Antenna Limitations and Q-factor Trade-off betweenParameters, Steps towards Optimal Antenna Design

SHUAI SHI

Doctoral Thesis in Electrical EngineeringKTH Royal Institute of Technology

Stockholm, Sweden 2018

TRITA-EECS-AVL-2018:89ISBN 978-91-7873-019-3

KTH Royal Institute of TechnologySE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i Elektroteknikonsdag den 05 december 2018 kl. 13:00 i Kollegiesalen, Brinellvägen 8, KungligaTekniska högskolan, Stockholm.

© 2018 Shuai Shi, unless otherwise noted.

Tryck: Universitetsservice US AB

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To my wife Meng and daughter Emma

Abstract

Sub-wavelength antennas have become ubiquitous in essential devices, such as mo-bile phones, sensors, Internet of things (IoT) and machine to machine (M2M) com-munication devices. Such antennas are often embedded as a small part of the devicechassis or their circuit-boards. The size assigned both to the antennas as well asthe device tends to shrink, while demands on antenna performance are increasing.In such a context knowledge of optimal performance is of increasing importance.The subject of this thesis is on the bounds of small antennas, in particular boundson impedance bandwidth performance.

The main tool to obtain bounds is antenna current optimization. The boundsare mainly focused on determining limits on the Q-factor for small antennas, andhence implicitly on the available bandwidth at a given reflection coefficient. Weinvestigate Q-factor bounds under a number of constraints including directivity,far-field radiation pattern, efficiency, and the embedded position of the antenna. Inthis process, we combine physical methods, mathematical tools, and antenna engi-neering. We use the Method of Moments (MoM) approach to solving the ElectricField Integral Equations (EFIE), in this context we formulate and solve antennaoptimization problems where the surface current density is an unknown variable,and we solve convex and non-convex quadratically constrained quadratic programs(QCQPs). For non-convex problems, we investigate different methods to obtain thesolution, but with the main focus on the semidefinite relaxation (SDR) technique.Different current optimization problems are solved for a range of shapes, where theQ-factor and the optimal surface current are determined; the results are comparedwith full-wave-simulation of antennas that approach the bounds.

To determine the Q-factor for an available space in the device is here proposed tobe an initial step of an antenna design procedure. The current optimization helps usto determine the optimal trade-off between the different performance parametersof a small antenna, and it can inspire antenna design with better performance.We furthermore show that a multi-position feeding strategy to realize an optimalcurrent successfully realize a non-standard far-field performance. As an example,we show that the desired radiation patterns are obtained with small costs of Q-factor. The thesis ends with a discussion of initial steps to a methodology with thegoal of obtaining a Q-factor optimal antenna. Here the current optimization playsan important role in the antenna synthesis and analysis stages of the process. Anapplication to the embedded antenna is discussed in detail.

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Sammanfattning

Elektriskt små antenner har blivit vanliga i väsentliga vardagsföremål, såsom mobil-telefoner, sensorer, internet av saker (IoT) och maskin till maskin (M2M) kommu-nikationsenheter. Sådana antenner är ofta inbäddade som en liten del av enhetensskal eller deras kretskort. Storlek som tilldelas både antennerna och den omgivandeenheten tenderar att minska, medan krav på antennprestanda ökar. I ett sådantsammanhang är kunskap om optimal prestanda allt viktigare. Ämnet för dennaavhandling ligger på begränsningar för små antenner, i synnerhet begränsningar avbandbreddsprestanda i utifrån antennens matningsport.

Huvudverktyget för att bestämma begränsningar för en antenn är en så kalladantennströmoptimering. Begränsningarna är huvudsakligen inriktade på att denlägsta möjliga Q-faktorn för små antenner, och följaktligen erhålla implicit denmaximala tillgängliga bandbredden för en given reflektionskoefficient. Vi under-söker en undre begränsning av Q-faktorn under sido-villkor på t.ex. direktivitet,strålningsmönster, effektivitet och antennens inbäddade position. I denna processkombinerar vi fysikaliska metoder, matematiska verktyg och antennteknik. Vi an-vänder en momentmetod för att lösa den de elektriska integralekvationerna (EFIE).I det här sammanhanget formulerar vi antennoptimeringsproblem där ytströmtä-theten är en okänd variabel och löser konvexa och icke-konvexa kvadratiskt medkvadratiska villkor. För de icke-konvexa problemen undersöker vi olika metoderför att erhålla lösningen, med ett fokus på en relaxation teknik kallad SDR. Olikaströmoptimeringsproblem löses för en rad geometrier, där Q-faktorn och den opti-mala ytströmmen bestäms. Resultaten jämförs i några fall med fullvågsimuleringav antenner.

Att bestämma den optimala Q-faktorn för ett givet utrymme i enheten börvara ett första steg i en antennkonstruktion. En strömoptimering kan hjälpa ossatt bestämma en avvägning mellan olika prestandaparametrar hos en liten antenn,en optimal ström kan ocksåge inspiration till bra antenndesigner. Vi visar vi-dare att en multi-positions matningsstrategi fungerar för att närma sig en optimalström för dipolantennen under olika begränsningar. Strålningsmönster med givnaicke-standard egenskaper kan uppnås med småkostnader för Q-faktor även för småantenner i vissa fall. Avhandlingen avslutas med en diskussion om initiala steg moten metod för att erhålla en Q-faktor optimal antenn. Här spelar strömoptimeringenen viktig roll i antennsyntesen och analysstadier av processen. En applikation meden inbäddad antenn diskuteras i detalj.

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Preface

This thesis is in partial fulfillment for the degree of Doctor of Philosophy at KTHRoyal Institute of Technology, Stockholm, Sweden.

The work in this thesis was carried at the Department of Electromagnetic En-gineering at KTH from September 2014 till December 2018.

Professor Lars Jonsson is the main advisor, and parts of this thesis have beencosupervised by Professor Fabien Ferrero with University Nice Sophia-Antipolis andProfessor Martin Norgren with KTH.

The thesis was supported by the Swedish Foundation for Strategic Researchfor the project “Convex Analysis and Convex Optimization for EM Design” (SSF/AM130011), and the Swedish Governmental Agency for Innovation Systems (Vin-nova) through the Center ChaseOn in the project “iAA” (ChaseOn/iAA). Mr.Shuai Shi is supported by the China Scholarship Council (CSC), which is gratefullyacknowledged.

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Acknowledgements

First, I would like to express my great gratitude to my supervisor Professor LarsJonsson. His patient guidance, valuable suggestions, and constant encouragementhas helped me to successfully complete this thesis. His conscientious academic spiritand modest, open-minded personality inspire me both in academic study and dailylife.

Furthermore, I wish to extend my deep gratitude to my co-supervisor ProfessorMartin Norgren, and to Associate Professor Oscar Quevedo Teruel for his reviewof this thesis.

I wish to thank my co-authors and co-workers including Professor Mats Gustafs-son with Lund University, Professor Fabien Ferrero with University Nice Sophia-Antipolis, Dr. Lei Wang with Hamburg University of Technology, Dr. Bo Xu withEricsson Research, Dr. Kun Zhao with Sony Mobile, and my office mate AndreiOsipov with KTH. I gratefully acknowledge to Professor Daniel Sjöberg from LundUniversity and Associate Professor Miloslav Capek from Czech Technical Universityin Prague for the fruitful discussion I had with them.

I would like to thank the department head Professor Rajeev Thottappillil for hisadministrative and strategic support, Ms. Carin Norberg and Ms. Brigitt Högbergin financial administration, Mr. Peter Lönn in maintaining computers and software,and Dr. Nathaniel Taylor for the great support of the Linux server.

I would also like to thank several former and current Ph.D. and licentiate stu-dents at the department for their companionship. These include Dr. MarianaDalarsson, Dr. Christos Kolitsidas, Dr. Elena Kubyshkina, Dr. Kexin Liu, Dr.Lipeng Liu, Dr. Bing Li, Dr. Per Westerlund, Dr. Jan Henning Jürgensen, FatemehGhasemifard, Mahsa Ebhrahimpouri, Qingbi Liao, Qiao Chen, Ahmad Emadeddin,Henrik Frid, Johan Malmström, Mauricio Aljure Rey, Patrick Janus, Björn Peters-son, Kateryna Morozovska, Sanja Duvnjak Zarkovic, Mrunal Parekh, Cong-ToanPham, Yue Cui, Priyanka Shinde, Yang Jiao, Jing Hao, Xianliang Zeng, amongmany others.

Although it is not possible to mention them all, I would also like to expressmy sincere gratitude to a number of people from other departments at KTH, whohave to various extents contributed to the success of my education and subsequentresearch. A special mention here is to Professor Anders Forsgren and Ph.D. stu-dent Han Zhang with Department of Mathematics, as well as Dr. Rong Du with

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xii ACKNOWLEDGEMENTS

Department of Network and Systems Engineering, for the discussions on convexoptimization and the SDR technique.

I am also grateful to the colleagues with International Relation Office at KTH,especially Xintong Zhang, Åsa Andersson, Caroline Mosson, Daniel Ståhl, ÅsaCarlsson and Dr. Xun Wang, with whom I spent a wonderful time working oninteresting projects.

I am very fortunate to have many friends when I just arrived in Sweden living inFlemingsberg. A special mention here is to Yansong Ren, Yushan Zhou, XiaojingWang, Zibo Liu and Jinzhi Lu, together with whom I enjoyed the wonderful fouryears study in Sweden.

I deeply appreciate having my invaluable friends outside the school, includingmy teammates with Albion FC and CFC group, my friends with CSSA and myNPU alumni.

Last but not least, I want to send all my thankfulness to my parents, my wifeMeng and my daughter Emma, for their endless love and support.

Shuai Shi

Stockholm, 2018

List of Publications

The thesis is based on the following papers:

Journal Papers

I. B. L. G. Jonsson, Shuai Shi, Lei Wang, Fabien Ferrero, and Leonardo Lizzi,“On Methods to Determine Bounds on the Q-factor for a Given Directivity”,IEEE Transaction of Antennas and Propagation, 2017, 65(11): 5686-5696.

II. Shuai Shi, Lei Wang, and B. L. G. Jonsson, “Antenna Current Optimiza-tion and Realizations for Far-Field Pattern Shaping”, IEEE Transaction ofAntennas and Propagation, submitted 2018.

III. Shuai Shi, B. L. G. Jonsson, Fabien Ferrero, and Lei Wang, “An Investiga-tion of the Bandwidth Optimal Position of an Embedded Antenna”, IEEETransaction of Antennas and Propagation, to be submitted 2018.

