Analysis and Design of Microwave Reconfigurable Filters in Esiw Technology

Home , Waveguide filter

FACULTY OF ENGINEERING AND SUSTAINABLE DEVELOPMENT

ANALYSIS AND DESIGN OF MICROWAVE RECONFIGURABLE FILTERS IN ESIW TECHNOLOGY

Mar´ıa Trinidad Julia´ Morte 910411-T308 Juan Rafael Sanchez´ Mar´ın 911111-T275

September 2015

Master’s Thesis in Electronics

“After climbing a great hill, one only ﬁnds that there are many more hills to climb.” Nelson Mandela

To my brother, wherever you are you will be always proud of me. JR

Acknowledgement

We would like to thank to our supervisor Prof. Carmen Bachiller for the continuous sup- port of our Master’s Thesis and related research, for her patience, motivation, and immense knowledge. Her guidance helped us in all the time of research and writing of this thesis. We could not have imagined having a better supervisor.

Furthermore, we reserve our biggest thanks to our families for encouraging us in our experience in Gavle.¨ They have been always supporting us throughout writing this thesis and in our life in general.

Juanra and Mar´ıa

Abstract

Microwave filters are essential components in high frequency communication systems. The features required for these devices are increasing due to the increase in frequency, caused by the saturation of the electromagnetic spectrum. Among these features, the need for low cost devices with a reduced mass and volume and the need to integrate them with the current planar technology are highlighted. In addition, it is interested to have reconfigurable devices, that is, they can adjust their frequency response, replacing the need of multiple devices. So far, metallic waveguide filters have been used, but lately Substrate Intregated Waveguide (SIW) technology has appeared, and within this family, the innovative Empty Substrate Integrated Waveguide (ESIW). ESIW technology is empty, so that, it can be filled with liquid crystal (LC). The LC is a dielectric material with variable permittivity, which makes the filter can change its center frequency and bandwidth. There are several commercial tools based on numerical methods that enable to carry out the analysis and design of these structures, but they require a very high computational time during the analysis process. This affects negatively the automated design of these structures. On the one hand, an efficient and accurate analysis tool is developed in the thesis by following a strategy that consists on dividing the device under analysis in simple building blocks: waveguides filled with dielectric material, change of medium, discontinuities between guides and dielectric discontinuities. All of them are canonical structures or sufficiently simple that can be analyzed with modal methods. The Generalized Scattering Matrix (GSM) of each block is obtained, and they are linked in cascade by using an efficient technique. The accuracy and effectiveness of this tool are checked using it to analyze multiple filters, comparing the result with a commercial software. Furthermore, an analysis about how the variation of the permittivity of the liquid crystal affects in the frequency response of the filter is made. On the other hand, a tool for designing filters for high frequency communications is developed, in order to integrate it into a Computer Aided Design (CAD) tool. To do this, the classical techniques for designing waveguide filters are followed by adapting them to the new topology under design. Different synthesis and optimization strategies are implemented. These strategies are based on the synthesis of a starting point, the segmentation of the structure under design and the hybridization of different optimization algorithms. Also, a tool for calculating the values of the permittivity of the LC that allows the reconfiguration of the filter is developed. These new permittivities enable to obtain a new desired frequency response. VIII

This tool has been used to design various microwave filters on H plane, using ESIW technology filled with liquid crystal. The accuracy and the effectiveness of this tool are checked. In addition, the reconfiguration tool designed is evaluated. Finally, a study of the manufacturing aspects is done, such as how to feed the filter and the solutions to polarize the liquid crystal. Taking into account all these considerations a prototype is manufactured and measured. Index

Abstract ...... VII

List of ﬁgures XI

List of tables XV

1 Introduction 1 1.1 Motivation and aims ...... 1 1.2 Structure of the thesis ...... 2 1.3 State of the Art ...... 4 1.3.1 Microwave filters ...... 4 1.3.2 Reconfigurable filters ...... 7 1.3.3 ESIW technology ...... 8

2 Analysis of reconfigurable filters 11 2.1 Analysis methods for waveguide filters ...... 11 2.1.1 Modal methods ...... 12 2.1.2 Numerical methods ...... 12 2.1.3 Hybrid methods ...... 13 2.2 Analysis methodology of reconfigurable filters ...... 13 2.2.1 Analysis method of the change of medium ...... 15 2.2.2 Analysis method of multiple discontinuities ...... 18 2.2.3 Analysis method of steps between waveguides ...... 24 2.2.4 Generalized Scattering Matrices connection method ...... 26 2.3 Validation of the analysis tool ...... 28 2.3.1 Filter analysis ...... 28 2.3.2 Permittivity variation of the liquid crystal ...... 31

3 Design of reconfigurable filters 43 3.1 Specifications ...... 45 3.2 Synthesis ...... 45 3.2.1 Lowpass prototype ...... 46 3.2.2 Scaling and frequency conversion ...... 48 3.2.3 Implementation of the filter ...... 50 X Index

3.3 Optimization ...... 53 3.3.1 Error function ...... 54 3.3.2 Optimization strategies ...... 55 3.3.3 Optimization strategy used ...... 63 3.4 Validation of the design tool ...... 64 3.4.1 Two cavities filter ...... 66 3.4.2 Four cavities filter ...... 68 3.5 Reconfiguration ...... 71 3.6 Validation of the reconfiguration tool ...... 73 3.6.1 Two cavities filter ...... 73 3.6.2 Four cavities filter ...... 75

4 Technological aspects of manufacturing 83 4.1 Filter feed ...... 83 4.2 Liquid crystal and its microwave applications ...... 87 4.2.1 Liquid crystal properties ...... 89 4.2.2 Liquid crystal encapsulation ...... 89 4.2.3 Liquid crystal polarization ...... 90 4.3 Choke Inductor ...... 92

5 Results and measurements 95 5.1 Redesign of the ﬁlter ...... 95 5.2 Manufacturing process ...... 100 5.3 Measurements ...... 108

6 Conclusions and future research lines 113

A Calculation of the Z matrices of multiple discontinuities 117

B Resolution of the connection method for Generalized Scattering Matrices (GSM)121

C Routines 125 C.1 Analysis routines ...... 125 C.2 Design routines ...... 129 List of Figures

1.1 ESIW filter filled with liquid crystal...... 2 1.2 Coupled cavities H-plane filter...... 5 1.3 Coupled cavities filter with dielectric resonators posts...... 5 1.4 SIW scheme...... 8 1.5 ESIW layout...... 9

2.1 N cavities filter filled with dielectric material...... 11 2.2 Filter sections for analysis...... 14 2.3 Change of medium and reference system...... 15 2.4 Multiple discontinuities and reference system...... 18 2.5 Detail of the current waves in the structure...... 20 2.6 Specific case for three discontinuities...... 22 2.7 Multiple discontinuities with voltage waves...... 23 2.8 Section step analyzed with Mode Matching...... 25 2.9 Cascade connection of N dispersion matrices with the new method...... 27 2.10 CST configuration to simulate the filter...... 29 2.11 Frequency response of the 2 cavities filter. (a)S11. (b)S21...... 30 2.12 Frequency response of the 4 cavities filter. (a)S11. (b)S21...... 31 2.13 Frequency response of the two cavities filter for case 1...... 32 2.14 Frequency response of the two cavities filter for case 2...... 33 2.15 Frequency response of the two cavities filter for case 3...... 34 2.16 Frequency response of the two cavities filter for case 4...... 35 2.17 Frequency response of the two cavities filter for case 5...... 36 2.18 Frequency response of the two cavities filter for case 6...... 37 2.19 Frequency response of the four cavities filter for case 1...... 37 2.20 Frequency response of the four cavities filter for case 2...... 38 2.21 Frequency response of the four cavities filter for case 3...... 38 2.22 Frequency response of the four cavities filter for case 4...... 39 2.23 Frequency response of the four cavities filter for case 5...... 39 2.24 Frequency response of the four cavities filter for case 6...... 40 2.25 Frequency response of the four cavities filter for case 7...... 40 2.26 Frequency response of the four cavities filter for case 8...... 41 2.27 Frequency response of the four cavities filter for case 9...... 41 XII LIST OF FIGURES

3.1 Design flow of a computer-aided design...... 44 3.2 Synthesis process flow...... 45 3.3 Chebyshev ideal response. (a) Two cavities. (b) Four cavities...... 46 3.4 Lowpass normalized prototype. (a) Prototype starting with a shunt element. (b) Prototype starting with a series element...... 47 3.5 Low-pass to bandpass prototype transformation. (a) Transformation of series inductive element. (b) Transformation of shunt capacitive element...... 49 3.6 Filter scheme using impedance inverters...... 50 3.7 Waveguide filter filled with dielectric...... 51 3.8 Analyzed section for the calculation of the coupling window width...... 52 3.9 Formation of the initial simplex...... 58 3.10 Ideal Chebyshev response. (a) Two cavities. (b) Four cavities...... 65 3.11 Two cavities prototype...... 66 3.12 Frequency response of the 2 cavities filter. (a) S11. (b) S21...... 69 3.13 Four cavities prototype...... 70 3.14 Frequency response of the 4 cavities filter. (a) S11. (b) S21...... 71 3.15 Frequency response of a filter of two cavities centered at 11.3 GHz...... 74 3.16 Frequency response of a filter of two cavities centered at 11.2 GHz...... 75 3.17 Frequency response of a filter of two cavities centered at 11.1 GHz...... 76 3.18 Frequency response of a filter of two cavities centered at 10.9 GHz...... 77 3.19 Frequency response of a filter of two cavities centered at 10.8 GHz...... 78 3.20 Frequency response of a filter of two cavities centered at 10.7 GHz...... 78 3.21 Frequency response of a filter of four cavities centered at 11.2 GHz...... 79 3.22 Frequency response of a filter of four cavities centered at 11.1 GHz...... 79 3.23 Frequency response of a filter of four cavities centered at 10.9 GHz...... 80 3.24 Frequency response of a filter of four cavities centered at 10.8 GHz...... 80 3.25 Frequency response of a filter of four cavities centered at 10.7 GHz...... 81

4.1 Electric ﬁeld distribution of the dominant mode. (a) Microstrip line. (b) Rectangular waveguide ESIW...... 84 4.2 Coplanar transition scheme...... 85 4.3 Linear taper transition scheme...... 85 4.4 Inverted taper transition scheme...... 86 4.5 Exponential transition scheme (3D view)...... 86 4.6 (a) Taper layout (top view). Dark gray is dielectric substrate; light gray represents the copper metallization on top layer; black represents the border metallization which has been used to close the ESIW; and white is air. (b) Detail of the taper end...... 87 4.7 Transition scheme (3D view)...... 88 4.8 (a) Transition layout (top view). Light gray is dielectric substrate; dark gray represents the copper metallization on top layer; black represents the border metallization which has been used to close the ESIW; and white is the liquid crystal. (b) Detail of the taper end...... 88 LIST OF FIGURES XIII

4.9 Liquid crystal polarization...... 89 4.10 An inverted-microstrip structure for a LC bandpass ﬁlter...... 90 4.11 Surface currents distribution of mode TE10 in a rectangular waveguide. . . . 91 4.12 Layout of an ESIW with the upper cover cut in small squares...... 91 4.13 Result of the simulation of the ESIW with the upper cover cut in small squares. 92 4.14 Admittance of the choke inductors over the frequency...... 93 4.15 Shape of the Choke inductor and dimensions...... 94 4.16 Shape of the cover for feeding the choke inductor. Light gray is the substrate and dark gray represents the copper metallization in the upper layer...... 94

5.1 Ideal and optimized frequency response of the redesigned filter...... 96 5.2 Layout of upper cover of the optimum filter with the dimensions on it. . . . . 97 5.3 Frequency response of the optimum filter...... 98 5.4 Frequency response of the optimum transition...... 99 5.5 Layout of the complete filter with the transition...... 100 5.6 Result of the simulation of the ESIW with the upper cover cut in small squares for εr = 2.4...... 101 5.7 Result of the simulation of the ESIW with the upper cover cut in small squares for εr = 3...... 102 5.8 Result of the simulation of the empty filter...... 102 5.9 View of the computer with the CircuitPro software connected to the milling machine ProtoMat S103...... 103 5.10 Steps of the manufacturing process in CST Studio...... 104 5.11 Layer of the polarization holes in the upper cover modeled in CST Studio. . . 104 5.12 Polarization holes in the upper cover...... 105 5.13 Removing the adhesive material with the milling machine...... 105 5.14 Metallization of the filter. (a) Metallization of sidewalls and holes. (b) Pro- Conduct Paste...... 106 5.15 Layer of the union holes modeled in CST Studio...... 106 5.16 Lower cover with the union holes...... 107 5.17 Layer of the squares modeled in CST Studio. (a) Uncoupling square. (b) Small squares...... 107 5.18 Squares. (a) Uncoupling square. (b) Small squares...... 108 5.19 Soldering the polarization holes...... 109 5.20 Layer of the isolating holes modeled in CST Studio...... 109 5.21 Layer of the contour of the filter modeled in CST Studio...... 110 5.22 Removing the adhesive material from the central body with the milling machine.110 5.23 Substrate in the oven to affix the ProConduct paste...... 111 5.24 Layer of the exponential transition modeled in CST Studio...... 111 5.25 Layer of the linear transition modeled in CST Studio...... 111 5.26 Central body manufactured...... 112 5.27 Complete manufactured filter...... 112 5.28 Result of the measurement of the empty filter...... 112

List of Tables

2.1 Computational cost comparative of a 2 cavities ﬁlter...... 29 2.2 Computational cost comparative of a 4 cavities ﬁlter...... 30

3.1 Characteristics of each step of the segmentation strategy...... 63 3.2 Stages in the optimization process of 2 cavities filter...... 68 3.3 Stages in the optimization process of 4 cavities filter...... 70 3.4 Values of the reconfiguration parameters for two cavities filter...... 73 3.5 Values of the permittivites for the two cavities filter...... 74 3.6 Values of the reconfiguration parameters for four cavities filter...... 76 3.7 Values of the permittivites for the four cavities filter...... 76

4.1 Characteristics of the inductance LQW18ANR16G0Z...... 94

5.1 Values of the diamaters of polarization holes...... 101

Chapter 1

Introduction

1.1 Motivation and aims

The purpose of this Master Thesis is the analysis, the design, and the manufacturing of reconfigurable microwave filters on H plane geometry, which can be used in applications of high frequency communications. A particular type of filters is studied: those based on Empty- Substrate Integrated Waveguide (ESIW) technology [1], on H plane, filled with dielectric with variable permittivity, since they meet the specifications of the filters that are commonly required in high frequency applications. The origin of this Master Thesis arises from the natural development of the research lines initiated in the Microwave Applications Group (GAM) attached to the Institute of Telecom- munications and Multimedia Applications (iTEAM) of the Polytechnic University of Valen- cia (UPV). On the one hand, the doctoral thesis by Dr. Carmen Bachiller Martin [2] analyzes waveguides with arbitrary cross section and waveguides with dielectric posts, improving techniques developed previously by members of that group [3] [4] and linking them efficiently. However, that developed method does not allow the analysis of transversal sections of the waveguide completely filled with dielectric with variable characteristics. On the other hand, the doctoral thesis by Dr. Jose´ Vicente Morro [5] allows to design, in an automated way and efficiently, different topologies of microwave filters with a Computer Aided Design (CAD) tool. The analysis techniques developed by the members of the same group [2], [3], [4] are integrated in this tool. However, the developed methods do not allow the design of filters with cavities filled with a dielectric of variable characteristics. This thesis has three main aims. The first one is to develop a tool for an efficient and accurate analysis for such reconfigurable filters in ESIW technology, it means, filters with waveguide sections completely filled with liquid crystal (LC) [6], [7], [8], [9], [10]. LC acts as a dielectric material with variable permittivity in the different filter areas. This topology is shown in Figure 1.1. Afterward, the second aim is to integrate the analysis tool in a CAD tool for designing reconfigurable filters in ESIW technology filled with dielectric. This process allows to obtain the geometric parameters of a filter given a frequency response. 2 Introduction

Figure 1.1: ESIW ﬁlter ﬁlled with liquid crystal.

Finally, the last aim is to manufacture the reconfigurable device and to measure its frequency response for different values of permittivity. Thus, the real measurements are compared to the simulated ones to check if the design specifications are met. There are two main benefits of this project, which make it particularly novel. Firstly, the analysis tool developed is faster than the current commercial analysis softwares. Secondly, it is possible to reconfigure the frequency response of the filter, due to ESIW technology filled with liquid crystal.

1.2 Structure of the thesis

In order to achieve the goals of this master thesis, the next work planning is followed.

• Study of the State of Art. It takes a comprehensive work of literature search, which helps to check the need and relevance of the work to be performed, and to ascertain the current techniques to analyze such devices. Specifically, a study of some different types of filters used in high frequency communications is performed, including classical rectangular waveguide technology, and Substrate Integrated Circuit (SIC) technology. The Substrate Integrated Waveguide (SIW) technology belongs to SIC group. A literature review about SIW technology is performed in order to know the possibilities to make a filter reconfigurable. The use of liquid crystal and its properties is further investigated, since one of the aims of this master thesis is the analysis of filters filled with this material. The use of liquid crystal in these filters entails the use of the innovative ESIW technology. Hence, it is also performed a research in depth of this technology. • Analysis of filters in ESIW reconfigurable technology. It is calculated the frequency response of the H plane cavities filter in rectangular waveguide, whose sections (cavities and inductive windows) are presented with different dielectric permittivities. For this purpose, a segmentation strategy is followed to analyze this type of structure. It consists on dividing the filter into several building blocks: waveguide sections filled 1.2 Structure of the thesis 3

with dielectric, dielectric changes in a waveguide with the same cross section, steps between different cross section waveguides, and waveguide sections with multiple dielectric discontinuities. Using a very efficient modal method [4], [11], [12] it can be obtained multimodal matrices of the scattering parameters for the steps. Multimodal matrices of waveguides filled with a dielectric material can be found in the theory [13], [14], since they are canonical structures. Furthermore, a method must be developed to get generalized multimodal scattering matrices (GSM) for both dielectric changes and multiple discontinuities sections. S21 and S11 parameters of the TE10 fundamental mode, i.e, the electrical response of the whole filter, are got efficiently by linking the GSMs of the different building blocks [2], [15]. The analyzed modes are TEm0, thus there will be matrices of S parameters in blocks, each one of size M × M, where M is the number of modes analyzed. Finally, some prototypes are analyzed to ensure that the proposed specifications are met. Its validity is checked by comparing the frequency response obtained with the simulation result of such structures using a commercial software (CST Microwave Stu- dio). In addition, their efficiency is verified by comparing the computational cost of the developed analysis tool and the commercial software.

• Design of reconfigurable filters in ESIW technology. A study is performed in order to get the process to follow when designing filters in waveguided technology. This study is necessary since for the ESIW design. The techniques commonly known for resonant cavities filters in rectangular waveguide can be used. Afterward, tools for designing filters filled with dielectric are developed. To do this, synthesis algorithms based on a circuit model have been used for calculating the starting point. To optimize the frequency response, the geometric parameters of the structure to make the response as close as possible to the ideal have to be found. In this process, some well known optimization algorithms as the method of direct search with coordinates rotation, the Downhill Simplex Method and the Method of Broyden Fletcher Goldfarb Shanno have been used. These methods have been modified to improve their performance. Finally, a tool for reconfiguring the filter filled with dielectric is developed. Once the filter is manufactured, its geometric parameters cannot be changed, but nevertheless a reconfiguration can be got, i.e, a change in the frequency response of the filter, the center frequency and bandwidth. This is achieved by the liquid crystal (LC) which ESIW is filled. Applying a bias voltage to the LC, it changes its dielectric permittivity, changing the frequency response of the filter. For obtaining the values of permittivities that give the desired frequency response a specific tool is developed. This allows reconfiguring the filter once manufactured.

