Stochastic Full-Wave Modeling for the Variability Analysis ... · Stochastic Full-Wave Modeling for the Variability Analysis of Textile Antennas Marco Rossi Promotoren: prof. dr

Stochastic Full-Wave Modeling for the Variability Analysis of Textile Antennas

Marco Rossi

Promotoren: prof. dr. ir. H. Rogier, prof. dr. ir. D. Vande Ginste

Proefschrift ingediend tot het behalen van de graad van

Doctor in de ingenieurswetenschappen: elektrotechniek

Vakgroep Informatietechnologie

Voorzitter: prof. dr. ir. D. De Zutter

Faculteit Ingenieurswetenschappen en Architectuur

Academiejaar 2016 - 2017

ISBN 978-90-8578-962-8

NUR 959

Wettelijk depot: D/2016/10.500/94

Stochastic Full-Wave Modeling for the Variability Analysis of TextileAntennas

Marco Rossi

Dissertation submitted to obtain the academic degree ofDoctor of Electrical Engineering

Publicly defended at Ghent University on December 19th, 2016

Supervisors:prof. dr. ir. H. Rogierprof. dr. ir. D. Vande GinsteElectromagnetics groupDepartment of Information TechnologyFaculty of Engineering and ArchitectureGhent UniversityTechnologiepark Zwijnaarde 15// B-9052Ghent, Belgiumhttp://emweb.intec.ugent.be

Members of the examining board:prof. dr. ir. R. Van de Walle (chairman) Ghent Universityprof. dr. ir. D. De Zutter (substitute chairman) Ghent Universitydr. ir. S. Agneessens (secretary) Ghent Universityprof. dr. ir. H. Rogier (supervisor) Ghent Universityprof. dr. ir. D. Vande Ginste (supervisor) Ghent Universityprof. dr. ir. A. Costanzo University of Bolognaprof. dr. ir. A. Tijhuis Eindhoven University of Technologyprof. dr. ir. L. Van Langenhove Ghent University

“Broaden your mind, Malcolm, broaden your mind!It takes all sorts to make a world [...]

If grace perfects nature it must expand all our naturesinto the full richness of the diversity which God intended when He made them,

and heaven will display far more variety than hell.”

C. S. Lewis, Letters to Malcolm

Acknowledgements

It is with the greatest pleasure that I add this last section to my thesis, as a sealthat rounds off what I consider to be in primis a deeply personal experience. Asfor arguably all new things, and especially those in which one puts a great dealof his expectations and ambition, time and experience steadily and inevitablydismount most certainties to gradually replace them with deeper and sounderones. Four years of Ph.D. were no exception to this rule. The focus on meretechnical achievements, which I thought was the main aspect of research and theprimary reason to pursue a Ph.D., gave way to a more subtle and comprehensiveunderstanding of what I was doing. Step by step, and not always following astraight path, I got to believe that nailing down results was only part of mywork, but maybe not even the main thing that a Ph.D. is supposed to bringalong. Instead, the whole magic lies in the small, essential lessons that have tobe learned on a daily basis, and that in the end shape you as a person. Learningto outline your work independently, to be consistent and clear, to have patienceand trust in your own means, to pay attention to all sorts of details, whicheventually make the difference between an average job and a very good one,as well as understanding how incredibly precious, rewarding and challenging isthe discussion with other people and the possibility to compare your ideas withtheirs. These are the aspects that I consider to be the most valuable and thatmake me think that it was really worth it. They have permanently changed myoutlook on everything.

As a matter of fact, many people contributed to this positive experience andthey rightly deserve to be thanked. First of all, Prof. Daniel De Zutter and myfirst supervisor, Prof. Hendrik Rogier, because they gave me the opportunityto join the EM group for a Ph.D. Moreover, Hendrik’s suggestions were alwaysprecious, and they certainly helped me polish up my approach to work, mypresentation skills and my written English. To my second supervisor, Prof.Dries Vande Ginste, goes my gratitude for his constant support during these fouryears, especially when it was most needed. Each time I asked him a question, hewould promptly answer that and anticipate the other 1.5 that would inevitablyfollow. I also wish to thank the members of the examination board for theirpositive feedback on my thesis and because it was a pleasure to defend it beforethem. Finally, many thanks to Isabelle for being very gentle and helpful toclear up bureaucracy and all sorts of small hassles.

The atmosphere in the workplace was always very positive and serene. There, Ihad the chance to meet a considerable number of smart and interesting people,with many of whom I also became good friend. Therefore, going to work wasactually a pleasure. Other five people started their Ph.D. in the same year as I.

iv

Among them, Giorgos is the first one that I got to know. As a foreigner with nosignificant international experience who moves his first, unsteady steps in a newcountry, getting acquainted with a novel environment was rather challenging forme. The interest that he took in me, his constant presence and his commitmentto involve me in all sorts of activities made me feel at home. Moreover, withhim I had the first chance to talk about things I was really interested about,and that I discovered could be the subject of long and fruitful conversations.

Zdravko came in second. I have always been sure that being a so-called ‘food-nazi’ (definition that should nevertheless be reconsidered, according to me) wasall for the best. Here is a very good example. After listening to the way heprepared (and probably still prepares) carbonara, after he mentioned the ninthingredient for the sauce (whereas there should only be four), I felt compelled tostop his narration and invite him over for lunch. That was the beginning of avery precious friendship. Many things I consider valuable of this man, amongwhich being very passionate about what he does, having always an honest,positive opinion about everybody, his genuine love for his family, for his country,for his people, from which we all should learn. I also wish to thank him forintroducing me to an entire part of the world that was completely unknownto me. Visiting Belgrade and that Easter lunch together were very gratifyingexperiences. There is something special about Serbia, and I am pleased that Igot in touch with it.

Gert-Jan, Thijs and Sam Lemey were the other three Ph.D. students from myyear. I would like to thank them for being all very kind and generous persons,helpful when any sort of issue showed up and enjoyable in the moments ofidleness and fun.

A central role during these years was played by the moka ritual at the coffeecorner, which offered me the opportunity to leave my computer for short andrefreshing breaks. That brought a group of colleagues together, which becameever closer thanks to those famous movies nights: Alessandro a.k.a. Manà,Joachim a.k.a. Gioacchino, Sam Agneessens and Arnaut a.k.a Arnoldo. I wouldlike to thank Alessandro for being enthusiastic about all sorts of things andfor the fun we had in the many evenings at his place talking, playing gamesand, unfortunately only lately, making beer. To Joachim, whose warm andwelcoming smile can fix on its own the worst of days, goes my gratitude forbeing the kindest person that I have ever met, whom I always like to see and tospend time with, playing snooker or drinking (his) good whiskeys. Sam is thecolleague from which I learned the most during my Ph.D, and to whom I amconnected the most as well. From the conference in Memphis to biking togetherto work, we shared many good moments and I am glad that a large part ofthe work that I did involved our mutual collaboration. Moreover, I appreciatehis sharp humour and keen standpoint whatever the topic of the conversationmay be. Finally, Arnaut is probably the person that I disappointed the most.Despite his incredible enthusiasm and his constant effort, he never got me to

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share his passion for good movies, good songs and ‘America’ (this last one tobe rigorously read with a long M, a badass intonation and picturing DwayneJohnson in the act of tearing apart chains in ‘Hercules’). Apart from jokes, Iwish to thank him for being an amusing presence that I most gladly meet, withhis amazing ‘algemeen Italiaans’ that perfectly sews together French, Englishand Italian from the Middle Ages, for the cultural tours of the Pajottenlandand for forming a very fun duo with Sam. With these four guys, I also wishto thank their respective girlfriends: Zusa, Romy, Laura and Liesa, because Itruly appreciate how in perfect harmony they are to their better halves.

The coffee corner, at the Technicum as well as in Zwijnaarde, was actually aplace that teemed with Italians: ‘il Dottore’ Luigi Vallozzi, Damiano, Daniele,Paolo and Domenico. Thanks for the fun and the jokes that we had and forforming a small community that (I hope) everybody got to like. In particular,most appreciated was the ‘Spirito degli Abruzzi’ that Luigi and Domenicobestowed on the EM group, and I will always be thankful to Domenico for thevery intimate (in a very heterosexual sense) week that we spent together at hisplace, which was also the opportunity to become friends. In addition, a specialacknowledgment to Paolo for being, so far, the only example of reasonable‘interista’ that I know, an inexplicable peculiarity which originates from hisgentle and fair temperament, and Damiano, for his drive and his overwhelmingpassion for the things that he likes (therewith I mean photography). Finally,thanks to Annelies, who often joined this colourful group with her gracefulpresence.

Also many thanks to the members of the office at the Technicum that have notbeen mentioned yet: Patrick, Sophian, Marijn, Eline, Katja, for the little thingsof everyday, for the many conversations we had and for the Glühwein eventsthat warmly welcomed Christmas.

Finally, I should certainly not forget that group of students that I shall nameas ‘De Jongens’, that is, all Ph.D. students that joined the EM group in theyears that followed mine: Martijn, Irven, Arne, Niels, Olivier, Dries, Michiel,Simon, Thomas and Quinten. It was a pleasure to meet them, especially thosewho had the luck to share their office with me and the privilege to listen tomy outstanding Dutch. Far from rejecting the craziness that Zdravko and Ibrought into the office, they gradually took part in it and contributed to createa warm atmosphere. I am also proud to say that, even though they probablydo not realize it yet, we helped them find their own path in the labyrinth ofresearch, which only through a wise combination of work, frivolity and interestin each other, can lead to success.

An added value to the years at Ghent University was certainly represented bythe Doctoral Training Program, which helps you build up your personality andyour expertise. And, in case the content of the course is not really a majorbreakthrough in your career, it still allows you to meet new people. I wouldtherefore like to thank Max for the time we spent together, for our weekly visits

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to the swimming pool, for the swimming competition and for his never tooappreciated kindness.

Learning Dutch was another important aspect of my stay in Belgium, and thelessons at Het Perspectief kept me busy for quite some time. Many thanks tothe persons that I met there: Alessia, Nuria, Marco, Emiljano, Yulia, Pabloand Fien, for making up for the boredom of those lessons, for the picnics andand for all the good moments that I had with them.

Voglio qui ringraziare mia madre Maria e mia sorella Chiara, per le quali nonè stato facile vedermi partire con la prospettiva che sarei rimasto lontano perquattro anni e, forse, anche di più. Nonostante le distanze, o proprio grazie aqueste, credo che in questi anni siamo cresciuti insieme e che il nostro legamesia ora più forte e meno scontato, e che un po’ alla volta nuovi modi di aiutarsi,di venirsi incontro e di volersi bene si siano delineati. Di questo sono grato. Unsincero ringraziamento va anche a tutta la mia famiglia, perché le difficoltà cihanno dato modo di ritrovarci uniti, a disposizione dell’altro, con semplicità, aconferma di come il peggiore dei momenti può essere il luogo della più grandeserenità e il più profondo affetto.

Un grazie sentito anche alla Poma Family, per l’entusiasmo e l’affetto con cuisono stato sempre accolto, per il calore che ho sempre trovato nella loro casa,che dopo tanti anni mi hanno portato a riscoprire sensazioni che pensavo fosseroandate perdute per sempre.

I will conclude with my girlfriend Martina, because eventually everything startsfrom her and to her it gets back. In the first place, she actively supported mydecision to go abroad at the end of my master, because we both knew that itwould be a great opportunity, despite the fact that, as I am bound to think,most girlfriends would never do anything of that kind. The perspective of livingapart with very little possibility to see each other was a big issue. However, itwill never strike me enough how we never thought much about how we coulddo it for so many years. Such light-hearted approach allowed us not to worryabout the future and to stay focused on the challenges of everyday. In theend, and this is the greatest achievement of these four years, we pulled it off.Thanks for the inexhaustible energy and patience with which you forgive mymistakes and help me become a better and better man, and for the magic offeeling that we are becoming more and more like one single being. With thegreatest enthusiasm I now look forward to what the future will bring to us.

I am grateful to God, for never giving up on me, for all the many good thingsthat this Ph.D. has brought about, for the beauty that always lies on my path.

Ghent, 19th december 2016Marco Rossi

Contents

Samenvatting ix

Summary xiii

List of Abbreviations xvii

List of Symbols xix

List of Publications xxi

1 Introduction 31.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Current state of the art . . . . . . . . . . . . . . . . . . . . . 61.4 Novel contributions and outline of the thesis . . . . . . . . . . 8

2 Stochastic Electromagnetic Modeling 172.1 The Monte Carlo method . . . . . . . . . . . . . . . . . . . . 172.2 Generalized Polynomial Chaos . . . . . . . . . . . . . . . . . 192.3 Stochastic Collocation Method . . . . . . . . . . . . . . . . . 202.4 Stochastic Testing . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Padé approximants . . . . . . . . . . . . . . . . . . . . . . . . 22

3 A Stochastic Framework for the Variability Analysis of TextileAntennas 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Stochastic analysis framework for textile antennas . . . . . . 303.3 Validation for a representative textile antenna . . . . . . . . . 323.4 Experimental validation . . . . . . . . . . . . . . . . . . . . . 353.5 Application to a dual-polarized antenna . . . . . . . . . . . . 403.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Non-Destructive Electromagnetic Characterization of FlexibleMaterials for Wearable Antennas 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Characterization process . . . . . . . . . . . . . . . . . . . . . 524.3 Inset-fed patch antenna-based characterization fixture . . . . 554.4 Aperture-coupled patch antenna-based characterization fixture 624.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

viii Contents

5 Stochastic Analysis of the Impact of Substrate Compression on thePerformance of Textile Antennas 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Stochastic analysis framework for compressible antennas . . . 735.3 Characterization of the statistics of compressible substrates . 785.4 Numerical validation for a textile GPS antenna . . . . . . . . 815.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Stochastic Analysis of the Efficiency of a Wireless Power TransferSystem Subject to Antenna Variability and Position Uncertainties 896.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . 926.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 Conclusions 1117.1 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Samenvatting

Tijdens de afgelopen decennia waren we getuige van een grote technologischevooruitgang. Een van de meest opmerkelijke gevolgen hiervan is dat er meer enmeer elektronische producten apparaten aanwezig zijn in onze maatschappij.Nadat de computer en mobiele telefoon werden geïntroduceerd, richtte hetonderzoek door wetenschappers en bedrijven zich op twee aspecten. Enerzijdswerden er grote inspanningen geleverd om de draagbaarheid en inzetbaarheidvan elektronische producten te vergroten. Anderzijds werden communicatienet-werken en –standaarden aangepast en verbeterd, zodoende te kunnen omgaanmet grotere datadebieten en de gebruiker een betere ervaring aan te bieden.Zoals steeds het geval is wanneer een bepaalde technologie meer maturiteitbegint te vertonen, neemt het aantal gebruikers toe en de prijs van de toestellenzakt. Vandaag hebben veel mensen een smartphone, een tablet, of zelfs eensmartwatch die kan connecteren met het Internet, wat de gebruikers toeganggeeft tot bijna ongelimiteerde diensten en informatie. De vraag is dan ook: watvolgt er nog?

Een mogelijk antwoord op de bovenstaande vraag komt er misschien dankzij hetzogeheten ‘Internet of Things’ (IoT) paradigma, dat voorspelt dat er binnenkortgeen barrières meer zullen zijn tussen computers en de wereld rondom ons. Inhet IoT-concept wordt dit bewerkstelligd door dagelijkse gebruiksvoorwerpente transformeren naar “slimme voorwerpen”. Daartoe zullen dus niet enkelmobiele telefoons, maar ook alle gebruiksvoorwerpen uitgerust worden metmicrochips, sensoren, actuatoren en communicatie-eenheden. Zodoende kunnenal deze voorwerpen met elkaar worden verbonden en interageren met computerswaardoor er een alomtegenwoordig netwerk, een ‘Internet’, ontstaat dat opautonome wijze data kan verzamelen en verdelen, alsook een grote mate vancontrole en interactie met onze omgeving mogelijk maakt.

Vanuit dit perspectief wordt er recent veel aandacht besteed aan draagbaresystemen, voorvloeiende uit de brede waaier aan mogelijke applicaties die dezesystemen mogelijk maken. In het bijzonder zijn persoonsgerichte systemen,die gebruik maken van textielantennes en ‘slimme’ textielmaterialen, heel in-teressant, want zij zijn licht, flexibel en kunnen op onzichtbare wijze wordenverwerkt in kledij waardoor de gebruiker er nauwelijks weet van heeft. Hetontwerp van dergelijke textielantennes brengt echter een aantal uitdagingenmet zich mee. Ten eerste moet de interactie tussen de antenne enerzijds enhet menselijk lichaam anderzijds worden vermeden. Dit kan immers leiden totgezondheidsrisico’s en het staat vast dat dergelijke interacties een nefast effecthebben op de werking van de antennes. Ten tweede brengen productietechnieken,van manuele of machinale aard, onvermijdelijk onzekerheden en toleranties met

x Samenvatting

zich mee waardoor de geproduceerde antenne afwijkt van het originele, nominaleontwerp. Daarenboven zorgt het gebruik van goedkope, gemakkelijk beschikbaretextielmaterialen tijdens de productie wel voor een aanzienlijke reductie van dekost, maar tevens werkt dit problemen op elektromagnetisch vlak in de hand.Aangezien deze materialen immers niet-uniforme eigenschappen vertonen, zalhet elektromagnetisch gedrag van de antennes suboptimaal zijn. Ten slottezal tijdens het dragen van de kledij de antenne worden gebogen, vervormd ensamengedrukt, want ze moet zich aanpassen aan de bewegingen en morfologievan de gebruiker.

In dit werk ligt de klemtoon op de ontwikkeling van een methodologie dieantenneontwerp toelaat waarbij de productietoleranties en –onzekerheden ende samendrukbaarheid van de materialen in rekening worden gebracht. Incombinatie met vroeger doctoraatsonderzoek, dat aandacht besteedde aan deisolatie van de antenne ten opzichte van het menselijk lichaam en aan devervorming van de antennes tijdens het dragen, draagt dit werk bij tot het totstand brengen van een betrouwbare, vermarktbare technologie.

Aangezien alle bovenvernoemde effecten (samendrukbaarheid, toleranties, . . . )een willekeurig karakter hebben, moet hun impact op de antennewerking bestu-deerd worden op stochastische wijze. Hoofdstuk 2 beschrijft daarom een aantalbelangrijke numerieke technieken die gebruikt worden in dit proefschrift. Denadruk wordt hierbij gelegd op de Monte Carlo (MC) methode en de Stochas-tische Collocatiemethode (SCM). Deze laatste is gebaseerd op veralgemeendePolynomiale Chaos (generalized Polynomial Chaos – gPC). Daarenboven wordenook Padé benaderingen en de Stochastic Testing (ST) methode besproken. Deeerste wordt typisch gebruikt om niet-lineaire systemen te modelleren en detweede is nuttig wanneer het aantal toevalsveranderlijken toeneemt.

In Hoofdstuk 3 worden productieonzekerheden bestudeerd. We beschouwen hier-toe twee antenneontwerpen, namelijk een vlakgelaagde microstrip textielantenneen een textielantenne met tweevoudige polarisatie. De gPC-gebaseerde SCMis toegepast op beide antennes en hun ingangsimpedantie is gemodelleerd alsfunctie van de breedte van de antenne. Dit is immers de parameter waaraan deimpedantie het gevoeligst is. Daarna wordt de eerste antenne gevalideerd aan dehand van een geschikt aantal prototypes, vervaardigd op een rigide hoogfrequentsubstraat. Een experimentele reconstructie van de kansdichtheidsfunctie van debreedte van de antenne verifieert de analyse van de tweede antenne.

De combinatie van metingen van textielsubstraten met veldsimulaties leidttot een snelle, niet-destructieve karakterisering van deze substraten, zoalsbeschreven in Hoofdstuk 4. Deze aanpak is gebaseerd op de gekende resonantie-perturbatie methode waarbij, in dit werk, vlakke resonerende structuren gebruiktworden die geschikt zijn voor de 2.45 GHz ISM frequentieband en de GPS L1band. Macromodellen van de antennes’ belangrijkste karakteristieken wor-den vergeleken met metingen om de materiaaleigenschappen te schatten. Ditleidt tot heel nauwkeurige resultaten die behaald worden met een beperkte

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computationele kost.

De stochastische methode, zoals beschreven in Hoofdstuk 3, wordt in Hoofdstuk5 verder uitgebreid om ook de samendrukbaarheid van antennesubstraten tekunnen in rekening brengen. Samendrukken heeft immers een invloed op dedikteen op de relatieve permittiviteit van het substraat. We analyseren een GPSantenne die is opgebouwd uit een samendrukbaar beschermend schuimmateriaal,geschikt voor integratie in de kledij van brandweermannen. De analyse gebeurtop basis van de SCM en gebruik makende van Hermite-Padé benaderingen,want de antennekarakteristieken hangen immers op niet-lineaire wijze af vande substraateigenschappen. De karakteriseringsmethode, zoals voorgesteldin Hoofdstuk 4, wordt aangewend om de analyse te valideren en daartoe be-studeren we de gezamenlijke kansverdeling van de hoogte van het substraaten de permittiviteit, en dit voor verscheidene monsters van het beschouwdeschuimmateriaal.

De betrouwbaarheid van textielantennes is een noodzaak wanneer we eencompleet draadloos systeem ontwerpen. In Hoofdstuk 6 bestuderen we eeneenvoudig draadloos vermogensoverdrachtsysteem waarin aan de ontvangstzijdeeen textielantenne, werkzaam in de 2.45 GHz ISM band, wordt geconnecteerdmet een gelijkrichter. Deze antenne wordt aan de zendzijde bestraald dooreen hoornantenne. Gebruik makende van gPC worden productietolerantiesgerelateerd aan de vermogensoverdrachtefficiëntie van het systeem. Met behulpvan een geschikt model dat de interactie tussen de verschillende onderdelenvan het systeem in het stralende nabije-veld kan analyseren, wordt tevens deinvloed van de niet-perfecte uitlijning tussen zender en ontvanger op een snelleen nauwkeurige manier in rekening gebracht.

Summary

The technological boost that took place in the last decades has produced theincredible diffusion of electronic devices which is the most peculiar aspect ofour society. After the groundbreaking introduction of personal computers andmobile phones, the progress led by researchers and enterprises has constantlymoved in a twofold direction. On the one hand, considerable efforts have beendevoted to the enhancement of the portability and the capabilities of electronicproducts. On the other hand, communication networks and standards havesteadily been upgraded to cope with the ever increasing amount of informationwhich is offered and exchanged. The practical result, as the prices inevitablyfall when a technology becomes mature, is that today everybody has got asmartphone, or a tablet, or maybe a smartwatch that connects to the Internetand gives access to virtually unlimited services and contents. The question nowis: what next?

A possible answer is envisioned by the so-called ‘Internet of Things’ (IoT)paradigm, which prophesies the removal of the physical barriers between com-puters and the world around us, achieved by transforming everyday objects, andnot only mobile phones, into ‘smart things’ equipped with sensors, microchips,communication units and actuators. Connected to each other and accessible bycomputers, they would become the key units of an ubiquitous and pervasivenetwork, or ‘Internet’, that would autonomously collect and share information,as well as offer us an unprecedented degree of control and interaction with oursurroundings.