Conference Papers

IV. Shuai Shi, Andrei Ludvig-Osipov, and B. L. G. Jonsson, “Examination ofBandwidth Bounds of Antennas using Sum Rules for the Reflection Coeffi-cient”, IEEE International Conference on Microwave and Millimeter WaveTechnology (ICMMT), Beijing, 2016, 1: 226-229.

Other papers by the author that are not included in the thesis:

Journal Papers

V. Bo Xu, Mats Gustafsson, Shuai Shi, Kun Zhao, Zhinong Ying and SailingHe, “Radio Frequency Exposure Compliance of Multiple Antennas for Cel-lular Equipment based on Semidefinite Relaxation”, IEEE Transactions onElectromagnetic Compatibility, 2018.

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xiv LIST OF PUBLICATIONS

Conference Papers

VI. B. L. G. Jonsson, Shuai Shi, and Andrei Ludvig-Osipov, “On Bounds of theQ-Factor as a Function of Array Antenna Directivity”, The 20th Interna-tional Conference on Electromagnetics in Advanced Applications (ICEAA),Cartagena, Colombia, 2018.

VII. Shuai Shi, Lei Wang, and B. L. G. Jonsson, “On Q-Factor Bounds for aGiven Front-to-Back Ratio”, IEEE International Symposium on Antennasand Propagation and USNC-URSI Radio Science Meeting (AP-S/URSI), SanDiego, California, USA, 2017: 151-152.

VIII. Bo Xu, Mats Gustafsson, Shuai Shi, Zhinong Ying, and Sailing He, “UpperBound Study of 5G RF EMF Exposure”, IEEE International Symposium onAntennas and Propagation and USNC-URSI National Radio Science Meeting(AP-S/URSI), Boston, Massachusetts, USA, 2018.

IX. Shuai Shi, B. L. G. Jonsson, Fabien Ferrero, and Lei Wang, “On the Q-Factor Optimal Position for Embedded Antennas in Small Devices”, The 40thProgress in Electromagnetics Research Symposium (PIERS), Toyama, Japan,2018, abstract.

X. Shuai Shi, B. L. G. Jonsson, and Lei Wang, “A Case Study for AntennaCurrent Optimization Problems with Quadric Constraints”, The Swedish Mi-crowave Days, Lund, Sweden, 2018, abstract.

XI. B. L. G. Jonsson, Shuai Shi, and Lei Wang, “Relaxation of Some Non-ConvexConstraints for Q-Factor Optimization”, URSI General Assembly and Scien-tific Symposium (URSI GASS), Montreal, Canada, 2017, abstract.

XII. Shuai Shi and B. L. G. Jonsson, “Investigation of the Optimal MatchingImpedance for Finite Array Antennas”, The 36th Progress in Electromagnet-ics Research Symposium (PIERS), Prague, Czech, 2015, abstract.

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Authors’ contributions to the included papers:

• In Paper I: My supervisor B. L. G. Jonsson and I solved the problem to-gether, i.e., minimizing Q-factor with total directivity constraints. I appliedthe semidefinite relaxation (SDR) method to solve it, while my supervisorintroduced an eigenvalue-based method. Both the methods work. My super-visor and Lei Wang worked on most of the numerical results. Fabien Ferreroand Leonardo Lizzi contributed mainly to the super-directive antenna design.

• In Paper II: I developed the concept, performed the simulations, preparedthe figures and the main body of the manuscript. Lei Wang helped me insome of the numerical implementations. My supervisor B. L. G. Jonsson gaveme great help in writing the manuscript and improving the paper.

• In Paper III: My supervisor B. L. G. Jonsson and I developed the concept,performed simulations and prepared the numerical results. Fabien Ferrerocontributed mainly to the antenna validation. I developed the main body ofthe manuscript. All the co-authors reviewed and edited the manuscript.

• In Paper IV: I developed the concept, performed the simulations, preparedmost of the figures and the main body of the manuscript. Andrei Ludvig-Osipov helped me in the case of the off-center feeding dipole. My supervisorB. L. G. Jonsson gave me great help in writing the manuscript and improvingthe paper.

xvi LIST OF PUBLICATIONS

Contents

Abstract v

Sammanfattning vii

Preface ix

Acknowledgements xi

List of Publications xiii

Contents xvii

List of Acronyms xix

1 Introduction 11.1 Motivation in Historical Perspective . . . . . . . . . . . . . . . . . . 11.2 Background and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theory 92.1 Time-Harmonic Maxwell Equations . . . . . . . . . . . . . . . . . . . 92.2 Green Function for Helmholtz Equations . . . . . . . . . . . . . . . . 102.3 Electric Field Integral Equation (EFIE) . . . . . . . . . . . . . . . . 112.4 Method of Moments (MoM) . . . . . . . . . . . . . . . . . . . . . . . 112.5 Convex Function and Convex Optimization . . . . . . . . . . . . . . 132.6 Semidefinite Relaxation (SDR) . . . . . . . . . . . . . . . . . . . . . 13

3 Antenna Terms 153.1 Bandwidth and Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Stored Energy and Radiated Power . . . . . . . . . . . . . . . . . . . 163.3 Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Radiation Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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xviii CONTENTS

4 Antenna Current Optimization 214.1 Minimizing Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Directivity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Pattern Shaping Constraints . . . . . . . . . . . . . . . . . . . . . . . 284.4 Radiation Efficiency Constraints . . . . . . . . . . . . . . . . . . . . 294.5 Embedded in Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Optimal Antenna Design 335.1 Antenna Geometry Generation . . . . . . . . . . . . . . . . . . . . . 345.2 Realization of the Optimal Current . . . . . . . . . . . . . . . . . . . 375.3 Comparison with the Bounds . . . . . . . . . . . . . . . . . . . . . . 43

6 Conclusions and Future Work 456.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Future Work and Extensions . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 49

List of Acronyms

2D Two-Dimensional3D Three-Dimensional5G Fifth Generation Wireless SystemsCEM Computational ElectromagneticsDoF Degrees of FreedomEFIE Electric Field Integral EquationEMC Electromagnetic CompatibilityEMF Electromagnetic FieldFBR Front-to-Back RatioFBW Fractional BandwidthFDTD Finite-Difference Time-Domain MethodFEM Finite Element MethodGPS Global Positioning SystemIEEE Institute of Electrical and Electronics EngineersIoT Internet of ThingsISM Band Industrial, Scientific, and Medical Radio BandLoRa A Wireless Data Communication Technology for IoTM2M Machine to MachineMoM Method of MomentsPEC Perfect Electric ConductorPFBR Power Front-to-Bank RatioPIFA Planar Inverted-F AntennaQCQP Quadratically Constrained Quadratic Program

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xx LIST OF ACRONYMS

RF Radio FrequencyRFID Radio Frequency IdentificationRWG Rao-Wilton-GlissonSDR Semidefinite RelaxationSRD Short-Range DeviceSigFox A Wireless Data Communication Technology for IoTWi-Fi A Wireless Local Area Networking Technology

Chapter 1

Introduction

According to the standard definition of IEEE [1], antennas are the devices “to ra-diate or to receive electromagnetic waves” in transmitting or receiving systems.During the last one and a half centuries, the theory and practice of antenna de-sign have experienced its birth and tremendous development. Nowadays, antennasare extremely widely used in our daily life and making the world increasingly con-nected. The application of antennas has covered radio, television, radar, mobilephones, satellite communications, RFID, Internet of things (IoT), Machine to Ma-chine (M2M) communication, etc.

The history of antennas is traced back to Maxwell and Hertz, see e.g. [2]. Thefirst antennas were built in a laboratory in 1888 by Hertz. Using these antennas,Hertz proved the existence of electromagnetic waves predicted by Maxwell equa-tions [3]. Since these results, in an increasing pace, antennas have been used fortransmitting signals, see e.g. [4, 5]. People keep pursuing high performance acrossdifferent sets of parameters while demanding antenna designs with low cost andlimited size. As a consequence, studies on how key antenna parameters competeor collaborate in trade-off have become an important area. In this thesis, we focuson the limitations and the optimal trade-off between parameters of small antennas.Mathematical tools like current optimization and sum rules are used to find thephysical bounds and investigate the Q-factor optimal antennas.

1.1 Motivation in Historical Perspective

The antenna design methods can be divided clearly into two main eras, i.e., theage before computers and the time after. Advances in computer science since the1960s have had a dramatic influence on the development of modern antenna tech-nology, and they are expected to make a greater difference on antenna design in thefuture [2, 6]. Meanwhile, people start to think about what is the best or optimalantennas with respect to different parameters and the trade-off between them.

1

2 CHAPTER 1. INTRODUCTION

History: Before and After Computers

In the time without computers, antenna designers had to develop theoretical modelsand solve them by hand. Antenna theory was analytically difficult because theelectromagnetic wave equations must be solved with different boundary conditionsover complex antenna shapes. In many cases, the exact analysis was impossibleso approximations were made. Nevertheless, the great scientists developed severalphenomenal types of antennas, which are classical and still used everywhere today.Prior to World War II, most of the antennas are of wire type (e.g., dipoles, helices,rhombuses, and Yagi-Uda antennas). Large aperture antennas (such as horns,reflectors, and lenses) came out between the 1940s and the 1960s because the opticalprinciples and diffraction theory were applied to predict the patterns [4].

Since the 1960s, when computers became commonly available, numerical meth-ods have been introduced to antenna engineering, and computational electromag-netics (CEM) has become a dominant tool in antenna design. For example, theMethod of Moments (MoM) enables the current on a small antenna to be predictedwith a much better accuracy than earlier. Finite-difference time-domain method(FDTD) and finite element method (FEM) are also used for antenna simulationby computers, which makes it possible to solve the electromagnetic wave equationsdirectly [2,6]. Computer simulation tools can give accurate predictions for antennasperformance before the prototype testing, and it increases humans’ confidence toanalyze and design antennas. As a result, the microstrip or patch antennas havebeen developed in the 1970s, and there has been much development with othertypes of antennas as well, see e.g. [7].