• Manufacturing aspects. Because of some technological problems in terms of the polarization of the liquid crystal, before manufacturing the designed ﬁlters, a design 4 Introduction

that solves these problems by keeping the dimensions of the ﬁlter is raised. This design is simulated with commercially available software, CST in this case, to verify that the solution is feasible, that is, that the ﬁlters have the desired response once manufactured.

• Results of the design, the analysis and the manufacturing of the filter. Some designed prototypes are analyzed to verify the proposed specifications are met. The validity of the tool is checked by comparing the frequency response obtained with the simulation result of such structures. The commercial software CST Microwave Stu- dio is used for this purpose. In addition, its efficiency is verified by comparing the computational costs of both the developed analysis tool and commercial software CST. The developed tool for designing prototypes is used. Specifically, two filters in ESIW technology filled with liquid crystal are designed, one of two cavities and another of four cavities. The frequency response obtained with the dimensions of the starting point, the optimized dimensions and the ideal response are compared, in order to validate the correct performance of the design tool. Finally, a re-design of the two cavities filter is done in order to have a filter with bigger dimensions. This filter is manufactured and measured. The measurement is carried out with the filter empty and filled with liquid crystal. The real measurements are compared with the simulations.

1.3 State of the Art

1.3.1 Microwave filters From the beginning, microwave filters are essential devices in whatever high frequency telecommunications system. Its basic function of radio-frequency (RF) processing is to allow the signal power transmission in a certain range of frequencies, and to delete, as much as possible, this signal transmission outside this range. The progressive increase in the electromagnetic spectrum saturation has resulted in a continuous increment of the operation frequency. This increment and the development of in- creasingly sophisticated high frequency applications require developing microwave circuits with advanced performances. Referring to filters, the possibility of high selectivity filtering on compact circuits has been always a key feature to efficiently remove noise and interfering signals outside the passband. Other features currently required for these electronic components, such as agile frequency reconfiguration, are a result of the most recent trends towards developing high frequency multifunction devices [16],[17]. Waveguide technology filters composed of metal only (Figure 1.2) has been used since the first telecommunications applications, as they allow excellent performances such as overall electromagnetic shielding (radiation losses are completely removed), low insertion losses, ability to carry high power signals and high quality factor [18]. However, this kind of filters is not appropriate to mass production due to the difficulty of assembly and operation. 1.3 State of the Art 5

Figure 1.2: Coupled cavities H-plane ﬁlter.

In order to meet requirements of reduced volume and low mass, on [2] it is proposed a new ﬁlter generation, which contains dielectric inside the cavities acting as resonators. They can be cylinders or square posts passing completely through the guide (Figure 1.3), with a stable permittivity for the operating bandwidth.

Figure 1.3: Coupled cavities ﬁlter with dielectric resonators posts.

At the operating frequency, most of the electromagnetic energy is stored within the dielectric. This causes the fields outside of dielectric tail off rapidly with distance. The resonance frequency is strongly controlled by both the dimensions and the electric permittivity of the dielectric material, whereas the quality factor, Q, is determined by the losses tangent of the dielectric. Another topology of rectangular waveguide filters consists on inserting screws or clear- ance posts in the waveguide. They can be metallic or dielectric. The screws work as the coupling windows, and the waveguide section between screws operates as resonators [2]. Evanescent H plane filters are a breakthrough in the design of high-performance filters. Such filters are small and provide significant improvements in eliminated bandwidth between the first and second resonance with respect to coupled cavities filters [19],[20]. The problem of these topologies is the difficulty of integrating the devices using microwave power circuits. In order to solve this problem and due to the need for low-cost technologies with high performance, and suitable for mass production, planar technology 6 Introduction

(printed circuits) was arose. Planar devices have a lot of advantages, since they reduce the volume, weight, and consumption of telecommunications equipments. However, they have high losses, low quality factor, and hence low filtering selectivity. In order to overcome weaknesses and limitations of both waveguide technology and planar circuit technology, and to make easy the integration between both technologies, Dominic Deslandes and Ke Wu [21] discovered and proposed an ingenious and revolutionary concept that has developed a new generation of high-frequency integrated circuits, known as Substrate Integrated Circuits (SIC). The foundation of the SIC is to synthesize non-planar structures with a planar dielectric substrate. Within this large family of integrated circuits, Substrate Integrated Waveguide (SIW) technology is a mixed technology waveguide- printed circuit which solves perfectly the shortcomings associated with traditional technologies. It integrates a rectangular waveguide in a planar substrate. This is an “artificial” waveguide synthesized by two parallel rows of metallic holes (vias), joining two planar conductors separated by a dielectric substrate of high permittivity and low losses, as shown in Figure 1.4. SIW technology respects the original specifications of the waveguide filters but getting integrated structures much smaller, significantly cheaper, and easier to manufacture. The SIW devices have much lower losses than their counterparts in traditional planar technologies, and a higher quality factor. In addition, the synthesis of SIW allows efficiently the realization of broadband transitions with planar circuits. So that, it can be achieved the integration of planar circuits and SIW circuits in the same substrate and using a planar technique production of low cost [22]. Recently, a new technology has appeared, the Empty Substrate Integrated Waveguide, or ESIW [1], which is an advance of SIW technology. In the case of ESIW, the side walls of the waveguide are continuous metal walls, instead of rows of metalized holes (see figure 1.5). This increases the quality factor and selectivity of the devices by a factor of 8, so that the performance of the ESIW is very close to that of the rectangular waveguide, while maintaining the advantages of low cost, easy manufacturing, and integration with planar circuits, since ESIW is integrated on a dielectric substrate. A common problem of these technologies is that the device may not have the desired frequency response after obtaining the designed and manufactured filter. This may be due to manufacturing tolerances or the electromagnetic characteristics of the materials used. There- fore, it is desirable to make a retuning of the device after manufacturing, that is, to adjust the center frequency and the bandwidth to meet the initial specifications. To do this, tuning screws are typically used. However, the major problem of all the filters mentioned so far is that, although they can be retuned, they are designed to operate at a single frequency and bandwidth. As stated above, the retuning only allows adjustment of the frequency response to meet specifications, i.e, they cannot change their response over a wide bandwidth, since they do not have the ability to reconfigure their operating parameters as the center frequency and bandwidth. It is useful that the filters can adjust in a controlled manner its characteristics such as the center frequency, bandwidth and/or selectivity, replacing the need for multiple devices and improving system reliability as a whole. In addition, this permits reductions in size, weight, 1.3 State of the Art 7 complexity and cost. That is, it has the ability of various microwave filters into one.

1.3.2 Reconfigurable filters The reconfiguration is one of the pillars of this project, since it allows to provide various utilities to one filter once it is made. Therefore, in this section, different possibilities to make a reconfigurable filter are studied. For reconfiguration in SIW technology different elements can be used:

• Ferrite Sections [23], [24], [25]. In the cavity of a filter, either in waveguide or SIW technology, a small section of ferrite is introduced, on which a direct current is applied. This produces a variation in the inductance of the ferrite and thus a variation in the resonance frequency of the cavity. Hence, filters can be reconfigured only by applying a direct current to the ferrite section. This reduces the complexity of the control system compared to traditional methodologies. These filters have numerous applications in advanced RF devices whose essence is stability and simple tuning.

• PIN Diodes [26]. Inside the cavities of SIW filters, posts are introduced to make a fully tunable filter. There is an optimum location for the posts that provides fine tuning and allows adaptation of several states. Such conditions are achieved by connecting and disconnecting the posts to the top metal layer of the SIW. The PIN diodes located in this layer are responsible for that switching task. To achieve this switching a direct current with different polarities is applied to the diodes. The activation of the posts in a symmetrical way achieves a greater tuning.

• Empty Ring [27]. A vacuum ring is inserted into the cavity of a SIW filter with a central metal post. This ring is connected or disconnected to ground. The insertion of capacitive posts results in bulky cavities and planar activation, which leads to reconfigurable filters with high quality factor, Q, and control conditions of simple tuning.

• Varactor Diodes [28], [29]. In the resonant cavity of the SIW ﬁlter, a varactor diode is inserted. This diode is connected with a variable DC potential difference. Depending on the potential difference applied to the diode, the capacity varies and, therefore, the resonator frequency.

• Combination of varactors and ferrites [30], [31]. Inside the cavities of the SIW ﬁlters a combination of varactors and ferrites is used. A potential difference in varactor and a current difference in ferrites are used. Thus, various parameters are improved at the same time, having greater control over parameters such as resonance frequency, bandwidth or the phase-shift of the resonator.

• Capsule with liquid crystal (LC) [7]. CL is located in a cavity of a SIC filter, in order to get the filter reconfiguration by only varying its dielectric permittivity. Applying an electric field, the permittivity of the LC varies and therefore parameters such as center frequency, f0, and bandwidth vary. This allows the reconfiguration of the filter. 8 Introduction

1.3.3 ESIW technology As shown in Figure 1.4, the SIW is a mixed technology waveguide-printed circuit that allows to synthesize an “artificial” waveguide. This technology is based on small cylindrical metallic vias (or fully metallic posts) joining two planar conductors separated by a dielectric substrate of low losses and high dielectric permittivity (the same used in the design and implementation of lines and planar devices) [21]. A suitable periodic distribution of the vias allows to “artificially” synthesize the sidewalls of an “equivalent rectangular waveguide” (see figure 1.4), confining the electromagnetic fields between the two rows of holes and the conductive layers. The conductive layers are deposited on the upper and lower side of the substrate.

Figure 1.4: SIW scheme.

For an electromagnetic wave, the SIW is similar to a waveguide filled with dielectric and reduced height. This height is reduced compared to the commonly relation used for a conventional rectangular waveguide of wide:height 2 : 1. It operates with only a single mode. This miniaturization results in a reduction of the impedance seen by the wave, thus increasing the capacitance value per unit length of the line [32]. The SIW topology improves the performance of planar circuits in terms of losses and quality factor; also it improves the performance of conventional rectangular waveguide due to its low cost manufacturing, printed circuit integration and greater compactness in the vertical dimension, because they are integrated into a printed circuit. In addition, due to the fact that they are filled with a dielectric substrate, also compaction occurs in the horizontal dimensions, since as shown in equation (1.1), for the same cutoff frequency, the presence of dielectric makes width, a, smaller. √ c / ε f = 0 r (1.1) cT E10 2a However, the electromagnetic analysis of the periodic structure is highly complex when compared with a conventional rectangular waveguide. Furthermore, compaction makes these devices more sensitive to manufacturing tolerances, which may become a serious problem 1.3 State of the Art 9 at high frequencies. The presence of the dielectric increases losses, resulting in a significant reduction of the quality factor compared with classical rectangular waveguides. In order to improve the performance of integrated circuits and given the continuous increase in the carrier frequency, a new method for manufacturing empty waveguides, without dielectric substrate, but also fully integrated in a planar substrate is proposed. This new structure, known as Empty Substrate Integrated Waveguide (ESIW), is a very promising alternative to SIW technology. It has better results in terms of losses and quality factor (quality factor measured is about 8 times higher than for the same filter in SIW technology), especially with increasing frequency [1]. In traditional SIW, electromagnetic waves are confined in a dielectric body enclosed by the metal walls of the upper and lower substrate, and lateral metallic circular holes, as shown in Figure 1.4. In ESIW, electromagnetic fields travel in vacuum, and are confined to the top, bottom and side metal walls. This structure is manufactured emptying a rectangular hole in a planar substrate. Next, the substrate is metallized using the same procedure as for the metallization through holes in SIW. Thus, the side walls of the empty waveguide are created. Finally, two thin metal walls, top and bottom are welded to the substrate (see Figure 1.5).

Figure 1.5: ESIW layout.

A broadband transition with return losses of more than 20 dB in the whole bandwidth allows the integration of the waveguide in the ﬂat substrate, so that the waveguide can be accessed directly with a microstrip line, thanks to this innovative transition [1]. This integration process is simple and inexpensive and is explained in detail in Chapter 4.

Chapter 2

Analysis of reconﬁgurable ﬁlters

The aim of this chapter is to present the analysis of reconfigurable filters in ESIW (Empty- Substrate Integrated Waveguide) technology filled with a dielectric material. It is necessary to analyze a rectangular waveguide filter with cavities and inductive windows filled with dielectric of different electric permittivities (view figure 2.1). Varying the permittivity in the sections allows to reconfigure the filter response.

Figure 2.1: N cavities ﬁlter ﬁlled with dielectric material.

Before proceeding to develop the analysis method of the reconﬁgurable ﬁlter, different analysis methods for the classic waveguide technology are studied. These methods can be applied to ESIW technology. Then, an analysis method for the aforementioned structure is developed in depth.

2.1 Analysis methods for waveguide ﬁlters

A waveguide is a structure that directs the propagation of an electromagnetic signal in a particular direction confining its energy. The waveguides consist of metal tubes with uniform section, typically circular and rectangular. Resonators can be constructed by using rectangular waveguides. They are key elements when designing filters, since they have quality factor, Q, much greater than the ones based on microstrip technology. 12 Analysis of reconfigurable filters

In order to achieve resonance in the waveguide, short-circuit boundary conditions are situated at distances of multiple half wavelength of the desired frequency. Dielectric posts may also be introduced into the waveguide to improve the characteristics of the resonator. In order to build up a bandpass filter with a waveguide, the physical structure of resonators separated by inverters must be implemented. The inverters are generated in a waveguide with inductive tuning elements, such as irises, inductive windows or metal screws [33]. High frequency communications applications require strict transfer functions with low losses in the passband, and extreme selectivity. The traditional way to get these responses has been the implementation of metallic waveguide filter structures with directly coupled cavities, as shown in figure 1.2. In these structures, the tuning inductive elements are coupling windows (or irises), and the resonators are the cavities located between these windows. There are various methods to deal with the analysis of this type of microwave structures. They are classified into three groups: analytical, numerical, and hybrid methods [2]. The numerical methods (also called spatial discretization methods) [34], [35], [36] are able to analyze problems with arbitrary geometries, but their disadvantage is the high consumption of memory and CPU time. The analytical methods (also called modal methods) [37], [11] provide very accurate results in an efficient way, i.e., with a low computational time. These methods are applicable to canonical structures with regular geometries. The hybrid methods [38], [39], [40] are a combination of the aforementioned methods. These methods enable the analysis of arbitrary geometries with higher efficiency than the numerical methods. Thanks to the strategy of segmentation of the structure under analysis, explained in further sections, modal methods are used in this project, since they are the most efficient. An enumeration and a short description of each of these methods is done below.

2.1.1 Modal methods Modal methods [12] are optimal to analyze straightforward waveguide structures. They provide a high accuracy and a low computational cost. They are also the oldest analysis methods. However, they do not allow the analysis of complex structures. The most important methods are mentioned below.

• Integral equation Method or Method of Green Function [41].

• Mode Matching Method [42].

• Generalized Immitance Matrix (GIM) Method [4],[12].

2.1.2 Numerical methods Numerical methods have been relevant when analyzing electromagnetic problems because of two main reasons. Firstly, a large number of researchers have been focused on them. Secondly, a lot of computational tools have been developed. 2.2 Analysis methodology of reconﬁgurable ﬁlters 13

These methods are classiﬁed into two groups, depending on the domain where the discretization is done. They are methods in the space-time domain, and methods in the frequency domain. The main numerical methods in the space-time domain are mentioned below.

• Finite Element Method (FEM) [43].

• Difference in the Temporal Domain Method (FDTD) [36].

• Boundary Element Method (BEM) [34].

• Transmission Line Method (T LM) [44].

When analyzing planar structures, those methods that perform the discretization in the Fourier domain are better than many numerical methods in the space-time domain. Some of them are mentioned below.

• Method of Moments (MoM) [45], [14], [46].

• Spectral Domain Analysis (SDA), and Spectral Domain Immitance Analysis (SDIA) Methods [42].

• Method of Segmentation in Planar Circuits, [47].

• Fast Multipole Method [48]

2.1.3 Hybrid methods Hybrid methods are combinations of numerical and modal ones, using the advantages of both methods, i.e, the analysis ﬂexibility of the numerical methods, and the computational efﬁciency of the modal methods. The most relevant hybrid methods are mentioned below.

• Boundary Contour Mode Matching Method (BCMM) [40].

• Boundary Integral-Resonant Mode Expansion Method (BI-RME) [49].

• New hybrid method for an efﬁcient analysis of complex ﬁlters [2].

2.2 Analysis methodology of reconﬁgurable ﬁlters

So far, several computer commercial tools based on numerical methods (such as An- soft HFSS [50] and CST Microwave Studio [51]) allow to analyze the following structures: waveguide filters in H-plane with resonant cavities, filled with dielectric materials. However, such tools require a very high computational time. This dramatically affects the automated design of the structures under analysis, because if the analysis tool and the simulation are not efficient, the automated design is too costly. 14 Analysis of reconfigurable filters

As a result of this, the development of more efficient methodologies to analyze these structures and then, to use them in Computer Aided Design (CAD) tools is considered a need for high frequency communication sector. In order to solve this problem, one of the objectives of this project is to develop a tool for efficiently and accurately analyzing ESIW filters filled out of liquid crystal sections. This results in dielectric substrates with variable characteristics in different zones of the filter. The aforementioned filter is analyzed by sections. A Generalized Scattering Matrix (GSM) is obtained for each section, and all of them are properly linked for achieved a global GSM which characterizes the filter response, i.e. the S21 and S11 parameters of the TE10 mode. The analyzed modes are TEm0, hence there are M × M blocks of S matrices. The analysis tool is entirely modal and analytical, since all the filter sections are canonical structures. In figure 2.2, it is shown a filter in H-plane filled out of dielectric. It has been segmented into the following simple building blocks.

• Waveguide sections ﬁlled with dielectric. The matrices S1 and SN are obtained analytically. They are called ﬁlled-waveguide matrices

• Dielectric changes in a waveguide sections with the same cross section. The matrices S2 and SN−1 are obtained. They are called change of medium matrices. • Steps between waveguides with different cross sections. These discontinuities give step or impedance change scattering matrices. They are S3, S5, and etcetera. • Waveguide sections with multiple dielectric discontinuities. Multiple discontinuities scattering matrices are obtained. They are S4, S6, and etcetera.

Figure 2.2: Filter sections for analysis.

The analysis of waveguide sections filled with dielectric does not have any problem and it is sufficiently discussed in the literature [13], [41]. The GSM of these waveguide sections may be obtained analytically by following the next equations. 2.2 Analysis methodology of reconfigurable filters 15

(m,n) (m,n) S11 = S22 = 0 (2.1)

e−γm l m = n S(m,n) = S(m,n) = (2.2) 21 12 0 m 6= n where γm is the propagation constant of the m-th mode. For the analysis of change the of medium in a waveguide with the same cross section and the waveguide sections with multiple discontinuities, analytical methods developed in this master thesis are used. They are described in depth in sections 2.2.1 and 2.2.2, respectively. On the other hand, in order to analyze discontinuities between waveguides, highly efficient analytical methods may be used. In this thesis, the mode matching method [52], [53], [54], [55], [56] explained in section 2.2.3 is used. Finally, using the aforementioned methods, the Generalized Scattering Matrix (GSM) of each section is obtained. These matrices have to be linked in order to obtain the global GSM by using an efficient method developed in [2], [15]. This method is briefly explained in section 2.2.4. It replaces the previously used method, which linked couples of GSM recursively by pairs.

