perturbation approach to reconstructions of boundary ...893763/fulltext01.pdf · fram metoder f¨or...

Perturbation approach to reconstructions of boundary

deformations in waveguide structures

MARIANA DALARSSON

Doctoral Thesis

Stockholm, Sweden 2016

TRITA-EE 2015:109ISSN 1653-5146ISBN 978-91-7595-801-9

Elektroteknisk teori och konstruktionOsquldas vag 6

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlaggestill offentlig granskning for avlaggande av teknologie doktorsexamen fredagen den12:e februari 2016 klockan 14.00 i sal F3, Lindstedtsvagen 26, Kungl Tekniskahogskolan, Valhallavagen 79, Stockholm.

c© Mariana Dalarsson, February 2016

Tryck: Universitetsservice US AB

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Abstract

In this thesis we develop inverse scattering algorithms towards the ultimategoal of online diagnostic methods. The aim is to detect structural changesinside power transformers and other major power grid components, like gen-erators, shunt reactors etc. Power grid components, such as large powertransformers, are not readily available from the manufacturers as standarddesigns. They are generally optimized for specific functions at a specific po-sition in the power grid. Their replacement is very costly and takes a longtime.

Online methods for the diagnostics of adverse changes of the mechanicalstructure and the integrity of the dielectric insulation in power transformersand other power grid components, are therefore essential for the continuousoperation of a power grid. Efficient online diagnostic methods can provide areal-time monitoring of mechanical structures and dielectric insulation in theactive parts of power grid components. Microwave scattering is a candidatethat may detect these early adverse changes of the mechanical structure orthe dielectric insulation. Upon early detection, proper actions to avoid fail-ure or, if necessary, to prepare for the timely replacement of the damagedcomponent can be taken. The existing diagnostic methods lack the abilityto provide online reliable information about adverse changes inside the activeparts. More details about the existing diagnostic methods, both online andoffline, and their limitations can be found in the licentiate thesis precedingthe present PhD thesis.

We use microwave scattering together with the inverse scattering algo-rithms, developed in the present work, to reconstruct the shapes of adversemechanical structure changes. We model the propagation environment as awaveguide, in which measurement data can be obtained only at two ends(ports). Since we want to detect the onset of some deformation, that onlyslightly alters the scattering situation (weak scattering), we have linearizedthe inverse problem with good results. We have calculated the scattering pa-rameters of the waveguide in the first-order perturbation, where they havelinear dependencies on the continuous deformation function. A linearizedinverse problem with a weak scattering assumption typically results in anill-conditioned linear equation system. This is handled using Tikhonov regu-larization, with the L-curve method for tuning regularization parameters.

We show that localized one-dimensional and two-dimensional shape de-formations, for rectangular and coaxial waveguide models, are efficiently re-constructed using the inverse scattering algorithms developed from the firstprinciples, i.e. Maxwell’s theory of electromagnetism. An excellent agree-ment is obtained between the reconstructed and actual deformation shapesfor a number of studied cases. These results show that it is possible to usethe inverse algorithms, developed in the present thesis, as a theoretical basisfor the design of a future diagnostic device. As a part of the future work,it remains to experimentally verify the results obtained so far, as well as tofurther study the theoretical limitations posed by linearization (first-orderperturbation theory) and by the assumption of the continuity of the metallicwaveguide boundaries and their deformations.

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Sammanfattning

I denna avhandling har vi utvecklat inversa spridningsalgoritmer i syfte att tafram metoder for onlinediagnostik av strukturforandringar inom krafttrans-formatorer och andra storskaliga komponenter inom det elektriska kraftnatetsom generatorer, shuntreaktorer etc. Dessa komponenter ar i regel inte till-gangliga som standardprodukter fran tillverkarna, utan ar istallet optimeradefor specifika funktioner pa specifika platser i kraftnatet, och att ersatta demar darfor bade dyrt och tidskravande.

Onlinemetoder for diagnostik av skadliga forandringar i den mekaniskastrukturen samt de dielektriska egenskaperna hos isolationsmaterialet, i kraft-transformatorer och andra stora kraftnatskomponenter, ar darfor vasentligafor optimal drift och underhall av kraftnatet. Effektiva metoder for online-diagnostik kan mojliggora realtidsovervakning av de mekaniska strukturernaoch isolationen i de aktiva delarna av kraftnatskomponenten sa att potentielltskadliga forandringar kan upptackas i ett tidigt skede. I sa fall kan man vidtalampliga atgarder for att undvika ett haveri eller (om det ar nodvandigt) kanman forbereda ett kontrollerat byte av den skadade komponenten i god tid tilllagre totalkostnad. De existerande diagnostiska metoderna, saval online somoffline, saknar formagan att ge fullt tillforlitlig information om begynnandeskadliga forandringar inom t.ex. krafttransformatorerna som ar i drift. En merdetaljerad beskrivning av existerande diagnostiska metoder, saval online somoffline, och deras begransningar finns i den tidigare framlagda licentiatavhan-dlingen som forsvarats innan den nu aktuella doktorsavhandlingen.

Vi betraktar spridningsmiljon som en vagledare, dar matningar av sprid-ningsparametrarna endast kan goras i tva andar (portar). Da vi onskar upptac-ka begynnande deformationer som medfor relativt sma andringar av spridnings-situationen (svag spridning) kan det inversa problemet linjariseras. Vagledarensspridningsparametrar far da ett linjart beroende av den kontinuerliga defor-mationsfunktionen. Ett linjart inverst problem som beskriver svag spridningresulterar ofta i ett illa konditionerat linjart ekvationssystem. Detta hanterasmed hjalp av Tikhonovregularisering med L-kurvametoden for att finjusteraregulariseringsparametrarna.

Vi aterskapar bade endimensionella och tvadimensionella lokaliseradedeformationer med god noggrannhet, i rektangulara och koaxiella vagledar-modeller. Till detta anvander vi inversa spridningsalgoritmer som utvecklatsfran grundprinciperna, d.v.s. Maxwells elektromagnetiska teori. En utmarktoverensstammelse mellan aterskapade och faktiska deformationer for ett antalstuderade fall redovisas. Dessa resultat visar att det ar mojligt att anvandade inversa algoritmerna, som utvecklats inom ramen for detta arbete, som enteoretisk bas for konstruktion av framtida diagnostiska verktyg. Som delarav framtida arbeten, aterstar att verifera vara resultat experimentellt, samtatt vidare studera de teoretiska begransningarna som hanger samman medlinjariseringen, samt antagandet om att vagledarens metalliska gransytor arkontinuerliga.

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Acknowledgments

It is my pleasure to express my gratitude to a number of people without whomthis work would not be possible. First of all, I would like to express my sinceregratitude to my supervisor, professor Martin Norgren, for providing the possibil-ity for me to first do my MSc thesis and subsequently to join the department ofElectromagnetic Engineering at the KTH Royal Institute of Technology as a PhDstudent. Thank you for your guidance, your feedback on articles and theses, andfor supporting me to not only develop as a researcher but also as a teacher inelectromagnetic theory.

I would also like to thank the Swedish Energy Agency, who funded the first halfof my PhD project through Project Nr 34146-1. My work has also been part ofEIT/KIC InnoEnergy through the CIPOWER innovation project.

I gratefully acknowledge professor Lars Jonsson as a reviewer of my PhD thesis,as well as Dr. Hans Edin as a reviewer of my preceding licentiate thesis. Thanks arealso due to my former fellow PhD student and co-author Dr. Alireza Motevasselian,as well as my co-author Mr. Seyed Mohamad Hadi Emadi.

I am generally grateful to all the colleagues at the School of Electrical Engi-neering, with whom I have a good fortune to interact. I would like to especiallythank professor Lars Jonsson, Dr. Daniel Mansson and Dr. Nathaniel Taylor formany interesting discussions related to the academic life, and the department headprofessor Rajeev Thottappillil for his administrative and strategic support. I alsoappreciated the support of Ms. Carin Norberg in financial administration and thesupport of Mr. Peter Lonn in maintaining computers and software. I would alsolike to thank several former and current PhD students at the department for theircompanionship. These include Dr. Shuai Zhang, Dr. Johanna Rosenlind, FatemehGhasemifard, Andrey Osipov, Christos Kolitsidas, Elena Kubyshkina, Shuai Shi,Kun Zhao, Mahsa Ebhrahimpouri, Lipeng Liu, Mengni Long, Patrick Janus, JanneNilsson, among many others.

Although it is not possible to mention them all, I would also like to expressmy sincere gratitude to a number of people from other departments of KTH Roy-al Institute of Technology, who have to various extent contributed to the successof my education and subsequent research. A special mention here is to associateprofessor emeritus Karim Daho, who fueled the very start of my academic careerby encouraging me to become a teaching assistant in mathematics. Together withother faculty and PhD students at the Department of Mathematics, he taught mevaluable skills that I could use in later research and teaching efforts. I believe thatthis quote by Henry Adams holds very true here: “A teacher affects eternity! Hecan never tell where his influence ends!”

I would also like to mention several people at the Alfven Laboratory. Thanksare due to professor Goran Marklund and professor emeritus Nils Brenning formany interesting lunch discussions, as well as to professor Jan Scheffel for valulable

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input on teaching. I am also grateful to Petter Strom, Richard Fridstrom, ArminWeckmann, Estera Stefanikova, Alvaro Garcia and Emmi Tholerus for adopting meon movie nights and other activities!

My time as a PhD student would not have been the same without the inter-actions with students in my electromagnetic theory classes. I would therefore liketo thank the students at Engineering Physics at KTH for their polite nature andmany bright questions! It made the long teaching hours pass in a breeze, and I wasalways able to leave the classroom with a smile on my face. I believe in you, andthat you will become the next technological leaders in Sweden in both the academiaand the industry.

I deeply appreciate having my invaluable friends outside the university includingRebecka, Anna, Maria, Mengxi, Olga, Johan, Kalle, Alex, Jakob, Johannes, Carl,Mattias, Oliver and Anna.

Last but not the least, I would like to thank my parents and siblings, whomotivated me to get my education and supported me in my achievements, and mydear husband Henrik, for always being by my side and brightening my day.

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List of papersThe thesis is based on the following peer-reviewed international journalpapers:

1. M. Dalarsson, A. Motevasselian and M. Norgren, “Online power transformer

diagnostics using multiple modes of microwave radiation to reconstruct wind-

ing conductor locations”, Inverse Problems in Science and Engineering, Vol.21, 2013, DOI 10.1080/17415977.2013.827182.

2. M. Dalarsson, A. Motevasselian and M. Norgren, “Using multiple modes to

reconstruct conductor locations in a cylindrical model of a power transformer

winding”, International Journal of Applied Electromagnetics and Mechanics,Vol. 41, No. 3, 2013, DOI 10.3233/JAE-121612.

3. M. Dalarsson and M. Norgren, “First-order perturbation approach to trans-

former winding deformations”, Progress In Electromagnetics Research Let-ters, Vol. 43, 2013, DOI 10.2528/PIERL13072307.

4. M. Dalarsson, S. M. H. Emadi and M. Norgren, “Perturbation approach to

reconstructing deformations in a coaxial cylindrical waveguide”, MathematicalProblems in Engineering, Vol. 2015, Article ID 915497, 2015,DOI 10.1155/2015/915497.

5. M. Dalarsson and M. Norgren, “Two-dimensional boundary shape reconstruc-

tions in rectangular and coaxial waveguides”, submitted to Wave Motion onDecember 14 2015.

The theory behind papers 1-3 has been presented in detail in the licentiate thesispreceding this PhD thesis:

• M. Dalarsson, “Online power transformer diagnostics using multiple modes ofmicrowave radiation”, KTH Royal Institute of Technology, Stockholm,Sweden, 2013.

No detailed presentation of this theory will therefore be given in the present thesis.For the interested reader, a download link to the licentiate thesis is provided inAppendix A.

The author’s contribution:I performed the main part of the work in the papers included in this thesis.

Martin Norgren suggested the topic and provided the initial theoretical basis forthe thesis. Thereafter, Martin Norgren proposed a number of valuable commentsand improvements on all the papers. Alireza Motevasselian provided the syntheticmeasurement data used in Papers 1 - 2. Seyed Mohamad Hadi Emadi provided thesynthetic measurement data used in Paper 4.

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Parts of the thesis work have also been presented at the following peer-reviewed international conferences:

6. M. Dalarsson, A. Motevasselian and M. Norgren (invited),“On using multiple

modes to reconstruct conductor locations in a power transformer winding”,PIERS 2012 Proceedings, Kuala Lumpur, Malaysia, March 27-30, pp. 516-523, 2012.

