Perturbation approach to reconstructions of boundary deformations in waveguide structures

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Perturbation approach to reconstructions of boundary deformations in waveguide structures

MARIANA DALARSSON

Doctoral Thesis

Stockholm, Sweden 2016

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TRITA-EE 2015:109 ISSN 1653-5146

ISBN 978-91-7595-801-9

Elektroteknisk teori och konstruktion Osquldas väg 6 SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillst˚ and av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 12:e februari 2016 klockan 14.00 i sal F3, Lindstedtsvägen 26, Kungl Tekniska högskolan, Valhallavägen 79, Stockholm.

Mariana Dalarsson, February 2016 c

Tryck: Universitetsservice US AB

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Abstract

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

Online methods for the diagnostics of adverse changes of the mechanical structure and the integrity of the dielectric insulation in power transformers and other power grid components, are therefore essential for the continuous operation of a power grid. Efficient online diagnostic methods can provide a real-time monitoring of mechanical structures and dielectric insulation in the active parts of power grid components. Microwave scattering is a candidate that may detect these early adverse changes of the mechanical structure or the dielectric insulation. Upon early detection, proper actions to avoid fail- ure or, if necessary, to prepare for the timely replacement of the damaged component can be taken. The existing diagnostic methods lack the ability to provide online reliable information about adverse changes inside the active parts. More details about the existing diagnostic methods, both online and offline, and their limitations can be found in the licentiate thesis preceding the present PhD thesis.

We use microwave scattering together with the inverse scattering algo- rithms, developed in the present work, to reconstruct the shapes of adverse mechanical structure changes. We model the propagation environment as a waveguide, in which measurement data can be obtained only at two ends (ports). Since we want to detect the onset of some deformation, that only slightly alters the scattering situation (weak scattering), we have linearized the inverse problem with good results. We have calculated the scattering pa- rameters of the waveguide in the first-order perturbation, where they have linear dependencies on the continuous deformation function. A linearized inverse problem with a weak scattering assumption typically results in an ill-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 first

principles, i.e. Maxwell’s theory of electromagnetism. An excellent agree-

ment is obtained between the reconstructed and actual deformation shapes

for a number of studied cases. These results show that it is possible to use

the inverse algorithms, developed in the present thesis, as a theoretical basis

for 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 to

further study the theoretical limitations posed by linearization (first-order

perturbation theory) and by the assumption of the continuity of the metallic

waveguide boundaries and their deformations.

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Sammanfattning

I denna avhandling har vi utvecklat inversa spridningsalgoritmer i syfte att ta fram metoder för onlinediagnostik av strukturförändringar inom krafttrans- formatorer och andra storskaliga komponenter inom det elektriska kraftnätet som generatorer, shuntreaktorer etc. Dessa komponenter är i regel inte till- gängliga som standardprodukter fr˚ an tillverkarna, utan är istället optimerade för specifika funktioner p˚ a specifika platser i kraftnätet, och att ersätta dem

¨ ar därför b˚ ade dyrt och tidskrävande.

Onlinemetoder för diagnostik av skadliga förändringar i den mekaniska strukturen samt de dielektriska egenskaperna hos isolationsmaterialet, i kraft- transformatorer och andra stora kraftnätskomponenter, är därför väsentliga för optimal drift och underh˚ all av kraftnätet. Effektiva metoder för online- diagnostik kan möjliggöra realtidsövervakning av de mekaniska strukturerna och isolationen i de aktiva delarna av kraftnätskomponenten s˚ a att potentiellt skadliga förändringar kan upptäckas i ett tidigt skede. I s˚ a fall kan man vidta lämpliga ˚ atgärder för att undvika ett haveri eller (om det är nödvändigt) kan man förbereda ett kontrollerat byte av den skadade komponenten i god tid till lägre totalkostnad. De existerande diagnostiska metoderna, s˚ aväl online som offline, saknar förm˚ agan att ge fullt tillförlitlig information om begynnande skadliga förändringar inom t.ex. krafttransformatorerna som är i drift. En mer detaljerad beskrivning av existerande diagnostiska metoder, s˚ aväl online som offline, och deras begränsningar finns i den tidigare framlagda licentiatavhan- dlingen som försvarats innan den nu aktuella doktorsavhandlingen.

Vi betraktar spridningsmiljön som en v˚ agledare, där mätningar av sprid- ningsparametrarna endast kan göras i tv˚ a ändar (portar). D˚ a vi önskar upptäc- ka begynnande deformationer som medför relativt sm˚ a ändringar av spridnings- situationen (svag spridning) kan det inversa problemet linjäriseras. V˚ agledarens spridningsparametrar f˚ ar d˚ a ett linjärt beroende av den kontinuerliga defor- mationsfunktionen. Ett linjärt inverst problem som beskriver svag spridning resulterar ofta i ett illa konditionerat linjärt ekvationssystem. Detta hanteras med hjälp av Tikhonovregularisering med L-kurvametoden för att finjustera regulariseringsparametrarna.