2.2.1 Analysis method of the change of medium The problem to deal with is illustrated if Figure 2.3, where 2 sections can be observed. To solve the problem, the nomenclature and the reference system shown in that figure are used. The origin of the reference system is located at the point where the change of medium in the waveguide occurs. The region (1) is defined in z < 0 and the region (2) in z > 0; the waveguide in the regions (1) and (2) are considered semi-infinite.

Figure 2.3: Change of medium and reference system.

Although there are infinite modes, for the analysis only MTEm0 modes are selected, so that the transverse fields of each section, i, in the waveguide can be calculated as follows: 16 Analysis of reconfigurable filters

M −→(i) X (i) −→(i) E t = Vm (z) e m (x, y) (2.3) m=1

M −→(i) X (i) −→(i) H t = Im (z) h m (x, y) (2.4) m=1 where s 2Z(i) mπx −→e (i) = −yˆ 0m sin (2.5) m a a s −→ 2Z(i) mπx h (i) =x ˆ 0m sin (2.6) m a a with

Z(1) = jωµ 0m γmI

r 2 2π m γmI = − 1 λ0 2a/λ0

Z(2) = jωµ 0m γmP (2.7) r 2 2π m γmP = − r λ0 2a/λ0 and

(i) (i) (i) +(i) −γm z −(i) +γm z Vm (z) = Vm e + Vm e (2.8)

+(i) −(i) V (i) V (i) (i) m −γm z m +γm z Im (z) = (i) e − (i) e (2.9) Z0m Z0m In the boundary, (z = 0), the continuity condition of transverse ﬁelds must be satisﬁed.

−→(1) −→(2) E t |z=0 = E t |z=0 (2.10)

−→(1) −→(2) H t |z=0 = H t |z=0 (2.11)

M M X (1) (1) X (2) (2) Vm em (x, y) = Vm em (x, y) (2.12) m=1 m=1 2.2 Analysis methodology of reconﬁgurable ﬁlters 17

M M X (1) (1) X (2) (2) Im hm (x, y) = Im hm (x, y) (2.13) m=1 m=1 −→ −→ Since e m(x, y) and e n(x, y) with m6=n are orthogonal, when the equations are scalarly −→ −→ multiplied by e n(x, y) and h n(x, y), it is obtained: q q (1) (1) (2) (2) Z0mVm |z=0 = Z0mVm |z=0 (2.14)

q q (1) (1) (2) (2) Z0mIm |z=0 = Z0mIm |z=0 (2.15)

q q +1 −1 1 +2 −2 2 (Vm + Vm ) Z0m = (Vm + Vm ) Z0m (2.16)

+1 −1 q +2 −2 q Vm Vm 1 Vm Vm 2 1 − 1 Z0m = 2 − 2 Z0m (2.17) Z0m Z0m Z0m Z0m +1 −1 +2 −2 In order to obtain the S matrix, Vm and Vm must be connected with Vm y Vm , so that computing and considering that:

−1 +1 −2 +2 +1 −2 Vm = S11Vm + S12Vm Vm = S21Vm + S22Vm (2.18) The relations (2.19) and (2.20) are obtained. 2 Doing (2.16)−Z0m·(2.17)

1 2 p 1 2 −1 Z0m − Z0m +1 2 Z0mZ0m −2 Vm = − 1 2 Vm + 1 2 Vm (2.19) Z0m + Z0m Z0m + Z0m

1 Doing (2.16)+Z0m·(2.17)

p 1 2 2 1 −1 2 Z0mZ0m +1 Z0m − Z0m −2 Vm = 1 2 Vm − 1 2 Vm (2.20) Z0m + Z0m Z0m + Z0m

In this manner, each mode of the matrix Sm is as follows:  q  (2) (1) (1) (2) Z − Z 2 Z0mZ0m  0m 0m   (1) (2) (1) (2)   Z0m + Z0m Z0m + Z0m  Sm =   (2.21)  q   2 Z(1)Z(2) (1) (2)   0m 0m Z0m − Z0m  (1) (2) (1) (2) Z0m + Z0m Z0m + Z0m Therefore, the multimodal S matrix is as shown in (2.22) 18 Analysis of reconﬁgurable ﬁlters

 (1,1) (1,N)  S1 0 ··· 0 S1 0 ··· 0  (1,1) (1,N)   0 S2 ··· 0 0 S2 ··· 0   . . . . .   . .. . | .. .     0 ··· S(1,1) 0 0 ··· S(1,N) 0   M−1 M−1   (1,1) (1,N)   0 ··· 0 SM 0 ··· 0 SM      S =   (2.22)      (N,1) (N,N)   S1 0 ··· 0 S1 0 ··· 0   (N,1) (N,N)   0 S2 ··· 0 0 S2 ··· 0   . . . . .   ......   . . | .   (N,1) (N,N)   0 ··· SM−1 0 0 ··· SM−1 0  (N,1) (N,N) 0 ··· 0 SM 0 ··· 0 SM This matrix has dimensions 2M × 2M, since each block are M × M matrices. Further- more, the blocks are diagonal matrices, so S will also be a diagonal block matrix.

2.2.2 Analysis method of multiple discontinuities The problem to deal with is illustrated in Figure 2.4. In order to solve it, the nomenclature and the reference system shown in that ﬁgure, are used. N different sections of different length li can be observed.

Figure 2.4: Multiple discontinuities and reference system.

Although there are inﬁnite modes, for the analysis, only MTEm0 modes are selected, so that the transverse ﬁelds of each section, i, in the waveguide can be calculated as follows:

M −→(i) X (i) −→(i) E t = Vm (z) e m (x, y) (2.23) m=1

M −→(i) X (i) −→(i) H t = Im (z) h m (x, y) (2.24) m=1 2.2 Analysis methodology of reconﬁgurable ﬁlters 19

−→(i) −→(i) where e m and h m are the m-th modes in the i-th section deﬁned by: s 2Z(i) mπx −→e (i) = −yˆ 0m sin (2.25) m a a s −→ 2Z(i) mπx h (i) =x ˆ 0m sin (2.26) m a a with

(i) jωµ Z0m = (i) γm  q r 2(i) mπ 2 (i) 2  jβm = j k − Propagation (i) fcm a γm = jk 1 − f = q 2(i) mπ 2  αm = j k − a Cutoff

and

(i) (i) (i) +(i) −γm zi −(i) +γm zi Vm (zi) = Vm e + Vm e (2.27)

+(i) −(i) V (i) V (i) (i) m −γm zi m +γm zi Im (zi) = (i) e − (i) e (2.28) Z0m Z0m For an odd i section, the dielectric permittivity of the medium is that of the vacuum, i.e. (i) εr = 1 and γm is as follows: s 2 (i) 2π m γmI = − 1 (2.29) λ0 2a/λ0 On the other hand, in even sections, there is a dielectric different to the vacuum, so that (i) εr 6= 1 and γm is as follows: s 2 (i) 2π m γP = − εr (2.30) λ0 2a/λ0 Thus:

( jωµ (i) For odd i (i) γmI Z0m = jωµ (2.31) (i) For even i γmP After the analysis of these equations, the next problem is obtained

+(i) −(i) • 2N unkwons, since there are Vm and Vm for i = 1, 2, . . .N 20 Analysis of reconﬁgurable ﬁlters

• 2(N − 1) equations, since transverse ﬁelds continuity is forced, Jc,i = 0, in N − 1 discontinuities

+(1) −(N) There are 2(N − 1) = 2N − 2 equations for 2N unkwons, so Vm and Vm have to be ﬁgured out. They are given by:

1 (1,1) (1) (1,N) (N) Vm = Zm Im + Zm Im (2.32)

N (N,1) (1) (N,N) (N) Vm = Zm Im + Zm Im (2.33) (1) (1) +(1) −(1) (N) (N) +(N) where Vm and Im depends on Vm and Vm , and Vm and Im depends on Vm and −(N) Vm in the following way:

(1) +(1) −(1) ) Vm (z1 = 0) = Vm + Vm +(1) −(1) (1) Vm −Vm (2.34) Im (z1 = 0) = (1) Z0m (N) +(N) −(N)  Vm (zN = 0) = Vm + Vm  +(N) −(N) (2.35) (N) Vm Vm Im (zN = 0) = − (1) − (1)  Z0m Z0m N Note that Im is incoming, as observed in ﬁgure 2.5. This is the reason for the negative sign in equation (2.35).

Figure 2.5: Detail of the current waves in the structure.

+(N) −(N) +(1) −(1) To obtain the S matrix, Vm and Vm have to be related with Vm and Vm . Forcing continuity of transverse ﬁelds:

−→(i) −→(i+1) E t |zi=li = E t |z(i+1)=0 (2.36)

−→(i) −→(i+1) H t |zi=li = H t |z(i+1)=0 (2.37)

M M X (i) −→ X (i+1) −→ Vm (zi = li) e m(x, y) = Vm (z(i+1) = 0) e m(x, y) (2.38) m=1 m=1 2.2 Analysis methodology of reconﬁgurable ﬁlters 21

M M X (i) −→ X (i+1) −→ Im (zi = li) h m(x, y) = Im (z(i+1) = 0) h m(x, y) (2.39) m=1 m=1 −→ −→ As e m(x, y) and e n(x, y) with m6=n are orthogonals, if they are scalarly muliplied by −→ −→ e n(x, y) or h n(x, y), it is obtained:

(i) (i+1) ) Vm (zi = li) = Vm (zi+1 = 0) (i) (i+1) (2.40) Im (zi = li) = Im (zi+1 = 0)

(i) (i) +(i) −γm li −(i) +γm li +(i+1) −(i+1) Vm e + Vm e = Vm + Vm (2.41)

+(i) −(i) +(i+1) −(i+1) V (i) V (i) V V m −γm li m +γm li m m (i) e − (i) e = (i+1) − (i+1) (2.42) Z0m Z0m Z0m Z0m (i+1) Doing (2.41) + (2.42)·Z0m

" (i+1) ! (i+1) ! # 1 (i) Z (i) Z V +(i+1) = e−γm li 1 + 0m V +(i) + e+γm li 1 − 0m V −(i) (2.43) m 2 (i) m (i) m Z0m Z0m

(i+1) Doing (2.41) − (2.42)·Z0m

" (i+1) ! (i+1) ! # 1 (i) Z (i) Z V −(i+1) = e−γm li 1 − 0m V +(i) + e+γm li 1 + 0m V −(i) (2.44) m 2 (i) m (i) m Z0m Z0m Thus:

+(i+1) ! +(i) ! Vm (i) Vm −(i+1) = A −(i) (2.45) Vm Vm where

 (i+1) ! (i+1) !  (i) Z (i) Z e−γm li 1 + 0m e+γm li 1 − 0m  (i) (i)   Z Z  1  0m 0m  A(i) =   (2.46) 2    (i+1) ! (i+1) !   (i) Z (i) Z   −γm li 0m +γm li 0m  e 1 − (i) e 1 + (i) Z0m Z0m The number of discontinuities is a multiple of 2 and N is odd, since the first and the last mediums are vacuum. Therefore, they can be grouped in the case of figure 2.6 as follows: 22 Analysis of reconfigurable filters

Figure 2.6: Speciﬁc case for three discontinuities.

+(3) ! +(2) ! Vm (2) Vm −(3) = A −(2) (2.47) Vm Vm

+(3) ! +(1) ! Vm (2) (1) Vm −(3) = A A −(1) (2.48) Vm Vm In the general case of N mediums shown in ﬁgure 2.4 it is:

+(N) ! 1 +(1) ! Vm Y (2k) (2k−1) Vm −(N) = A A −(1) (2.49) Vm N−1 Vm k= 2 | {z } A The product of the A matrix is as follows: A = A(N−1)A(N−2) A(N−3)A(N−4) ... A(2)A(1) | {z } | {z } | {z } Last dielectric First dielectric where

P P  P γ P γ  −γml2k m +γml2k m e 1 + I e 1 − I  γm γm  1   A(2k)A(2k−1) =   · (2.50) 4    P P  P γ P γ  −γml2k m +γml2k m  e 1 − I e 1 + I γm γm I I  I γ I γ  −γml2k−1 m +γml2k−1 m e 1 + P e 1 − P  γm γm      ·    I I  I γ I γ  −γml2k−1 m +γml2k−1 m  e 1 − P e 1 + P γm γm so that

+(N) ! +(1) ! ( +(N) +(1) −(1) Vm Vm Vm = A11Vm + A12Vm −(N) = A −(1) =⇒ −(N) +(1) −(1) (2.51) Vm Vm Vm = A21Vm + A22Vm 2.2 Analysis methodology of reconﬁgurable ﬁlters 23

In Appendix A, calculations to work out Zm are found but, what really matters is the multimodal matrix S, so it is better to solve Sm. By deﬁnition and considering the diagram in ﬁgure 2.7, the values of the S parameters depending on the input and output waves can be expressed as:

Figure 2.7: Multiple discontinuities with voltage waves.

−(1) +(1) −(N) Vm = S11Vm + S12Vm +(N) +(1) −(N) (2.52) Vm = S21Vm + S22Vm By operating on the expressions (2.51) in order to obtain similar equations to (2.52), it is obtained that:

−(1) −1 +(1) −1 −(N) Vm = −A22 A21 Vm + A22 Vm (2.53) | {z } |{z} S11 S12

+(N) −1 +(1) −1 −(N) Vm = A11 − A12A22 A21 Vm + A12A22 Vm (2.54) | {z } | {z } S21 S22

By comparing with the equations (2.52) the values of the matrix Sm may be determined, as indicated in the above expressions.Thus, each mode of the aforementioned matrix is expressed as follows:

 −1 −1  −A22 A21 A22 Sm =   (2.55) −1 −1 A11 − A12A22 A21 A12A22 The multimodal S matrix can be written as: 24 Analysis of reconﬁgurable ﬁlters

 (1,1) (1,N)  S1 0 ··· 0 S1 0 ··· 0  (1,1) (1,N)   0 S2 ··· 0 0 S2 ··· 0   . . . . .   . .. . | .. .     0 ··· S(1,1) 0 0 ··· S(1,N) 0   M−1 M−1   (1,1) (1,N)   0 ··· 0 SM 0 ··· 0 SM      S =   (2.56)      (N,1) (N,N)   S1 0 ··· 0 S1 0 ··· 0   (N,1) (N,N)   0 S2 ··· 0 0 S2 ··· 0   . . . . .   ......   . . | .   (N,1) (N,N)   0 ··· SM−1 0 0 ··· SM−1 0  (N,1) (N,N) 0 ··· 0 SM 0 ··· 0 SM This matrix is 2M × 2M, since each block is M×M matrices. Moreover, they are diagonal matrices, so S is also diagonal in blocks.

2.2.3 Analysis method of steps between waveguides To solve the problem of the analysis of the steps between waveguides with different cross- sections, Mode Matching is used [52], [54], [55], [56]. By using Mode Matching, the generalized Scattering matrix of an abrupt step in H plane between two rectangular waveguides is obtained. As can be seen in ﬁgure 2.8, access 1 has the narrow waveguide of width a and access 2, has the widest waveguide of width d. It is considered that waveguide 1 is centered and completely contained in waveguide 2 and, the coordinate axes are located at the discontinuity. Given these considerations the transverse ﬁeld of the waveguide 1 in the discontinuity can be calculated as follows:

M M X (1) (1) X (1) (1) E1 = am em + bm em (2.57) m=1 m=1 M M X (1) (1) (1) X (1) (1) (1) H1 = − am Y0m em + bm Y0m em (2.58) m=1 m=1 where: s 2Z(1) mπ e(1) = − 0m sin (y − y ) (2.59) m a a 0

(1) 1 ωµ Z0m = (1) = (1) (2.60) Y0m βx 2.2 Analysis methodology of reconﬁgurable ﬁlters 25

Figure 2.8: Section step analyzed with Mode Matching.

( pk2 − ( mπ )2 if k > mπ β(1) = q a a x mπ 2 2 mπ (2.61) −j a − k if k < a Similarly the transverse ﬁeld in the waveguide 2, also in the interface, can be calculated obtaining: N N X (2) (2) X (2) (2) E2 = an en + bn en (2.62) n=1 n=1 M N X (2) (2) (2) X (2) (2) (2) H2 = an Yn en − bn Y0n en (2.63) n=1 n=1 where [73]: s 2Z(2) nπ e(2) = − 0n sin y (2.64) n d d

(2) 1 ωµ Z0n = (2) = (2) (2.65) Y0n βx ( pk2 − ( nπ )2 si k > nπ β(2) = q d d x nπ 2 2 nπ (2.66) −j d − k si k < d As the transverse fields must be continuous at the interface [52], [54], [55], the following equations may be established: M M N N X (1) (1) X (1) (1) X (2) (2) X (2) (2) am em + bm em = an en + bn en (2.67) m=1 m=1 n=1 n=1 26 Analysis of reconfigurable filters

M M M N X (1) (1) (1) X (1) (1) (1) X (2) (2) (2) X (2) (2) (2) − am Y0m em + bm Y0m em = an Yn en − bn Y0n en (2.68) m=1 m=1 n=1 n=1 Note that in equation (2.67) the total ﬁeld at the interface is projected on the modes of the largest guide. It is assumed that the small waveguide modes are null outside the interface:

Z d Z y0+a (1) (2) (1) (2) em en dy = em en dy = Amn (2.69) 0 y0 So the electric ﬁeld in the junction plane, outside the small waveguide will be zero, which is expected in a transverse incident ﬁeld on a perfect conductive wall. The equations are:

M M X (2) (1) X (2) (1) (2) (2) Y0n Amnam + Y0n Amnbm = an + bn (2.70) m=1 m=1

N N (1) (1) X (2) (2) X (2) (2) − am + bm = AmnY0n an − AmnY0n bn (2.71) n=1 n=1 After a series of operations described in detail in Appendix A of [2] the following expressions are got: T −1 T S11 = (I + DD ) (I − DD ) T −1 S12 = (I + DD ) (2D) T −1 T (2.72) S21 = (I + D D) (2D ) T −1 T S22 = −(I + D D) (I + D D) Where D is a M × N matrix whose elements are in the equation (2.73).

 sin nπ y −(−1)m sin nπ (y +a) mπ ( d 0) ( d 0 )  a cos a y0 + mπ + nπ  a d   m n v  si = u (2)  a d u Y0n  Dmn = t (1) · (2.73) Y0m ad   sin nπ y −(−1)m sin nπ (y +a) sin nπ y −(−1)m sin nπ (y +a)  ( d 0) ( d 0 ) ( d 0) ( d 0 )  mπ − nπ + mπ + nπ  a d a d   m n si a 6= d

2.2.4 Generalized Scattering Matrices connection method By using a new method developed in [2], [15] the global scattering matrix is obtained in a more direct way. Figure 2.9 shows the names assigned to the progressive voltage waves and the regressive voltage waves in the input and output ports of each network. ai is the progessive wave in the input port of the network i, and bi is the regressive wave. In the output 2.2 Analysis methodology of reconﬁgurable ﬁlters 27

Figure 2.9: Cascade connection of N dispersion matrices with the new method.

port of the last network, the convention is changed, and bN+1 is the progressive wave and aN+1 is the regressive. In the case of multimodal voltage waves, ai and bi are vectors of M elements, being M the number of guided modes. If the analysis is monomode, M = 1. In this case, scattering matrices Si have 2 M × 2 M elements, and each block Si,δ γ (with δ, γ = 1, 2) has M × M elements. The objective is to obtain the scattering parameters of the global network (SG) relating the incident and reﬂected waves in the input and output ports:

b1 = SG,11a1 + SG,12aN+1 (2.74)

bN+1 = SG,21a1 + SG,22aN+1 (2.75)

After a series of operations described in detail in Appendix B the global scattering matrix is obtained, which is expressed in (2.76)

 2 !  Y  S1,11 + S1,12 B2 SN,21 Ai     i=N    SG =   (2.76)  2 !   Y   SN,21 Ai S1,11 + S1,12 B2  i=N being

Ci = Si,11 + Si,12 Bi+1 (2.77) −1 Ai = (I − Si−1,22 Ci) Si−1,21 (2.78)

Bi = Ci Ai (2.79)

The equations (B.24) to (B.28) are valid for all the values of i (i = [2, ··· ,N]), except for CN , since the correct value is CN = SN,11. The procedure requires N −1 matrix inversions and 5N −3 matrix products. Furthermore, it has been verified that for electromagnetic problems, the matrices that should be inverted with this method are well conditioned and do not have problems of numerical instability. 28 Analysis of reconfigurable filters

2.3 Validation of the analysis tool

2.3.1 Filter analysis In order to evaluate the validity and efficiency of the analysis tool, some filters designed in the thesis (see section 3.4) are analyzed. The analysis tool will allow to calculate the frequency response of the filter. The two designed filters will be analyzed with this tool in order to check both the performance of the filter and the tool. It is expected to have the response as similar as possible to the ideal Chebyshev response. Different studies are carried out by using various analysis tools and comparing different criteria. On the one hand, the following tools are used:

• Analysis tool developed in this project and implemented in Matlab [57] that allows efficiently calculate the frequency response (parameter S11 and S21) of a reconfigurable microwave filter, symmetrical, of N cavities and filled with liquid crystal.