In spite of the help of computer simulations, antenna design is still far awayfrom an easy skill. The design process typically includes physical modeling, thumbrules, computer simulation, optimization, and experimental verification. This istime-consuming and laborious and requires a lot of expertise and experience. As aconsequence, people expect that the computer can play a greater role in antennaengineering. Different optimization methods have been applied into antenna de-sign process since the 1990s. For example, heuristic optimization methods, suchas genetic algorithms and particle swarms, have been used to improve antennaperformance by including different parameters as optimization targets, and manynew antennas with good performance are designed, see e.g. [8–17]. As computerscan help us design better antennas, people begin to care about the limitations ofantenna performance, such as: What is the optimal radiation pattern? What isthe maximum bandwidth or highest gain, if the antenna size is given? How can Idesign optimal antennas with regard to these parameters?

Future: Optimal Antenna Design

Antenna design is often restricted by its size, geometry or topology, while the designrequirements often combine different parameters such as bandwidth, directivity,radiation pattern and radiation efficiency. As mentioned above, it is desirable to

1.2. BACKGROUND AND SCOPE 3

design antennas that perform better while being smaller in size. Unfortunately, theperformance parameters of an antenna, e.g., bandwidth, directivity, and efficiency,are in strict opposition to its electrical size. Small (sub-wavelength) antennas usu-ally have narrow bandwidth, low radiation efficiency and limited types of radiationpatterns [4, 18]. Thus we want to find an optimal trade-off or the fundamentalphysical limitations of antennas.

The limitations or bounds are the best performance characteristics when an-tenna designs are restricted, for example, by size [19–25]. Fundamental physi-cal limitations give a priori estimates on antenna design possibilities with respectto different antenna evaluation parameters, e.g., bandwidth, directivity, and effi-ciency [22]. For example, studying antenna limitations for different basic geome-tries can provide theoretical experience for us to choose antenna shapes for design.Besides, some optimization methods can give the optimal current for specified con-straints, such as size, bandwidth, gain, and so on, and thus a wide range of antennadesign issues can be translated into specific current implementation problems. Asshown in this thesis, the combination of antenna engineering skills with appliedmathematical tools and numerical optimization methods help us to find the opti-mal or near-optimal solutions for antenna design.

1.2 Background and Scope

There are different approaches to find the physical limitations of small antennas,e.g., circuit models, vector models, sum rules and stored energy [19, 20, 22–30],see Figure 1.1. In this thesis, we use antenna current optimization based on storedenergy to find the fundamental Q-factor bounds for antennas with limited size. Herethe Q-factor is a good estimation for the bandwidth of small antennas [21–25, 29–31]. Parameters like directivity, far-field radiation pattern, losses, and embeddedpositions can be included as trade-off constraints for optimizations.

Q-factor for Small Antennas

For small antennas, the Q-factor is a good approximation for the impedance band-width [21–25,29–31], as shown in Figure 1.2, and discussed in detail in Section 3.1.The Q-factor furthermore provides interesting and useful information about an-tenna design possibilities [21–25]. The first limitation in terms of Q-factor for smallantennas bounded by sphere was given by Wheeler and by Chu in 1940s [19, 32],where they transferred the complicated electromagnetic problem into a circuit prob-lem. In recent years, the Q-factor limitations on small antennas have been developedto tighter bounds for arbitrarily shaped antennas, based on stored energy and usingcurrent optimization, see e.g. [22, 33], Paper I and Paper II.

4 CHAPTER 1. INTRODUCTION

Figure 1.1: The approaches to find the antenna physical bounds.

Figure 1.2: The comparison of a dipole antenna model and a resonant Q model.

1.2. BACKGROUND AND SCOPE 5

Antenna Current OptimizationThe idea behind and usage of antenna current optimization was first publishedin [22], solving a range of convex Q-related problem with various constraints.Semidefinite relaxation (SDR) technique is first applied to non-convex quadrati-cally constrained quadratic programs (QCQPs) in Paper I, where the current op-timization problems are solved for total directivity. In Paper II, a class of far-fieldpattern shaping problems is considered, and the validation of the optimal currentis investigated using a multi-position feeding strategy. Antennas with a groundplane and embedded in a device are studied in [22, 34, 35] and in Paper III, wheresimplifications are made for the current optimization problems.

Generally, antenna current optimization is to find an optimal current distri-bution in a given antenna geometry or space with regard to the selected antennaperformance parameters. Here the current distribution is not restricted to anyspecific feed-point or antenna placement constraints in the design region, thus thecurrent distribution from any antenna in the space is a possible candidate or a fea-sible solution to the optimization problem [22, 36]. Therefore, the optimal currentmethod can be used to determine the physical bounds for antennas restricted tothe design region.

For example, in Paper I, we consider two rectangular plates of size x = 32 mm,y = 44 mm, and with a 40 mm space between them along the z-axis, see Fig-ure 1.3 (a). The optimization problem is minimizing the Q-factor with a directivityconstraint near f = 868 MHz (the LoRa and SigFox ISM-band in Europe), whichgives practical Q-factor bounds. Figure 1.3 (b) depicts the antenna design, a high-directivity parasitic element antenna of the shape fits within the left two-rectanglestructure, for which more detailed realization and discussion see [37, 38]. We willdiscuss in detail on this example later on in this thesis.

Sum Rules ApproachBesides the current optimization approach, sum rules are also used to investigatethe bandwidth limitations for one-port antennas [39–43] and arrays [44]. For losslessantennas, we have an integral type of bounds for the reflection coefficient in terms ofthe low-frequency behavior of the input impedance (capacitance and inductance).These identities are a consequence of that the antenna system is passive, linear andtime-invariant, and based on the properties of positive real (PR) functions. A shorttheory and numerical investigations are illustrated and discussed in Paper IV.

6 CHAPTER 1. INTRODUCTION

Figure 1.3: The two-plate design region and the model of parasitic superdirectiveantenna. (© 2017 IEEE)

1.3 Thesis Structure

The organization of this thesis is as follows.In Chapter 2, we introduce the physical theory and mathematical tools which

are used in the antenna problems in this thesis. We start with Maxwell equations,Helmholtz equations, and their Green function. Then we introduce the ElectricField Integral Equation (EFIE) and present the general procedure of solving EFIEusing the Method of Moments (MoM). We show that many antenna current op-timization problems belong to the quadratically constrained quadratic programs(QCQPs). For the non-convex QCQPs, the semidefinite relaxation (SDR) tech-nique is used to relax them to linear problems which can be solved by efficientmethods.

In Chapter 3, we state the antenna parameters used in the following optimiza-tion and design. Firstly Q-factor is discussed in detail in terms of stored energies.Then the energies and power are expressed in their integral forms in terms of thesurface current density. Radiation pattern, directivity, and radiation efficiency arealso introduced since these parameters will be used as design requirements or op-timization constraints. All the parameters are also presented within MoM matrixforms so that they can be used for antenna current optimization.

In Chapter 4, different antennas current optimization problems are presented,including minimizing Q-factor, with constraints of directivity, pattern shaping, andlosses. We also present the corresponding physical Q-factor limitations for someantennas as case studies. Embedded antennas have an advantage for current opti-mization problems because they can be solved faster than the whole devices.

In Chapter 5, we summarize how the antenna bounds and current optimization

1.3. THESIS STRUCTURE 7

help on the motivation, i.e., optimal antenna design. Examples in the papers areused to demonstrate the concept of optimal or near optimal design. For instance,before antenna design and computer simulation, we can investigate the optimalgeometry and the embedded position of the antenna, and the optimal current cangive us a good prediction for the design. By comparing the antenna performancewith the physical bounds, we can have a general idea of how good the antenna isand the trade-off between antenna parameters.

The thesis ends with a conclusion, future work, and a bibliography.

Chapter 2

Theory

In this chapter, the physical theory and mathematical tools for antenna currentoptimization are introduced. We start with the classical Maxwell Equations andthen state a procedure that solves Electric Field Integral Equation (EFIE) usingMethod of Moments (MoM). This way we obtain the antenna current produced bythe incident wave or the port excitations. Thereafter, the mathematical optimiza-tion problems are introduced, and the semidefinite relaxation (SDR) technique ispresented to solve a certain class of non-convex quadratically constrained quadraticprograms (QCQPs). Most of the antenna current optimization problems in the fol-lowing chapters belong to this class. The theory and methods in this chapter arenot original by the author, but they are important prior knowledge for this thesis.

2.1 Time-Harmonic Maxwell Equations

For time-harmonic electromagnetic fields, the Maxwell equations with a time-factorejωt are

O×E = −jωB, (2.1)O×H = J + jωD, (2.2)

O ·B = 0, (2.3)O ·D = ρ, (2.4)

where D = εE is the electric field, B = µH is the magnetic field, ρ and Jare charge and current density distributions, µ and ε are homogeneous isotropicmaterial coefficients. From (2.2) and (2.4) we can find the continuity equation forcharge density and current density

O · J = −jωρ. (2.5)

We can also reduce Maxwell equations to Helmholtz equations

(∆ + k2)E = 1εOρ+ jωµJ , (2.6)

9

10 CHAPTER 2. THEORY

(∆ + k2)H = −O× J , (2.7)which determine the fields for given charge and current distribution. Here k = ω

√µε

is the wave number.

2.2 Green Function for Helmholtz Equations

In electromagnetism, the free space Green function helps us to solve Helmholtzequations. Considering the Helmholtz equation with radiation conditions in time-harmonic media

(∆ + k2)G(r1, r2, k) = −δ(r1 − r2), (2.8)where r1 ∈ Rn is the observation point, and r2 ∈ Rn is the source point, thesolution of (2.8) is

G(r1, r2, k) = e−jk|r1−r2|

4π|r1 − r2|, (2.9)

which represents the spherical wave out from r = r2.For example, if we consider generated electromagnetic fields by source ρ and J ,

both the scalar potential φ and the vector potential A satisfy Helmholtz equations:

(∆ + k2)A = −µJ , (2.10)

(∆ + k2)φ = −ρε. (2.11)

Their respective solutions are

A(r1) = µ

4π

∫V

e−jk|r1−r2|

|r1 − r2|J(r2) dV2, (2.12)

φ(r1) = 14πε

∫V

e−jk|r1−r2|

|r1 − r2|ρ(r2) dV2. (2.13)

Then we can derive the electric field E and magnetic field H generated by thesources from these potentials

E = −Oφ− jωA, (2.14)

H = 1µO×A. (2.15)

Besides, we can also obtain the fields using Green functions from Helmholtz equa-tions (2.6) and (2.7), and the solutions are

E(r1) = − 14π

∫V

e−jk|r1−r2|

|r1 − r2|

(O2ρ(r2)ε

+ jωµJ(r2))

dV2, (2.16)

H(r1) = 14π

∫V

e−jk|r1−r2|

|r1 − r2|O2 × J(r2) dV2, (2.17)

which are the same fields as we determined in (2.14) and (2.15) with help of thepotentials.

