(GA)
200
400 600 f (MHz)
800
1000
(b)
0
200
400 600 f (MHz)
800
1000
(c)
Figure 4.10 Shape optimization results for cost function 1 with four modes considered: (a) The best fit antenna at the 3rd, 15th, 100th, and 800th generations, (b) characteristic modal Qs of an early stage (3rd generation) structure, compared with those of the whole plate, and (c) characteristic modal Qs of an intermediate stage structure (100th generation), compared with those of the whole plate [3].
64
4.10 shows the optimization results using the cost function in (4.16), with four modes considered. As Figure 4.10(a) shows, the antenna starts with a random shape, but gradually converges to the whole plate as the optimization proceeds. Figures 4.10(b) and (c) show the characteristic modal Qs of substructures at the 3rd and 100th generation. The markers represent the characteristic modal Qs of the substructures, while the solid lines represent those of the complete plate. The Qs of the 800th generation substructure are very close to those of the whole plate and are not shown explicitly. These results demonstrate that the minimum modal Qs are obtained by allowing currents to occupy the entire aperture when no other constraints are applied. While the mother plate is found to have the lowest Qs in all the different modes considered, it is not a practical solution as it is difficult to effectively excite the plate. However, the result from this cost function is helpful in understanding the limits in MIMO antenna shape optimization. We observe that the modal Qs of the optimized substructures tend to converge to the Qs of the complete plate, implying that the Qs of all substructures cannot be reduced below the Qs of the plate. This observation inspired our work in Chapter 6, where we theoretically investigated the bounding relation between the Q factor of a structure and its substructures from a mathematical perspective.
4.4.2
Minimize the Sum of the Normalized Modal Qs and Occupied Area
In the previous optimization results based on cost functions 1, it appears that the modal Qs of the substructures are bounded by the modal Qs of the whole plate. Because of the different magnitudes of the modal Qs (ranging from 13.4 to 1563), cost functions 1 emphasize the minimization of the higher order modal Qs at the expense of the low Q modes. To counter this problem, a cost function based on normalized modal Qs is defined as Qodd1 Qeven1 cost2 = WQ ∗ ( + ) + WA ∗ A (4.17) Qodd1 P late Qeven1 P late for two-port MIMO antennas, and cost2 = WQ ∗ (
QOE QOE P late
+
QEO QEO P late
+
QOO QOO P late
65
+
QEE QEE P late
) + WA ∗ A
(4.18)
(a)
(b)
200
200
QOE(Plate)
Qodd1(Plate)
QEO(Plate)
Qeven1(Plate)
150
QOO(Plate)
150
Qodd1(GA)
QEE(Plate)
100
Q
Q
Qeven1(GA)
QOE(GA)
100
QEO(GA) QOO(GA)
50
0
QEE(GA)
50
200
400 600 f (MHz)
800
1000
(c)
0
200
400 600 f (MHz)
800
1000
(d)
Figure 4.11 The optimization results based on cost function 2 for (a) two-port case and (b) four-port case; and their characteristic modal Qs compared with the modal Qs of the whole plate: (c) two-port case, (d) four-port case [3].
for four port MIMO antennas. The normalized antenna area A is included in the cost function to simplify the end geometry for easier excitation. The weight coefficient WA on antenna area is chosen as 50 for both the two-port and the four-port cases. An additional weight coefficient, WQ , is set to 20 based on optimization trials. Figures 4.11(a) and (b) show the optimized structures based on cost function 2 in (4.17) and (4.18), and Figures 4.11(c) and (d) show the characteristic modal Qs of these structures, compared with the Qs of the whole plate. The lowest two modal Qs of the shape in Figure 4.11(a) and the lowest four modal Qs of the shape in Figure 4.11(b) at the optimization frequency, 500 MHz, are given in Table 4.1.
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Table 4.1 The Modal Qs of the Optimized Two-Port and Four-Port Substructures at 500 MHz Using Cost Function 2.
Shape
QOE (Qodd1 ) QEO (Qeven1 )
QOO
QEE
Area
full plate
13.4
27.5
52.9
1563
100%
two port
15.5
30.6
N/A
N/A
38.6%
four port
15
31.6
57.3
1784
32%
It is interesting to see that the simple geometry as shown in Figures 4.11(a) and (b) has almost the same Q factors as the complete structure, implying that we have great freedom in manipulating the antenna shapes for optimal bandwidth. However, it is also enlightening to observe from optimization results from cost function 1 and 2 that all the substructures has characteristic modal Q factors lower than the complete structure. The advantage of the optimized antenna aperture over the complete plate is their easy excitation and extraction of characteristic modes. Interested readers can refer to [3] for the excitation of characteristic modes on the optimized apertures.
4.5
Summary
In this Chapter, we have considered feed-independent shape synthesis of symmetric MIMO antennas. We first showed that for this type of antennas, its characteristic modes can be extracted using simple decoupling networks, reducing MIMO antenna design problem to the design of multiple characteristic modes. Assuming random 3D uniform scattering environments, lossless antennas and ideal matching networks, the only limiting factor in MIMO antenna performance is its characteristic modal Q factors. We then employed genetic algorithm for antenna shape synthesis. Different cost functions are proposed and it’s found that the minimization of only Q factor always ends up with the trivial result of the complete geometry, implying certain bounding phenomenon of the modal Q factors. However, by including area in the cost function, we end up with some very simple antenna apertures that has almost the same modal Q factors as the complete geometry, demonstrating great freedom in manipulating the antenna shapes for optimal bandwidth.
67
Chapter 5 Systematic Design of Planar MIMO Antennas In Chapter 3, using CMT we have developed efficient circuit models for MIMO antennas with arbitrary geometry and arbitrary number of feed positions by modeling the behavior of individual characteristic modes. The validity of the circuit model implies the possibility to separate the antenna shape and feed design, and inspired us to consider feed-independent antenna shape synthesis, as was pursued in Chapter 4. In this chapter, we want to combine these techniques and demonstrate their usefulness for systematic design of MIMO antennas.
5.1
Shape-first Feed-next Design Methodology
From the modal point of view, the bandwidth of each mode is fundamentally determined by the physical antenna aperture, and different feeds only serve to couple to the modes differently. Therefore, in order to have the optimal design, the first step will be to find the geometry that offers the optimal bandwidth. Once the optimal geometry is obtained, the optimal feed positions that offer good matching and isolation can be located. This concept is what we call shape-first feed-next design methodology, as is illustrated in Figure 5.1. This methodology separates the antenna shape and feed design, and significantly reduces the complexity of the MIMO antenna design. For example, if there are M shapes
68
feed specification
shape synthesis
given aperture
optimal shape
optimal feeds
Figure 5.1 The concept of shape-first feed-next design methodology.
to be searched for the optimal antenna shape and for each shape there are N potential feeding schemes (N increases rapidly with the number of ports), a brute force search will result into M × N searches. However, by separating the shape and the feed design, only M + N searches are necessary, making the antenna design orders of magnitude faster. In fact, with our circuit models described in Chapter 3, we can visually observe the optimal feed position almost immediately after one characteristic mode analysis, making the feed design a trivial work.
5.2
Design Example 1: Two-Port Planar MIMO Antenna on Air Substrate
Following the design methodology in Section 5.1, the first step in designing the MIMO antenna is to find the optimal antenna aperture, and the next step is specifying the optimal feed positions. We will design an antenna based on this methodology and the developed techniques in Chapter 3 and 4.
5.2.1
Shape Synthesis of Self-Resonant MIMO Antennas
In Chapter 4, we have demonstrated some antenna shape optimization for optimal bandwidth, but without dealing with input matching. There we assumed that a matching network exists that can perfectly decouple and match the antennas. Therefore, in Chapter 4, antenna Q factor is considered as the main design parameter. However, in many real applications, it is usually preferred to have the antenna being resonant and well matched
69
by itself. In this section, we will explore the optimization of MIMO antennas that have both low Q factor (broad bandwidth) and self-resonant behavior, removing the necessity to add a matching network. There will be trade-off in such kind of optimization, depending on the weight on different parameters. As demonstration, a rectangular patch (12 cm × 7.6 cm, height h = 5 mm) is chosen as a study case. An infinite ground plane is assumed and modeled in the Green’s function. The optimization frequency is chosen as 1.2 GHz, at which the electrical size of the patch is 0.48λ × 0.3λ × 0.02λ (Note that it is a patch antenna instead of a plate). Conventional half-wavelength resonant antenna design method won’t apply here because of the small electrical size. To drive the optimization algorithm to make the antenna being self-resonant, we have to include that into the cost function. The cost function for the optimization is chosen as cost = C1 + C2 + 10 ∗ A
(5.1)
where Cn = 200∗(1−MSn )+Qn , and MSn = 1/|1+jλn | and Qn are the modal significance and modal Q factor of the n-th mode. A is the normalized antenna area. At self-resonance, the MSn will reach its maximum of 1. This cost function will facilitate the selection of the structures that are self-resonant (λn = 0) and broadband, and eliminate the unnecessary area. Similar to the shape synthesis technique introduced in Chapter 4, we use the binary genetic algorithm, and only MoM Z matrix at the design frequency is required. The characteristic modal Q factors are calculated at a single frequency using Vandenbosch’s source formulation [63] (see Section 2.2.4 and Appendix A for details on Q calculation). To get the modal significance and modal current, the characteristic eigenvalue equation has to be solved at the design frequency as well, which could be time consuming if the antenna is densely meshed to ensure accuracy. In order to make the optimization process more time efficient, an iterative mesh refinement technique is developed here. Generally, the antenna optimization starts with a coarse mesh and after a certain number of generations (in this case, 80 generations with 40 populations in each generation), the optimized geometry in the previous iteration is fed as the starting substructure into the next iteration, where the entire geometry mesh is
70
cost = 223.1
cost = 132.0
cost = 243.0
cost = 275.4
cost = 134.8
Figure 5.2 The evolution of the shape optimization with three iterative mesh and 80 generations in each iteration [4].
refined by a factor of four. Since the later iteration starts with the optimal geometry from previous iteration, it does not take long for it to reach convergence. The refinement of the mesh is conducted by dividing one triangle into four smaller triangles by connecting the centers of the three edges. Figure 5.2 shows the evolution of the geometry as the optimization progresses. Three iterative meshes are used in this case. The blue arrow indicates the evolution of the optimization on the same mesh and the red arrow indicates the iterative mesh refinement and the mapping of the optimized geometry from previous iteration to next iteration. It is interesting to see how the final optimized geometry resembles the optimized geometry in the first iteration of optimization before any mesh refinement. Because the optimization starts from a relatively coarse mesh, and the optimization in the refined mesh starts with the optimal geometry from last iteration, this iterative optimization procedure is very time efficient, compared with the direct optimization that starts in the very beginning with the most refined mesh. The necessity of the 3rd iteration can be observed from the costs shown in Figure 5.2. The cost of the optimized geometry in 2nd iteration changes from 134.8 to 243.0 when mesh is refined, demonstrating that the mesh in the 2nd iteration is not enough to accurately model the antenna response. Technically, you can repeat this
71
Figure 5.3 The modal significance of the optimized two port patch antenna.
process to get better accuracy, but we here only considered three iterations, because of the computational cost for very fine meshes. The final optimized geometry is the one shown at the bottom left in Figure 5.2. Figure 5.3 shows the modal significance of the first two characteristic modes of the optimized antenna aperture. Obviously, we have two modes resonant at the same frequency, as expected.
5.2.2
Feed Specification
After obtaining the optimal antenna aperture, the next step is to specify the optimal feed positions on the antenna so that both good impedance matching and port isolation are achieved. It is not a trivial task if without the circuit models developed in Chapter 3. Referring to the Chapter 3, the input self and mutual impedances of a planar antenna over a ground plane are expressed as Zii =
X 1 2 + jωLpi + γi,m Zm jωCdc m
72
(5.2)
Zji =
X 1 + jωMji + γj,m γi,m Zm jωCdc m
(5.3)
where Zm is the modal impedance of each mode at the reference point, and γi,m the transforming ratios between the port i and the mth mode. Cdc is the DC capacitance of the geometry, and Lpi the probe inductance at port i and Mji the mutual impedance between port i and j. (Refer to Chapter 3 for more details on the calculation of these parameters) Based on this circuit model, the S11 for all the possible feeds under the patch are easily calculated and plotted as a heat map in Figure 5.4 (a). The blue region represents the well matched positions. From these positions, any single feed can be selected to excite the aperture. We here choose P1 (0.01125 m, 0.007125 m, z) as the first feed position, and sweep the second feed under the patch to generate the S21 map that shows the coupling level between the two ports, as shown in Figure 5.4 (b). Based on Figure 5.4 (a) and (b), we can specify a second feed P2 (-0.0075 m, 0.007125 m, z) that is both well matched (from Figure 5.4 (a)) and isolated from port P1 (from Figure 5.4 (b)). Figure 5.4 (c) shows the predicted S11 , S22 and S21 (solid lines) between the two ports, and compares them with that from full wave simulation in FEKO (markers). Good agreement is obtained over the wide frequency range, validating our design approach.
5.2.3
Fabrication and Measurement
Following the steps in previous two Sections, we now have designed a two port antenna that is resonating at the design frequency, with both ports being well matched and isolated from each other. To experimentally verify the design method, an antenna prototype is fabricated and measured. In this design, the patch antenna is modeled without substrate; therefore, the fabrication and measurement is a little bit difficult. The antenna patch is fabricated on a copper sheet first, and then mounted on a large aluminum ground plane. Some supporting pillars with the exact height of the patch antenna (5 mm) is placed under the four corners of the patch. Two SMA feed probes are soldered on the patch at the specified feed positions (P1 and P2). Figure 5.5 (a) shows the fabricated two port antenna prototype, and Figure 5.5 (b) shows the corresponding two port S parameter from measurement using VNA. As expected
73
Find the answer for yourself if you really want to know, instead of casually asking for an explanation. Think by and for yourself.
Find the answer for yourself if you really want to know, instead of casually asking for an explanation. Think by and for yourself.
