UNIVERSIDADE DO PORTO Inverse Scattering Techniques for the Synthesis of Microwave Structures by M´ario Rui Silveira Pereira A thesis submitted in candidature for the degree of Doctor of Philosophy in the Faculdade de Engenharia Departamento de Engenharia Electrot´ecnicae de Computadores February 2014 Thesis Identification Title: Inverse Scattering Techniques for the Synthesis of Microwave Structures Keywords: Transmission Line, Inverse Scattering, Layer-Peeling, Group Delay, Wide-band, Microstrip, Directional Couplers, Filters, Impedance Transformers Start: October 2006 Candidate information: Name: M´arioRui Silveira Pereira e-Mail: [email protected] Curriculum Vitae: http://www.fe.up.pt/ dee06002/cv.pdf e Supervisor information: Name: Henrique Manuel de Castro Faria Salgado e-Mail: [email protected] Curriculum Vitae: http://www.fe.up.pt/ hsalgado/cv.pdf e Educational Establishment: Faculdade de Engenharia da Universidade do Porto Research Institution: INESC Porto Co-Supervisor information: Name: Jos´eRocha Pereira e-Mail: [email protected] Curriculum Vitae: http://www.it.pt/ jrp/cv.pdf e Educational Establishment: Universidade de Aveiro Research Institution: Instituto de Telecomunica¸c~oes iii Statement of Originality The work presented in this thesis was carried out by the candidate. It has not been presented previously for any degree, nor is it at present under consideration by any other degree awarding body. Candidate: (M´arioRui Silveira Pereira) Advisor: (Henrique Manuel de Castro Faria Salgado) v Statement of Availability I hereby give consent for my thesis, if accepted, to be available for photocopying and for interlibrary loan, and for the title and summary to be made available to outside organizations. Candidate: (M´arioRui Silveira Pereira) vii Acknowledgements First, I would like to express my sincere gratitude to my supervisor, Professor Hen- rique Salgado and co-supervisor Professor Jos´eRocha Pereira for the continuous support of my research and for their patience, motivation and immense knowledge. I am also grateful to Instituto de Engenharia de Sistemas e Computadores do Porto for providing me the technical conditions necessary to perform my research work. I acknowledge the support from Funda¸c~aopara a Ci^enciae a Tecnologia for the PhD scholarship. I also would like to thank my friends Lu´ısPessoa, Jo~aoOliveira, and Qi Luo for their support and all fruitful discussions. Last, but certainly not least, I am indebted to my family for their encouragement and constant support. ix Summary Today, due to the ever-increasing data-rate needs there is a high demand for faster, more reliable and ubiquitous microwave systems. Consequently, broad-band and multi- band applications are renewing the interest in the synthesis and design of novel planar ultra-wideband devices with enhanced capabilities. In this context, inverse scattering techniques for the synthesis of high performance passive microwave devices, emerge as a pertinent and timely subject. Inverse scattering theory is widely used throughout many areas of physics and deals with determining the characteristics of a medium from the limited knowledge of the fields propagating through that medium. In the particular case of electromagnetic theory, the inverse scattering problem is concerned with finding the electrical properties of a medium from the associated scattering data such as, the reflection or transmission coefficients. It is the objective of this thesis to present a solid working base regarding direct scat- tering and inverse scattering subjects and also to push the current boundary regarding the synthesis of high performance structures by introducing innovative methods/proce- dures, tools and novel microwave passive devices based on the inverse scattering theory. This thesis addresses an in-depth analysis of numerical and analytical direct scatter- ing methods as well as a complete analysis regarding inverse scattering procedures and methods. As a major contribution from this work, a new theoretical framework regard- ing the parallelism between discrete nonuniform transmission lines and digital filters is established opening new frontiers on the synthesis of microwave devices through inverse scattering. Finally, several microwave passive devices have been designed employing the pro- posed inverse scattering techniques and validated through both numerical and electro- magnetic simulations as well as by experimental measurements. xi Sum´ario Nos dias de hoje s~aocada vez maiores os requisitos de d´ebito dos sistemas de comu- nica¸c~oessem fios exigindo o recurso a tecnologias de micro-ondas cada vez mais r´apidas, fi´aveis e dispon´ıveis. Neste sentido, tem crescido o n´umerode aplica¸c~oesde banda-larga e multi-banda que exigem a s´ıntese e projeto de