In this view, wearable systems has drawn considerable attention, owing to theplethora of possible applications that they offer. In particular, body-centricsystems that rely on textile antennas and ‘smart’ textiles are most interestingas they are light-weight, inconspicuous and flexible, so the user is hardly awareof wearing of them. However, the design and the implementation of textileantennas confront designers with a number of serious challenges. First of all,the interaction between the antenna and the human body has to avoided, as itmay result in health issues and it certainly compromises the performance of thesystem. Second, production techniques, whether manual or relying on machines,inevitably introduce uncertainties in the manufactured antenna. In addition,the application of cheap, off-the-shelf materials, despite considerably reducingproduction costs, aggravates this issue as the electromagnetic properties oftextiles are typically non-uniform. Finally, the deployment of the antenna onbody entails substrate bending, as it has to adapt to the movements and themorphology of the user, and compression.

xiv Summary

This work is focused on the development of a thorough framework that allowsdesigners to assess the performance of a given antenna design with respect toproduction and material uncertainties, as well as substrate compression. Incombination with previous works, which deal with the isolation of the antennafrom the body and with substrate bending, it answers the need for reliability thata given technology must have to acquire sufficient maturity and be marketed.As all the addressed effects occur in a random fashion, their impact on theperformance of the antenna has to be investigated by means of stochasticmethods. Therefore, Chapter 2 introduces the main stochastic numericaltechniques applied in this manuscript. Special attention has been devoted to thedescription of Monte Carlo and Stochastic Collocation Methods (SCMs) basedon generalized Polynomial Chaos (gPC) expansions. Padé approximants, usedto model highly non-linear system, and the Stochastic Testing (ST) method,which is beneficial as the number of variables increases, are also discussed.In Chapter 3, production uncertainties are dealt with. Two antenna designs areconsidered, being a representative inset-fed patch textile microstrip antenna anda dual-polarized textile antenna. A SCM based on gPC expansions is appliedto both and their input impedance is modeled as a function of their patchwidth, which is found to be the most sensitive design parameter. Then, theimplementation of a suitable number of prototypes on a rigid high-frequencylaminate validates the approach for the first design, whereas the experimentalreconstruction of the probability density function of the patch width verifiesthe analysis of the second antenna.The possibility to combine measurements on textile substrates with full-wavesimulations to outline a fast non-destructive characterization technique is treatedin Chapter 4. The method relies on a well-known resonance-perturbationmethod based on planar resonating fixtures operating in the vicinity of the2.45 GHz Industrial, Scientific and Medical (ISM) and the Global PositioningSystem (GPS)-L1 frequency bands. The implementation of a macromodel of theantenna’s figures of merit that are compared to the measured ones to estimatethe properties of the materials yield accurate results at the expense of a limitedcomputational time.In Chapter 5, the stochastic framework introduced in Chapter 3 is extended tothe analysis of substrate compression, which has an impact on both the heightand the relative permittivity of the antenna substrate. In this case, a GPSantenna implemented on a compressible protective foam for the integration intofirefighters’ gears is analyzed by means of both a SCM based on both polynomialchaos expansions and Hermite-Padé approximants, given the highly non-lineardependence of the antenna’s figures of merit on variations in the substrate’sproperties. The characterization technique described in Chapter 4 validates theanalysis by providing the joint probability distribution of substrate height andpermittivity, for several compressed samples of the considered material.The reliability of textile antennas is certainly interesting not only if considered

xv

in itself but also if related to a more elaborate context, such as the incorporationin a complete wireless system. Therefore, in Chapter 6, a simple Wireless PowerTransfer (WPT) link is studied in which a textile antenna operating in the 2.45GHz ISM band and connected to a rectifying circuit is fed by a transmittinghorn antenna. By relying on gPC expansions, the production uncertainties onthe textile antenna are linked to the Power Transfer Efficiency (PTE) of thesystem. Then, an efficient model for the interaction between devices in theradiative near-field is applied and the misalignments between the antennas arealso accounted for in a very precise and fast way.

List of Abbreviations

AC alternating currentADS Advanced Design SystemAR axial ratioCDF Cumulative Distribution FunctionCST Computer Simulation TechnologyDC direct currentDOA direction of arrivalgPC generalized Polynomial ChaosGPS Global Positioning SystemIoT Internet of ThingsISM Industrial, Scientific and MedicalMUT material under testPDF Probability Distribution FunctionPTE Power Transfer EfficiencyRF radio frequencyRFID radio frequency identificationRV random variableSBO surrogate-based optimizationSCM Stochastic Collocation MethodSGH standard gain hornSIW Substrate Integrated WaveguideST Stochastic TestingSVD singular value decompositionWPT Wireless Power Transfer

List of Symbols

< ·, · > inner product.T transpose matrix.∗ complex conjugate|·| magnitude of a complex or a real number‖·‖ Euclidean norm

ε permittivity of a mediumεr relative permittivityμ permeability of a medium1

μ0 vacuum permeabilitytan δ loss tangentZ characteristic impedance of free space

μ mean value2

σ standard deviationΓ function support

j imaginary unit� set of complex numbers

Zin input impedanceRin input resistanceXin input reactanceVg generator voltageZg generator impedanceZL load impedanceIsc short-circuit current

fr resonance frequencyω angular frequencyλ wavelength

Ω Ewald sphereθ inclination angleφ azimuthal angleα, β, γ Euler angles

xx List of Symbols

· field vector· unit field vectorT (·) translation operatorh2

l (·) l-order spherical Hankel function of the second kindPl(·) Legendre polynomial of degree lAR

pq, BRpq spherical harmonics coefficients

F spherical harmonics expansionF−1 inverse spherical harmonics expansiondr

pq Wigner small d-matrixYpq(·) orthonormalized scalar spherical harmonic of degree p

and order qR rotation in the spatial domainRSH rotation in the spherical harmonics domain

List of Publications

Articles in International JournalsRelated to the work presented in this Ph.D. thesis

• M. Rossi, A. Dierck, H. Rogier, and D. Vande Ginste, “A stochastic frame-work for the variability analysis of textile antennas”, IEEE Transactionson Antennas and Propagation, vol. 62, no. 12, pp. 6510–6514, 2014.

• M. Rossi, S. Agneessens, H. Rogier, and D. Vande Ginste, “Stochasticanalysis of the impact of substrate compression on the performance oftextile antennas”, IEEE Transactions on Antennas and Propagation, vol.64, no. 6, pp. 2507–2512, 2016.

• M. Rossi, G.-J. Stockman, H. Rogier, and D. Vande Ginste, “Stochasticanalysis of the efficiency of a wireless power transfer system subject toantenna variability and position uncertainties”, Sensors, vol. 16, no. 7,2016.

• M. Rossi, S. Agneessens, H. Rogier, and D. Vande Ginste, “Assembly-line-compatible electromagnetic characterization of wearable antennasubstrates”, Accepted for publication in IEEE Antennas and WirelessPropagation Letters, 2016.

Other journal contributions

• S. Prashant, M. Rossi, I. Couckuyt, D. Deschrijver, H. Rogier, and T.Dhaene, “Constrained multi-objective antenna design optimization usingsurrogates”, Submitted to International Journal of Numerical Modelling:Electronic Networks, Devices and Fields, Jul. 2016.

Articles in Conference Proceedings

• F. Boeykens, M. Rossi, L. Vallozzi, D. Vande Ginste, and H. Rogier,“A stochastic framework to model bending of textile antennas”, in In-ternational Symposium on Antennas and Propagation and USNC-URSINational Radio Science Meeting, 2014, pp. 9–10.

xxii List of Publications

• M. Rossi, S. Agneessens, H. Rogier, and D. Vande Ginste, “Dedicatedstochastic framework for the variability analysis of compressible textileantennas”, in International Symposium on Antennas and Propagation andUSNC-URSI National Radio Science Meeting, 2016, pp. 1923–1924.

• M. Rossi, S. Agneessens, H. Rogier, and D. Vande Ginste, “Enhancingthe design of textile antennas with a polynomial chaos-based stochas-tic framework”, in URSI Commission B International Symposium onElectromagnetic Theory, 2016, pp. 522–525.

• M. Rossi, H. Rogier, and D. Vande Ginste, “Generalized polynomial chaosparadigms to model uncertainty in wireless links”, Accepted at the 11thEuropean Conference on Antennas and Propagation, 2017.

Book Chapters

• C. Loss, M. Rossi, S. Agneessens, R. Goncalves, H. Rogier, P. Pinho,and R. Salvado, “Electromagnetic characterization of textile materials forthe design of wearable antennas and systems”, in Wearable Technologiesand Wireless Body Sensor Networks for Healthcare, F. J. Velez and F. D.Miyandoab, Eds., Stevenage, SG1 2AY, UK: IET, in press, ch. 6.

Awards

• URSI Commission B Young Scientist Award, URSI Commission B Inter-national Symposium on Electromagnetic Theory (EMTS 2016), Espoo,Finland, August 2016International Union of Radio Science, 2016

Stochastic Full-Wave Modeling for the VariabilityAnalysis of Textile Antennas

1Introduction

1.1 ContextIn 1999, Kevin Ashton introduced the concept of Internet of Things (IoT) inrelation to his application of radio frequency identification (RFID) tags tothe management of Procter & Gamble’s supply chain for the tracking andhandling of products in its warehouses [1]. The idea that he was trying to putforward was very simple: even though Internet has drastically changed theshape of our society and a gigantic amount of information has readily becomeavailable to everybody, most or all of these data are still gathered and sharedby human beings. But what would happen if computers could autonomouslycollect information without people’s supervision and even make decisions basedon it?

The possibilities offered by the implementation of such a principle would radi-cally affect an immense range of aspects of our life [2–4] and considerably help usfacing the challenges that our society is confronted with. Think about the globalwarming, for instance. As we are increasingly asked to rationalize the consump-tion of energy and resources, we could greatly benefit from a pervasive networkof sensors that, deployed in each house and in each neighbourhood, would allowcollecting and sharing the data about power usage to optimize the supply to thepower grid. Or consider healthcare. As the world population is growing older,hospitalization for everybody will become practically unattainable. Therefore,the possibility of assisting patients and monitoring their condition from a dis-tance would allow home hospitalization without sacrificing the effectiveness andthe responsiveness of treatments.

But how are we going to turn such alluring scenarios into reality? First ofall, the physical barriers between computers and the world around us need tobe removed. To this end, a great variety of everyday objects will have to be

4 Chapter 1. Introduction

transformed into ‘smart objects’ or ‘things’ by equipping them with sensorsto collect information, microchips and communication components to processand transmit it, and actuators to respond to specific instructions [5, 6]. Next,a network that allows sharing the collected data in an effective way must beimplemented, as well as a service layer through which the services required bythe users are managed. Finally, an interface through which the user can activelyinteract with the whole system is necessary [7, 8]. Once these challenges will beovercome, the resulting degree of interaction and control with our sorroundingswill be immense.

For the time being, the advances in RFID and sensor technology, promptedby the progress undergone by silicon technology, have enabled the large-scaleproduction of cheap, realiable and small-size components for easy integrationinto everyday objects for sensing and wireless communication between eachother [9, 10]. Therefore, even though a clear and definitive solution for thedevelopment and the implementation of a network protocol has not yet beenfound [7], and although a number of issues related to privacy and security [11]still need to be addressed, the IoT is likely to become a reality in the nextdecade.

In this context, considerable attention has been devoted to the human body andthe possibility to enhance its capabilities by leveraging an ever closer interactionwith ‘smart’ sorroundings. The widespread diffusion of smartphones and, lateron, the introduction of smartwatches testify how beneficial portable devicescan be in everyday life. In this respect, the complete on-body integration of awireless system capable of different functionalities has to be seen as the naturalfurther step towards a perfect interdependence between man and computer.The applications envisioned by researchers for such body-centric systems aremanifold: patient monitoring and rehabilitation [12–14], augmented reality [15,16], and localization during rescue operations [17–19], only to mention a few. Inparticular, a most interesting class of wearable systems rely on textile antennasto provide pervasive wireless communication to the user [20–23] or even completefunctionality by embedding the electronics into the antenna substrate [24]. Beinglight-weight, inconspicous and flexible, such ‘smart’ textiles are bound to playa central role within the future IoT framework.

1.2 MotivationAmong the conditions that any technology needs to meet in order to reach asufficient degree of maturity, reliability comes first. This generally enforcedrequirement is particularly stringent for wearable systems, since most of theapplications for which they are designed are critical. Home hospitalization isfully beneficial only if patients are constantly monitored and can be promptlyassisted. In the same way, the position of the rescuers during an emergencyhas to be accessible at any moment. As far as the transmission of data is

1.2. Motivation 5

concerned, two main aspects of a wireless system require special attentionfrom the designers. First, the communication units have to be reliable andas robust as to withstand the deployment in adverse scenarios. Second, thecommunication link has to be designed such that it guarantees a stable data flowin any operating condition. Therefore, the aim of this dissertation is to addressthe reliability of textile antennas for body-centric communication systems.

Textile antennas are typically designed and produced in a patch antenna topology.More specifically, a textile material serves as antenna substrate, whereas aconductive patch and a ground plane are glued on it at either side. In othercases, both the patch and the ground plane are directly embroidered on thesubstrate [25, 26]. In this way, the ground plane counteracts the high absorptionrate of the electromagnetic fields by the body and the radiation is directedtowards broadside [27]. However, the overall precision of the entire process isarguably low, as the patch is cut and glued by hand and embroidering machinesyield no better accuracy than manual cutting. Moreover, the antenna feed isalso placed and soldered by hand. As a result of these uncertainities, the actualantenna may very well differ from the designed one in terms of performance.

In this respect, another issue arises from the choice of textile materials for theantenna. As previously mentioned, textiles benefit from their light weight, theirflexibility and the possibility to integrate them into everyday clothing. In thisway, the comfort of the user is not sacrificed and the entire wearable systembecomes hardly noticeable. In addition, the use of off-the-shelf materials consid-erably reduces the production costs. However, cheap general purpose textilescannot be expected to exhibit stable electromagnetic properties, in contrastto expensive rigid high-frequency laminates. Conventionally, textile antennasare designed by relying on a nominal average value for the electromagneticcharacteristics of the substrate. As a result, any non-uniformity in the appliedmaterials may entirely compromise a design that would otherwise be correct.

The intrinsic characteristics of textile antennas also have a considerable impacton their figures of merit, when they are deployed in a real scenario. Beingflexible, they necessarily bend to adapt to the morphology and the movements ofthe body, especially if they are placed on the limbs of the user. Moreover, somespecial materials, such as the flexible closed-cell expanded rubber protectivefoam applied in [19] for textile antennas integrated into firefighters’ garments, arecompressible. As antenna compression may always occur in critical situations,it is paramount to establish whether it may result in the incorrect operation ofthe communication system.

In a nutshell, the impact of production uncertainties, non-uniformity in theapplied materials, bending and compression have to be carefully considered toassess the robustness of any antenna design. Since these effects appear in arandom way, a precise statistical evaluation of their influence on the performanceof the antenna is needed. On the one end, this requires to figure out a strategy todetermine the exact probabilistic distribution of the variations in the antenna’s


design parameters. On the other hand, the overall impact of such variationson the figure’s of merit of the antenna has to be rigorously quantified. Sincetime-consuming full-wave solvers are conventionally used in antenna design,traditional techniques, such as Monte Carlo analysis, are of little avail here, asthey are accurate but require to simulate thousands of realizations. Therefore,another approach has to be adopted.

Finally, it is certainly interesting to take a broader view on how all the afore-mentioned effects may affect the performance of the system in which textileantennas are deployed. In particular, one of the pillars of the IoT consists inthe replacement of active RFID and sensors, which necessarily rely on batteriesfor activation and operation, by passive components. Therefore, we can picturea scenario in which a transmitter powers on-body sensors via textile antennas,which embed the electronics to convert the impinging electromagnetic radiationinto direct current (DC) power. In this case, the efficiency of the WirelessPower Transfer (WPT) system depends on the impedance matching betweenthe antenna and the embedded rectifying circuit, as well as on the relativepositions of transmitter and receiver. Since the antenna’s radiation impedancechanges due to the uncertainties in its design parameters, the results availablefrom variability analyses performed on single antenna designs can be applied togain further insight into the operation and the efficiency of wearable systems.

1.3 Current state of the artIndependent from the application, it is well known that high simulation timeforms the Achilles’ heel of Monte Carlo-based formulations for stochastic prob-lems, since the accuracy of the solutions converges only with the inverse ofthe square root of the number of processed realizations [28]. To overcomethis limitation, Stochastic Collocation Method (SCM) based on generalizedPolynomial Chaos (gPC) expansions have recently been introduced [29–31].More specifically, these techniques relate the figures of merit of the system understudy to its random design parameters by means of polynomial functions. Suchpolynomials are defined to achieve exponential convergence of the approximatedfigure of merit for the input Probability Density Functions (PDFs) accordingto which the design parameters undergo variations. Therefore, for this kindof applications SCMs are usually preferred over general purpose determinisiticsurrogate modeling techniques, such as Neural Networks or Kriging. Moreover,as only a relatively small number of realizations need to be evaluated to con-struct such functions, a significant speed-up is achieved with respect to theMonte Carlo approach, without making any sacrifice in terms of the solution.Finally, the non-intrusive versions of this paradigm, which will be explored inthis dissertation, can easily be combined with full-wave solvers.

Polynomial chaos expansions have been applied to address a remarkable varietyof problems: variability analysis of interconnects and lumped circuits [32–35],

1.3. Current state of the art 7

as well as of multiport systems [36, 37], uncertainty quantification relatedto scattering problems [38, 39], direction of arrival (DOA) estimation [40],to mention a few. Nevertheless, for antenna design their potential has longbeen overlooked, even though variability analysis is not less important herethan in other fields. Recently, the influence of substrate bending on thetextile antenna’s resonance frequency has been addressed with a stochasticframework that combines the polynomial chaos expansion with a dedicatedcavity model [41]. In this case, the probability density function of the antenna’scurvature radius has been obtained from a measurement campaign involvingover seven thousands people, during which the radius of each person’s armhas been measured. However, the proposed framework is not applicable to theeffects due to variations in material characteristics, substrate compression andproduction uncertainties, and this for two reasons. First, different random designparameters are considered, which affect several figures of merit of the antenna,besides its resonance frequency. Second, full-wave solvers must be applied toinvestigate the influence of each design parameter on the antenna’s figures ofmerit. Therefore, dedicated strategies are developed in this manuscript.

In particular, the non-uniformity of textile substrates may be accounted for intwo ways. On the one hand, the probability density function of the material’selectromagnetic characteristics can be determined and combined with polyno-mial chaos expansions to calculate the corresponding statistics of the antenna’sfigures of merit. On the other hand, each antenna substrate may be carefullycharacterized before the antenna is produced in order to avoid discrepanciesbetween design and implementation. In either case, a reliable characterizationtechnique for textiles is required. In this respect, several approaches are avail-able in literature, among which non-resonating broadband transmission linemethods [42, 43] and planar antenna resonator techniques [44, 45] are the mostcommon. However, despite providing a very precise characterization of substratematerials, they are generally time-consuming and destructive, meaning thatthe characterized sample cannot be reused after its properties are estimated.Therefore, in case very inhomogeneous materials are applied with non-uniformelectric characteristics, their characterization turns out to be practically useless,because the results found for one substrate sample cannot be assumed to stillbe valid for another one. As a result, a fast accurate non-destructive approachis needed.

Finally, different WPT systems have been studied in literature as a result ofthe recent emphasis on passive radio frequency (RF) components in the contextof the IoT [46–50]. However, near-field WPT can achieve high Power TransferEfficiency (PTE) only if source and target are in close proximity and well aligned,whereas far-field schemes are not very efficient. In contrast, radiative near-field(Fresnel region) WPT is found to yield an acceptable trade-off between efficiencyand operational range. Several numerical techniques are available that addressthe performance of near-field WPT schemes [51–56]. Nevertheless, they donot consider potential variations in the radiation characteristics of the actually


deployed antennas resulting from production uncertainties, nor do they estimatehow the PTE varies for a given system configuration when the position of thedevices deviates from the nominal one. Therefore, they would considerablybenefit from the combination with gPC techniques.

1.4 Novel contributions and outline of the thesisThis dissertation is primarily focused on assessing the impact of productionuncertainties, non-uniformity in textiles and substrate compression on theperformance of textile antennas. The resulting contributions, published ininternational peer-reviewed journals and presented at international conferences,are entirely novel. Along with a previous work addressing substrate bending [41],they provide antenna designers with a complete and reliable framework to verifythe robustness of a given design with respect to manufacturing and deploymentin real scenarios. Moreover, the extension of the results and the methodologyto the analisys of a simple wireless system operating in the radiative nearfield provides the first statistical insights into the efficiency of a WPT strategyrelying on textile antennas.

In Chapter 2, the main numerical techniques applied throughout the entiremanuscript are outlined. In particular, both Monte Carlo techniques and SCMsare discussed, with special attention devoted to gPC expansions. Moreover,both the Padé approximants, which are found to be more efficient than gPCexpansions for highly non-linear relationships between figures of merit andrandom variables [57], and the Stochastic Testing (ST) method, which reducesthe number of collocation points necessary to compute the gPC expansions [58],are presented.

In Chapter 3, the effect of production uncertainties on the performance of arepresentative inset-fed patch textile microstrip antenna and a dual-polarizedtextile antenna are addressed. First, a sensitivity analysis is performed onboth designs to select the antenna’s geometry and material parameter whosevariations have the highest influence on the antenna’s input impedance. Then,for each antenna, two gPC expansions are introduced to relate the statisticaldistribution of both the real and the imaginary part of the input impedance tothe distribution of these design parameters. For the first design, the approachis validated by implementing a suitable number of prototypes on a rigid high-frequency laminate. For the dual-polarized textile antenna, 100 patches arecut and measured to reconstruct the probability density functions of the mostsensitive design parameter, being the patch width. Then, a Monte Carlo analysisproves the efficiency and the accuracy of the technique.

In Chapter 4, a novel procedure for the non-destructive electomagnetic charac-terization of textile substrates is presented. More specifically, the propertiesof a material under test (MUT) are estimated by comparing simulations andmeasurements performed with a resonance-perturbation method based on pla-

1.4. Novel contributions and outline of the thesis 9

nar resonating fixtures operating at 2 GHz and in the vicinity of the GlobalPositioning System (GPS)-L1 frequency band ([1.56342,1.58742] GHz). Foreach structure, the approach is validated on materials whose electromagneticcharacteristics are well known, after which the method is applied to textiles ofinterest. The technique allows precise and fast material characterization.

In Chapter 5, substrate compression is taken into account. To this end, the jointheight and relative permittivity probability density function of a compressiblesubstrate is rigorously quantified by means of the characterization techniquedescribed in Chapter 4. Then, for the obtained distribution, an SCM based onboth polynomial chaos expansions and Hermite-Padé approximants is definedto model the effect of substrate compression on the performance of a probe-fedGPS textile antenna. Hermite-Padé approximants prove to converge faster thangPC expansions, given the highly nonlinear relationship between height andpermittivity and the figures of merit of the antenna. As in Chapter 3, a MonteCarlo simulation validates the analysis and proves its superior efficiency.

In Chapter 6, a WPT system consisting of a transmitting horn antenna and areceiving textile antenna operating in the Industrial, Scientific and Medical (ISM)band at 2.45 GHz is considered. The gPC theory is combined with an efficientmodel for the interaction between devices in the radiative near-field to investigatethe impact of randomness in the design parameters of the textile antennas andtheir relative position on the PTE of the system. In a first stage, the radiationcharacteristics of the textile antenna are modeled as functions of the designparameters exhibiting uncertainties. Then, position variaibility is included andthe overall PTE of the system is accurately assessed. The technique proves tobe more efficient and flexible than Monte Carlo.

Conclusions and future research are summarized in Chapter 7.

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[39] Z. Zubac, D. de Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the multilevel fast mul-tipole method (MLFMM) with the stochastic galerkin method (SGM)”,IEEE Antennas and Wireless Propagation Letters, vol. 13, pp. 1275–1278,2014.

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[41] F. Boeykens, H. Rogier, and L. Vallozzi, “An efficient technique based onpolynomial chaos to model the uncertainty in the resonance frequency oftextile antennas due to bending”, IEEE Transactions on Antennas andPropagation, vol. 62, no. 3, pp. 1253–1260, 2014.

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2Stochastic Electromagnetic

Modeling

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Since textile antennas are typically designed with the aid of full-wave solvers, stochastic variations in their design parameters cannotbe related to their performance by means of deterministic equa-tions. Therefore, their variability analysis necessarily relies onnon-intrusive approaches, which reconstruct the statistics of the fig-ures of merit of the antenna under study from the results of thefull-wave simulations performed on a suitable set of realizations.The two main non-intrusive methods applied in this manuscriptare the Monte Carlo method, which serves as the validation tech-nique for each stochastic analysis, and the Stochastic CollocationMethod (SCM) based on generalized Polynomial Chaos (gPC) ex-pansions. In addition, Padé approximants, which outperform gPCexpansions for highly non-linear dependencies of the antenna’s fig-ures of merits on the random variables, and the Stochastic TestingStochastic Testing (ST) method, which is adopted when a high num-ber of random variables compromise the efficiency of gPC expansions,are also discussed.