0.04
0.04 -5
-2
-8
-10
-0.02
0
-10
-15
P1
P2
-20
S21 (dB)
-6
P1
0
y (m)
0.02
-4
S11 (dB)
y (m)
0.02
-0.02 -25
-12
-0.04 -0.06
-0.04
-0.02
0 x (m)
0.02
0.04
-0.04 -0.06
0.06
-30
-0.04
-0.02
(a)
0.02
0.04
0.06
(b) 12
10
0 -5
S Parameter (dB)
0 x (m)
S11(model) S12(model)
-10
S22(model) S11(FEKO)
-15
S12(FEKO) S22(FEKO)
-20 -25 -30 1.05
1.1
1.15
1.2 1.25 freq (GHz)
1.3
1.35
(c)
Figure 5.4 (a) the S11 map at 1.2GHz, (b) the S21 map at 1.2GHz with one fixed feed at P1(0.01125m,0.007125m,z), and (c) the S parameter curve based on the chosen fixed feed P1 and the chosen 2nd feed at P2(0.0075m, 0.007125m, z) in (b), compared with the simulation result based on FEKO (markers) [4].
74
(a)
0
S Parameter (dB)
-5
S11 S12 S21 S22
-10 -15 -20 -25 -30 1.05
1.1
1.15
1.2 f (GHz)
1.25
1.3
1.35
(b)
Figure 5.5 (a) The prototype of the 2-port MIMO antenna over air substrate, and (b) the measurement result (S12 overlaps with S21 due to reciprocity).
75
from design, the two ports of the antenna are all radiating around the design frequency (1.2 GHz). However, the mutual coupling is higher than the simulation result. We believe this is due to the fact that the patch antenna is fabricated with no substrate. Any perturbation such as tilt or warp of the patch would cause such deviation, an effect verified through HFSS simulation. Therefore, this suggests that a better controlled fabrication is required. This inspires our work in the following few Sections, where this design methodology is extended to the antennas over dielectric substrates.
5.3
Modeling of Antennas on Dielectric Substrates
So far, the shape synthesis and circuit modeling techniques we developed are all based on antennas over an air substrate. In real application, patch antennas are mostly designed over a dielectric substrate. Moreover, the fabrication and measurement of the two-port antenna in Section 5.2 shows that it is difficult to have an air-substrate-based antenna even fabricated and measured. We therefore extend our design methods to antennas on dielectric substrates. To extend the design methodology in Section 5.2 to antennas on dielectric substrates, we need to calculate the MoM Z matrix and characteristic modal Q factors for antenna shape synthesis, and calculate the vertical E field for the circuit models. It is not an easy task because of the Dyadic Green’s Functions (DGF) required to model a microstrip substrate. In this Section, we will explain how a microstrip substrate can be modeled using Dyadic Green’s Functions, and how the planar antenna circuit model in Chapter 3 can be extended to microstrip antennas.
5.3.1
Spectral Domain DGF of Microstrip Substrate
Consider a grounded infinite dielectric slab as shown in Figure 5.6. Because of the presence of the dielectric substrate, the medium is no longer homogeneous for electromagnetic waves. Therefore, a different Green’s function that takes into account the dielectric substrate is needed. The modeling the effect of dielectric substrates on wave propagation has been investigated by many researchers using field Green’s functions [81, 82] and potential Green’s functions [83, 84]. Since mixed potential integral equation (MPIE) is
76
z unit dipole 𝐽ҧ
𝜖𝑟
ℎ
x
Figure 5.6 A single layer microstrip substrate.
more generally used for characteristic mode analysis, we here will consider the formulation based on potential Green’s functions. When there are both horizontal and vertical currents, the definition of scalar potential Green’s function is a little bit involved, but has been fully solved. Detailed information can be found in [85, 86]. However, when there are only horizontal currents, the definition of scalar potential Green’s function is quite straightforward [87, Chapter 3]. As we are only considering horizontal currents on the antenna, we will stick with this simple case, where the E field can be directly related to vector and scalar potentials as E = −jωµ0
Z S
¯ · JdS 0 − ∇ G A
Z S
q Gq dS 0 0
(5.4)
¯ and G are the spatial domain Green’s functions for vector and scalar potentials. where G A q In order to get the desired spatial domain Green’s functions in (5.4), we need to first solve the microstrip substrate DGFs in spectral domain, and then convert the spectral domain DGFs to their spatial domain counterparts. Using the Sommerfeld potentials, the ˜ A ) has the following form [87]: spectral domain magnetic vector potential DGF (G
˜ xx 0 G 0 A xx ˜ ˜ GA = 0 GA 0 zz ˜ zx G ˜ zy ˜ G G A A A and the electric scalar potential Green’s function Gq is a single scalar.
77
(5.5)
Table 5.1 Dyadic Green’s Functions of Single Layer Microstrip Substrate
Spectral GF
Expression
Region
˜ xx G A
1 sin(kzd z) DT E sin(kzd h)
0≤z≤h
˜ zx G A
jkx (r −1) cos(kzd z) DT E DT M cos(kzd h)
0≤z≤h
˜q G
jkz0 −kzd tan(kzd h) sin(kzd z) DT E DT M sin(kzd h)
0≤z≤h
DT E = jkz0 + kzd cot kzd h,DT M = jr kz0 − kzd tan kzd h q
q
kz0 = −j kρ2 − k02 ,kzd = −j kρ2 − r k02 ˜ xx ˜ yy G A =GA ,
˜ zx G A jkx
=
˜ zy G A jky
After some mathematical derivation using Maxwell’s Equations, the spectral domain DGFs for a single layer microstrip substrate is obtained and shown in Table 5.1. Interested readers can refer to Appendix C for necessary steps to get these results. As we only ˜ zz is not needed here. In the next consider horizontal currents on the planar antenna, G A section, we will show how the spatial domain DGFs can be obtained from the spectral domain DGFs.
5.3.2
Spatial Domain DGF Using DCIM
The spatial domain DGF can be obtained from the spectral domain DGF through the following transform [56], known as Sommerfeld Integrals: Gxx A (ρ) =
1 Z +∞ ˜ xx GA (kρ )J0 (kρ ρ)kρ dkρ 2π 0
78
(5.6)
1 Z +∞ ˜ Gq (kρ )J0 (kρ ρ)kρ dkρ (5.7) Gq (ρ) = 2π 0 1 Z +∞ ˜ zx Gzx (ρ) = GA (kρ )J0 (kρ ρ)kρ dkρ (5.8) A 2π 0 Since the integrands of these Sommerfeld integrals are not analytic, they cannot be directly evaluated in analytical forms. However, direct numerical evaluation suffers from slow convergence issue because of the oscillatory behavior of the integrand. In 1991, Chow was able to get the closed-form expression for microstrip dyadic Green’s functions using a method later on known as Discrete Complex Image Method (DCIM) [5]. DCIM soon found favor in many other researchers [88, 89, 90] and was widely studied and enhanced since Chow’s first introduction. Because of some of the approximations in DCIM, sometimes people still prefer direct numerical evaluation, and there are also efforts on this track [91, 92]. In this dissertation, we will employ DCIM for the the calculation of Gxx A (ρ) and Gq (ρ), and the specific steps directly follows the approach in [5, 56] (The treatment of Gzx A (ρ) is slightly different and will be explained afterwards). Without going into all the details, we will here briefly explain the steps in DCIM. First, extract out the analytic terms that can be analytical evaluated, such as the quasi-static term and the surface wave term. The remaining terms are then approximated as weighted sum of complex exponentials using Prony’s/Matrix-Pencil method [93] (we here use MPM), which are then directly evaluated using Sommerfeld Identity [56]. Following these steps, the resulted spatial dyadic Green’s function can finally be expressed as ci GA = Gstatic + Gsw A A + GA
(5.9)
ci where Gstatic , Gsw A A and GA represents respectively the contribution from the quasi-static terms, the surface wave terms and the complex image terms. The calculation of Gzx A is a little bit different, since it can not be directly approximated ˜ zx ˜ zx in terms of complex exponentials because of the kx term in G A . Luckily, GA /jkx can be directly approximated by Prony’s/Matrix-Pencil method, leading to a closed form zx expression for the integral of Gzx A [94]. The spatial representation of GA is simply found as the differentiation of the latter with respect to x. In our implementation, instead of following the approach proposed in [94], where static and surface wave terms are extracted,
79
˜ zx /jkx as complex exponentials, which can be converted into we directly approximated G A spatial domain as Z Nci X e−jkRi Gzx dx ≈ a (5.10) i A 4πRi i=1 q
where Ri = ρ2 − b2i , and ai and bi are the parameters obtained from complex image extraction [5, 94]. The Gzx A is then found as follows by taking differentiation of the above expression Gzx A (ρ) =
Nci Nci cos φ X 1 ∂ X e−jkRi ρ −jk = − 2 )e−jkRi ai ai ( ∂x i=1 4πRi 4π i=1 Ri Ri Ri
(5.11)
˜ zx ˜ zy Recalling that G A /jkx = GA /jky , the approximation in (5.10) is also the approximaR tion for Gzy A dy. Therefore, Gzy A (ρ) =
Nci Nci ∂ X e−jkRi sin φ X ρ −jk 1 ai = ai ( − 2 )e−jkRi ∂y i=1 4πRi 4π i=1 Ri Ri Ri
(5.12)
where φ is the angle between x axis and the vector from source point to observation point (projected on the same horizontal plane).
5.3.3
Numerical Examples of Dyadic Green’s Functions
Based on the above mentioned process, the spatial domain DGFs for microstrip substrates have been implemented, and integrated into our customized MoM code. As a demonstration of its accuracy, we here give some numerical examples. The first example is a direct comparison of DGFs with results published in literature. In [5], the Gxx A and Gq for a microstrip substrate with thickness h = 1mm and relative permittivity r = 12.6 is presented. We here calculated the Gxx A and Gq for such an example using our implemented code, and the results are shown in Figures 5.7 (b) and 5.8 (b), which agree well with the results in literature [5], shown in Figures 5.7 (a) and 5.8 (a). In the second example, the broadband frequency response of a bowtie antenna is calculated using our customized MoM code with DGFs, and is compared with the full wave
80
12 6 h=l mm
E,=
-____
1.00
Measurement of D Microstr
4
1
\\
J
-1.00
-
\
f = 1 0 GHz
c,=12.6 h = l mm
-----3.00
for horizontal electric dip Elec. Eng., vol. 135, pt. H [6] J. A. Stratton, Electroma 1941, p. 576, eq. (17). [7] J. Dai and Y. L. Chow, charges for study MMIC‘ Conf. (Beijing, China), Oc [8] J. Duncan, The Elements 1968. [9] R. W. Hamming, Numer New York: Dover, 1973, p
numerical integrotitm closed form Green’s
numerical integration closed form Green‘s function
P. A. Bern
Abstract -A new approach materials by means of a mi method is used in conjunction capacitance of a multilayer m the effective permittivity and The results are compared wi guide cavity by cavity perturb
I.
~ ~ ~ ~ , , , , , 1 , , , , , , , , , , , , , , , , , , , 1
(a)
quasi-dynamic images (i.e., eq. (lo)), three terms for the complex images (i.e., Table I), and two terms for the surface waves. The savings in computer time can be more than tenfold, with the error less than 1%compared with the numerical integration of the Sommerfeld integrals. Finally, it should be pointed out that the closed-form equation (18) applies to all source-to-field distances on the substrate surface. As discussed in the Introduction, it is in the form A + B + C , representing the contributions of the quasi-dynamic images, the complex images, and the surface wave. At different distances, certain contributions may be small and can be dropped without causing much error in the spatial Green’s function. ACKNOWLEDGMENT The authors thank the reviewers for many constructive comments. (b)
Microstrip ring resonat crowave integrated circuit rectangular, circular, and r ied in oscillators and filter that is several wavelength effects [5] and problems microstrip resonators. Wh Q factors of about 250, co The method presented effective permittivity will c modified by placing a di substrate, thereby changin The variational calculation to compute the effective crostriplike ring resonator several test materials. It is well known that th quency. Since we are very dence, the test material
Manuscript received Februa The authors are with the Physique, Laboratoire De Phy 351, Cours de la Liberation, 3 IEEE Log Number 904194
Figure 5.7 The example spatial DGF for Gxx A from (a) literature [5], and (b) our DGF 0018-9480/91/0300-0592$01.00 01991 IEEE implementation (normalized in the figure for easy comparison).
81
592
1E;EE TRANSACTIONS ON MICROWAVE THEORY AND TEC
R
[l] Y. L. Chow, “An approxim three dimensions for finit Microwave Theory Tech., vo [2] J. R. Mosig and F. E. Ga microstrip structures,” in Physics, vol. 59. London: [3] P. B. Katehi and N. G. Sommerfeld integrals with J. Math. Phys., vol. 24, pp. [4] E. Alanen and I. V. Lindel tion in horizontally layere plane,” Proc. Inst. Elec. E [5] D. G. Fang, J. J. Yang, an for horizontal electric dipo Elec. Eng., vol. 135, pt. H, [6] J. A. Stratton, Electromag 1941, p. 576, eq. (17). [7] J. Dai and Y. L. Chow, charges for study MMIC‘s, Conf. (Beijing, China), Oct [8] J. Duncan, The Elements 1968. [9] R. W. Hamming, Numeric New York: Dover, 1973, pp
1 .oo
f=10 GHz
- 1 .oo 12 6 h=l mm
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numerical integrotitm form Green’s
- _ _ _ _ closed
(a)
1.00
Measurement of D Microstri
4
1
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-1.00
-
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f = 1 0 GHz
c,=12.6 h = l mm numerical integration closed form Green‘s function
P. A. Berna
Abstract -A new approach f materials by means of a mic method is used in conjunction capacitance of a multilayer mi the effective permittivity and h The results are compared wit guide cavity by cavity perturba
I. IN
~ ~ ~ ~ , , , , , 1 , , , , , , , , , , , , , , , , , , , 1
(b)
Figure 5.8 The example spatial DGF for Gq from (a) literature [5], and (b) implementation (normalized in the figure for easy comparison).
quasi-dynamic images (i.e., eq.82 (lo)), three terms for the complex images (i.e., Table I), and two terms for the surface waves. The savings in computer time can be more than tenfold, with the error less than 1%compared with the numerical integration of the Sommerfeld integrals. Finally, it should be pointed out that the closed-form equa-
Microstrip ring resonato crowave integrated circuit rectangular, circular, and ri ied in oscillators and filters our thatDGF is several wavelength effects [5] and problems c microstrip resonators. Wha Q factors of about 250, com The method presented effective permittivity will ch modified by placing a die substrate, thereby changing The variational calculation to compute the effective
12
10 5 12
Figure 5.9 The broadband response of a bowtie antenna calculated using our customized MoM code (dashed line) and the commercial MoM solver FEKO (solid line).
simulation in FEKO. The results are shown in Figure 5.9. Again, very good agreement is observed over the wide frequency range of 100 MHz - 3 GHz. All these examples demonstrate clearly the accuracy of our implementation of microstrip DGFs.