Vi ˚ aterskapar b˚ ade endimensionella och tv˚ adimensionella lokaliserade deformationer med god noggrannhet, i rektangulära och koaxiella v˚ agledar- modeller. Till detta använder vi inversa spridningsalgoritmer som utvecklats fr˚ an grundprinciperna, d.v.s. Maxwells elektromagnetiska teori. En utmärkt

¨ overensst¨ammelse mellan ˚ aterskapade och faktiska deformationer f¨or ett antal

studerade fall redovisas. Dessa resultat visar att det är möjligt att använda

de inversa algoritmerna, som utvecklats inom ramen f¨or detta arbete, som en

teoretisk bas f¨or konstruktion av framtida diagnostiska verktyg. Som delar

av framtida arbeten, ˚ aterst˚ ar att verifera v˚ ara resultat experimentellt, samt

att vidare studera de teoretiska begr¨ansningarna som h¨anger samman med

linjäriseringen, samt antagandet om att v˚ agledarens metalliska gränsytor är

kontinuerliga.

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Acknowledgments

It is my pleasure to express my gratitude to a number of people without whom this work would not be possible. First of all, I would like to express my sincere gratitude 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 of Electromagnetic Engineering at the KTH Royal Institute of Technology as a PhD student. Thank you for your guidance, your feedback on articles and theses, and for supporting me to not only develop as a researcher but also as a teacher in electromagnetic theory.

I would also like to thank the Swedish Energy Agency, who funded the first half of my PhD project through Project Nr 34146-1. My work has also been part of EIT/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 are also 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 especially thank professor Lars Jonsson, Dr. Daniel M˚ ansson and Dr. Nathaniel Taylor for many interesting discussions related to the academic life, and the department head professor Rajeev Thottappillil for his administrative and strategic support. I also appreciated the support of Ms. Carin Norberg in financial administration and the support of Mr. Peter L¨onn in maintaining computers and software. I would also like to thank several former and current PhD students at the department for their companionship. These include Dr. Shuai Zhang, Dr. Johanna Rosenlind, Fatemeh Ghasemifard, Andrey Osipov, Christos Kolitsidas, Elena Kubyshkina, Shuai Shi, Kun Zhao, Mahsa Ebhrahimpouri, Lipeng Liu, Mengni Long, Patrick Janus, Janne Nilsson, among many others.

Although it is not possible to mention them all, I would also like to express my 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 success of my education and subsequent research. A special mention here is to associate professor emeritus Karim Daho, who fueled the very start of my academic career by encouraging me to become a teaching assistant in mathematics. Together with other faculty and PhD students at the Department of Mathematics, he taught me valuable skills that I could use in later research and teaching efforts. I believe that this quote by Henry Adams holds very true here: “A teacher affects eternity! He can never tell where his influence ends!”

I would also like to mention several people at the Alfven Laboratory. Thanks

are due to professor G¨oran Marklund and professor emeritus Nils Brenning for

many 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 Str¨om, Richard Fridstr¨om, Armin Weckmann, Estera Stefanikova, Alvaro Garcia and Emmi Tholerus for adopting me on 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 like to thank the students at Engineering Physics at KTH for their polite nature and many bright questions! It made the long teaching hours pass in a breeze, and I was always able to leave the classroom with a smile on my face. I believe in you, and that you will become the next technological leaders in Sweden in both the academia and the industry.

I deeply appreciate having my invaluable friends outside the university including Rebecka, 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, who

motivated me to get my education and supported me in my achievements, and my

dear husband Henrik, for always being by my side and brightening my day.

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List of papers

The thesis is based on the following peer-reviewed international journal papers:

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”, Mathematical Problems 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 on December 14 2015.

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

• M. Dalarsson, “Online power transformer diagnostics using multiple modes of microwave 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 in Appendix 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 for

the thesis. Thereafter, Martin Norgren proposed a number of valuable comments

and improvements on all the papers. Alireza Motevasselian provided the synthetic

measurement data used in Papers 1 - 2. Seyed Mohamad Hadi Emadi provided the

synthetic 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 of ICIPE 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, Fr¨osundavik, 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 Power Transformers, Workshop KTH, ABB and Indian Institute of Technology, June 5, 2012, V¨aster˚ as.

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 applications in the power industry, April 23-24 2013, Linnaeus University, V¨axj¨o.

17. M. Dalarsson and M. Norgren (invited), “Reconstruction of boundary pertur- bations in a coaxial model of a transformer winding”, AntennEMB, March 11-12, 2014, Gothenburg, Sweden.

Comments:

• (*) The author received the “YSA - Young Scientist Award” awarded to distinguished 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 of the 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 PIERS 2012 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 Research Letters, 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.

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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 cables and/or transmission lines. Introduction of smart electric grids and increased use of renewable energy sources, increases the need for novel and more accurate on- line diagnostic methods. Service outages for critical power-grid components can be reduced using diagnostic tests designed to detect internal damages. A more exten- sive discussion of these aspects and a review of possible degradation mechanisms related to power transformers can be found in the licentiate thesis preceding this PhD 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 methods have a major disadvantage since they involve a non-service stress of a component and financial loss of revenue during the tests. Some theoretical proposals how to migrate from offline to online winding deformation diagnostics using FRA include e.g. [4]. However, the FRA method, even applied online, is a differential method measuring 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 partial discharge (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 sensors mounted 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 sample can be used to identify an on-going fault [7]. However, DGA is an integral method and it does not provide a reasonable prediction of the remaining lifetime of the transformer [8]. The abovementioned techniques can also be used for other power grid components like power generators, as described in [9]. A diagnostic method

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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 generally termed Electrical Motor Diagnostics (EMD) [11]. In summary however, there are currently 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 applied inverse problems by using microwave scattering, with the objective to reconstruct the shapes of perfect electrically conducting (PEC) objects from observations of the 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 power grid components, such as power transformers or generators, based on microwave scattering exist on the market today.