2.1 The Monte Carlo methodConsider a given textile antenna design and suppose that its generic figure ofmerit G (such as, for example, its return loss |S11|dB) is affected by randomvariations in N design parameters. Each of these parameters is now associated

18 Chapter 2. Stochastic Electromagnetic Modeling

with an input independent orthonormal random variable (RV) xi, which varieson a support Γi according to a given Probability Distribution Function (PDF)pXi

(xi). We collect all these RVs in a single vector x = [x1, x2, ..., xN ], with

corresponding multivariate PDF PX =N∏i

pXi(xi) and support Γ =

N⋃i=1

Γi. Each

specific value xr ∈ Γ of the random vector x, generated according to the PDFPX , is called a realization. The realization xr is related to the vector x′

r in theactual domain of the antenna’s randomly varying design parameters x′ by thefollowing transformation of RVs:

x′r = T xr + μ, (2.1)

where T is the upper triangular matrix resulting from the Cholesky decomposi-tion of the covariance matrix Σ corresponding to x′ and μ is the mean vectorof x′. As such, the relation (2.1) also accounts for any potential correlationamong the antenna’s design parameters.

The traditional technique to collect statistical information about the antenna’sfigure of merit G is by means of the Monte Carlo method [1]. More specifically,Mr realizations xr are randomly generated according to the PDF PX , trasformedby (2.1) and simulated by a full-wave solver to compute the corresponding Mr

values of G. This set of solutions is then used to determine the output PDF ofthe figure of merit G, corresponding to the variations in the antenna’s designparameters. In particular, the mean and the variance of the output PDF aregiven by:

μG =∑MR

i=1 Gi

MR(2.2)

σ2G =

∑MR

i=1(Gi − μG)2

MR − 1 , (2.3)

where Gi denotes the i-th Monte Carlo realization of G.

Being robust and easy to implement, the Monte Carlo method is one of the mostknown and widely applied mathematical techniques. However, the accuracyof the solutions that it provides only converges with the inverse of the squareroot of Mr, independent from the number N of the considered RVs. Therefore,very large sets of realizations have to be processed to reach an acceptableaccuracy. Consequently, in case the evaluation of a single realization requiresconsiderable time and computational resources, as is the case with full-wavesimulations, Monte Carlo turns out to be a sub-optimal choice. Yet, when thenumber of RVs is very high, it may still represent the most convenient option.In this manuscript, the variability analyses performed on textile antennas and aWireless Power Transfer (WPT) system typically only involve a limited numberof RVs. Therefore, Monte Carlo provides the reference solution through whichthe results of more sophisticated and efficient methods are validated.

2.2. Generalized Polynomial Chaos 19

2.2 Generalized Polynomial ChaosA more efficient alternative to Monte Carlo techniques is provided by thegPC method [2]. Instead of iteratively sampling the figure of merit G forMr realizations to constructs its statistics, the following expansion of G isintroduced:

G = f(x) =K∑

k=0ykϕk(x), (2.4)

where yk are the expansion coefficients, yet to be determined, whereas ϕk(x)are multivariate polynomial basis functions. These are constructed to beorthonormal to the multivariate PDF PX , as follows:

< ϕk(x), ϕl(x) >=∫

Γ

ϕk(x)ϕl(x)PX(x)dx = δkl, (2.5)

with δkl = 0 if k �= l, δkl = 1 if k = l, (k, l = 0, ..., K) and the PDF PX acts asweighting function in this inner product. Since the considered N RVs xi aremutually independent, the PDF PX is defined as the product of the PDFs pXi(xi)that correspond to the single input RVs. Therefore, the multivariate basisfunctions ϕk(x) are defined as products of univariate orthonormal polynomialsφik

(xi) associated to the single random variable xi:

ϕk(x) =N∏

i=1φik

(xi), (2.6)

where ik is the degree of the (k, i)-th polynomial. In particular, the polynomialsφik

(xi) are orthonormal to the PDF pXi(xi):

< φik(xi), φil

(xi) >=∫

Γi

φik(xi)φil

(xi)pXi(xi)dxi = δkl. (2.7)

In this way, the optimal convergence of the approximation (2.4) is achieved. Forwell-known PDFs, such as the Gaussian or the uniform PDF, the polynomialsφik

(xi) are selected by following the Wiener-Askey scheme [3]. Alternatively,the modified-Chebyshev algorithm can be leveraged for more unconventionalinput PDFs [4]. The total degree P of a multivariate basis functions ϕk(x) isgiven by:

P =N∑

i=1ik. (2.8)

When using only basis functions with a total degree that is smaller than orequal to P , and for N RVs, the total number K + 1 of polynomials in (2.4) isgiven by:

K + 1 = (N + P )!N !P ! . (2.9)


The gPC expansion (2.4) serves as a macromodel of the antenna under study.Upon its knowledge, the output PDF of the figure of merit G is calculatedby evaluating (2.4) for a suitable set of realizations, generated according tothe input PDF PX . Since calculating the value of G for a single realizationby simply evaluating this polynomial function is incommensurably faster thanusing a full-wave solver, gPC expansions are expected to be much more efficientthan the Monte Carlo method. Moreover, it is easily shown that the meanand the variance of the output PDF are readily computed from the expansioncoefficients yk:

μG = y0 (2.10)

σ2G =

K∑k=1

|yk|2. (2.11)

2.3 Stochastic Collocation MethodThe gPC coefficients yk in (2.4) are yet to be determined. In this dissertation,we focus on non-intrusive methods, since textile antennas are simulated bymeans of full-wave solvers and no deterministic equations are available toanalytically carry out the stochastic analysis of the figure of merit G. Therefore,the coefficients yk are calculated by sampling the values of G using a suitableset of realizations called collocation points. Galerkin weighting is applied to(2.4) and, by leveraging (2.5), the following equation is obtained:

yk =< f(x), ϕk(x) >=∫

Γ

f(x)ϕk(x)PXdx. (2.12)

The most straightforward way to calculate the integral (2.12) is to resort to aGaussian quadrature rule:

yk ≈Q1∑

q1=1...

QN∑qN =1

wq1 ...wqNφ1k

(xq1)...φNk(xqN

)f(x′q1

, ..., x′qN

) (2.13)

where the quadrature points xq1 , ..., xqNand their corresponding weights wq1 , ...,

wqNare derived by the Golub-Welsch algorithm [5] according to the input PDF

PX . This preserves the convergence of the solution. The points x′q1

, ..., x′qN

resultfrom the transformation of the points xq1 , ..., xqN

according to (2.1). Finally,the values f(x′

q1, ..., x′

qN) are evaluated by means of a full-wave solver. It is

important to point out that the total number of quadrature points xq1 , ..., xqN,

necessary to compute the coefficients yk, is:

Q =N∏

i=1Qi. (2.14)

2.4. Stochastic Testing 21

Therefore, when the total order of expansion and the number of RVs increase,the implementation of a SCM requires the processing of an increasing numberof quadrature points. This issue is called the curse of dimensionality. As aresult, for some applications, gPC expansions may not yield clear benefits withrespect to Monte Carlo or may even be less efficient.

2.4 Stochastic TestingSeveral methods are available in literature to tackle the curse of dimensionalityin stochastic collocation methods, such as Smolyak’s rules [6] and Stroudcubatures [7]. Their common strategy consist is based on a subset of points froma tensor product rule and, in this way, reducing the number of quadrature pointsthat are needed to compute the integrals (2.12). However, these techniquesstill become quite inefficient for problems involving very high number of RVsand they also entail a loss of accuracy. Recently, a novel ST algorithm hasbeen proposed which outperforms the aforementioned approaches [8]. Thenon-intrusive implementation of this method starts by defining a completeset of Q quadrature points xq = [xq1 , ..., xqN

] constructed by means of the N -dimensional tensor product Gaussian quadrature rule applied in (2.13). Then,a number M = K � Q of quadrature points tm is selected from this set asfollows. First, the set of points xq is sorted according to their correspondingweight wq = wq1 ...wqN

, in a decreasing order, and the first point x0 is selectedas first collocation point t0. Next, the following M × 1 matrix V is computed:

V = ϕ(t0)‖ϕ(t0)‖ , (2.15)

where ϕ(t0) = [ϕ0(t0), ..., ϕK(t0)]T . Finally, all the other M − 1 collocationpoints are selected by means of an iterative procedure. More specifically, apoint xq is selected as a new collocation point tm if:

‖v(xq)‖‖ϕ(xq)‖ > χ, (2.16)

where χ is a threshold, in this thesis selected to be 10−3, and v(xq) is given by

v(xq) = ϕ(xq) − V V T ϕ(xq). (2.17)

Then, the matrix V is updated by including the normalized vector

v(xq)‖v(xq)‖ (2.18)

as a new column. The algorithm ends when M collocation points have beenselected. As a final step, both a matrix A, whose elements are defined as


amk = ϕk(tm), and its inverse B are computed. The coefficients yk in (2.4) arethen calculated as in [9]:

yk =M∑

m=0bmkf(tm), (2.19)

where bmk is the mk-th element of matrix B and f(tm) is evaluated in tm.

2.5 Padé approximantsIn some applications, despite the number of RVs is low and, thus, the curse ofdimensionality is of little importance, the relationships between the figure ofmerit G and the input RVs may be highly non-linear. As a result, polynomialapproximations such as the gPC expansions may lead to the Runge phenomenon,where the interpolating polynomial exhibits an oscillating behavior at the edgesof the doamin Γ . A higher order of expansion does not necessarily guaranteethat these oscillations will disappear. In these scenarios, a robust alternativeto gPC expansions is required and Padé approximants are an alternative [10].Thereto, the figure of merit G is expressed as:

G = f(x) = PS(x)UL(x) , (2.20)

where PS(x) and UL(x) are polynomials of degrees S and L, respectively.Moreover, UL(x) > 0 has to hold in the entire domain Γ of f(x). Thenumerator PS and the denominator UL are defined as follows:

PS =K1∑

k=0pkϕk(x), (2.21)

UL =K2∑

k=0ukϕk(x), (2.22)

which are entirely equivalent to expansion (2.4) and use the same basis func-tions ϕk(x). In order to determine the expansion coefficients pk and uk in (2.21)and (2.22), an integer T ≥ 1 is chosen and J basis functions ϕk(x) are selected,whose total degree is higher than S but lower than S + T + 1. By leveragingthe orthonormality of the polynomials ϕk(x), we find that:

< PS(x), ϕk(x) >=< f(x)UL(x), ϕk(x) >= 0,

k = K1 + 1, ..., K1 + J. (2.23)

2.5. Padé approximants 23

The second equality in (2.23) corresponds to the following linear system:

Cu =

⎡⎢⎣

< fϕ1, ϕK1+1 > · · · < fϕK2 , ϕK1+1 >...

. . ....

< fϕ1, ϕK1+J > · · · < fϕK2 , ϕK1+J >

⎤⎥⎦

⎡⎢⎣

u1...

uK2

⎤⎥⎦ = 0, (2.24)

where u is the vector of the coefficients uk in (2.22). The elements of thematrix C are computed by means of the quadrature rule (2.13). Since, typically,for a given T ≥ 1, J is higher than K2, the system (2.24) is overdetermined.Therefore, the vector u is chosen to be the optimal solution of the system (2.24)in the least-squares sense, which means that it is found by solving the followingminimization problem:

min‖u‖=1

‖ Cu ‖, (2.25)

where ‖ · ‖ is the L2-norm. A singular value decomposition (SVD) of thematrix C provides the solution to (2.25), which is easily and efficiently computed,since the values of S, L and T are relatively small. Finally, the following equalityyields the coefficients pk of the numerator PS :

pk =< PS(x), ϕk(x) >=< f(x)UL(x), ϕk(x) >, k = 0, 1, ..., K1, (2.26)

where the value of < f(x)UL(x), ϕk(x) > is calculated using the same quadra-ture rule and the same quadrature points as in (2.24).

References[1] G. Fishman, Monte Carlo: Concepts, algorithms, and applications. Springer

Science & Business Media, 2013.[2] D. Xiu, “Fast numerical methods for stochastic computations: A review”,

Commun. Comput. Phys, vol. 5, no. 2-4, pp. 242–272, Feb. 2009.[3] D. Xiu and G. E. Karniadakis, “The Wiener–Askey polynomial chaos for

stochastic differential equations”, SIAM Journal on Scientific Computing,vol. 24, no. 2, pp. 619–644, 2002.

[4] W. Gautschi, Orthogonal Polynomials - Computation and Approximation.Oxford, UK: Oxford University Press, 2004, p. 76.

[5] G. Golub and J. Welsch, “Calculation of Gauss quadrature rules”, Mathe-matics of Computation, vol. 23, no. 106, pp. 221–230, Apr. 1969.

[6] S. A. Smolyak, “Quadrature and interpolation formulas for tensor productsof certain classes of functions”, in Dokl. Akad. Nauk SSSR, vol. 4, 1963,p. 123.

[7] D. Xiu and J. S. Hesthaven, “High-order collocation methods for dif-ferential equations with random inputs”, SIAM Journal on ScientificComputing, vol. 27, no. 3, pp. 1118–1139, 2005.


[9] P. Manfredi, D. Vande Ginste, D. De Zutter, and F. Canavero, “General-ized decoupled polynomial chaos for nonlinear circuits with many randomparameters”, IEEE Microwave and Wireless Components Letters, vol. 25,no. 8, pp. 505–507, 2015.


3A Stochastic Framework for the

Variability Analysis of TextileAntennas

Marco Rossi, Arnaut Dierck, Hendrik Rogier, Dries Vande Ginste

Based on the article published in IEEE Transaction on Antennas andPropagation

� � �

A novel framework to accurately quantify the effect of stochasticvariations of design parameters on the performance of textile anten-nas is developed and tested. First, a sensitivity analysis is appliedto get a rough idea about the effect of these random variations onthe textile antenna’s performance. Next, a more detailed view isobtained by a generalized Polynomial Chaos (gPC) technique thataccurately quantifies the statistical distribution of the textile an-tenna’s figures of merit, for a given range over which geometry andmaterial parameters vary statistically according to a given distribu-tion. The method is validated both for a simple inset-fed patch textilemicrostrip antenna and for a dual-polarized textile antenna. For theformer, we experimentally test the proposed stochastic framework byconstructing a suitable number of prototypes on a reliable substrate.For the latter, the probability density function corresponding to itsmost sensitive design parameter, being the width, is experimentally

28 Chapter 3. Variability Analysis of Textile Antennas

estimated by means of measurements performed on 100 patches. AKolmogorov-Smirnoff test proves that, for all considered examples,the results are as accurate as those obtained via Monte Carlo analy-sis, while the new technique is much more efficient. Indeed, speedupsup to a factor 1667 are demonstrated.

3.1 IntroductionDuring the past years, wearable textile antennas have acquired a lot of interestas suitable devices for deployment in critical operations such as rescue missions,military interventions and e-health applications. Although they have thepotential to provide high gain and large radiation efficiency [1–6], even inproximity of the human body [7], their antenna characteristics are typicallymore subject to variations than observed in conventional rigid planar-circuit-board antennas [8, 9]. Two causes may be identified that lie at the origin ofthese deviations from the nominal antenna characteristics. First, the productionprocess of the applied materials and of the final antenna assembly is lessaccurate compared to antennas defined on high-frequency laminates. Second,the operating conditions combined with the flexibility and compressibility oftextile antennas may modify the textile antennas’ geometry as well as materialcharacteristics, thereby changing their radiation characteristics [2].

Quite recently, a novel approach [10], based on the combination of a dedicatedcavity model with the Stochastic Collocation Method (SCM), was proposed toobtain a gPC expansion [11, 12] that relates the textile antenna’s resonancefrequency to its bending radius. This method enables the textile antennadesigner to account for the effect of random variations in bending radius onthe antenna performance. Yet, this technique is dedicated to one specificadverse operating condition only, being bending. Currently, all other randomeffects the wearable antenna is subjected to in its production and operatingphases, are accounted for during the design phase by overspecification of theantenna requirements. In addition, a simple sensitivity analysis is typicallyperformed during the computer-aided design phase to verify whether variationsdue to production tolerances are acceptable or not. A posteriori measurementson prototypes deployed in a variety of adverse conditions are then carriedout to verify whether antenna performance remains satisfactory in real-lifeconditions [13].

It is clear that the above-described design process is suboptimal and may turn outto be uneconomical. In addition, the textile antenna may need redesign after theprototyping phase to ensure the required performance in the actual application.Thus, a more precise stochastic framework to support textile antenna designersis outlined in this work. The conventional sensitivity analysis is taken as astarting point to get a rough idea of the effect of variations on the textile

3.1. Introduction 29

antenna’s performance. Yet, to obtain a more precise statistical characterizationduring the computer-aided design process, we introduce a non-intrusive SCMbased on gPC expansions, which we apply as a more efficient method thanconventional Monte Carlo analysis to quantify random variations in the textileantenna’s figures of merit. In contrast to a posteriori experiments, whichonly demonstrate the aggregate effect of variations in all design parameters,this stochastic design framework enables the efficient analysis of performancevariations due to uncertainty in a single design parameter.

The polynomial chaos expansion was introduced in a SPICE simulation envi-ronment to quantify variability in lumped circuits and in distributed intercon-nects [14–17], as well as in multiport systems [18, 19]. The method was also usedto model the statistics of composite media [20] as well as for quantifying theeffect of geometric and material variations in scattering problems [21–24]. Up tonow, however, its application to antenna design remains limited [25], althoughthe importance of taking into account variations in the figures of merit of textileantennas due to adverse deployment conditions and fabrication tolerances hasbeen stressed before [1, 2, 4, 8, 9, 26]. As already mentioned, a stochasticframework combining the polynomial chaos expansion with a dedicated cav-ity model has recently been presented to study the influence of the substratebending on the antenna resonance frequency [10]. However, the approach onlyaims to quantify statistical variations in one particular application scenario,being antenna bending, and the method is not suited to study the effects due tovariations of material characteristics, substrate compression and uncertaintiesin the production process. Therefore, the development of a rigorous and moregeneral approach to characterize the performance of textile patch antennas isof paramount interest.

This chapter is organized as follows. First, Section 3.2 outlines the stochasticframework that extends sensitivity analysis with an SCM to accurately quantifythe statistics. Next, in Section 3.3 the computer-aided design process based onthis stochastic framework is illustrated for a representative textile antenna. It isshown that the gPC technique describes the variations in the figures of merit ofthe antenna with excellent accuracy, also providing a high efficiency in terms ofCPU time. Then, the new formalism is experimentally validated in Section 3.4by manufacturing a minimum number of prototypes, serving as a starting pointfor the statistical analysis. It is demonstrated that complete and accuratestochastic data are obtained with this limited set of prototypes and that this setis only slightly larger than what would be required by the sensitivity analysis.Finally, in Section 3.5 the applicability of the new stochastic framework isdemonstrated by studying the Industrial, Scientific and Medical (ISM) bandtextile antenna presented in [27], for which the distribution of geometricalvariations corresponding to the antenna’s production process is experimentallydetermined. Conclusions are summarized in Section 3.6.


3.2 Stochastic analysis framework for textileantennas

Consider a textile microstrip inset-fed patch antenna, as shown in Fig. 3.1.This antenna is designed to exhibit a sharp resonance peak and a specifiedantenna input impedance Zin at the resonance frequency fr. Note that thischoice of antenna topology and the focus on Zin at fr as antenna figure ofmerit is by no means restrictive and the subsequent analysis may be extendedto any kind of topology and to any figure of merit. In addition, the influenceof variations in the antenna’s design parameters on the antenna’s radiationpattern is addressed in Chapter 6. Our aim is to test the effect of stochastic

Figure 3.1: Representative textile antenna exhibiting a microstrip inset-fedpatch antenna topology for which the stochastic framework is developed.

variations of different design parameters on the input impedance Zin of theantenna at the nominal operating frequency fr. More specifically, in the presentanalysis we consider both the real part Rin and the imaginary part Xin of Zin

as two distinct parameters. The conventional approach consists of performing

3.2. Stochastic analysis framework for textile antennas 31

a sensitivity analysis where Rin and Xin are evaluated for the nominal designand at the two extremities of the range over which the random variable isvarying [2]. This method, however, does not provide all relevant statistical datato the antenna designer. Therefore, following the Wiener-Askey scheme [12],we introduce a SCM to carefully investigate the relation between the input andthe output distributions. More specifically, we relate the real and the imaginarypart of Zin to the random variable y, being the design parameter under study,by means of polynomial expansions:

Rin = f1(y) =P1∑

k=0t1kφY

k (y), (3.1)

Xin = f2(y) =P2∑

k=0t2kφY

k (y), (3.2)

where φYk (y) are suitably chosen polynomial basis functions of degree k, which

are normal with respect to the following inner product:

< u(y), v(y) >=∫

Γ

u(y)v(y)pY (y)dy, (3.3)

where the input Probability Distribution Function (PDF) pY (y), with supportΓ , is taken as a weighting function. Such an orthonormal set of polynomialsφY

k (y) can be computed by means of the modified-Chebyshev algorithm [28].The corresponding expansion coefficients t1k, t2k are as yet unknown.

We now exploit the orthonormality of the basis functions φYk (y), to calculate

the coefficients t1k, t2k:

t1k =< f1(y), φYk >=

∫Γ

f1(y)φYk (y)pY (y)dy, (3.4)

t2k =< f2(y), φYk >=

∫Γ

f2(y)φYk (y)pY (y)dy. (3.5)

We compute integrals (3.4), (3.5) by means of the following N1 and N2-pointsquadrature rules:

t1k =N1∑i=1

wif1(yi)φYk (yi), (3.6)

t2k =N2∑j=1

wjf2(yj)φYk (yj), (3.7)

where the quadrature points yi, yj and the corresponding weights wi, wj arederived by the Golub-Welsch algorithm [29]. Finally, the values f1(yi) and f2(yj),in (3.6), (3.7), are evaluated by means of the Advanced Design System (ADS)Momentum full-wave electromagnetic field solver.


Finally, we test the accuracy of the statistical data resulting from the SCManalysis by generating a representative population of realizations by means of aMonte Carlo simulation, to which we apply the Kolmogorov-Smirnoff test [30].This test enables us to reject the null hypothesis that the sample set matchesthe distribution imposed by the SCM analysis, with a significance level α. Morespecifically, the maximum distance D between the two Cumulative DistributionFunctions (CDFs) generated by the Monte Carlo samples and resulting fromthe SCM analysis, respectively, is compared to a threshold distance Dα. IfD > Dα, the null hypothesis is rejected with a significance level α, otherwise itis accepted.

3.3 Validation for a representative textile antenna3.3.1 Sensitivity analysisConsider the design of a representative textile antenna on a flexible closed-cellexpanded rubber protective foam substrate with a height h equal to 3.94 mm,permittivity εr equal to 1.52 and a loss tangent given by 0.012, characterizedwith the same procedure as in [31]. A computer-aided design procedure wascarried out to match the antenna impedance to 50 Ω at 2.45 GHz. The designparameters considered are the permittivity εr, the thickness of the substrateand the geometrical dimensions of the patch and the inset. Table 3.1 providesthe optimized dimensions of the antenna (see also Fig. 3.1). The correspondinginput impedance at the frequency of 2.45 GHz is 50.29 + 1.028j Ω.

Table 3.1: Nominal values of the antenna geometrical parameters (Fig. 3.1) ona protective foam substrate.

parameter nominal value (mm)patch length L 48.01patch width W 55.3inset length Lin 14inset width Win 2.27feed length Lf 20feed width Wf 15.43

We now first investigate the statistical variation in Zin due to a variation ofa single parameter by means of a sensitivity analysis. Since the closed-cellexpanded rubber foam exhibits a compression set of 30%, after being compressedby 50% at 23◦ for 72 h, we assume that the substrate height h varies between70% and 100% of the nominal value, being [2.758 mm, 3.94 mm]. For all otherdesign parameters, we set a variation of ±5% with respect to the nominal value.The corresponding values of Zin are reported in Table 3.2.