5.3.4
Extension of Planar Antenna Circuit Model to Microstrip Antennas
The other very important reason we need to implement this DGFs is that we want to calculate the vertical E field under the planar microstrip antennas, which is required for zy the circuit model in Section 3.2. With the ability to calculate Gzx A and GA , we can now calculate the vertical E field, which is the third spatial component of the E field in (5.4). Now we have everything we need to generate the circuit model for microstrip antennas. Here is a brief review of the necessary steps to get the circuit model: (some are different from those in Section 3.2) • Conduct CMA to get J and λ. (we can get the resonant frequency from eigenvalue
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spectrum) • Since the source formulation for antenna Q factor calculation considers only free space Green’s function, it cannot be directly used here. Q factor is calculated as Q ≈ f2 dλ [76] (we therefore need to conduct CMA at two or three frequencies around df the design frequency in order to apply finite difference). • Calculate vertical E field at the reference port to find Rrad from (3.40) and then calculate the RLC circuit elements for each mode from Rrad , Qres and fres . • Calculate the transformation ratios between the observation port and the reference port. • DC capacitance is calculated as Cddc ≈ r C0dc , where C0dc is the DC capacitance in free space. Calculations of other DC parameters remain the same. • Assemble the circuit model based on Equations (5.2) and (5.3). We here demonstrate our circuit modeling technique using a two-port rectangular patch antenna as an example. The two-port patch antenna has the size of 12cm × 7.6cm, placed on top of a 5 mm thick substrate with the permittivity of r = 4. The two ports are arbitrarily chosen as (0.035 m, 0.014 m) and (0.011 m, -0.018 m). Figure 5.10 shows the two-port S parameters of the rectangular patch. The solid lines are the results from our circuit model, and the dash lines are the results from full wave simulation in FEKO. Very good agreement is observed over the wide frequency range. The circuit model also enables us to visualize the input S parameters under the patch as a heat map. Shown in Figure 5.11 is the S11 heat map under the rectangular patch, at its first resonance. The blue region will be the region where a low reflection coefficient can be achieved if placing the feed there.
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Figure 5.10 Modeling of an arbitrary two-port rectangular microstrip antenna using our circuit model (solid lines), in comparison with the result from full wave simulation in FEKO (dashed lines).
Figure 5.11 S11 heat map of the microstrip rectangular patch at its first resonant frequency generated using our circuit model.
S11 heat map at Freq=6.1678 e8 Hz 85
5.4
Design Example 2: Two-Port Microstrip MIMO Antenna
Now with the capability to model a microstrip antenna, we can easily extend our design method in Section 5.2 to a microstrip patch antenna. Still the first step is to conduct feed-independent antenna shape synthesis.
5.4.1
Shape Synthesis
For demonstration, a two-port planar MIMO antenna operating around 2.4GHz is considered. The dielectric substrate is Rogers RT/duroid 5880 with relative permittivity r = 2.2 and height h = 3.175 mm. The complete antenna aperture before optimization has the dimension of 3.8 cm × 2.5 cm, the electrical size of which at the design frequency is 0.45λd × 0.2966λd . Conventional resonant antenna design would require a square patch with a dimension of about 0.5λd × 0.5λd . Different from Design Example 1 in Section 5.2, here the modal Q factors are approximated as the frequency differentiation of eigenvalues, we therefore need to do eigendecomposition at three frequencies (using central finite difference for the differentiation operation). The cost function is still chosen as cost = C1 + C2 + 10 ∗ A
(5.13)
where Cn = 200∗(1−MSn )+Qn , and MSn = 1/|1+jλn | and Qn are the modal significance and modal Q factor of the nth mode. A is the normalized antenna area. Again, to accelerate the optimization convergence, iterative shape optimization technique is applied. Figure 5.12 shows the evolution of the antenna optimization with the costs given in the figure. It can be clearly seen that the mesh refinement is necessary in that the cost changes significantly when the antenna mesh is refined, demonstrating that the coarse mesh does not model the antenna behavior very accurately. It is found that the modal resonance frequencies change more significantly than the antenna modal Q factors. Technically, you can repeat the iterative optimization process to get better accuracy, but we stopped at the third iteration, considering the computational cost with
86
cost = 181.1
cost = 159.5
cost = 370.9
cost = 339.2
cost = 152.1
Figure 5.12 Iterative shape optimization of two-port patch antenna over microstrip substrate.
Modal Significance
1 Mode 1 Mode 2
0.8 0.6 0.4 0.2 0 2.3
2.35
2.4
2.45
2.5
freq (GHz) Figure 5.13 Modal significance of the final optimized antenna aperture.
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such fine mesh. The final optimized geometry is the one at the bottom right. Figure 5.13 shows the modal significance of the first two modes of the optimized aperture. Obviously, we have two modes resonant at the design frequency.
5.4.2
Feed Specification
With the optimal antenna aperture, the next step is to find the optimal feed position that offers good input matching and isolation. Using our circuit model technique developed in Section 5.3.4, we here generate the S11 heat map under the patch at its resonance frequency, as shown in Figure 5.14 (a). We pick the first feed position at P1 (4.76 mm, 0) and move the second feed position around to generate the S21 heat map, as shown in 5.14 (b). Combining S11 and S21 heat maps, we specify the second feed position at P2 (0, 3.108 mm), where both good input matching and low coupling with respect to port 1 is achieved. The predicted two-port S parameters using our circuit model is given in Figure 5.15, in comparison with the full wave simulation in FEKO. Similar to the result of design example 1 in Figure 5.4 (c), the self terms agrees well with simulation, while the mutual terms are higher than the full wave simulation result, because of the simple treatment of higher order modes. Essentially by adding more modes, the accuracy will be improved. It is worth noting that though the two modes are optimized to be self-resonant at the design frequency (2.4 GHz), the extra probe inductance and higher order modes all contribute inductive reactance, and cause the resonant frequency of the physical probe-excited antenna to be shifted to a higher frequency.
5.4.3
Fabrication and Measurement
The design in the above Sections using Method of Moment (MoM) is based on ideal infinite ground plane, lumped gap-voltage source on wire probes and lossless materials. Therefore, it cannot be considered as a final design, but only an initial design. In order to model the antenna system with realistic physical parameters and structures, we conducted a full wave simulation in ANSYS HFSS, where finite ground plane, loss in conductor and dielectrics, as well as physical 3D model of the RF connectors are considered. The results are shown as dashed lines in Figure 5.16 (b). In getting the simulation results in
88
0.015 -1
0.01 -2
y (m)
0.005
-3
P1
0
-4
-0.005
-5
-0.01
-6
-0.015 -0.02
-7
-0.01
0
0.01
0.02
x (m) (a)
0.015 -5
0.01
-10 P2
y (m)
0.005
-15 P1
0
-20 -25
-0.005
-30
-0.01 -0.015 -0.02
-35
-0.01
0
0.01
0.02
x (m) (b)
Figure 5.14 (a) S11 heat map of the optimized antenna at 2.4 GHz, and (b) S21 heat map of the optimized antenna at 2.4 GHz with the first port being fixed at P1 (4.76 mm, 0).
89
0
S Parameter (dB)
-10 -20 -30 S 11
-40 -50
S 12 S 22
circuit model: solid lines FEKO simulation: marked lines
-60 2.3
2.35
2.4
2.45
2.5
freq (GHz) Figure 5.15 Comparison of the antenna input S parameters from our circuit model (solid lines) and full wave simulation in FEKO (marked lines).
Figure 5.16 (b), the antenna is also scaled up by 1% in the y dimension to move down the resonant frequency of the second mode, and the feed positions are adjusted to P1’ (4.5 mm, 0) and P2’ (0, 2.5 mm) for better input matching. Though the operating frequency is slightly shifted, the overall behavior of the antenna is very close to the predicted response in Figure 5.15. As verification of the proposed design, an antenna prototype is fabricated on Rogers RT/duroid 5880 using a milling machine and is shown in Figure 5.16 (a). Because of the close spacing (about 5 mm) between the two ports, the small MMCX connectors are used as feeds, and MMCX-to-SMA adapters and converters are used for calibration and measurement. The solid lines in Figure 5.16 (b) shows the measured two-port S parameters from Vector Network Analyzer (VNA). Though the operating frequency is offset by about 2% from the simulation result in HFSS, the overall frequency behavior of the fabricated antenna is very similar to simulation results in dash lines. The offset in operating frequency between measurement and simulation is tolerable considering the rough fabrication using milling machine. This antenna prototype demonstrates the validity of our proposed shape-first feed-next
90
y Port 2
x Port 1
(a)
0
S Parameter (dB)
-10 -20 S 11-sim S 21-sim
-30
S 22-sim S 11-meas
-40 -50 2.3
S 21-meas S 22-meas
2.35
2.4
2.45
2.5
2.55
2.6
f (GHz) (b)
Figure 5.16 (a) The prototype of the 2-port MIMO antenna over microstrip substrate, and (b) the simulation result in HFSS (dashed lines) and measurement result (solid lines).
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design methodology. This design methodology is not only time-efficient but also general for antennas of arbitrary geometry (symmetry is not required, though employed here for simplicity).
5.4.4
Antenna Pattern and Correlation
To characterize the antenna’s far field behavior, the 3D field patterns are measured at the Wireless Research Center of North Carolina (WRCNC). Figure 5.17 shows the normalized measured 2D patterns (E and H planes) of port 1 in comparison with the normalized simulation result in HFSS, while Figure 5.18 shows that of port 2. The compared data are based on the values at their center frequencies (2.485 GHz for the measured data and 2.435 GHz for the simulated data). Very good agreement is observed between simulation and measurement. The measured realized gain (which includes mismatch loss) of port 1 and port 2 at center frequency 2.485 GHz are 7.2 dBi and 6.6 dBi respectively. Table 5.2 compares the measured realized gain with the antenna gain from full wave simulation, and they closely agree with each other. In order to characterize the correlation between the two ports of the antenna, the envelope correlation coefficient (ρe ) is calculated based on antenna patterns and S parameters respectively [95] as R | Ω F1 (θ, φ) · F∗2 (θ, φ)dΩ|2 R ρe = R (5.14) 2 2 Ω |F1 (θ, φ)| dΩ Ω |F2 (θ, φ)| dΩ ρe =
∗ ∗ S22 |2 S12 + S21 |S11 (1 − (|S11 |2 + |S21 |2 ))(1 − (|S22 |2 + |S12 |2 ))
(5.15)
The results are shown in Figure 5.19, where the solid line is the result calculated based on measured 3D antenna pattern, while the marker represents the result calculated based on measured S parameters [95]. The two results are supposed to be exactly the same for lossless antennas. However, because of the different measurement environment (lab for S parameters and anechoic chamber for patterns) and some loss effects, there is some deviation between the two results. Overall, because of the low mutual coupling, the correlation coefficient is very low throughout the interested band. In the above correlation coefficient calculation, we have assumed a rich scattering environment with 3D random uniform scattering. In real application, where more com-
92
Solid Lines: Meas. Dash Lines: Sim. 30 Red: E3 Blue: E?
0 0 -10 -20
60
Solid Lines: Meas. Dash Lines: Sim. 30 Red: E3
30 60
Blue: E?
0
-10
30
-20
60
-30
60
-30
90
90
120
90
120 150
0
90
120
150
120 150
180
E Plane (?=0) (a)
150 180 H Plane (?=90) (b)
Figure 5.17 Normalized radiation patterns of port 1: (a) E plane (φ = 0), and (b) H plane (φ = 90); the solid lines are the measurement results at center frequency (2.485 GHz) and the dashed lines are the simulation results at center frequency (2.435 GHz).
Solid Lines: Meas. Dash Lines: Sim. 30 Red: E3 Blue: E?
0
0 -10 -20
60
Solid Lines: Meas. Dash Lines: Sim. 30 Red: E3
30
Blue: E?
60
0
-10
90
90
120
120 150
60
-30 90
120
30
-20
60
-30 90
0
120 150
150 180 H Plane (?=0) (a)
150 180 E Plane (?=90) (b)
Figure 5.18 Normalized radiation patterns of port 2: (a) H plane (φ = 0), and (b) E plane (φ = 90); the solid lines are the measurement results at center frequency (2.485 GHz) and the dashed lines are the simulation results at center frequency (2.435 GHz).