• Analysis tool developed in this project and implemented in Matlab using an innovative strategy of fast sweep to calculate the S parameters of the ﬁlter. This strategy has been implemented by the Microwave Applications Group (GAM) of the Polytechnic University of Valencia where this project has been carried out.

• Electromagnetic commercial software CST Microwave Studio [51].

The developed tool, explained above in this chapter, uses modal methods, while CST uses numerical methods. Furthermore, the following criteria for the different tools will be compared:

• Frequency response of the designed ﬁlters, S11 and S21 parameters. • Computational cost of the tool used.

Two cavities filter The analysis of a 2 cavities filter designed in section 3.4.1 is carried out (see figure 3.11), between 10.5 GHz and 11.5 GHz, by using 1001 frequency points. Moreover, CST simulation will use the configuration shown in figure 2.10. In figure 2.11, the next frequency responses are shown: ideal, calculated with the analysis tool developed with and without fast sweep, and simulated with CST. It is observed that the frequency response is equal for the three tools, being almost the ideal one. It can be said that all the tools are accurate. In addition to the frequency response, it has made a study of the computational cost involved in the analysis of these filters, that is, the time spent by each of the tools used. The simulations have been conducted on an Intel Core 2 Quad Q6600 processor with 8GB of RAM. Table 2.1 shows the comparative of times. 2.3 Validation of the analysis tool 29

Figure 2.10: CST conﬁguration to simulate the ﬁlter.

Developed tool Developed tool+Fast CST Time 14.8 s 3.3 s 55 s

Table 2.1: Computational cost comparative of a 2 cavities ﬁlter.

It can be seen that the simulation time of the tool with fast sweep is much smaller compared to other tools. This tool is very useful because, in addition to being fast, provides a good frequency response. It is hoped that the tools designed in this Thesis are more efficient and accurate than commercial tools, because they are designed for this type of filters in particular, using only modals methods. CST, although it is an accurate tool, has a much lower efficiency, because time is about 16 times higher than the developed tool. The advantage of this software is that it allows the analysis of any type of structure at the expense of the increased computational time. 30 Analysis of reconfigurable filters

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.11: Frequency response of the 2 cavities ﬁlter. (a)S11. (b)S21.

Four cavities filter Now, the analysis of the four cavities filter designed in section 3.4.2 is carried out (see figure 3.13). As for the case of two cavities filter, it is proceeded to the analysis of this filter between 10.5 GHz and 11.5 GHz, using 1001 frequency points. Furthermore, for simulation in CST the same configuration will be used as shown above in Figure 2.10. In Figure 2.12, the next frequency responses are shown: the ideal, calculated with the analysis tool developed with and without fast sweep, and simulated with CST. It can be said the same as in the case of two cavities, all the tools are equally accurate. For this case, it has also been made a study of computational costs associated with the analysis of the filters, with the same processor as in the case of two cavities. The comparison of times is shown in the table 2.2.

Developed tool Developed tool+Fast CST Time 30.6 s 8.6 s 267 s

Table 2.2: Computational cost comparative of a 4 cavities ﬁlter.

Related to the computational cost, it can be concluded the same that in the previous case. The tool with fast sweep is more accurate and efﬁcient, the commercial software is the slow- est, and the developed tool without fast sweep is in an intermediate point. 2.3 Validation of the analysis tool 31

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.12: Frequency response of the 4 cavities ﬁlter. (a)S11. (b)S21.

2.3.2 Permittivity variation of the liquid crystal As explained in previous chapters, to get reconfigurable filters ESIW technology filled with liquid crystal is used. This material acts as dielectric with variable characteristics. To see the effect of the variation of the liquid crystal in the frequency response of the filter, an analysis varying the relative permittivity in different sections is performed, using the symmetry of the filter, that is, the permittivity is varied in one section and its symmetrical. This is intended to understand how the material works and the effect it has on the filters under analysis. The variation of the permittivity of the liquid crystal is between two extreme values, εr = 2.4 and εr = 3.2, with a step of 0.2, i.e, four tests are performed. The permittivities of the sections that remain unchanged are left fixed at εr = 2.8, which is the midpoint of the possible permittivities of LC and the value at which the filter has been designed.

Variation of the permittivity of the ﬁlter of two cavities For the two cavities ﬁlter the following cases are studied:

• Case 1: Only the value of the permittivity in the two resonant cavities is varied. The result is shown in Figure 2.13. It is observed that if the permittivity of LC increases, the center frequency decreases, resulting in an increase of the bandwidth and a decrease in the return losses, ranging from -25 dB for εr = 2.8 to -13 dB for εr = 3.2, approximately. On the other hand, if the permittivity in the cavities is smaller than 2.8 the filter 32 Analysis of reconfigurable filters

response loses its shape.

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.13: Frequency response of the two cavities ﬁlter for case 1.

• Case 2: The value of the permittivity in the two resonant cavities and the input and output waveguides is varied. The result is shown in Figure 2.14. As in the previous case, it is observed that if the permittivity of LC increases, the center frequency decreases, resulting in an increase in bandwidth, and a decrease in the return losses varying from -25 dB for εr = 2.8 to -17 dB for εr = 3.2, approximately. Also, if the permittivity in the cavities is smaller than 2.8 the filter response loses its shape. It can be concluded that this case is better than the first one, since the return losses are better and the rest of parameters vary as well. • Case 3: The value of the permittivity in the first and third coupling windows is varied. The result is shown in Figure 2.15. In this case it is observed that the variation of the permittivity of LC in the windows does not affect the filter center frequency. In addition, if the permittivity decreases the bandwidth increases and the return losses increase as well, ranging from -25 dB for εr = 2.8 to -13 dB for εr = 3.2. On the other hand, when the permittivity reaches a value higher than 2.8, the filter response loses its shape. • Case 4: The value of the permittivity in the central coupling window is varied. The result is shown in Figure 2.16. In this case, increasing the permittivity of LC, the bandwidth increases and the return losses increase, ranging from -25 dB for εr = 2.8 to -18 dB for εr = 3.2. If the permittivity decreases both the bandwidth and return losses decrease, reaching the extreme case εr = 2.4, where the filter response loses its shape. 2.3 Validation of the analysis tool 33

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.14: Frequency response of the two cavities ﬁlter for case 2.

• Case 5: The value of the permittivity in the three coupling windows is varied. The result is shown in Figure 2.17. In this case it is observed that if the permittivity of LC decreases both the bandwidth the return losses increase, varying from -25 dB for εr = 2.8 to -17 dB for εr = 2.4. If the permittivity increases, the bandwidth decreases and in the extreme case of εr = 3.2 the ﬁlter response loses its shape. • Case 6: The value of the permittivity in all cavities and coupling windows (including input and output waveguides) is changed. The result is shown in Figure 2.18. By varying the permittivity of LC in all sections of the ﬁlter, whether increases as if decreases, the return losses are constant. However, when the permittivity increases, the bandwidth and the center frequency decreases and vice versa.

Variation of the permittivity of the filter of four cavities • Case 1: The value of the permittivity in cavities one and four is varied. The result is shown in Figure 2.19. It is noted that the filter response only roughly matches with the ideal response when the permittivity of LC takes its central value, εr = 2.8, with return losses of about -25 dB. For other values of permittivity, the filter response completely loses its shape.

• Case 2: The value of the permittivity in the cavities two and three is varied. The result is shown in Figure 2.20. As in the previous case, the filter response only approximately matches the ideal response when the permittivity of LC takes its central value, εr = 34 Analysis of reconfigurable filters

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.15: Frequency response of the two cavities ﬁlter for case 3.

2.8, with return losses approximately of -25 dB. For other values of permittivity, the ﬁlter response completely loses its shape.

• Case 3: The value of the permittivity in all resonant cavities is varied. The result is shown in Figure 2.21. In this case, if the permittivity increases, the center frequency decreases and the return losses get worse ranging from -25 dB for εr = 2.8 to -15 dB for εr = 3.2 and, when the value is εr = 3.2, the response loses a pole. When the permittivity decreases, the center frequency increases but the ﬁlter response loses its shape. The bandwidth remains constant for both cases.

• Case 4: The value of the permittivity in all cavities is varied, including input and output lines. The result is shown in Figure 2.22. In this case, the ﬁlter response is very similar to the two cavities case. If the permittivity increases, the center frequency decreases and vice versa, and the ﬁlter response loses its shape. The bandwidth remains constant.

• Case 5: The value of the permittivity in the coupling window one and five is varied. The result is shown in Figure 2.23. In this case, the filter response only retains its shape in the central value of the permittivity εr = 2.8. • Case 6: The value of the permittivity in the coupling windows two and four is varied. The result is shown in Figure 2.24. In this case, the filter center frequency does not change. In the extreme values of permittivity, the filter response loses its shape. The bandwidth is increasing or decreasing if the permittivity increases or decreases, 2.3 Validation of the analysis tool 35

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.16: Frequency response of the two cavities ﬁlter for case 4.

respectively. Finally, regarding the return losses, these get worse when both increase or reduce the value of the permittivity. • Case 7: The value of the permittivity in the center coupling window is varied. The result is shown in Figure 2.25. It can be seen that the filter response only loses its shape in the extreme value of permittivity εr = 2.4. Whether the permittivity increases or decreases, the filter center frequency is unchanged and the return losses get worse. Related to the bandwidth, as in the previous case, it is increasing or decreasing when the permittivity increases or decreases, respectively. • Case 8: The value of the permittivity varies in all windows. The result is shown in Figure 2.26. As in previous cases, the filter response only retains its shape in the central value of the permittivity εr = 2.8. • Case 9: The value of the permittivity in all cavities and all coupling windows (including input and output) is changed. The result is shown in Figure 2.27. In this case, if the value of the permittivity increases, the center frequency and bandwidth decrease and vice versa. Related to the return losses, these remain constant for all values of permittivity.

It can be concluded that the value of the permittivity in the resonant cavities, both for two and four cavities filters, cannot be smaller than the permittivity in the coupling windows, since the filter response loses its shape. By changing the permittivity in the coupling windows the bandwidth of the filter varies. By changing the permittivity in the resonant cavities, 36 Analysis of reconfigurable filters

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.17: Frequency response of the two cavities filter for case 5. the filter center frequency varies. Thus, when all the permittivities change, both the cental filter frequency and bandwidth are varied, maintaining the relative bandwidth. The center frequency, in this case, ranges from about 10.3 GHz and 11.9 GHz for both filters. For this reason, it is better to fill the whole filter and to polarize the liquid crystal at the same time in the coupling windows and the resonant cavities. The response is the best that can be achieved. There are limits to this reconfiguration, i.e, for εr outside a certain range, the filter response loses its shape. It must be kept in mind that the LC has a value of permittivity ranging from εr = 2.4 and εr = 3.2. In addition, even using materials with a larger variation range than the LC, this could not get any response, that is, this could not reconfigure the filter in any range. There is a limitation given by the structure. 2.3 Validation of the analysis tool 37

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.18: Frequency response of the two cavities ﬁlter for case 6.

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.19: Frequency response of the four cavities filter for case 1. 38 Analysis of reconfigurable filters

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.20: Frequency response of the four cavities ﬁlter for case 2.

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.21: Frequency response of the four cavities ﬁlter for case 3. 2.3 Validation of the analysis tool 39

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.22: Frequency response of the four cavities ﬁlter for case 4.

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.23: Frequency response of the four cavities filter for case 5. 40 Analysis of reconfigurable filters

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.24: Frequency response of the four cavities ﬁlter for case 6.

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.25: Frequency response of the four cavities ﬁlter for case 7. 2.3 Validation of the analysis tool 41

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.26: Frequency response of the four cavities ﬁlter for case 8.

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 2.27: Frequency response of the four cavities ﬁlter for case 9.

Chapter 3

Design of reconﬁgurable ﬁlters

A typical ﬂowchart of a CAD process for a microwave device is shown in ﬁgure 3.1. The design process, regardless of the structure, consists of three main steps:

• Specifications of the design. It is desired to design a filter with a specific frequency response, so that the specification is that frequency response. Some values as the physical dimensions of the filter and the permittivities values (εr) must be selected to obtain the desired frequency response. Some parameter are fixed and other have to be optimized. This is done because in the technological processes there are some variables that cannot change (or is better to maintain them as fixed). For instance, the dimensions of the drill bit limit the length of the coupling windows and the availability of a specific dielectric material limits the permittivity of the substrate.

• Synthesis of the starting point. In order to have a starting structure of the filter, the well known Chebysev lowpass prototype is chosen [58]. This prototype is implemented in a waveguide filter with resonant cavities. The number of poles and consequently, the number of cavities has to be chosen to satisfy the specifications. Afterward, the synthesis of the prototype will give the starting point of the filter topology (dimensions of the cavities and the coupling windows) that meets the specifications.

• Optimization. Once the starting point has been calculated, the simulation (analysis) tool is used to analyze the structure with the dimensions obtained in the previous step. Then, it is verified whether the filter meets with the specifications (required frequency response) or not given these physical characteristics. If the results do not verify the design specifications, the initial structure must be modified until obtaining a new point by a process of optimization. The response of the new structure is obtained and it is verified if it meets design specifications. This process of analysis, comparison to specifications, and modification of the design parameters, will be repeated until the specifications are met.

When the designed structure meets the specifications, the filter is manufactured and measured. If the measured data meets the specifications, the design process is completed. 44 Design of reconfigurable filters

The designed filter is filled with liquid crystal by sections allowing a double reconfiguration. On the one hand, it allows to meet the specifications after manufacturing by changing the permittivity value εr, it means, it allows to readjust the inaccuracies produced in the frequency response during the manufacturing process. On the other hand, change the permittivity εr allows to change the frequency response at will, i.e, change the central frequency and the bandwidth in a certain range meeting new specifications. However, if the filter does not meet the initial specifications after this reconfiguration, the device must be manufactured again, since it has been a fail in the manufacturing process.

Design Specifications

Starting Point Synthesis

EM Simulation Optimization

Are specifications NO met?

YES

Manufacturing

Are specifications NO Are specifications Reconfiguration met? met?

YES YES OK OK

Figure 3.1: Design ﬂow of a computer-aided design.

Below, a review of the basic techniques for microwave frequencies bandpass filter design will be done, specifically the insertion loss method. The insertion loss method [13], [14], is based on network synthesis techniques to allow the design of filters with a specific type of frequency response. It offers a high control degree over the variation of the amplitude and phase response of the filter, both inside and outside the passband, with a systematic way to synthesize the desired response. Furthermore, this method improves the response of the filter in a direct manner, as a con- sequence of a higher filter order. The higher the filter order, the better the filter performance in terms of selectivity, but with a larger size, increasing the losses. 3.1 Specifications 45

The design flow that must be followed in order to obtain a filter using this technique is shown in the figure 3.2. First of all, the specifications that must be met are set. Once the required features have been defined, the low pass prototype has to be designed. This prototype is normalized in impedance and frequency to simplify the design. Afterward, an impedance scaling and a frequency transformation are done to convert the lowpass design to a bandpass one. After this operation, the required model is obtained by using lumped elements. Finally, this circuit model is implemented by using distributed elements, since the lumped elements may not be manufactured at microwave frequencies.

Figure 3.2: Synthesis process ﬂow.

Next, a detailed description about the most important aspects of each step is carried out. Finally, the aspects related with the optimization process are explained in detail.

3.1 Speciﬁcations

The main design goal is to obtain a frequency response as much similar as possible to a Chebysev ideal filter. Chebysev filter, also known as a constant ripple filter, is characterized by having a constant ripple at the passband and a maximally flat response at the stopband. The purpose is to obtain a design in X band, centered at 11 GHz, with a bandwidth of 300 MHz and return losses greater than 25 dB, using two and four cavities, as shown in figure 3.3.

3.2 Synthesis

In this section a structure able to meet the given specifications is found. It is desired to design a reconfigurable filter using Empty Substrate Integrated Waveguide (ESIW) [1] technology. This filter is filled with a dielectric material of variable characteristics, specifically, liquid crystal (LC), [8], [10]. However, the design must be started by designing a waveguide 46 Design of reconfigurable filters

S parameters (2 cavities filter) S parameters (4 cavities filter)

Magnitude(dB) Magnitude (dB) Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 3.3: Chebyshev ideal response. (a) Two cavities. (b) Four cavities. resonant cavities filter filled with dielectric by sections. It is vital to find a good starting point for the design process. This starting point requires the calculation of resonant cavities lengths and inductive windows widths. For the search of the starting point, a theoretical study of the process to follow is taken [33], [59], [60]. For designing a filter by using the insertion loss method, a lowpass prototype normalized in frequency and impedance is designed. This normalization simplifies the design of the filter for any impedance, frequency and type. Afterward, an impedance and frequency transformation is done in order to obtain the lumped elements circuit. To work at microwave frequencies, lumped elements must be replaced by distributed elements. This process is detailed below.

3.2.1 Lowpass prototype Once the specifications of the filter have been defined, the next step is to obtain the lowpass prototype. The transfer function for a Chebysev lowpass prototype is given by the equation (3.1).

2 1 |S21(jΩ)| = 2 (3.1) 1 + ε TN (Ω) where ε is a constant related to the ripple LAR in dB, following the next expression. q LAR ε = 10 10 − 1 (3.2) and TN (Ω) is the N order Chebysev polynomial. 3.2 Synthesis 47

This prototype is defined as a lowpass filter made up with lumped elements whose values are normalized for a source impedance equal to one, expressed as g0 = 1 and an angular cutoff frequency Ωc = 1. This prototype is shown in figure 3.4.

Figure 3.4: Lowpass normalized prototype. (a) Prototype starting with a shunt element. (b) Prototype starting with a series element.