3.3. Validation for a representative textile antenna 33

Table 3.2: Zin values corresponding to all geometrical parameters variations.

nominal value - 5% nominal value + 5%L 15.398 − 26.443j Ω 150.678 − 68.855j ΩW 38.454 + 3.291j Ω 62.998 − 7.474j ΩLin 42.331 + 5.96j Ω 58.177 − 9.89j ΩWin 49.462 + 2.025j Ω 50.995 + 0.038j ΩLf 49.822 + 1.128j Ω 50.557 + 0.675j ΩWf 56.699 + 0.491j Ω 44.342 + 1.265j Ωεr 26.506 − 13.446j Ω 98.58 − 20.153j Ω

70 % nominal value nominal valueh 19.917 − 5.933j Ω 50.29 + 1.028j Ω

We notice that Zin is appreciably affected by a variation of all parametersunder study, except for the width of the inset and the length of the feed line.Therefore, these parameters may be discarded.

3.3.2 SCM-based full statistical analysisBased on the sensitivity analysis, we now further focus on the patch lengthL, which has the most profound effect on Zin. Hence, the random variable yintroduced in Section 3.2 equals the design parameter L. Then, we relate Rin

and Xin, on the one hand, and the input random variable, on the other hand,by means of the gPC expansions (3.1), (3.2), constructed using the SCM. Bothfor a truncated Gaussian distribution and a uniform distribution of the lengthL, we generate the CDFs of Rin and Xin, based on sample points computedwith the full-wave simulator ADS Momentum. We gradually increase the orderof the gPC expansion until the Kolmogorov-Smirnov test confirms that stableCDFs have been found. We then validate these CDFs by comparing them withthe ones obtained by a Monte Carlo full-wave simulation with a sample set of10000 points.

We first assume that the random variable L is distributed following a truncatedGaussian distribution with mean value μ equal to 48.01 mm and standarddeviation σ equal to 0.8 mm, i.e. one third of the half of the variation interval,which spans a support Γ of ±5% with respect to its nominal value (see Fig. 3.2).First, we construct a set of normal polynomials φK

k (y), with respect to thetruncated Gaussian distribution as a weighting function, by means of the ModifiedChebyshev algorithm. More specifically, the algorithm takes the Hermitepolynomials, which are conventionally used for the closely related Gaussiandistribution of infinite support, as a starting point to generate a new set oforthogonal polynomials that have the truncated Gaussian distribution as aweighting function. Based on this set of polynomials, we adaptively construct


Figure 3.2: Truncated Gaussian distribution used as input PDF pY (y) forthe random variable y = L. Bottom axis: absolute variation of L. Top axis:percentage variation of L.

the gPC expansions (3.1), (3.2) up to orders of expansion P1 = 5 and P2 = 8,for which we obtain convergence for both f1(y) and f2(y). Both Rin and Xin

as functions of L are reported in Fig. 3.3.

Finally, we apply the Kolmogorov-Smirnov test to verify whether the CDFsobtained by the SCM and Monte Carlo techniques correspond to the samedistribution. If we set the significance level α to 0.05, we obtain that Dα isequal to 0.019233. We find that the maximum distances DRin

and DXinare

equal to 0.0052 and 0.0182, respectively, proving that the CDF obtained by theSCM is a good approximation for the output CDF, considering a significancelevel of 5%. The CDFs resulting from the Monte Carlo full-wave simulation andfrom the SCM are shown in Fig. 3.4.

Second, we turn our attention to a uniform distribution. We assume thatthe distribution of the patch length L has a mean value μ equal to 48.01mm and varies within a support of ±5% with respect to its nominal value.We adaptively construct two gPC expansions of order P1 = 6 and P2 = 10,respectively, to relate the input and output distributions. Then, we againapply the Kolmogorov-Smirnov test to verify if the CDFs obtained by the SCMtechnique and the Monte Carlo approach correspond to the same distribution.The maximum distances DRin and DXin are now equal to 0.0169 and 0.0093.Hence, even in this case, the SCM allows to accurately approximate the outputCDF, considering a significance level of 5%. All the computed CDFs are reported

3.4. Experimental validation 35

Figure 3.3: Xin and Rin as function of the patch length L according to theStochastic Collocation Method (SCM).

in Fig. 3.5.

As a final remark, we point out that the time required to process a single pointwith ADS Momentum is about 6 s (Intel i7 CPU, 16 Gb RAM). Thus, we needabout 17 hours to process the complete sample set of 10000 points, whereas onlya few seconds are required to obtain the values f1(yi) and f2(yj), to performthe quadrature rules (3.6), (3.7), to construct the basis and the weights in (3.1),(3.2), and to complete the gPC expansions. Therefore, it is clear that the SCMis able to correctly reconstruct the CDFs of both Rin and Xin, in a significantlymore efficient manner than the full-wave Monte Carlo simulation, reachingspeedup factors up to 1667. Note that, specifically for microstrip inset-fed patchantennas, approximate empirical formulas may be used to compute both Rin

and Xin as a function of the patch length L [32], which would considerablyspeed up our analysis. However, the procedure outlined in this paper aims tobe generally applicable for any antenna topology and for all figures of merit, forwhich such approximate empirical formulas may not be available. Therefore,our analysis is based on an SCM combined with an accurate full-wave simulator.

3.4 Experimental validationIn order to experimentally validate the stochastic framework and to demonstratethat it may also be applied during the production phase, we now construct thestatistics of the antenna radiation characteristics based on measurements on a


Figure 3.4: Comparison between the CDFs constructed with the StochasticCollocation Method (SCM) and the Monte Carlo (MC) simulations, assuminga truncated Gaussian distribution of L.

limited set of antenna prototypes. In addition, the SCM introduced in Section3.3.2 generates a complete statistical description through the construction of aminimal number of prototypes. Specifically, each prototype has a patch lengthcorresponding to one of the points used in the quadrature rules (3.6), (3.7),respectively, to construct the coefficients of expansions (3.1), (3.2). However,when applying the formalism based on measurements of prototype realizations,two complications occur. First, fabrication tolerances introduce an additionaldegree of uncertainty. Second, in contrast to the numerical experiments, itis impossible to separate the statistical variations induced by one particularrandom variable from the stochastic effects generated by other variations.Specifically, even when we only deliberately vary the patch length L, we willalso measure variations originating from fabrication tolerances and materialinhomogeneities. These effects will be particularly important when applyingthe stochastic framework based on textile antenna prototype realizations. Toaccurately validate our new approach, we therefore verify our technique usinginset-fed microstrip patch antenna prototypes implemented on a RO4350B rigidsubstrate, which exhibits very stable material properties, with εr = 3.66 andtan δ = 0.003, and allows patterning the antenna by etching with fabricationtolerances of 20 μm. For a substrate height of 1.524 mm, a computer-aideddesign procedure was carried out to match the antenna impedance to 50 Ω at


Figure 3.5: Comparison between the CDFs constructed with the StochasticCollocation Method (SCM) and the Monte Carlo (MC) simulations, assuminga uniform distribution of L.

2.45 GHz. The nominal values of the dimensions of the antenna are reportedin Table 3.4, yielding an input impedance of 49.5026 + 2.31438j Ω. This newnominal antenna was taken as a starting point for the stochastic frameworkthat relies on a sensitivity analysis supplemented by the SCM, if required.

Table 3.3: Nominal values of the antenna geometrical parameters (Fig. 3.1) ona RO4350B substrate.

parameter nominal value (mm)L 32.01W 32.09Lin 11.75Win 1.94Lf 10Wf 3.82

We investigate how antenna performance changes when perturbing the patchlength L. As the rigid antenna substrate exhibits a higher permittivity anda smaller thickness, the antenna implemented on the high-frequency laminate


will be more sensitive to length variations than encountered in textile antennas.Therefore, we limit the variation on L to ±2%. This leads to a significantvariation in the antenna impedance Zin ranging from 13.4237 − 41.6178j Ωto 243.399 − 89.6536j Ω. We introduce a gPC expansion to relate input andoutput distribution and we find that both f1(y) and f2(y) converge from aminimum order of expansion equal to 5. We then perform a full-wave MonteCarlo simulation with a set of 10000 points exhibiting a truncated Gaussiandistribution with a mean value μ corresponding to the nominal value of L anda standard deviation σ equal to one third of the half of the variation interval,which spans a support of ± 2 % from the nominal value of the patch length.The resulting CDFs are reported in Fig. 3.6.

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

Rin (Ohm)

CDF

-50 -25 0 25 50 75

Xin (Ohm)

Rin, SCM

Rin, MC

Xin, SCM

Xin, MC

Figure 3.6: Comparison between the CDFs constructed with the StochasticCollocation Method (SCM) and the Monte Carlo (MC) simulations.

We notice that the two CDFs are overlapping. Furthermore, the maximumdistances DRin and DXin are now equal to 0.0119 and 0.0101, respectively,proving that the SCM accurately approximates the output CDF, consideringa significance level of 5%. Next, we construct a single set of 6 prototypes toexperimentally validate the technique for both Rin and Xin, corresponding tothe quadrature points necessary to construct the expansion.

We have measured the input impedance Zin for each of the prototypes atthe operating frequency of 2.45 GHz and we have de-embedded the phaseshift introduced by the connector soldered to the feed. The resulting CDFs


(curve labeled with “no shift") are reported in Fig. 3.7, compared to that oneconstructed by means of SCM.

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

Rin (Ohm)

CDF

-50 -25 0 25 50 75

Xin (Ohm)

Rin, SCM

Rin, 3 MHz shift

Rin, no shift

Xin, SCM

Xin, 3 MHz shift

Xin, no shift

Figure 3.7: Comparison between the CDFs constructed with the StochasticCollocation Method (SCM) and the measured values, with and without frequencyshift.

We obtain a reasonable agreement between the measured and the simulatedCDFs. Yet, a closer look at the simulated and the measured values of the S11parameter of every prototype reveals that they differ by a global frequencyshift of 3 MHz, probably due to uncertainties on the value of the permittivityεr of the substrate. Taking into account this shift, and therefore using themeasured values of Rin and Xin at the frequency of 2.447 GHz, we obtain anew CDF, also shown in Fig. 3.7. In this case, we obtain a far better matchbetween simulation and measurement. The small remaining discrepancies aredue to the fabrication process, which introduces unavoidable inaccuracies onthe parameters whereas the simulations refer to an ideal design. Nevertheless,the obtained result is more than satisfactory, as it demonstrates the potentialto perform a complete and reliable stochastic analysis based on a very limitednumber of constructed prototypes of the antenna under study.


3.5 Application to a dual-polarized antenna3.5.1 Sensitivity analysisWe now apply the developed stochastic framework to the textile antennapresented in [27] to demonstrate its applicability to any antenna configuration.More specifically, the antenna considered is a dual-polarized probe-fed patchantenna operating in the 2.4-2.4835 GHz ISM band (see Fig. 3.8). The antenna’ssubstrate again consists of a flexible closed-cell expanded rubber protectivefoam with a height h equal to 3.94 mm, permittivity εr equal to 1.53 and a losstangent given by 0.012. A computer-aided design procedure was carried out tomatch the antenna impedance, which is equal for both feed 1 and feed 2, to50 Ω, and to have an isolation −20log|S21| better than 15 dB within the entireISM band. Table 3.4 provides the optimized dimensions of the antenna. Thecorresponding input impedance at the frequency of 2.45 GHz is 49.91 − 1.93j Ω.

Figure 3.8: Schematic of the dual polarized probe-fed ISM band textile antennaunder study.

We repeat the procedure outlined in Section 3.3. Therefore, we carry outa sensitivity analysis on the antenna, in order to investigate the statistical

3.5. Application to a dual-polarized antenna 41

variation of Zin and −20log|S21| corresponding to the variation of a singleparameter. We find that the patch width W has the most profound effect onboth Zin and −20log|S21|. Therefore, we decide to proceed with our analysisby considering only W .

Table 3.4: Nominal values of the antenna geometrical parameters (Fig. 3.8) ona protective foam substrate.

parameter nominal value (mm)patch length L 44.46patch width W 45.32slot length Ls 14.88slot width Ws 1

feed points (±xf , yf ) (±5.7,5.7)

3.5.2 Estimation of the input PDFIn order to estimate the real distributions of the patch width W and lengthL, and to give a practical validation to the presented framework, 100 patchescorresponding to the dual-polarized textile antenna shown in Fig. 3.8 werehand-cut and measured. To ensure the randomness of the manual cuttingprocess, the patches were carefully prepared with the highest possible accuracyby several people. The textile patches were then measured by means of a NikonVeritas VM-250V optical setup with an accuracy of 0.4 μm. Measurements yielda marginal distribution for the values of W , which fits a Gaussian distributionwith mean value μ equal to 45.385 mm and standard deviation σ equal to 0.1268mm (see Fig. 3.9).

As for the length L, we find it to be fitted by a Gaussian distribution withwith mean value μ equal to 44.515 mm and standard deviation σ equal to0.1627 mm. The infinite support of this distribution is then truncated to ±4σwith respect to the mean value. Moreover, as for the joint distribution, thecorrelation between the measured values of W and L equals -0.0089. Therefore,W and L are not correlated, and we will further focus on W in the next section.

3.5.3 SCM-based full statistical analysisWe now apply the SCM analysis to this distribution for the patch width W . Onthe one hand, for the interval of variation assumed, we find that the isolation−20log|S21| is always larger than 15 dB. As for Rin and Xin, on the other hand,the variations remain not negligible. Therefore, we directly relate Rin and Xin

to the patch width W by means of the gPC expansions (3.1), (3.2), with ordersof expansion P1 = 2 and P2 = 2, where now y = W . Following the adaptivefitting procedure, we obtain convergence for both f1(y) and f2(y), with orders


Figure 3.9: Results of the measurements of the patch width W and fittedGaussian distribution.

Figure 3.10: Xin and Rin as a function of the patch width W according to theStochastic Collocation Method (SCM).

of expansion P1 = 2 and P2 = 2. Both Rin and Xin, as functions of W , arereported in Fig. 3.10. From the comparison between the CDFs obtained by a

3.6. Conclusions 43

Figure 3.11: Comparison between the CDFs constructed with the StochasticCollocation Method (SCM) and the Monte Carlo (MC) simulations for themeasured truncated Gaussian distribution of W .

Monte Carlo analysis based on 10000 samples and those resulting from (3.1),(3.2), we derive that the maximum distances DRin

and DXinare equal to 0.0083

and 0.0075, respectively. Therefore, the CDF obtained by the SCM is a goodapproximation for the output CDF, considering a significance level of 5%. TheCDFs resulting from the Monte Carlo full-wave simulation and from the SCMare shown in Fig. 3.11.

3.6 ConclusionsA novel stochastic framework for the variability analysis of a general figure ofmerit of textile patch antennas was presented. First, a representative antennais considered and a gPC analysis is performed to accurately reconstruct thestochastic behavior of the figure of merit under study. Then, the analysis isvalidated by constructing and measuring a limited number of prototypes ofthe antenna under study. Next, in order to prove its general applicability andreliability, the proposed stochastic framework is repeated on a textile antennaknown from literature, also experimentally quantifying the uncertainty on theantenna’s geometrical dimensions. The results show an excellent agreement andsuperior efficiency with respect to the standard Monte Carlo approach, used to


validate the method.

References[1] A. Tronquo, H. Rogier, C. Hertleer, and L. Van Langenhove, “Robust

planar textile antenna for wireless body LANs operating in 2.45 GHz ISMband”, Electronics Letters, vol. 42, no. 3, pp. 142–143, Feb. 2006.


[3] M. Klemm and G. Troester, “Textile UWB antennas for wireless bodyarea networks”, IEEE Trans. on Antennas and Propagation, vol. 54, no.11, pp. 3192–3197, Nov. 2006.

[4] I. Locher, M. Klemm, T. Kirstein, and G. Troster, “Design and charac-terization of purely textile patch antennas”, IEEE Trans. on AdvancedPackaging, vol. 29, no. 4, pp. 777–788, Nov. 2006.

[5] R. Moro, S. Agneessens, H. Rogier, and M. Bozzi, “Wearable textileantenna in substrate integrated waveguide technology”, Electronics Letters,vol. 48, no. 16, pp. 985–987, Aug. 2012.

[6] T. Kaufmann and C. Fumeaux, “Wearable textile half-mode substrate-integrated cavity antenna using embroidered vias”, IEEE Antennas andWireless Propagation Letters, vol. 12, pp. 805–808, 2013.

[7] P. Salonen, Y. Rahmat-Samii, and M. Kivikoski, “Wearable antennas inthe vicinity of human body”, in IEEE Antennas and Propagation SocietyInternational Symposium, vol. 1, Monterey, CA, Jun. 2004, pp. 467–470.

[8] P. Salonen and Y. Rahmat-Samii, “Textile antennas: Effects of antennabending on input matching and impedance bandwidth”, in EuCAP 2006.First European Conference on Antennas and Propagation, Nice, France,pp. 1–5.

[9] Q. Bai and R. Langley, “Crumpling of PIFA textile antenna”, IEEE Trans.on Antennas and Propagation, vol. 60, no. 1, pp. 63–70, Jan. 2012.





[13] M. L. Scarpello, D. Vande Ginste, and H. Rogier, “Design of a low-coststeerable textile antenna array operating in varying relative humidityconditions”, Microwave and Optical Technology Letters, vol. 54, no. 1,pp. 40–44, Jan. 2012.


[15] K. Strunz and Q. Su, “Stochastic formulation of SPICE-type electroniccircuit simulation with polynomial chaos”, ACM Trans. Model. Comput.Simul., vol. 18, no. 4, pp. 1–23, Sep. 2008 Sep. 2008.

[16] I. Stievano, P. Manfredi, and F. Canavero, “Stochastic analysis of mul-ticonductor cables and interconnects”, IEEE Trans. on ElectromagneticCompatibility, vol. 53, no. 2, pp. 501–507, May 2011.



[19] A. Austin and C. Sarris, “Efficient analysis of geometrical uncertainty inthe FDTD method using polynomial chaos with application to microwavecircuits”, IEEE Trans. on Microwave Theory and Techniques, vol. 61, no.12, pp. 4293–4301, Dec. 2013.

[20] Jianxiang Shen, Heng Yang, and R. Chen, “Analysis of electrical propertyvariations for composite medium using a stochastic collocation method”,IEEE Trans. on Electromagnetic Compatibility, vol. 54, no. 2, pp. 272–279,Apr. 2012.

[21] J. Ochoa and A. Cangellaris, “Macro-modeling of electromagnetic domainsexhibiting geometric and material uncertainty”, Applied ComputationalElectromagnetics Society Journal, vol. 27, no. 2, pp. 80–87, Feb. 2012.

[22] C. Chauviere, J. Hesthaven, and L. Lurati, “Computational modeling ofuncertainty in time-domain electromagnetics”, in CEM-TD 2005. Work-shop on Computational Electromagnetics in Time-Domain, Atlanta, GA,pp. 32–35.

3.6. Conclusions 47

[23] Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the method of momentswith the stochastic galerkin method”, IEEE Transactions on Antennasand Propagation, vol. 62, no. 9, pp. 4852–4856, 2014.


[25] P. Sumant, Hong Wu, A. Cangellaris, and N. Aluru, “Reduced-order mod-els of finite element approximations of electromagnetic devices exhibitingstatistical variability”, IEEE Trans. on Antennas and Propagation, vol.60, no. 1, pp. 301–309, Jan. 2012.

[26] P. Nepa and G. Manara, “Design and characterization of wearable anten-nas”, in ICEAA 2013. International Conference on Electromagnetics inAdvanced Applications, Turin, Italy, pp. 1168–1171.

[27] L. Vallozzi, H. Rogier, and C. Hertleer, “Dual polarized textile patchantenna for integration into protective garments”, IEEE Antennas andWireless Propagation Letters,, vol. 7, pp. 440–443, May 2008.

[28] W. Gautschi, Orthogonal Polynomials - Computation and Approximation.Oxford, UK: Oxford University Press, 2004, p. 76.


[30] A. Kolmogorov, “Sulla determinazione empirica di una legge di dis-tribuzione”, G. Ist. Ital. Attuari, vol. 4, pp. 83–91, 1933.

[31] F. Declercq, I. Couckuyt, H. Rogier, and T. Dhaene, “Environmental highfrequency characterization of fabrics based on a novel surrogate modellingantenna technique”, IEEE Trans. on Antennas and Propagation, vol. 61,no. 10, pp. 5200–5213, Oct. 2013.

[32] C. A. Balanis, Antenna Theory - Analysis and Design. John Wiley &Sons Canada, 1996, p. 734.

4Non-Destructive Electromagnetic

Characterization of FlexibleMaterials for Wearable Antennas

Marco Rossi, Sam Agneessens, Hendrik Rogier, Dries Vande Ginste

Based on the article accepted for publication in IEEE Antennas and WirelessPropagation Letters

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The assessment of the electromagnetic characteristics of the materi-als on which wearable components are implemented is of paramountimportance for the design and the production of reliable devices.Therefore, we propose a novel approach that compares simulationsand measurements performed with a resonance-perturbation methodbased on planar resonating fixtures to quickly estimate the propertiesof a given substrate sample. First, the resonance frequency of thesimulated fixture, or the two frequency values at which its return losspeak crosses a given threshold value, are modeled as polynomial func-tions of the relative permittivity and the loss tangent of the materialunder test. Then, the comparison between modeled and measuredfrequencies yields the electromagnetic properties of the material. Themeasurement fixtures under study are an inset-fed patch antennaoperating at 2 GHz and an aperture-coupled patch antenna operatingin the vicinity of the Global Positioning System (GPS)-L1 frequency

50 Chapter 4. Electromagnetic Characterization of Flexible Materials

band ([1.56342,1.58742] GHz). For each structure, the approachis first tested and validated for a material whose electromagneticcharacteristics are very stable and well-known. Then, the method isapplied to characterize the properties of several textile materials ofinterest. It is shown that the proposed approach allows a precise andfast characterization of the materials under test. Moreover, the testis non-destructive and after characterization all involved samplescan still be used for the production of the actual antennas.

4.1 IntroductionA decade ago, the integration of electronic devices into our everyday outfit wasintroduced as an intriguing concept that envisioned a broad field of applicationsbut which still needed to be fully investigated. Nowadays, the ongoing advancesin wearable technology have demonstrated the potential of smart textiles andthe interest for intelligent clothing is growing every year [1–4]. The success oftextile integrated systems has largely to be credited to the fact that they arelightweight, inconspicuous and designed to be conformal to the body shape.In this way, the movements and the comfort of the user are not disturbedduring deployment. Moreover, wearable components are usually implementedon inexpensive general purpose textile materials, which serve as substrates, withelectrotextiles used as conductive parts.

Although the use of off-the-shelf materials is beneficial in terms of productioncosts, it has a significant impact on the performance of the manufactured devices.On the one hand, the properties of electrotextiles, such as their conductivity,may differ from their plain metal equivalents [5]. On the other hand, theelectromagnetic characteristics of textiles, such as electrical permittivity andloss tangent, are not readily available and are expected to be more inhomo-geneous than in the conventional high-frequency laminates [6]. As a result,the estimation of the electromagnetic properties of the materials is crucial forwearable component designers.

The properties of electrotextiles have been thoroughly investigated in litera-ture [5, 7]. As to substrate materials, several techniques have been proposedfor their characterization. They can be subdivided into non-resonating andresonating methods. In the first group, broadband transmission line methodsrepresent a common approach [8–10]. These methods extract the material’s per-mittivity and loss tangent from the scattering parameter measurements of twotextile microstrip lines of different length. In the second group, planar antennaresonator techniques represent a convenient choice, as their implementation istypically simple and, despite being narrowband, they are more accurate andsensitive than non-resonating methods [11]. Moreover, they account for theimpact of glue and electrotextile. In general, resonating methods are based


on the comparison between the measurements of the resonance frequency ofa textile antenna and that same frequency simulated by full-wave solvers orempirical formulas [12, 13]. Recently, a novel approach has been introduced,which is based on a surrogate-based optimization (SBO) of a cost functionthat quantifies the difference between measured and modeled antenna. Suchoptimization leads to the estimation of the properties of both the substratematerial and the electrotextile [14].