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Table 5.2 Comparison of the Realized Gains (dBi) at the Center Frequencies From Simulation (2.435 GHz) and Measurement (2.485 GHz)
Port Number
Simulation
Measurement
Port 1
6.8
7.2
Port 2
7.1
6.6
Envelope Correlation
0.1 ;e-Measured Pattern
0.08
;e-Measured S Parameter
0.06 0.04 0.02 0 2.35
2.4
2.45
2.5
2.55
2.6
freq (GHz) Figure 5.19 The envelope correlation coefficient calculated from the measured 3D radiation pattern and the S parameters.
plex scattering environments are considered, the envelope correlation coefficient can be calculated as [96, 97]:
ρe = R Ω
| X E E∗ P 1+X 1θ 1θ θ
+
R X Ω
E E∗ P + 1+X 1θ 2θ θ
1 E E∗ P 1+X 1φ 1φ φ
dΩ
1 E E ∗ P dΩ|2 1+X 1φ 2φ φ R X 1 ∗ ∗ × Ω 1+X E2θ E2θ Pθ + 1+X E2φ E2φ Pφ dΩ
(5.16) where Pθ and Pφ are the power distribution probability of the incoming signals in θ and φ directions, and X is the cross polarization power ratio and is defined as the mean received power in the vertical polarization to the mean received power in the horizontal
94
polarization. Eθ and Eφ are the θ and φ components in the antenna field pattern.
5.5
Summary
In this Chapter, an efficient and systematic design methodology is proposed for planar MIMO antennas of arbitrary geometry. The first antenna prototype over air substrate is not well controlled in fabrication and measurement, and inspired our work to extend this design methodology to microstrip MIMO antennas. We implemented microstrip DGFs in our customized code, and successfully extended the methodology to microstrip antennas. One antenna prototype around 2.4GHz is design and fabricated. Good agreement between simulation and measurement is observed in this case, verifying our proposed design methodology.
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Chapter 6 Physical Limits of Antennas of Arbitrary Geometry In Chapter 4, we have conducted antenna shape optimization for minimum characteristic modal Q factors. There, we have empirically observed some bounding relation between the characteristic modal Q factors of a given structure and its arbitrary substructures. In this chapter, we will investigate the antenna Q factor limit from a theoretical perspective. Antenna Q factor is an important parameter in antenna design because of its inverse relationship to the matched fractional bandwidth [21, Eq. 87]. Though it has been an important research topic since the pioneering work of Wheeler [98] and Chu [20], there are still questions to be answered in terms of its calculation and physical limits. Antenna tuned Q factor is defined as Q=
2ω max{W e , W m } Prad
(6.1)
where W e and W m are stored electric and magnetic energies, and Prad is the radiated power. Though the definition is obvious, its calculation is quite involved for even a single dipole antenna. The calculation of antenna Q factors is complicated by the fact that the total electric/magnetic energy in space is composed of both energy in evanescent waves and energy in propagating waves, the latter of which propagates into infinite 3D space, resulting into infinite total energy after integration over the entire space. To understand this, imagine an
96
infinite long waveguide or transmission line, in which the energy of the propagating wave will be infinite. Chu [20] used a circuit approach to model the far field propagating wave as an impedance model, thus avoiding the calculation of the energy in the propagating wave. However, Chu’s approach only applies to spherical modes, because of the absence of analytical circuit models for arbitrary antennas. Collin [99], Fante [100] and Mclean [101] used a field approach, where the energy in the propagating wave is subtracted from the total energy. Though this approach can be applied to arbitrary antennas, there is no easy analytical solution for EM fields of an arbitrary antenna, and this subtraction approach remains controversial [102]. Recently, a new approach based on integration of sources (current and charge) on the antenna aperture is proposed [63, 103, 104]. In [63], Vandenbosch defined a modified stored energy by subtracting the contribution from the propagating radiation energies (similar to that in [21]), and further related the field to the antenna sources by employing magnetic and electric potentials and applying a series of Green’s function identities. As explained in detail in Appendix A, based on the source formulation [63, 47], antenna tuned Q factor can be calculated as Q(I) =
max{IH Xe I, IH Xm I} 2ω max{W e , W m } = Prad IH RI
(6.2)
where the calculation of the radiation power (Prad ) and the stored electric and magnetic energy (W e ,W m ) are simplified as quadratic operations on antenna current as e Wvac =
1 H I Xe I 4ω
(6.3)
m Wvac =
1 H I Xm I 4ω
(6.4)
1 Prad = IH RI (6.5) 2 and Xe and Xm are respectively the operators for the stored electric and magnetic energy, whose definition can be found in Appendix A. R is the real part of the MoM Z matrix. As pointed out by some researchers [104, 105], the positive definiteness of the operators
97
(Xe ,Xm ) fails to hold for electrical large structures, but holds true for electrically small geometries (typically smaller than λ/3). Since antenna Q factor is mostly meaningful for electrically small antennas, which is the case for the antennas we are studying, we find these operators valid and useful for the problems we looked at.
6.1
Problem of Q Limit
The ability to calculate the Q factor of arbitrary antennas using simple matrix operations inspires us to look at the Q limit problem again. Chu’s limit on antenna Q factors based on spherical modes is the most well-known result in antenna community, and has inspired many broadband small antenna designs using spherical geometries [106, 107]. However in real applications, antennas of spherical geometry are rarely used. Often, antennas are designed around a predefined device platform. This deviation from a canonical geometry leads to a realized Q-factor much higher than the bound predicted by Chu. In other words, Chu’s limit on antenna Q-factor is too optimistic for many practical one- and two-dimensional antennas and is less useful for practical antenna design on compact devices. This shortcoming has led to extensive research in deriving lower bounds for the Q-factor of antennas of arbitrary geometries [108, 109, 110]. The results on fundamental limit on Q-factors of arbitrary geometries [110], though promising, only provide the minimum Q-factor for currents confined to a fixed region. Lacking is a strong connection between the bound for a given geometry and that of an arbitrary subset of that geometry. In this Chapter, we will demonstrate that the minimum ¯ will always be greater than or equal Q-factor of currents contained within a region Ω ¯ as a subset, as illustrated in to the corresponding bound for a region which contains Ω Figure 6.1.
6.2
Relation Between Impedance Matrices of A Structure and Its Substructures
¯ that is contained Figure 6.1 illustrates a structure Ω and its one arbitrary substructure Ω within Ω. Using basis transformation, we here offer a mathematical relation between the
98
¯⊆Ω Ω ¯ J ¯ Z, Ω Z, J ¯ ⊆ Ω. Figure 6.1 An illustrative example of a structure Ω and its one arbitrary substructure Ω
¯ The reasoning here follows impedance matrix of the structure Ω and its substructure Ω. ¯ be the MoM impedance matrix of substructure Ω ¯ created using the that in [22]. Let Z ¯ j }. A current on the substructure Ω ¯ can be written in this basis as basis {W ¯
J¯ =
K X
¯j . I¯j W
(6.6)
j=1
As will be shown later on, this same physical current can also be represented using the bases {Wi } of the complete structure Ω. The two bases are related as ¯j = W
K X
mij Wi
(6.7)
j=1
where we denote M = {mij }. Substituting (6.7) into (6.6) and exchanging the summation operator, we get J¯ =
¯ K X K X
(
mij I¯j )Wi =
i=1 j=1
K X
Ii Wi .
(6.8)
i=1
¯ can also be expanded in terms of the bases of the Therefore, J¯ on the substructure Ω complete structure Ω, with coefficient I = M¯I. Whatever bases we use for the current expansion, the total power of the system
99
from the antenna should be the same for the same current. Therefore, the system power ¯ ¯I, should be the same as the power for a current ¯I on the substructure, which is ¯IH Z caused by the expansion of the same current on the complete structure I = M¯I, which is IH ZI = ¯IH MH ZM¯I. Therefore, we get ¯ = MH ZM. Z
(6.9)
This relation further implies that all constituent matrices of Z also follow this form for transformation onto the substructure basis, i.e.,
6.3
¯ = MH RM R
(6.10)
¯ e = M H Xe M X
(6.11)
¯ m = MH Xm M. X
(6.12)
Substructure Eigenvalue Bound
Consider a general eigenvalue problem using matrices derived from the complete structure Ω, AI = λBI. (6.13) Here A and B are chosen to be real symmetric matrices, such as those on the left hand sides of (6.10)-(6.12). Here we assume B is positive definite (e.g. R matrix of an antenna 1 ), while A could be any matrix or linear combination of matrices that satisfies the sub-full-structure relation in (6.10)-(6.12). ¯ Now consider that governing equation over a substructure Ω, ¯B ¯ ¯I = λ ¯ ¯I A
(6.14)
¯ is real symmetric and positive definite, a unique principal factorization Because B 1
R is positive definite for general radiating antennas
100
¯ 12 [111, Theorem 1.29] exists such that we can rewrite (6.14) as ¯ =B ¯ 21 B B ¯ B ¯ − 21 A ¯B ¯ − 12 (B ¯ 12 ¯I) ¯ 12 ¯I) = λ( B
(6.15)
¯ − 12 A ¯B ¯ − 12 and B ¯ −1 A ¯ have the same eigenvalue Because of the real symmetry property, B [112, Theorem 1.3.22.]. Therefore (6.14) and (6.15) have the same eigenvalues. ¯ = MH AM and Invoking the sub-full structure relation in (6.10)-(6.12), we obtain A ¯ = MH BM. We can then rewrite the matrix on the left hand side of (6.15) as B ¯ =B ¯ − 21 A ¯B ¯ − 12 = (MH BM)− 21 (MH AM)(MH BM)− 12 C
(6.16)
Following the idea in [22], we here define 1
1
U = B 2 M(MH BM)− 2 ,
(6.17)
¯ the matrices B± 12 and B ¯ ± 12 and noticing that due to the Hermitian nature of B and B, are also Hermitian, it follows that UH U = Im×m such that the right hand side of (6.16) can be rewritten in terms of U as ¯ = (MH BM)− 12 (MH B 21 B− 21 AB− 12 B 21 M)(MH BM)− 12 = UH B− 12 AB− 12 U = UH CU C (6.18) ¯ and C are both arranged in algebraic ascending order and the If the eigenvalues of C Poincar´e Separation theorem [112, 2nd edition, Chapter 4, Corollary 4.3.37] is applied, we get ¯ k ≤ λk+K−K¯ , 1 ≤ k ≤ K, ¯ λk ≤ λ (6.19) ¯ k are respectively the k th eigenvalue of C and C ¯ in ascending order, and where λk and λ are respectively the same as that of (6.13) and (6.14). Most important here is the relation that ¯1. λ1 ≤ λ (6.20) Poincar´e Separation Theorem[112, 2nd edition, Chapter 4, Corollary 4.3.37]: Let MK be the space of all square matrices of dimension K, and C ∈ MK be Hermitian. ¯ ≤ K, and let U1 , · · · , UK¯ be orthogonal, namely UH U = IK¯ . Let Suppose that 1 ≤ K
101
the eigenvalues of C and UH CU be arranged in algebraic ascending order. Then ¯ λi (C) ≤ λi (UH CU) ≤ λi+K−K¯ (C), i = 1, · · · , K
6.4
(6.21)
Limit on Minimum Tuned Q Factor
The lower bound on tuned Q-factor achievable by currents on a PEC structure Ω was formulated using an explicit dual solution in [110]. There, the process of finding the mimimum Q-factor Qlb is reduced to solving Qlb = max Q1ν , 0 ≤ ν ≤ 1 ν
(6.22)
In the above expression Q1ν is the lowest-order solution to ((1 − ν)Xm + νXe )Inν = Qnν RInν
(6.23)
where Q1ν ≤ Q2ν ≤ ...QN ν . In practice, (6.22) can be solved using a simple bisection algorithm [113] such that the eigenvalue problem in (6.23) is solved at only a small set of values of the parameter ν. Using the relations in (6.10)-(6.12), the analogous problem can be defined on a ¯ For any value of the parameter ν, the governing eigenvalue problem substructure Ω. ¯ take the form of (6.13) and in (6.23) for the full structure Ω and the substructure Ω (6.14), respectively. Therefore, through the manipulations outlined in Section 6.3 and the subsequent relation in (6.20), we arrive at the inequality ¯ 1ν . Q1ν ≤ Q
(6.24)
Though the value of the parameter ν which solves (6.22) may be different for the complete ¯ the relation in (6.24) guarantees that structure Ω and its substructure Ω, ¯ lb Qlb ≤ Q
(6.25)
¯ lb are the lower bounds on Q-factor for the structure Ω and substructure where Qlb and Q
102
l Example: lectrically small angular aperture ≈ 0.628
¯ respectively. Ω,
6.4.1
Numerical Examples
As a numerical example, a rectangular plate with dimension of L × L2 is considered, with the electrical size of kL ≈ 0.628. Several substructures of this aperture is examined, as shown in the first column in Figure 6.2. The minimum Q factor of each substructure and the optimal ν that achieves the minimum Q factor are searched using the bisection algorithm as suggested in [110]. As shown in Figure 6.2, all the substructures has minimum Q factors larger than the complete plate, agreeing with the result in (6.25).
Antenna Geometry 𝐿𝐿
substructures minimum Q n the complete
𝐿𝐿 2
Minimum Q
𝝂𝝂
101.42
0.819
128
0.865
101.56
0.819
110
0.884
116
0.931
`
Figure 6.2 An example of bound on minimum tuned Q factors: all the considered substructures have minimum Q factors higher than the complete structure.
103
8
6.5
Limit on Untuned Modal Q Factors
In the previous Section, we have demonstrated the bounding relation between the minimum tuned Q factor of a structure and that of its substructures. However, the formulation in (6.23) has only physical meaning for minimum tuned Q factors, and the interpretation for higher order modes becomes difficult. In this Section, we will apply the Poincar´e Separation Theorem to the untuned Q factors of an antenna [114], and demonstrate some bounds for the fundamental and the higher order untuned modal Q factors. Employing the source formulation of stored energy in Appendix A, the untuned Q-factor [114] can be expressed as IH Xm I + IH Xe I ω(W e + W m ) = (6.26) Qu (I) = Prad 2IH RI which is a Rayleigh quotient. Therefore in this case, the problem of finding the minimum untuned Q-factor and its corresponding current is equivalent to solving the following generalized eigenvalue problem 1 (Xm + Xe )Inu = Qnu RInu 2
(6.27)
and taking the lowest-order solution corresponding to I1u and Q1u , where Q1u ≤ Q2u ≤ ...QN u . Higher-order modes can be studied as well. Namely, the second minimum untuned Q-factor mode that has current orthogonal with the first minimum Q-factor mode is the second eigenmode corresponding to Q2u and I2u ; and so on for higher-order modes. Equation (6.27) has the same form as (6.13), and its equivalent formulation on a ¯ will have the same form as (6.14). Therefore, we can apply the eigenvalue subregion Ω bound in (6.19) on this problem, and claim the following result: ¯ K ¯ ku ≤ Qk+K− ¯ Qku ≤ Q , 1 ≤ k ≤ K, u
(6.28)
¯ is always larger stating that the k-th minimum untuned Q-factor of a substructure Ω than or equal to the k-th minimum untuned Q-factor of the complete structure Ω.