Both circuital schemes have similar behavior and response. The number of reactive elements is determined by the filter order N. Moreover, it should be noted that the values gi for i = 1 to N represent the inductance of the series inductors or the capacitance of the shunt capacitors. If g1 is a shut capacitor, then g0 is defined as the source resistance and gN+1 as the load resistance. On the other hand, if g1 is a series inductor, g0 and gN+1 are defined as the source conductance and load conductance, respectively. The different lowpass prototype values, gi, for a constant ripple LAR and a cutoff frequency Ωc = 1 are calculated using the expressions (3.3)

g0 = 1

2 π g1 = γ sin 2N

(2i−1)π (2i−3)π 1 4sin( 2N )sin( 2N ) (3.3) gi = (i−1)π for i=1,2,3,...n gi−1 2 2 γ +sin ( N )

1 for odd n gN+1 = 2 β coth ( 4 ) for even n where 48 Design of reconﬁgurable ﬁlters

L β = ln coth AR 13.37 β γ = sinh 2N Finally, the ﬁlter order should be determined. The minimum order is determined by the ripple in the band pass LAR dB and the reject in the stop band LAS at the frequency Ω = Ωs q cosh−1 100.1LAS −1 100.1LAR −1 N ≥ −1 (3.4) cosh ΩS

3.2.2 Scaling and frequency conversion

Only lowpass prototype filters with normalized impedance g0 = 1 and cutoff frequency Ωc = 1 have been considered so far. However, the purpose is to obtain the frequency response and the values of the elements for a bandpass filter. These filters are based on the lowpass prototype, so that a scaling and a frequency conversion is done. The first step is the impedance denormalization. This will be done in all the elements. For this denormalization, a constant known as impedance scaling factor is defined, γ0. Z0/g0 if g0 is a resistance γ0 = (3.5) g0/Y0 if g0 is a conductance where Z0 and Y0 = 1/Z0 are the source impedance and the source admittance, respectively. Applying the impedance scaling factor in the following way:

L → γ0L

C → C/γ0 (3.6) R → γ0R

G → G/γ0 there is no change in the frequency response. This is because, in the resistive elements, only the impedance transformation is done; however, in the reactive elements a frequency transformation is needed. This transformation is detailed in (3.7), assuming that the bandpass ﬁlter has a ω2 − ω1 passband, where ω1 and ω2 indicate the cutoff angular frequencies: Ω ω ω Ω = c − 0 (3.7) ∆ ω0 ω with 3.2 Synthesis 49

ω − ω ∆ = 2 1 ω0 √ ω0 = ω1ω2 where ω0 indicates the central frequency and ∆ is the relative bandwidth. Applying this transformation to the prototype coefficients g , it is obtained: Ω g 1 Ω ω g jΩg → jω c + c 0 (3.8) ∆ω0 jω ∆ This implies that a series inductive element g is transformed to a series resonant circuit LC as shown in figure 3.5(a) with the following values of the elements: Ωc Ls = γ0g ∆ω0 (3.9) 1 Cs = 2 ω0 Ls On the other hand, a shunt capacitive element g is transformed in a shunt resonant circuit LC as shown in figure 3.5(b) with the following values of the elements: Ωc g Cp = ∆ω0 γ0 (3.10) 1 Lp = 2 ω0 Cp

Figure 3.5: Low-pass to bandpass prototype transformation. (a) Transformation of series inductive element. (b) Transformation of shunt capacitive element.

The indicated transformations are performed to lumped elements. Nevertheless, these lumped elements work well at low frequencies, but some problems arise at microwave frequencies. Lumped elements are available only for a limited range of values and can be dif- 50 Design of reconﬁgurable ﬁlters

ﬁcult to implement at microwave frequencies. Furthermore, at microwave frequencies the distances between ﬁlter components are not negligible.

3.2.3 Implementation of the filter For avoiding the aforementioned problems, a different method is utilized, in which the circuit is implemented with a unique resonator type, series or shunt, by using immittance inverters, i.e. impedance K or admittance J. In this case, for the starting design, a waveguide filter will be chosen by using impedance inverters and half wavelength λg0/2 waveguide sections. λg0 is the wavelength at the central frequency. The schematic of the filter is shown in figure 3.6.

Figure 3.6: Filter scheme using impedance inverters.

The impedance inverters behaviour is similar to the one of a transmission line of length λ/4 with characteristic impedance K for all the frequencies. The values of K are calculated with the expressions (3.11): q Z0∆xi K0,1 = Ωcg0g1

∆ q xixi+1 Ki,i+1 = for i=1,... N-1 (3.11) Ωc gigi+1

q ∆xixnZn+1 KN,N+1 = Ωcgngn+1

Where ∆, gi y Ωc parameters have been deﬁned above and xi is the resonator slope parameter at the central frequency, ω0 (3.12): ω dX (ω) x = 0 i | = ω (3.12) i 2 dω ω 0 where Xi(ω) is the resonator reactance. In this case, the utilized resonator is the waveguide section of half wavelength (λg0/2), with characteristic impedance Z0, whose input impedance when the waveguide is terminated with short circuit is:

Zin = jxi = jZ0tg(βl) (3.13) 3.2 Synthesis 51

Therefore, the slope parameter value can be calculated as:

ω d[Z tg(βl)] x = 0 0 | (3.14) i 2 dω ω=ω0 After doing some calculations the ﬁnal expression is shown in the equation (3.15):

2 πZ0 λg0 xi = (3.15) 2 λ0 where

λ0 λg0 = q (3.16) 1 − fc f0 The expression (3.15) can be evaluated easily at the design central frequency, for obtaining the values of the impedance inverters, K. Once the prototype values have been determined, a starting point can be calculated for the resonant cavities filter filled with dielectric, whose schematic is shown in figure 3.7.

Figure 3.7: Waveguide ﬁlter ﬁlled with dielectric.

Knowing the filter structure that is intended to design, some limitations must be taken into account, among others: the irises width must be smaller than the cavities, the cavities lengths will determine the total filter length and the permittivities will have technological limitations. However, the design of all the parameters at the same time is inefficient due to the complexity of the structure. For that reason, the segmentation strategy explained in following sections is carried out. The inverters are implemented by using coupling windows as shown in figure 3.8. All the windows will have the same length t, decided in the design process. On the other hand, the width of each window must meet the inversion constant K calculated for each iris. For the design of a given coupling window, firstly a minimum and a maximum values for the width of the iris, w, must be set. The minimum value is fixed by a technological limitation and the maximum value by the adjacent cavities widths, since the coupling window width must be smaller than the cavities. 52 Design of reconfigurable filters

Figure 3.8: Analyzed section for the calculation of the coupling window width.

Afterward, the S11 parameter is calculated for the ﬁlter section shown in ﬁgure 3.8. This value is calculated by using the segmentation technique described above that consists in calculating the S parameters matrices of the section blocks under study and join them for a global scattering matrix of the entire section. These blocks are described below:

• Filled waveguides: S1 y S7

• Change of medium: S2 y S6

• Impedance changes: S3 y S5

• Multiple discontinuities: S4

The S11 parameter calculation depends on diverse parameters, among others, the window width, w. In this process, that width varies in the whole range of some established values, and the section is simulated for each value w. The process ends when the condition set in (3.17) is met.

1 − |S11| 2 − Ki,i+1 = 0 (3.17) 1 + |S11| where Ki,i+1 is the normalized inversion constant of the window and S11 is the parameter obtained in the simulation of each section, with different window width w. In conclusion, the goal of this process consists on obtaining the value of w, which gets that the S11 parameter meet the aforementioned condition. For that, the Matlab function fzero is used. It should be mentioned that the obtained iris does not have the same response as the inverter, they only match in absolute value. This is because at both sides of the inverter a transmission line section (rectangular waveguide) is added that only affects to the response phase. In order to achieve the same phase response between the iris and the inverter, the 3.3 Optimization 53 adjacent cavities lengths must change. For that, the way to proceed is as follows: ﬁrstly, for a given cavity, the left window formed by the elements shown in ﬁgure 3.8 must be simulated, in order to obtain the S parameters as in the previous case. Thus, the phase shift due to this window should be calculated. This phase shift is:

Φ = ( S11 + 2βl0 + π)/2 (3.18) The same must be done with the right window, in order to know the phase shift due to this one. This value is obtained by using the expression (3.18). Hence, the total cavity length can be calculated. For that, the cavity length is added to the calculated value. The length value is expressed in (3.19):

l = l0 + Φ/β (3.19) where l is the ﬁnal length of the resonant cavity, l0 is the starting length of the cavity calculated as λg0/2.

3.3 Optimization

Once the starting point has been calculated, the next step is the optimization of the design variables. Generally, the optimization process of microwave devices is carried out with computer aided design (CAD) tool. The complexity of the current structures makes impossible the manual design. The most used ways to proceed differ in the degree of participation of the designer during the design process. The classical techniques are based on analysis tools, in which the designer proposes a certain structure and analyze it by any analysis tool. If the result is not desired, then the designer modifies the structure and re-analyze with the analysis tool. This iterative process ends when the results meet the initial specifications, which sometimes is very tough. This approach is difficult to apply when the number of design variables increases, since it is the designer who must propose modifications to the structure according to his intuition and check if the result is better or not, which causes the process is very slow. In addition, it presents the disadvantage that the designer cannot hit with the changes to be made in the device, making impossible to obtain reasonably good results. Currently, CAD tools are used in order to avoid the problems of manual design. These tools replace manual modifications of the dimensions of the filter, through an optimization process performed by the computer, from the initial design and using some algorithms for the variation of the design parameters automatically. There are several methodologies for the optimization within a CAD environment [61], [62], allowing the designers to choose the method that best fits the topology and filter characteristics. All of them make an optimization of all the design parameters at the same time. This results in a heavy and costly optimization when designing complex structures involving a large number of design parameters. 54 Design of reconfigurable filters

For that reason, in order to improve the velocity and robustness of the optimization process the structure is divided with the segmentation process proposed in [63] and [64]. This new technique takes the advantage of the symmetry of the designing structure, dividing the optimization process into some steps, each one with a small number of design parameters. Once the different blocks have been optimized, the design process ends with a last step, where all parameters are slightly adjusted at the same time. This is done by an optimization from the starting point given by the dimensions obtained in the previous steps. However, if the design contains cavities or resonant elements, there is the risk that exists coupling among all the cavities, not only between the adjacent. Thus, the filter is not properly designed, if these couplings are not taken into account. This causes that the last step of refinement of the dimension is very long, because the starting point provided by the design of individual blocks is far from the optimal values, by not considering all existing couplings. Therefore, intermediate refinement steps are usually made once a certain number of blocks have been designed, in order to improve the starting point of the last optimization. The fundamental idea of these techniques of automated optimization is the definition of an error function that indicates the difference between the desired and simulated response. The simulation is done with the different design parameters values in each iteration. Once defined this error function, the purpose is to find the value of the variables where the function takes the minimum value by applying some optimization algorithms, i.e. the goal is to find the physical topology of the device for which the real response is as much similar as possible to the ideal response.

3.3.1 Error function The performance of a microwave filter is usually expressed as a frequency-dependent specification set, although time-dependent specifications could also be defined. In practice, a discrete set of m frequency points is considered. These points must be representative enough of the whole interest frequency range. Thus, the error function is defined as the difference between the given specifications and the electromagnetic response in each of those frequency points, as follows [65]:

ei(x) = wi|Rj(x) − Si| i = 1, 2, , m. (3.20) where x ∈ Rn is the vector of parameters to be designed, R(x) ∈ Rm×1is a vector with the electromagnetic response of the device in the m frequency points, S ∈ Rm×1 is a vector with the design speciﬁcations in the m frequency points, and w ∈ Rm×1 is a vector with positive weights, which allows to emphasize the importance of some parts of the device response according to the design needs. From the deﬁnition of the error function, the device optimization problem ca be mathe- matically expressed as:

min U(x) (3.21) x∈Rn being U(x) the objective function. 3.3 Optimization 55

Function U(x) is generally subject to m restrictions, both equality and inequality ones, which must be fulfilled during the optimization process and also by the optimum solution. Thanks to the use of the objective function U(x), a parameter extraction problem can be converted in a multidimensional minimal search problem. In fact, its definition is essential, as it determines the shape of the multidimensional surface in which the optimization algorithm must find the minimum. U(x) is usually defined on the basis of an lp of e(x). The properties of the different norms used in the design of microwave devices will be described in the next paragraphs.

lp-norm This norm [66] provides with a scalar value which makes it possible to estimate the devi- ation between the device response and the speciﬁcations. It is deﬁned as:

" m #1/p X p kekp = |ei| (3.22) j=1 The most common used norms are the values p = 1 and p = 2.

Huber norm

This norm [67] combines the properties of l1 and l2 norms, low order and continuity:

m X kekH = H(e) = ρk(ei) (3.23) j=1 where

2 ei /2 if |ei| ≤ k ρk(ei) = 2 k|ei| − k /2 if |ei| > k

It can be observed that the Huber norm coincides with the l2 norm for ei ∈ [−k, k], and with l1 norm out of that range. So the l1 norm is used when the optimum point is far, while the l2 norm is used when it is close.

3.3.2 Optimization strategies The variables of the problem function, which is known as objective function or error function, may be subject to some restrictions, limiting the search area of the solution. This changes the way of treating the problem, thus distinguishing two types of optimization with and without restrictions, The methods applied are different [68]. There are also advanced optimization algorithms using evolutionary structures, hybrid, spatial mapping or multi-objective optimization. 56 Design of reconﬁgurable ﬁlters

Non-restrictions Optimization The non-restrictions optimization aims to ﬁnd the minimum of the error function, which depends on a real variable, whose value is not subject to any restrictions. Methods to minimize multidimensional functions when they are not subject to any restrictions are many and varied, as the chances to ﬁnd the minimum of a function are vast. However, all methods for minimization of functions have generally the same structure. It starts from an initial point x0, introduced by the user, and successive approximations of the solution are taken, obtaining a sequence of points x(k) converging to the minimum of the function (where the index k shows the number of the current iteration). The way to approach the minimum in each iteration is what distinguishes the different methods because the information to approach to the minimum may be based on both the value of the function and the value of some of its derivatives, and even the previous iterations. This iterative process ends when a termination criterion is met, for which there are many possibilities too, being able to use the most appropriate in each case. For example, if the value of the function at the optimum point x∗ is known, the criterion applied is

∗ f(x(k)) − f(x ) ≤ ftol (3.24) where ftol is the tolerance value introduced by the user. However, if the value of the function at x∗ is not known it is possible to deﬁne several criteria, the most common are:

• If between two successive iterations, the distance between the two obtained points x is less than a threshold xtol deﬁned by the user, the algorithm ends, i.e:

|x(k + 1) − x(k)| ≤ xtol (3.25)

• If between two consecutive iterations, the difference between the values taken by the function in the points is less than a certain tolerance ftol (deﬁned by user), the algorithm ends, i.e: |f(x(k)) − f(x(k + 1))|| ≤ ftol (3.26)

Depending on the information of the function and the derivatives used by the method to approximate the solution, these methods can be classiﬁed into the following three types:

• Direct Search Methods. These methods only evaluate the objective function at various points. The search for new points is determined from the previously calculated values. This group includes the Direct Search with Coordinates Rotation Method and Downhill Simplex Method, which are suitable for high linearity functions or with discontinuities. Such methods are very robust, but have a slower convergence than the others [68], [69].

• Gradient Methods. These methods use the value of the gradient function. These algorithms are appropriate when the information of the derivatives can be easily obtained, since the calculation of the derivatives can be computationally expensive. This 3.3 Optimization 57

group includes the Maximum Slope Method, Method of Conjugated Gradients or any so-called Quasi-Newton methods, such as Fletcher Goldfarb Broyden Shanno (BFGS). Its efﬁciency is much higher when the gradient is continuous [68], [69].

• Methods of higher order. These methods need to calculate higher order derivatives, which is a great disadvantage because of the high computational cost needed for this calculation. However, in most cases, it has the advantage of its fast convergence. In this group, the most representative method is the Newton Method [68], [69].

Most of the methods that require gradient information or other higher derivatives have a similar performance. In each iteration, the objective function is approached by a quadratic function whose origin is at x(k) by using the Taylor Theorem. It is important to say that, in most cases, the ﬁrst and second order derivatives are calculated numerically, because its analytical expression is not usually had. For this, the Finite Difference method is applied. This method approximates the value of the derivative (of any order) of a function at a point, from various evaluations of the objective function. More information about other methods that do not consider restrictions is available in [68] and [69]. Among all the unconstrained optimization methods that have been exposed, the Direct Search with Coordinates Rotation Method, the Downhill Simplex Method and the Method of Fletcher Goldfarb Broyden Shanno are used to optimize the ﬁlters designed in this work, since they provided the best results in terms of convergence, robustness, and speed. The basic characteristics of the algorithms mentioned above are explained below. It is also explained some properties implemented in [70] that improve their performance.

Direct search with coordinate rotation This simple algorithm gives a quick approach to the whereabouts of the minimum, but converges very slowly compared to other methods when we are very close to such minimum. It is used when there is not a good starting point. A variation of the method described in [71] y [72] has been implemented. The new features are:

• The step length after each coordinate rotation iteration is set according to the distance advanced in that iteration. If no advance has been made, the step length is increased in order to explore further regions.

• In the direct search iteration across a line, only steps that reduce the error function are accepted. If a step leads to a worse point, the step length is reduced and a new movement is tried until the step length is below some threshold value.

Downhill Simplex Method The Downhill Simplex Method, created by Nelder and Mead [73], it is a method of geometric nature that only conducts evaluations of the objective function, and not their derivatives. This makes this method has a slow convergence, not being very efficient in terms of 58 Design of reconfigurable filters number of function evaluations, and therefore, in terms of computational cost. However, it is one of the most robust methods for finding the minimum of a multidimensional function, since, as it can be seen below, its operation prevents, in most cases, that it is trapped in a local minimum. This method is based on a structure called simplex. A simplex is a geometric figure of n dimensions and n + 1 vertices, where n is the number of variables of the function to be minimized. Thus, if the function depends on two variables (n = 2), the simplex is a triangle; if the function depends on three variables, the simplex will be a triangular pyramid, and so on. A simplex does not have to be regular, but can not be a degenerate form, i.e, multiple vertices may not match at the same point. The algorithm starts at an initial simplex, i.e, it needs n+1 points representing the vertices. ~ The easiest way to get these points is the following: starting from an initial point P0 (which is a vector of n variables) and the n remaining vertices of the simplex are obtained by moving one certain amount in each of the n spatial directions, i.e, the rest of the simplex vertices correspond to the following expression: ~ ~ Pi = P0 + λeî, ∀i ∈ [1,N] (3.27) where each eî is a unit vector of n-dimensional space, and where λ is a constant (could be different for each direction, but it is not common). Graphically this can be seen in figure 3.9 for an optimization problem of 3 variables.

Figure 3.9: Formation of the initial simplex.