However, even though all aforementioned methods provide a very precise char-acterization of substrate materials, their appositeness is limited because thecharacterized sample cannot be reused after its properties are estimated. Infact, if the material under test (MUT) is very non-uniform, as for off-the-shelftextiles, its electromagnetic characteristics may significantly change from sampleto sample. As a result, the value of permittivity or loss tangent obtained forone sample may not be assumed valid for another one. Moreover, in caseresonating methods are applied, the extraction of the parameters may be verytime-consuming, as it involves full-wave simulations of the measured antenna.Therefore, we propose a novel approach based on the comparison between simu-lations and measurements of a resonating structure interacting with the MUT.Thereto, first, the return loss |S11|dB of the measurement fixture is measured.Next, the structure is simulated by varying the value of the relative permittivityεr of the MUT and by comparing the simulated and the measured resonancefrequencies to define the variation interval of εr, whereas the variation intervalof the loss tangent tan δ is defined a priori based on practical considerations.Then, for these variation intervals, we model the resonance frequency of themeasurement fixture as a polynomial function of εr. Alternatively, we describethe frequency values for which its return loss |S11|dB near the resonance peakcrosses a given threshold value as a polynomial function of εr and tan δ. Finally,the electromagnetic properties of the MUT are found as the numerical solution ofthe equation or the system of equations resulting from the comparison betweenthe models and the measured frequencies. Only few full-wave simulations arerequired to estimate the variation interval of εr and to construct the model for agiven material. Then, the extraction of εr and tan δ is carried out at negligiblecomputational cost for several samples of the same material. Moreover, sincethe samples are not altered during the characterization process, they can bereused in the subsequent production stage.

This chapter is organized as follows. First, in Section 4.2 the proposed approachis fully discussed and outlined. Next, in Section 4.3 we apply it to a measurementfixture consisting of an inset-fed patch antenna operating at 2 GHz. Then, inSection 4.4, we introduce an aperture-coupled patch antenna operating in theGPS-L1 frequency band ([1.56342,1.58742] GHz) as a resonating structure. Foreach structure, we validate our method by characterizing several substrates ofinterest, including a Teflon reference substrate and a protective foam substrate.Conclusions are drawn in Section 4.5.


4.2 Characterization processThe measurement fixtures discussed in this chapter are manifactured in planarmicrostrip technology, which is a convenient choice for low-loss substrate materi-als with permeability equal to μ0 [11]. The proposed characterization process isbased on the patch antenna topology. A patch antenna is typically designed toresonate around a given frequency fr, which corresponds to a resonance peak inthe return loss |S11|dB at that frequency. If an MUT interacts with the antenna,the frequency fr is expected to shift towards lower or higher frequencies and themagnitude of |S11|dB changes, depending on the electromagnetic characteristicsof the material and its structure. These variations can be quantified and usedto estimate the properties of the MUT.

The characterization process is outlined in Fig. 4.1. It starts with the acquisitionof the measured return loss |S11|dB of the antenna interacting with a given MUT.Then, the variation interval of the relative permittivity Δεr of the MUT isdetermined (see red frame in Fig. 4.1). To this end, we leverage the independenceof the resonance frequency of the antenna from variations in the tan δ of theMUT, which is valid for low-loss materials, such as typical textile substratesapplied in antenna design, as we know from previous analyses [10], [14], [15].Therefore, we first simulate the structure with a value of permittivity εsim

r ofthe MUT for which the simulated resonance frequency fsim

r is higher than themeasured one fmeas

r . More specifically, if no information is available on theelectromagnetic characteristics of the MUT, as for textile substrates, the value ofεsim

r is set to 1. In contrast, if the properties of the MUT are known or providedby the manifacturer, the value of εsim

r is chosen close to the known value. Then,we increase the value of εsim

r by a value Δεsimr until the simulations yield a

value of fsimr that is lower than fmeas

r . In all simulations, the value of theloss tangent tan δsim of the MUT remains fixed and can be chosen arbitrarilysmall, as, for the considered materials, its value does not affect the simulatedresonance frequency. The variation interval of εr is finally defined as:

Δεr = [max(εsimr − 2Δεsim

r , 1), εsimr + 2Δεsim

r ], (4.1)

where εsimr is the last value of permittivity used in the simulations. The value

of Δεsimr is chosen large enough such that Δεr accounts for the non-uniformity

of the MUT and only few simulations are needed to determine it. In all cases,Δεsim

r =0.1 has proved to be a suitable choice.

Next, the algorithm proceeds with the extraction of the electromagnetic prop-erties of the MUT (see blue frame in Fig. 4.1). First, we a priori define thevariation interval Δtan δ for the loss tangent. Since all textile materials consid-ered in previous works exhibit a loss tangent tan δ lower than 0.04 [10], [14],[15], we set Δtan δ to [0,0.1], which is expected to be valid for our scope. Next,we set a threshold value tdB to half the peak magnitude of the measured |S11|dB .This choice yields stable and accurate material properties. Then, we model thetwo frequencies f tdB

sim1 and f tdBsim2 at which the antenna’s simulated return loss

4.2. Characterization process 53

Acquire measurement |S11|dB

New material?

Simulate fixture with εsimr and tan δsim

Compare resonance frequencies:is fsim

r < fmeasr ?

εsimr =εsimr +Δεsimr

Δεr=[max(εsimr − 2Δεsimr , 1), εsimr + 2Δεsimr ]

Define Δtan δ

Define threshold tdB that crosses |S11|dB

Model crossing frequencies f tdBsim1, f

tdBsim2

as functions of εr and tan δ

Compare measured f tdBmeas1, f

tdBmeas2 with f tdB

sim1, ftdBsim2

to extract εr, tan δ

Enlarge Δεr, Δtan δεr ∈ Δεr

tanδ ∈ Δtan δ?

End

Find variation interval permittivity Δεr

Extract εr and tan δ

yes

no

no

yes

no

yesεr, tan δ

Figure 4.1: Schematic representation of the characterization process.


|S11|dB crosses the threshold tdB as functions of εr and tan δ by means of theexpansions:

f tdBsim1(x1, x2) =

P1∑k1=0

P2∑k2=0

yk1k2φX1k1

(x1)φX2k2

(x2) (4.2)

f tdBsim2(x1, x2) =

P1∑k1=0

P2∑k2=0

tk1k2φX1k1

(x1)φX2k2

(x2), (4.3)

where x1, x2 ∈ [−1, 1] are two orthonormal variables corresponding to εr andtan δ, respectively, whereas φXi

ki(xi) are Legendre polynomials of degree ki cor-

responding to the i-th variable. The expansion coefficients yk1k2 and tk1k2 arethe unknowns to be determined. The total order P of expansions (4.2) and(4.3) is equal to P1 + P2, where P1 and P2 are the maximum values that can beachieved by k1 and k2, respectively. The variables x1 and x2 in (4.2) and (4.3)are related to εr and tan δ by the following relations:

εr = x1εsim

r + 2Δεsimr − max(εsim

r − 2Δεsimr , 1)

2 +

εsimr + 2Δεsim

r + max(εsimr − 2Δεsim

r , 1)2 , (4.4)

tan δ = Δtan δ

2 x2 + Δtan δ

2 . (4.5)

The normalized Legendre polynomials φXi

ki(xi) are orthonormal with respect to

the following inner product [16]:

< u(xi), v(xi) >=∫

[−1,1]u(xi)v(xi)dxi. (4.6)

The coefficients yk1k2 and tk1k2 are computed by applying Galerkin weightingto (4.2) and (4.3):

yk1k2 = < f tdBsim1, φX1

k1φX2

k2>=

∫ ∫[−1,1]×[−1,1]

f tdBsim1φX1

k1φX2

k2dx1dx2 (4.7)

tk1k2 = < f tdBsim2, φX1

k1φX2

k2>=

∫ ∫[−1,1]×[−1,1]

f tdBsim2φX1

k1φX2

k2dx1dx2.(4.8)

We compute integrals (4.7), (4.8) by the following R1 ×R2-point Gauss-Legendrequadrature rules:

yk1k2 ≈R1∑

l1=0

R2∑l2=0

wl1wl2φX1k1

(xl1)φX2k2

(xl2)f tdBsim1(xl1 , xl2), (4.9)

tk1k2 ≈R1∑

l1=0

R2∑l2=0

wl1wl2φX1k1

(xl1)φX2k2

(xl2)f tdBsim2(xl1 , xl2), (4.10)

4.3. Inset-fed patch antenna-based characterization fixture 55

where xl1 ,xl2 are the quadrature points and wl1 ,wl2 are the correspondingweights. The values f tdB

sim1(xl1 , xl2) and f tdBsim2(xl1 , xl2) in (4.9), (4.10) are eval-

uated by a full-wave electromagnetic field solver, such as Advanced DesignSystem (ADS) Momentum. Finally, the measured values of f tdB

meas1 and f tdBmeas2,

at which the measured |S11|dB crosses the threshold tdB , are compared to theapproximations (4.2), (4.3) of f tdB

sim1 and f tdBsim2. The solution of the resulting

system of equations yields the values of εr and tan δ of the MUT:P1∑

k1=0

P2∑k2=0

yk1k2φX1k1

(x1)φX2k2

(x2) = f tdBmeas1

P1∑k1=0

P2∑k2=0

tk1k2φX1k1

(x1)φX2k2

(x2) = f tdBmeas2

(4.11)

In case the values of εr and tan δ do not belong to the variation intervals definedat the beginning, a larger value of Δεsim

r is substituted into (4.1) to enlargethe variation interval Δεr and a larger interval Δtan δ is considered. Then, wereconstruct the expansions (4.2), (4.3) and we reprocess the measured data untila valid solution is found. Moreover, if the MUT has already been characterizedand a model is already available for it, the newly measured values of f tdB

meas1 andf tdB

meas2 are directly introduced into the algorithm used to determine the valuesof εr and tan δ. In that case, the properties of the material are readily computedwithout resorting to full-wave simulations. Finally, the outlined procedure caneasily be modified in case only εr has to be found. Then, only the interval Δεr

is determined at the beginning of the process and the resonance frenquencyfsim

r of the structure is modeled as a function of εr. Given the value of themeasured resonance frequency fmaes

r , the solution of a simple equation in onevariable yields the value of εr.

As a final remark, we point out that the entire characterization process may beaffected by uncertainties in MUT thickness. To address this issue, the heighth of the MUT can be included as a third variable x3 in (4.2), (4.3), measuredduring the characterization process and then substituted for x3 before solving(4.11). However, we found that a variation as large as 25% in h results inabsolute errors of less than 1% and 5% for εr and tan δ, respectively. Therefore,we carry out the validation of the advocated technique by simply consideringthe MUT’s nominal thickness.

4.3 Inset-fed patch antenna-basedcharacterization fixture

The characterization process outlined in Section 4.2 is first applied to a mea-surement fixture consisting of an inset-fed patch antenna, such as the oneshown in Fig. 4.2, which is manifactured on a rigid I-Tera MT40 high-frequency


laminate, with relative permittivity εr=3.56, loss tangent tan δ=0.0035 andthickness h=0.508 mm. The patch is etched in a copper layer with thickness 35μm. The antenna has been designed in ADS Momentum to resonate at 2.25GHz, thus allowing material characterization in the proximity of the [2.4, 2.5]GHz Industrial, Scientific and Medical (ISM) frequency band, which is veryimportant for on-body communication. The design parameters of the antennaare reported in Table 4.1.

Figure 4.2: Schematic of the inset-fed patch antenna resonating at 2.25 GHz.

This particular configuration of the proposed characterization technique isapplicable to thin substrates for εr up to 10 and tan δ up to 0.1, and forfrequencies below 10 GHz. Moreover, it is suitable for integration into anassembly-line (Fig. 4.3), where substrate sample are first quickly characterizedbefore being used in the production of the actuall antennas. More specifically,during the characterization procedure, the MUT is unrolled on top of theinset-fed patch antenna (Fig. 4.2). At each sample, the assembly line stops


and a Styrofoam block presses the substrate onto the patch antenna to avoidair gaps without interfering with the return loss |S11|dB measurement of thefixture. Then, the process described in Section 4.2 is carried out to extractthe εr and the tan δ of the MUT. The extraction algorithm is implemented inMatlab. The structure is modeled with the full-wave solver ADS Momentum.

Figure 4.3: Assembly line with the MUT unrolled on top of the inset-fed patchantenna for characterization.

Table 4.1: Dimensions of the patch antenna at 2.25 GHz (see Fig. 4.2).

parameter at 2.25 GHz value (mm)patch length L 36patch width W 36.5inset length Lin 10inset width Win 3feed length Lf 10feed width Wf 1.2

4.3.1 Characterization of a high-frequency laminateFirst, as a validation, we measure the |S11|dB of the antenna without any MUTand we extract the εr and the tan δ of the I-Tera MT40 laminate at around2 GHz. Based on the values provided by the manifacturer, we set tan δsim=tan δ=0.0027 and we initialize the values of εsim

r and Δεsimr in the algorithm

to 3 and 0.1, respectively. Moreover, since the laminate is expected to exhibitvery stable material properties and very low losses, in this case the variationinterval Δtan δ is set to [0, 0.01] and not [0, 0.1]. After 7 full-wave simulations,the interval Δεr is found to be equal to [3.4,3.8]. Then, a threshold tdB = −8dB is selected and the expansions (4.2) and (4.3) are constructed. To this end,9 full-waves simulations of the structure are carried out. We find that bothexpansions converge with orders of expansion P1 = P2 = 2. Figs. 4.4 and 4.5


show f tdBsim1 and f tdB

sim2 as functions of εr and tan δ.

Figure 4.4: Approximation of f tdBsim1 as a function of the relative permittivity εr

and the loss tangent tan δ.

Figure 4.5: Approximation of f tdBsim2 as a function of the relative permittivity εr

and the loss tangent tan δ.


Then, the values f tdBmeas1 and f tdB

meas2 are extracted from the measurement and thesolution of the system (4.11) yields εr = 3.547 and tan δ = 0.005627. Therefore,the results are consistent with those provided by the manifacturer, even thoughthe loss tangent tan δ is slightly overestimated. Finally, in ADS Momentum wereplace the values of εr and tan δ provided by the manifacturer with those foundwith the proposed characterization strategy and we simulate the structure again.The measured and the simulated return losses |S11|dB are reported in Fig. 4.6.

2 2.1 2.2 2.3 2.4 2.5-40

-30

-20

-10

0

f (GHz)

|S11| dB

simulationmeasurement

Figure 4.6: Comparison between the measured |S11|dB and the simulated onewith the extracted εr and tan δ for the patch antenna operating at 2 GHz.

We notice that the simulated and the measured curves differ only in themagnitude of the peak in the |S11|dB , whereas the resonance frequency fr andthe frequency bandwidth at the threshold tdB = −8 dB are the same.

As a final remark, we point out that each simulation in ADS Momentum took81 s. As a result, the estimation of the interval Δεr took 9 min 27 s, whereasthe construction of the the expansions required a simulation time of 12 min 9 s.After this setup time, the solution of the system (4.11) for a single measurementonly took a fraction of second.

4.3.2 Characterization of Textile SubstratesThe proposed approach was then applied to the characterization of six substratesof potential interest for antenna production. The nominal characteristics of theconsidered materials are reported in Table 4.2. Four materials, being 3D white,


corktex, EVA green and fake leather are shown in Fig. 4.7.

(a) 3D white (b) corktex

(c) EVA green (d) fake leather

Figure 4.7: Photographs of four of the materials characterized with the advo-cated method.

In particular, 3D fabrics are applied in the manufacturing of protective clothing,e.g., for firefighters, whereas fake leather and EVA substrates are popular infashion design and sport clothing, respectively. In contrast, cork is becomingmore and more important in the production of components for cars and othervehicles. A sample of all materials was placed on top of the patch antenna andthe corresponding reflection loss |S11|dB was measured. Then, for each newmaterial the characterization procedure described in Section 4.2 was carried out.Since the materials were expected to exhibit a low permittivity, εsim

r was set to1, whereas Δεsim

r was kept equal to 0.1 and the loss tangent tan δsim was set tothe mean value of the variation interval Δtan δ, being 0.05. The threshold tdB

was kept equal to -8 dB and all the models converged with orders of expansionP1 = P2 = 2.

For each material, a square sample was measured by positioning it such thateach time one different side is aligned with the microstrip feed of the antennaand by repeating the procedure by turning it upside down. The mean valueand the variance of the results for both the permittivity and the loss tangent,as well as the setup time, are reported in Table 4.3. For each material, theextraction of all 8 results took 1.28 s.


Tabl

e4.

2:N

omin

alch

arac

teris

tics

ofth

em

ater

ials

char

acte

rized

with

the

advo

cate

dm

etho

d.

mat

eria

lm

anifa

ctur

erre

fere

nce

com

posit

ion

h(m

m)

area

dens

ity(g

·m−

2 )3D

blac

kLe

andr

oM

anue

lAra

újo

3003

100%

poly

este

r2.

65-

3Dgo

ldLe

andr

oM

anue

lAra

újo

3013

100%

poly

este

r3.

068

350

3Dw

hite

Lean

dro

Man

uelA

raúj

o30

1510

0%po

lyes

ter

4.14

363

cork

tex

Cor

ticei

raA

mor

im82

4510

0%co

rk1.

1-

EVA

gree

n-

-10

0%Et

hyle

neV

inyl

Ace

tate

2.01

-fa

kele

athe

r-

-10

0%po

lyes

ter

0.83

1-


Table 4.3: Results of the characterization process.

material setup time εr σεr tan δ σtan δ

3D black 14 min 52 s 1.12 1.74e-2 0.0063 2.36e-33D gold 14 min 52 s 1.12 1.70e-2 0.0061 2.25e-33D white 14 min 52 s 1.13 6.16e-3 0.037 6.83e-4corktex 16 min 13 s 1.30 1.32e-2 0.046 1.07e-3

EVA green 14 min 52 s 1.10 1.15e-3 0.040 8.25e-4fake leather 16 min 13 s 1.27 4.63e-3 0.045 1.81e-3

We notice that the overall CPU time required to find the interval Δεr, tocompute the models of f tdB

sim1 and f tdBsim2 and to extract the values of εr and tan δ

is even lower than that required to characterize the board itself, hence allowingfor a quick characterization of all the considered materials.

As a final validation, two prototypes of a probe-fed textile patch antennas weredesigned with CST Microwave Studio to resonate between 2 GHz and 3 GHz.They were manufactured with the 3D black and the 3D gold substrates. Acommercial e-textile with resistivity ρs = 0.18 Ω/sq and thickness 80 μm wasapplied for both the ground plane and the patch. The first antenna consists ofa 45.37 mm-long and 35 mm-wide rectangular patch, with the feed positionedat 8.5 mm from the center towards the short edge of the patch. In contrast,the second prototype exhibits a square patch length of 52.5 mm and feed at15 mm from the center of the patch. Both measured and simulated |S11|dB ofboth antennas are shown in Fig. 4.8. We notice that a very good agreementbetween measurements and simulations is found for the prototype manufacturedwith the 3D gold substrate. The measured resonance peak of the prototypemanufactured with the 3D black material is slightly shifted with respect to thesimulated one. Such a shift corresponds to a difference in the material’s εr assmall as 0.04 with respect to the value obtained by the characterization process.This we ascribe to the presence of the glue applied to attach both the patch andthe ground plane to the substrate [10]. Such an effect may also be accountedfor during antenna fabrication.

4.4 Aperture-coupled patch antenna-basedcharacterization fixture

The second measurement fixture to which the advocated characterization pro-cedure is applied is based on the aperture-coupled patch antenna principle, asshown in Fig. 4.9 and sketched in Fig. 4.10, with dimensions given in Table 4.4.The structure is composed of two planar circuit boards, with the MUT in be-tween, which are supported at four sides by means of eight metal posts screwedon a bread-board in order to provide the necessary high-precision aligmentbetween the lower and the upper part. The two boards are implemented on a

4.4. Aperture-coupled patch antenna-based characterization fixture 63

2 2.4 2.8 3.2 3.6 4-30

-20

-10

0

f (GHz)

|S11| dB

3D black simulation3D black measurement3D gold simulation3D gold measurement

Figure 4.8: Measured versus simulated |S11|dB of the two 2.45GHz probe-fedpatch antennas.

rigid RO4350B high-frequency laminate, which exhibits very stable materialproperties, with εr=3.66, tan δ=0.003 and thickness equal to 1.524 mm. Thelower board contains a microstip line on the bottom side, which tapers into awaveguide in Substrate Integrated Waveguide (SIW) technology. On the otherside, the ground plane features an aperture through which the SIW waveguidecouples to the patch. A microstrip-to-SIW transition is adopted to minimizethe back radiation through the slot. To position the antenna patch properly ontop of the MUT, the upper board is used. In this way, the alignment betweenthe patch and the slot is very precise and the MUT is perfectly kept adherent tothe boards. Finally, the upper board is designed such that the patch is centeredabove the slot, optimizing the coupling between them.

The choice for an aperture-coupled patch antenna is motivated by the followingconsiderations [17]:

• the ground plane isolates the feed structure and the radiating element.Moreover, it minimizes the interference of spurious radiation by thefeed. In addition, the bottom substrate is usually implemented in highpermitivity materials, whereas a thick low-dielectric material is applied forthe top one, corresponding to the materials typically used in the designof textile antennas;

• measurements are performed very quickly, since the MUT can readily bechanged;


Figure 4.9: Photographs of the measurement fixture, with and without upperboard.

• the strongest electromagnetic fields are confined within the MUT, therebyallowing a better characterization of its properties.

Moreover, it is easy to deploy other circuit boards for other frequency rangesand materials, thus creating a modular measurement setup.

This particular aperture-coupled antenna is designed for a flexible closed-cellexpanded rubber protective foam as MUT, whose nominal permittivity εr andloss tangent, previously determined by adopting the procedure described in [18],are equal to 1.56 and 0.012, respectively, and whose thickness is 3.94 mm.This material is very commonly used in the production of textile antennas.Moreover, the complete fixture has been designed in Computer SimulationTechnology (CST) Microwave Studio’s frequency domain solver to resonate inthe GPS-L1 frequency band ([1.56342,1.58742], a common frequency of interestfor antenna designers.

4.4. Aperture-coupled patch antenna-based characterization fixture 65

Figure 4.10: Exploded view of the measurement fixture, as simulated in Com-puter Simulation Technology (CST) Microwave Studio. The dotted lines repre-sent the feed and the patch implemented on the bottom sides of the lower andthe upper board, respectively. (Dimensions are given in Table 4.4).

Table 4.4: Dimensions of the antenna measurement fixture in (Fig. 4.10).

parameter value (mm) parameter value (mm)patch length LP 62 patch width WP 80slot length LS 3 slot width WS 55

SIW length LSIW 28 SIW width WSIW 70feed length LF 22 feed width WF 2.8

tapering length LT 32 tapering width WT 25


4.4.1 Measurement principle and resultsThe resonance frequency of the aperture-coupled patch antenna changes withthickness and permittivity of the MUT. Therefore, given the exactly knownspacing between the two boards, the value of the measured resonance frequencycan be used to estimate the permittivity of the sample. The procedure describedin Section 4.2 has been modified since only the resonance frequency fr

sim of themeasurement fixture is considered. To this end, we model fr

sim as a function ofthe relative permittivity εr of the MUT alone:

fsimr (x) =

P∑k=0

ykφXk (x). (4.12)

Therefore, given the measured value fmeasr of the resonance frequency of the

structure, the solution of a simple equation yields the value of εr of the MUT.The validity of the presented technique was tested by characterizing a teflonlaminate, a substrate material whose electromagnetic characteristics are verystable and well-known. The teflon had a height h equal to 3.16 mm and arelative permittivity εr equal to 2.1. First, the proposed procedure was carriedout. In the algorithm, the values of εsim

r and Δεsimr are set to 1.9 and 0.1,

respectively. The value of loss tangent tan δsim is chosen to be the one providedby the manifacturer, being 0.00028. After 3 full-wave simulations, the intervalΔεr is found to be [1.9,2.3]. Then, the expansion (4.12) is constructed. To thisend, 3 full-wave simulations of the structure are carried out. We find that theexpansion converge for an order of expansion P = 2. Fig. 4.11 shows fsim

r asfunction of εr. Then, the value of fmeas

r is extracted from the measurementand the value εr = 2.0634 is found. The small relative error of 1.74% withrespect to the value provided by the supplier of the material is acceptable,clearly validating the presented approach.Then, 25 samples of the flexible closed-cell expanded rubber have been char-acterized in the same way. Since the materials were expected to have a lowpermittivity, initially εsim

r was set to 1, whereas Δεsimr was kept equal to 0.1

and the loss tangent tan δsim was set equal to its nominal value of 0.012. Sevensimulations were necessary to find Δεr = [1.4, 1.8]. The model (4.12) of fsim

r hasproved to converge for an order of expansion P = 2. The histogram in Fig. 4.12reports the values εr found with the advocated procedure. We notice that, asexpected, the material has very non-uniform electromagnetic characteristics.Moreover, the nominal value of εr, which was adopted for the design of textileantennas with this material, is found to be correct only for a limited number ofthe characterized samples.As a final remark, we point out that each simulation in CST Microwave Studiotook 7 min 30 s. As a result, the estimation of the interval Δεr for the blackfoam took 52 min 30 s, whereas the construction of the the expansions requireda simulation time of 22 min 30 s. Then, the characterization of all 25 samplesof the MUT was readily carried out.