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6.5.1
Numerical Examples
To demonstrate the above result of bound on modal untuned Q-factors of substructures, we here use a rectangular plate as a numerical example. The electrical size of the rectangular plate is ka = 0.25, where a is the radius of the smallest sphere that encloses the plate, and k is the wavenumber. Generally, we employ binary genetic algorithm (GA) to search evolutionarily a large ¯ will number of substructures and then observe if all the investigated substructures (Ω) have untuned modal Q factors that are bounded below by that of the complete structure (Ω), as predicted by (6.28). Five optimizations are conducted independently, each with the objective of minimizing the n-th untuned modal Q-factor. The cost function for the each optimization is defined as ¯n cost = Q (6.29) u ¯ nu is the n-th minimum untuned modal Q-factor of the substructure Ω. ¯ The BGA where Q optimization is conducted with 500 generations and 100 individuals in each generation.
106
untuned Modal Qs
Q1u Q2u
105
Q3u Q4u
104
Q5u
103 102
0
100
200
300
400
500
generation Figure 6.3 Convergence plot of the five independent shape optimization, each targeting at the minimization of one individual untuned modal Q factor.
105
The mutation rate is 0.02. In each generation, the BGA examines a large number of substructures against the cost function and finds the best-fit geometry with the minimum cost. Figure 6.3 shows the convergence plot of the five separate optimizations (plotted together). As can be clearly seen, the five untuned modal Q-factors of all the substructures are bounded from below by the corresponding modal untuned Q-factor of the complete structure (horizontal dash lines), agreeing with the theoretical result in (6.28).
6.5.2
Implications
The bound we established in Section 6.5 is for untuned modal Q factors. In antenna community, people speak more about tuned Q factors because of its inverse relationship to matched fractional bandwidth. The relation between tuned Q factor and untuned Q factors can be explained using the diagram shown in Figure 6.4. The untuned Q factor is the ratio of the stored energy of the antenna over the radiated energy in one cycle, while the tuned Q factor takes into account of not only the stored energy of the antenna but also that of an external element that tunes the antenna to resonance. Therefore the tuned Q factor and untuned Q factor can be related as [114]: Q(I) = Qu (I) + Qext (I)
(6.30)
where
|IH XI| (6.31) 2IH RI and X = Xm − Xe is the imaginary part of the MoM Z matrix. It is obvious that the tuned Q factor is always larger than the untuned Q factor. Considering the bound we established in (6.28), we claim the following result Qext (I) =
¯ ku ≤ Q ¯ k , 1 ≤ k ≤ K, ¯ Qku ≤ Q
(6.32)
which states that the untuned k-th modal Q factor of a structure always lower bounds the tuned k-th modal Q factor of its arbitrary substructures. This is true not only for the fundamental mode, but also for the higher order modes. In this sense, the untuned modal Q factors of a general antenna aperture serve as lower bound for the tuned modal
106
Xseries ZAnt Qtuned
Quntuned
Figure 6.4 Relation between the tuned Q factor and the untuned Q factor of an antenna.
Q factors of its arbitrary substructures. One application of this Q bound for different order of modes is guidance of multi-mode antenna designs. We can use the lower bounds to examine whether a certain specification can be met for a given electrical size aperture.
6.6
Summary
Employing a source formulation on antenna stored energy and Poincar´e Separation Theorem in matrix analysis, we showed that the minimum tuned Q factor of an arbitrary geometry Ω lower bounds the minimum tuned Q factor of any arbitrary substructure ¯ contained within it, namely Qlb ≤ Q ¯ lb . This minimum Q bound is tighter than Chu’s Ω Q limit based on spherical modes, and applies to antennas of arbitrary geometry. When applying the same analysis to untuned Q factors, we proved that all the modal untuned Q factors of an arbitrary geometry lower bounds those of its substructures, namely ¯ ku , 1 ≤ k ≤ K. ¯ Considering the relation that Q ¯ ku ≤ Q ¯ k , we find that the untuned Qku ≤ Q modal Q factors of the complete structure also lower bounds the tuned Q factors of its ¯k ≤ Q ¯k. substructures, namely Qku ≤ Q u
107
Chapter 7 Decoupling Networks for MIMO Antennas In Chapter 5, we discussed MIMO antenna design techniques, and how low mutual coupling could be achieved by exciting orthogonal modes, such as the one demonstrated in Section 5.4. However, in practice, most MIMO antennas typically encounter certain level of mutual coupling, because of the changing near field loading in different user scenarios [115]. In situations where mutual coupling is significant, it is possible to combat the mutual coupling effect through decoupling networks. As illustrated in Figure 7.1, the purpose of the decoupling network is that when connecting a multiport antenna to the decoupling network, the load will observe a decoupled system (The coupled two-port antenna impedance ZAnt is decoupled into two uncoupled eigen port impedance Λ11 and Λ22 in this case). The decoupling network by itself should be a lossless multiport network. The design of decoupling network is not an easy task, primarily because of the difficulty in synthesizing a multiport network. Though quite a lot of work on decoupling network has been reported in literature [39, 37, 38, 41, 40], only a few [41, 40] are general enough to be useful for arbitrary loads, but even those are just for decoupling at a single frequency. Another difficulty in analyzing the decoupling network is the lack of metrics for performance characterization and comparison. Some researchers have tried to address this issue by defining new metrics, such as total reflection coefficient [41, 116], but so far, no
108
Λ11
Decoupling Network
ZAnt
Λ22 Figure 7.1 The concept of decoupling network.
generally accepted metrics have been established yet. Since most of the reported DNs only achieves optimal decoupling at the single design frequency, the resulted DN may not be optimal in terms of the broadband system performance. It is therefore enlightening to look at DN optimization for optimal broadband system performance. In this Chapter, we will try to address these questions, and also explore the possibility of more general and broadband decoupling networks.
7.1
A Decoupling Network Based on Characteristic Port Modes
7.1.1
Mathematics and Topology of the DN
In Appendix B, we have proved that the multiport transformer network serves as an impedance transformation network. For convenience, a figure similar to that in Appendix B is given here in Figure 7.2. As is shown in Appendix B, if loading port A with ZL , the input admittance at port B is B Yin = AYL AT
109
(7.1)
+ VB2
IB2
1:α21 α21IA1
IB1
VB1
-
1:α11 α11IA1
1:α22 +
α21VB2 -
-
+
α22IA2
+
+
α22VB2 -
α12IA2
1:α12
+
α11VB1
α12VB1
-
-
IA1 + VA1 -
IA2 + VA2 -
Figure 7.2 The multi-port transformer network.
where YL = (ZL )−1 and A is the matrix of transformation ratios. If loading port B with ZL , the input impedance at port A is T ZA in = A ZL A
(7.2)
It is interesting to see that the above analysis implies that if the coupling between an N-port antenna can be considered as a multiport impedance transformation of N independent modes (i.e., A is a square N × N matrix), the above two transformation inverses the transformation behavior of each other. For example, if an N-port antenna can be expressed as YAnt = AYm AT
(7.3)
which was demonstrated possible in Chapter 3 using characteristic modes (see (3.8)),
110
considering this antenna as a load (at port B) to a multiport TF network with behavior of T ZA in = A ZL A
(7.4)
T T T −1 ZA in = A ZL A = A (AYm A ) A = Zm
(7.5)
results in perfect decoupling
−1 where Zm = Ym is the diagonal modal impedance matrix. A straightforward way to understand this transformation and inverse transformation behavior of the TF network is through the generalized ABCD parameter. In this case, we can define a generalized ABCD parameter for the multiport TF network. For the transformation relation in (7.2), its corresponding generalized ABCD parameter is
A B AT 0 = C D 0 A−1
(7.6)
whereas the corresponding generalized ABCD parameter of the transformation relation in (7.1) is −T A B A 0 = (7.7) C D 0 A Figure 7.3 (a) gives a straightforward illustration of this decoupling network using ABCD parameters. If the antenna can be modeled as the transformation of modal impedances as shown in the dash box, then the inverse of the transformation network decouples the multiport antenna system. Figure 7.3 (b) can be considered as the implementation details of the decoupling network for a two port system. If the reader recalls, in Chapter 3, we have developed circuit models for MIMO antennas that has exactly the same form as shown in the dash box. Therefore, ideally, the inverse of the TF network in the circuit model in Figure 3.1 will decouple the antenna system. However, in real application, this does not work quite well, because: 1) there are an infinite number of structural modes in the antenna system. A truncated DN would not decouple the higher order modes, which would result into some coupling at the input of the DN; and 2) The transformation ratios are not absolutely frequency independent.
111
Antenna Model
Decoupling Network
𝐷 𝑇
𝑨 0
0 𝑨−1
𝐷𝐴 −𝑇
𝑨 0
0 𝑨
𝑡
1 ``
𝑚
𝒁
= 𝒁𝑚 (a)
Antenna Model
Decoupling Network α21:1
α22:1
1:α21
1:α22
1:α11
1:α12
B2
α12:1
α11:1 B1
A1 + Port 2 -
+ Port 1 -
Y1
A2 Y2
(b)
Figure 7.3 (a) Illustration of the antenna decoupling network based on characteristic port mode using ABCD parameters; (b) details within the DN.
112
Fortunately, we can formulate the relation in (7.3) using characteristic port modes of the port impedance matrix as well, which has equal number of modes as the number of ports. For a N port antenna, the characteristic port mode is defined as Xport Jn = λn Rport Jn
(7.8)
where Xport and Rport are the imaginary and real part of the port impedance matrix, and λn and Jn are the nth eigenvalue and eigencurrent. The eigencurrent are orthogonal with respect to Rport and Xport as JTm Rport Jn = δmn , and JTm Xport Jn = λn δmn . The input impedance can then be expanded as Zport = J−T (1 + jΛ)J−1
(7.9)
where J = [J1 , J2 , J3 , · · · , JN ], Λ = diag[λ1 , λ2 , λ3 , · · · , λN ]. If taking a port (e.g. port P) as reference port, Zport = A−T Zm A−1
(7.10)
where the m-th diagonal term of Zm is Zm = (1 + jλm )/J2m (P ), and A is the J matrix normalized to the current at the reference port. Here A is also frequency dependent. Therefore generally we can only achieve perfect decoupling at a single frequency using the fixed transformer-based DN. The decoupling frequency depends on the frequency you choose for the transformation ratios. Though we independently developed this DN topology, it is worth mentioning that a similar topology of multiport transformer network was later on found in the appendix in [117]. However, we believe our formulation of the decoupling concept using characteristic port modes and our explanation using ABCD parameters are unique, and as far as we know, this is the first time this DN is systematically explained and applied to general antenna decoupling problems.
113
7.1.2
Numerical Examples
The DN we proposed in Section 7.1.1 is simple in concept. In this section, we will demonstrate with numerical examples that it has better performance than some other available techniques. As an example, consider two parallel λ/4 monopoles, spaced by λ/8, as shown in Figure 7.4 (a). The frequency response of such a two-port antenna is shown in Figure 7.4 (b). Obviously, the two ports are strongly coupled. For comparison purpose, we first demonstrate the decoupling performance of the lumped LC π network proposed in [41], the topology of which is shown in Figure 7.5 (a). The DN S parameter block of the LC π network is given in [41] as:
svd SDN
1
−V V H U ∗ ΛL V T V (I − Λ2L ) 2 ULH = ∗ L L2 1 T UL (I − ΛL ) 2 V VL ΛL ULH
(7.11)
where UL , ΛL and VL are the matrices from singular value decomposition of the load S parameter SL at the design frequency, and V is an arbitrary unitary matrix. As explained in [41], there is an infinite number of implementations for this decoupling network depending on the unitary matrix V employed in (7.11). Figure 7.5 (b) demonstrates the decoupled S parameter of the two parallel monopoles based on one arbitrary implementation of the lumped LC π network. Figure 7.6 (a) shows the decoupling network topology based on our TF network. Since the TF network only achieves decoupling at the design frequency, single-port matching networks are needed for each decoupled port. Figure 7.6 (b) shows decoupled S parameters using our characteristic port mode based DN. It clearly offers much broader decoupling and matching bandwidth than the lumped LC π network proposed in [41].
114
0
S Parameter (dB)
-5
𝜆 4
-10 -15
S11 S12 S21 S22
-20 -25 -30 250
𝜆 8
300
350
f (MHz)
(a)
(b)
Figure 7.4 (a) two parallel λ/4 monopoles, spaced by λ/8; (b) S parameters of the parallel monopoles (S11 overlaps with S22 due to symmetry and S12 overlaps with S21 due to reciprocity).
0 Y3 Y13
1
-10 3
Y12
Y34
ZA
S parameters
Y1
-20 S 11
-30
S 12 S 22
-40 2
4 Y24
-50 2.5
Y4
Y2
3
freq (a)
3.5 # 108
(b)
Figure 7.5 (a) the topology of the DN based on LC π networks; (b) corresponding decoupled S parameters of the parallel λ/4 monopoles.
115
1:α21
1:α22
0
ZA S parameters
-10 1:α12
1:α11
-20
S 22 S 12
-30
S 11
-40 MN
MN
-50 2.5
3
freq (a)
3.5 # 108
(b)
Figure 7.6 (a) the topology of the DN based on CM port modes; (b) corresponding decoupled S parameters of the parallel λ/4 monopoles (S21 is very low, thus not shown in this scale).