2.3. ELECTRIC FIELD INTEGRAL EQUATION (EFIE) 11

2.3 Electric Field Integral Equation (EFIE)

Considering a scattering problem for a perfect electric conducting (PEC) antennasituated in free-space with an excitation or an incident field Ei(r1), a currentdensity J(r2) is produced on the surface of the antenna, which generates a scatteredelectric field Es(r1). In this case, the total field is the sum of the incident andscattered fields

Etotal(r1) = Ei(r1) +Es(r1). (2.18)

Then the scattered field can be determined by the boundary condition on the con-ductor surface

n×Etotal(r1) = n× (Ei(r1) +Es(r1)) = 0. (2.19)

From the combination of (2.5), (2.12), (2.13) and (2.14), we obtain the scatteredelectric field given by the current:

Es(r1) = −jωµ4π

∫S

e−jk|r1−r2|

|r1 − r2|J(r2) dS2 −

j

4πεωO1

∫S

e−jk|r1−r2|

|r1 − r2|O2 · J(r2) dS2,

(2.20)where S2 is the antenna surface. Since on the surface n×Ei(r1) = −n×Es(r1),we obtain the relation between the incident field Ei(r1) (known) and the producedsurface current J(r2) (unknown) which is

n×Ei(r1) = n×(LJ)(r1)

= n×(jωµ

4π

∫S

e−jk|r1−r2|

|r1 − r2|J(r2) dS2 + j

4πεωO1

∫S

e−jk|r1−r2|

|r1 − r2|O2 · J(r2) dS2

).

(2.21)

where we introduce a linear operator L for the integral relation. In electromag-netism, an integral equation like (2.21) with unknown current is called an ElectricField Integral Equation (EFIE).

2.4 Method of Moments (MoM)

Method of Moments (MoM) is one of the main techniques to solve integral equa-tions. For antenna design, MoM enables the current on a small antenna to bepredicted with a good accuracy. Here considering the EFIE (2.21), where the vec-tor E(r1) is known, we expand the unknown functions J(r2) as

J(r2) ≈N∑n=1

Inψn(r2), (2.22)

where a current vector I ∈ CN with elements In is introduced to simplify thenotation. So In is an unknown expansion coefficient, and ψn is a known functioncalled a basis function. Solving the equation (2.21) with the help of (2.22) thus

12 CHAPTER 2. THEORY

reduces to finding the expansion coefficients for all n. To this end substituting(2.22) into the equation (2.21), we have

n×N∑n=1

In(Lψn

)(r1) ≈ n×E(r1). (2.23)

Thereafter we use the inner product with ωm(r) on both sides of (2.23), whichleads to

N∑n=1

In〈ωm(r1),(Lψn

)(r1)〉 ≈ 〈ωm(r1),E(r1)〉. (2.24)

Here the definition of the inner product is

〈ωm(r),ψn(r)〉 =∫S

ω∗m(r) ·ψn(r) dS, (2.25)

where “∗” denotes the complex conjugate and ωm(r1) is the test basis function. InMoM, (2.24) can be written in a matrix form

ZI ≈ V, (2.26)

where Z ∈ CN×N is the impedance matrix, and V ∈ CN is the excitation columnvector. Individual elements of Z and V are

Zmn = 〈ωm(r),L(ψn(r))〉

= jη

4π

∫S

∫S

e−jk|r1−r2|

|r1 − r2|

(kωm(r1) ·ψn(r2)− 1

kO1 · ωn(r1)O2 ·ψn(r2)

)dS1 dS2

(2.27)

and

Vm = 〈ωm(r1),E(r1)〉 =∫S

ωm(r1) ·E(r1) dS1, (2.28)

respectively. Here η =√µ/ε is the free-space impedance. The expansion coeffi-

cients, or the current density vector I can be solved as I = Z−1V. This methodworks for both scattering and radiation problems, by defining the excitation V asthe incident wave or voltage feedings respectively.

In this thesis, we use an in-house MoM code where Rao-Wilton-Gilson (RWG)edge elements are chosen, see [45]. The Galerkin’s Method, i.e., using the samebasis functions for ψn(r) and ωn(r), are applied, which makes Z symmetric andthus reduces the computation [4].

The current optimization approach in this thesis is formulated in terms of thecurrent vector I, and the required matrices are determined using methods similarto the calculation of Z. A more detailed discussion on the stored energies is givenin Section 3.2.

2.5. CONVEX FUNCTION AND CONVEX OPTIMIZATION 13

Figure 2.1: Convex function.

2.5 Convex Function and Convex Optimization

A convex function f(x) : Rn → R satisfies the inequality [36]:

f(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y), (2.29)

see Figure 2.1. For convex functions, the minimum is always a global minimum [36].We use the standard notation of [36] for the mathematical optimization prob-

lems:minimize f0(x)

subject to fi(x) ≤ 0,hj(x) = 0,

(2.30)

where i = 1, 2, ...,M, j = 1, 2, ...,N indicates the number of different constraints.An optimization problem is convex if and only if the objective function f0(x) and in-equality constraint functions fi(x) are convex functions, and the equality constraintfunctions hj(x) are affine (a sum of a linear function and a constant, with the formof hj(x) = Ajx+ bj). In this thesis, we consider minimization problems expressedin a linear form f(I) = bI and a quadratic form f(I) = IHCI, where bH, I ∈ CNis N -vectors with complex coefficients, C is N × N matrices, and the superscript“H” denotes the Hermitian transpose. For the quadratic form, a matrix C must bepositive semidefinite and Hermitian to make (2.30) a convex optimization problem.

2.6 Semidefinite Relaxation (SDR)

In recent years, the semidefinite relaxation (SDR) has been known as an efficienthigh-performance approach in the area of signal processing and communications, forinstance, MIMO detection, sensor network localization, and phase retrieval [46–48].It is also used to find the physical limitation of MIMO antennas, and the upperbound of 5G RF electromagnetic field exposure [49,50], and in wireless power trans-fer [51]. Roughly speaking, SDR is a powerful, computationally efficient approxi-mation technique for a host of very difficult optimization problems. In particular, itcan be applied to many non-convex quadratically constrained quadratic programs

14 CHAPTER 2. THEORY

(QCQPs) [46,48,52,53]. These include the following type of problems:

minimizeI∈CN

IHCI

subject to IHMiI = ai,

IHNjI ≥ bj ,

(Q)

where C, Mi and Nj are N × N matrices, ai and bj are constants. Most of theantenna current optimization problems belong to this class of problems, since thestored energy and radiated power are both the quadratic forms of current vectors.

The problem (Q) is here in general non-convex for matrices C, Mi and Nj .Certain non-convex problem can be relaxed into a convex problem, see e.g. [36].Here we apply the semidefinite relaxation (SDR) technique to relax (Q) into aconvex, semidefinite problem [46, 48, 52–54], see also [55–57]. The procedure is asfollows:

IHMiI = tr (IHMiI) = tr (MiIIH), (2.31)

and similarly for Nj, C. We can thus rewrite (Q) as

minimizeK∈HN×N ,K≥0

tr (CK)

subject to tr (MiK) = ai,

tr (NjK) ≥ bj .

(R)

where K = IIH is an N × N Hermitian matrix. The problem (R) together withrank(K) = 1 is equivalent with (Q) [46,48,52,53]. We observe that the constraintsin (R) are affine functions in the matrix K, and that K = IIH is equivalent tothat K is a rank one semidefinite Hermitian matrix. If we drop the constraintrank(K) = 1, we get the relaxed problem (R), which is known as an SDR problemof (Q). (R) is convex (linear), and it can be solved by many efficient methods. Inthis thesis, we use an efficient solver called CVX [36, 58, 59], which can be utilizedwithin e.g. Matlab. We can thus get a globally optimal solution K∗ to problem (R),that is clearly a lower bound for the problem (Q). If K∗ is of rank one, we determineI∗ from K = I∗IH

∗ , and I∗ will be a feasible and optimal solution to the originalproblem. It is shown in [60], see also [57], that if the number of the constraintsM+N is small, then there exists a solution with rank(K∗) = 1. If rank(K∗) > 1,we need to convert K∗ into a feasible approximation I to solve problem (Q) usingthe methods from e.g. [46,48].

Besides the SDR technique, an eigenvalue-based method can also be used tosolve the non-convex problems (Q), see e.g. [61] and Paper I. The idea of theeigenvalue method is to find the Pareto front for the dual problems by sweepingthe relative parameters.

Chapter 3

Antenna Terms

As mentioned above, this thesis investigates the methods to determine the limi-tations and the optimal trade-off between different antenna parameters. In thischapter, several performance parameters of small antennas are introduced, includ-ing bandwidth, Q-factor, stored energy, radiation pattern, directivity, gain, andradiation efficiency. As a preparation for current optimization in the followingchapters, we present the antenna parameters within both physical expression andMoM matrix forms.

3.1 Bandwidth and Q-factor

The antenna bandwidth is defined as “the range of frequencies within which theperformance of the antenna, with respect to some characteristic, conforms to aspecified standard” [4]. For small antennas, we often use the fractional impedancebandwidth which expressed as a ratio in percentage:

FBW = f2 − f1

f0, (3.1)

where f0 = (f1 + f2)/2 is the center frequency. The reflection coefficient Γ of aone-port antenna is defined as

Γ = Zin − Z0

Zin + Z0, (3.2)

where Z0 is the characteristic impedance of the antenna port and Zin is the inputimpedance.

For a single resonant antenna, the fractional bandwidth is related to the Q-factor, see e.g. [21, 62]:

FBW ≈ 2|Γ0|Q√

1− |Γ0|2, (3.3)

15

16 CHAPTER 3. ANTENNA TERMS

where |Γ0| denotes the maximally allowed reflection coefficient. It has been shownin [21] that the Q-factor is a good estimation of the bandwidth. Usually for Q & 5,we want to maximize the bandwidth, which corresponds to minimizing the antennaQ, when we design an optimal antenna.