Early attempt at microwave diagnostics

One early numerical study of transformer diagnostics using microwave scattering can be found in [14]. In this conference paper, a radar antenna operating at 9.5 GHz 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 a parametric study of the dependence of a scattering parameter (S 11 ) on the relative axial position of the two windings, as well as on the axial position of the measuring antenna. However, this study does not provide a general framework for detection of deformations of transformer windings in both radial and axial directions, although this issue is mentioned as a possibility in the conclusions. Furthermore, the method described in [14] does not provide any automated method for obtaining the axial positions of the windings, but is based on manual comparisons with pre-calibrated scattering 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 mentioned but 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 and

an inverse problem approach, was made in our paper [15]. The power transformer

environment was modeled as a waveguide with a set of conducting obstacles, repre-

senting segments of the transformer winding. Using microwave scattering, changes

in 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 my

licentiate thesis [1].

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1.2. MOTIVATION FOR THE PROJECT 3

Techniques to handle both surface impedance and shape reconstruction

The reconstruction of an inhomogeneous surface impedance of a two-dimensional cylindrical scatterer located above a PEC plane is considered in [19]. Such a method can be used to reconstruct not only the shape but also the surface impedance of the scatterer. A reconstruction method in which an inaccessible PEC scatterer is modeled as a surface impedance was developed in [20], with the aim to reconstruct the 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 related to this project, since radial variations of the cross section are indicators of boundary deformations. For the TEM-mode, equivalent circuit models as T- or Π-networks can represent waveguides with abruptly changing radii [21]. Waveguides of generic shapes, undergoing continuous changes in the axial direction only, can also be ap- proximated as a cascade of invariant sections and treated semi-analytically using the 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 the literature, but most of them employ various discrete methods. A few publications do present the proper continuous solutions for some special cases, often inspired by analogies with similar quantum-mechanical models (either exact or using the Wentzel-Kramers-Brillouin (WKB) approximation) [23]. However, a proper theory where waveguide boundaries can be modeled as arbitrary continuous functions, in the context of an inverse problem approach to microwave shape reconstruction, has not 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 that are not directly accessible for visual inspection or direct measurements. The results from the present thesis can be used in a new online diagnostic method for power grid components such as e.g. power transformers, with the potential to detect the effects of various deterioration processes in a more accurate way, with less risks of adverse consequences of the measurement process for the normal operation.

Microwave measurements, up to a few GHz, is appropriate for online diagnostics since it has wavelengths comparable to the dimensions of the mechanical structures and potential adverse deformations in the active parts of e.g. power transformers.

The analysis of measured microwave signals and their relations to the structure

parameters, being the signatures of mechanical deformations, is an inverse electro-

magnetic problem. The idea can be practically realized by inserting antennas inside

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4 CHAPTER 1. INTRODUCTION

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

The main advantage of the diagnostic method described here, is that it does not require the disconnection of a monitored component from the power grid. Compared to some existing online methods, our method proposed does not require connection of test voltages to high-voltage terminals of the monitored component. In other words, the test signals (high-frequency microwaves) are completely independent of any low-frequency signals applied to the monitored component in the course of its regular operation. Furthermore, our diagnostic method can actually give a snapshot of the present state of the power grid component, i.e. it can detect the actual deformations measured in suitable length units (e.g. mm) at the very instant when 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 problems based on electromagnetic waves propagating within waveguide structures. In prac- tical diagnostic situations, we often want to characterize objects in their working environments that are often far from optimal for solving the inverse problems. One typical situation is that one cannot generate input signals and measure response signals in the vicinity of the object. Instead, the measurement can only be carried out at certain locations (ports), while the object itself remains “hidden” inside the structure, as e.g. the winding structure of a power transformer. On their way back and forth to the object, the microwaves undergo scattering and diffraction due to other obstacles than the studied object itself. However, for many diagnostic purposes it is justified to neglect some properties of the complex actual structure and to model the system as an object inserted into a waveguide, where we have a well-defined electromagnetic field pattern. As a generic model of the propaga- tion environment, we therefore use a waveguide, where measurement data can be obtained only in a limited number of regions.

In a waveguide, there may be several different kinds of objects and properties to 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.

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1.4. PAPERS 1 - 3 5

1.4 Papers 1 - 3

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

1.5 Thesis disposition

Chapter 1 starts with an overview of the aim of the thesis, followed by a survey of

the 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 the

direct electromagnetic (EM) scattering problem at hand. It contains the detailed

perturbation theory for one-dimensional deformations in the thin coaxial waveg-

uide, as well as the detailed perturbation theory for two-dimensional deformations

in rectangular and thin coaxial waveguides. Chapter 3 describes the inverse electro-

magnetic (EM) scattering problem. The chapter starts with the background theory

of Tikhonov regularization and the L-curve method used to choose the regular-

ization parameters, and then proceeds to describe the details of choosing penalty

terms in the regularization, the discretization of the problem in the 1D and 2D

case and the resulting minimization problem. Thereafter, Chapter 4 consists of

a short overview and discussion of the main results of papers 1-5. Note that all

graphical or tabular reconstruction results for the investigated cases are presented

in 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 are

discussed.