7.2
Decoupling Network Synthesis Using TLs
In [41], a decoupling network using lumped elements is proposed based on the analysis of generalized π network. However, its implementation using lumped elements is not quite suitable for many RF and microwave systems because of the parasitic effects and loss in lumped elements. We therefore look for a general network synthesis technique using transmission lines. In [118], a network synthesis technique using transmission lines is proposed, where transmission lines are modeled as ABCD parameters and for a given topology a series of equations can be obtained by selectively shorting some of the ports. Though generic, the paper [118] fails to propose a network topology that could synthesize arbitrary networks, and only conventional ring hybrid and directional couplers are considered in the given examples. Moreover, the solution approach depending entirely on ABCD parameters makes it cumbersome for different systems with different number of ports, as new equations has to be derived once the number of ports are changed.
116
TL11
TL33
TL13
1
3
TL12
TL34
2
4 TL24
TL22
TL44
Figure 7.7 The topology of the generalized TL π network (two-port case)
In this section, we will demonstrate that a TL-based generalized π network can be used to synthesize an arbitrary passive, reciprocal and lossless network, and the analysis using Y parameters simplifies the analysis greatly. The same set of equations can be scaled to analyze networks with arbitrary number of ports.
7.2.1
Mathematical Analysis
The generalized π network of transmission lines is shown in Figure 7.7 (a). Similar as in [118], we choose to parameterize each TL branch as
AT L
a jb = jc d
(7.12)
Considering symmetric TLs (a = d), the Y parameter of one TL branch can be represented as [119] a
jb YT L = −1 jb
−1 jb a jb
(7.13)
Following the analysis in [41], the Y parameter of a N-port generalized π network can
117
be assembled from the Y parameters of each branch as P N
YΠ =
a1k k=1 jb1k −1 jb21 −1 jb31
.. .
−1 jbN 1
−1 jb12 PN a2k k=1 jb2k −1 jb32
−1 jb13 −1 jb23 PN a3k k=1 jb3k
−1 jbN 2
−1 jbN 3
.. .
.. .
... ... ... .. . ...
−1 jb1N −1 jb2N −1 jb3N
.. .
aN k k=1 jbN k
PN
(7.14)
where bnk = bkn . The degree of freedom in a reciprocal N −port network is (N 2 + N )/2 [41], whereas the number of unknowns we need to decide is (N 2 + N ) (two unknowns for each of the (N 2 + N )/2 branches). This might suggest that we don’t need so many branches. But in fact, to guarantee a transmission line implementation, we have to force |a| ≤ 1 (a is cos θ in TL ABCD parameter), which is not always guaranteed if with a simpler topology. Therefore, to ensure a TL implementation and generality, we first specify all the a in the transmission lines to be some value in the range of −1 and 1, and then equate YΠ with the admittance of the to-be-synthesized N −port network to get the solution of the remaining unknowns (all the b’s). With the values of a and b on each TL branch, we can convert that to the electrical length and characteristic impedance of the TL branch using the following relation
AT L
a jb cos θ jZ0 sin θ = = jc d jZ0 / sin θ cos θ
(7.15)
Namely, a = cos θ
(7.16)
b = Z0 sin θ
(7.17)
where Z0 and θ are the characteristic impedance and electrical length of a transmission line branch. Similar to other decoupling networks, this network only achieves decoupling at the design frequency. Though it is always desirable to have a small θ to make the TL shorter, sometimes the resulted b being negative requires that sin θ has to be negative. In this case, the θ derived
118
y
P1
P2 x
0.12 m (a)
(b)
Figure 7.8 (a) the geometry of the two-port square patch antenna, (b) the original S parameter behavior of the two-port patch (S12 overlaps with S21 ).
from (7.16) has to be modified as 2π − θ to meet the requirement. All these analysis above has been written in a compact Matlab code and is given in Appendix E.
7.2.2
Numerical Examples
As verification of the above mentioned TL network synthesis technique, we here give an example of decoupling network synthesis for a two-port patch antenna. As shown in Figure 7.8 (a), a square patch with arbitrary two ports (P1 (0.025,0.021) and P2 (0.044,0.009)) is used as an example. The frequency response of the antenna before decoupling is shown in Figure 7.8 (b). Strong mutual coupling exists between the two ports. Here the DN S parameter in (7.11) is used for the DN synthesis. Following the analysis in Section 7.2.1, we can get the solution for all the unknowns of the TL branches. Table 7.1 shows the corresponding electrical length and characteristic impedance of each TL branch. Figure 7.9 shows the corresponding DN topology in ADS and Figure 7.10 shows the S parameter of the two-port antenna after inserting the synthesized TL-based DN. As expected, the two ports are decoupled at the design frequency.
119
Figure 7.9 Simulation of the synthesized TL-DN in Keysight ADS.
0 S11 S12 S21 S22
S Parameter (dB)
-10 -20 -30 -40 -50 1.1
1.15
1.2
1.25
1.3
f (GHz) Figure 7.10 The decoupled S parameters of the coupled two-port patch antenna using the TL-based DN (S12 overlaps with S21 ).
120
Table 7.1 Characteristic Impedance and Electrical Length of All the TL Branches
TL Branch
Z0 (Ω)
Electrical Length (degree)
T L11
417.2
257.922
T L12
12003
268.1825
T L13
201.88
246.598
T L14
318.137
28.4123
T L22
122.765
159.4238
T L23
75.8
71.7682
T L24
207.39
153.8
T L33
126.98
323.4072
T L34
56
296.57
T L44
214.84
149.1196
This network synthesis technique could not only be used for DN design, but also design of other microwave networks, such as arbitrary power dividers and directional couplers.
7.3
Broadband Mulitport Matching Network Optimization
7.3.1
An Information-theoretic Metric for DN Characterization
In previous sections, we have introduced different decoupling network synthesis techniques. As discussed earlier and also implied in [41], some decoupling techniques can provide an infinite number of DN implementations. Because of the multiple ports in MIMO systems, it is not quite clear which metric to use to evaluate the performance of a DN. Since the purpose of having a DN in our case is to increase the MIMO system capacity, we find MIMO capacity a suitable metric to use. However, the time consuming Monte Carlo
121
𝐛0
𝐛1
𝐚2
𝐚1
𝐛2
0
𝐒𝐴
𝐒𝑀 =
𝐒11 𝐒21
𝐒12 𝐒22 𝐒
Antenna
1
Matching Network
2
0
Load
Figure 7.11 A simple MIMO system in S parameters.
simulation in capacity calculation complicates the analysis. We here propose a simplified metric based on MIMO capacity that avoids the Monte Carlo simulation. Consider a simple MIMO system in Figure 7.11, where receiver load is just terminal impedance Z0 , and between the load and the antenna, there is a multiport MN. b0 is the available source wave vector at the antenna terminals when the antennas are terminated with terminal impedance Z0 . Following the derivation in [14], the signal at the load is b2 = S21 a1 = S21 (I − SA S11 )−1 b0
(7.18)
Throughout this section, we will use boldface uppercase letters to represent matrices and boldface lowercase letters to represent vectors. The letters not in boldface represent scalars. Here we consider the simple case of uniform 3D random scattering environment, where the signal arrives as incident plane waves with independent random direction, complex phasors, and polarization. Following the work in [120, 121, 122], we here model the incoming signal to the antenna as 1
b0 = Rs2 Hw x
122
(7.19)
where Rs = c(I − SA SH A)
(7.20)
is the available source signal covariance matrix (antenna is terminated with reference impedance) [122], Hw denotes a NnR × NnT channel matrix with i.i.d. CN (0, 1) entries, and c is a real constant. The observed signal is the voltage across terminal load Z0 with noise: r = v2 + n
(7.21)
where 1
1
v2 = Z02 b2 = Z02 S21 (I − SA S11 )−1 b0
(7.22)
and n ∼ CN (0, σ 2 I). Here we consider channel information at the receiver side only. Therefore, the optimal solution is equal power distribution at the transmitter side [121]. E(xxH ) = Σx =
P I N
(7.23)
The system capacity in this case is
H1
1
−H H 2 Z0 Σx HH S21 S21 (I − SA S11 )−1 Rs2 Hw w Rs (I − SA S11 ) C = E log2 det(InT + ) σ2 (7.24) in To simplify the notation, we define another correlation matrix Rs as H Rin s = c(I − Sin Sin )
(7.25)
where Sin is the input S parameters looking into the matching network from the load side, as marked in Figure 7.11. For convenience, we here define the normalized correlation ¯ in as matrix R s ¯ in = (I − Sin SH ) R (7.26) s in ¯ in and Rin s = cR s .
123
H1
1
−H H 2 It is shown in Appendix D that Rin S21 S21 (I − SA S11 )−1 Rs2 s and Rs (I − SA S11 ) have the same eigenvalues. As MIMO capacity only depends on the eigenvalue of the SNR matrix [12], we can rewrite the capacity as
C = E log2 det(InT
Z0 P + HH Rin Hw ) NnT σ 2 w s
ρ ¯ in Hw ) HH R NnT w s
= E log2 det(InT +
(7.27)
where ρ = cZσ02P . ¯ in is full rank, and SNR is high, C can approximated as [123, Chapter If NnT = NnR , R s 9] ρ H ¯ in ) C ≈ E log2 det( Hw Hw ) + log2 det(R (7.28) s NnT where only the second term is related to antenna and matching network, and its maximum value is 0, which is achieved when system is perfectly decoupled and matched. When ¯ in ) is negative, causing capacity system is not perfectly decoupled or matched, log2 det(R s ¯ in ), we here loss. Since to maximize capacity is equivalent to minimizing − log2 det(R s define it as the information mismatch loss Γinfo = log2
1 ¯ in ) det(R s
!
(7.29)
which in linear scale is det(1R¯ in ) . This information mismatch loss can be used as a single s metric to measure how well a DN is, information-wise, for a coupled antenna system at every frequency. In the following Section, we will not only use this metric to compare different DNs, but also use that to optimize available DN for broadband system performance. The information mismatch loss in (7.29) should be differentiated from the total reflection coefficient in [124, 125], as the information mismatch loss can be easily extended to complex situations where maximum power transfer may not be optimal. For example, in the MIMO system studied in [15] where more complex noise correlations are considered, it is proved that minimum noise figure matching (instead of power matching) maximizes system capacity. In this case, we can extend our definition of information mismatch loss as ¯ in ) in (7.29) with the determinant the loss from noise figure mismatch, and replace det(R s
124
of the normalized SNR matrix in [15].
7.3.2
Broadband MN Optimization
With the metric (information mismatch loss, Γinfo ) we defined in (7.29), we can easily characterize a decoupling network and also conduct DN optimization. In this Section, we will demonstrate a broadband DN optimization using this single metric. The example we considered here is a pair of monopoles in parallel (quarter wavelength long and spaced by one eighth of a wavelength), the same example used in Section 7.1.2. The frequency response of this two antenna system is shown in Figure 7.4 (b). Obviously, strong mutual coupling exist between the two ports. We consider this two-port MIMO antenna for a MIMO-OFDM system operating from 270 MHz to 330 MHz, with 20 subbands. Because of the strong mutual coupling, decoupling network is designed and included between the antenna and the load, as illustrated in Figure 7.11. The first decoupling network we consider is a single frequency decoupling network proposed in [41]. Following [41], we generated an arbitrary DN based on generalized LC π network that achieves the decoupling at center frequency, 300 MHz. Shown in Figure 7.12 (a) is the S parameter of the antennas after inserting this decoupling network. The broadband capacity of the MIMO-OFDM system is calculated by taking weighted sum of the capacity at all the subbands: K 1 X ρ H ¯ in,k C= H R Hw ) E log2 det(InT + K k=1 NnT w s
(7.30)
¯ in,k is the corresponding value evaluated at the k-th subband. The capacity of where R s each subband is obtained through Monte Carlo simulation. Figure 7.12 (b) shows the plot of the mismatch loss defined in (7.29) over the interested band. The resulting capacity is shown in Figure 7.12 (c). Now we consider the optimization of the multiport matching network by employing genetic algorithm, and use the mismatch loss defined in (7.29) as the cost function. To facilitate optimization, each branch of the generalized π-network is represented by a series of inductor and capacitor, and the component values are subject to optimization. The entire band is divided into 20 subbands, and the mismatch loss at all the sub-bands are
125
100
Mismatch Loss (linear)
S parameters
0
-10 S 11
-20
S 12 S 22
-30
-40 2.7
2.8
2.9
3
3.1
3.2
freq
80 60 40 20 0 2.7
3.3 # 108
2.8
2.9
3
freq
(a)
3.1
3.2
3.3 # 108
(b) 14 LC-single-freq-match iid
Ergodic Capacity (bps/Hz)
12 10 8 6 4 2 0 0
5
10
15
20
SNR (dB)
(c)
Figure 7.12 The performance of a DN based on generalized π network, designed at a single frequency: (a) decoupled S parameters, (b) mismatch loss in linear scale and (c) resulting capacity of the MIMO system with this DN.
added as an equally weighted sum and are used as cost function for DN optimization. cost =
K 1 X 1 ¯ in,k K k=1 det(R s )
126
(7.31)
12
Mismatch Loss (linear)
S parameters
0
-10
-20
S 11 S 12
-30
-40 2.7
S 22
2.8
2.9
3
3.1
3.2
freq
10 8 6 4 2 0 2.7
3.3 # 108
2.8
2.9
3
freq
(a)
3.1
3.2
3.3 # 108
(b) 14 Optimized LC-MN iid
Ergodic Capacity (bps/Hz)
12 10 8 6 4 2 0 0
5
10
15
20
SNR (dB)
(c)
Figure 7.13 The performance of the optimized broadband DN based on 2nd-order LC-π network: (a) decoupled S parameters, (b) mismatch loss in linear scale and (c) resulting capacity of the MIMO system with this DN.