novos dispositivos planares de banda muito larga contendo carater´ısticasmais exigentes. E´ neste contexto que surge de forma pertinente o t´opicoda s´ıntese de dispositivos passivos de micro-ondas de elevado desem- penho recorrendo a t´ecnicasde dispers~aoinversa. A teoria da dispers~aoinversa ´eamplamente usada em diversas ´areasda f´ısica e refere-se ao c´alculodas carater´ısticas de um determinado meio atrav´esdo conhecimento limitado dos campos que se propagam nesse meio. No caso particular da teoria eletro- magn´etica,a teoria da dispers~aoinversa dedica-se `adetermina¸c~aodas propriedades el´etricasde um meio atrav´esde informa¸c~aode dispers~aoassociada, seja atrav´esde par^ametrosde reflex˜aoou transmiss~ao. Pretende-se nesta tese apresentar uma base de trabalho s´olida sobre os assuntos de dispers~aodireta e inversa bem como fazer avan¸caro atual limite de conhecimento sobre a s´ıntese de dispositivos micro-ondas de elevado desempenho, ao introduzir novos m´etodos, procedimentos, e ferramentas baseados na teoria da dispers~aoinversa. Em particular, realiza-se uma an´alisedetalhada de m´etodos num´ericose anal´ıticosde dispers~aodireta bem como uma an´alisecompleta sobre procedimentos e m´etodos de dispers~aoinversa. Como contribui¸c~aodestacada deste trabalho, refere-se a proposta de uma nova base te´oricaque permite um paralelismo entre a teoria de linhas de transmiss~aodiscretas n~ao uniformes e filtros digitais, abrindo novas fronteiras no campo da s´ıntese de dispositivos de micro-ondas atrav´esde dispers~aoinversa. Finalmente, s~aoprojetados v´ariosdispositivos passivos de micro-ondas recorrendo `ast´ecnicasde dispers~aoinversa propostas, e apresentada a sua valida¸c~aoatrav´es de simula¸c~oesnum´ericase electromagn´eticasbem como atrav´esde medi¸c~oesexperimentais. xiii Contents 1 Introduction1 1.1 Thesis organization...............................3 1.2 Contributions..................................4 2 An overview on transmission line theory and network analysis7 2.1 Introduction...................................7 2.2 General solutions for electromagnetic waves.................8 2.2.1 Maxwell equations...........................8 2.2.2 Wave equation.............................9 2.2.3 Plane waves in lossless mediums................... 10 2.2.4 Plane waves in lossy mediums..................... 11 2.2.5 General guided wave solutions.................... 12 2.2.5.1 TE waves........................... 16 2.2.5.2 TM waves.......................... 17 2.2.5.3 TEM waves.......................... 18 2.3 Transmission lines............................... 20 2.3.1 Waves on a transmission line..................... 23 2.3.2 The lossless line............................ 25 2.4 The scattering matrix............................. 26 2.4.1 S-parameters and time average power................ 28 2.4.2 Generalized scattering parameters.................. 31 2.5 The wave-amplitude transmission matrix................... 34 2.6 Summary.................................... 35 3 Direct scattering 37 3.1 Introduction................................... 37 3.2 Coupled mode theory............................. 37 3.2.1 The cross-section method....................... 39 3.2.1.1 Uniform waveguides mode expansion........... 40 3.2.1.2 Closed Waveguides...................... 41 3.2.1.3 Open Waveguides...................... 41 3.2.1.4 Coupled-mode equation for nonuniform waveguides... 42 3.2.1.5 Coupled-mode equations for nonuniform transmission lines 44 3.2.2 Riccati equation for NTLs....................... 49 3.2.3 Weak coupling............................. 52 3.2.4 Constant coupling........................... 55 xv Contents xvi 3.2.5 Periodic coupling............................ 59 3.2.6 Iterative solution............................ 66 3.3 Transmission matrix method......................... 70 3.3.1 Constant coupling discretization................... 71 3.3.2 Constant impedance discretization.................. 73 3.4 Summary.................................... 75 4 Inverse scattering 77 4.1 Introduction................................... 77 4.2 Inverse scattering................................ 77 4.2.1 The Schr¨odingerequation inverse scattering problem........ 79 4.2.1.1 Realizability conditions................... 84 4.2.1.2 Analytical solutions for the potential function...... 86 4.2.2 The Zakharov-Shabat inverse scattering problem.......... 97 4.2.2.1 Analytical solutions for the impedance profile....... 109 4.2.2.2 Series solution - iterative algorithm............ 115 4.3 The layer-peeling algorithm.......................... 117 4.3.1 Continuous layer-peeling........................ 118 4.3.2 Discrete layer-peeling......................... 129 4.4 Summary.................................... 142 5 Microwave applications of inverse scattering