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Chapter 2

Direct EM scattering problem

We define the “direct problem” as solving for the scattered EM fields given a known waveguide model geometry and incident fields. On the contrary, the “inverse prob- lem” amounts to determining the waveguide model geometry given known scattered and incident EM fields. The description of the direct problem theory is given in this chapter, while the inverse problem is described in Chapter 3. The direct problem is 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 in rectangular and thin coaxial waveguides. This theory is used as basis for papers 4 and 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 of a winding structure within a power grid component is typically based on a coaxial waveguide, or to a certain approximation on a parallel-plate waveguide. Although a rectangular waveguide is in general not a suitable candidate to model an inher- ently cylindrical winding structure, there is a significant mathematical similarity between the results obtained in a rectangular waveguide model with some of the corresponding results obtained in a thin coaxial waveguide model. As a conceptual introduction, it is therefore worthwhile to study the rectangular waveguide model in some detail, prior to proceeding with a thorough study of the more realistic thin coaxial waveguide model. To this end, we consider the wave propagation within a rectangular waveguide with a cross-section shown in Fig. 2.1(a). We assume a shallow rectangular waveguide where b ≤ a/5, such that the first four leading modes of microwave propagation are the TE n -modes (TE n0 -modes), where 1 ≤ n ≤ 4.

Based on the standard theory of rectangular waveguides [26], it is convenient to introduce the following basis functions

ψ n (x) = 2 r Z TE

ab sin nπx a

, Z TE = k η k zn

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

7

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8 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

a

b z y

x

ˆ

n ₀ = −ˆy

(a)

a

b z y

x bδg(x, z) ˆ n

(b)

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

where k = ω√µ , η = pµ/ and k zn = pk ² − n ² π ² /a ² . Hereafter, and µ denote respectively the effective permittivity and permeability of the medium within the waveguide and we adopt the convention exp(jωt) for the time dependence of all the fields. The TE n -fields within the rectangular waveguide, propagating in (±z)- direction, are then given by

E ^± _n (r) = P n ψ n (x) exp(∓jk zn z) ˆ y ,

H ^± _n (r) = P n

kη [∓k zn ψ n (x) ˆ x + jψ ⁰ _n (x) ˆ z ] exp(∓jk zn z) , (2.2) where P n is a dimensional constant, which explicit form is not essential for the present 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, to calculate the first order scattered fields arising from the small deformation g(x, z).

Here we recall that, for a given operating frequency f, the electromagnetic

fields in any hollow waveguide are described by the infinite set of modes with their

respective cutoff frequencies f c . Far away from any localized source of the fields, like

a boundary deformation discussed here or an aperture in the waveguide boundary,

the fields are relatively simple propagating modes. However, in the vicinity of a

deformation or an aperture, the fields are generally superpositions of many different

modes [26]. The fields near a localized source are typically assumed to be the

suitable expansions of the regular fields far away from the localized source. The

problem at hand is to determine the amplitudes of the fields in the vicinity of

a localized source in terms of the amplitudes of the unperturbed fields, far away

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2.1. 2D DEFORMATIONS IN A RECTANGULAR WAVEGUIDE 9

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

E(x, aδg, z) =

∞

X

p=0

∞

X

m=0

∂ ^m E p

∂y ^m (x, 0, z) (ag) ^m

m! δ ^m+p . (2.3)

To the first order of perturbation, with δ = 1, the fields can be written as E(x, z) = E 0 (x, z) + E 1 (x, z), where we assume the following expression for the unperturbed (zeroth-order) fields

E 0 (x, z) = X

n

{c ⁺ n [E Tn (x) + E zn (x)ˆ z]e ^−jk

^zn

^z + c ⁻ _n [E Tn (x) − E zn (x)ˆ z]e ^+jk

^zn

^z } , (2.4)

H 0 (x, z) = X

n

{c ⁺ n [H Tn (x)+H zn (x)ˆ z]e ^−jk

^zn

^z +c ⁻ _n [−H Tn (x)+H zn (x)ˆ z]e ^+jk

^zn

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

E 1 (x, z) = X

m

{d ⁺ m [E Tm (x)+E zm (x)ˆ z]e ^−jk

^zm

^z +d ⁻ _m [E Tm (x)−E zm (x)ˆ z]e ^+jk

^zm

^z } , (2.6)

H 1 (x, z) = X

m

{d ⁺ m [H Tm (x)+H zm (x)ˆ z]e ^−jk

^zm

^z +d ⁻ _m [−H ^Tm (x)+H zm (x)ˆ z]e ^+jk

^zm

^z } , (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 the first-order perturbation fields (2.6)-(2.7). The non-negative integers n and m 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) describe the electric field generated by the deformation that exists in the interval 0 ≤ z ≤ d and are therefore valid outside the deformation region, i.e. for z ≤ 0 and z ≥ d. On the other hand, the equations (2.4)-(2.5) are valid in the entire studied waveguide region. Using the excitation theorem ([26], sec. 8.12), we can write the following expression for the d ^± _m coefficients

d ^± _m = R

S [ ˆ n 0 × E 1 (x, 0, z)] · H ^± m dS 2 R

S [E Tm × H ^Tm ] · ˆzdS . (2.8)

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10 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