The optimized component values are shown in Table 7.2. Very small inductors and very large capacitors can be considered as short circuits. Figure 7.13 (a) shows the decoupled S parameter using the broadband optimized DN, and it is clear that some decoupling is sacrificed for broadband matching. Figure 7.13 (b) shows plot of the metric over the
127
Table 7.2 Component values of the optimized 2nd order LC π matching network
LC Branch
Lopt (uH)
Copt (pF)
LC11
1.4729
0.24106
LC12
0.64001
0.32554
LC13
1.6353
0.21911
LC14
1.0448×10−14
17.584
LC22
0.91059
0.41845
LC23
1.0425×10−14
15.786
LC24
0.7449
0.48124
LC33
0.38121
0.94465
LC34
1.0432×10−14
3.9653
LC44
0.11132
1010
band of interest, which is considerably lower than that in Figure 7.12 (b). Figure 7.13 (c) shows the broadband capacity of this two-port system with the optimized decoupling networks. Compared with the result in Figure 7.12 (c), the system capacity is increased by about 1.5 bps/Hz at SNR = 20 dB, or 90 Mbps considering the 60 MHz bandwidth. This example demonstrates the usefulness of the defined metric in (7.29) for multiport MN characterization and broadband multiport MN optimization.
7.4
Summary
In this Chapter, we proposed a decoupling network based on characteristic port modes, which can be synthesized as transformer networks. This DN is much simpler than that based on the generalized LC − π networks. Inspired by a recent work on TL network synthesis, we proposed a decoupling network synthesis technique using TL-π network. Considering the loss and parasitics in lumped elements, the TL implementation of the DN may be appealing to RF and microwave engineers. At last, in order to have optimal
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broadband system performance, we proposed an information-theoretic metric for DN optimization. In one example case, the optimized DN using genetic algorithm was able to improve the broadband MIMO system capacity by 1.5 bps/Hz, or 90 Mbps considering the 60 MHz bandwidth.
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Chapter 8 Conclusion and Future Directions In this Chapter, we will give a summary of our research work and contributions in this dissertation, and then provide some recommendations for future work.
8.1
Summary of Dissertation
In Chapter 3, based on the characteristic modal expansion of antenna admittance and impedance, we developed circuit models for general wire and planar MIMO antennas. These circuit models are very efficient in terms of time consumption in that only with one characteristic mode analysis, the input parameters at arbitrary feed positions for arbitrary number of ports can be obtained immediately. These circuit models enables us to visually locate the optimal feed positions on the antenna aperture, a useful feature for antenna feed design. We have demonstrated the validity and accuracy of these circuit models using different antenna examples. However, on the other hand, these circuit models have their own limitations in terms of accuracy. The accuracy of the wire MIMO antenna circuit model can be improved by employing more complex modal circuit templates (e.g. higher order HP circuits), allowing frequency dependent transformation ratios (αim ) in the transformer network and including more modes in the circuit model. Due to the approximate nature of the planar antenna circuit model, its accuracy may not be improved even if the model impedance Zm is modeled by more accurate circuit templates (a phenomenon we observed in our research work). However, including more modes and
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allowing frequency dependent transformation ratios should improve the accuracy. Because the planar antenna circuit model assumes uniform vertical E field, the accuracy may reduce when the thickness of the patch antenna makes this assumption invalid. Overall, from the examples we have examined, these circuit models demonstrate great accuracy. To the best of the author’s knowledge, this is the first time a general multiport antenna can be systematically and efficiently modeled using simple circuits. Though rational fitting techniques for multiport systems [66, 67] exist, it remains difficult to synthesize a matrix of rational functions. Moreover, our antenna circuit model based on the modal approach allows quick access to input parameters for arbitrary feed combinations, whereas the traditional fitting/interpolation techniques require a priori knowledge of the input parameters and therefore can only provide model for given fixed feeds. In Chapter 4, we developed feed-independent shape synthesis techniques for symmetric MIMO antennas using CMT. We demonstrated that the characteristic modes of general antennas with one plane and two planes of symmetry can be simply extracted using frequency independent decoupling vectors, and the decoupled ports behave similar to the individual characteristic modes. We therefore focused on antenna shape synthesis using modal parameters, eliminating the necessity to include physical feeds. The shape optimization results demonstrate certain bounding phenomenon that the characteristic modal Q factors of substructures are lower bounded by that of the complete structure. It also shows that the lower bounding characteristic modal Q factors can be approached using very simple antenna geometries, such as cross dipoles and loops. In Chapter 5, combining the feed-independent shape synthesis technique in Chapter 4 and the antenna modeling technique in Chapter 3, we proposed a shape-first feed-next design methodology for planar MIMO antennas. First, the optimal antenna aperture is synthesized using a genetic algorithm, and then the optimal feed positions are specified using our circuit models developed in Chapter 3. We also extended this design methodology to substrate-based planar antennas and implemented DGFs for microstrip substrates. A two-port microstrip antenna with an electrical size of 0.45λd × 0.2966λd is designed and measured. Other than the 2% shift in operating frequency due to the rough fabrication method, the overall behavior agree well with our design, validating our design methodology. Mutual coupling as low as -45 dB is achieved in this design.
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In Chapter 6, we investigated the Q limit problem for an arbitrary antenna geometry. Using newly developed source formulation and Poincar´e Separation Theorem, we proved that the complete antenna aperture has the minimum tuned Q factor, Qlb , and any of its ¯ lb ≥ Qlb . We also looked at Q limit substructure will have a minimum tuned Q factor Q for untuned Q factors, where we get the bounding relation that the modal untuned Q factors of the complete structure lower bounds that of its arbitrary substructures, namely ¯ ku , 1 ≤ k ≤ K, ¯ not only for the fundamental mode, but also for the higher order Qku ≤ Q modes. In Chapter 7, we developed new decoupling network synthesis techniques based on multiport transformer networks and transmission line networks. The TF-based DN is found to have a simpler implementation than most available decoupling techniques (e.g. the LC-π network). The TL-based DN is an alternative to the LC-π network at high frequencies in that it is more suitable for RF and microwave applications where parasitics of lumped elements cannot be neglected. Based on information capacity of a simple MIMO system, we proposed a simple metric, information mismatch loss (Γinfo ) for DN characterization and broadband optimization. In the example of parallel λ/4 monopoles, the optimized DN is found able to improve the system capacity by 1.5 bps/Hz at SNR = 20 dB, which corresponds to 90 Mbps considering the 60 MHz system bandwidth, demonstrating the usefulness of this metric for MN optimization and the limitation of the single-frequency decoupling techniques for broadband systems.
8.1.1
Contributions
This body of research has made several important contributions to the field of MIMO antenna theory, design and decoupling. Specifically, we have: • Developed a novel broadband circuit model for general wire and wire-fed MIMO antennas, which to the best of the author’s knowledge, is the first circuit model that can be applied to general MIMO antennas of arbitrary geometry. We believe this analytical circuit model will facilitate the investigation of MIMO antenna design, broadband matching, and system level performance evaluations. • Contributed a pioneering work on efficient and general modeling of arbitrary planar
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MIMO antennas by employing virtual feed concept and characteristic modal response expansion. This is the first time an arbitrary planar MIMO antenna aperture can be efficiently and accurately modeled, independent of physical feed probes and valid for arbitrary number of ports. In [2], we showed with a simple rectangular patch that our circuit model can generate the S21 heat map at a single frequency over 400 times faster than the conventional full wave simulation. The general applicability and the time efficiency of this modeling technique makes it a powerful tool for planar MIMO antenna design. • Developed a feed-independent shape synthesis technique for MIMO antennas using genetic algorithm and CMT, which extended the application of CMT for MIMO antenna shape synthesis and numerically demonstrated some bounding phenomena on antenna characteristic modal Q factors. • Proposed a novel shape-first feed-next design methodology for planar MIMO antennas and verified this method with a two-port microstrip MIMO antenna. This design method will facilitate the optimal design of both antenna aperture and feed positions, significantly simplifies the MIMO antenna design problem. • Mathematically established the minimum tuned Q factor bound for antennas of arbitrary geometry, which is tighter than Chu’s spherical Q limit and shows that any increase in antenna aperture area reduces the minimum tuned Q factor whereas any reduction in antenna aperture area increases the minimum tuned Q factor. • Independently developed a multiport transformer-based DN and for the first time applied it to general antenna decoupling problems; Introduced a scalable TL-based DN synthesis technique, which is an alternative to the lumped LC-based decoupling networks at high frequencies. • Proposed a new information-theoretic metric for multiport DN characterization and conducted broadband DN optimization based on this metric, demonstrating with results the usefulness of this new metric and also room for system performance improvement by designing broadband DNs.
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We believe this body of research work will shed new insight on design and analysis of compact MIMO antennas. In particular, our work on efficient circuit models for MIMO antennas of arbitrary geometry is new in antenna community and will find more application as people pursue efficient design methodologies for compact MIMO antennas. The establishment of the bounding relation between Q factor of a structure and its substructures will also provide insight and guideline for antenna design based on arbitrary geometries.
8.2
Future Directions
Several future research avenues could be pursued based on this research work, as suggested in the following paragraphs. • MIMO Antenna Feeding Techniques In this dissertation, we have demonstrated MIMO antenna designs with two ports, where optimal feed positions can be visually located based on S11 and S21 heat maps. However, feed specification for MIMO antennas with more than two ports might be challenging as it becomes hard to place the feed of one mode at the nulls of all the other modes. A more systematic feeding technique will be needed. One example of such techniques can be found in [29], where parasitic ports are introduced and terminated with reactive loading for resonance-tuning/decoupling purposes. We believe that our circuit models using characteristic mode theory could extend this method to antennas of arbitrary geometry, and also offers better accuracy. • Antenna Pattern Synthesis In Chapter 3, we proposed circuit models for arbitrary wire and planar MIMO antennas, and investigated quick access to input parameters of arbitrary feed combinations. One aspect that is missing is the investigation of antenna radiation patterns. Technically, we can obtain the radiation pattern for different excitation ports through modal expansion as well. This will enable us to specify feed positions that provide certain radiation pattern or gain response if so desired. This is not difficult for wire MIMO antennas, but could be difficult for planar MIMO antennas
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in that virtual feed probes are employed in the circuit model instead of physical feeds. • Shape Synthesis of Periodic Structures Though we only investigated shape synthesis for single aperture antennas, the synthesis technique could be extended to periodic structures as well once the periodic Green’s functions are included in the MoM code. This will enable CMT to be used for design of periodic open structures, such as frequency selective surface, high-impedance surface and reflectarrays. • MIMO Antenna Array In this dissertation, we only investigated MIMO antenna design on a compact single aperture. It is possible to design MIMO antenna arrays based on our proposed multimode single elements as well. Compared with conventional antenna array design, a compact multimode antenna array could achieve a significant antenna size reduction. For example, in [126], a 54% reduction in antenna size is observed when using four-port single elements in the array. This significant size reduction in antenna array makes multimode antennas useful for large-scale array design as well, such as massive MIMO. • Physical Limit on Tuned Q Factor of Multimode Antennas In Chapter 6, we have established the bounding relation between the minimum Q factor of a structure and that of its arbitrary substructures. However, the bound only applies to the lowest minimum Q factor. In order to apply this to MIMO system, a bound for higher order modes will be needed. Based on our observation, the establishment of such a bound will require the simultaneous diagonalization of the following matrices: R, X and Xe + Xm , which is only true for special matrices. Another option is to relate the tuned Q factors to untuned Q factors, and try to establish a bound for tuned modal Q factors based on the bound on untuned modal Q factors given in Section 6.5. We have some observation on this in Section 6.5.2, but some tighter bounds than what we discussed in Section 6.5.2 may be more desirable.
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APPENDICES
147
Appendix A Source Formulation of Antenna Stored Energy and Q Factor From [63], the modified stored electric and magnetic energies of an antenna is found able to be represented analytically as integration over sources (current and charge distribution) on the antenna as: e Wvac
m Wvac
Prad
Z Z 1 cos(k0 r21 ) = (∇1 · J1 )(∇2 · J∗2 ) dV1 dV2 − 2 16πω 0 V1 V2 r21 ! k0 Z Z 2 ∗ ∗ k (J1 · J2 ) − (∇1 · J1 )(∇2 · J2 ) sin(k0 r21 )dV1 dV2 (A.1) 2 V1 V2 0
Z Z cos(k0 r21 ) 1 2 k0 (J1 · J∗2 ) dV1 dV2 − = 2 16πω 0 r21 V1 V2 ! k0 Z Z 2 ∗ ∗ k (J1 · J2 ) − (∇1 · J1 )(∇2 · J2 ) sin(k0 r21 )dV1 dV2 (A.2) 2 V1 V2 0
=
sin(k r ) 1 Z Z 2 0 21 k0 (J1 · J∗2 ) − (∇1 · J1 )(∇2 · J∗2 ) dV1 dV2 (A.3) 8πω0 V1 V2 r21
If expanding the current in terms of MoM basis (J(r) =
148
PN n
In ψn (r)), similar to the
aforementioned MoM formulation in Chapter 2, the above equations can be reduced to the following matrix operation [127]: e Wvac =
1 H I Xe I 4ω
(A.4)
m Wvac =
1 H I Xm I 4ω
(A.5)
1 Prad = IH RI 2
(A.6)
where each entry of the matrix Xe is Xemn
Z Z 1 cos(k0 r21 ) = dV1 dV2 − (∇1 · ψm (r1 ))(∇2 · ψn∗ (r2 )) 4πω0 V1 V2 r21
! k0 Z Z 2 ∗ ∗ k (ψm (r1 ) · ψn (r2 )) − (∇1 · ψm (r1 ))(∇2 · ψn (r2 )) sin(k0 r21 )dV1 dV2 (A.7) 2 V1 V2 0
and similar expressions can be derived for Xm . It was later on discovered that the above representation of stored energies can be directly related to MoM matrix as [127] !
e Wvac
1 ∂X X 1 H ≈ IH − I= I Xe I 8 ∂ω ω 4ω
m Wvac
1 1 H ∂X X ≈ IH + I= I Xm I 8 ∂ω ω 4ω
(A.8)
!