There are different approaches to determine the Q-factor for antennas, seee.g. [21,31]. Utilizing stored energies we can define the Q-factor as the ratio betweenthe time-average stored energy and the dissipated power

Q = 2ωmax(We,Wm)Prad + Ploss

, (3.4)

where ω is the angular frequency, We and Wm are the stored electric and magneticenergy, Prad and Ploss are the total radiated power and ohmic losses respectively.The total radiated power Prad of the antenna is defined as

Prad =∫

S2U(r) dΩ = 1

2η limR0→∞

∫|r|=R0

|E|2 dS, (3.5)

where S2 is the unit sphere in R3, U(r) is the radiation intensity in the directionr, dΩ = sin θ dθ dφ, where we let θ be the polar angle and φ be the azimuth angle.Here E is the radiated electric field, dS = r2 dΩ, and η is the free space impedance.For lossless antennas, Ploss = 0.

If the antenna input impedance is known, we have

Zin = Rin + jXin = 2Prad + 4jω(Wm −We)|Iin|2

. (3.6)

where Rin and Xin are the real and imagary part of the input impedance Zinrespectively. Then we can estimate the Q-factor, QZ , from Zin by the relationin [21]

QZ =√

(ωR′in)2 + (ωX ′in + |Xin|)2

2Rin. (3.7)

Both (3.4) and (3.7) can be used to calculate the Q-factor for small antennas.Utilizing (3.7) we can determine QZ when we have a realized antenna. The defini-tion (3.4) provides Q-factor bounds for all antenna geometries within a given region,e.g., before the antenna is realized. For realizable antennas with large enough Q-factor values, Q ≈ QZ , see e.g. [31].

3.2 Stored Energy and Radiated Power

Stored energy of antennas has been extensively examined in [5,21,23,27–31,40,62–67]. The stored electric and magnetic energies can be determined by the integral

3.3. RADIATION PATTERN 17

of surface current densities, see e.g. [29, 68]:

We = η

4ω

∫S

∫S

(∇1 · J1)(∇2 · J∗2 )cos(kr12)4πkr12

−(k2J1 · J∗2 −∇1 · J1∇2 · J∗2

) sin(kr12)8π dS1 dS2, (3.8)

Wm = η

4ω

∫S

∫S

k2J1 · J∗2cos(kr12)4πkr12

−(k2J1 · J∗2 −∇1 · J1∇2 · J∗2

) sin(kr12)8π dS1 dS2, (3.9)

where we use the notation r1, r2 ∈ R3, J1 = J(r1), J2 = J(r2), r12 = |r1 − r2|.Similarly, the total radiated power is

Prad = η

2

∫S

∫S

(k2J1 · J∗2 −∇1 · J1∇2 · J∗2 ) sin(kr12)4πkr12

dS1 dS2. (3.10)

Noticing thatWe,Wm largely correspond to components in the electric field integralused in the Method of Moments (MoM), we find the matrix formulations of thestored energy and the radiated power as well as the power intensity, following thenotation of [68]:

We ≈18IH(∂X

∂ω− Xω

)I = 1

4ω IHXeI, (3.11)

Wm ≈18IH(∂X

∂ω+ Xω

)I = 1

4ω IHXmI, (3.12)

Prad ≈12IHRI, (3.13)

where R = Re (Z), X = Im (Z), and Z is the MoM impedance matrix. It is clearthat all the above antenna quantities are quadric forms of the current I.

3.3 Radiation Pattern

An antenna radiation pattern is defined as “a mathematical function or a graphicalrepresentation of the radiation properties of the antenna as a function of space co-ordinates. In most cases, the radiation pattern is determined in the far-field regionand is represented as a function of the directional coordinates” [1, 4]. Normally,the patterns are normalized to their maximum values, resulting in the normalizedfield and power patterns. In addition, the power patterns are usually plotted on alogarithmic scale (in dB), see e.g. Figure 4.6.

18 CHAPTER 3. ANTENNA TERMS

We can express the far field radiation pattern using surface current density J .The far-field radiation intensity is

U(r) = η

32π2

∫S

∫S

k2J1 · J∗2 ejk(r1−r2)·r dS1 dS2 ≈12η IH(FHF

)I = 1

2η |FI|2,

(3.14)where FI = F(r)I ≈ |F (r)|, and in the far field rE(r1)ejkr → F (r) as r →∞.

3.4 Directivity

Directivity is an essential antenna design parameter. The directivity D(r) is theratio of the radiation intensity in a direction r to the average radiation intensity [1]:

D(r) = U(r)U

= 4πU(r)Prad

≈ 4πη

IH(FHF)I

IHRI . (3.15)

The partial directivity D(r, e) is defined as the the directivity in one direction rwith one polarization e, see e.g. [1, 22].

Theoretically, the directivity of an antenna can be infinitely high, which hasbeen shown in [69–71]. However, the price for a high directivity is a high Q-factor.Antennas with superdirectivity have been studied in [28, 72–76]. In [77, 78], theexplicit relation between the possible bandwidth and the directivity was initiallyinvestigated for arrays. Small antennas with given partial directivity were studiedin [22,68] using convex current optimization. There are also other approaches to ex-plicitly obtain the trade-off between bandwidth and directivity, e.g., the eigenvaluemethods [37,57,61], and the degrees of freedom (DoF) method [73].

3.5 Radiation Efficiency

The radiation efficiency e is used to take into account losses (usually conduction anddielectric) within the structure of the antenna, see e.g. [4]. Besides, the radiationefficiency also relates the gain and directivity: G(r) = eD(r). For lossless antennas,we have G(r) = D(r).

For small antennas of metal, the losses are the ohmic losses or joule heatingcaused by the current density flowing on a non-perfect conductor. In this case, theradiation efficiency is

e = Prad

Prad + Ploss= 1

1 + δ≈ IHRI

IH(R + RΩ)I , (3.16)

where δ is the dissipated factor [79,80]

δ = Ploss

Prad= 1− e

e. (3.17)

3.5. RADIATION EFFICIENCY 19

Here the MoM impedance matrix become

Z = (R + RΩ) + j(Xm −Xe), (3.18)

where we ignored the reactive contribution. For pure metal antenna with resistivesheets, the ohmic losses is [80]

Ploss ≈12IHRΩI = Rs

2 IHΨI, (3.19)

where Rs =√ωµ0/2σ is the resistance, σ is the conductivity, µ0 is the free space

permeability, and Ψ is the Gram matrix with elements [80]

Ψmn =∫S

∫S

ψm(r1) · δ(r1 − r2) · ψn(r2) dS1 dS2 =∫S

ψm(r) · ψn(r) dS. (3.20)

Above we have introduced certain important antenna performance parametersutilized in this thesis. In the next chapters, we present how to investigate thetrade-off between them and their physical bounds using current optimization.

Chapter 4

Antenna Current Optimization

As stated in the introduction, current optimization based on stored energy is oneof the main approaches to find antenna limitations. The idea is to find an optimalcurrent distribution on the antenna surface with regard to certain performanceparameters. Here the optimization problems are minimizing the Q-factor, andconstraints are added in terms of directivity, radiation pattern and/or radiationefficiency. We also consider embedded antenna optimization, which is an importanttopic for IoT applications. We also show that embedded cases reduce the size ofthe minimization problem.

4.1 Minimizing Q-factor

To determine the minimum Q-factor for lossless antennas, we consider all possiblecurrents for which We <∞, Wm <∞, Prad > 0 that minimize the Q-factor:

Qmin = minJ

2ωmax(We,Wm)Prad

. (4.1)

We can rewrite the current optimization problem (4.1) into the stand QCQP formbecause of the scaling invariance:

minimizeI∈CN

maxIHXeI, IHXmI

subject to IHRI = 1,(4.2)

where we use the power normalization Prad = 2 or IHRI = 1. Problem (4.2)is in the class of QCQPs (Q), thus it can be solved by the SDR technique. Withadditional linear or quadric constraints, we can formulate different kinds of antennacurrent optimization problems, to determine the lowest Q-factor with requirementson directivity, radiation pattern, etc. A small Q-factor ensures through (3.3) a largefractional bandwidth at a given worst reflection coefficient.

21

22 CHAPTER 4. ANTENNA CURRENT OPTIMIZATION

As an example in Paper II, we study an infinitely thin strip dipole with thegeometry lx = 0.02lz, as shown in the right of Figure 4.1. The frequency is chosenaccording to the size, such that lz = 0.48λ, or klz = 0.48 · 2π. In the MoM, weuse a mesh with Nx = 2 in x-direction and Nz = 100 in z-direction. Meshesnot fine enough gives higher Q-factors, since there would not be sufficient manychoices with current, which will be discussed later on. With the minimizing Qproblem (4.2), we find the Q-factor for the classical half-wavelength dipole, whichis about 5 depending on the geometry of the dipole.

4.2 Directivity Constraints

The problem of the partial directivity over Q-factor (maximizing D(r, e)/Q) canbe reformulated as a convex optimization problem, as shown in [22], where theauthors choose F(r, e)I = −j as a constraint. A minimizing Q problem with partialdirectivity at least Dp0 is

minimizeI∈CN

maxIHXeI, IHXmI

subject to F(r, e)I = −j,

IHRI = 1 ≤ 4πηDp0

,

(4.3)

which has been shown to predict the behavior of good superdirective antennas [22,68].

The method solving non-convex QCQPs in this thesis makes it possible to con-sider the total directivity D(r). To design an antenna with directivity of at leastD0 in the r direction, i.e., D(r) ≥ D0, we have the optimization problem with twoquadric constraints [57]:

minimizeI∈CN

maxIHXeI, IHXmI

subject to IHRI = 1,

IHMI ≥ ηD0

4π ,

(4.4)

whereM = F(r)HF(r) = F(r, e)HF(r, e) + F(r, h)HF(r, h). (4.5)

Here [r, e, h] is a right-hand coordinate system. Note that when we take M =F(r, e)HF(r, e), it become a partial directivity constraint, then physically (4.4) isequivalent to the problem (4.3).

Half-wavelength DipoleThe lowest Q-factor bounds with partial and total directivity constraints are com-pared for the half-wavelength dipole in Figure 4.1. We investigate the bandwidth

4.2. DIRECTIVITY CONSTRAINTS 23

Figure 4.1: The lower bounds on the Q-factor for a given desired partial or totaldirectivity at r1 : θ = 90, φ = 0 and r2 : θ = 10, φ = 45 for the strip dipole.