4.5. Conclusions 67

1.85 1.95 2.05 2.15 2.25 2.351.35

1.40

1.45

1.50

εr

fsim

r(G

Hz)

Figure 4.11: Approximation of fsimr as a function of the relative permittivity

εr.

1.4 1.425 1.45 1.475 1.5 1.525 1.55 1.575 1.60

2

4

6

8

10

εr

Figure 4.12: Histogram of the relative permittivity εr of the 25 black foamsamples characterized by the advocated procedure.

4.5 ConclusionsA novel technique was proposed for the characterization of the electromagneticproperties of substrate materials for textile antenna design. Two differentresonating structures, being an inset-fed patch antenna and an aperture-coupled


patch antenna, interact with a material of interest and affect the measured returnloss |S11|dB . Then, the same structures are simulated by means of commercialfull-wave solvers. Both the frequency values for which the resonance peak of thereturn loss |S11|dB crosses a given threshold value and the resonance frequencyare modeled as functions of the parameters that need to be determined, such asthe relative permittivity εr and the loss tangent tan δ. Finally, the comparisonbetween these functions and the measured data yields the electromagneticproperties of the material under test. The proposed approach has proved tobe fast, accurate and flexible. Moreover, the measured samples can be reusedafter being characterized, thus considerably improving the effectiveness of theproduction of textile antennas, in case the considered material exhibits largenon-uniformities.

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[14] F. Declercq, I. Couckuyt, H. Rogier, and T. Dhaene, “Environmental highfrequency characterization of fabrics based on a novel surrogate modellingantenna technique”, IEEE Transactions on Antennas and Propagation,vol. 61, no. 10, pp. 5200–5213, 2013.

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5Stochastic Analysis of the Impactof Substrate Compression on thePerformance of Textile Antennas

Marco Rossi, Sam Agneessens, Hendrik Rogier, Dries Vande Ginste

Based on the article published in IEEE Transactions on Antennas andPropagation

� � �

One of the many adverse effects modifying the performance of tex-tile antennas in real operating conditions is substrate compression.Therefore, we present a stochastic collocation method that eitherrelies on the generalized Polynomial Chaos (gPC) expansion or ona novel Hermite-Padé approximant. The method is introduced torigorously quantify the effect of random variations in the height andthe permittivity of the substrate on the figures of merit of a textileantenna. Next, the joint height and permittivity probability distri-bution of a compressible substrate is characterized by means of anew measurement setup based on a resonant-perturbation technique.Finally, the method is validated for a probe-fed Global PositioningSystem (GPS) textile antenna. It is shown that Hermite-Padé ap-proximants model the highly nonlinear relationship between thesesubstrate random variables (RVs) and the figures of merit of the

72 Chapter 5. Impact of Compression on the Performance of Textile Antennas

antenna more efficiently than the gPC. Moreover, a Kolmogorov-Smirnoff test proves that the resulting distributions of the antenna’sfigures of merit are as accurate as those obtained by means of aMonte Carlo analysis, with demonstrated speedup factors up to 123.

5.1 IntroductionOver the last years, the design of devices in wearable technology has acquireda significant maturity and smart textiles have witnessed a growing interest ina broad spectrum of applications. On the one hand, their potential has beendemonstrated in several domains, such as rescue operations [1], integrationin firefighters’ garments for positioning and tracking [2] and telemedicine [3,4]. On the other hand, the available know-how allows designers to cope withmany problems arising from the deployment of these components in real-lifescenarios. More specifically, the choice of patch-antenna topologies has suc-cessfully counteracted the absorption of radiated fields by the human body [5],whereas accurate and efficient stochastic models exist to account for the effectof bending and the impact of uncertainties in the antenna’s design parame-ters, due to the inevitable inaccuracy of the production process [6, 7]. Yet,up to now, such a model does not exist to quantify the effect of substratecompression, modifying substrate height, permittivity and loss tangent. Inrealistic deployment conditions, these variations will occur simultaneously withother adverse effects, such as bending, crumpling and stretching. The antennadesigner needs to understand the isolated effect of compression on antennaperformance to take appropriate countermeasures during design. Separatingthe effect of substrate compression from other adverse effects may only bedone in simulation. Unfortunately, at present such a model is currently notavailable since no reproducible experiments are available in literature to studythe correlated variation of substrate thickness and dielectric properties undercontrolled compression. Moreover, a dedicated framework is currently unavail-able to quantify the relationship between correlated substrate parameters andthe textile antenna’s figures of merit.

Conventionally, a flexible substrate is characterized by a compression set, whichis a percentage that indicates the permanent deformation loss of the materialfrom its starting thickness after being compressed for a set time at a settemperature. For example, the closed-cell expanded rubber protective foamconsidered throughout the manuscript has an average compression set of 14%after being compressed by 50% at 23 ◦C for 22 h, measured according to normASTM D 1056-78. At present, the effect of compression is accounted for inthe design phase by checking the performance of the antenna for two differentvalues of the thickness of the substrate, being the nominal one and the onecorresponding to the compression set [2]. In the latter case, the permittivity of

5.2. Stochastic analysis framework for compressible antennas 73

the compressed material is typically adjusted, since the material density and,hence, its dielectric constant, increase under compression [2].

However, this approach is far from rigorous and not supported by any ex-perimental result. Moreover, since only two extreme substrate conditions areconsidered, it fails to assess what happens for other values of thickness orpermittivity, also given the lack of information about the correlation betweenthose two parameters. Therefore, we extend the formalism previously intro-duced in [6, 7] to a stochastic framework for the characterization of textileantennas’ performance in the presence of substrate compression. Starting frommeasurements performed on several samples of the aforementioned protectivefoam, a material commonly used in the design of textile patch antennas [2, 8],we carry out a precise statistical evaluation of the effect of variations in thepermittivity and the thickness of the substrate on the antenna’s figures of merit.Specifically, we introduce a non-intrusive Stochastic Collocation Method (SCM)based on gPC [9] and also on Hermite-Padé approximation [10], as more efficientalternatives to the brute-force Monte Carlo method.

Even though gPC has proved to be reliable in a wide range of applications,including the modeling of circuits, interconnects and multiport systems [11–15],we expect some difficulties to arise from the nonlinear relationship between therandom substrate characteristics and the variations of the antenna’s figuresof merit. Therefore, we apply the Hermite-Padé approximation to efficientlymodel highly nonlinear or discontinuous systems, where other methods, such asgPC, lead to the well-known Runge phenomenon.

This chapter is organized as follows. First, in Section 5.2, we relate thejoint probability distribution of the correlated random design variables to theantenna’s figures of merit. Next, in Section 5.3, the measurement setup used tocharacterize the input Probability Distribution Function (PDF) correspondingto the permittivity and the thickness of a compressible substrate is described,including measurement results of a Teflon reference substrate and a protectivefoam substrate. Finally, in Section 5.4, the application of this framework toa GPS textile antenna is carried out for the input PDF resulting from thesubstrate measurements. Conclusions are drawn in Section 5.5.

5.2 Stochastic analysis framework forcompressible antennas

Consider a textile microstrip probe-fed compressible patch antenna, such asthe textile GPS antenna shown in Fig. 5.1. This antenna consists of a squarepatch with two truncated corners and is fed in the top right corner by means ofa coaxial probe to excite a right hand circular polarization. The geometricalparameters of the antenna are optimized (see Table 5.2 in Section 5.4) tomatch the requirements of the GPS-L1 standard, i.e. a return loss |S11| lower


than 0.316 (-10 dB) and an axial ratio (AR) (defined as the ratio between theamplitudes of the orthogonal components composing the circularly polarizedfield) smaller than 1.41 (3 dB) in the [1.56342,1.58742] GHz frequency band.Our aim is to test the effect of stochastic variations in the substrate permittivity

Figure 5.1: Representative textile antenna exhibiting a microstrip probe-fedGPS patch antenna topology to which the stochastic framework is applied.

εr and height h on AR and the magnitude of the S11 at the nominal operatingfrequency fr=1.57542 GHz. First, we introduce a SCM and we relate both ARand |S11| to the antenna’s parameters under study by means of gPC expansions.Next, we investigate the use of Hermite-Padé approximants as an alternative togPC expansions, for figures of merit, such as AR, that are expected to dependin a highly nonlinear manner on the antenna substrate characteristics.

5.2.1 Stochastic Collocation MethodFollowing the Wiener-Askey scheme [9], we consider both AR and |S11| asfunctions of N input RVs x1, x2, ..., xN , by means of polynomial expansions:

AR = f1(x1, x2, ..., xN ) =P1∑

k1=0

P2∑k2=0

...

PN∑kN =0

yk1k2...kN×

φX1k1

(x1)φX2k2

(x2)...φXN

kN(xN ), (5.1)


|S11| = f2(x1, x2, ..., xN ) =Q1∑

k1=0

Q2∑k2=0

...

QN∑kN =0

tk1k2...kN×

φX1k1

(x1)φX2k2

(x2)...φXN

kN(xN ), (5.2)

where φXi

ki(xi) are suitably chosen polynomial basis functions of degree ki

corresponding to the i-th variable, and the expansion coefficients yk1k2...kN

and tk1k2...kNare the unknowns to be determined. The total orders Σki of

expansions (5.1) and (5.2) are equal to P · N and Q · N , respectively, as wechoose P = P1 = P2 = ... = PN and Q = Q1 = Q2 = ... = QN , which are themaximum values that can be achieved by k1, k2, ..., kN .

It is important to stress that the input RVs x1, x2, ...xN in (5.1) and (5.2) are as-sumed to be mutually independent and the corresponding PDFs pX1(x1), pX2(x2),..., pXN

(xN ) are normalized to zero mean and unit variance by proper transla-tion and scaling. Therefore, polynomials φXi

ki(xi) are introduced as functions

of a single RV xi. They are orthonormal with respect to the following innerproduct:

< u(xi), v(xi) >=∫

Γi

u(xi)v(xi)pXi(xi)dxi, (5.3)

where Γi is the support of the PDF pXi(xi), acting as a weighting function.

Such an orthonormal set of polynomials φXi

ki(xi) is selected according to the

Wiener-Askey scheme for commonly used PDFs [9]. The independent RVsx1, x2, ..., xN in (5.1), (5.2) are related to the correlated antenna design param-eters x′

1, x′2, ..., x′

N by the following relation:

[x]′ = [T ][x] + [μ], (5.4)

where [T ] is the upper triangular matrix resulting from the Cholesky decomposi-tion of the covariance matrix [Σ] (such as (5.20)) corresponding to x′

1, x′2, ..., x′

N ,whereas [μ] is the vector of the mean values of x′

1, x′2, ..., x′

N (such as (5.19)).This transformation (5.4) accounts for any existing correlation among thevariations on the antenna’s design parameters.

Next, we exploit the independence of the RVs xi and the orthonormality of thebasis functions φXi

ki(xi) to calculate the coefficients yk1k2...kN

and tk1k2...kN:

yk1k2...kN=< f1, φX1

k1φX2

k2...φXN

kN>

=∫

...

∫Γ

f1φX1k1

φX2k2

...φXN

kNPX1X2...XN

dx1dx2...dxN (5.5)

tk1k2...kN=< f2, φX1

k1φX2

k2...φXN

kN>

=∫

...

∫Γ

f2φX1k1

φX2k2

...φXN

kNPX1X2...XN

dx1dx2...dxN (5.6)


where Γ =⋃

i Γi is the N -dimensional support of the joint input PDF PX1X2...XN

=∏

i PXi(xi). We compute integrals (5.5), (5.6) by means of the followingR1 × R2 × · · · × RN and V1 × V2 × · · · × VN -points Gaussian quadrature rules:

yk1k2...kN≈

R1∑l1=0

R2∑l2=0

...

RN∑lN =0

wl1wl2 ...wlN×

φX1k1

(xl1)φX2k2

(xl2)...φXN

kN(xlN

)f1(x′l1

, x′l2

, ...x′lN

) (5.7)

tk1k2...kN≈

V1∑m1=0

V2∑m2=0

...

VN∑mN =0

wm1wm2 ...wmN×

φX1k1

(xm1)φX2k2

(xm2)...φXN

kN(xmN

)f2(x′m1

, x′m2

, ...x′mN

) (5.8)

where the quadrature points xl1 ,xl2 ,..., xlN, xm1 ,xm2 ,..., xmN

and the corre-sponding weights wl1 ,wl2 ,..., wlN

, wm1 ,wm2 ,..., wmNare derived by the Golub-

Welsch algorithm [16]. Finally, the values f1(x′l1

, x′l2

, ..., x′lN

) and f2(x′m1

, x′m2

,..., x′

mN) in (5.7), (5.8) are evaluated by means of the Advanced Design Sys-

tem (ADS) Momentum full-wave electromagnetic field solver. Note thatthe quadrature points xl1 ,xl2 ,..., xlN

, xm1 ,xm2 ,..., xmNare modified accord-

ing to (5.4) before being used to compute the values f1(x′l1

, x′l2

, ..., x′lN

) andf2(x′

m1, x′

m2, ..., x′

mN). In this way, the information about the correlation among

the antenna’s design parameters is correctly embedded into the expansion coef-ficients yk1k2...kN

and tk1k2...kN.

5.2.2 Hermite-Padé approximantWe introduce the following least-square N -dimensional Hermite-Padé approxi-mants [10] for both AR and |S11|:

AR = f1(x1, x2, ..., xN ) = PM1(x1, x2, ..., xN )QL1(x1, x2, ..., xN ) , (5.9)

|S11| = f2(x1, x2, ..., xN ) = PM2(x1, x2, ..., xN )QL2(x1, x2, ..., xN ) , (5.10)

where PM1 , PM2 , QL1 , QL2 are polynomials of degrees M1, M2, L1, L2, respec-tively, with QL1 , QL2 > 0 in the entire domain of f1 and f2. With M andL the orders of expansion of the numerator PM and the denominator QL ina Hermite-Padé approximant, first, we define a number c(M) and c(L) ofpolynomials:

Φj(x1, x2, ..., xN ) = φX1k1

(x1)φX2k2

(x2)...φXN

kN(xN ),

k1 + k2 + ... + kN ≤ M(L, respectively), (5.11)

where φXi

ki(xi) are the same orthonormal polynomials computed through the

modified-Chebyshev algorithm as used in the SCM of Section 5.2.1. Then, we


expand PM and QL as:

PM =c(M)∑j=1

pjΦj (5.12) QL =c(L)∑j=1

qjΦj , (5.13)

with pj and qj the expansion coefficients to be determined.

Next, we choose an integer K ≥ 1 and following the same procedure as in (5.11),we compute the c(M + K) − c(M) polynomials whose total degrees are higherthan M and lower than M + K + 1. By exploiting the orthonormality of thepolynomials Φj , we find that:

< PM , Φj >=< fQL, Φj >= 0,

j = c(M) + 1, ..., c(M + K), (5.14)

where f is the function that we want to approximate. The second equality in(5.14) leads to the following linear system:

[A][q] =

⎡⎢⎣

< fΦ1, Φc(M)+1 > · · · < fΦc(L), Φc(M)+1 >...

. . ....

< fΦ1, Φc(M+K) > · · · < fΦc(L), Φc(M+K) >

⎤⎥⎦

×

⎡⎢⎣

q1...

qc(L)

⎤⎥⎦ = 0, (5.15)

where [q] is the vector of the coefficients qi in (5.13). The elements of the matrix[A] are computed by means of the quadrature rule used in (5.7), (5.8). Wenotice that [A] is a (c(M + K) − c(M)) × c(L) matrix. Since, typically, thefollowing condition is satisfied for K ≥ 1:

c(M + K) − c(M) > c(L), (5.16)

the system (5.15) is overdetermined. Thus, we choose the vector [q] to be theoptimal solution of the system (5.15) in the least-squares sense, which meansthat it has to be solution of the following minimization problem:

min‖q‖=1

‖ [A][q] ‖, (5.17)

where ‖ · ‖ is the L2-norm. This problem is solved by means of a singularvalue decomposition (SVD) of the matrix [A], which is easily and efficientlycomputed, given that the values of L, M and K are small. Finally, we computethe coefficients pi of the numerator PM by means of the following equality:

pi =< PM , Φi >=< fQL, Φi >, i = 1, 2, ..., c(M) (5.18)


In the sequel, we will test the validity of the SCM and the Hermite-Padé approx-imant analysis by applying the Kolmogorov-Smirnoff test to a representativepopulation of realizations generated by means of a Monte Carlo simulation.Thereto, the maximum distance D between the cumulative distribution func-tions (CDFs) for AR and |S11| generated by the Monte Carlo samples, on theone hand, and computed from the expansions (5.1), (5.2) and the approximants(5.9), (5.10), on the other hand, is compared to a threshold distance Dα. ForD < Dα, the Kolmogorov-Smirnoff test accepts the null hypothesis that boththe sample sets correspond to the same distribution, with a significance level α.This proves the validity of the gPC or the Hermite-Padé approximant analysis.

5.3 Characterization of the statistics ofcompressible substrates

5.3.1 Measurement fixtureSeveral techniques to estimate the electromagnetic properties of materials havebeen studied and are available in literature. Among them, resonant-perturbationmethods implemented in microstrip planar technology guarantee a high accuracyand sensitivity, and they are very suitable for characterizing low-loss substrateswhose permeability is μ0, such as the compressible protective foam understudy [17]. Therefore, our measurement fixture is based on the aperture-coupledpatch antenna principle, as sketched in Fig. 5.2, with dimensions given inTable 5.1.

Table 5.1: Dimensions of the antenna measurement fixture in (Fig. 5.2).

parameter value (mm) parameter value (mm)patch length LP 62 patch width WP 80slot length LS 3 slot width WS 55

SIW length LSIW 28 SIW width WSIW 70feed length LF 22 feed width WF 2.8

tapering length LT 32 tapering width WT 25

The structure is composed of two planar circuit boards, with the material undertest (MUT) in between, which are supported and kept perfectly aligned bymeans of eight metal posts screwed on a bread-board. More specifically, the twoboards are implemented on a rigid RO4350B high-frequency laminate, whichexhibits very stable material properties, with εr=3.66, tanδ=0.003 and thicknessequal to 1.524 mm. The lower board contains a microstip line on the bottomside, which tapers into a waveguide in Substrate Integrated Waveguide (SIW)technology and couples to the patch through an aperture in the ground planeon the other side. The microstrip-to-SIW transition is adopted to minimize the

5.3. Characterization of the statistics of compressible substrates 79

back radiation through the slot. The upper board keeps the MUT perfectlyadherent to the boards. The patch antenna on the upper board, place on top ofthe MUT, is perfectly centered above the slot, as such optimizing the couplingbetween them.

Figure 5.2: Exploded view of the measurement fixture, as simulated in ComputerSimulation Technology (CST) Microwave Studio. The dotted lines representthe feed and the patch implemented on the bottom sides of the lower and theupper board, respectively.

The aperture-coupled patch antenna topology allows performing measurementsvery quickly, since the MUT can readily be changed. Moreover, the groundplane isolates the feed from the radiating element and minimizes the interfer-ence of spurious radiation by the feed. The strongest electromagnetic fieldsremain confined within the MUT, thus allowing a better characterization of


its properties [18]. Finally, it is easy to deploy other circuit boards for otherfrequency ranges and materials, thus creating a modular fixture.

In this manuscript, we consider a flexible closed-cell expanded rubber protectivefoam as MUT, whose nominal permittivity εr and loss tangent, previouslydetermined by following the procedure described in [19], are equal to 1.56 and0.012, respectively, and whose thickness is 3.94 mm. Moreover, we aim atcharacterizing the properties of textile antennas in the GPS-L1 frequency band([1.56342,1.58742] GHz). Therefore, the complete fixture has been designed inComputer Simulation Technology (CST) Microwave Studio’s frequency domainsolver to resonate in that frequency band with the aforementioned protectivefoam as MUT.

5.3.2 Measurement principle and resultsIn a resonant-perturbation method, the MUT interacts with a resonator andchanges its resonance frequency and quality factor. In our case, the changesin the resonance frequency of the aperture-coupled patch antenna are inducedby variations in the thickness and permittivity of the MUT. Therefore, ifthe thickness of the substrate is known, the value of the measured resonancefrequency can be used to estimate the permittivity of the sample. This procedureis carried out by simulating the measurement fixture in CST Microwave Studio’sfrequency domain solver with the measured value of thickness of the MUTand by varying the value of the permittivity of the MUT until the value ofthe simulated resonance frequency matches the measured one. The presentedtechnique was validated by using a teflon laminate, a substrate material withstable and well-known electromagnetic characteristics, whose height h andrelative permittivity εr were equal to 3.16 mm and 2.1, respectively. For thismaterial, the proposed characterization technique returned a value of relativepermittivity εr equal to 2.064 and thus led to a small relative error of 1.71%with respect to the value provided by the supplier of the material.

Next, we applied this technique to estimate the PDF of the relative permittivityεr and the thickness h of the protective foam. Therefore, 25 samples wererepeatedly compressed up to 70% of the nominal value of h [2] by means of aHotronix Air Fusion Heat Press to achieve 10 different values of thickness, whichwere measured by means of a Dino-Lite AD7013MTL digital microscope, witha resolution of 8 μm. From the measurements performed using the fixture ofSection 5.3.1, the 10 corresponding values of relative permittivity were collectedfor each sample. The resulting histogram of the 250 tuples (h,εr) is reported inFig. 5.3.

Finally, a bivariate normal distribution was fitted on the measured data. Thefollowing vector [μ] of the mean values μh and μεr of the thickness h and thepermittivity εr, respectively, as well as the covariance matrix [Σ] of the PDF,

5.4. Numerical validation for a textile GPS antenna 81

Figure 5.3: 3D histogram of data collected from the measurements of theprotective foam substrate.

were obtained:

[μ] =[

μh

μεr

]=

[3.35821.5809

](mm) (5.19)

[Σ] =[

σ2h σhσεr ρ

σhσεrρ σ2

εr

]=

[0.1137 -0.01132

-0.01132 0.002409

](mm2) (5.20)

with σh and σεrbeing the standard deviations of h and εr, respectively, whereas

ρ is the correlation value between these two parameters. We notice that εr andh are correlated, since the matrix [Σ] is not diagonal. Moreover, with ρ equalto -0.68, the correlation between them is strong. The minus sign confirms thatsubstrate compression yields higher permittivity values.

As a final remark, we point out that potential variations in the loss tangent arenot included in our analysis, since these only have a second-order influence onthe textile antenna’s figures of merit compared to εr and h. Indeed, the setup’sQ factor is stable for all the measurements, demonstrating that variations inloss tangent are negligible. Therefore, in the remainder of this chapter, we willassume the loss tangent to be constant and equal to its nominal value.