7. M. Dalarsson, A. Motevasselian and M. Norgren (invited), “On-line power

transformer diagnostics using multiple modes of microwave radiation to re-

construct winding conductor locations”, Sixth International Conference on In-verse Problems: Modeling and Simulation, May 21-26, 2012, Antalya, Turkey.

8. M. Norgren and M. Dalarsson, “Reconstruction of boundary perturbations in

a waveguide”, URSI-EMTS 2013 Proceedings, Hiroshima, Japan, May 20-24,pp. 934-937, 2013.

9. (*) M. Dalarsson and M. Norgren, ”First-order perturbation approach to ellip-

tic winding deformations”, URSI-EMTS 2013 Proceedings, Hiroshima, Japan,May 20-24, 2013.

10. M. Dalarsson and M. Norgren (invited), “Conductor Locations Reconstruc-

tion in a Cylindrical Winding Model”, PIERS 2013 Proceedings, Stockholm,Sweden, August 12-15, 2013.

11. M. Dalarsson, S. M. H. Emadi and M. Norgren,“Reconstruction of Continuous

Mechanical Deformations in Power Transformer Windings”, Proceedings ofICIPE 2014, May 12-15 2014, Krakow, Poland.

12. (*) M. Dalarsson, S. M. H. Emadi and M. Norgren,“Reconstruction of Contin-

uous Deformations in a Coaxial Cylindrical Waveguide using Tikhonov Reg-

ularization”, Proceedings of URSI-GASS 2014, August 16-23 2014, Beijing,China.

13. M. Dalarsson and M. Norgren, “Inverse scattering of two-dimensional bound-

ary deformations in waveguide structures”, Radio and Antenna Days of the In-dian Ocean 2015 Proceedings, Belle Mare, Mauritius, September 21-24, 2015,DOI 10.1109/RADIO.2015.7323404.

Parts of the thesis work have also been presented at the following work-shops:

14. M. Dalarsson, A. Motevasselian and M. Norgren (invited), “On using multiple

modes to reconstruct conductor locations in a power transformer winding”,AntennEMB, March 6-8, 2012, Frosundavik, Sweden.

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15. M. Dalarsson and M. Norgren, “Reconstruction of Conductor Locations in

a Power Transformer Winding”, Modelling, Design and Monitoring of PowerTransformers, Workshop KTH, ABB and Indian Institute of Technology, June5, 2012, Vasteras.

16. M. Dalarsson and M. Norgren, “Using Multiple Modes to Reconstruct Con-

ductor Locations in a Cylindrical Model of a Power Transformer Winding”,Workshop on Mathematical Modelling of Wave Phenomena with applicationsin the power industry, April 23-24 2013, Linnaeus University, Vaxjo.

17. M. Dalarsson and M. Norgren (invited), “Reconstruction of boundary pertur-

bations in a coaxial model of a transformer winding”, AntennEMB, March11-12, 2014, Gothenburg, Sweden.

Comments:

• (*) The author received the “YSA - Young Scientist Award”awarded todistinguished young scientists, for these conference papers.

Other publications by the author (not related to the thesis topic):

18. M. Dalarsson and P. Tassin,“Analytical Solution for Wave Propagation Through

a Graded Index Interface Between a Right-Handed and a Left-Handed Mate-

rial”, Optics Express, Volume 17, Issue 8, pp. 6747-6752, 2009.

19. M. Dalarsson, Z. Jaksic and P. Tassin, “Exact Analytical Solution for Oblique

Incidence on a Graded Index Interface Between a Right-Handed and a Left-

Handed Material”, Journal of Optoelectronics and Biomedical Materials, Vol-ume 1, Issue 4, pp. 345-352, 2009.

20. M. Dalarsson, Z. Jaksic and P. Tassin, “Structures Containing Left-Handed

Metamaterials with Refractive Index Gradient: Exact Analytical Versus Nu-

merical Treatment”, Microwave Review, Volume 15, Number 2, pp. 2-5, 2009.

21. M. Dalarsson and M. Norgren, “Exact Solution for Lossy Wave Transmission

through Graded Interfaces between RHM and LHM Media”, Proceedings ofthe Metamaterials 2010 Conference, Karlsruhe, Germany, September 12-16,pp. 854-856, 2010.

22. M. Dalarsson, M. Norgren and Z. Jaksic, “Lossy gradient index metamateri-

al with sinusoidal periodicity of refractive index: case of constant impedance

throughout the structure”, Journal of Nanophotonics, Volume 5, Issue 1, pp.051804-, 2011.

23. M. Dalarsson, M. Norgren and Z. Jaksic, “Lossy Wave Propagation through a

Graded Interface to a Negative Index Material - Case of Constant Impedance”,Microwave Review, Volume 17, Number 2, pp. 2-6, 2011.

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24. M. Dalarsson, M. Norgren, N. Doncov and Z. Jaksic, “Lossy gradient index

transmission optics with arbitrary periodic permittivity and permeability and

constant impedance throughout the structure”, Journal of Optics, Volume 14,Issue 6, pp. 065102-, 2012.

25. M. Dalarsson, M. Norgren and Z. Jaksic, “Lossy Gradient Index Metamaterial

with General Periodic Permeability and Permittivity: The Case of Constant

Impedance throughout the Structure”, Proceedings of the PIERS 2012 Confer-ence, Kuala Lumpur, Malaysia, March 27-30, pp. 190-194, 2012.

26. M. Dalarsson, M. Norgren, T. Asenov and N. Doncov, “Gradient Index Meta-

material with Arbitrary Loss Factors in RHM and LHM Media: The Case

of Constant Impedance throughout the Structure”, Proceedings of the PIERS2012 Conference, Moscow, Russia, August 19-23, pp. 1390-1394, 2012.

27. M. Dalarsson, M. Norgren, and Z. Jaksic, “Lossy Wave Transmission Through

Graded Interfaces Between RHM and LHM Media - Case of different Loss

Factors in the two Media”, Proceedings of the Metamaterials 2012 Conference,Saint Petersburg, Russia, September 17-22, 2012.

28. M. Dalarsson, M. Norgren, T. Asenov and N. Doncov, “Arbitrary loss fac-

tors in the wave propagation between RHM and LHM media with constant

impedance throughout the structure”, Progress In Electromagnetics ResearchLetters, Volume 137, pp. 527-538, 2013.

29. M. Dalarsson, M. Norgren, T. Asenov, N. Doncov and Z. Jaksic,“Exact analyt-

ical solution for fields in gradient index metamaterials with different loss fac-

tors in negative and positive refractive index segments”, Journal of Nanopho-tonics, Volume 7, Issue 1, pp. 073086-, 2013.

Contents

Contents xi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation for the project . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Aim of the present thesis . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Papers 1 - 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Thesis disposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Direct EM scattering problem 72.1 2D deformations in a rectangular waveguide . . . . . . . . . . . . . . 72.2 Field theory of coaxial waveguides . . . . . . . . . . . . . . . . . . . 112.3 2D deformations in a thin coaxial waveguide . . . . . . . . . . . . . . 17

2.4 1D deformations in a thin coaxial waveguide . . . . . . . . . . . . . . 28

3 Inverse EM scattering problem 31

3.1 Theory of regularization . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 The L-curve method . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Discretization of the 1D inverse problems . . . . . . . . . . . . . . . 333.4 Discretization of the 2D inverse problems . . . . . . . . . . . . . . . 35

3.5 Regularization approach in the 1D case . . . . . . . . . . . . . . . . 373.6 Regularization approach in the 2D case . . . . . . . . . . . . . . . . 39

4 Results 434.1 Paper 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Paper 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Conclusions and future work 455.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 49

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xii Contents

A Licentiate thesis 53

Chapter 1

Introduction

1.1 Background

The electric power grid consists of a number of components like power generators,power transformers, power switchyards (substations) as well as of power cablesand/or transmission lines. Introduction of smart electric grids and increased useof renewable energy sources, increases the need for novel and more accurate on-line diagnostic methods. Service outages for critical power-grid components can bereduced using diagnostic tests designed to detect internal damages. A more exten-sive discussion of these aspects and a review of possible degradation mechanismsrelated to power transformers can be found in the licentiate thesis preceding thisPhD thesis [1].

Most of the existing diagnostic methods are offline methods, such as e.g. fre-quency response analysis (FRA) [2] and dielectric spectroscopy [3]. Offline methodshave a major disadvantage since they involve a non-service stress of a componentand financial loss of revenue during the tests. Some theoretical proposals how tomigrate from offline to online winding deformation diagnostics using FRA includee.g. [4]. However, the FRA method, even applied online, is a differential methodmeasuring the frequency response of the entire machine, and therefore cannot pro-vide a tomographic image of the winding deformations.

Online methods that are currently being used by power utilities include partialdischarge (PD) diagnostics applied by means of external measurement equipment,where the sparks from partial discharges are detected using acoustic techniques [5].The “acoustic triangulation” method for fault localization utilizes pressure sensorsmounted outside the tank [6]. Another online method is Dissolved Gas in oil Anal-ysis (DGA), where absolute and relative amounts of dissolved gases in an oil samplecan be used to identify an on-going fault [7]. However, DGA is an integral methodand it does not provide a reasonable prediction of the remaining lifetime of thetransformer [8]. The abovementioned techniques can also be used for other powergrid components like power generators, as described in [9]. A diagnostic method

1

2 CHAPTER 1. INTRODUCTION

for power generator windings quantifying moisture absorption was proposed in [10].Testing methods and instrumentation designed for electrical and mechanical diag-nosis of rotating as well as coil-wound machinery in power generators is generallytermed Electrical Motor Diagnostics (EMD) [11]. In summary however, there arecurrently no diagnostic methods that give a reliable real-time diagnosis of the ac-tual state of the internal structure of critical power grid components (transformers,generators, shunt reactors etc.).

There exist many practical investigations in the literature about solving appliedinverse problems by using microwave scattering, with the objective to reconstructthe shapes of perfect electrically conducting (PEC) objects from observations ofthe scattered electromagnetic fields. Some applications where this is of interest,include ground penetrating radar [12] as well as non-destructive testing and eval-uation [13]. However, to the best of our knowledge, no diagnostic tools for powergrid components, such as power transformers or generators, based on microwavescattering exist on the market today.

Early attempt at microwave diagnostics

One early numerical study of transformer diagnostics using microwave scatteringcan be found in [14]. In this conference paper, a radar antenna operating at 9.5GHz and located in the wall of the tank, has been used to detect axial displace-ments of two windings, modeled as solid metal cylinders. The paper is essentially aparametric study of the dependence of a scattering parameter (S11) on the relativeaxial position of the two windings, as well as on the axial position of the measuringantenna. However, this study does not provide a general framework for detection ofdeformations of transformer windings in both radial and axial directions, althoughthis issue is mentioned as a possibility in the conclusions. Furthermore, the methoddescribed in [14] does not provide any automated method for obtaining the axialpositions of the windings, but is based on manual comparisons with pre-calibratedscattering data. Although our work is by no means based on or related to this study,it can be considered a major generalization and extension of the ideas mentionedbut not investigated in [14].

Detection of mechanical deformations in power transformers

The first step towards an online diagnostic method for detection of mechanical de-formations in power transformer winding structures, using microwave scattering andan inverse problem approach, was made in our paper [15]. The power transformerenvironment was modeled as a waveguide with a set of conducting obstacles, repre-senting segments of the transformer winding. Using microwave scattering, changesin the winding geometry could be detected and the positions of the displaced wind-ing segments reconstructed. The results of this method of reconstructing the po-sitions of individual conductors have been reported in our papers [16–18] and mylicentiate thesis [1].

1.2. MOTIVATION FOR THE PROJECT 3

Techniques to handle both surface impedance and shapereconstruction

The reconstruction of an inhomogeneous surface impedance of a two-dimensionalcylindrical scatterer located above a PEC plane is considered in [19]. Such a methodcan be used to reconstruct not only the shape but also the surface impedance ofthe scatterer. A reconstruction method in which an inaccessible PEC scatterer ismodeled as a surface impedance was developed in [20], with the aim to reconstructthe shape of the scatterer from far field measurements.

Field theory of waveguides with varying cross section

The analysis of waveguides with continuously varying cross section is closely relatedto this project, since radial variations of the cross section are indicators of boundarydeformations. For the TEM-mode, equivalent circuit models as T- or Π-networkscan represent waveguides with abruptly changing radii [21]. Waveguides of genericshapes, undergoing continuous changes in the axial direction only, can also be ap-proximated as a cascade of invariant sections and treated semi-analytically usingthe mode-matching technique [22]. In other words, it is possible to find a num-ber of publications concerned with the subject of non-uniform waveguides in theliterature, but most of them employ various discrete methods. A few publicationsdo present the proper continuous solutions for some special cases, often inspiredby analogies with similar quantum-mechanical models (either exact or using theWentzel-Kramers-Brillouin (WKB) approximation) [23]. However, a proper theorywhere waveguide boundaries can be modeled as arbitrary continuous functions, inthe context of an inverse problem approach to microwave shape reconstruction, hasnot been reported in the literature prior to the present study.