In the numerator of the theorem (2.8), we have the scattered first-order perturbation field E 1 integrated over the aperture where the deformation is located. Using equations (2.4)-(2.7), we obtain (for detailed derivation for the analogous case of a thin coaxial waveguide, see Section 2.3 below)

ˆ

n 0 × E ¹ (x, 0, z) = aˆ n 0 ×

∇g ˆ n 0 · E ⁰ (x, 0, z) − g ∂E 0

∂y (x, 0, z)

, (2.9)

which is used in the numerator. The denominator on the other hand, is the integral of the unperturbed (zeroth-order) modes over the waveguide cross section, and it is 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 ⁻

=

M ^PP M ^PM M ^MP M ^MM

c ⁺ c ⁻

, (2.10)

where c ^± and d ^± are vectors of coefficients c ^± _n and d ^± _m respectively, while M ^PP , M ^PM , M ^MP and M ^MM are so-called mode-conversion matrices. These matrices are related to the familiar two-port scattering matrices. Thus, the equation (2.8) finally yields the reflection coefficients S _mn ¹¹ and S _mn ²² in a rectangular waveguide with a perturbation on the lower horizontal boundary as follows

S ¹¹ _mn = j 2a √

k zn k zm

Z a 0

dx Z z

2

z

¹

dz g(x, z)e ^−j(k

^zm

^+k

^zn

^)z ×

× k ² + k xm k xn + k zm k zn cos[(k xm + k xn )x]−

− k ² − k xm k xn + k zm k zn cos[(k xm − k xn )]x

, (2.11)

S ²² _mn = j 2a √

k zn k zm

Z a 0

dx Z z

²

z

¹

dz g(x, z)e ^+j(k

^zm

^+k

^zn

^)z ×

× k ² + k xm k xn + k zm k zn cos[(k xm + k xn )x]−

− k ² − k xm k xn + k zm k zn cos[(k xm − k xn )x]

, (2.12)

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2.2. FIELD THEORY OF COAXIAL WAVEGUIDES 11

as well as the transmission coefficients S _mn ¹² and S _mn ²¹ in a rectangular waveguide with a perturbation on the lower horizontal boundary as follows

S ¹² _mn = δ mn + j 2a √

k zn k zm

Z a 0

dx Z z

²

z

1

dz g(x, z)e ^−j(k

^zm

^−k

^zn

^)z ×

× k ² + k xm k xn − k zm k zn cos(k xm + k xn )x−

− k ² − k xm k xn − k zm k zn cos(k xm − k xn )x

, (2.13)

S ²¹ _mn = δ mn + j 2a √

k zn k zm

Z a 0

dx Z z

2

z

1

dz g(x, z)e ^+j(k

^zm

^−k

^zn

^)z ×

× k ² + k xm k xn − k zm k zn cos(k xm + k xn )x−

− k ² − k xm k xn − k zm k zn cos(k xm − k xn )x

. (2.14)

In (2.11)-(2.14) we also have k xl = lπ/a and k zl = pk ² − k ² xl with l = {m, n}. The uppercase index indicates the port number, thus providing information whether these are reflection or transmission coefficients. The lowercase indices mn denote the mode numbers of the reflection/transmission coefficients, i.e. they generally describe the reflection/transmission of the mode of number n into the different mode of number m. It is important to note that waveguide scattering parameters normally depend on the shape of the deformation function g(x, z) in a nonlinear way. However, due to the linearized (first-order perturbation) theory employed, the scattering parameters here are linear functions of g(x, z), as is evident from the equations (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 cylindrical

winding structure within an active part of e.g. a power transformer. We consider a

TM wave, traveling in the z-direction through a coaxial waveguide, with inner radius

R I and outer radius R O , as shown in Fig. 2.2. In the licentiate thesis [1] preceding

this PhD thesis, a detailed derivation of the expressions for the electromagnetic

fields in a coaxial waveguide is presented. It is therefore only briefly summarized

in this section.

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12 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

r z y

x R I

R O

ϕ

ˆ r ˆ

ϕ ˆ z

(a)

z y

x R I

R O

ˆ n

ˆ n 0 = −ˆr

(b)

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

TM waves in a coaxial waveguide

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

E z (r, ϕ, z) = A

J n (k T r) − J n (k T R I )

N n (k T R I ) N n (k T r)

sin(nϕ) cos(nϕ)

e ^∓jk

^z

^·z , (2.15)

where r and ϕ are the usual radial and azimuthal cylindrical coordinate variables, such that R I < r < R O . In (2.15), J n (u) is the Bessel function of integer order, while N n (u) is the Neumann function of integer order. We also define the longitudinal wave vector component k z = pω ² µ − k ² T , and the discrete set of values for the transverse component of the wave vector k T = k T(m,n) is obtained from the cross- product condition (3.23) in [1] (with R O = λR I and λ > 1), such that the values for k T(m,n) are given by [28]

k T(m,n) = w + α

w + β − α ²

w ³ + γ − 4αβ + 2α ³

w ⁵ + . . . , (2.16) with [28]

w = mπ

λ − 1 , α = 4n ² − 1

8λ , β = (4n ² − 1)(4n ² − 25)(λ ³ − 1) 384λ ³ (λ − 1) ,

γ = (4n ² − 1)(16n ⁴ − 456n ² + 1073)(λ ⁵ − 1)

5120λ ⁵ (λ − 1) . (2.17)