5.4 Numerical validation for a textile GPSantenna

The textile GPS antenna shown in Fig. 5.1 and presented in [8] is designed onthe flexible closed-cell expanded rubber protective foam substrate considered


in Section 5.3. A computer-aided design procedure was carried out to matchthe antenna impedance to 50 Ω at 1.57542 GHz. Table 5.2 summarizes theoptimized dimensions of the antenna. The corresponding values of AR and|S11| at the operating frequency are 1.098 (0.812 dB) and 0.1986 (-14.04 dB),respectively, thus satisfying the requirements of the GPS-L1 standard.

Table 5.2: Nominal values of the protective foam substrate antenna’s geometricalparameters (Fig. 5.1).

parameter nominal value (mm)patch length L 73.5patch width W 69.5

inset side length c 5feed point (xf , yf ) (8.5,14.5)

First, since the estimated input PDF is a bivariate normal distribution, the setsof orthonormal polynomials φX1

k1(x1), φX2

k2(x2) simply comprise conventional

Hermite polynomials. Then, based on these sets of polynomials, we relateboth AR and |S11| to the normalized variables x1 and x2 by means of the gPCexpansions (5.1), (5.2), on the one hand, and the Hermite-Padé approximants(5.9), (5.10), on the other hand. Here, the variables x1 and x2 correspondto the thickness h and the relative permittivity εr, respectively. The ordersof the expansions (5.1), (5.2) and the approximants (5.9), (5.10), for whichconvergence is achieved, are reported in Table 5.3.

Table 5.3: Expansion orders for gPC expansions and Hermite-Padé approxi-mants.

AR |S11|gPC PA gPC PA

P = 10 M1=4, K1=4, L1=4 Q = 11 M2=4, K2=5, L2=4

The maximum orders of the gPC expansions correspond to P = 10 and Q = 11for AR and |S11|, respectively, whereas for the Hermite-Padé approximants wehave to compute the polynomials Φj until a maximum order corresponding toM + K, being 8 and 9 for AR and |S11|, respectively. Moreover, the elements ofthe matrix [A] in (5.15) and the expansion coefficients yk1k2...kN

and tk1k2...kNin

(5.1), (5.2) are computed by extending a one-dimensional numerical integrationto the 2D case with a tensor product rule, using the same number of quadraturepoints in each dimension. As a result, we only need (M + K + 1)2 quadraturepoints (being 81 and 100 for AR and |S11|, respectively) to compute the elementsof the matrix [A] in (5.15), compared to the (P + 1)2 = 121 and (Q + 1)2 = 144

5.4. Numerical validation for a textile GPS antenna 83

points necessary to compute the expansion coefficients yk1k2...kNand tk1k2...kN

in (5.1), (5.2), respectively. Therefore, for both AR and |S11|, the Hermite-Padéapproximation will be carried out faster, as further detailed below. Figs. 5.4and 5.5 show AR and |S11| as functions of the variables x1 and x2, in a spanof ±3, corresponding to a normalized interval of ±3σ. The requirements ofthe GPS-L1 standard are indicated by means of a grey plane. We notice thatvariations in h and εr have a significant impact on AR, whereas their influenceon |S11| is less important.

Figure 5.4: Padé approximation of AR as a function of the normalized thicknessh (x1) and the normalized relative permittivity εr (x2). The GPS-L1 thresholdof 1.41 is indicated in grey.

Next, we perform a Monte Carlo analysis on a sample set of 10000 points, whichrequires 10000 full-wave simulations in ADS Momentum with values of h andεr drawn according to their pertinent PDF. We then apply the Kolmogorov-Smirnov test to verify whether these CDFs of AR and |S11| and those found bymeans of the gPC expansions (5.1), (5.2) and the Hermite-Padé approximants(5.9), (5.10) correspond to the same distribution. If we set the significancelevel α to 0.05, Dα equals 0.019233. The computed values of DAR and D|S11|equal 0.0187 and 0.011, respectively, for the gPC, and 0.0113 and 0.0114,respectively, for the Hermite-Padé approximants. Therefore, the null hypothesisof equality between the CDFs is validated with a significance level of 5%. TheCDFs resulting from the Monte Carlo analysis and from the Hermite-Padéapproximants are shown in Figs. 5.6 and 5.7.

We notice that all the CDFs are perfectly overlapping. Moreover, we observe


Figure 5.5: Padé approximation of |S11| as a function of the normalized thicknessh (x1) and the normalized relative permittivity εr (x2). The GPS-L1 thresholdof 0.316 is indicated in grey.

1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

AR=1.41

AR

CDF

PAMC

Figure 5.6: Comparison between the CDFs of AR constructed with the Hermite-Padé approximants (PA) and the Monte Carlo (MC) simulations.

that in 97, 58% of the cases, |S11| is within the threshold value of 0.316. Incontrast, the AR requirement is satisfied only in 38, 68% of the cases, making

5.5. Conclusions 85

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

|S11|=0.316

|S11|

CDF

PAMC

Figure 5.7: Comparison between the CDFs of |S11| constructed with the Hermite-Padé approximants (PA) and the Monte Carlo (MC) simulations.

this a much more critical figure of merit.

As a final remark, we point out that the time required to perform one full-wavesimulation in ADS Momentum is about 16 s (Intel i7 CPU, 16 GB RAM).Thus, we need about 44 hours to generate the complete samples set of 10000points. In contrast, only 32 and 38 minutes are necessary to obtain the valuesf1(x′

l1, x′

l2, ...x′

lN) and f2(x′

m1, x′

m2, ...x′

mN) required to perform the quadrature

rules (5.7), (5.8), respectively, whereas 1.093 s and 1.481 s are required toconstruct the basis and the weights in (5.1), (5.2) and complete the gPCexpansions. As for the Hermite-Padé approximation, 21 and 26 minutes arerequired to collect the samples of f1 and f2, respectively, whereas 2.341 sand 2.715 s are needed to compute the Hermite-Padé approximants (5.9),(5.10). Therefore, both the gPC analysis and the Hermite-Padé approximationoutperform the full-wave Monte Carlo simulation. As expected, the Hermite-Padé proves to be a more efficient alternative to gPC, yielding speedup factorsup to 123 with respect to Monte Carlo.

5.5 ConclusionsA stochastic framework for the analysis of textile antennas’ figures of merit inthe presence of substrate compression was presented. First, the variations in thepermittivity of the substrate corresponding to different values of compressionare experimentally quantified by means of a set of measurements performed on


different samples with a resonant-perturbation technique. Then, based on theresults of the measurements, the stochastic behavior of the figures of merit ofa GPS antenna known from the literature is reconstructed by means of bothgPC and Hermite-Padé approximants. Compared to the standard Monte Carloanalysis, the method shows excellent agreement and superior efficiency.

References[1] L. Vallozzi, W. Vandendriessche, H. Rogier, C. Hertleer, and M. L.

Scarpello, “Wearable textile GPS antenna for integration in protectivegarments”, in 4th European conference on Antennas and Propagation(EuCAP 2010), IEEE, 2010.


[3] S. Park and S. Jayaraman, “Enhancing the quality of life through wearabletechnology”, IEEE Engineering in Medicine and Biology Magazine, vol.22, no. 3, pp. 41–48, 2003.

[4] P. Bonato, “Wearable sensors/systems and their impact on biomedicalengineering”, IEEE Engineering in Medicine and Biology Magazine, vol.22, no. 3, pp. 18–20, 2003.

[5] P. Salonen, Y. Rahmat-Samii, and M. Kivikoski, “Wearable antennas inthe vicinity of human body”, in 2004. IEEE Antennas and PropagationSociety International Symposium, vol. 1, 2004, pp. 467–470.


[7] M. Rossi, A. Dierck, H. Rogier, and D. Vande Ginste, “A stochastic frame-work for the variability analysis of textile antennas”, IEEE Transactionson Antennas and Propagation, vol. 62, no. 12, pp. 6510–6514, 2014.

[8] L. Vallozzi, W. Vandendriessche, H. Rogier, C. Hertleer, and M. Scarpello,“Design of a protective garment GPS antenna”, Microwave and OpticalTechnology Letters, vol. 51, no. 6, pp. 1504–1508, 2009.






[13] P. Sumant, Hong Wu, A. Cangellaris, and N. Aluru, “Reduced-order mod-els of finite element approximations of electromagnetic devices exhibitingstatistical variability”, IEEE Trans. on Antennas and Propagation, vol.60, no. 1, pp. 301–309, Jan. 2012.


[15] P. Li and L. J. Jiang, “Uncertainty quantification for electromagneticsystems using ASGC and DGTD method”, IEEE Transactions on Elec-tromagnetic Compatibility, vol. 57, no. 4, pp. 754–763, 2015.


[17] L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan, and V. K. Varadan, Mi-crowave Electronics: Measurement and Materials Characterization. JohnWiley & Sons, 2004, p. 40.



6Stochastic Analysis of the

Efficiency of a Wireless PowerTransfer System Subject to

Antenna Variability and PositionUncertainties

Marco Rossi, Gert-Jan Stockman, Hendrik Rogier, Dries Vande Ginste

Based on the article published in Sensors

� � �

The efficiency of a Wireless Power Transfer (WPT) system in theradiative near-field is inevitably affected by the variability in thedesign parameters of the deployed antennas and by uncertainties intheir mutual position. Therefore, we propose a stochastic analysisthat combines the generalized Polynomial Chaos (gPC) theory withan efficient model for the interaction between devices in the radiativenear-field. This framework enables us to investigate the impact ofrandom effects on the Power Transfer Efficiency (PTE) of a WPTsystem. More specifically, the WPT system under study consists of atransmitting horn antenna and a receiving textile antenna operatingin the Industrial, Scientific and Medical (ISM) band at 2.45 GHz.First, we model the impact of the textile antenna’s variability on

90 Chapter 6. WPT Efficiency in Presence of Antenna and Position Variability

the WPT system. Next, we include the position uncertainties of theantennas in the analysis in order to quantify the overall variations inthe PTE. The analysis is carried out by means of polynomial-chaos-based macromodels, whereas a Monte Carlo simulation validates thecomplete technique. It is shown that the proposed approach is veryaccurate, more flexible and more efficient than a straightforwardMonte Carlo analysis, with demonstrated speedup factors up to 2500.

6.1 IntroductionWith the advent of the Internet of Things (IoT), radio frequency identification(RFID) systems and, in general, radio frequency (RF) sensors and actuators haveacquired significant importance. Distributed in our surroundings in a pervasiveand inconspicuous way, these elementary components interact with each other tocollect, process and exchange data [1], [2]. The potential applications leveragedby the IoT paradigm are manifold and range from transportation over logisticsto healthcare and from infrastructure monitoring to emergency services. As aresult, assessing the correct operation of the systems involved is of paramountimportance.

One of the main requirements of the IoT is that RFID components and RFsensors need to be small, low cost and not limited in lifetime by the durationof a battery [1]. Therefore, they are often designed to be passive and theyrely on WPT for their activation and operation [3, 4]. Up to now, WPTin the reactive near-field and in the far-field has been widely investigated inliterature [5–8]. However, the proposed solutions can be considered sub-optimalfor several reasons. On the one hand, WPT operating in the reactive near-fieldcan achieve high PTE, but it requires source and target to be very close toeach other [9]. Moreover, variations in distance between the devices stronglyaffect the resonance frequency and the load impedance of the system. On theother hand, the PTE achieved by far-field schemes is often too low. As a result,research efforts have recently shifted to WPT in the radiative near-field (Fresnelregion), where the PTE, although rapidly decreasing with distance, can still behigh enough to power sensor networks [10–12].

The most effective way to assess the performance of a WPT system is to studyits transfer efficiency, since random fluctuations may alter the incoming power atthe receiving element, whereas the output voltage level delivered by the powermanagement system is typically first stabilized by a voltage regulator to providea constant supply voltage needed to feed electronic circuits. To this end, severalnumerical techniques may be leveraged to assess the efficiency of a WPT systemin the radiative near-field, such as source reconstruction methods [13], multipoleexpansion of the electromagnetic field [14], model reduction combined withfull-wave solvers [15] and domain decomposition methods [16]. A very efficient


approach has recently been proposed in [17, 18], where the electromagneticinteraction among arbitrarily positioned radiating devices is modeled by means oftheir measured or simulated radiation patterns, allowing for device repositioningwithout requiring new simulations or measurements. Moreover, no constraint isenforced on the antenna configurations considered, as long as their radiationpattern is provided.

Although this approach efficiently quantifies the PTE of a WPT system fordifferent positions and rotations of the antennas, it cannot assess how the PTEvaries for a given system setup when devices undergo small random rotations orwhen their mutual position is affected by uncertainties. Moreover, the methodis based on a single simulation of the radiation pattern of each element ofthe system, which typically corresponds to its nominal design. However, theradiation characteristics of the actually deployed antennas deviate from thenominal ones due to the inevitable uncertainties on the design parametersarising during the production process. As a result, the antenna variability hasto be included in the WPT model to correctly estimate the PTE of the system.

In this chapter, we propose a Stochastic Collocation Method (SCM) basedapproach, leveraging on gPC expansions [19, 20]. The formalism overcomes thelimitations mentioned in the previous paragraph as it efficiently allows assessingthe impact of position uncertainties and antenna variability on the PTE of aWPT system. More specifically, we start from given PDFs according to whichthe antennas’ design parameters vary. Next, we introduce a gPC expansionfor each antenna configuration to model the corresponding variations in itsradiation characteristics. Further, on a higher level, a second gPC expansionassesses the impact of both antenna variability and position uncertainties onthe PTE of the system. In this way, the analysis of the WPT system requires alower number of simulations compared to a single gPC expansion accountingfor all variations.

The proposed approach is demonstrated on a simple WPT system consisting of atransmitting horn antenna and a receiving ISM textile antenna operating at 2.45GHz, which is connected to a rectifier circuit to deliver direct current (DC) powerto a load [18]. For this receiving antenna, we use experimentally determinedprobability density functions of the design parameters [21, 22]. The results arefound to be as accurate as those obtained by the traditional Monte Carlo method,here used to validate our technique. However, owing to the considerably lowernumber of simulations needed to construct a gPC expansion compared to thenumber required for Monte Carlo method to converge, the proposed approachproves to be much more efficient and flexible. The polynomial chaos expansionhas been applied to model lumped circuits and distributed interconnects [23,24], multiport systems [25], the effect of geometric and material variations inscattering problems [26, 27], in direction of arrival (DOA) estimation [28] andin antenna design [21, 22]. However, to our best knowledge, the application ofuncertainty quantification to a WPT system is completely new.


This chapter is organized as follows. In Section 6.2, we briefly describe boththe WPT model [17] and the SCM. Then, in Section 6.3, the results for a WPTconsisting of a transmitting horn antenna and a receiving ISM textile antennaconnected to a rectifier circuit are presented and then discussed in Section 6.4.Conclusions are summarized in Section 6.5.Notations: We denote field vectors by underlined letters, e.g. v, and unit vectorswith a "hat", e.g. v. All sources and fields are assumed to be time harmonicwith angular frequency ω and time dependencies ejωt are suppressed. Vectorelements and arrays are represented by boldface characters, e.g. x. For a givenarray x ∈ �, xT denotes its transpose, whereas ‖x‖ denotes its Euclidean norm.

6.2 Materials and Methods6.2.1 Wireless Power Transfer System ModelConsider a simple WPT system consisting of a transmitting antenna TX anda receiving antenna RX. We assume, for simplicity, that both TX and RXare one-port devices with radiation impedances Ztx and Zrx, respectively, andwe represent them by means of two equivalent circuits [29] as in Figure 6.1.More specifically, the transmitter is driven by means of a Thévenin generatorcomposed of a sinusoidal voltage source Vg and an internal impedance Zg,whereas the receiving antenna is modeled as a Norton equivalent with a shortcircuit current Isc and load impedance ZL. The receiver also includes a matchingcircuit and a rectifier to transform the alternating current (AC) power receivedby the antenna into the DC power delivered to the load RL. The short circuit

Vg

Zg

Ztx ZrxIsc MatchingZL

D

CL RL Vout

Figure 6.1: Equivalent circuit of a WPT link, i.e. a transmit antenna and areceive antenna with rectifier (rectenna).

current Isc is computed as:

Isc = − 1V0

∫VRX

einc(r) · j(r)dr, (6.1)

where V0 is a pertinent normalization factor [30], VRX is the volume of RX,einc(r) is the electric field incident on the receiver, j(r) is the current densityimpressed on in VRX and r is the position vector. It was shown in [17] that in

6.2. Materials and Methods 93

the radiative near-field Isc can be expressed as follows:

Isc = − 1ZV0

∫Ω

T (rtx,rx, k)F tx(k) · F rx(−k) dk, (6.2)

where we integrate over the Ewald sphere Ω. Further, F tx(k) and F rx(k) are theradiation patterns of the transmitter and the receiver, respectively, Z = μ

ε is thewave impedance of the background medium, k = sin θ cos φx+sin θ sin φy+cos θz

is the wave vector in spherical coordinates, T (rtx,rx, k) is a translation operatorthat allows efficient translations between the two antennas and rtx,rx is therelative position between the phase centers of the two devices. The operatorT (rtx,rx, k) is calculated as follows:

T (rtx,rx, k) ≈L∑

l=0(2l + 1)j−lh

(2)l (k‖rtx,rx‖)Pl(k · rtx,rx), (6.3)

where j is the imaginary unit, h(2)l (·) is the l-th order spherical Hankel function

of the second kind, and Pl(·) is the Legendre polynomial of degree l. Thenumber L determines the accuracy of the approximation (6.3) and traditionalguidelines are followed to select it [31]. The relation (6.2) is valid only as longas the two antennas are not positioned in each other’s reactive near-field, thuswith ‖rtx,rx‖ at least equal to a sixth of the wavelength.

The rotation of the devices around their phase center is included in the describedformalism by applying the appropriate rotation R to the radiation patternsF tx(k) and F rx(k) in the spherical harmonics domain, as shown in Figure 6.2.Thereto, first, the radiation pattern F (k) = Fθ(k)θ + Fφ(k)φ is expanded intospherical harmonics Apq and Bpq by means of the transformation F , givenby [32]:

{Apq

Bpq

}= − 1

p(p + 1)

2π∫φ=0

π∫θ=0

[q

{jFφ(θ, φ)−Fθ(θ, φ)

}Y ∗

pq(θ, φ)

+ sin θ

{−Fθ(θ, φ)jFφ(θ, φ)

}dY ∗

pq(θ, φ)dθ

]dθdφ, (6.4)

where the unit vector k has been replaced by (θ, φ), Ypq(θ, φ) and their complexconjugates Y ∗

pq(θ, φ) are the orthonormalized scalar spherical harmonics. Then,the coefficients Apq and Bpq are rotated in the spherical harmonics domain(RSH) by means of Wigner D-matrices. The rotated coefficients AR

pq and BRpq

are calculated as [33]{

ARpq

BRpq

}=

{Apq

Bpq

} ∑|r|≤p

e−jqγdrpq(β)e−jrα, (6.5)


with drpq(β) the Wigner small d-matrix, given by [34]

drpq(β) = (−1)r−q

√(p + r)!(p − r)!(p + q)!(p − q)! ·

∑s

(−1)s

×(

cos β2

)2(p−s)+q−r(sin β

2)2s−q+r

(p + q − s)!s!(r − q + s)!(p − r − s)! . (6.6)

The range of s is determined such that all factorials are nonnegative. α, β andγ in (6.5), (6.6) are the standard Euler angles that define the rotation using thez − y − z convention in a right-handed frame. The angles (α, β, γ) are relatedto the desired inclination and azimuthal angles θ and φ by choosing α = φ,β = θ and γ = 0. Finally, the rotated radiation pattern FR(θ, φ) is found fromthe rotated coefficients AR

pq and BRpq by means of the transformation F−1 given

by [32]:

{FR

θ (θ, φ)FR

φ (θ, φ)

}=

P∑p=0

∑|q|≤p

[{AR

pq

jBRpq

}dYpq(θ, φ)

dθ+{

BRpq

jARpq

}qYpq(θ, φ)

sin θ

], (6.7)

where P is a parameter that sets the accuracy, which for practical reasons maybe chosen equal to L in (6.3) [35].

F (θ, φ) (Apq, Bpq)

FR(θ, φ) (ARpq, BR

pq)

F

RSHR

F−1

Figure 6.2: Rotation of F (θ, φ) to FR(θ, φ) using the spherical harmonicsdomain.

We notice from (6.2) that only the (measured or simulated) radiation patternsof the two devices are needed to calculate the influence of the transmitter on thereceiver. Moreover, the combination of the rotation mechanism of Figure 6.2and the use of the translation operator T (rtx,rx, k) in (6.2) allow computing theshort circuit current Isc for any set of rotations and positions of the devices. Asa result, this formalism is more efficient and flexible than traditional simulationstools or measurements, which require a new computation or measurement forevery rotation and repositioning.

The PTE of the WPT system considered consists of three contributions. First,we need to calculate the wireless link efficiency ηlink, which is defined as theratio between the power Prx delivered to the receiving antenna’s load ZL and


the power Ptx emitted by the transmitter:

ηlink = PrxPtx

=

12

(ZL |Isc|2

∣∣∣ ZrxZrx+ZL

∣∣∣2)

12Z

∫Ω

|F tx(k)|2 dk. (6.8)

Then, the efficiencies ηmatch and ηrect of the matching and the rectifying circuitsare included. Since the input impedance ZL of a nonlinear circuit depends on theincoming power, ηmatch has the same dependency. Therefore, we simulate it withthe commercial tool Advanced Design System (ADS) by Keysight Technologies.As to the efficiency of the rectifier, it is calculated as ηrect = Pinc/PDC , wherePinc = ηmatch · Prx and PDC are the AC power injected into the rectifier andthe DC power delivered to the load RL, respectively. The DC power PDC isgiven by:

PDC = V 2out/RL, (6.9)

where the DC output voltage Vout is also computed by ADS. Finally, the overallPTE of the system is given by:

PTE = PDC

Ptx= ηlink · ηmatch · ηrect. (6.10)

6.2.2 Stochastic Collocation MethodConsider a generic system output G (such as the PTE of the WPT system)and N input random variables (RVs) x1, x2, ..., xN that affect it. We assumethese variables to be actually independent and we collect them in the vectorx = [x1, x2, ..., xN ]. Then, following the Wiener-Askey scheme [19], we relate Gto x by means of a polynomial chaos expansion

G = f(x) =K∑

k=0ykϕk(x), (6.11)

where ϕk(x) are suitably chosen multivariate polynomial basis functions and theexpansion coefficients yk are the unknowns to be determined. The polynomialsϕk(x) are constructed to be orthonormal with respect to the PDF PX , whichdescribes the likelihood of the input x. Thus:

< ϕj(x), ϕk(x) >=∫

Γ

ϕj(x)ϕk(x)PX(x)dx = δjk, (6.12)

with δjk = 0 if i �= j, δjk = 1 if i = j, and Γ being the support of PX .Consequently, PX acts as a weighting function. Since the considered inputRVs are independent, the PDF PX is defined as the product of the PDFscorresponding to the single input variables. Therefore, the polynomials ϕk(x)


are constructed as products of N univariate polynomials, each one associated toa single input RV. Further, the multivariate polynomials have a total degree ofmaximally Q, meaning that the sum of the orders of the univariate polynomialsis at most Q.

Finally, by means of the Stochastic Testing (ST) algorithm [36], we select M = Kcollocation points xm and we compute both a matrix A, whose elements aredefined as amk = ϕk(xm), and its inverse B. The coefficients yk in (6.11) arethen calculated as in [37]:

yk =M∑

m=0bmkf(xm), (6.13)

where bmk is the mk-th element of matrix B and f(xm) is the function to beapproximated, evaluated in xm.

6.2.3 Wireless Power Transfer Uncertainty QuantificationThe efficiency of the WPT system under study is affected by two main issues. Onthe one hand, both the transmitter and the reicever may undergo uncertaintiesin their design parameters, since, for example, the production process yieldsdevices that do not perfectly correspond to their nominal design. As a result,random variations may be expected on both radiation impedances Ztx and Zrx,and on the antennas’ radiation patterns. On the other hand, the system maybe conceived to operate in a given configuration and undesired small randomrotations or variations in the mutual position of the antennas affect the influenceof the transmitter on the receiver.