1.2 Motivation for the project

The main motivation for the project is to investigate the possibility of the develop-ment of online diagnostic tools that may enable the reconstruction of objects thatare not directly accessible for visual inspection or direct measurements. The resultsfrom the present thesis can be used in a new online diagnostic method for powergrid components such as e.g. power transformers, with the potential to detect theeffects of various deterioration processes in a more accurate way, with less risks ofadverse consequences of the measurement process for the normal operation.

Microwave measurements, up to a few GHz, is appropriate for online diagnosticssince it has wavelengths comparable to the dimensions of the mechanical structuresand potential adverse deformations in the active parts of e.g. power transformers.The analysis of measured microwave signals and their relations to the structureparameters, being the signatures of mechanical deformations, is an inverse electro-magnetic problem. The idea can be practically realized by inserting antennas inside

4 CHAPTER 1. INTRODUCTION

the components. They can be placed above and below the metallic surface to bediagnosed, and measure the resulting scattering parameters.

The main advantage of the diagnostic method described here, is that it does notrequire the disconnection of a monitored component from the power grid. Comparedto some existing online methods, our method proposed does not require connectionof test voltages to high-voltage terminals of the monitored component. In otherwords, the test signals (high-frequency microwaves) are completely independentof any low-frequency signals applied to the monitored component in the courseof its regular operation. Furthermore, our diagnostic method can actually give asnapshot of the present state of the power grid component, i.e. it can detect theactual deformations measured in suitable length units (e.g. mm) at the very instantwhen the measurement is performed.

1.3 Aim of the present thesis

The aim of the present thesis work is to expand the knowledge on inverse problemsbased on electromagnetic waves propagating within waveguide structures. In prac-tical diagnostic situations, we often want to characterize objects in their workingenvironments that are often far from optimal for solving the inverse problems. Onetypical situation is that one cannot generate input signals and measure responsesignals in the vicinity of the object. Instead, the measurement can only be carriedout at certain locations (ports), while the object itself remains “hidden” inside thestructure, as e.g. the winding structure of a power transformer. On their wayback and forth to the object, the microwaves undergo scattering and diffractiondue to other obstacles than the studied object itself. However, for many diagnosticpurposes it is justified to neglect some properties of the complex actual structureand to model the system as an object inserted into a waveguide, where we havea well-defined electromagnetic field pattern. As a generic model of the propaga-tion environment, we therefore use a waveguide, where measurement data can beobtained only in a limited number of regions.

In a waveguide, there may be several different kinds of objects and propertiesto reconstruct. One can reconstruct e.g.

1. local deformations in the wall geometry, like indentations and extrusions.

2. dielectric and/or magnetic properties of bulk obstacles inside the structure.

3. local changes of surface impedances (material properties of boundary walls).

4. cracks and appearance/disappearance of ducts in the waveguide walls.

Our main focus in the present thesis is on the reconstructions of the local deforma-tions in the wall geometry, like indentations and extrusions.

1.4. PAPERS 1 - 3 5

1.4 Papers 1 - 3

The underlying theory behind papers 1-3 [16–18] can be found thoroughly presentedin the licentiate thesis [1] preceding this PhD thesis. A download link to [1] isprovided in Appendix A. Therefore, no detailed presentation of this theory willbe given in the present thesis, and an interested reader is referred to [1] for moreinformation.

1.5 Thesis disposition

Chapter 1 starts with an overview of the aim of the thesis, followed by a survey ofthe field and a brief summary of papers 1-3. For papers 4 and 5 [24,25], a more de-tailed theoretical description is given in Chapters 2 and 3. Chapter 2 describes thedirect electromagnetic (EM) scattering problem at hand. It contains the detailedperturbation theory for one-dimensional deformations in the thin coaxial waveg-uide, as well as the detailed perturbation theory for two-dimensional deformationsin rectangular and thin coaxial waveguides. Chapter 3 describes the inverse electro-magnetic (EM) scattering problem. The chapter starts with the background theoryof Tikhonov regularization and the L-curve method used to choose the regular-ization parameters, and then proceeds to describe the details of choosing penaltyterms in the regularization, the discretization of the problem in the 1D and 2Dcase and the resulting minimization problem. Thereafter, Chapter 4 consists ofa short overview and discussion of the main results of papers 1-5. Note that allgraphical or tabular reconstruction results for the investigated cases are presentedin the enclosed papers 1-5, and are therefore not repeated in this chapter. Finally,in Chapter 5, conclusions and short- and long term proposals for future work arediscussed.

Chapter 2

Direct EM scattering problem

We define the “direct problem”as solving for the scattered EM fields given a knownwaveguide model geometry and incident fields. On the contrary, the “inverse prob-lem”amounts to determining the waveguide model geometry given known scatteredand incident EM fields. The description of the direct problem theory is given in thischapter, while the inverse problem is described in Chapter 3. The direct problemis solved by means of first-order perturbation theory for one-dimensional deforma-tions in the thin coaxial waveguide, as well as for two-dimensional deformations inrectangular and thin coaxial waveguides. This theory is used as basis for papers 4and 5 [24, 25].

2.1 2D deformations in a rectangular waveguide

As argued in the licentiate thesis [1] preceding this PhD thesis, a realistic model ofa winding structure within a power grid component is typically based on a coaxialwaveguide, or to a certain approximation on a parallel-plate waveguide. Althougha rectangular waveguide is in general not a suitable candidate to model an inher-ently cylindrical winding structure, there is a significant mathematical similaritybetween the results obtained in a rectangular waveguide model with some of thecorresponding results obtained in a thin coaxial waveguide model. As a conceptualintroduction, it is therefore worthwhile to study the rectangular waveguide modelin some detail, prior to proceeding with a thorough study of the more realistic thincoaxial waveguide model. To this end, we consider the wave propagation withina rectangular waveguide with a cross-section shown in Fig. 2.1(a). We assume ashallow rectangular waveguide where b ≤ a/5, such that the first four leading modesof microwave propagation are the TEn-modes (TEn0-modes), where 1 ≤ n ≤ 4.Based on the standard theory of rectangular waveguides [26], it is convenient tointroduce the following basis functions

ψn(x) = 2

√

ZTE

absin(nπx

a

)

, ZTE =k η

kzn

, n = 1, 2, . . . , (2.1)

7

8 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

a

b

z

y

x

n0 = −y(a)

a

b

z

y

x bδg(x, z)

n(b)

Figure 2.1: Geometry of rectangular waveguide: (a) without boundary deformation,(b) with boundary deformation.

where k = ω√εµ , η =

√

µ/ε and kzn =√

k2 − n2π2/a2. Hereafter, ε and µ denoterespectively the effective permittivity and permeability of the medium within thewaveguide and we adopt the convention exp(jωt) for the time dependence of allthe fields. The TEn-fields within the rectangular waveguide, propagating in (±z)-direction, are then given by

E±n (r) = Pn ψn(x) exp(∓jkznz) y ,

H±n (r) =

Pn

kη[∓kznψn(x) x + jψ′

n(x) z] exp(∓jkznz) , (2.2)

where Pn is a dimensional constant, which explicit form is not essential for thepresent discussion, as they cancel out in the scattering formulae. Let us now con-sider a rectangular waveguide with a boundary perturbation as shown in Fig. 2.1(b).The continuous perturbation function can be written in the form y = bg(x, z) with|g(x, z)|max � 1 over the intervals 0 ≤ x ≤ a and 0 ≤ z ≤ d, and zero elsewhere.Inspired by the study in [27], we seek to apply first-order perturbation theory, tocalculate the first order scattered fields arising from the small deformation g(x, z).

Here we recall that, for a given operating frequency f , the electromagneticfields in any hollow waveguide are described by the infinite set of modes with theirrespective cutoff frequencies fc. Far away from any localized source of the fields, likea boundary deformation discussed here or an aperture in the waveguide boundary,the fields are relatively simple propagating modes. However, in the vicinity of adeformation or an aperture, the fields are generally superpositions of many differentmodes [26]. The fields near a localized source are typically assumed to be thesuitable expansions of the regular fields far away from the localized source. Theproblem at hand is to determine the amplitudes of the fields in the vicinity ofa localized source in terms of the amplitudes of the unperturbed fields, far away

2.1. 2D DEFORMATIONS IN A RECTANGULAR WAVEGUIDE 9

from the localized source. A boundary deformation, in the first-order perturbationtheory, can be described as an equivalent aperture in the lower boundary of therectangular waveguide with a specified magnetic current density. Such an equivalentaperture serves as a source of the perturbation fields over the intervals 0 ≤ x ≤ aand 0 ≤ z ≤ d. The electric field can be expanded into perturbation and Taylorseries around the unperturbed surface (y = 0) as follows

E(x, aδg, z) =

∞∑

p=0

∞∑

m=0

∂mEp

∂ym(x, 0, z)

(ag)m

m!δm+p . (2.3)

To the first order of perturbation, with δ = 1, the fields can be written as E(x, z) =E0(x, z)+E1(x, z), where we assume the following expression for the unperturbed(zeroth-order) fields

E0(x, z) =∑

n

{c+n [ETn(x) + Ezn(x)z]e−jkznz + c−n [ETn(x) −Ezn(x)z]e+jkznz} ,

(2.4)

H0(x, z) =∑

n

{c+n [HTn(x)+Hzn(x)z]e−jkznz +c−n [−HTn(x)+Hzn(x)z]e+jkznz} ,

(2.5)Outside the deformation region, the first-order perturbation fields are

E1(x, z) =∑

m

{d+m[ETm(x)+Ezm(x)z]e−jkzmz +d−m[ETm(x)−Ezm(x)z]e+jkzmz} ,

(2.6)

H1(x, z) =∑

m

{d+m[HTm(x)+Hzm(x)z]e−jkzmz+d−m[−HTm(x)+Hzm(x)z]e+jkzmz} ,

(2.7)where c±n are the coefficients in the mode expansion of the unperturbed (zeroth-order) fields (2.4)-(2.5), while d±m are the coefficients in the mode expansion of thefirst-order perturbation fields (2.6)-(2.7). The non-negative integers n andm are la-bels of the waveguide modes in the zeroth- and first-order perturbation expansions,respectively. It should also be noted here that the equations (2.6)-(2.7) describethe electric field generated by the deformation that exists in the interval 0 ≤ z ≤ dand are therefore valid outside the deformation region, i.e. for z ≤ 0 and z ≥ d. Onthe other hand, the equations (2.4)-(2.5) are valid in the entire studied waveguideregion. Using the excitation theorem ([26], sec. 8.12), we can write the followingexpression for the d±m coefficients

d±m =

∫

S[n0 × E1(x, 0, z)] ·H±

mdS

2∫

S[ETm × HTm] · zdS

. (2.8)


In the numerator of the theorem (2.8), we have the scattered first-order perturbationfield E1 integrated over the aperture where the deformation is located. Usingequations (2.4)-(2.7), we obtain (for detailed derivation for the analogous case of athin coaxial waveguide, see Section 2.3 below)

n0 × E1(x, 0, z) = an0 ×[

∇gn0 · E0(x, 0, z)− g∂E0

∂y(x, 0, z)

]

, (2.9)

which is used in the numerator. The denominator on the other hand, is the integralof the unperturbed (zeroth-order) modes over the waveguide cross section, and itis proportional to the normalization constant of these modes, defined by equation(2.2).