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2.2. FIELD THEORY OF COAXIAL WAVEGUIDES 13

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

H T = −jω

k ² _T

∂E z

∂r ϕ ˆ + 1 r

∂E z

∂r ˆ r

, (2.18)

E T = ∓ jk z

k ² _T

∂E z

∂r r ˆ + 1 r

∂E z

∂ϕ ϕ ˆ

. (2.19)

where ˆ r and ˆ ϕ are the unit vectors in cylindrical coordinates. The wave impedance for the TM-waves (Z TM ) is given by

Z TM = 1 ω

q

ω ² µ − k ² T . (2.20)

The actual transverse components of the electric and magnetic fields, in terms of Bessel 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 TE waves propagating in the ±z-direction, is given by

H z (r, ϕ, z) = A

J n (k T r) − J ⁰ _n (k T R I )

N ⁰ _n (k T R I ) N n (k T r)

sin(nϕ) cos(nϕ)

e ^∓jk

^z

^·z , (2.21)

where the discrete set of values for k T = k T(m,n) is now obtained from the following cross-product condition (with R O = λR I and λ > 1),

J ⁰ _n (k T R I )

N ⁰ _n (k T R I ) = J ⁰ _n (k T R O )

N ⁰ _n (k T R O ) , (2.22)

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

w = mπ

λ − 1 , α = 4n ² + 3

8λ , β = (16n ⁴ + 184n ² − 63)(λ ³ − 1) 384λ ³ (λ − 1) , γ = (64n ⁶ + 2906n ⁴ − 8212n ² + 1899)(λ ⁵ − 1)

5120λ ⁵ (λ − 1) . (2.23)

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

H T = ∓ jk z

k _T ²

∂H z

∂r ˆ r + 1 r

∂H z

∂ϕ ϕ ˆ

, (2.24)

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14 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

E T = −jωµ k ² _T

∂H z

∂r ϕ ˆ + 1 r

∂H z

∂r r ˆ

. (2.25)

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

Z TE = ωµ

pω ² µ − k ² T

. (2.26)

The actual transverse components of the electric and magnetic fields, in terms of 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 modes in 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 for each k T = k T(m,n) , since for each azimuthal eigenvalue (n ≥ 1) there can be either a cos(nϕ) or a sin(nϕ)-dependence [29]. All linearly independent combinations of cos(nϕ) and sin(nϕ) can be used as well, but {cos(nϕ), sin(nϕ)} is a suitable choice of orthogonalized modes. Thus, the difference between the two degenerate modes is a 90 ^◦ rotation of the field-pattern. Unlike the rectangular waveguide, where the occurrence of degeneracy depends partly on the relation between the side lengths, the occurrence of degeneracy in the circular waveguide is independent of the radius [29].

TEM waves in a coaxial waveguide

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

E r (r, z) = A 0

r e ^∓jk

^z

^·z , H ϕ (r, z) = ± A 0

ηr e ^∓jk

^z

^·z , η = r µ

, (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 the coaxial waveguide is thin, in a sense that the thickness of the waveguide, defined as a = R O − R I , is small compared to the average radius of the waveguide cavity

b

2 = ¹ ₂ (R I + R O ), i.e.

R O − R I R I + R O

2 ⇔ a b

2 . (2.28)

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

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2.2. FIELD THEORY OF COAXIAL WAVEGUIDES 15

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

E z (r, ϕ, z) = A sin mπ(r − R ^I ) R O − R I

sin(nϕ) cos(nϕ)

e ^∓jk

^z

^z . (2.29) Analogously, for TE-modes we obtain

H z (r, ϕ, z) = A cos mπ(r − R ^I ) R O − R I

sin(nϕ) cos(nϕ)

e ^∓jk

^z

^z . (2.30) For thin coaxial waveguides it is convenient to introduce a new radial coordinate such that it is equal to zero on the inner boundary of the waveguide, i.e. to define a differential radial coordinate ρ = r − R ^I . Using the definitions of a, b and ρ, we can rewrite the expression for the longitudinal electric field in case of the TM-modes (2.29) as follows

E z (r, ϕ, z) = A sin mπρ a

sin(nϕ) cos(nϕ)

e ^∓jk

^z

^z . (2.31) as well as the expression for the longitudinal magnetic field in case of the TE-modes (2.30) as follows

H z (r, ϕ, z) = A cos mπρ a

sin(nϕ) cos(nϕ)

e ^∓jk

^z

^z . (2.32) The wave impedance for TM mn -modes can now be rewritten as

Z TM = k z

ω = k z

ω√µ r µ

= k z η

k , (2.33)

while the wave impedance for TE mn -modes becomes Z TE = ωµ

k z

= ω√µ k z

= kη k z

. (2.34)

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

k _T(m,n) ² = m ² π ²

(R O − R ^I ) ² + 4n ²

(R O + R I ) ² = m ² π ² a ² + 4n ²

b ² , (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

k z(m,n) = r

ω ² µ − m ² π ² a ² − 4n ²

b ² . (2.36)

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16 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