2 2.5 3 3.5 4 4.5 54

10

20

30

4050

100

10 log10(D)

(ka)3Q

90

60

50

403020

10100

0a

x

y

Figure 4.2: The lower bounds on the Q-factor for the rectangular plate with a givendesired partial or total directivity when sweeping the direction r(θ, φ) for the polarangle θ ∈ [0, 90] and φ = 0. The total directivity optimization curves are linesmarked with small dots and are depicted above for θ ∈ [0, 10, 20, 90]. (© 2017IEEE)

24 CHAPTER 4. ANTENNA CURRENT OPTIMIZATION

cost associated with an increase of the directivity in the r direction by optimizingover all possible currents. It is well-known that the directivity for a half-wavelengthelectric dipole is about 1.64, see e.g. [4]. As the desired directivity goes up, the min-imum Q increases rapidly, which corresponds to a smaller relative bandwidth, asshown in Figure 4.1. The partial directivity for polarization θ and the total direc-tivity are compared at r1 : θ = 90, φ = 0 and r2 : θ = 10, φ = 45. Asshown in the result, at r1, the polarization is exactly θ, so the curves of partial andtotal overlap. At r2, the polarization is the combination of both θ and ϕ, thereforewe find that the total directivity gives a lower Q-bound than the partial directivity.By comparison, we see that the two methods give the same result if we know andchoose the polarization in advance. As expected, we see that a small increase indirectivity rapidly reduces the best available bandwidth.

The stability of the solutions is investigated here by increasing the mesh size.The solid lines are with Nx = 2 and Nz = 100. We have also used Nx = 3 andNz = 150 for the total directivity cases, with only a few points inserted in Figure 4.1(the red triangle or diamond points). We see that the points coincide with theircorresponding two lines respectively, indicating that the solution has converged.

Rectangular PlateAnother case study comparing partial and total directivity is for an infinitely thinrectangle of size 32 mm × 44 mm located in the xy-plane, at the frequency f =900 MHz (in Paper I). The geometry and its corresponding Q-factor bounds areshown in Figure 4.2, where we use the (ka)3 scaling of Q-factor to make the resultless frequency dependent. In the calculation, we use an N = 10 × 14 regular gridmesh on the rectangle. We observe that with total directivity constraint it has alower Q-bound for θ ≤ 10 as compared with the partial directivity optimizationproblem (e = ϕ = y), in particular for D > 2.1 and D > 2.5 respective for θ = 0and θ = 10. For θ ∈ [20 ∼ 90] both minimizations yield the same minimum inthe considered range of Q-values. These behaviors agree with the investigation andinterpretation of the dipole above. Apart from that, we see the total directivityproblems find solutions also for lower directivity, as is indicated as the horizontal-tail of θ = 90 in light blue, in the lower right-hand corner of the graph.

Two-plate StructureIn Paper I, for the two rectangular plates shown in Figure 1.3 (a), we investigatethe radiation patterns of partial directivity problems in different directions. Itshows that the minimization problem (4.3) does not state that we obtain the peakpartial directivity in the given direction r, but only that the directivity in thedesired direction is larger than Dp0. We can have the same statement for the totaldirectivity problem (4.4) as well. Here the partial directivity optimization problemwith polarization e = ϕ = y is solved at 900 MHz for a range of different directionsr(θ, φ), where θ ∈ [0, 90] and φ = 0. The radiation patterns corresponding to

4.2. DIRECTIVITY CONSTRAINTS 25

the optimal solutions are shown in Figure 4.3, where the blue curve is for constraintDp0 = 5.4, and the red is for minimizing Q problems without directivity constraints.We observe in Figure 4.3 (b) that the peak directivity of the red curve is not atthe aimed angle for 36, but at θ ≈ 70. It is because that a current slightlyperturbed with the optimal current with low-Q radiation at 90 (see Figure 4.4)can ensure that the direction 36 has large enough far-field amplitude, resultingin peak radiation direction at θ ≈ 70. In this case, it is more costly in Q-factor(smaller bandwidth) to have the peak directivity at 36 than it is to scale up aslightly perturbed radiation pattern with peak directivity at 90. By increasingthe demands on partial directivity to Dp0 ≈ 5.4, the peak direction moves moretowards 36, see the blue line in Figure 4.3 (b). It is because it costs more in Q-factor to use a peak directivity in another direction since high directivity amountsto less radiation in non-peak directions. We see the same phenomena for the desireddirectivity at 63 in Figure 4.3 (c).

The lower bounds on Q for a given partial directivity (4.3) are depicted in bluishcolors in Figure 4.4, and here the red envelope in Figure 4.4 is the smallest Q insolving (4.3) sweeping θ ∈ [0, 90]. In essence, the red curve is a description of thelowest Q-factor increase with demands on directivity. The green lines consists of thefrequencies f = [0.88, 0.89, 0.90] GHz for θ = 90. Due to the (ka)3-normalizationof the Q-factor, we find that they essentially are coinciding. Thus these boundsare frequency independent over a small range of frequency. In addition, we alsocompare the solutions by SDR and the eigenvalue method, which shows that bothmethods work for this case, and the SDR method is faster with given sufficientmemory.

26 CHAPTER 4. ANTENNA CURRENT OPTIMIZATION

Figure 4.3: The normalized radiation patterns correspond to a r(θ, φ)-goal directionof θ ∈ [0, 36, 63, 90], φ = 0 for the two-plate structure. The red lines are forthe minimizing Q problem (4.3) without directivity constraints, and the blue areradiation patterns with Dp0 = 5.4 for the same desired r-directions. (© 2017 IEEE)

4.2. DIRECTIVITY CONSTRAINTS 27

Figure 4.4: The lower bounds on the Q-factor for the two-plate structure with agiven desired partial directivity when sweeping the direction r(θ, φ) for the polarangle θ ∈ [9, 18, 54, 63, 90] and φ = 0 at 900 MHz (the blue curves). The dark-red envelope at the bottom of the figure is corresponding to the lowest Q sweepingθ ∈ [0, 90]. The green-hue colors corresponds to the envelope of the lowest Q-values from the partial directivity optimization (4.3) for frequencies [0.88, 0.89, 0.9]GHz. The line marked parasitic corresponds to a Parasitic element antenna, seeFigure 1.3. (© 2017 IEEE)

28 CHAPTER 4. ANTENNA CURRENT OPTIMIZATION

4.3 Pattern Shaping Constraints

Far-field pattern-shaping is a common requirement in the design of large antennasand arrays, see e.g. [4,81–83]. Here similar measures are applied to small antennas incurrent optimization problems. A pattern shaping constraint can be formulated asa requirement on the ratio of radiated power in the angular region Ω0 as comparedwith the total radiated power:

α = PΩ0

Prad. (4.6)

The radiated power in the region Ω0(θ, φ) is derived by

PΩ0 =∫

Ω0

U(r) dΩ ≈ 12η IH( ∫

Ω0

FHF dΩ)I. (4.7)

Consequently, to minimize the Q-factor with α ≥ α0 as a far-field constraint, weformulate the optimization problem as:

minimizeI∈CN

maxIHXeI, IHXmI

subject to IHRI = 1,

IH( ∫Ω0

FHF dΩ)I ≥ ηα0.

(4.8)

Besides, we define another special pattern shaping problem with power front-to-bank ratio (PFBR) constraints, where the PFBR is the ratio of the radiated powerbetween the front and back hemispheres:

PFBR =

∫Ω+

U(r) dΩ∫Ω−

U(r) dΩ, (4.9)

and we use the notation:

Ω+(u) = r ∈ R3 : u · r > 0, (4.10)

Ω−(u) = r ∈ R3 : u · r < 0, (4.11)where r is the observation direction, and u indicates the “forward” directions inthe space. In many cases the PFBR is a better description on the isolation betweenantennas or systems than using the classic front-to-back ratio (FBR) [1, 84, 85].Here the optimal Q-factor problem with the PFBR constraint is:

minimizeI∈CN

maxIHXeI, IHXmI

subject to IHRI = 1,IH( ∫

Ω+FHF dΩ

)I

IH( ∫

Ω−FHF dΩ

)I≥ PFBR.

(4.12)

4.4. RADIATION EFFICIENCY CONSTRAINTS 29

Figure 4.5: The results with power pattern shaping problems for the strip dipole.(a) is the lower bounds on the Q-factor for problem (4.8). (b) is the upper boundson the fractional bandwidth FBW (%) for the PFBR problem (4.12), |Γ0| = 1/

√2.

In Paper II, the above pattern shaping problems are illustrated for the dipoleantenna. For the power pattern shaping problem (4.8), we consider three sectorsΩκ are sectors, κ = 1, 2, 3 where

Ω1 = r ∈ R3 : 75 < θ < 105, (4.13)

Ω2 = r ∈ R3 : 70 < θ < 110, (4.14)

Ω3 = r ∈ R3 : 60 < θ < 120, (4.15)

and the lower bounds on the Q-factor are shown in Figure 4.5 (a). For the PFBRproblem (4.12), we study two different constrain choices of Ω±: Ω±(z) and Ω±(x),and the upper bounds of bandwidth are illustrated in Figure 4.5 (b). For some cer-tain optimal current, the 2D and 3D radiation patterns are presented in Figure 4.6and Figure 4.7. We observe that the pattern shaping algorithm works well withcurrent optimization even for the dipole structure.

4.4 Radiation Efficiency Constraints

For antennas with losses, the Q-factor is

Q = 2ωmax(We,Wm)Prad + Ploss

= e2ωmax(We,Wm)

Prad= eQrad, (4.16)

where we use the normalized Q-factor with efficiency Qrad = Q/e, in order to focuson the radiated part of the Q-factor.

30 CHAPTER 4. ANTENNA CURRENT OPTIMIZATION

Figure 4.6: The 2D radiation pattern with optimal currents together with the powerpattern shaping problems for the strip dipole. (a) is for the power pattern shapingproblem (4.8). (b) is for the PFBR problem (4.12).

Figure 4.7: The 3D radiation pattern with optimal currents together with the powerpattern shaping problems for the strip dipole. (a) is for the classical dipole. (b)is for the power pattern shaping problem (4.8), Ω1, α0 = 0.5. (c) is for the PFBRproblem (4.12), PFBR=5, Ω±(z). (d) is for the PFBR problem (4.12), PFBR=2,Ω±(x).

4.5. EMBEDDED IN DEVICES 31

For pure metal antennas with ohmic losses, from the numerical representationof radiation efficiency (3.16), we can introduce the constraint e ≥ e0 to (4.2) as

(1− e0)IHRI− e0IHRΩI ≥ 0, (4.17)

and we also change the optimized function from Q to Qrad. If we use the nor-malization IHRI = 1, constraint (4.17) is equivalent to IHRΩI ≤ δ0, whereδ0 = (1 − e0)/e0 is the dissipated factor. So the minimizing Qrad problem foran antenna with ohmic losses is:

minimizeI∈CN

maxIHXeI, IHXmI

subject to IHRI = 1,IHRΩI ≤ δ0.