The theorem (2.8) relates the expansion coefficients d±m of the scattered, m-labeled modes arising due to the boundary perturbation, to the expansion coeffi-cients c±n of the unperturbed, n-labeled modes. Performing the integrations indi-cated in (2.8), results in the matrix equation

[

d+

d−

]

=

[

MPP MPM

MMP MMM

] [

c+

c−

]

, (2.10)

where c± and d± are vectors of coefficients c±n and d±m respectively, while MPP,MPM, MMP and MMM are so-called mode-conversion matrices. These matricesare related to the familiar two-port scattering matrices. Thus, the equation (2.8)finally yields the reflection coefficients S11

mn and S22mn in a rectangular waveguide

with a perturbation on the lower horizontal boundary as follows

S11mn =

j

2a√kznkzm

∫ a

0

dx

∫ z2

z1

dz g(x, z)e−j(kzm+kzn)z×

×{[

k2 + kxmkxn + kzmkzn

]

cos[(kxm + kxn)x]−

−[

k2 − kxmkxn + kzmkzn

]

cos[(kxm − kxn)]x}

, (2.11)

S22mn =

j

2a√kznkzm

∫ a

0

dx

∫ z2

z1

dz g(x, z)e+j(kzm+kzn)z×

×{[

k2 + kxmkxn + kzmkzn

]

cos[(kxm + kxn)x]−

−[

k2 − kxmkxn + kzmkzn

]

cos[(kxm − kxn)x]}

, (2.12)

2.2. FIELD THEORY OF COAXIAL WAVEGUIDES 11

as well as the transmission coefficients S12mn and S21

mn in a rectangular waveguidewith a perturbation on the lower horizontal boundary as follows

S12mn = δmn +

j

2a√kznkzm

∫ a

0

dx

∫ z2

z1

dz g(x, z)e−j(kzm−kzn)z×

×{[

k2 + kxmkxn − kzmkzn

]

cos(kxm + kxn)x−

−[

k2 − kxmkxn − kzmkzn

]

cos(kxm − kxn)x}

, (2.13)

S21mn = δmn +

j

2a√kznkzm

∫ a

0

dx

∫ z2

z1

dz g(x, z)e+j(kzm−kzn)z×

×{[

k2 + kxmkxn − kzmkzn

]

cos(kxm + kxn)x−

−[

k2 − kxmkxn − kzmkzn

]

cos(kxm − kxn)x}

. (2.14)

In (2.11)-(2.14) we also have kxl = lπ/a and kzl =√

k2 − k2xl with l = {m, n}. The

uppercase index indicates the port number, thus providing information whetherthese are reflection or transmission coefficients. The lowercase indices mn denotethe mode numbers of the reflection/transmission coefficients, i.e. they generallydescribe the reflection/transmission of the mode of number n into the differentmode of number m. It is important to note that waveguide scattering parametersnormally depend on the shape of the deformation function g(x, z) in a nonlinearway. However, due to the linearized (first-order perturbation) theory employed, thescattering parameters here are linear functions of g(x, z), as is evident from theequations (2.11)-(2.14).

2.2 Field theory of coaxial waveguides

We now turn to a coaxial waveguide being more suitable as a model of a cylindricalwinding structure within an active part of e.g. a power transformer. We consider aTM wave, traveling in the z-direction through a coaxial waveguide, with inner radiusRI and outer radius RO, as shown in Fig. 2.2. In the licentiate thesis [1] precedingthis PhD thesis, a detailed derivation of the expressions for the electromagneticfields in a coaxial waveguide is presented. It is therefore only briefly summarizedin this section.


r

z

y

x

RI

RO

ϕ

r

ϕz

(a)

z

y

x

RI

RO

n

n0 = −r

(b)

Figure 2.2: Geometry of a thin coaxial waveguide (a) without boundary deformationand (b) with boundary deformation.

TM waves in a coaxial waveguide

The longitudinal electric field, for TM waves propagating in ±z-direction, is givenby Eq. (3.2), with (3.18) and (3.23) in [1], i.e.

Ez(r, ϕ, z) = A

[

Jn(kTr) −Jn(kTRI)

Nn(kTRI)Nn(kTr)

]{

sin(nϕ)cos(nϕ)

}

e∓jkz·z , (2.15)

where r and ϕ are the usual radial and azimuthal cylindrical coordinate variables,such thatRI < r < RO. In (2.15), Jn(u) is the Bessel function of integer order, whileNn(u) is the Neumann function of integer order. We also define the longitudinalwave vector component kz =

√

ω2µε− k2T, and the discrete set of values for the

transverse component of the wave vector kT = kT(m,n) is obtained from the cross-product condition (3.23) in [1] (with RO = λRI and λ > 1), such that the valuesfor kT(m,n) are given by [28]

kT(m,n) = w +α

w+β − α2

w3+γ − 4αβ + 2α3

w5+ . . . , (2.16)

with [28]

w =mπ

λ− 1, α =

4n2 − 1

8λ, β =

(4n2 − 1)(4n2 − 25)(λ3 − 1)

384λ3(λ − 1),

γ =(4n2 − 1)(16n4 − 456n2 + 1073)(λ5 − 1)

5120λ5(λ− 1). (2.17)


where m and n are nonnegative integers. The transverse electric and magnetic fieldsET and HT are obtained using (A.38-A.39) from [1], with Hz = 0 (for TM-waves),such that

HT =−jωε

k2T

(

∂Ez

∂rϕ +

1

r

∂Ez

∂rr

)

, (2.18)

ET = ∓ jkz

k2T

(

∂Ez

∂rr +

1

r

∂Ez

∂ϕϕ

)

. (2.19)

where r and ϕ are the unit vectors in cylindrical coordinates. The wave impedancefor the TM-waves (ZTM) is given by

ZTM =1

ωε

√

ω2µε − k2T . (2.20)

The actual transverse components of the electric and magnetic fields, in terms ofBessel and Neumann functions, are then calculated by substituting (2.15) into (2.17)and (2.18) respectively.

TE waves in a coaxial waveguide

Analogously to the case of TM-waves above, the longitudinal magnetic field for TEwaves propagating in the ±z-direction, is given by

Hz(r, ϕ, z) = A

[

Jn(kTr) −J′n(kTRI)

N′n(kTRI)

Nn(kTr)

]{

sin(nϕ)cos(nϕ)

}

e∓jkz·z , (2.21)

where the discrete set of values for kT = kT(m,n) is now obtained from the followingcross-product condition (with RO = λRI and λ > 1),

J′n(kTRI)

N′n(kTRI)

=J′n(kTRO)

N′n(kTRO)

, (2.22)

such that the values for kT(m,n) are again given by (2.16), but with [28]

w =mπ

λ − 1, α =

4n2 + 3

8λ, β =

(16n4 + 184n2 − 63)(λ3 − 1)

384λ3(λ − 1),

γ =(64n6 + 2906n4 − 8212n2 + 1899)(λ5 − 1)

5120λ5(λ− 1). (2.23)

The transverse electric and magnetic fields are obtained using (A.38-A.39) from [1],with Ez = 0 (for TE-waves), such that

HT = ∓ jkz

k2T

(

∂Hz

∂rr +

1

r

∂Hz

∂ϕϕ

)

, (2.24)


ET =−jωµ

k2T

(

∂Hz

∂rϕ +

1

r

∂Hz

∂rr

)

. (2.25)

The wave impedance for the TE-waves (ZTE) is given by

ZTE =ωµ

√

ω2µε− k2T

. (2.26)

The actual transverse components of the electric and magnetic fields, in termsof Bessel and Neumann functions, are then calculated by substituting (2.21) into(2.24) and (2.25) respectively. It should be noted that the higher-order modesin a complete coaxial waveguide (0 ≤ ϕ ≤ 2π), both TM-modes (2.15) and TE-modes (2.21), are degenerate and there are two possible field configurations foreach kT = kT(m,n), since for each azimuthal eigenvalue (n ≥ 1) there can be eithera cos(nϕ) or a sin(nϕ)-dependence [29]. All linearly independent combinationsof cos(nϕ) and sin(nϕ) can be used as well, but {cos(nϕ), sin(nϕ)} is a suitablechoice of orthogonalized modes. Thus, the difference between the two degeneratemodes is a 90◦ rotation of the field-pattern. Unlike the rectangular waveguide,where the occurrence of degeneracy depends partly on the relation between theside lengths, the occurrence of degeneracy in the circular waveguide is independentof the radius [29].

TEM waves in a coaxial waveguide

The coaxial waveguide allows also for the propagation of TEM-modes. Here thelongitudinal components of both electric and magnetic fields vanish (Ez = 0 andHz = 0). In such a case the transverse fields, for waves propagating in ±z-direction,have the simple form

Er(r, z) =A0

re∓jkz·z , Hϕ(r, z) = ±A0

ηre∓jkz·z , η =

√

µ

ε, (2.27)

where η is the TEM wave impedance for the medium within the waveguide.

Thin coaxial waveguide model

In a number of practical diagnostic situations, it is reasonable to assume that thecoaxial waveguide is thin, in a sense that the thickness of the waveguide, defined asa = RO − RI, is small compared to the average radius of the waveguide cavityb2 = 1

2 (RI +RO), i.e.

RO − RI �RI +RO

2⇔ a� b

2. (2.28)

Under this assumption, the curvature effect on the field distribution is small. This


implies that Bessel and Neumann functions, to the first order in a/b, can be ap-proximated by trigonometric functions [30]. Thus we obtain for TM-modes

Ez(r, ϕ, z) = A sin

[

mπ(r − RI)

RO −RI

]{

sin(nϕ)cos(nϕ)

}

e∓jkzz . (2.29)

Analogously, for TE-modes we obtain

Hz(r, ϕ, z) = A cos

[

mπ(r − RI)

RO − RI

]{

sin(nϕ)cos(nϕ)

}

e∓jkzz . (2.30)

For thin coaxial waveguides it is convenient to introduce a new radial coordinatesuch that it is equal to zero on the inner boundary of the waveguide, i.e. to define adifferential radial coordinate ρ = r−RI. Using the definitions of a, b and ρ, we canrewrite the expression for the longitudinal electric field in case of the TM-modes(2.29) as follows

Ez(r, ϕ, z) = A sin(mπρ

a

)

{

sin(nϕ)cos(nϕ)

}

e∓jkzz . (2.31)

as well as the expression for the longitudinal magnetic field in case of the TE-modes(2.30) as follows

Hz(r, ϕ, z) = A cos(mπρ

a

)

{

sin(nϕ)cos(nϕ)

}

e∓jkzz . (2.32)

The wave impedance for TMmn-modes can now be rewritten as

ZTM =kz

ωε=

kz

ω√εµ

√

µ

ε=kzη

k, (2.33)

while the wave impedance for TEmn-modes becomes

ZTE =ωµ

kz

=ω√εµ

kz

=kη

kz

. (2.34)

The discrete set of values for kT = kT(m,n) is now obtained from a simple approxi-mate formula [30]

k2T(m,n) =

m2π2

(RO − RI)2+

4n2

(RO + RI)2=m2π2

a2+

4n2

b2, (2.35)

valid in the same order of approximation as the results for the longitudinal fields(2.29) and (2.30). Thus the longitudinal component of the wave vector becomes

kz(m,n) =

√

ω2µε − m2π2

a2− 4n2

b2. (2.36)


The cutoff frequencies fc, under which the modes cannot propagate, for eitherTMmn-modes or TEmn-modes, are then given by

fc(m,n) =v

2π

√

m2π2

a2+

4n2

b2, (2.37)

where v is the wave propagation speed in the isotropic medium, which in case of thefree space (ε = ε0 and µ = µ0) is equal to the speed of light v = c = 299 792 458 m/s.Hereafter, in all numerical examples, we will assume that v = c. As explained in theprevious section, the coaxial waveguide allows for propagation of a TEM-mode withm = n = 0 which can propagate at all frequencies, since its cutoff frequency is zero.Thus, the TEM-mode is a dominant mode for a coaxial waveguide. Regarding thehigher-order modes, we note that by assumption we have a � b/2, which impliesthat the second term on the right-hand side of the equation (2.35) is typically muchsmaller than the first term, i.e. 4n2/b2 � m2π2/a2, for m being of the same orderof magnitude as n. Furthermore, we note that higher-order TM-modes cannotexist for m = 0, since the longitudinal component of the electric field (2.29) isidentically equal to zero for m = 0. On the other hand, the longitudinal componentof the magnetic field (2.30) for higher-order TE-modes does not vanish for m = 0,and the higher-order TE-modes do exist for m = 0. Thus, after the dominantTEM-mode, the first few higher-order modes are TE0n-modes with approximatecutoff-frequencies

fc(0,n) =nc

πb. (2.38)

TE0n-modes (TEn-modes)

Here we consider the special case of TE0n-modes (m = 0), with the fields

Hz(ρ, ϕ) = H0 cosnϕ e∓jkz·z , (2.39)

Er =jωµb

2nH0 sinnϕ e∓jkz·z , (2.40)

Hϕ = ∓ jkzb

2nH0 sinnϕ e∓jkz·z . (2.41)

It is now convenient to introduce the following base functions

ψn(ϕ) =

√

2ZTE

πabsinnϕ =

√

2kη

πabkz

sinnϕ , (2.42)

which are normalized to ZTE, i.e.