The cutoff frequencies f c , under which the modes cannot propagate, for either TM mn -modes or TE mn -modes, are then given by

f c(m,n) = v 2π

r m ² π ² a ² + 4n ²

b ² , (2.37)

where v is the wave propagation speed in the isotropic medium, which in case of the free space ( = 0 and µ = µ 0 ) 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 the previous section, the coaxial waveguide allows for propagation of a TEM-mode with m = 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 the higher-order modes, we note that by assumption we have a b/2, which implies that the second term on the right-hand side of the equation (2.35) is typically much smaller than the first term, i.e. 4n ² /b ² m ² π ² /a ² , for m being of the same order of magnitude as n. Furthermore, we note that higher-order TM-modes cannot exist for m = 0, since the longitudinal component of the electric field (2.29) is identically equal to zero for m = 0. On the other hand, the longitudinal component of 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 dominant TEM-mode, the first few higher-order modes are TE 0n -modes with approximate cutoff-frequencies

f c(0,n) = nc

πb . (2.38)

TE _0n -modes (TE n -modes)

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

H z (ρ, ϕ) = H 0 cos nϕ e ^∓jk

^z

^·z , (2.39)

E r = jωµb

2n H 0 sin nϕ e ^∓jk

^z

^·z , (2.40) H ϕ = ∓ jk z b

2n H 0 sin nϕ e ^∓jk

^z

^·z . (2.41) It is now convenient to introduce the following base functions

ψ n (ϕ) = r 2Z TE

πab sin nϕ = r 2kη

πabk z

sin nϕ , (2.42)

which are normalized to Z TE , i.e.

Z R

^O

R

^I

rdr Z 2π

0 dϕ ψ ² _n (ϕ) = r ² 2

R

^O

R

I

2Z TE

πab Z 2π

0 sin ² (nϕ)dϕ =

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2.3. 2D DEFORMATIONS IN A THIN COAXIAL WAVEGUIDE 17

1 2 (R ² _O − R ² I ) · 2Z TE

πab · Z 2π

0 1 − cos 2nϕ

2 dϕ = ab Z TE

ab 1 2π

Z 2π 0

dϕ = Z TE · 2π

2π = Z TE , (2.43) where we used R ² _O − R ² I = (R O − R I )(R O + R I ) = ab. Thus we can define the base fields as follows

E Tn = P n ψ n (ϕ)ˆ r , (2.44)

H Tn = ∓ P n

Z TE

ψ n (ϕ) ˆ ϕ = ∓ k zn

kη ψ n (ϕ) ˆ ϕ , (2.45)

H zn = 2 jωµb

∂E Tn

∂ϕ = − 2j b · 1

ω√µ 1 p _µ

ψ ⁰ _n (ϕ) = − 2j

bkη ψ ⁰ _n (ϕ) , (2.46) where P n is a dimensional constant, which explicit form is not essential for the present 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 shown in Fig. 2.2(b). The perturbation function can here be written as

ρ = r − R I = ag(ϕ, z) , |g(ϕ, z)| max 1 , (2.47) Introducing here the book-keeping perturbation parameter δ, to be set equal to unity at the end, we can write

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

The outwardly directed unit normal vector ˆ n on the perturbed metallic surface is given by

ˆ

n = − ∇[aδg − ρ]

|∇[aδg − ρ]| ∝ ∇[aδg − ρ] , (2.49) or

ˆ

n ∝ aδ∇g − ˆρ = ˆ n 0 + aδ∇g , (2.50)

where ˆ n 0 = −ˆρ is the normal unit vector on the unperturbed surface, as indicated in Fig. 2.2(b). The boundary condition on the perturbed metallic surface is

ˆ

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

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18 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

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

E(ρ, ϕ, z) =

∞

X

m=0

∂ ^m E

∂ρ ^m (0, ϕ, z) ρ ^m

m! . (2.52)

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

∞

X

m=0

∂ ^m E

∂ρ ^m (0, ϕ, z) (ag) ^m

m! δ ^m . (2.53)

Next, we introduce the perturbation series E =

∞

X

p=0

E _p δ ^p , (2.54)

such that the equation (2.53) becomes

E(aδg, ϕ, z) =

∞

X

p=0

∞

X

m=0

∂ ^m E _p

∂ρ ^m (0, ϕ, z) (ag) ^m

m! δ ^m+p . (2.55)

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

∞

X

p=0

∞

X

m=0

(ag) ^m

m! δ ^m+p (ˆ n 0 + aδ∇g) × ∂ ^m E p

∂ρ ^m (0, ϕ, z) = 0 . (2.56) To the zeroth order in δ, this gives the unperturbed boundary condition

ˆ

n 0 × E ⁰ (0, ϕ, z) = 0 . (2.57)

To the first order in δ, we obtain ˆ

n 0 × E 1 (0, ϕ, z) = −a[g ˆ n 0 × ∂E 0

∂ρ (0, ϕ, z) + ∇g × E 0 (0, ϕ, z)] . (2.58) Using here a × (b × c) = b · (a · c) − c · (a · b), i.e.