(A.9)
Employing the matrix expressions for stored energies, the antenna tuned Q factor can be expressed as Q(I) =
2ω max{W e , W m } max{IH Xe I, IH Xm I} = Prad IH RI
149
(A.10)
Appendix B Multi-port Transformer Network In this appendix, mathematical proof is provided for the proposed circuit models in Chapter 3. Figure B.1 shows an example multiport transformer network. In the figure, the current and voltage transformation relations has been given and shown explicitly. It is obvious that we can write the following set of equations:
VA1 = −(α11 VB1 + α21 VB2 + α31 VB3 )
(B.1)
VA2 = −(α12 VB1 + α22 VB2 + α32 VB3 )
(B.2)
IB1 = α11 IA1 + α12 IA2
(B.3)
IB2 = α21 IA1 + α22 IA2
(B.4)
IB3 = α31 IA1 + α32 IA2
(B.5)
or in matrix form
V A1 VA2
α11 α21 = − α12 α22
VB1 α31 VB2 α32 VB3
(B.6)
I α α12 B1 11 IA1 IB2 = α21 α22 IA2 IB3 α31 α32
150
(B.7)
+ VB3
IB3
1:α31 α31IA1
IB2
1:α21 α21IA1
IB1
1:α22 +
α21VB2 -
1:α11 α11IA1
VB1
+
α32VB3 -
α22IA2
+
+
1:α32
α31VB3 -
+ VB2
α32IA2
+
+
α22VB2 -
α12IA2
1:α12
+
α11VB1
α12VB1
-
-
-
IA2 + VA2 -
IA1 + VA1 -
Figure B.1 The multi-port transformer network.
In the circuit model in Figure 3.1, port A is loaded with modal admittance Ym . Namely,
I Y V A1 = − 1 A1 IA2 Y2 VA2
(B.8)
Plugging (B.8) into (B.7), and then applying (B.6), we get
V I α α12 B1 11 B1 Y α α α 1 21 31 11 IB2 = α21 α22 VB2 Y2 α12 α22 α32 IB3 α31 α32 VB3
151
(B.9)
Therefore, we get YB = AYm AT
(B.10)
which has the same form as equation (3.8), and A is the matrix of transformation ratios in the transformer network. Similarly, in the circuit model in Figure 3.10, port B is loaded with modal impedances. Namely,
V Z I B1 1 B1 VB2 = − IB2 Z2 VB3 Z3 IB3
(B.11)
Plugging (B.11) into (B.6), and then applying (B.7), we get
V A1 VA2
Z 1
α11 α21 α31 = α12 α22 α32
Z2
α α12 11 IA1 α21 α22 IA2 Z3 α31 α32
(B.12)
Therefore, we get ZA = AT Zm A
(B.13)
which has the same form as equation (3.32) if we define Γ = AT . Equation (B.10) and (B.13) demonstrate that the multiport transformer network functions as a multiport impedance/admittance transformer.
152
Appendix C Dyadic Green’s Function of Microstrip Substrate We here give a brief description of necessary steps to get the Dyadic Green’s functions for a single layer microstrip substrate, with relative permittivity r and height h. The Green’s function we try to seek are those for vector and scalar potentials, which can be derived from the field Green’s functions. So the first step is to find the field Green’s functions.
z unit dipole 𝐽ҧ
𝜖𝑟
ℎ
x
Figure C.1 A single layer microstrip substrate with an x-oriented unit dipole.
153
C.1
Field Green’s Function
The derivation here follows that in [128]. All the horizontal field components can be expressed in terms of the vertical field components. We therefore only need to solve the vertical components. The vertical Ez and Hz of a microstrip substrate with a point horizontal source J = xˆδ(x)δ(y)δ(z − h), as shown in Figure C.1, are found as [128] kz0 kx η0 cos(kzd z), 0 < z < h E¯zx = k0 Tm
(C.1)
¯ zx = − jky sin(kzd z), 0 < z < h H Te
(C.2)
Te = kzd cos(kzd h) + jkz0 sin(kzd h)
(C.3)
Tm = r kz0 cos(kzd h) + jkzd sin(kzd h)
(C.4)
2 kzd = r k02 − kρ2
(C.5)
2 kz0 = k02 − kρ2
(C.6)
kρ2 = kx2 + ky2
(C.7)
where
To ensure physicality, the required restriction for real kρ is kz = −j|kz | (i.e., Imkz < 0 when |kρ | > k) [86]. We therefore define kzd and kz0 as q
kzd = −j kρ2 − r k02 q
kz0 = −j kρ2 − k02
(C.8) (C.9)
The results in (C.1) and (C.2) are the corresponding Green’s functions for the horizontal point source. Namely, ˜ zx = kz0 kx η0 cos(kzd z) G (C.10) E k0 Tm jky ˜ zx G sin(kzd z) H = − Te
(C.11)
The field for the yˆ-directed dipole can be obtained from the above through the
154
substitutions kx → ky and ky → −kx . The other field components can be expressed in terms of the vertical components, but is not needed in our study.
C.2
Potential Green’s Function
To get the Green’s function for the potentials, we have to find the relation between potential and field. Here we employ the popular Sommerfeld potential form for the vector potentials dyadic Green’s function ˜ xx ˜ zx ˜ yy ˜ zy ˜ zz ˜ A = (ˆ G ex G ˆz G ex + (ˆ ey G ˆz G ey + eˆz G ˆz A +e A )ˆ A +e A )ˆ Z e
(C.12)
The potential and field can be related as [87]:
˜ zx kρ2 G A
˜ zx ˜ xx = − GH G A jky
(C.13)
˜˙ zx kx G H zx ˜ = jωGE + ky
(C.14)
Gzy H ˜ yy = G A jkx ˜ zy ˜ zy kρ2 G A = jωGE −
(C.15) ˜˙ zy ky G H kx
˜ zz = jωG ˜ zz kρ2 G A E
(C.16) (C.17)
˜˙ = ∂ G . where G ∂z Plugging (C.10) and (C.11) into (C.13) and (C.14) respectively, we get: ˜
sin(kzd z) 1 sin(kzd z) ˜ xx G = A = Te DT E sin(kzd d)
(C.18)
(r − 1)kx sin(kzd d) j(r − 1)kx cos(kzd z) ˜ zx G cos(kzd z) = A = Tm Te DT E DT M cos(kzd d)
(C.19)
DT E = jkz0 + kzd cot(kzd d)
(C.20)
where
155
DT M = jr kz0 − kzd tan(kzd d)
(C.21)
The other components can be obtained from the following properties[87, 86]: ˜ yy ˜ xx G A = GA
(C.22)
˜ xx ˜ zy G G A = A jky jkx
(C.23)
The scalar potential can also be expressed in terms of the Field potentials as [87] ˜˙ zx ˜ zx jω0 G k G jkz0 − kzd tan(kzd h) sin(kzd z) ˜ Gq = 2 ( E ) − 0 ( )2 ( E ) = kρ jkx kρ jky DT E DT M sin(kzd h)
156
(C.24)
Appendix D Eigenvalue Equivalence H1
Here is a proof of the property used in Section 7.3, that A = Rs 2 (I−SA S11 )−H SH 21 S21 (I− 1 −1 2 H SA S11 ) Rs and B = c(I − Sin Sin ) have the same eigenvalues, where c is a real constant. Before the proof, we will list several properties that we will use throughout the proof. Since the MN is lossless, we have H SH 11 S11 + S21 S21 = I
(D.1)
H SH 11 S12 + S21 S22 = 0
(D.2)
H SH 12 S11 + S22 S21 = 0
(D.3)
H SH 12 S12 + S22 S22 = I
(D.4)
For general square matrices C and D of the same dimension, we have the following properties [112, Chapter 1] eig(CD) = eig(DC) (D.5) eig(I + DDH ) = eig(I + DH D)
(D.6)
For a Hermittian matrix D, we have the following property eig(D) = eig(D∗ )
157
(D.7)
Proof: H1
1
−1 2 eig(A) = eig(Rs 2 (I − SA S11 )−H SH 21 S21 (I − SA S11 ) Rs ) −1 = eig((I − SA S11 )−H SH 21 S21 (I − SA S11 ) Rs )
= eig(S21 (I − SA S11 )−1 Rs (I − SA S11 )−H SH 21 )
(D.8)
−H H = eig(cS21 (I − SA S11 )−1 (I − SA SH S21 ) A )(I − SA S11 )
where all these steps are obtained by applying (D.5). From Figure 7.11, it is obvious that Sin = S22 + S21 SA (I − S11 SA )−1 S12 . Therefore, we have eig(B) =eig(c(I − Sin SH in )) =eig(c(I − SH in Sin )) =eig(c[I − (S22 + S21 SA (I − S11 SA )−1 S12 )H (S22 + S21 SA (I − S11 SA )−1 S12 )]) −1 H −1 H H H =eig(c[SH 12 S12 − S22 S21 SA (I − S11 SA ) S12 − S12 (I − S11 SA ) SA S21 S22 − −H H H SH SA S21 S21 SA (I − S11 SA )−1 S12 ]) 12 (I − S11 SA ) −1 H =eig(cSH + (I − S11 SA )−1 SH 12 [I + S11 SA (I − S11 SA ) A S11 − H −1 (I − S11 SA )−H SH A S21 S21 SA (I − S11 SA ) ]S12 ) −1 H =eig(cSH + (I − S11 SA )−1 SH 12 [I + S11 SA (I − S11 SA ) A S11 − H −1 (I − S11 SA )−H SH A (I − S11 S11 )SA (I − S11 SA ) ]S12 )
(D.9)
where from step 1 to step 2, (D.6) is applied, from step 3 to step 4, (D.4) is applied H (replacing I − SH 22 S22 with S12 S12 ), and from step 4 to step 5, (D.1)-(D.3) is applied H H H H H (replacing I − SH 21 S21 with S11 S11 , S21 S22 with S11 S12 , and S22 S21 with S12 S11 ). Noting that I + S11 SA (I − S11 SA )−1 = (I − S11 SA )−1 and expanding the last term in
158
(D.9), we get −1 H eig(B) =eig(cSH + (I − S11 SA )−1 SH 12 [(I − S11 SA ) A S11 − −1 (I − S11 SA )−H SH A SA (I − S11 SA ) + H −1 (I − S11 SA )−H SH A S11 S11 SA (I − S11 SA ) ]S12 ) −H H =eig(cSH SA SA )(I − S11 SA )−1 + 12 [(I − (I − S11 SA ) H −1 (I − S11 SA )−H SH A S11 (I − S11 SA ) ]S12 ) −H −1 =eig(cSH (I − SH 12 [(I − S11 SA ) A SA )(I − S11 SA ) ]S12 )
(D.10)
where from step 1 to step 2, the 2nd and the 4th terms are grouped together, and the property that I + S11 SA (I − S11 SA )−1 = (I − S11 SA )−1 is applied again. From step 2 to H −H step 3, the property that I + (I − S11 SA )−H SH is applied. A S11 = (I − S11 SA ) From reciprocity of the antenna and matching network, we know that S12 = ST21 , S11 = ST11 , and SA = STA . We can then rewrite the above equation as −1 T eig(B) =eig(cS∗21 [(I − S11 SA )−H (I − SH A SA )(I − S11 SA ) ]S21 ) T T −T =eig(cS∗21 [(I − STA ST11 )−∗ (I − SH ]ST21 ) A SA )(I − SA S11 )
=eig(cS21 [(I − STA ST11 )−1 (I − STA S∗A )(I − STA ST11 )−H ]SH 21 ) −H H =eig(cS21 [(I − SA S11 )−1 (I − SA SH ]S21 ) A )(I − SA S11 )
=eig(A)
(D.11)
where from step 2 to step 3, property (D.7) is applied. From step 3 to step 4, properties S11 = ST11 and SA = STA are applied.
159
Appendix E Code on N-port Network Synthesis Using TLs 1
% A Demonstration of the TL - based DN
2 3 4 5
clear all ; [ filename , pathname ] = uigetfile ( ’ *. s2p ’ , ’ Select the snp file to open ’) ; DataFileName = strcat ( pathname , filename ) ;
6 7 8 9
hS = sparameters ( DataFileName ) ; S = hS . Parameters ; freq = hS . Frequencies ;
10 11 12 13 14 15
FreqToMatch =1.2 e9 ; [ dummy , F0 ]= min ( abs ( FreqToMatch - freq ) ) ; omega0 =2* pi * freq ( F0 ) ; SL (: ,:) = S (: ,: , F0 ) ; SL =( SL + SL . ’) /2;
16 17
S_DN = SVD_DN ( SL ) ;
18 19
1 2
[ Z0 , E_length ]= Netw ork_S ynth esis_ TLs ( S_DN ) % This script synthesizes an arbitrary lossless S matrix using TL Pi - network % by Binbin Yang , 9/10/2016
3 4
function [ Z0 , E_length ]= Net work_ Synt hesis _TLs ( S )
5 6 7 8
N = size (S ,1) ; Y = s2y ( S ) ; % here the theta ( or a ) is a randomly generated symmetric matrix . It can also be specified , e . g . all being pi /2.
160
9 10 11 12
% theta = ones ( N ) * pi /3; theta = rand ( N ) * pi ; theta =( theta + theta . ’) /2; a = cos ( theta ) ;
13 14 15 16 17
a_offdiag =a - diag ( diag ( a ) ) ; a_diag = diag ( a ) ; Y_offdiag =Y - diag ( diag ( Y ) ) ; Y_diag = diag ( Y ) ;
18 19 20 21 22 23 24
% find the off - diagonal term of b matrix b_offdiag =1 i ./ Y_offdiag ; for i =1: N b_offdiag (i , i ) =0; end b_offdiag ( find ( abs ( b_offdiag ) >1 e8 ) ) =0;
25 26 27 28
% find the diagonal term of b matrix Y11 = sum ( Y_offdiag .* a_offdiag + diag ( diag ( Y ) ) ,2) ; b_diag = -1 i * a_diag ./ Y11 ;
29 30
31 32 33 34 35 36
% If the self Y_kk is zero , then that branch is not needed . We here force its Z0 being zero . for i =1: N if abs ( Y11 ( i ) )