∫ RO

RI

rdr

∫ 2π

0

dϕ ψ2n(ϕ) =

[

r2

2

]RO

RI

2ZTE

πab

∫ 2π

0

sin2(nϕ)dϕ =

2.3. 2D DEFORMATIONS IN A THIN COAXIAL WAVEGUIDE 17

1

2(R2

O −R2I ) ·

2ZTE

πab·∫ 2π

0

1 − cos 2nϕ

2dϕ = ab

ZTE

ab

1

2π

∫ 2π

0

dϕ = ZTE · 2π

2π= ZTE ,

(2.43)

where we used R2O −R2

I = (RO −RI)(RO +RI) = ab. Thus we can define the basefields as follows

ETn = Pnψn(ϕ)r , (2.44)

HTn = ∓ Pn

ZTEψn(ϕ)ϕ = ∓kzn

kηψn(ϕ)ϕ , (2.45)

Hzn =2

jωµb

∂ETn

∂ϕ= −2j

b· 1

ω√εµ

1√

µε

ψ′n(ϕ) = − 2j

bkηψ′

n(ϕ) , (2.46)

where Pn is a dimensional constant, which explicit form is not essential for thepresent discussion, as these constants cancel out in the scattering formulae.

2.3 2D deformations in a thin coaxial waveguide

Let us now consider a thin coaxial waveguide with the boundary perturbation shownin Fig. 2.2(b). The perturbation function can here be written as

ρ = r − RI = ag(ϕ, z) , |g(ϕ, z)|max � 1 , (2.47)

Introducing here the book-keeping perturbation parameter δ, to be set equal tounity at the end, we can write

ρ = aδg(ϕ, z) ⇒ aδg(ϕ, z) − ρ = 0 . (2.48)

The outwardly directed unit normal vector n on the perturbed metallic surface isgiven by

n = − ∇[aδg− ρ]

|∇[aδg− ρ]| ∝ ∇[aδg− ρ] , (2.49)

orn ∝ aδ∇g − ρ = n0 + aδ∇g , (2.50)

where n0 = −ρ is the normal unit vector on the unperturbed surface, as indicatedin Fig. 2.2(b). The boundary condition on the perturbed metallic surface is

n × E = 0 ⇒ n0 × E + aδ∇g × E = 0 . (2.51)


Let us now develop the electric field vector E(ρ, ϕ, z) into Taylor series about theunperturbed boundary (ρ = 0) as follows

E(ρ, ϕ, z) =

∞∑

m=0

∂mE

∂ρm(0, ϕ, z)

ρm

m!. (2.52)

On the perturbed surface, we have ρ = aδg, such that (2.52) becomes

E(aδg, ϕ, z) =

∞∑

m=0

∂mE

∂ρm(0, ϕ, z)

(ag)m

m!δm . (2.53)

Next, we introduce the perturbation series

E =

∞∑

p=0

Epδp , (2.54)

such that the equation (2.53) becomes

E(aδg, ϕ, z) =

∞∑

p=0

∞∑

m=0

∂mEp

∂ρm(0, ϕ, z)

(ag)m

m!δm+p . (2.55)

Substituting (2.55) into the boundary condition (2.51), we obtain

∞∑

p=0

∞∑

m=0

(ag)m

m!δm+p(n0 + aδ∇g) × ∂mEp

∂ρm(0, ϕ, z) = 0 . (2.56)

To the zeroth order in δ, this gives the unperturbed boundary condition

n0 × E0(0, ϕ, z) = 0 . (2.57)

To the first order in δ, we obtain

n0 × E1(0, ϕ, z) = −a[gn0 ×∂E0

∂ρ(0, ϕ, z) + ∇g × E0(0, ϕ, z)] . (2.58)

Using here a × (b × c) = b · (a · c) − c · (a · b), i.e.

n0 × (n0 × E0) = n0 · (n0 · E0) − E0 · (n0 · n0) , (2.59)

we can writeE0 = n0 · (n0 · E0) − n0 × (n0 × E0) . (2.60)


From (2.57), we know that n0 × E0(0, ϕ, z) = 0, such that

E0(0, ϕ, z) = n0(n0 · E0) . (2.61)

Substituting (2.61) into (2.57), we obtain

n0 ×E1(0, ϕ, z) = −a[

gn0 ×∂E0

∂ρ(0, ϕ, z) + ∇g × n0(n0 · E0(0, ϕ, z))

]

, (2.62)

or rearranging (∇g × n0 = −n0 ×∇g)

n0 × E1(0, ϕ, z) = an0 ×[

∇gn0 · E0(0, ϕ, z) − g∂E0

∂ρ(0, ϕ, z)

]

. (2.63)

In the next three subsections, we will analyze the scattering between the differentmodes of a thin coaixal waveguide. According to Section 2.1, the dominant mode inthe thin coaxial waveguide is the TEM-mode while the first few higher-order modesare TE0n-modes. Thus, the scattering processes of interest include scattering fromthe TEM mode to TEM mode, the TEM mode to the TE0n-modes and possibly theTEm-modes to TEn-modes. Each of these possible scattering situations is presentedin a dedicated subsection.

TEm-modes to TEn-modes

Let us now consider the TEn-modes with base fields

E±n (r) = E±

Tn(r) = Pnψn(ϕ) e∓jkz·zρ , (2.64)

H±n = H±

Tn(r) +H±znz =

Pn

kη

[

∓kznψn(ϕ)ϕ − 2j

bψ′

n(ϕ)z

]

e∓jkz·z , (2.65)

with

ψn(ϕ) =

√

2kη

πabkz

sinnϕ , (2.66)

where again Pn is a dimensional constant, which explicit form is not essential forthe present discussion, as these constants cancel out in the scattering formulae. Infact, it is also legitimate to set Pn = 1, since it will not affect the results of thepresent study. To the first order of perturbation, with δ = 1, the fields can bewritten as

E(ϕ, z) = E0(ϕ, z) + E1(ϕ, z) , (2.67)


H(ϕ, z) = H0(ϕ, z) + H1(ϕ, z) . (2.68)

The zeroth-order fields are

E0(ϕ, z) =∑

n

{c+n [ETn(ϕ) +Ezn(ϕ)z]e−jkznz + c−n [ETn(ϕ) − Ezn(ϕ)z]e+jkznz} ,

(2.69)

H0(ϕ, z) =∑

n

{c+n [HTn(ϕ)+Hzn(ϕ)z]e−jkznz+c−n [−HTn(ϕ)+Hzn(ϕ)z]e+jkznz} ,

(2.70)

Outside the deformation region, the first-order perturbation fields are

E1(ϕ, z) =∑

m

{d+m[ETm(ϕ)+Ezm(ϕ)z]e−jkzmz+d−m[ETm(ϕ)−Ezm(ϕ)z]e+jkzmz} ,

(2.71)

H1(ϕ, z)=∑

m

{d+m[HTm(ϕ)+Hzm(ϕ)z]e−jkzmz+d−m[−HTm(ϕ)+Hzm(ϕ)z]e+jkzmz} .

(2.72)

where c±n are the coefficients in the mode expansion of the unperturbed (zeroth-order) fields (2.69)-(2.70), while d±m are the coefficients in the mode expansion ofthe first-order perturbation fields (2.71)-(2.72). The non-negative integers n andm are labels of the waveguide modes in the zeroth- and first-order perturbationexpansions, respectively. It should also be noted here that the equations (2.71)-(2.72) describe the electric field generated by the deformation that exists in theinterval 0 ≤ z ≤ d and are therefore valid outside the deformation region, i.e. forz ≤ 0 and z ≥ d. On the other hand, the equations (2.69)-(2.70) are valid in theentire studied waveguide region.

By means of the excitation theorem for hollow PEC waveguides ([26], section8.12), we can write the following expression for the d±m coefficients

d±m =

∫

S[n0 × E1(0, ϕ, z)] · H±

mdS

2∫

S[ETm × HTm] · zdS

. (2.73)

For TEn-modes, we have (2.64),

E±0n = c±nψn(ϕ) e∓jkzn·zρ , (2.74)


such that∂E0

∂ρ(0, ϕ, z) = 0 , n0 = −ρ . (2.75)

Using (2.75) in (2.63) gives

n0 × E1(0, ϕ, z) = a(ρ ×∇g)ρ · E0(0, ϕ, z) , (2.76)

or using (2.74)

n0 × E1(0, ϕ, z) = a(ρ×∇g)c±nψn(ϕ) e∓jkzn·z . (2.77)

Let us now use the vector identity

(a × b) · c = (c × a) · b = (b × c) · a , (2.78)

to calculate(ρ ×∇g) · H±

m = (H±m × ρ) · ∇g . (2.79)

Using here (2.65), we can calculate

H±m × ρ =

1

kη[∓kzmψn(ϕ)ϕ × ρ− 2j

bψ′

m(ϕ)z × ρ] e±jkzm·z . (2.80)

Since ϕ × ρ = −z and z × ρ = ϕ, we obtain

(H±m × ρ) · ∇g = − 1

kη∇g[ 2j

bψ′

m(ϕ)ϕ ∓ kzmψn(ϕ)z] e±jkzm·z . (2.81)

Using (2.77) with (2.81), we obtain

[n0×E1(0, ϕ, z)]·H±m = −c±n

a

kηψn(ϕ)ej(±kzm∓kzn)·z

[

2j

bψ′

m(ϕ)ϕ ∓ kzmψn(ϕ)z

]

·∇g .

(2.82)On the other hand, from (2.64) and (2.65), we see that

ETm = ψm(ϕ)ρ , (2.83)

HTm = +kzm

kηψm(ϕ)ϕ , (2.84)

such that

ETm × HTm = +kzm

kη[ψm(ϕ)]2ρ× ϕ =

kzm

kη[ψm(ϕ)]2z , (2.85)


and

(ETm × HTm) · z =kzm

kη[ψm(ϕ)]2 . (2.86)

Thus we can use (2.66) to calculate

2

∫

S

(ETm × HTm) · zdS =2kzm

kη

∫ RO

RI

rdr

∫ 2π

0

2kη

πabkzm

sin2 nϕdϕ =

2ab

2· 2

πab

∫ 2π

0

1 − cos 2nϕ

2dϕ = 2 . (2.87)

In (2.82), we have a product

ψn(ϕ)ψ′m(ϕ) =

(

√

2kη

πabkzn

sinnϕ

)(

m

√

2kη

πabkzm

cosmϕ

)

=

2kη

πab· m√

kznkzm

sinnϕ cosmϕ . (2.88)

Using here the trigonometric formula

sinα cos β =1

2[sin(β + α) − sin(β − α)] , (2.89)

we can write

sinnϕ cosmϕ =1

2[sin(m+ n)ϕ− sin(m− n)ϕ] , (2.90)

and

ψn(ϕ)ψ′m(ϕ) =

kη

πab· m√

kznkzm

[sin(m+ n)ϕ− sin(m− n)ϕ] . (2.91)

In (2.82), we also have a product

ψn(ϕ)ψm(ϕ) =2kη

πab√kznkzm

sinnϕ sinmϕ . (2.92)

Using here the trigonometric formula

sinα sinβ =1

2[cos(β − α) − cos(β + α)] , (2.93)


we obtain

ψn(ϕ)ψm(ϕ) =kη

πab√kznkzm

[cos(m− n)ϕ − cos(m+ n)ϕ] . (2.94)

Substituting (2.91) and (2.94) into (2.82), we obtain

[n0 × E1(0, ϕ, z)] · H±m = −c±n

a

kηej(±kzm∓kzn)·z·

kη

πab√kznkzm

{

2jm

b[sin(m+ n)ϕ− sin(m− n)ϕ]ϕ

±kzm[cos(m+ n)ϕ − cos(m− n)ϕ]z}

∇g , (2.95)

or finally

[n0 × E1(0, ϕ, z)] · H±m = −c±n

1

πab√kznkzm

ej(±kzm∓kzn)·z·

{

2jm

b[sin(m+ n)ϕ− sin(m− n)ϕ]ϕ ± kzm[cos(m+ n)ϕ− cos(m− n)ϕ]z

}

∇g .

(2.96)Substituting (2.87) and (2.96) into (2.73) gives

d±m = −1

2c±n

1

πb√kznkzm

b

2

∫ 2π

0

dϕ

∫ z2

z1

dzej(±kzm∓kzn)·z·

{

2jm

b[sin(m+ n)ϕ − sin(m− n)ϕ] · 2

b

∂g

∂ϕ± kzm[cos(m+ n)ϕ− cos(m− n)ϕ]

∂g

∂z

}

.