The next step of the algorithm is to sort the vertices of the simplex. For this, the value of the error function is calculated in each vertex and they are sorted in ascending order based on these calculated values. Thus, the vertex where the error function takes the lesser value is ~ ~ named as P0, and they are named in this order until the vertex Pn, where the function takes the greater value. From this initial structure, the iterations of the method start in order to advance to the minimum. Thus in each iteration, the worst vertex (where the error function takes the highest value) is replaced by another where the function value is less. This is what is called simplex movement. Thus, the simplex is moving toward the minimum of the function. 3.3 Optimization 59

As there is no progress in the direction of better vertex, but the worst is removed, the method does not converge too quickly. However, the aim of this strategy is to gain robustness, because with this kind of movement it is prevented that the method is trapped in a local minimum. ~ Therefore, in each iteration, the vertex Pn is substituted by another point, which is reached by a series of movements. The various possible movements are:

• Reﬂection

• Expansion

• External contraction

• Internal contraction

• Multidimensional contraction

The manner to chose a movement or another one, for each iteration, is explained below. Firstly, the direction for searching a best point is set. It is interesting to be as far away ~ as possible from the worst point (Pn), the direction chosen is the one that connects this point with the midpoint of the remaining vertices of the simplex. The midpoint has the following expression:

PN−1 P~ P~ = i=0 i (3.28) m N ~ ~ ~ Therefore, the moving direction of the simplex is described by the vector d = Pn · Pm Once defined this direction, the method begins trying a reflection movement. In this case, the reflected point corresponds to

~ ~ ~ Pr = Pm + ρd (3.29) where ρ is an adjustable parameter, although it is usually ρ = 1. Once the reﬂected point is obtained, the value of the error function is calculated at that ~ ~ point. If f(Pr) < f(Pn−1), that is, if the new point is better than the worst of the n − 1 remaining, the reﬂected point is accepted. If, in addition, this new point is better than the ~ ~ rest, that is, f(Pr) < f(P0) the simplex has moved to a right direction. Therefore, what is done afterward is to try an expansion movement to see if moving to this direction, an even better point is obtained. The expression of the expanded point is:

~ ~ ~ Pe = Pm + ψρd (3.30) where ψ is another adjustable parameter. This is adjusted following the particular necessities of each function. Its value will be always grater than 1 (usually 2). However, if the first test is not met, that is, the reflected point is no better than the worst ~ ~ ~ ~ ~ of the n − 1 remaining points but better than Pn (i.e, f(Pr) > f(Pn−1) but f(Pr) < f(Pn)), it is assumed to have gone too far in the chosen direction. Therefore, it must go back and see what happens with an external contraction movement. The expression of this new point is: 60 Design of reconfigurable filters

~ ~ ~ PCi = Pm + φρd (3.31) where φ is another characteristic parameter of the method, whose value will be always lower than 1 (usually 0,5). ~ Another possibility is that the reﬂected point Pr is not better than the worst of all points ~ ~ Pn. In this case, it makes no sense move away from Pn, but trying to get closer by an internal contraction movement. The expression of the internal contraction point is:

~ ~ ~ PCi = Pm − φρd (3.32) In the expression, the parameter φ is the same that above, whose value is usually 0,5. ~ Finally, if the movement of internal contraction do not provide a better point than Pn−1, a multidimensional contraction movement is performed. With this movement, the vertices of ~ the simplex are approached to the best of them, P0. So the probability that the function values are better in the next iteration increases. In the multidimensional contraction movement all ~ vertices, except P0, are modiﬁed as follows:

~ ~ ~ ~ Pi = P0 − σP0Pi (3.33) All mentioned veriﬁcations are performed in each iteration to determine which movement is applied. The method ends when a termination criterion is met, or if the maximum number of iterations or function evaluations of the target is exceeded. As discussed above, the quality of this method is its robustness. The strategy explained above to determine the movement that must be performed makes that the probability that the simplex being trapped in a local minimum is minimized. However, the convergence speed is sacriﬁced.

Broyden Fletcher Goldfarb Shanno (BFGS) This a gradient method, suitable to reach the minimum efﬁciently once it is close. The Broyden-Fletcher-Goldfarb-Shanno [74] combines the advantages of the Steepest descent and the Newton-Rapshon methods. This method requires the estimation of the inverse of the Hessian matrix H−1 of the error function. The initial guess for this matrix is the identity matrix, and its value is updated in each iteration according to the Broyden iteration. In the automated design procedure a restart has been included each time that (3.34) or (3.35) is below a small threshold

∇U j+1 − ∇U j (3.34)

(xj+1 − xj)(∇U j+1 − ∇U j) (3.35) where ∇U j is the gradient of the error function in the j-th iteration and xj is the position (parameter vector) of the error function in the j-th iteration. 3.3 Optimization 61

This restart has been added to the method because there are some practical situations in ﬁlter design where the error function is locally a tilted plane, and the advance direction determined by the BFGS algorithm is a constant line across that plane. In that situation, a move forward or backward supposes no change in the error function, and no change (or very small) in its gradient (∇U j+1 = ∇U j), though the minimum has not been reached yet. In that case the standard BFGS stops. In the implementation used in this CAD procedure, the restart enables the advance in the steepest descent direction in the next iteration, going down the tilted plane.

Optimization with restrictions Optimization with restrictions aims to find the minimum of a function that depends on real variables, whose values are subject to a number of restrictions that describe an allowed region. This is the case that more often occurs in practice, because usually some restrictions are imposed so that the variables of the problem are possible values in reality, for example, if the variables represent distances, they can not be negative. It can be seen that the method to be applied depends on the nature of the function and the constraints. Therefore, the methods are classified according to the complexity of the problem that can solve. In this classification, three types of problems and three kinds of methods are considered. First, it considers the case in which the objective function is quadratic and the restricions are linear, known as Quadratic Programming. In this issue, first equality constraints are only taken, having three methods: the Direct Variables Elimination Method, the Generalized Variables Elimination Method, and the Method of Lagrange Multipliers [68] . In the following case, the inequality restrictions are added to increase the scope of the problem. This situation is called Quadratic Programming with inequality constraints, and it is usually solved by the Active Set Method [68]. Finally, the case called Nonlinear Programming, in which both the objective function and the constraints can be nonlinear. This situation is the most general and it is considered in this work because in the design of microwave devices, the error function to minimize is usually a nonlinear function. In addition, in the multi-objective optimization that has been carried out, nonlinear functions are also handled [75]. Inside Nonlinear Programming, the most efficient and used method is the method of Lagrange-Newton [75], also known as Quadratic Programming Sequential Method. This method uses the aforementioned methods: at each iteration, the objective function is approx- imated by a quadratic function and the constraints are approached by linear functions. Thus, with these approaches the problem can be simplified because solving it directly raises too many difficulties. To summarize, the steps of the method of sequential quadratic programming (SQP) are explained, although the reader is referred to [75] for a more exhaustive explanation. Basically, the method consist on minimizing, at each iteration i, the function:

q(~s(i)) = (1/2)~s(i)T W (i)~s(i) + ~g(i)T ~s(i) (3.36)

where 62 Design of reconﬁgurable ﬁlters

• ~g(i) is the gradient of the error function.

• ~s(i) is the increase of the vector variable that it wants to be optimized, that is, from one iteration to another point it varies as follows ~x(i+1) = ~x(i) + ~s(i), where that increase is subject to the restrictions that limit the valid region.

• W (i) is the Hessian of Lagrange that is approached by using the Broyden Flectcher Goldfarb Shanno (BFGS) method, technique used in the Quasi Newton Method, since its calculation by using ﬁnite-differences require a very high number of evaluations of the objective function [68], [69].

Hybrid optimization These algorithms combine several optimization strategies; the aim is to use in each case the most suitable algorithm, since all strategies are not valid for all optimization steps nor all problems. There are problems when either there is not a good starting point or the minimum search space is too large, i.e. it is not well bounded. This usually happens at the beginning of the optimization algorithm, so the process starts with direct search algorithms, genetic algorithms or space mapping, because of their robustness. In other cases, the minimum of the error function is bounded and it is really easy to reach it. In this case, a gradient algorithm can be used to accelerate the convergence. Thereby, convergence speed and robustness is increased. In addition to select the algorithms, some mechanisms to switch from one to another must be set. These mechanisms, on the one hand, will limit the number of evaluations of the goal function in each method and, on the other hand, they will set thresholds for the goal function or the progressive increments. There are different methods to combine the several optimization algorithms. In this case, a segmentation strategy is used. It is explained below.

Segmentation A segmentation strategy is proposed in [63], [64] for the design of some filter structures, such as H-plane direct-coupled cavity filters composed of N resonant cavities and N + 1 coupling windows. This strategy consists on designing at each step i only the parameters related to the i-th cavity (dimensions of that cavity, as well as dimension of the two adjacent coupling windows and of the previous cavity), and using the values obtained in the previous iterations for the first i − 1 cavities dimensions. At each step i, only the response of the first i cavities is simulated and compared with an objective response for that particular part of the structure. This segmentation technique transforms a slow multidimensional design process into several efficient and robust design steps, where a small number of parameters are designed at the same time. Still, there is the risk that the coupling among all cavities (not just among adjacent ones) is not properly designed by means of this classical segmentation approach. 3.3 Optimization 63

In order to solve this problem, new steps have been added to the original segmentation strategy in [70]. The resulting segmentation strategy designs the ﬁlter through the following steps:

• Ordinary step. The parameters related to the i-th cavity are designed by simulating the first i cavities and using the values obtained in previous iterations for the rest of the parameters of the first i − 1 cavities. The error function is computed by comparing the response of the i first cavities with their ideal response.

• Coupling step. Every time three consecutive cavities are designed, a new optimization process adjusts, at the same time, all the design parameters of the cavities previously designed. This step provides the small changes required in the parameter values due to the couplings from near cavities.

• Central cavity step. For symmetric filters, when the central cavity is reached, and the first half of the filter has been designed step by step, a new optimization is performed considering the whole structure of the filter, but only the dimensions of the central cavity are finely adjusted, thus considering the coupling among all the cavities.

• Full structure step. A final step is made in order to refine the design and to take into account all possible interactions among cavities. The whole filter is simulated and all the dimensions are refined at the same time, being the starting point being the result of the previous steps.

Table 3.1 summarizes the characteristics of each step in the implemented segmentation strategy. When the optimum point is far (ordinary, coupling and central steps), the transmission parameter, S21, is used to calculate the objective function. Whereas in the full structure step, the reﬂection parameter, S11, is used, as the minimum is close.

Step Simulated structure Design parameters Error function Performed

Ordinary first i cavities cavity i S21 for each cavity Coupling first i cavities first i cavities S21 each three cavities Central Cavity whole central cavity S21 once Full Structure whole all the cavities S11 once

Table 3.1: Characteristics of each step of the segmentation strategy.

3.3.3 Optimization strategy used The optimization strategy used in this thesis is based on the combination of the segmentation strategy and the use of a hybrid optimization tool. Both the efficiency and the robustness of the optimization process can be improved by using an adequate combination of some optimization algorithms instead of using a single one. If only one gradient method is used, it may fail to reach the optimum if the starting point 64 Design of reconfigurable filters is far from it. On the other hand, the use of a robust method such as the simplex method or a genetic algorithm ensures convergence but at the cost of a low efficiency. In this case, in the design process the following combination of optimization algorithms is used. At the beginning robust non-gradient methods (Direct search and simplex) are used and, after some iterations, an efficient gradient algorithm (Broyden Fletcher Goldfarb Shanno (BFGS)) is used to refine the solution when the minimum is close. The most suitable combination of the optimization algorithms depends on the proximity of the starting point to the minimum. Therefore, a different hybridization strategy has been adopted for each step of the segmentation procedure.

• Ordinary step. In this step the parameters related to the i-th cavity are designed. Since this is the ﬁrst time that these parameters are designed, there is a small probability that the starting point is very close to the minimum. Therefore, the optimization starts with a genetic algorithm or the direct search with an algorithm of coordinates rotation in order to approach the minimum. The search is then continued with the simplex method. At this point the minimum is close, and the BFGS method is used.

• Coupling step. Every time that three consecutive cavities are designed, the parameters of the ﬁrst i cavities are all re-designed at the same time. Here the minimum should be close, so only simplex and BFGS are used.

• Central cavity step. When the central cavity is reached, the whole structure is simulated, but only the parameters related to the central cavity are designed. Since it is the ﬁrst time that we simulated the whole structure, we might be far from the optimum so three different algorithms are used (direct search with coordinates rotation, simplex and BFGS)

• Full structure step. In this ﬁnal step, the whole structure is simulated and the parameters of all the cavities are re-designed together. In this case, only the simplex and BFGS algorithms are used to minimize the error.

3.4 Validation of the design tool

Once the design tool has been implemented, the dimensions of two filters (two and four cavities) in ESIW technology may be calculated. The design tool will allow to calculate the starting point of the design parameters in ESIW technology (lengths of resonant cavities and widths of coupling windows), and to optimize these parameters. Firstly, the specifications of the frequency response must be known. The goal is to get a Chebyshev design filter in X band, centered at 11 GHz, 300 MHz of bandwidth and return losses greater than 25 dB, for two and four cavities. The ideal responses are shown in figure 3.10. On the other hand, it must be taken into account that the filter is filled with liquid crystal (LC) in both cavities and coupling windows, changing the dimensions of the filters. This 3.4 Validation of the design tool 65

S parameters (2 cavities filter) S parameters (4 cavities filter)

Magnitude(dB) Magnitude (dB) Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 3.10: Ideal Chebyshev response. (a) Two cavities. (b) Four cavities.

material has an electric permittivity varying between εr = 2.4 and εr = 3.2, so the design is made in the intermediate value, i.e. the filters are designed as if they are filled with dielectric material of permittivity εr = 2.8. See section 4.2 for more information concerning liquid crystal. To calculate the dimensions of the filter, some design routines implemented in Matlab are used. These routines use the analysis and design tools developed in the project. See Appendix C for more information about the software routines. The starting point routines need some parameters, detailed below: Parameters related to the structure:

• Ncav, number of cavities in the structure. • a, width of the input/output waveguides and resonant cavities.

• b, structure height.

• t, length of the coupling windows.

• M, number of modes in the input and output waveguide.

• εri , relative dielectric permittivity in each section of the waveguide.

• µr, relative magnetic permeability.

Parameters related to the frequency response: 66 Design of reconﬁgurable ﬁlters

• fci, lower cutoff frequency .

• fcs, higher cutoff frequency.

• LAR, ripple in the bandpass in dB.

With these parameters, the tool calculates the next parameters:

• li, lengths of the resonant cavities.

• wi, widths of the coupling windows.

This allows to have a starting point of the filter dimensions that meets with the specifications indicated in the input parameters. Once the starting point has been obtained, the parameters are optimized following the method explained above with the objective that the final response is as similar as possible to the ideal response. The analysis tool is used in this optimization process to calculate the response in each iteration. First of all, a filter of two cavities is designed and afterward, a filter of four cavities.

3.4.1 Two cavities filter In this section, the filter shown in figure 3.11 will be designed. For designing a filter of two cavities the input parameters needed are:

Figure 3.11: Two cavities prototype.

• Ncav = 2 resonant cavities.

• εri = 2.8 for all i.

• µr = 1.

• a = 19√.05 = 11.3846 mm, εr 3.4 Validation of the design tool 67

• b = 0.5 mm. • t = 2 mm. • M = 11 modes.

• fci = 10.85 GHz.

• fcs = 11.15 GHz.

− RL • LAR = −10log(1 − 10 10 ) = 0.01375 dB. With these parameters, the obtained starting point is:

• li = [8.0660 8.0660] mm, lengths of the resonant cavities.

• wi = [7.2191 5.7972 7.2191] mm, widths of the coupling windows. The tool spends t = 0.605535 s to calculate the starting point. This time is calculated using a AMD Quad-Core A10-5757M processor with Turbo Core technology up to 3.5 GHz and 8 GB of RAM memory. These geometric values are optimized in 3 stages exploiting the symmetry of the filter. In the first stage the length of the first cavity (parameter 2) and the widths of its adjacent windows, a left and right (parameters 1 and 3 respectively) are optimized by comparing the frequency response of the first cavity, the parameter S21, with the ideal response in the cavity, that is, calculating the error at each frequency point. In the following stages, the purpose is to adjust all the filter parameters at once. The filter only has two cavities, so the parameters to optimize are the same parameters that have been optimized in the previous stage. However, in this case, the whole filter frequency response is compared, i.e, the parameter S21 in stage 2 and the S11 in the last stage are compared with the ideal response shown in figure 3.10(a). In Table 3.2 a summary of the steps followed in each stage of the optimization strategy is shown. The number of parameters to be optimized (N Parameters), which parameters (Parameters) are optimized, and the optimization method used (Method) are shown in the table. There are other key parameters that allow to change from one method to another, such as tolerance of the variable to optimize (Xtol), the tolerance of the error function (Ftol) and the maximum number of evaluations of the function (N Evaluations). Finally, the norm used (Norm) and its order (k) are shown. After the optimization, in topt = 42.272077 s, using the processor described above, the following values are obtained:

opt • li =[8.0565 8.0565] mm, lengths of the resonant cavities. opt • wi =[7.2220 5.7989 7.2220] mm, widths of the coupling windows. In figure 3.12 the ideal response, the starting point response and the response with the optimized dimensions are shown. It is observed that the optimized response matches to with the specifications, since it is almost the ideal one. 68 Design of reconfigurable filters

Stage 1 N Parameters Parameters Method Xtol Ftol N Evaluations Norm k 1 2 Simplex 10−6 10−4 100 Huber 5 3 2 3 1 Simplex 10−6 10−7 400 Huber 5 3 2 3 1 BFGS 10−6 10−7 400 Huber 5

Stage 2 N Parameters Parameters Method Xtol Ftol N Evaluations Norm k 3 2 3 1 Simplex 10−6 10−8 400 Huber 5 3 2 3 1 BFGS 10−6 10−8 400 Huber 5

Stage 3 N Parameters Parameters Method Xtol Ftol N Evaluations Norm k 3 3 2 1 Simplex 10−6 10−4 400 Huber 5 3 3 2 1 Simplex 10−6 10−4 600 Huber 5 3 3 2 1 BFGS 10−6 102 1 Huber 5

Table 3.2: Stages in the optimization process of 2 cavities ﬁlter.

3.4.2 Four cavities filter In this section, the filter shown in figure 3.13 is designed. For designing a four cavities filter, the input parameters are:

• Ncav = 4 resonant cavities.

• εri = 2.8 for all i.

• µr = 1.

• a = 19√.05 = 11.3846 mm. εr • b = 0.5 mm.

• t = 2 mm.

• M = 11 modes.

• fci = 10.85 GHz.

• fcs = 11.15 GHz.

− RL • LAR = −10log(1 − 10 10 ) = 0.01375 dB.

With these parameters, the obtained starting point is:

• li =[8.8734 10.1433 10.1433 8.8734] mm, lengths of the resonant cavities. 3.4 Validation of the design tool 69

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 3.12: Frequency response of the 2 cavities ﬁlter. (a) S11. (b) S21.

• wi =[6.7729 4.8258 4.4838 4.8258 6.7729] mm, widths of the coupling windows.

The tool spends t = 1.105315 s to calculate the starting point. The same processor is used. These geometric values are optimized in 4 stages, exploiting the symmetry of the filter. In the first stage the length of the first cavity (parameter 2) and the widths of its adjacent windows, left and right (parameters 1 and 3 respectively), are optimized by comparing the frequency response of the first cavity, the parameter S21, with the ideal response of the cavity, that is calculating the error at each frequency point. In the next stage, the purpose is to adjust the length of the second cavity (parameter 4) and the widths of its adjacent windows (parameters 3 and 5) by comparing the frequency response of the first two cavities, the parameter S21, with the ideal response. Finally, in the last two stages the design is refined taking into account all iterations between cavities, simulating the whole filter and adjusting all parameters at once. In this case, the frequency response of the whole filter is compared, the parameter S21 in step 3 and S11 in the last stage, with the ideal response shown in figure 3.10(b). Table 3.3 shows a summary of the steps followed in each stage of the optimization strategy. The number of parameters to be optimized (N Parameters), which parameters (Parameters) are optimized, the optimization method used (Method) are shown in the table. There are other key parameters that allow to change from one method to another, such as 70 Design of reconfigurable filters

Figure 3.13: Four cavities prototype. tolerance of the variable to optimize (Xtol), the tolerance of the error function (Ftol) and the maximum number of evaluations of the function (N Evaluations). Finally, the norm to be used (Norm) and its order (k) are shown.