Conventionally, these problems are expected to arise simultaneously. Therefore,the variations in the WPT efficiency corresponding to both antenna variabilityand position uncertainies are investigated by means of gPC expansions (6.11),which are defined as functions of the design parameters of the antennas andof the WPT system. However, the use of a single gPC expansion to carryout the complete analysis is sub-optimal. This is understood as follows. Theconstruction of a gPC expansion requires to select and process M collocationpoints xm to compute the coefficients yk in (6.11). Even though translationsand rotations are efficiently accounted for by the WPT model described inSection 6.2.1, both the antennas’ radiation impedances and their radiationpatterns are usually computed by means of full-wave solvers, such as ADSMomentum, whose simulations are typically time-consuming. As a result,accounting for the variability of the antennas requires a high number of full-wave simulations. Moreover, if the efficiency of the WPT system is evaluated fordifferent system configurations or different position uncertainties, a new gPCexpansion has to be constructed for each configuration and for each position.As a result, for each such expansion a new set of collocation points has to beselected and processed, again requiring full-wave simulations, which significantly


decrease the efficiency of the method with respect to more naive approaches,such as Monte Carlo.

In order to overcome these limitations and drastically improve efficiency, theconstruction of the gPC expansions that model the efficiency of the wholesystem is preceded by an intermediate step. More specifically, for each antennaconfiguration undergoing variability, we introduce gPC expansions (6.11) ofthe real and the imaginary parts of both its radiation impedance Z and thespherical harmonic coefficients Apq and Bpq in (6.4):

Zre =KZre∑k1=0

yk1ϕk1(xvar) (6.14)

Zim =KZim∑k2=0


Arepq =

KArepq∑

k3=0yk3ϕk3(xvar) (6.16)

Aimpq =

KAim

pq∑k4=0


Brepq =

KBrepq∑

k5=0yk5ϕk5(xvar) (6.18)

Bimpq =

KBim

pq∑k6=0

yk6ϕk6(xvar), (6.19)

where xvar is the vector of the antenna’s design parameters that are affected byuncertainties. These gPC expansions can now be interpreted as macromodelsof the considered antennas. Once constructed, they allow accurately computingboth an antenna’s radiation impedance and radiation pattern for a value of xvar

without having to resort to full-wave simulations. After this intermediate step,we model both the link efficiency ηlink and the overall PTE of the system bymeans of the following gPC expansions:

ηlink =Kηlink∑k7=0

yk7ϕk7(xwpt) (6.20)

PTE =KPTE∑k8=0

yk8ϕk8(xwpt), (6.21)

where xwpt is the vector of all the parameters in the WPT system subject tovariations, which comprises all the vectors xvar and the variables corresponding


to position uncertainties. The procedure is now as follows. First, Mηlink andMpte collocation points xwpt

m are selected to construct the gPC expansions(6.20),(6.21), respectively. Next, for each collocation point, the gPC expansions(6.14)-(6.19) are used to rapidly compute both the radiation impedances andthe radiation patterns of the antennas undergoing variability. Finally, withthe calculated antennas’ radiation characteristics, the WPT model describedin Section 6.2.1 processes the values of xwpt

m corresponding to the positionuncertainties in the system and computes the values of ηlink and PTE in thosecollocation points required to calculate the coefficients yk7 ,yk8 in (6.20),(6.21).

The benefits of this approach are twofold. First, since for each antenna thenumber of design parameters undergoing variations is expected to be significantlylower than the total number of parameters affecting the system, the amountof full-wave simulations necessary to construct all antenna macromodels isexpected to be substantially lower than what is required to compute a single gPCexpansion that models the entire WPT system. Second, once these macromodelsare available, the analysis of the WPT system can be repeated for any givensystem configuration at a negligible computational cost, since no full-wavesimulations are required to calculate the antennas’ radiation characteristics.

6.3 Results6.3.1 Validation example setupThe proposed approach is demonstrated on the WPT system shown in Figure 6.3.The transmitting device is a standard gain horn (SGH) antenna radiating at2.45 GHz with a power of 10 dBm. The receiving device is a 2.4-2.4835 GHzISM band textile microstrip probe-fed dual-polarized patch antenna [38], using aflexible closed-cell expanded rubber foam as substrate, and shown in Figure 6.4.The antennas are placed at a separation distance d and their phase centers arealigned.

The radiation impedance Ztx of the SGH antenna is 50 Ω and its radiationpattern is computed by analytical expressions (see the appendix in [17], wherea = 368.6 mm, b = 273.1 mm, ρ1 = 343.9 mm, ρ2 = 363.4 mm). Instead, the2.45 GHz ISM band textile antenna is designed and simulated by means ofADS Momentum. Its nominal design paramenters are reported in Table 6.1. Itsnominal radiation impedance Zrx, which is equal for both feed 1 and feed 2, is49.91 − 1.93j Ω at the frequency of 2.45 GHz. An isolation −20log|S21| betweenthe two feed ports of more than 15 dB is achieved within the entire ISM band.

The wireless link efficiency ηlink between the SGH and the textile antenna iscomputed by means of the formalism described in Section 6.2.1. The parametersL and P in (6.3) and (6.7), respectively, are both set to 5, leading to 36coefficients Apq and Bpq. Next, a rectifier is attached to the patch antennato form the rectenna shown in Figure 6.5, which is designed and simulated in

6.3. Results 99

y

x

d

θ

φ

Figure 6.3: Simulation setup where an SGH acts as transmitter and a patchantenna as receiver.

Figure 6.4: Schematic of the dual polarized probe-fed ISM band textile antennaunder study. Top panel: topview. Bottom panel: side view. Antenna parametersare indicated in Table 6.1.


Table 6.1: Nominal values of the antenna parameters (Figure 6.4).

parameter nominal value

patch length L 44.46 mmpatch width W 45.32 mmslot length Ls 14.88 mmslot width Ws 1 mm

feed points (±xf , yf ) (±5.7,5.7) mmsubstrate height h 3.94 mm

permittivity εr 1.53loss tangent tanδ 0.012

ADS. More specifically, the rectenna consists of a matching network, a voltagedoubler and a rectifier. The matching circuit is given by an inductor Lm = 5 nH,whereas the voltage doubler and rectifier circuit itself consist of two HSMS-2850Schottky diodes, for which a pertinent SPICE model is used, together with theirpackage parasitics Cp = 0.08 pF and Lp = 2 nH. The capacitors C1 and C2 areequal to 100 pF and the load resistance is equal to RL = 100Ω. The matchingcircuit is designed to have optimal matching when Prx = −10 dBm. However,because of the nonlinear diodes in the voltage doubler and rectifier circuit, theload impedance ZL, and, therefore, the matching efficiency ηmatch, depend onthe incoming power Prx. This impedance is computed for different incomingpowers by using a harmonic balance simulation in ADS and the matchingefficiency is then calculated as:

ηmatch = 1 − |Γ |2 = 1 −∣∣∣∣ZL(Prx) − Z∗

rxZL(Prx) + Zrx

∣∣∣∣2

. (6.22)

Finally, the efficiency ηrect of the voltage doubler and rectifier is calculated as in(6.9), and the total efficiency ηtot of the rectenna is given by ηtot = ηmatch ·ηrect.

ZrxIsc

Lm C1 Lp D

CpLp

DCp

C2 RL Vout

Patch Antenna Matching Voltage Doubler and Rectifier Load

Figure 6.5: The complete schematic of a rectenna element as designed andsimulated in ADS.

6.3. Results 101

6.3.2 Antenna VariabilityIn the example under study, we only consider antenna variability of the 2.45 GHzISM band textile antenna, since a SGH antenna is not expected to deviate fromits nominal characteristic. Textile antennas, however, usually exhibit variationsin their design parameters, which significantly alter their figures of merit. Onthe one hand, the production process inevitably introduces uncertainties inthe geometrical dimensions of the antenna. On the other hand, the flexibleclosed-cell expanded rubber protective foam used as antenna substrate is a verynon-uniform material. As a result, the value of its electrical permittivity εr maybe quite different from the nominal one.

In [21], it is shown that variations in the patch length L, the patch width Wand the electrical permittivity εr of the antenna have a significant impact on itsradiation impedance Zrx, whereas the influence of other parameters is negligible.Moreover, a quick sensitivity analysis confirms that only variations in L, W andεr affect the radiation pattern of the antenna. Therefore, we relate both the realand the imaginary part Zre

rx and Zimrx , respectively, of the radiation impedance

Zrx of the receiving antenna, as well as the real and the imaginary part of thecoefficients Apq and Bpq in (6.4), to the parameters L, W and εr by means ofthe gPC expansions (6.14)-(6.19) where thus, xvar = [L, W, εr]. As we knowfrom [21], [22] that these parameters are independent and vary according toGaussian distributions, according to the Wiener-Askey scheme [19], the multi-variate polynomials ϕk1(xvar),ϕk2(xvar),ϕk3(xvar),ϕk4(xvar),ϕk5(xvar),ϕk6(xvar)in (6.14)-(6.19) consist of products of Hermite polynomials. Finally, the PDFPX in (6.12) is given by the product of three univariate Gaussian distributions.Its mean vector μ and covariance matrix Σ are given by:

μ =

⎡⎣ μL

μW

μεr

⎤⎦ =

⎡⎣45.385

44.5151.526

⎤⎦ (mm) (6.23)

Σ =

⎡⎣ σ2

L σLσW ρLW σLσεrρLεr

σW σLρLW σ2W σW σεr

ρW εr

σεrσLρLεr

σεrσW ρW εr

σ2εr

⎤⎦

=

⎡⎣0.0161 0 0

0 0.0265 00 0 0.00102

⎤⎦ (mm2), (6.24)

where σL, σW and σεrare the standard deviations of L, W and εr, respectively,

whereas ρLW = ρLεr = ρW εr = 0 indicates that there is no correlation betweenthe design parameters.

The gPC expansions of Zrerx and Zim

rx are found to converge for orders ofexpansion with a total degree QZre

rxand QZim

rxequal to 2 and 3, respectively.

In contrast, the real and the imaginary part of the coefficients Apq and Bpq


exhibit a more intricate behavior. Therefore, a total degree QApq = QBpq = 6is necessary to accurately catch the variations determined by L, W and εr onthese parameters. Since both the radiation impedance Zrx and the radiationpattern of the antenna are computed during the same full-wave simulation, thetotal degree of all gPC expansions (6.14)-(6.19) is set to Qvar = 6. In this way,only one set of collocation points is selected and simulated in ADS to computethe coefficients yk1 , yk2 , yk3 , yk4 , yk5 , yk6 in (6.14)-(6.19) and to construct all thegPC expansions, which is clearly beneficial in terms of computation time. Inthis case, a number Mvar = 84 of collocation points xvar

m has been selected bymeans of the ST algorithm and simulated in ADS.

6.3.3 Position UncertaintiesNominally, the WPT system shown in Figure 6.3 is constructed to operate inthe radiative near-field with a distance d between the two antennas equal to0.6 m ≈ 5λ and with their phase centers aligned at coordinates x = y = 0.Further, the radiating surface of the SGH antenna and the patch antenna arealigned with the xy plane, which means that the rotation angles θ and φ areboth equal to 0◦. In practice, the antennas are perturbed by small randomrotations and variations in their mutual position. For the sake of conciseness,the SGH antenna is stationary and all the variations are cumulated in theposition and the rotation of the 2.45 GHz ISM band antenna. Finally, sinceno experimental data are available to estimate the PDFs corresponding to theconsidered parameters, we suppose that d, x, y, θ and φ are independent andvary according to Gaussian distributions, whose mean values and standarddeviations are reported in Table 6.2:

Table 6.2: Mean values and standard deviations of the geometrical parametersof the link (Figure 6.3).

parameter mean value μ standard deviation σ 3σ

d 0.6 m 0.01666 m 0.05 mx 0 m 0.00666 m 0.02 my 0 m 0.00666 m 0.02 mθ 0◦ 10◦ 30◦

φ 0◦ 10◦ 30◦

More specifically, we assume that the variations in the position of the ISMpatch antenna in the xy plane are limited to intervals x = ±2 cm and y = ±2cm, which correspond to a displacement of about half the width W and thelength L of the patch. As for d, θ and φ, the variations are assumed to be largeenough to account for a potential displacement in a real scenario.

6.3. Results 103

6.3.4 Wireless Power Transfer EfficiencyThe two gPC expasions (6.20),(6.21) are introduced to relate both the linkefficiency ηlink and the overall PTE to the parameters L, W , εr, d, x, y,θ and φ. All parameters are independent and vary according to Gaussiandistributions. As a result, the multivariate polynomials ϕk7(xwpt),ϕk8(xwpt)of both the gPC expansions are Hermite polynomials, as in Section 6.3.2.Note, however, that upon the availability of other experimental data, otherdistributions for these parameters can equallt be dealt with by means of theadvocated gPC-based approach. We find that both expansions converge foran order of expansion corresponding to a total degree Qηlink and Qpte equalto 4. A number Mηlink = Mpte = 495 of collocation points xlink

m is selected bymeans of the ST algorithm and processed with the WPT model and the antennamacromodels in order to compute the coefficients yk7 ,yk8 in (6.20),(6.21).

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

ηlink

CDF

SCMMC

Figure 6.6: Comparison between the CDFs of ηlink constructed with the ad-vocated Stochastic Collocation Method (SCM) and the Monte Carlo (MC)simulations.

Next, we perform a Monte Carlo analysis of both ηlink and PTE by processing asample set of 10000 realizations of L, W , εr, d, x, y, θ and φ, drawn according totheir pertinent PDFs. Then, we compute the CDFs of ηlink and PTE based onboth the Monte Carlo simulation and the SCM analysis. The curves are shownin Figures 6.6 and 6.7. We notice that the CDFs are perfectly overlapping.Finally, in order to validate our analysis, we apply the Kolmogorov-Smirnov testto verify whether the CDFs of ηlink and PTE and those found by means of theSCM analysis correspond to the same distribution. In particular, the maximum


distance D between them is compared to a threshold distance Dα. For D < Dα,the Kolmogorov-Smirnoff test accepts the null hypothesis that both the samplesets correspond to the same distribution, with a significance level α. If we setthe significance level α to 0.05, Dα equals 0.019233. The computed values ofDηlink and DP T E are equal to 0.0067 and 0.0085, respectively. Therefore, thenull hypothesis of equality between the CDFs is validated with a significancelevel of 5%.

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.2

0.4

0.6

0.8

1

PTE

CDF

SCMMC

Figure 6.7: Comparison between the CDFs of the PTE of the WPT systemconstructed with the advocated Stochastic Collocation Method (SCM) and theMonte Carlo (MC) simulations.

6.4 DiscussionThe proposed approach allows quantifying the variations of both the linkefficiency ηlink and the PTE of a WPT system in a more efficient and flexibleway than both an SCM analysis based on a single gPC expansion and a MonteCarlo analysis. As shown in Table 6.3, both the number of full-wave simulationsand the overall CPU time necessary to perform the analysis are greatly reducedfor both ηlink and PTE.

More specifically, the simulation of a single realization of the 2.45 GHz ISM bandantenna in ADS Momentum requires about 15 s, whereas the construction ofthe gPC expansions that model its radiation impedance Zrx and the coefficientsApq and Bpq takes 3.72 s, once the Mvar = 84 antenna realizations have beensimulated in ADS. As a result, the construction of the gPC-based macromodels

6.5. Conclusions 105

Table 6.3: Simulation data for the analysis of the ηlink and the PTE of theWPT with different methods.

Number of Full-Wave Simulations Overall CPU TimeMethodηlink PTE ηlink PTE

gPC +macromodels

84 84 21 min 53 s 22 min

single gPC 495 495 2 h 4 min35 s

2 h 4 min42 s

Monte Carlo 10000 10000 41 h 48 min25 s

41 h 50 min46 s

for Zrx and the radiation pattern of the antenna requires about 21 min. Then,Mηlink = Mpte = 495 collocation points have to be processed to construct thegPC expansions of both ηlink and PTE. In order to obtain 495 samples of thelink efficiency ηlink by using the constructed macromodel of the antenna andthe WPT model of Section 6.2.1, about 25 s are required. In contrast, about32 s are necessary to collect 495 samples of the overall PTE, which include thesimulations of the rectifier. Finally, the construction of both the gPC expansionsof ηlink and PTE takes about 25 s. Therefore, a first complete analysis of theWPT system requires about 22 min. Once the antenna characteristics havealready been modeled, additional analyses of other WPT systems using thisantenna or for other distributions of the position parameters take only about 1min. In comparison, an analysis based on a single gPC, which requires to directlysimulate 495 antenna realizations in ADS, takes more than 2 hours. Moreover,this operation has to be repeated each time any difference is introduced in thedistributions according to which the parameters of the WPT system vary. Asfor the Monte Carlo procedure, the simulation of 10000 realizations by meansof ADS and the WPT model of Section 6.2.1 requires more than 41 hours. Asa result, the proposed approach greatly outperforms both the Monte Carlotechnique and an SCM analysis based on a single gPC expansion.

6.5 ConclusionsAn SCM analysis of the efficiency of a WPT system in the radiative near-fieldsubject to antenna variability and position uncertainties has been presented.More specifically, a first SCM analysis is carried out to account for the impactof uncertainties in the design parameters of the antennas on their radiationcharacteristics. The resulting gPC expansions are used as macromodels thatallow computing the antennas’ radiation characteristics in a more efficient waythan full-wave solvers. Then, a second SCM analysis quantifies the impact ofboth the variability of the deployed antennas and the uncertainties in theirmutual position on the efficiency of the WPT system. This is done by leveraging


the previously computed gPC-based macromodels and a very efficient modelfor WPT systems in the radiative near-field. Finally, the proposed approachis validated by means of a WPT system consisting of a SGH antenna and a2.45 GHz ISM band textile antenna. Compared to an SCM analysis based on asingle gPC expansion, as well as to a standard MC analysis, the method showsexcellent agreement and superior efficiency.

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7Conclusions

7.1 General conclusionsIn the next years, millions of everyday objects will be equipped with electroniccomponents to realize a pervasive network of ‘smart things’ that will extendpeople’s reach beyond the physical barriers that drastically limit their capabili-ties. Wearable systems and smart textiles are expected to be one of the tiles ofsuch a multifaceted mosaic.

In order to pave the way for such a promising future, engineers need to outlinethe best strategies and solutions to design and implement the technologies oftomorrow. In this context, the reliability of the novel devices and systemsbecomes more crucial than ever.

This work has aimed at targeting some of the most critical reliability problemsof wearable antennas, being the uncertainties arising during the productionprocess, often due to the inhomogeneity in the applied materials, and theeffect of potential substrate compression occurring in real scenarios. Each ofthese aspects has been thoroughly investigated in one of the chapters of thismanuscript by leveraging on the combination of Stochastic Collocation Methods(SCMs), the Advanced Design System (ADS) Momentum full-wave solver andmeasurements performed on real antenna prototypes.

In Chapter 3, the impact of the variations in the geometrical parameters oftextile antennas on their input impedance has been addressed. The realizationsof several prototypes on a rigid substrate and the measurements on 100 manuallycut patches have assessed the soundness of the application of SCMs and havevalidated the analysis with actual data, respectively.

In Chapter 4, a novel approach to the characterization of textile materials hasbeen introduced to overcome the limitations of already established techniques

112 Chapter 7. Conclusions

when non-uniform materials are examined. Being non-destructive, accurate andfast, the devised method represents a valid alternative to such techniques.

In Chapter 5, the methodology outlined in Chapter 3 has been extended tostudy the effect of substrate compression on the axial ratio and the return lossessof a Global Positioning System (GPS) textile antenna. The characterizationtechnique described in Chapter 4 has provided the information on the probabilitydistribution of substrate permittivity and height. Compression has proven tohave a critical impact on the axial ratio of the antenna.

Finally, in Chapter 6, tha variability analsys of a Wireless Power Transfer (WPT)system in the radiative near-field relying on textile antennas has been carriedout. The influence of the uncertainties in the antenna’s design parameters andin the relative position of the devices on the Power Transfer Efficiency (PTE) ofthe system has been assessed by combining the Stochastic Testing (ST) methodwith an efficient model for the interaction between devices. The results arefound to be accurate and the approach efficient.

7.2 Future researchThe results shown in this manuscript are not definitive and, even though severalissues were addressed in this work, many improvements can still be made.Moreover, as in the future new devices will be designed and new applicationsenvisioned, their robustness with respect to production and environmental fac-tors will require constant attention and novel dedicated strategies for variabilityanalysis will have to be developed.

The proposed approach for the analysis of the impact of stochastic uncertaintieson the performance of textile antennas and systems in which they are deployedis based on the combination of commercial solvers, such as ADS Momentum,or dedicated models, with SCMs. As a result, the simulation time requiredfor a single analysis is drastically reduced with respect to other traditionaltechniques, such as the Monte Carlo method. Moreover, the implementationof the software that performs the analysis is comparatively straightforward.However, the measurements campaigns that were carried out to estimate thePDFs according to which the design parameters of textile antennas undergovariations were demanding, since hundreds of measurements had to be performed.As such, they represent the bottleneck of the advocated strategies and a majorobstacle to their widespread adoption by design engineers. Therefore, it wouldcertainly be interesting to refer to or to develop numerical techniques thatwould allow to accurately reconstruct the properties of a given input PDF byrelying on a significantly lower number of measurements. In addition, the samemethods could be applied to select a limited amount of prototypes that, oncemanufactured and measured, could provide an insight on the real output PDFsof the antennas’ figures of merit.

7.2. Future research 113

In the last few years, the implementation of Substrate Integrated Waveguide(SIW) components on textile substrates has introduced a plethora of newpossibilities in terms of antenna design. Being robust, compact and high-performance, SIW antennas are going to play an important role in futurewearable systems. Conventionally, the design of such antennas relies on thecavity-backed topology. More specifically, this cavity is realized by rows ofconductive cylinders or vias that connect two conductive plates deployed on theopposite sides of a textile substrate. Therefore, uncertainties in the positionof those vias and the ever-present inhomogeneity in the applied materials mayresult in a suboptimal performance of the antennas. However, SIW componentscan only be accurately simulated by relying on 3D full-wave solvers, whichrequire several minutes to process a single realization. Moreover, the number ofrandom variables (RVs) to account for is high, as the position of tens of viashas to be considered. As a result, the extension of the procedure developedfor textile patch antennas to SIW components is not trivial, as even the mostefficient SCM turns out to be uneconomical. Moreover, the use of 3D solversmay introduce interpolation problems due to the re-meshing of the structureperformed before every simulation. Therefore, a new strategy has to be outlined.

As the number of communication devices is bound to exponentially increasein the next years, designers are turning their attention to the unlicensed 60GHz band to meet the future demands of wireless data traffic. The designand the implementation of textile antennas at this frequency need to accountfor additional effects that are negligible at lower frequencies, such as highersubstrate losses, the roughness of e-textiles and the impossibility of applying lasercutting to them with the required high-precision. Such issues have been partiallycounteracted by replacing e-textiles with copper foils. However, the efficiencyof the antenna is still compromised by the high losses of textile substrates.In addition, the benefits that the use of textiles yields at lower frequencies,such as flexibility, improved comfort for the user and easy integration of theantenna into everyday clothing, can hardly be considered an added value at60 GHz, where the dimensions of the antennas are very small. Therefore, theapplication of low-loss rigid laminates becomes more suitable at high frequencies.Nevertheless, the presence of production tolerances remains an important issueeven for antennas manufactured on rigid substrates. Moreover, the full systeminto which antennas and components will be integrated is still made of textile.Therefore, the assessment of the reliability of devices and the characterization ofmaterials at high frequencies are aspects that are certainly worth investigating.

Finally, the WPT analysis carried out in this thesis must be considered onlyas a preliminary result, since it only considers a static system in which onetransmitter and one receiver are deployed. The next logical steps range fromaddressing a more complex system involving more than two communicationunits to the analysis of a dynamic power transfer scenario. This could beparticularly interesting for a potential future implementation of WPT strategiesfor sensors and actuators positioned on moving targets and users. In addition,

114 Chapter 7. Conclusions

the results could be extended to the analysis of systems that focus on datatransmission.

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