ˆ

n 0 × (ˆ n 0 × E ⁰ ) = ˆ n 0 · (ˆ n 0 · E ⁰ ) − E ⁰ · (ˆ n 0 · ˆ n 0 ) , (2.59)

we can write

E 0 = ˆ n 0 · (ˆ n 0 · E 0 ) − ˆ n 0 × (ˆ n 0 × E 0 ) . (2.60)

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2.3. 2D DEFORMATIONS IN A THIN COAXIAL WAVEGUIDE 19

From (2.57), we know that ˆ n 0 × E 0 (0, ϕ, z) = 0, such that

E 0 (0, ϕ, z) = ˆ n 0 (ˆ n 0 · E 0 ) . (2.61)

Substituting (2.61) into (2.57), we obtain ˆ

n 0 × E ¹ (0, ϕ, z) = −a

g ˆ n 0 × ∂E 0

∂ρ (0, ϕ, z) + ∇g × ˆ n 0 (ˆ n 0 · E ⁰ (0, ϕ, z))

, (2.62)

or rearranging (∇g × ˆ n 0 = −ˆ n 0 × ∇g) ˆ

n 0 × E 1 (0, ϕ, z) = aˆ n 0 ×

∇g ˆ n 0 · E 0 (0, ϕ, z) − g ∂E 0

∂ρ (0, ϕ, z)

. (2.63)

In the next three subsections, we will analyze the scattering between the different modes of a thin coaixal waveguide. According to Section 2.1, the dominant mode in the thin coaxial waveguide is the TEM-mode while the first few higher-order modes are TE 0n -modes. Thus, the scattering processes of interest include scattering from the TEM mode to TEM mode, the TEM mode to the TE 0n -modes and possibly the TE m -modes to TE n -modes. Each of these possible scattering situations is presented in a dedicated subsection.

TE m -modes to TE n -modes

Let us now consider the TE n -modes with base fields

E ^± _n (r) = E ^± _Tn (r) = P n ψ n (ϕ) e ^∓jk

^z

^·z ρ ˆ , (2.64)

H ^± _n = H ^± _Tn (r) + H _zn ^± z ˆ = P n

kη

∓k zn ψ n (ϕ) ˆ ϕ − 2j b ψ _n ⁰ (ϕ)ˆ z

e ^∓jk

^z

^·z , (2.65)

with

ψ n (ϕ) =

r 2kη πabk z

sin nϕ , (2.66)

where again P n is a dimensional constant, which explicit form is not essential for the present discussion, as these constants cancel out in the scattering formulae. In fact, it is also legitimate to set P n = 1, since it will not affect the results of the present study. To the first order of perturbation, with δ = 1, the fields can be written as

E(ϕ, z) = E 0 (ϕ, z) + E 1 (ϕ, z) , (2.67)

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20 CHAPTER 2. DIRECT EM SCATTERING PROBLEM

H(ϕ, z) = H 0 (ϕ, z) + H 1 (ϕ, z) . (2.68)

The zeroth-order fields are E 0 (ϕ, z) = X

n

{c ⁺ n [E Tn (ϕ) + E zn (ϕ)ˆ z ]e ^−jk

^zn

^z + c ⁻ _n [E Tn (ϕ) − E zn (ϕ)ˆ z]e ^+jk

^zn

^z } , (2.69)

H 0 (ϕ, z) = X

n

{c ⁺ n [H Tn (ϕ)+H zn (ϕ)ˆ z]e ^−jk

^zn

^z +c ⁻ _n [−H Tn (ϕ)+H zn (ϕ)ˆ z]e ^+jk

^zn

^z } , (2.70) Outside the deformation region, the first-order perturbation fields are

E 1 (ϕ, z) = X

m

{d ⁺ m [E Tm (ϕ)+E zm (ϕ)ˆ z]e ^−jk

^zm

^z +d ⁻ _m [E Tm (ϕ)−E zm (ϕ)ˆ z]e ^+jk

^zm

^z } , (2.71)

H 1 (ϕ, z) = X

m

{d ⁺ m [H Tm (ϕ)+H zm (ϕ)ˆ z]e ^−jk

^zm

^z +d ⁻ _m [−H Tm (ϕ)+H zm (ϕ)ˆ z]e ^+jk

^zm

^z } . (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 of the first-order perturbation fields (2.71)-(2.72). The non-negative integers n and m are labels of the waveguide modes in the zeroth- and first-order perturbation expansions, 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 the interval 0 ≤ z ≤ d and are therefore valid outside the deformation region, i.e. for z ≤ 0 and z ≥ d. On the other hand, the equations (2.69)-(2.70) are valid in the entire studied waveguide region.

By means of the excitation theorem for hollow PEC waveguides ([26], section 8.12), we can write the following expression for the d ^± _m coefficients

d ^± _m = R

S [ ˆ n 0 × E 1 (0, ϕ, z)] · H ^± m dS 2 R

S [E Tm × H ^Tm ] · ˆzdS . (2.73) For TE n -modes, we have (2.64),

E ^± _0n = c ^± _n ψ n (ϕ) e ^∓jk

^zn

^·z ρ ˆ , (2.74)

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2.3. 2D DEFORMATIONS IN A THIN COAXIAL WAVEGUIDE 21

such that

∂E 0

∂ρ (0, ϕ, z) = 0 , ˆ n 0 = −ˆρ . (2.75) Using (2.75) in (2.63) gives

ˆ

n 0 × E ¹ (0, ϕ, z) = a(ˆ ρ × ∇g)ˆρ · E ⁰ (0, ϕ, z) , (2.76)

or using (2.74) ˆ

n 0 × E 1 (0, ϕ, z) = a(ˆ ρ × ∇g)c ^± n ψ n (ϕ) e ^∓jk

^zn

^·z . (2.77)

Let us now use the vector identity

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

(ˆ ρ × ∇g) · H ^± m = (H ^± _m × ˆρ) · ∇g . (2.79) Using here (2.65), we can calculate