(2.97)

Following a similar procedure as the one for rectangular waveguides, we finallyobtain

d±m = c±nj

4π√kznkzm

∫ 2π

0

dϕ

∫ z2

z1

dz ej(±kzm∓kzn)·zg(ϕ, z)·

{[

4m2

b2+

4mn

b2+ k2

zm ± kzm · (∓kzn)

]

cos(m+ n)ϕ−


[

4m2

b2− 4mn

b2+ k2

zm ± kzm · (∓kzn)

]

cos(m− n)ϕ

}

, (2.98)

such that

S11mn =

j

4π√kznkzm

∫ 2π

0

dϕ

∫ z2

z1

dz g(ϕ, z)e−j(kzm+kzn)·z·

{[

k2 +4mn

b2+ kzmkzn

]

cos(m+ n)ϕ−[

k2 − 4mn

b2+ kzmkzn

]

cos(m− n)ϕ

}

,

(2.99)

and

S22mn =

j

4π√kznkzm

∫ 2π

0

dϕ

∫ z2

z1

dz g(ϕ, z)ej(kzm+kzn)·z·

{[

k2 +4mn

b2+ kzmkzn

]

cos(m+ n)ϕ−[

k2 − 4mn

b2+ kzmkzn

]

cos(m− n)ϕ

}

,

(2.100)

S12mn = δmn +

j

4π√kznkzm

∫ 2π

0

dϕ

∫ z2

z1

dz g(ϕ, z)e−j(kzm−kzn)·z·

{[

k2 +4mn

b2− kzmkzn

]

cos(m+ n)ϕ−[

k2 − 4mn

b2− kzmkzn

]

cos(m− n)ϕ

}

,

(2.101)

and

S21mn = δmn +

j

4π√kznkzm

∫ 2π

0

dϕ

∫ z2

z1

dz g(ϕ, z)ej(kzm−kzn)·z·

{[

k2 +4mn

b2− kzmkzn

]

cos(m+ n)ϕ−[

k2 − 4mn

b2− kzmkzn

]

cos(m− n)ϕ

}

,

(2.102)

where, according to (2.36), we have

k2 = k2zm + k2

Tm = k2zm +

4m2

b2. (2.103)


TEM to TEm-modes

Let us now consider the case of TEm-modes generated by the lowest TEM modeexcitation, with the transverse electric field

E0(ρ, z) = ±c±0 ψ0(ρ) e∓jkzρ , (2.104)

and transverse magnetic field

H0(ρ, z) = ±c±01

ηψ0(ρ) e∓jkzϕ , (2.105)

where we put P0 = 1 for simplicity, such that

ψ0(ρ) =

√

η

2π lnλ· 1

RI + ρ=

√

η

2π lnλ

1

r· 1

r, (2.106)

with

λ =RI + a

RI= 1 +

a

RI, η =

√

µ

ε. (2.107)

Thus we can calculate [26]

n0 × E1(0, ϕ, z) = ±a(ρ ×∇g)c±0 ψ0(ρ)e∓jkz . (2.108)

Using here (2.81), we obtain

[n0 × E1(0, ϕ, z)] · H±m = ± a

kηc±0 ψ0(ρ)e

j(±kzm∓k)z·

[

2j

bψ′

m(ϕ)ϕ ∓ kzmψm(ϕ)z

]

· ∇g . (2.109)

Thus we can calculate

d±m = ± a

2kηc±0 ψ0(0)

∫ 2π

0

dϕ

∫ z2

z1

dzej(±kzm∓k)·z·

[

4j

b2ψ′

m(ϕ)∂g

∂ϕ∓ kzmψm(ϕ)

∂g

∂z

]

. (2.110)

Now by (2.106),

ψ0(0) =1√

2πlnλ· 1

RI, (2.111)


and by (2.66),

ψ′m(ϕ) = m

√

2kη

πabkz

cosmϕ , (2.112)

ψm(ϕ) =

√

2kη

πabkz

sinnϕ , (2.113)

such that (2.110) can be rewritten as follows

d±m = ±c±0λ− 1√

lnλ

1√ab · k · kzm

1

2π

∫ 2π

0

dϕ

∫ z2

z1

dz ej(±kzm∓k)·z·

[

4jm

b2cosmϕ

∂g

∂ϕ∓ kzm sinmϕ

∂g

∂z

]

, (2.114)

or upon integrating by parts

d±m = ±c±0λ − 1√

lnλ

1√ab · k · kzm

1

2π

∫ 2π

0

dϕ

∫ z2

z1

dz ej(±kzm∓k)·zg(ϕ, z) · sinmϕ·

[

4m2

b2± kzm(±kzm ∓ k)

]

, (2.115)

or finally

d±m = ±c±0λ− 1√

lnλ

1√ab · kkzm

[k2 ± kzm(∓k)]·

1

2π

∫ 2π

0

dϕ sinmϕ

∫ z2

z1

dzej(±kzm∓k)·zg(ϕ, z) , (2.116)

which gives

S11m0 = j

λ − 1√lnλ

1√ab · kkzm

[k2 + kkzm]·

a

2π

∫ 2π

0

dϕ sinmϕ

∫ z2

z1

dze−j(kzm+k)·zg(ϕ, z) , (2.117)

S22m0 = −j

λ − 1

lnλ

1√ab · kkzm

[k2 + kkzm]·


a

2π

∫ 2π

0

dϕ sinmϕ

∫ z2

z1

dze+j(kzm+k)·zg(ϕ, z) , (2.118)

S12m0 = j

λ − 1

lnλ

1√ab · kkzm

[k2 − kkzm]·

a

2π

∫ 2π

0

dϕ sinmϕ

∫ z2

z1

dze−j(kzm−k)·zg(ϕ, z) , (2.119)

S21m0 = −j

λ − 1

lnλ

1√ab · kkzm

[k2 − kkzm]·

a

2π

∫ 2π

0

dϕ sinmϕ

∫ z2

z1

dze+j(kzm−k)·zg(ϕ, z) , (2.120)

where we have used that δm0 is always zero, since m 6= 0.

TEM to TEM-modes

Finally, let us briefly consider the case of TEM-modes generated by the lowestTEM mode excitation. The scattering matrix elements S11

00 and S2200 are then a

straightforward generalization of the one-dimensional results, and read

S1100 = − jka

RI lnλ

1

2π

∫ 2π

0

dϕ

∫ z2

z1

dz g(ϕ, z)e−j2kz , (2.121)

S2200 = − jka

RI lnλ

1

2π

∫ 2π

0

dϕ

∫ z2

z1

dz g(ϕ, z)e+j2kz , (2.122)

while the scattering matrix elements S1200 and S21

00 are of the second order in theperturbation g(z),

S12 = 1 + O(max |g2|) , S21 = 1 + O(max |g2|) , (2.123)

and up to the first-order of perturbation can be considered as equal to one.


2.4 1D deformations in a thin coaxial waveguide

In a special case when a boundary deformation is axially symmetric and therebyindependent on the angular coordinate ϕ, we can consider a thin coaxial waveguidewith outer radius RO and inner radius RI as shown in Fig. 2.3. The approximateanalysis in this section is also based on the assumption (2.28). We utilize themathematical analysis pertinent to TM modes, since they include the dominantTEM mode. In the end, we limit our analysis only to the case of scattering fromTEM modes to TEM modes.

⊗

⊙

0

r

r

z

z1

z1

z2

z2

RI

RI

RO

RO

aδg(z)

aδg(z)

n

nn0

n0

Figure 2.3: Geometry of a thin coaxial waveguide with a boundary deformationalong the propagation direction of the waveguide.

The one-dimensional continuous perturbation function can be written in the form

r − RI − aδg(z) = 0 , (2.124)

with |g(z)|max � 1 over the interval z1 ≤ z ≤ z2, and zero elsewhere. Again,δ is just a book-keeping parameter to be set equal to unity (δ = 1) after theperturbation analysis. The boundary perturbation (2.124) acts as a localized sourceof the fields in an otherwise unperturbed coaxial waveguide. As before, in thefirst-order perturbation theory, the boundary deformation can be described as anequivalent localized aperture in the inner wall of the coaxial waveguide with aspecified magnetic current density. Such an equivalent aperture in the inner wallof the waveguide (r = RI) serves as source of the perturbation fields between z1and z2, and its shape is shown in Fig. 2.4. It turns out that for axially symmetricdeformations, we can reuse all the results obtained in the two-dimensional analysis


z

z1 z2

RI

dA

n0

Figure 2.4: Geometry of the aperture serving as source of the perturbation fields.

and simply integrate out the angular coordinate ϕ, using the trivial identity

1

2π

∫ 2π

0

= 1 . (2.125)

Using the more general results (2.121)-(2.123), we readily obtain the scatteringmatrix elements in the case of scattering from TEM modes to TEM modes, asfollows

S1100 = − jka

RI lnλdϕ

∫ z2

z1

dz g(z)e−j2kz , (2.126)

S2200 = − jka

RI lnλdϕ

∫ z2

z1

dz g(z)e+j2kz , (2.127)

while the scattering matrix elements S1200 and S21

00 are of the second order in theperturbation g(z),

S12 = 1 + O(max |g2|) , S21 = 1 + O(max |g2|) , (2.128)

and up to the first-order of perturbation can be considered as equal to one. Thetransmission data are therefore assumed to be too sensitive for measurement errors,and was consequently not included in the analysis. Here, as before, k = ω

√εµ is

the wavenumber of the isotropic medium.

Chapter 3

Inverse EM scattering problem

We define the “inverse problem” in the present thesis as reconstructing the waveg-uide model geometry (e.g. conductor shapes and locations), given some known scat-tered and incident EM fields. This chapter describes the theoretical background ofthe inverse problem studied in this thesis. Reconstruction results for investigatedcases are presented in the enclosed papers 1-5, and they are therefore not includedin this chapter. The reader is referred to the papers 1-5 for these results.

In papers 1-3, the inverse problem of reconstructing the position of individualconductor obstacles in a waveguide model was solved by means of an optimizationmethod. The details of this method are extensively described in the licentiate the-sis [1] preceeding this PhD thesis, and are therefore only briefly addressed here.Generally speaking, optimization methods may be classified into local or globalmethods [31]. Local methods, although effective in terms of convergence speed,generally require a “domain knowledge”, since for nonlinear and multi-minima op-timization functions the initial trial solution must lie in the so-called “attractionbasin” of the global solution to avoid the convergence solution being trapped intolocal minima (i.e. wrong solutions of the problem at hand).

In contrast, global methods with stochastic algorithms are potentially able tofind the global optimum of the functional whatever the initial point(s) of the search.Regarding the choice of the optimization method, it is important that the optimiza-tion algorithm is efficient enough such that the best possible agreement between thetheoretically obtained scattering data and corresponding measured scattering datais obtained. In other words, the optimization method must provide for a correctreconstruction of the target geometry.

In the rest of this thesis, the inverse problem is to reconstruct the shapes ofdeformations of waveguide boundaries. Since we have used first-order perturbationtheory to linearize the direct problem, the corresponding inverse problem is lin-earized as well, and can be solved by direct inversion of a linear matrix equation.Due to the ill-posedness of the problem, Tikhonov regularization [32] with the L-curve method [33] is used to regularize the inverse problem. We begin with a general

31

32 CHAPTER 3. INVERSE EM SCATTERING PROBLEM

account of Tikhonov regularization and the L-curve method, before proceeding tothe specific application of these methods in papers 4-5.

The reconstruction methods in this thesis have primarily been tested using syn-thetic measurement data generated by commercial simulation software, in order toavoid the inverse crime [34]. By inverse crime, we mean that the same theoreticalcalculation is used to synthesize as well as to invert data in an inverse problem.Such procedures appear to produce better results than they should, once actualmeasurements are used as input [34] and should therefore be avoided. Therefore, inpapers 1-3, synthetic measurement data was generated using the commerical soft-ware Ansys HFSS [35], while in papers 4-5, the synthetic measurement data wasgenerated using the software CST Microwave Studio [36].

3.1 Theory of regularization

The method of solving an ill-posed inverse problem by providing some additionalinformation is known as regularization [37]. The additional information has theform of suitable penalty terms, such as restrictions for smoothness or bounds on thevector space norm. One widely used method for regularization of ill-posed inverseproblems is Tikhonov regularization [32]. If a general matrix equation Ax = b

is ill-posed, the standard least squares method gives an over- or underdeterminedsystem of equations. Furthermore, when x is reconstructed by means of the inverseproblem approach, the least squares method puts all elements of the reconstructedversion of x that are in the null-space of A to zero. The least squares methodsminimizes the sum of squared deviations ‖Ax − b‖2

2, where ‖X‖2 is the Euclideannorm of X. In order to improve the solution of the matrix equation Ax = b, apenalty term ‖Lx‖2 can be added to the minimization target term such that weminimize the expression

‖Ax − b‖22 + λ2‖Lx‖2

2 , (3.1)

where L is referred to as the Tikhonov matrix. This matrix is often chosen as ascalar multiple of the identity matrix (L = νI), but other choices are sometimesalso desirable, depending on the application. The regularization process is a wayto condition the inverse problem at hand, and thereby to allow for an explicitnumerical solution for x, as follows

x = (AT A + λ2LT L)−1AT b . (3.2)

Regularization can be fine-tuned by varying the scale parameter λ, and for λ = 0we recover the usual least squares solution without regularization, whenever thequadratic matrix AT A is invertible.