Stage 1 N Parameters Parameters Method Xtol Ftol N Evaluations Norm k 1 2 Simplex 10−6 10−4 100 Huber 5 3 2 3 1 Simplex 10−6 10−7 400 Huber 5 3 2 3 1 BFGS 10−6 10−7 400 Huber 5

Stage 2 N Parameters Parameters Method Xtol Ftol N Evaluations Norm k 1 4 Simplex 10−6 10−4 100 Huber 5 4 4 5 3 2 Simplex 10−6 10−7 400 Huber 5 4 4 5 3 2 BFGS 10−6 10−7 400 Huber 5

Stage 3 N Parameters Parameters Method Xtol Ftol N Evaluations Norm k 4 4 5 3 2 Rotation 10−6 10−4 100 Huber 5 4 4 5 3 2 Simplex 10−6 10−8 400 Huber 5 4 4 5 3 2 BFGS 10−6 10−8 400 Huber 5

Stage 4 N Parameters Parameters Method Xtol Ftol N Evaluations Norm k 5 5 4 3 2 1 Rotation 10−6 10−4 400 Huber 5 5 5 4 3 2 1 Simplex 10−6 10−1 600 Huber 5 5 5 4 3 2 1 Simplex 10−6 10−4 600 Huber 5 5 5 4 3 2 1 BFGS 10−6 102 1 Huber 5

Table 3.3: Stages in the optimization process of 4 cavities ﬁlter.

For the optimization process, the processor described in the previous section is used, obtaining in topt = 514.771370 s the following geometric values: opt • li = [8.8616 10.1331 10.1331 8.8616] mm, lengths of the resonant cavities. 3.5 Reconﬁguration 71

opt • wi = [6.7792 4.8380 4.4992 4.8380 6.7792] mm, widths of the coupling windows.

Figure 3.14 shows the difference between the ideal response, the starting point response, and the response with the optimized dimensions. Clearly, it is observed that the optimized response matches perfectly to the speciﬁcations, since it is almost the ideal one.

Parameter S11 of the filter Parameter S21 of the filter

)

(dB

Magnitude Magnitude

Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 3.14: Frequency response of the 4 cavities ﬁlter. (a) S11. (b) S21.

3.5 Reconﬁguration

Since the main objective of this thesis is the design of reconfigurable microwave filters using ESIW technology, a tool that allows that reconfiguration must be developed. The reconfiguration is achieved by filling of liquid crystal (LC) the ESIW sections. When a low frequency bias voltage is applied to the LC, this one changes its dielectric permittivity between two values, εr⊥ and εrk. This change in the dielectric permittivity allows to change some values of the filter frequency response, as the central frequency and the bandwidth. It is necessary to develop a tool that calculates the permittivity value that the cavities and the coupling windows must have in order to reach the desired frequency response change, and thus, the bias voltage that must be applied to the LC to achieve that permittivity. Following the same procedure that in the optimization process, an error function is defined, which must be minimized. The defined error function is: 72 Design of reconfigurable filters

Calculated m Calculated m 1/m X |li − li| X |wi − wi| error = + (3.37) li wi where

• li is the vector with the lengths values of the real resonant cavities, it means, the values designed before and, therefore, manufactured.

• wi is a vector with the widths values of the real coupling windows, it means, the values designed and optimized and, therefore, manufactured.

Calculated • li is a vector with the lengths of the resonant cavities, calculated in each iteration of the reconﬁguration process.

Calculated • wi is a vector with the widths of the coupling windows, calculated in each iteration of the reconﬁguration process.

• m is the norm order.

For some new given specifications, it means, a new central frequency and a new bandwidth, thus new values for lengths of the resonant cavities and widths of the coupling windows, the goal is to minimize the error function. In order to achieve this purpose, the dielectric permittivities of the LC that make that the calculated lengths of the cavities and the widths of the windows are equal to the real (manufactured) are calculated. First of all, it must be established a minimum and maximum value between which the permittivity of LC can vary. These values are set by a technological limitation, since the permittivity of the LC ranges between εr⊥ = 2.4 and εrk = 3.2 [7], [10]. Once these values are set, the reconfiguration process consists on calculating the lengths and widths that would be needed for the given specifications. This is reach by varying the value of the permittivity εr in both windows and cavities. These permittivities will vary until the error is less than a certain tolerance. This error will be smaller when the calculated dimensions are as close as possible to the actual dimensions. To minimize the error by following the process described above two different algorithms implemented in Matlab [57] can be used:

• fminsearch uses the Downhill Simplex algorithm described in section 3.3.2. This algorithm uses a simplex of n+1 points for n-th vectors x. First, it makes a simplex around the initial point x0 by adding 5 % of each component x0(i) to x0, using the n vectors as elements of the simplex and x0. Then, the algorithm modiﬁes the simplex following the process described in section 3.3.2.

• fminunc uses gradient algorithms, such as maximum slope or the BFGS, described in section 3.3.2. These methods are most effective when the function to be minimized is continuous in its ﬁrst derivative. 3.6 Validation of the reconﬁguration tool 73

3.6 Validation of the reconﬁguration tool

Once the filter is designed and meets the specifications, it can be manufactured. Once the filter has been manufactured, it may be reconfigured, i.e to change its center frequency and bandwidth, by using the tool developed in the thesis. The tool allows to obtain the dielectric permittivities that makes the filter meet the new desired response. The routine for calculating the permittivities requires as input parameters some initial values of permittivity εr0i and the actual dimensions of the filter lengths of the cavities, li, and widths of the coupling windows, wi. The filter center frequency is going to be change in steps of 100 MHz, keeping constant the other parameters such as bandwidth, 300 MHz, and return losses, 25 dB. This is made to know the maximum range of reconfiguration that allows the liquid crystal (LC). The LC, as noted in previous chapter, can only vary between εr⊥ = 2.4 and εrk = 3.2 , so the reconfiguration will be limited by these values.

3.6.1 Two cavities filter The desired center frequency is incremented and decremented keeping constant the bandwidth. The values of the dielectric permittivity that the LC should have in each section of the filter are obtained, using the symmetry of the filter. Therefore, four values of permittivity are needed to know, the calculated values are:

• Input waveguide length that will be equal to the output waveguide length.

• First and third coupling windows widths (windows 1 and 3).

• Resonant cavities lengths.

• Central coupling window width.

To calculate these values, the reconﬁguration tool developed is used. The following key parameters such as termination tolerance variable (Xtol), tolerance termination of the error function (Ftol), the maximum number of iterations (Maxiter) and the maximum number of evaluations of the error function (N Evals). The value of these parameters is shown in Table 3.4.

Xtol Ftol MaxIter N Evals 10−8 10−8 400 500

Table 3.4: Values of the reconﬁguration parameters for two cavities ﬁlter.

Table 3.5 shows the frequencies values for which the permittivities have been calculated together with these permittivities. The table shows that, considering the range of variation of the permittivity of the LC, the frequency range in which the center frequency may vary, 74 Design of reconﬁgurable ﬁlters

f0, keeping the bandwidth, 600 MHz. It is a symmetrical range, since the highest center frequency is 11.3 GHz and the minimum 10.7 GHz. A higher or lower center frequency require values of permittivity that the LC cannot cover or a change in some speciﬁcations of the ﬁlter, i.e more or less bandwidth, and more return losses.

Frequencies (GHz) Permittivities f0 fci fcs Input/Output waveguide Windows 1 and 3 Resonant cavities Central Window 11.3 11.15 11.45 2.6687 2.5640 2.6687 2.4242 11.2 11.05 11.35 2.7124 2.6366 2.7124 2.5388 11.1 10.95 11.25 2.7572 2.7117 2.7572 2.6573 10.9 10.75 11.05 2.8502 2.8694 2.8502 2.9071 10.8 10.65 10.95 2.8985 2.9522 2.8985 3.0387 10.7 10.55 10.85 2.9480 3.0379 2.9480 3.1751

Table 3.5: Values of the permittivites for the two cavities ﬁlter.

Figures 3.15, 3.16, 3.17, 3.18, 3.19, 3.20 show the frequency response for the different permittivity values calculated above. These frequency responses are compared to the ideal ones.

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.15: Frequency response of a ﬁlter of two cavities centered at 11.3 GHz.

It can be seen that all frequency responses perfectly fit to the ideal response at the same frequency, so it can be concluded that the reconfiguration tool works properly. 3.6 Validation of the reconfiguration tool 75

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.16: Frequency response of a ﬁlter of two cavities centered at 11.2 GHz.

3.6.2 Four cavities filter In this case, the same procedure that for the two cavities filter is followed. The desired center frequency is incremented and decremented keeping constant the bandwidth. The value of dielectric permittivity that the LC should have in each section of the filter is obtained, using the symmetry of the filter. Therefore, in this case, six values of permittivity are needed to calculate (see figure 3.13). The values in the following sections are needed:

• Input waveguide length that will be equal to the output waveguide length.

• Coupling windows widths, 1 and 5.

• Coupling windows widths, 2 and 4.

• Central coupling window width (3).

• Resonant cavities lengths, 1 and 4.

• Resonant cavities lengths, 2 and 3.

To calculate these values, the developed reconfiguration tool is used, in the same way that for the two cavities filter. However, in this case, the maximum number of iterations and the maximum number of evaluations of the error function should be increased. It is a more complex structure, so more iterations are required for the method to converge. Therefore, the parameters are shown in Table 3.6. 76 Design of reconfigurable filters

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.17: Frequency response of a ﬁlter of two cavities centered at 11.1 GHz.

Xtol Ftol MaxIter N Evals 10−8 10−8 800 1000

Table 3.6: Values of the reconﬁguration parameters for four cavities ﬁlter.

Table 3.7 shows the frequencies values for which the permittivities have been calculated together with those permittivities. The table shows that, considering the range of variation of the permittivity of the LC, the frequency range in which the center frequency may vary, f0, keeping the bandwidth, 500 MHz. In this case, it is not a symmetrical range, the highest center frequency is 11.2 GHz and the minimum 10.7 GHz. A higher or lower center frequency require values of permittivity that the LC cannot cover or a change in some speciﬁcations of the ﬁlter, i.e more or less bandwidth, and more return losses.

Frequencies (GHz) Permittivites f0 fci fcs I/O waveguide Windows 1/5 Windows 2/4 Window 3 Cavities 1/4 Cavities 2/3 11.2 11.05 11.35 2.7078 2.6293 2.7054 2.4376 2.7074 2.3727 11.1 10.95 11.25 2.7546 2.7028 2.7530 2.5672 2.7543 2.5110 10.9 10.75 11.05 2.8520 2.8574 2.8519 2.8409 2.8520 2.8030 10.8 10.65 10.95 2.9026 2.9386 2.9034 2.9852 2.9028 2.9572 10.7 10.55 10.85 2.9546 3.0225 2.9564 3.1349 2.9549 3.1171

Table 3.7: Values of the permittivites for the four cavities ﬁlter. 3.6 Validation of the reconﬁguration tool 77

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.18: Frequency response of a ﬁlter of two cavities centered at 10.9 GHz.

Figures 3.21, 3.22, 3.23, 3.24, 3.25 show the frequency response for the different permittivity values calculated above. These frequency responses are compared to the ideal ones. It can be seen that the range to increase the center frequency is less than to decrease it. Also in the figures it can be seen that at f0 = 11.2 GHz the frequency response differs from the ideal one, as well as at f0 = 10.7 GHz. In the case of two cavities all frequency responses perfectly fitted to the ideal responses. However, although in this case it is not equal to the ideal, it can be said that the reconfiguration tool also works correctly for a four cavities filter. 78 Design of reconfigurable filters

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.19: Frequency response of a ﬁlter of two cavities centered at 10.8 GHz.

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.20: Frequency response of a ﬁlter of two cavities centered at 10.7 GHz. 3.6 Validation of the reconﬁguration tool 79

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.21: Frequency response of a ﬁlter of four cavities centered at 11.2 GHz.

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.22: Frequency response of a filter of four cavities centered at 11.1 GHz. 80 Design of reconfigurable filters

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.23: Frequency response of a ﬁlter of four cavities centered at 10.9 GHz.

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.24: Frequency response of a ﬁlter of four cavities centered at 10.8 GHz. 3.6 Validation of the reconﬁguration tool 81

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 3.25: Frequency response of a ﬁlter of four cavities centered at 10.7 GHz.

Chapter 4

Technological aspects of manufacturing

Once the ﬁlters have been designed, they can be manufactured. However, this task is not as easy as it seems. There are important aspects to consider before proceeding to manufacture, including, the study of how the liquid crystal (LC) is polarized and how the ESIW is fed. Such considerations are explained in detail below.

4.1 Filter feed

One of the drawbacks associated with both classical waveguide technology and ESIW technology is the complexity associated with design and manufacturing of transitions to ease their integration with planar devices and systems. Over the last decades, several studies have proposed and validated (analytical and experi- mentally) different transitions that efﬁciently facilitates the integration of rectangular waveguides and microstrip lines [76, 77, 78, 79]. However, the structures have the following drawbacks:

• All of them are really complex three-dimensional structures to design and manufacture.

• The integration schemes always consist of two or more separate pieces, whose coupling implies a delicate assembly process as well as the emergence of undesired parasitic effects.

• The dielectric substrate sheet has to be cut in a speciﬁc way, so it is essential to use accurate techniques of manufacturing.

• Because of manufacturing tolerances, the spectral features of the transition can be se- riously degraded, so it will usually be necessary to provide ingenious mechanisms to facilitate the performance setting (“tuning”).

Therefore, the integration of planar and non-planar devices in a global communications system becomes a complex and expensive task. 84 Technological aspects of manufacturing

The microstrip technology has a great reception for the synthesis of microwave circuits, for its easy design, low manufacturing cost and good performance. Furthermore, it synthe- sizes a transmission line very suitable to excite the dominant mode of ESIW (fundamental mode TE10), as the distributions of the electric field in a cross section of both structures are oriented approximately in the same direction, while sharing the same profile, as shown in figure 4.1.

Figure 4.1: Electric ﬁeld distribution of the dominant mode. (a) Microstrip line. (b) Rectan- gular waveguide ESIW.

Thanks to this agreement in the field distributions, it is possible to excite the ESIW structure by a microstrip line connected directly to the center of the ESIW waveguide, since the electric field of the fundamental mode reaches its maximum value at this point. Therefore, for all the transitions related to the ESIW technology, it is required that the fundamental mode TE10 of the ESIW is adapted to the fundamental mode of the microstrip transmission line. Among the types of transitions for feeding a filter in ESIW technology, the most important are listed below.

• Transition from coplanar waveguide (CPW) to SIW [80]. As shown in ﬁgure 4.2 this transition is made up of a coplanar waveguide section with 90o elbows in each slot. The coplanar line adds a stub to mark the transition and the rectangular waveguide is deﬁned by posts. This transition has low losses in the microstrip line, but it does not cover the entire usable bandwidth of the SIW, and therefore the ESIW.

• Transition from microstrip to SIW in a linear taper [21]. It was the ﬁrst one to appear due to the similarity between the fundamental mode of the SIW and the microstrip waveguide, and the fact that the microstrip line is commonly used in planar circuits. The main reasons of its appearance are its simple structure and very low losses. This transition covers the full bandwidth of the SIW, and for similarity, also of the ESIW. It uses a microstrip line with a spindle shape for exciting the mode, as shown in ﬁgure 4.3. Although its behavior it is better compared to other microstrip or coplanar transitions. If the characteristic impedance of the ESIW is very high compared with the microstrip, it is better to use the exponential transition. 4.1 Filter feed 85

Figure 4.2: Coplanar transition scheme.

Figure 4.3: Linear taper transition scheme.

• Transition from microstrip to ESIW in inverted taper [81]. It is known that the electric field mode TE10 in a rectangular waveguide and in a ESIW reaches periodically (in the direction of wave propagation) its minimum value at intervals of λg/2. Because of this, the operating principle of this transition, shown in figure 4.4, is based on forcing the connection of the ground plane of the microstrip line to the ESIW at a distance λg/4 from the beginning of the section of the ESIW. This ensures that at the input, the electric field reaches its maximum value. Furthermore, this transition efficiently performs its function in a really wide frequency range.

• Exponential transition from microstrip to ESIW [1]. It is shown in figure 4.5 and also in figure 4.6, where all dimensions are detailed. It can be considered as a transition of two steps. In the first step there is a transition from the microstrip line to the ESIW partially filled with a dielectric block. This block has the same microstrip substrate, so its permittivity is also equal. An iris of width wir improves the transition from 86 Technological aspects of manufacturing

Figure 4.4: Inverted taper transition scheme.

microstrip to the ESIW partially ﬁlled with dielectric. Due to the strong similarity between the fundamental modes of the microstrip line and the ESIW with the dielectric block, the transition has low losses. Immediately after the microstrip, the width of the block of the ESIW is reduced exponentially in order to match a waveguide completely empty, i.e, the ESIW; this narrowing that constitutes the second step is called taper. To have certain mechanical stability in the taper, the length is limited to a length lt, and its end is rounded. The width of the taper decreases exponentially with distance, reaching its ﬁnal value, wtf . This transition is broadband, allowing the excitement of the ESIW. It presents low losses across the bandwidth of the ESIW, allowing its integration into the planar substrate.

Figure 4.5: Exponential transition scheme (3D view).

• New transition from microstrip to ESIW. This transition is entirely designed in this thesis, by joining the transition in linear taper way and the exponential transition. It is shown in ﬁgure 4.7, and all its dimensions are detailed in ﬁgure 4.8. It is a transition of two steps.

In the ﬁrst step there is a transition from the microstrip line of width wms to a ESIW 4.2 Liquid crystal and its microwave applications 87

Figure 4.6: (a) Taper layout (top view). Dark gray is dielectric substrate; light gray represents the copper metallization on top layer; black represents the border metallization which has been used to close the ESIW; and white is air. (b) Detail of the taper end.

completely ﬁlled with the same dielectric of the microstrip sheet. This transition ends in a microstrip line of width wt, the value which matches the impedances of the microstrip and the ESIW. The width of the microstrip vary linearly over a length lt, which is optimized for minimum losses.

An iris of width wir improves the transition from microstrip to ESIW filled with dielectric. The dimension of the iris along the propagation direction is just the width of the metallization layer. The widths wir and wpi are optimized for minimum reflection. There is little reflection in this step because there is a strong similarity between the fundamental modes of the microstrip line and the full ESIW. Immediately after the step from microstrip to ESIW, the width of the slab is exponentially reduced in order to match to a ESIW filled of LC. Since the taper cannot be of infinite length, and in order to provide it with certain mechanical stability, it is limited to a length lp, and the end is rounded. The final width of the exponential taper is wpf . This transition is broadband, allowing the excitement of the filled ESIW. It presents low losses across the bandwidth of the ESIW, allowing its integration into the planar substrate. Therefore, this transition is the most convenient to feed the topology under analysis in this thesis.

4.2 Liquid crystal and its microwave applications

The growing interest in developing reconﬁgurable devices for using in applications in the microwave frequency range has led to search for various technologies for designing these devices [7]. Thus, there are many studies on the use of piezoelectric materials in tunable phase devices and the use of ferroelectric materials in resonators and voltage controlled oscillators 88 Technological aspects of manufacturing

Figure 4.7: Transition scheme (3D view).

Figure 4.8: (a) Transition layout (top view). Light gray is dielectric substrate; dark gray represents the copper metallization on top layer; black represents the border metallization which has been used to close the ESIW; and white is the liquid crystal. (b) Detail of the taper end.

[82]. Also, it has been developed designs of tunable filters based on the use of MEMS [83] or varactor diodes [84]. During the last two decades, the use of liquid crystal (LC) for designing tunable devices at microwave frequencies has been a promising alternative [8] in the research field due to the dielectric anisotropy of the LC molecules. On the other hand, the reduced consumption, the reduced operating voltage and the sim- plicity of manufacturing the devices with LC [6] lead this technology over other alternatives in the case of designing tunable devices. For example, varactor diodes require high voltages, up to 30 V, to achieve large tuning range at frequencies greater than 1 GHz [85]. On the other hand, devices based on ferrite technology [86] require a magnetic field for tuning adjustment, which leads to problems in terms of size and power consumption. The LC need lower polarization voltages to achieve similar ranges of reconfiguration. Examples of designs based on LC technology for using in microwave devices are capac- 4.2 Liquid crystal and its microwave applications 89 itors [87], resonators [10], phase tunable devices [88], and variable frequency antennas [89]. There are also very recent contributions in tunable bandpass filters with microstrip technology [90].