(4.18)

4.5 Embedded in Devices

Small antennas are usually embedded as small parts of the device chassis or theircircuit-boards. This type of antennas is called embedded antennas. Bounds onembedded antennas can be obtained quicker than bounds on the whole device.Furthermore, they have many interesting applications, e.g., IoT, as shown in e.g. [22]and Paper III. Considering the device model in Paper III, a 5 cm × 5 cm infinite thinmetal plate, see Ω2 (the green region) in Figure 4.8). It is a good approximationof board metalization which is the basic component of many IoT devices. Theembedded antenna is allowed to design within the 2 cm × 2 cm Ω1, the brownregion in Figure 4.8. The center position of the embedded antenna in the device isgiven by (x, y). The idea behind this is that we optimize over currents I1 withinthe Ω1-region, utilizing the induced currents to determine the performance for thewhole device.

To modify the lossless minimizing Q-factor problem (4.2) for the embeddedantenna shown in Figure 4.8, we decompose the current vector as IT = [IT

1 IT2 ],

where Ik is the current vector in region Ωk, k = 1, 2. Meanwhile, Z, the MoMimpedance matrix associated with the PEC geometry Ω1 ∪ Ω2, is written as

Z =[Z11 Z12Z21 Z22

].

Similarly, we rewrite the excitation as VT = [VT1 VT

2 ], where V2 = 0 because onlyinduced current is in Ω2. According to the EFIE,[

Z11 Z12Z21 Z22

] [I1I2

]=[V10

],

then we have the relationZ21I1 + Z22I2 = 0. (4.19)

32 CHAPTER 4. ANTENNA CURRENT OPTIMIZATION

5 cm

5 cm

2 cm

2 cm

Ω2, I2

Ω1, I1

center x

center y

Figure 4.8: The embedded antenna (brown) within the device (green).

Therefore, we can express the induced current I2 with I1:

I2 = −Z22−1Z21I1 = ZrI1, (4.20)

and rewrite the problem (4.2) as

minimizeI1∈CN

maxIH1 XeI1, IH

1 XmI1

subject to IH1 RI1 = 1.

(4.21)

where Xe, Xm and R satisfy

X = X11 + X12Zr + ZHr X21 + ZH

r X22Zr, (4.22)

R = R11 + R12Zr + ZHr R21 + ZH

r R22Zr. (4.23)

We note that each of the matrices Xe, Xm and R are reduced smaller square matri-ces acting on I1. Consequently, the minimization problem (4.21) for the embeddedcurrents has the same form as (4.2), but now the minimization is over the embeddedcurrents. This approach is discussed in detail in e.g. [22, 86,87].

In addition to the above type of embedded antennas, another simplification forcurrent optimization can be done for antennas with ground plane using the imagetheory [4], see e.g. [34, 35].

Chapter 5

Optimal Antenna Design

In the above chapters, antenna theory and mathematical methods are combined tofind fundamental limitations for small antennas. These antenna bounds have beencompared with the performance properties of previous antenna designs [33, 37, 76,88], which encourages humans to design optimal antennas. As mentioned above,the antenna current optimization method can not only used for determine physicalbounds but also give good prediction for optimal antennas in terms of geometryand topology. In particular, we can design optimal or near optimal antennas byrealizing the obtained optimal currents. In this chapter, we use examples to statethe concept of optimal antenna design by current optimization.

Figure 5.1: The flowchart of optimal antenna design using current optimization.

33

34 CHAPTER 5. OPTIMAL ANTENNA DESIGN

5.1 Antenna Geometry Generation

The initial requirements on antenna design usually include the design space anddifferent performance parameters. In the first step, i.e., geometry generation, an-tenna current optimization can be used to find the best shape among a set of shapesand the optimal positioning of an antenna.

Optimal Shape in the Family of Rectangles

In Paper II, we investigate of the Q-factor with a given PFBR vary over a rangerectangular shapes with different length to width ratios. A range of rectangularPEC plates that fit within ka = 1 are considered. Here we assume that our antennadesign is restricted by the circle ka = 1. We also constrain the power front to backratio (PFBR) where the radiation between the front and the back region by Ω+(x)has PFBR=3. By solving the optimization problem (4.12), we find the trade-offbetween shape-ratio and the Q-factor for all possible candidates. In Figure 5.2 wedetermine the lowest Q-factor for three different values of the PFBR=1, 3, 8 whilesweeping the rectangular side-ratio lz/lx in the interval [10−2, 102]. For a givenPFBR constraint, the Q-bound varies with the geometry as depicted. With Ω+(x)PFBR constraints, we can see that we get the lowest Q-value for Ω+(x) PFBR=3when lz/lx ≈ 1.6 (if we only consider lz ≥ lx), which agrees with the similar resulton D/Q in [40]. This is expected since the PFBR constrained optimization coincidewith the unconstrained optimization. For lz/lx = 1.6 and Ω+(x) PFBR=3, we plotthe current and radiation pattern in Figure 5.3. Note that in the range betweenPFBR=3 and PFBR=8 the lowest Q-factor position shift from lz/lx = 1.6 to lz/lx =1, we note that the optimal shape for higher demands on PFBR approaches thesquare in shape for the lowest possible Q-factor.

This example shows that we can find the best shape among all the consideredshapes by current optimization, assuming that we can achieve the optimal current.

Optimal Space between Array Elements

Another case study in Paper II is to examine a small array with two short dipolesat a distance d from each other along the x-direction, see the inset in Figure 5.4. Inthis case, it is interesting to compare how the inter-element distance d interact withincreased demands on a minimum directivity D0 in the x direction. The elementshave size of length of lz/2 = 0.24λ and width of lx = 0.02lz. We find as expectedthat the lowest possible Q-factor depends on the inter-element distance for a givenD0, see the result in Figure 5.4. For a certain directivity requirement, e.g., D0 = 5,we find the first d with a local minimum Q at d ≈ 0.2λ, which agrees with theclassical two-element Yagi-Uda antenna. The normalized current distribution andthe radiation pattern for D0 = 5, d = 0.2λ is depicted in Figure 5.5. Since we canchoose any type of array elements, we state that current optimization is a goodgeneral method to investigate unknown types of arrays.

5.1. ANTENNA GEOMETRY GENERATION 35

Figure 5.2: The lower bounds on the Q-factor for rectangular plates ka = 1, whenΩ+(x) PFBR=3, PFBR=8, and without PFBR constraint.

Figure 5.3: The optimal current distribution (normalized magnitude in dB) on theplate lz/lx = 1.6, for Ω+(x) PFBR=3, and its corresponding radiation pattern.

36 CHAPTER 5. OPTIMAL ANTENNA DESIGN

Figure 5.4: The lower bounds on the Q-factor for different D(x) ≥ D0 for the twoshort dipoles structure.

Figure 5.5: The optimal current distribution on the elements the array, when d =0.2λ, D0 = 5, and its corresponding radiation pattern. The current is normalizedand the magnitude is expressed in dB and plotted in color on the structure, andthe phase of each element is shown in left and right insets respectively.

5.2. REALIZATION OF THE OPTIMAL CURRENT 37

Optimal Embedded PositionIn this part, we focus on the maximally available impedance bandwidth of smallembedded antennas as a function of their position within the device. The geometryof the embedded antenna in Paper III is shown in Figure 4.8, and our questionis where the best position is for a maximal bandwidth within the device on thefrequency 900 MHz. The search for the optimal position is illustrated in Figure 5.6.From the optimization result, it is clear that the bandwidth optimal position is anyof the corner positions of the device and that the edges are the second best positions.This certainly agrees with the design practice for small antennas and phones, wherethe embedded antenna often is placed at the outer edges of the chassis. To gain aphysical understanding as to why the corner is the optimal position, we study anoptimal current distribution in the device, and its associated radiation pattern, seeFigure 5.7.

A further investigation about the bandwidth optimal position for different fre-quencies between [0.9, 2.0] GHz is illustrated in Figure 5.8, whereas at higherfrequencies the best position moves from the corners to the center of the edge asthe center wavelength decrease in size.

Another study is investigating the size of an embedded antenna in the device,as shown in Figure 5.9. We observe that the bandwidth is a function of the heightof the controllable region h. The bandwidth grows as the size increases, but thegrowth rate is not linear. The effect of size is most noticeable when the h is small.This example illustrates how we can determine the embedded region for antennadesign, which is also a trade-off between the size and antenna performance.

The above examples demonstrate how to investigate the best embedded posi-tions or regions for antennas in devices. Since the current optimizing problemscan be solved fast for embedded antennas, it is very convenient to investigate theoptimal position and size before the designs.

5.2 Realization of the Optimal Current

When we find the best geometry for antenna design using current optimization, wealso obtain the optimal current distribution. It is a great and interesting challengemoving from the optimal current to the physical antenna design. Similar ideas havebeen studied for other cases in e.g. [33]. In Paper II, we present a multi-positionfeeding strategy for realizing the optimal dipole antenna with a desired radiationpattern. The narrower beam-shaping pattern and the tilted PFBR pattern arerealized with low Q-factors Q ≤ 1.61Qoptimal.

In these antennas, the current path is strongly constrained by the geometry,essentially one-dimensional. The optimal current shapes for the two consideredcases are depicted in Figure 5.10. We put three fed ports at the positions of crestsand troughs by Figure 5.10, and optimize the excitation at each port to satisfy thefar-field constraint and determines the corresponding Q-factor. The optimizationof ports excitation is similar to the embedded case (4.21). We choose the ports as

38 CHAPTER 5. OPTIMAL ANTENNA DESIGN

Figure 5.6: The color indicate the fractional bandwidth for the reflection coefficient|Γ| = 1/

√2 at 900 MHz for each center position of the embedded antenna.

Figure 5.7: The optimal current distribution for a corner embedded device at 900MHz (color is intensity of magnitude normalized in dB, arrows indicate the direc-tion) and its associated radiation pattern.