3.2. THE L-CURVE METHOD 33

0 20 40 600

0.5

1

1.5

2

λ = 2

|| A x b ||2 = 6.8574

|| x ||2 = 4.2822

0 20 40 600

0.5

1

1.5

2

λ = 0.02

|| A x b ||2 = 0.062202

|| x ||2 = 7.9112

0 20 40 600

0.5

1

1.5

2

λ = 0.003

|| A x b ||2 = 0.060594

|| x ||2 = 8.2018

Figure 3.1: An example of oversmoothing, appropriate smoothing, and under-smoothing of a regularized inverse problem. The exact solution is plotted usingthin lines, while the Tikhonov-regularized solution is plotted using thick lines. Theillustrated three values of λ correspond to oversmoothing (left), appropriate smooth-ing (center), and under-smoothing (right). Figure reprinted with courtesy of Prof.P. C. Hansen [33].

3.2 The L-curve method

When solving inverse problems, Tikhonov regularization involves a trade-off be-tween the quality of the reconstructed solution x as compared with the quality ofthe given data, and the“size” of the stable solution [33]. An excessive regularization(damping) of a solution spoils the fit to the given data and ‖Ax− b‖2

2 remains toolarge. Very little or no regularization provides a better fit to the data, but thesolution is polluted by data errors. In such a case, ‖Lx‖2 is too large. This point isillustrated, for Tikhonov regularization, by the examples shown in Fig. 3.1. Sincethe two norms in (3.1) are very important for optimal regularization, it is customaryto plot their logarithms against each other. This typically results in an L-shapedcurve called the L-curve. An L-curve example is shown in Fig. 3.2. The L-curverepresents a tradeoff between two quantities that both need to be controlled. Sim-ilar trade-off curves can also be found in other areas of engineering. Some of theearliest applications of the L-curve include [38, 39]. Thereby, the L-curve methodis useful for illustrating the trade-off between the fit ‖Ax − b‖2

2 of a regularizedsolution to the given data set, and its size ‖Lx‖2. It is often used as an aid inchoosing an appropriate regularization parameter λ for the given data. The opti-mal λ according to the L-curve method is found at the the “corner” of the L-curve,i.e. at the point of maximal curvature defined in Eq. (18) of [33], which can easilybe identified in Fig. 3.2.

3.3 Discretization of the 1D inverse problems

Let us now consider the perturbation function g(z) as for example the one shownin Fig. 2.3. Since we require that g(z1) = g(z2) = 0, any such function can be


101

100

101

100

101

102

103

The L curve for Tikhonov regularization

Residual norm || A xλ b ||

2

So

lutio

n n

orm

|| x

λ || 2

λ = 1

λ = 0.1

λ = 0.0001

λ = 1e 005

Figure 3.2: Example of L-curve plot. Figure reprinted with courtesy of Prof. P. C.Hansen [33].

expanded into the Fourier sine series of the form

g(z) =

∞∑

n=1

gnφn(z) , φn(z) = sin

(

nπz − z1z2 − z1

)

, (3.3)

where we use the infinite set of mutually orthogonal sine functions φn(z) satisfyingthe required conditions φn(z1) = φn(z2) = 0. The coefficients gn in (3.3) are realnumbers and they are not functions of z. With reference to (2.126) and (2.127), wenow apply the k-transforms defined by

G(k) =

∫ z2

z1

g(z)ej2kzdz , G∗(k) =

∫ z2

z1

g(z)e−j2kzdz , (3.4)

3.4. DISCRETIZATION OF THE 2D INVERSE PROBLEMS 35

to both sides of the equation (3.3), to obtain

G(k) =

∞∑

n=1

gnΦn(k) , G∗(k) =

∞∑

n=1

gnΦ∗n(k) , (3.5)

where Φn(k) are obtained through explicit integration

Φn(k) =

∫ z2

z1

φn(z)ej2kzdz =nπ(z2 − z1)(e

j2kz1 − (−1)nej2kz2)

n2π2 − 4k2(z2 − z1)2. (3.6)

In order to be able to perform the inversion numerically, we approximate the contin-uous inverse problem with a discrete inverse problem, where the deformation g(z)is expanded into a finite set of functions, whereby we truncate the infinite series in(3.5) to a finite number of terms denoted by N , as follows

G(k) =

N∑

n=1

gnΦn(k) , G∗(k) =

N∑

n=1

gnΦ∗n(k) . (3.7)

Since {gn}Nn=1 are real-valued, it is convenient to treat the real and imaginary parts

of (3.7) as separate equations

GR(k) =

N∑

n=1

gnΦnR(k) , GI(k) =

N∑

n=1

gnΦnI(k) , (3.8)

where G(k) = GR(k) + jGI(k) and Φn(k) = ΦnR(k) + jΦnI(k), with GR(k), GI(k),ΦnR(k) and ΦnI(k) being real-valued functions of k. The two functions G(k) andG∗(k) are proportional to the scattering matrix parameters S22(k) and S11(k) re-spectively as described in [24].

In papers 4 and 5, we perform the reconstructions using contributions fromboth S11 and S22. To formulate the minimization problem, the coefficients {gn}N

n=1

in (3.8) are collected into the vector g. From measurements of G(k) at severalvalues of k (frequencies), the right hand sides of (3.8) are collected into the vectorG = [GR GI ]

T, while the evaluations of Φn(k) are collected into the matrix withelements Φnk = Φn(k), such that

Φ = [ΦR ΦI]T , G = [GR GI ]

T . (3.9)

The unknown coefficients pn can then be obtained from the least square equationmin ||Φg−G||22, which enables us to reconstruct the unknown deformation functiong(z) using (3.3).

3.4 Discretization of the 2D inverse problems

Rectangular waveguide model

In order to use the reflection coefficients S11mn and S22

mn from the equations (2.11)-(2.14) for reconstruction of the unknown deformation function g(x, z) in the 2D


case, we represent g(x, z) as a two-dimensional Fourier series

g(x, z) =

∞∑

p=0

∞∑

q=0

gpq sin(pπx

a

)

sin(qπz

d

)

, (3.10)

Next we truncate the series (3.10) to finite upper limits N , and collect the N ×Ncoefficients gpq into a vector g. Note that the choice to truncate both series (overp and over q) to the same upper limit N is made for convenience and does notinfer any loss of generality of the present results. Any choice of upper limits (forexample 0 ≤ p ≤ M and 0 ≤ q ≤ N with M 6= N) is also possible and can behandled by the method used here. We thereafter collect the measured values ofthe reflection coefficients into a corresponding vector S. Thus we obtain a matrixequation Ag = S, where the elements of the matrix A can be obtained by explicitintegration by substituting (3.10) into (2.11)-(2.14). The unknown coefficients gpq

can then be obtained from the minimization condition min ||Ag − S||22 , whichenables us to reconstruct the shape of the unknown deformation function g(x, z)using (3.10).

Thin coaxial waveguide

In order to use the reflection coefficients from equations (2.99-2.100) and (2.117-2.120) for reconstruction of the unknown deformation function, we expand g(ϕ, z)into the following two-dimensional Fourier series

g(ϕ, z) =

∞∑

p=0

∞∑

q=0

gpq sin(2pϕ) sin(qπz

d

)

, (3.11)

where the deformation occurs on the intervals 0 ≤ z ≤ d and 0 ≤ ϕ ≤ π/2. Thelatter limitation of the angular interval is arbitrarily chosen to illustrate a case ofnon-symmetric angular deformation. As there is no particular reason to restrict theoccurrence of the deformation to any specific angular interval, this choice seems topose an unnecessary limitation to our model. However, it is merely a choice made toillustrate a special case. Ultimately, of course, an arbitrary deformation anywhereon the interval 0 ≤ ϕ ≤ 2π must be taken into consideration, in which case thetwo-dimensional Fourier expansion must be modified to reflect the entire availableangular interval. Otherwise, it would imply that we have an a priori knowledgeabout the potential risks for the deformation, which we normally do not have inrealistic diagnostic situations. It should be noted though that actual deformations,observed in decommissioned power transformers, usually are localized to a portionof the total circumference [40].

Following the same approach as for the rectangular waveguide, we can truncatethe series (3.11) to finite upper limits N , and collect the N×N coefficients gpq intoa vector g. We then collect the measured values of the reflection coefficients intoa corresponding vector S and again, obtain a matrix equation Ag = S, where the

3.5. REGULARIZATION APPROACH IN THE 1D CASE 37

elements of the matrix A can be obtained by explicit integration. The unknowncoefficients gpq can be then obtained from the optimization condition min ||Ag−S||22,which enables us to reconstruct the unknown deformation function g(ϕ, z) using(3.11).

3.5 Regularization approach in the 1D case

Computation of the L-matrix

In the present work, the Tikhonov matrix is chosen such that ||Lg||22 is proportionalto the r-th derivative of the deformation function g(z). The reason for this choiceis that the resulting L matrix puts a greater penalty on the higher order elementsin g. In other words, the coefficients g associated with the higher order harmonicsin the Fourier series expansion (3.3) are suppressed, and the profile is “smoothed”out. This is in contrast to using e.g. an identity matrix, which equally suppressesall the coefficients of g, and where over-regularization can lead to a greater underes-timation of the amplitude of the deformation functions. In this subsection, a shortderivation behind the L matrix will be outlined. For the r:th order derivative ofthe deformation g(z), (3.3) implies

drg

dzr=

N∑

n=1

gn

drφn

dzr. (3.12)

We seek to compute the L2 norm of this derivative, so we need

(

drg

dzr

)2

=

(

N∑

n=1

gn

drφn

dzr

)(

N∑

m=1

gm

drφm

dzr

)

=

N∑

n=1

N∑

m=1

gngm

drφn

dzr

drφm

dzr. (3.13)

For r = 2k + 1 odd (k = 0, 1, 2, . . .), we have

drφn

dzr= (−1)k

(

nπ

z2 − z1

)2k+1

cos

[


]

, (3.14)

For r = 2k even (k = 0, 1, 2, . . .), we have

drφn

dzr= (−1)k

(

nπ

z2 − z1

)2k

sin

[


]

, (3.15)

For both r even and r odd, the integral

∫ z2

z1

(

drg

dzr

)2

dz =

N∑

n=1

N∑

m=1

gngm

∫ z2

z1

drφn

dzr

drφm

dzrdz , (3.16)


is non-zero only when m = n, such that

∫ z2

z1

(

drg

dzr

)2

dz =

N∑

n=1

g2n(−1)2k

(

nπ

z2 − z1

)2r

·

∫ z2

z1

cos2(

nπ z−z1

z2−z1

)

dz , r odd∫ z2

z1

sin2(

nπ z−z1

z2−z1

)

dz , r even

=

N∑

n=1

g2n

(

nπ

z2 − z1

)2r

·∫ z2

z1

1 ± cos(

2nπ z−z1

z2−z1

)

2dz

=1

2

N∑

n=1

g2n

(

nπ

z2 − z1

)2r

·[

(z2 − z1) ±∫ z2

z1

cos

(

2nπz − z1z2 − z1

)

dz

]

=

=(z2 − z1)

2

N∑

n=1

g2n

(

nπ

z2 − z1

)2r

=π2r

2(z2 − z1)2r−1

N∑

n=1

n2rg2n =

=π2r

2(z2 − z1)2r−1

N∑

n=1

(nrgn)2 =π2r

2(z2 − z1)2r−1(Lg)2 ∝ ||Lg||22 , (3.17)

where

Lg =

1r 0 0 . . .0 2r 0 . . .0 0 3r . . ....

......

. . .

0 0 0 . . . N r

g1g2g3...gN

=

1rg12rg23rg3

...N rgN

=⇒ (Lg)2 =

N∑

n=1

(nrgn)2 .

(3.18)

1D regularization

In order to handle the illposedness of the inverse problem, we invoke Tikhonovregularization by adding a penalty term to the least square equation min ||Φg−G||22.From (3.18), we know that

∥

∥

∥

drg

dzr

∥

∥

∥

2

2=

∫ z2

z1

(

drg

dzr

)2

dz ∝N∑

n=1

n2rg2n = ||Lg||22 , (3.19)


where the matrix L = [diag{1, 2, . . . , N}]r. We define r as the order of the regular-ization, in the regularized problem

min{||Φg− G||22 + λ2||Lg||22} , (3.20)

where λ is the regularization parameter. The coefficient vector p that solves (3.19)is obtained from

(ΦTΦ + λ2LTL)g = ΦTG . (3.21)

Finally, when solving (3.21) parameterized by λ, we plot log10 ||Φg − G||2 andlog10 ||Lg||2 against each other, to find the appropriate value of the regularizationparameter λ by means of the L-curve method [33].

3.6 Regularization approach in the 2D case

Computation of the L matrix

Analogously to the 1D case, the Tikhonov matrix is chosen such that ||Lg||22 isproportional to the gradient of the deformation function g(x, z). We start with theFourier series expansion (3.10) and seek its derivatives

∂g

∂x=

N∑

p=0

N∑

q=0

gpq

(pπ

a

)

cos(πx

a

)

sin(qπz

d

)

, (3.22)

∂g

∂z=

N∑

p=0

N∑

q=0

gpq

(qπ

d

)

sin(πx

a

)

cos(qπz

d

)

. (3.23)

where the length parameters a and d have been defined in section 2.1. Since weseek to compute the L2 norm of the gradient, we need

(

∂g

∂x

)2

=

N∑

p=1

N∑

q=1

g2pq

p2π2

a2cos2

(pπx

a

)

sin2(qπz

d

)

+ cross terms , (3.24)

(

∂g

∂z

)2

=N∑

p=1

N∑

q=1

g2pq

q2π2

d2sin2

(pπx

a

)

cos2(qπz

d

)

+ cross terms . (3.25)

Thus we obtain

|∇g(x, z)|2 =

(

∂g

∂x

)2

+

(

∂g

∂z

)2

=


N∑

p=1

N∑

q=1

g2pqπ

2

[

p2

a2cos2

(pπx

a

)

sin2(qπz

d

)

+q2

d2sin2

(pπx

a

)

cos2(qπz

d

)

]

+ cross terms .

(3.26)

Now we can compute the L2 norm as follows

||∇g||22 =

∫ a

0

dx

∫ d

0

dz|∇g(x, z)|2 , (3.27)

where using (3.26), we obtain

||∇g||22 =

N∑

p=1

N∑

q=1

g2pqπ

2

[

p2

a2

∫ a

0

cos2(pπx

a

)

dx

∫ d

0

sin2(qπz

d

)

dz+

+q2

d2

∫ a

0

sin2(pπx

a

)

dx

∫ d

0

cos2(qπz

d

)

dz

]

+

∫ a

0

∫ d

0

(cross terms) dxdz . (3.28)

The last term on the right-hand side vanishes, and we have

||∇g||22 =N∑

p=1

N∑

q=1

g2pqπ

2 1

4

[

p2

a2

∫ a

0

dx

∫ d

0

dz +q2

d2

∫ a

0

dx

∫ d

0

dz

]

=

=

N∑

p=1

N∑

q=1

g2pq

(π

2

)2[

p2

a2ad+

q2

d2ad

]

=

N∑

p=1

g2pq

(π

2

)2[

d

ap2 +

a

dq2]

=

=(π

2

)2 N∑

p=1

N∑

q=1

[

g2pq

√

d

ap2 +

a

dq2

]2

∝ ||Lg|22 . (3.29)

This concludes the proof that the diagonal matrix L puts the weight of

weight(p, q) =

√

d

sp2 +

s

dq2 , (3.30)

on each expansion coefficient gpq. Note that s = a for the rectangular waveguide asdescribed above, while for the thin coaxial waveguide model, we analogously haves = π

2RI.


2D regularization

Analogously to the 1D case, for the 2D case we introduce a penalty term ||∇g||22 ∝||Lg||22 for the rectangular waveguide model according to (3.27), and analogouslyfor the coaxial waveguide model with x → ϕ and a → π/2. On each expansioncoefficient gpq, the diagonal matrix L puts the weight (3.30) and the coefficientsgpq are obtained using the regularized problem described by

(AT A + λ2LT L)g = AT S , (3.31)

where λ is the regularization parameter.

Chapter 4

Results

The reconstruction of our model with discrete deformations is described in detailin the licentiate thesis [1] preceding this PhD thesis. The focus of the presentationhere has been mainly on the shape reconstructions of continuous deformations ofmetallic boundaries in rectangular and coaxial waveguide models, reported in pa-pers 4 and 5. We considered air-filled waveguides with perfect electrical conductor(PEC) walls. The assumption of small perturbations of the boundaries enables usto use that the scattering parameters of the waveguide have a linear dependence onthe continuous deformation function. Thus, the corresponding inverse problem islinearized as well, and our developed algorithm employs direct inversion to obtainthe shape parameters of the deformation. To regularize the algorithm solutions,we use Tikhonov regularization with the L-curve method. A number of differentheights and shapes of deformation functions were investigated, and reconstructionresults for one-dimensional and two-dimensional localized shape deformations inrectangular and coaxial waveguide boundaries were presented. Note however thatno graphical or tabular reconstruction results are included in this chapter, but theycan be found in the appended papers.

4.1 Paper 4

The model in paper 4 considered one-dimensional deformations in a lower coaxialwaveguide boundary. Our simple and computationally efficient first-order pertur-bation method was used, together with synthetic measured reflection data from thedominant mode only. Reconstruction results were obtained with a good agreementbetween the reconstructed and true continuous deformations of waveguide bound-aries. The cases presented in this paper show that the method can reconstructboth continuous and abrupt discontinuous deformation functions. The methodalso works for a variety of heights and lengths of the boundary deformations. Anadditional important result is that deformation functions which vary only in thelongitudinal dimension (along the propagation direction of the waveguide), can bereconstructed using scattering parameters from the dominant mode only, if the datais sampled over a range of frequencies.

43

44 CHAPTER 4. RESULTS

4.2 Paper 5

The model in paper 5 is a generalization of paper 4, to also take two-dimensionaldeformations into consideration. When it comes to two-dimensional deformations,with variation in both the transverse and longitudinal directions of the waveguide,higher order modes are needed for reconstructions of sufficient accuracy. It wastherefore of interest to investigate the effect of adding scattering data from thehigher order modes in the waveguide model, over the same frequency band used forthe dominant mode. Since the same frequency range is used for both the studieswith the dominant mode and with the higher order mode information added, wecan get an indication of the specific effect from including the higher order modes.

The main result of the two-dimensional deformation study in paper 5 is thatself-reflection data is sufficient to reconstruct basic shape properties of the defor-mation, symmetric about the midpoint of the deformation region (e.g. the heightof the deformation). However, cross-reflection data between the dominant modeand the higher order modes is required for accurately capturing non-symmetricproperties, such as for example an offset in peak position. Furthermore, addingadditional terms to the mode expansion functions for the electromagnetic fields, weget a more accurate description of the fields and thereby improved resolution of thereconstruction results, which is particularly important when sharper peaks/edgesof the deformation shape are reconstructed.

Chapter 5

Conclusions and future work

5.1 Conclusions

In this thesis, we presented a theoretical study of a novel method for shape recon-structions in waveguide models of active parts of power grid components and othersimilar mechanical structures. The study covered both discrete conductor defor-mations (papers 1-3) and continuous deformations of metallic boundaries (papers4-5). The results are based on first-order electromagnetic perturbation theory. Westarted our investigations by employing a waveguide model where a set of parallelconductors between two waveguide boundaries were displaced from their ordinarypositions. The objective was to reconstruct the locations of the displaced discreteconductors. As a first model, we investigated the parallel-plate waveguide geom-etry [1]. Thereafter we proceeded to investigate coaxial waveguide models, wherethe conductors were envisioned as circular rings, using a similar approach as theparallel-plate waveguide study. In connection to that work, we extended the re-construction capabilities to include not only radial displacements but also ellipticaland wave-shaped (as defined in paper 3) deformations of the conductor rings. Thediscrete conductor model turned out to be computationally demanding for largernumbers of individual conductors, while at the same time, the results were onlyapplicable to a specific conductor configuration. In other words, detailed knowl-edge of the number and sizes of the conductors/rings was required for successfulreconstructions. If applied as a diagnostic tool for a power grid component, thetool would have to be tailor-made for each design.

In order to be able to develop a more general and less computationally complexmethod, we developed a method based on continuous waveguide models. Here, thedeformations can be described by means of a suitable class of continuous functions.This way, perturbation theory for waveguide conductor boundaries could be usedand the computation complexity reduced, as compared to the approach used inpapers 1-3. Furthermore, the method used in papers 4 and 5 is applicable to anykind of small deformation functions that weakly alter the scattering situation. In

45

46 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

conclusion, our method successfully fulfilled the objectives of being more general andless computationally complex. Furthermore, in our method it is easy to incorporatescattering data from both the lowest order dominant mode and the higher orderwaveguide modes, since the scattering parameters for all modes can be handledusing the same analytical approach. We can even obtain the transmission datausing the same principle, although they have not been used in the present research,and their usefulness remains to be investigated. A discussion of potential limitationsof our continuous model and the possible further improvements in order to achievea closer resemblance to more realistic scattering situations encountered in realisticpower grid components, is presented in the next section.

5.2 Future work

The work presented in this thesis constitutes an initial theoretical study of the pos-sibilities to develop a future online microwave-based monitoring system for powergrid components. The reconstruction method was therefore tested using syntheticmeasurement data generated by commercial simulation software, in order to avoidthe inverse crime. An obvious future development is to build an actual waveguidestructure and use measured scattering parameters from the laboratory, as an inputto the developed reconstruction algorithms. The geometries considered in this workhave shapes and dimensions similar to realistic shapes and dimensions observed ine.g. certain types of power transformers. Thus, the sizes used in the present studycould serve as an input to such future experimental investigations.

The abovementioned conclusions open for several possible paths for future theo-retical work as well. In the short term, paper 5 can be extended by not introducingthe thin coaxial approximation, but instead using the more exact solutions for elec-tromagnetic fields based on the Bessel functions of integer order. Further studiesof the systematic use of different waveguide modes to improve the reconstructionscan also enhance the results. Apart from that, the present thesis only consideredreflection data as input to the reconstruction method. However, incorporation ofthe transmission data is in principle possible and may prove to be a useful tool forfurther improvement.

The study of elliptical and wave-shaped (as defined in paper 3) deformations ofwindings can be further expanded. It can be done by comparing the numericallycomputed reflection coefficients for e.g. the wave-shaped winding deformations,with analytically computed reflection coefficients for the obtained equivalent radiigeometry. Matching aspects between the deformed and undeformed regions of thewinding can also be considered.

As an interesting further development in the longer term is to consider thefeasibility of using second order perturbation terms. Thereby we can investigatethe analytical and computational complexity of the second order perturbation ap-proach as well as the potential benefits in terms of improved accuracy of the shapereconstructions.

5.2. FUTURE WORK 47

Based on the results reported by other groups on reconstructing both impedanceand shape changes, approaches based on modeling the boundary deformations usingeffective material properties can also be considered. In other words, the possibilityto replace a waveguide with variable cross-section radius r(z) by a waveguide withconstant cross-section radius and e.g. a variable effective permittivity ε(z) or vari-able effective wave impedance Z(z) can be investigated. In this way, the Maxwellequations can be solved analytically for certain classes of continuous ε = ε(z) func-tions, without using a generic perturbation approach. However, for more complexε = ε(z) functions, approximate methods like WKB [23] or perturbation approachmay still be needed, but at a different level of computation. We have extensiveexperience from this in previous unrelated work [41–43]. An important question isthe frequency range in which this approach is valid, as well as what would be anappropriate continuous model variation of ε(z) over this frequency band.

Finally, in the present thesis, the winding was modeled as a smooth metal-lic surface, which is somewhat generic and neglects the aspects of anisotropy andinhomogeneity of actual structures that are modeled (e.g. power transformer wind-ings etc.). To that effect, one could consider sheath helix [44] or other impedanceboundary conditions [45, 46], in order to have a more accurate model. Such modelsbecome important when analyzing resonances, in order to determine usable fre-quency regions when solving the inverse problem.

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[26] J. D. Jackson, “Classcial Electrodynamics”, 3rd ed., Wiley, 1999.

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52 BIBLIOGRAPHY

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Appendix A

Licentiate thesis

The licentiate thesis preceding this PhD thesis can be found at the link below:

• M. Dalarsson, “Online power transformer diagnostics using multiple modes ofmicrowave radiation”, KTH Royal Institute of Technology, Stockholm,Sweden, 2013.http://www.diva-portal.org/smash/get/diva2:653608/FULLTEXT01.pdf.

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