5.2. REALIZATION OF THE OPTIMAL CURRENT 39

1 1.5 2 2.5

1

1.5

2

2.5

0.9GHz

1 1.5 2 2.5

1

1.5

2

2.5

1GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.1GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.2GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.3GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.4GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.5GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.6GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.7GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.8GHz

1 1.5 2 2.5

1

1.5

2

2.5

1.9GHz

1 1.5 2 2.5

1

1.5

2

2.5

2GHz

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.8: The optimal bandwidth position as a function of frequency. The colorcoding is the fractional bandwidth (FBW) normalized with with the maximumFBW at each frequency. Each color box within the above region corresponds tothat center-position’s normalized fractional bandwidth.

the “embedded” parts with I1, and the current on the dipole can be determinedby (4.20). Such a strategy is also applied to feeding optimization for a larger arraycase in the author’s paper [89] (Paper VI).

The surface current distribution generated by the multi-position feedings isshown in Figure 5.11. We observed that the surface current of the three-portfeeding structure is roughly similar to the optimal current, as shown in Figure 5.10.The associated radiation patterns for the multi-position feedings and for the opti-mal cases are compared in Figure 5.12, where the similarity between the radiationpatterns is very high since the deviation is barely visible.

We have illustrated above that it is possible to realize the required radiationpattern with rather simple methods, but at the cost of a higher Q-factor and withthree feedings. For the beam-forming associated with the Ω+(x)-case, we noticethat even for small PFBR, the Q factor is very high, so we expect it to be difficultto realize this case.

40 CHAPTER 5. OPTIMAL ANTENNA DESIGN

0 1 2 3 4 5

1

2

3

4

5

6

7

h, height of controllable region [cm]

FBW

[%]@

-3dB

h

5cm

3cm

N = 20

30

40 ⇔ 4363 unknowns

Simulated Lossless

Figure 5.9: The fractional bandwidth bound for the embedded antenna with respectto different height of controllable regien.

5.2. REALIZATION OF THE OPTIMAL CURRENT 41

Figure 5.10: The optimal current distributions (with normalized magnitude andphase) for the dipole with different radiation patterns. (a) is the optimal currentfor a beam-shaping pattern, with a narrower pattern. (b) is a PFBR-case with atilted pattern corresponding to the Ω+(z) case.

42 CHAPTER 5. OPTIMAL ANTENNA DESIGN

Figure 5.11: The current distributions (with normalized magnitude and phase) forthe dipole by optimal multi-position feedings, with different radiation patterns.(a) is the current for a beam-shaping pattern, with a narrower pattern. (b) is aPFBR-case with a tilted pattern corresponding to the Ω+(z) case.

Figure 5.12: The 2D radiation patterns (normalized in logarithmic scale) com-parison between by optimal current distributions and by optimal multi-positionfeedings. (a) is the current for a beam-shaping pattern, with a narrower pattern.(b) is a PFBR-case with a tilted pattern corresponding to the Ω+(z) case.

5.3. COMPARISON WITH THE BOUNDS 43

5.3 Comparison with the Bounds

In the previous section, we talked about the role of current optimization in theantenna synthesis phase in the early stages of antenna design. In the antennasimulation and analysis phase, we can compare the antenna performance with thetheoretical limitations, so we can know how good the antenna is and where we canimprove it. It is very interesting to investigate and design optimal antennas whichare close to the physical limitations.

For example in this thesis, the Q-factor or corresponding bandwidth of the multi-position fed dipoles in the above section are marked in Figure 4.5, where we seethey are close to the bound. (As we mentioned, Q ≤ 1.61Qoptimal). In the two-platecase see Figure 1.3, the parasitic element antenna is compared with the Q-bound inFigure 4.4. We note that this antenna performance is close to the bounds obtainedby the optimal current for D ≈ 3.8, which implies that it is a “good” antenna, i.e.,it has a fairly high directivity while keeping the Q-factor low. However, it can stillbe possible to realize antennas with better performance, closer to the bound. Inother words, this comparison tells us that the optimal Q-factor versus directivityfor the antenna, and the bandwidth will decrease if you need a higher directivity inthis direction for this antenna, but also for the bound. Another comparison is forthe embedded case in Figure 5.9, where a lossless simulation result is marked.

For the embedded antenna case given in Figure 4.8, we have found the Q-optimalembedded position as a function of frequency. A meandered planar inverted-F an-tenna (PIFA) is investigated as a realization. For this family, we note that the corneris the best embedded position, see Figure 5.13. Similarly, we note that the Q-factorbound is lowest (highest bandwidth) at the corner at these low frequencies. Therestill remains a gap for the PIFA antennas to reach the optimum bandwidth. Thefractional bandwidth for an antenna with ohmic losses in the metal is also depictedin Figure 5.13, where we use an infinitely thin layer with the sheet conductivity ofthe copper (with conductivity σ = 5.8× 107 S/m).

In addition to the examples in this thesis, the performance properties of manyfundamental small antennas have been examined and compared with their physicallimitations, including the dipoles, loops, folded spherical and cylindrical helices,and planar meander line antennas, see e.g. [88, 90]. Such comparisons give us agood understanding of a class of small antennas. The tools illustrated in this thesisare straightforward to use and can provide antenna engineers with a new tool topredict antenna behavior.

44 CHAPTER 5. OPTIMAL ANTENNA DESIGN

Figure 5.13: The comparison between the upper bound and the realized PIFAantenna for different edge positions at 900 MHz.

Chapter 6

Conclusions and Future Work

6.1 Conclusions

In this thesis, we study the physical limitations and the optimal trade-off betweendifferent parameters for small antennas using current optimization. Small antennasare now widely used in our daily lives. They appear in almost all the electronicdevices, such as mobile phones, sensors, and IoT devices. As the number of antennasincreases, the design space (for example, size) of each antenna becomes smaller andsmaller. At the same time, people expect high antenna performance, e.g., highgain, low losses, and low interference. However, the size of an antenna is inverselyproportional to most of these performance properties. As a consequence, in mostcases we have to study the trade-off between these antenna parameters, for example,a higher directivity requirement usually gives a narrower bandwidth if the antennasize is limited.

In order to study the optimal trade-off between these parameters, we use math-ematical tools like convex optimization, SDR, and MoM, which are presented inChapter 2. We introduce the antenna parameters and reformulate them in theMoM matrix forms in Chapter 3. Antenna current optimization problems areformulated as minimizing Q-factor with constraints, e.g., directivity, radiation pat-tern, and radiation efficiency in Chapter 4. Using these mathematical tools, theQ-factor bounds and optimal current are obtained for certain antenna problems.Current optimization has an advantage in solving problems of antennas embeddedin devices because we can optimize the current only in the antenna regions insteadof the whole devices. Through current optimization, we can obtain a-priori infor-mation how the Q-factor depends on different shapes and on the embedded positionof the antenna within the device. In addition, the optimal current can in certaincases give us a good prediction for an antenna design. Some examples of case stud-ies and attempts to validating the optimal current are presented in Chapter 5 inorder to illustrate how our methods help with the goal of optimal antenna design.

45

46 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

The pioneering contributions of this thesis are summarized as follows: We showthat certain very interesting non-convex antenna optimization problems can besolved with the SDR technique. As a result, we can, for example, determine theminimizing Q-factor as a function of the total directivity. Far-field radiation pat-tern shaping constraints are proposed for antenna current optimization, and wehave solved such problems for simple types of antennas. We apply current opti-mization to predict the best possible Q-factor for a range of geometries. Givena particularization of the geometry, we can determine which of the parameterizedgeometries that are optimal with respect to the Q-factor and antenna constraints.Using a multi-position feeding strategy, we have realized optimal currents for thedipole antenna with non-standard radiation patterns and a low increase in the Q-factor. We also compare our bounds with the performance of several small andembedded antennas.

6.2 Future Work and Extensions

Antenna limitation and optimal antenna design is a field that is open for investiga-tions and new discoveries. There are numerous possible ways to continue this workas extensions.

• From Current to Antenna DesignWe have repeatedly stressed that the optimal current can help us predictthe design of the best antennas. This thesis emphasizes that the current-optimization tools and the a-priori Q-factor estimates can be used to predictantenna behavior and fundamental bounds. In section 5.2, we present a multi-position feeding strategy to realize the optimal current for the half-wavelengthdipole antenna. Similar ideas have also been studied for other cases in e.g. [33].In the process of realizing the current, we can also combine the numericalmethod with antenna engineering. One possible direction of further work isto utilize an optimal current, or perhaps a family of optimal current and theQ-factor limitation together with the tools of machine learning to connect thegeometrical shape of a “good” antenna towards the goal of automatic antennadesign. Certain efforts using genetic algorithms have already been tested, seee.g. [17].

• 3D Large Mesh ProblemsIn this paper, we present some new ideas for current optimization and validatethem on some simple antenna structures. However, our current MoM programis limited in its implementation of acceleration methods, and hence it can bedifficult and time-consuming to solve big mesh problems. The SDR methodhas high requirements on the memory of the computers. How to developmore memory efficient algorithms that can solve 3D, large mesh and complexstructural problems is another research direction.

6.2. FUTURE WORK AND EXTENSIONS 47

• Near-Field ConstraintsIn this thesis, we have proposed and solved the far-field pattern shaping prob-lems by current optimization. In practical engineering applications, the near-field radiation of the antennas is also an interesting and important indicatorof concern. It is therefore of interest to study how antenna performance isaffected by constraints on the near-field strength, which has been first intro-duced in [91].For example, an increasing number of antennas are now integrated into thebase stations, automobiles, etc., and research on electromagnetic compati-bility (EMC) of systems has been paid a lot of attention. By realizing theshaping of the near-field radiation patterns, we can investigate the mutualcoupling problems for the antenna systems or arrays. Another interestingapplication for the near-field constraints study is the compliance with theelectromagnetic field (EMF) exposure limits, see e.g. [92]. This complianceis for the antennas embedded in or placed in close proximity of the humanbody, e.g., body area networks, mobile phones, and medical devices. In thisarea, the SDR technique is used to find the upper bound of near-field EMFexposure for 5G MIMO antennas by the author in [50].

• Dielectric Losses for Patch AntennasAll the antennas studies in this thesis are of metal and among them most arelossless. Although the ohmic losses of the metal have been discussed in thesection of the radiation efficiency constraints (4.4), more types of losses stillneed to be studied for different types of antennas, for example, patch anten-nas with the dielectric substrate, which are widely used in electric devices inour daily life. As an early theoretical investigation, the antenna current op-timization problems in lossy (dispersive and heterogeneous) media have beenintroduced by Gustafsson in e.g. [91, 93, 94], and there are many interestingproblems to solve within this area.

In summary, current optimization and optimal antenna design are new researchdirections and have great research prospects. There are many interesting questionsin this area that are still unknown and need to be explored.

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