4.2.1 Liquid crystal properties The dielectric anisotropy of LC molecules makes that the electrical of the LC properties depend on the orientation of these molecules [8],[7],[10]. This orientation can be changed by applying an external electric field. Specifically, the dielectric permittivity of LC can be tuned between two extreme values, εr⊥ and εrk, applying an external electric field to the sample. As shown in figure 4.9, in the absence of an electric field (V = 0), the LC molecules remain aligned parallel to the microstrip line and the microwave signal experiences the perpendicular permittivity, εr⊥. When the saturation voltage (V = Vsat) in the LC is reached, the molecules stand perpendicular to the microstrip line and the permittivity in the path of the microwave signal is the parallel permittivity εrk. It can be seen that as voltage increases, the permittivity increases as well.

Figure 4.9: Liquid crystal polarization.

The center frequency of a bandpass ﬁlter (f0) depends on the effective permittivity and the dimensions of the structure, so the continuous adjustment of the center frequency is generated as the voltage is applied between the two extreme values of permittivity (εr⊥ and εrk). This f0 decreases when the permittivity increases, that is, increasing the voltage. This allows that devices based on LC to be controlled by a voltage.

4.2.2 Liquid crystal encapsulation Another aspect to consider to polarize the liquid crystal is the conditioning of the cavity where it will be confined. The LC is a fluid material, which needs to be confined in a sealed 90 Technological aspects of manufacturing cavity within the device. This cavity should not have any crack, i.e, must be perfectly sealed. The side walls of the cavity must be of dielectric substrate, and the upper and lower layers must be conductors. The conductor must be specially treated so that the liquid crystal (LC) molecules align properly when the cavity is filled with the material [10]. During the process of filling the cavity, the liquid crystal is at a higher temperature than room temperature. To align the molecules and in order to recover their properties, the material must be cooled. This is the reason why the LC has limitations in terms of operating temperature, since its variation modifies the properties of the material. This makes the LC not suitable for applications where the temperature range varies widely, for example space communication applications.

4.2.3 Liquid crystal polarization As it has been mentioned above, the use of LC as dielectric substrate is the reason that enables the ﬁlter to be reconﬁgurable.

Figure 4.10: An inverted-microstrip structure for a LC bandpass ﬁlter.

To polarize the LC, it must be sited between two armatures physically separated, where a voltage difference is applied. This bias signal is a low frequency AC signal (1 kHz) ranging from 0 Vrms to 15 Vrms, which causes a variation in the dielectric permittivity of the LC. This is a technological problem, since the ESIW has one conductor and the LC must be polarized independently from the common structure. In [7] some reconfigurable filters have been manufactured using the LC. This is a fluid material, so it needs to be confined in a cavity inside the device and afterward sealed. There- fore, these filters are implemented using an inverted microstrip structure, as shown in figure 4.10. This structure easily enables the polarization of the LC, since it has two separate conductors, the low frequency signal can be applied at the same time that the RF signal. In order to try to solve the problem that arises in the case of the ESIW, it has raised the idea of cutting the top and bottom covers of the waveguide. For this, it is necessary to study how the currents are distributed in a ESIW. 4.2 Liquid crystal and its microwave applications 91

When a mode is set in a waveguide, some currents associated with that mode are generated at the same time, as shown in ﬁgure 4.11.

Figure 4.11: Surface currents distribution of mode TE10 in a rectangular waveguide.

If some slots are made in the upper waveguide cover in order to have independent armatures that allow to polarize the LC, then currents are interrupted, resulting in the occurrence of radiation. To avoid cutting the currents and the radiation, a new structure is designed which, instead of having a single cut, the upper cover is divided in several smaller squares. Thus, current can flow through the grid created between the squares, preventing interruption by the discontinuity. The two cavities filter designed in section 3.4 is modified to make the small squares in the upper cover as shown in figure 4.12.

Figure 4.12: Layout of an ESIW with the upper cover cut in small squares.

The voltage difference is applied between each small inner square and the rest of the structure. It is expected that this solution, even being a radiating structure, have a proper performance. This structure is simulated in CST electromagnetic commercial software, and the result of this simulation can be seen in ﬁgure 4.13. From the simulation it can be observed that the response of the ﬁlter is shifted 400 MHz to lower frequencies, but keeping perfectly the shape. It can be produced by the capacitance 92 Technological aspects of manufacturing

S parameters of the filter

)

(dB

Magnitude

Frequency (GHz)

Figure 4.13: Result of the simulation of the ESIW with the upper cover cut in small squares. introduced by the squares. The gap between the structure and the inner small squares can be modeled as a capacitor that affects the filter frequency response. It is not a big problem, since the filter can be designed taking into account these considerations. However, there are some limitations for manufacturing the filter, the size of the squares and the holes to polarize the liquid crystal. They cannot be as small as desired since the milling machine has some limitations in the accuracy and the diameter of the bit. For that reason, the filter is re-designed at a lower frequency (f0 = 6.5 GHz and BW=100 MHz) in order to obtain bigger dimensions. This is carried out in the next chapter.

4.3 Choke Inductor

Another aspect that has to be considered is the decoupling between the polarization signal (very low frequency) and RF signal. This is done to avoid that the RF signal follows the very low frequency track, producing losses by the radiation. This can be solved by using a Choke inductor [91] to decouple both signals. This inductor acts as a high impedance resistor at high frequency (RF) and as a short circuit or low impedance resistor at DC o low frequency, respectively. This is clearly seen in the equation (4.1). The higher the frequency the higher the impedance and viceversa.

Z = jωL (4.1) where ω = 2πf The following goal is to obtain the value of the inductance that meets the aforementioned behaviour. In the webpage of Murata [92] there are several series of inductors that can be 4.3 Choke Inductor 93 used with the present purpose. Two of them have been chosen (models LQW18ANR10GOZ [93] and LQW18ANR16GOZ [94] of Murata). Their admittances have been measured in the whole frequency range using a commercial software (AWR Microwave Office [95]) and the .s2p file given by the manufacturer (Murata). The result is shown in figure 4.14.

Figure 4.14: Admittance of the choke inductors over the frequency.

It can be observed that the pink trace (Inductance LQW18ANR16GOZ) has a near linear behaviour at the center frequency of the filter (6.5 GHz) and around it, while the blue line (In- ductance LQW18ANR10GOZ) presents some non-linearities near 6.6 GHz. The admittance is too low (high impedance) at this frequency and high enough (low impedance) at very low frequency (1 kHz). Therefore, the pink choke inductor is chosen, corresponding with the model LQW18ANR16GOZ. A picture of the inductance and its dimensions are shown in figure 4.15. Moreover, some characteristics of this inductance are shown in table 4.1. Finally, in order to use the Choke inductance, a small technological problem must be solved to solder the inductance to the cover. For that reason, the square shown in figure 4.16 is done in the upper cover. The light gray zone is where the copper metallization of the upper layer has been removed by using the milling machine. There will not be electrical contact between both the pins of the choke inductance. The bias voltage (low frequency) is fed by the pin 1. Thus, the inductance will allow that the low frequency signal goes through it to the rest of the structure biasing the liquid crystal. However, the RF signal cannot pass through the inductance (from pin 2 to pin 1) avoiding the losses and the radiation of the desired signal. 94 Technological aspects of manufacturing

Figure 4.15: Shape of the Choke inductor and dimensions.

Shape L size 1.6 0.2 mm W size 0.8 0.2 mm T size 0.8 0.2 mm Speciﬁcations Inductance 160 nH 2 % @ 100 MHz Rated current 150 mA Maximum of DC resistance 2.1 Ω Q (minimum) 32 @ 150 MHz Self resonance frequency (min.) 1350 MHz Operating temperature range -55 to 125º C

Table 4.1: Characteristics of the inductance LQW18ANR16G0Z.

Figure 4.16: Shape of the cover for feeding the choke inductor. Light gray is the substrate and dark gray represents the copper metallization in the upper layer. Chapter 5

Results and measurements

5.1 Redesign of the ﬁlter

The re-design is done at 6.5 GHz and BW=100 MHz. This enables to have a filter with dimensions that can be manufactured with the available milling machine in the research group. In addition, the dielectric permittivity that has been chosen to design the filter is the same one in the whole filter. The chosen value is εr = 2.4, which is the minimum value of the liquid crystal (LC). This value is produced when bias voltage is not applied. Once the filter is filled with the LC, its frequency response can be measured before starting the reconfiguration without applying any bias voltage. Taking into consideration the aforementioned problem about the frequency shift caused by the realization of squares in the upper cover, it is proceeded to perform the design at 7 GHz. The filter is designed with the developed tool. Hence, this also allows to validate the use of this tool for designing filters at any frequency. Both the ideal frequency response and the optimized frequency response achieved with the aforementioned tool are shown in Figure 5.1. Furthermore, the filter dimensions are shown below.

• Ncav = 2 resonant cavities.

i • εr = 2.4 for all i.

• µr = 1. • a = 19 mm,

• b = 0.5 mm.

• t = 1 mm.

• M = 11 modes.

• fci = 6.95 GHz.

• fcs = 7.05 GHz. 96 Results and measurements

− RL • LAR = −10log(1 − 10 10 ) = 0.01375 dB.

• li = [16.9166 16.9166] mm, length of the resonant cavities.

• wi = [9.5577 6.5364 9.5577] mm, width of the coupling windows.

Figure 5.1: Ideal and optimized frequency response of the redesigned ﬁlter.

Afterward, the ﬁlter is modeled and simulated in CST for a random number of squares, but this squares must meet certain characteristics.

• The squares on the input and output waveguides form an inverted triangle, i.e. no cuts are made in the area where the exponential taper is located.

• The length of the outer square in the windows is the same as the width of these windows, t.

• The squares in the cavities do not have major restrictions.

Once the ﬁrst simulation is done, it is necessary to optimize the response. The parameters that are optimized are detailed below.

• Distances between squares in both the longitudinal (dl) and transversal (da) dimensions. • Size of internal squares in the dimensions x and y. 5.1 Redesign of the ﬁlter 97

• External squares. They change their size proportionally to the inner squares since the width of the gap between them is constant (g = 150 µm). This is the minimum drill bit size of the milling machine.

• It also has to be considered that the dimensions in windows (aclW 1 and aclW 2) and input and output waveguides (acaio and aclio) are optimized independently of the cavities (aca and acl).

Figure 5.2: Layout of upper cover of the optimum ﬁlter with the dimensions on it.

On the one hand, considering these parameters, it is proceeded to the optimization using a genetic algorithm because a good starting point is not available. This algorithm performs the search to this optimum point throughout the set range. After a time t = 25.3 h, the software gets an optimization of the squares dimensions and the distances, obtaining the following dimensions and number of squares.

• Input and output waveguides: 8 columns of squares and 4 rows with a different number of squares in each row (2 in ﬁrst row, 4 in second one, 6 in third one, and 8 in last one). The dimensions are acaio = 1.4225 mm and aclio = 1.2375 mm. • Coupling windows 1 and 3: 4 squares in one row. The length of the squares is the same that the coupling window aclW = t = 1 mm and the width has been optimized being acaW 1 = 1.3644 mm

• Resonant cavities: 8 × 8 squares with dimensions aca = 1.4225 mm and acl = 1.1733 mm. 98 Results and measurements

• Central coupling window: 2 squares in one row. As explained above the length of the squares is the same that the coupling window aclW = t = 1 mm and the width has been optimized being acaW 2 = 2.0982 mm. • Distances between the squares. The distances are equal for the cavities, input and output waveguides and all the coupling windows. The resulting values are da = 0.58 mm and dl = 0.57 mm.

The filter layout in CST and the dimensions are shown in figure 5.2. The simulation of the optimized design is shown in figure 5.3. It can be observed that the frequency response is centered at 6.4 GHz, 600 MHz shifted from the expected center frequency. The bandwidth, the selectivity and the return losses of the filter have kept constant. Moreover, a small resonance has appeared around 6.79 GHz.

Figure 5.3: Frequency response of the optimum ﬁlter.

On the other hand, the transition must be also optimized to get the minimum reflection feeding the filter. The desired response is that S11 is as small as possible to avoid the reflec- tions. For that reason the following parameters of the transition are simulated and optimized with CST software [51] to obtain this purpose. These parameters are detailed below (further details in figure 4.8):

• wt, width of the end of the linear taper.

• lt, length of the linear taper.

• wir, width of the iris.

• wpi, width of the beginning of the exponential taper.

• lp, length of the exponential taper.

• wpf , width of the end of the exponential taper. 5.1 Redesign of the ﬁlter 99

After a time t = 8 h, the software gets an optimization of the aforementioned dimensions with the Downhill Simplex method, obtaining the following dimensions.

• wt = 7.917 mm.

• lt = 5.586 mm.

• wir = 12.540 mm.

• wpi = 10.773 mm.

• lp = 5.669 mm.

• wpf = 1.087 mm.

Moreover, the obtained simulation is shown in ﬁgure 5.4, which shows that the transitions presents losses under -20 dB in the range from 5.9 GHz to 6.7 GHz.

Figure 5.4: Frequency response of the optimum transition.

The next step is to join both the filter design and the transition and simulate it. The layout of the complete filter is shown in figure 5.5 and the simulation is shown in figure 5.6. It can be observed that thanks to the effect of the transition, the frequency response of the filter has centered at 6.5 GHz. It is noted that the return loss remains below -20 dB but the bandwidth is reduced. On the other hand, the transmission coefficient presents small losses in the passband due to the transition. The small resonance has lightly shifted and increased by the effect of the transition. Furthermore, in order to check if the filter can change its center frequency by changing the dielectric permittivity of the liquid crystal, the design is simulated for εr = 3. The simulation is shown in Figure 5.7. Although the LC can reach a dielectric permittivity of εr = 3.2 it will never be achieved since the LC is not polarized in the whole surface. This simulation shows that the filter keeps its filtering capacity centered at 5.9 GHz, that is 400 MHz of reconfiguration. The return losses are lightly above to -10 dB and the BW has increased at the cost of worse performance of the transmission coefficient. 100 Results and measurements

Figure 5.5: Layout of the complete ﬁlter with the transition.

Moreover, this filter can also act as an empty filter, with a different frequency response. The frequency response of the filter has changed, shifting to a higher frequency. The shifting corresponds to the equation (5.1). On the other hand, the frequency response is not optimum but it still maintain the shape of a bandpass filter as shown in Figure 5.8. √ New √ 9 f0 = f0 εr = 6.5 · 10 2.4 = 10.07 GHz (5.1)

5.2 Manufacturing process

The manufacturing process is carried out by using the milling machine ProtoMat S103 of LPKF [96] and this process is divided in two different steps: manufacture of the covers and of the central body. Before proceeding with the manufacturing process, the layout of the filter must be transformed into a compatible format by the software used by the milling machine, LPKF Circuit- Pro [97]. This software imports different kind of formats, but the most common is Gerber data. The different steps that the milling machine has to drill are modeled in CST Studio [51] as different layers. This CST file is exported into a Gerber file that the CircuitPro software can read and process to control the circuit board plotter (see figure 5.9). There are eight layers or steps, shown in figure 5.10 and explained in detail in each step that the milling machine has to follow. First, the following steps are done to get the upper and lower covers. The upper and lower cover are manufactured on a substrate ROGERS RO4003C [98] metallized on its both sides. This substrate has low dielectric tolerance and low loss, so that it has excellent electrical performance and it allows applications with higher operating frequencies. Furthermore, this substrate has a thickness of 0.8 mm giving the required consistency to the filter.

• Fiducials markers. First of all, the milling machine makes marking drills, which 5.2 Manufacturing process 101

Figure 5.6: Result of the simulation of the ESIW with the upper cover cut in small squares for εr = 2.4.

are small marks to properly make the drill bit holes. These marking drills are always done before making any hole. After that, four holes called ﬁducials markers are made. These markers are done in the corners of the substrate and they will allow to the milling machine to position the substrate plate when it is removed from the machine and later it is placed back again. At the end of this step, it is necessary to sand the substrate to remove the metal burrs. This is done for both upper and lower covers.

• Polarization holes. Before making the polarization holes the milling machine makes marking drills. There are 178 holes of different diameters. They correspond to each section of the filter, where the small squares have different sizes. In the table 5.1, the values of the different diameters are shown. This step is modeled in CST Studio as shown in figure 5.11 and the manufactured result is shown in figure 5.12. At the end of this step, as in the previous step, it is necessary to sand the substrate to remove the metal burrs. This is only done in the upper cover.

Section Diameter (mm) I/O waveguide 0.7 Coupling windows 0.4 Resonant cavities 0.6

Table 5.1: Values of the diamaters of polarization holes.

• Adhesive material. In this step, an adhesive material is stuck to the substrate. This is necessary because only the sidewalls of the waveguide are metallized. Therefore, to avoid the metallization of the upper and lower sides, they must be covered by the adhesive material. 102 Results and measurements

Figure 5.7: Result of the simulation of the ESIW with the upper cover cut in small squares for εr = 3.

Figure 5.8: Result of the simulation of the empty ﬁlter.

If the sidewalls are not metallized, the microstrip line would derive some of its power by the line of parallel plates formed in the upper and lower layers of the cover. Fur- thermore, since the substrate is covered by an adhesive material and it is desired to metallized both the outline of the ﬁlter and the polarization holes, it is necessary to drill again the polarization holes, and the outline of the ﬁlter to remove the adhesive material in those zones. This process is shown in Figure 5.13.

• Metallization. In order to metallize the sidewalls and the polarization holes, the LPKF ProConduct paste [99] is used (Figure 5.14(b)). It is a new technology for producing conductive through-holes without potentially hazardous chemical processing. This paste is applied on the polarization holes and in the outline of the covers. In the lower cover, the paste is only applied on the edge of the cover. After applying the ProConduct, in order to have it along the whole surface of the po- 5.2 Manufacturing process 103

Figure 5.9: View of the computer with the CircuitPro software connected to the milling machine ProtoMat S103.

larization holes, the substrate is placed over the vacuum table and suctioned. This produces that the paste goes along the holes covering the whole walls. This is shown in Figure 5.14(a). Last part of this step is to remove the adhesive material from the substrate and introduce the device in a hot-air oven (ProtoFlow S [100]) during 30 minutes to cure the ProConduct paste to the substrate.

• Union holes. Eight holes are done in every layer to join the whole filter after manufacturing using screws. Once the holes have been done the lower covers may be separated from the whole substrate. The layer of this step modeled in CST Studio is shown in figure 5.15 and the manufactured result is shown in figure 5.16. The upper cover needs more stages.

• Small squares and the uncoupling square. A big square is done on the upper side of the substrate in order to uncouple the zone where the holes are located and the rest of the structure. This will allow to polarize the holes independently of the rest of the structure as has been explained before. The layer of this step modeled in CST Studio is shown in figure 5.17(a) and the result is shown in figure 5.18(a). Furthermore, on the lower side of the substrate, the small squares are done around each polarization hole by using the bit of 150 µm of diameter. These squares will create a grid where the currents can flow. This is done only in the upper cover. This step is shown in figure 5.17(b) and the result is shown in figure 5.18(b). 104 Results and measurements

Figure 5.10: Steps of the manufacturing process in CST Studio.

Figure 5.11: Layer of the polarization holes in the upper cover modeled in CST Studio.

Once all the squares are done, the upper covers are separated from the whole substrate.

• Soldering the polarization holes. If the polarization holes are left opened the liquid crystal will overflow through them. For that reason, the holes are soldered as shown in Figure 5.19. Two holes are left without soldering to fill the filter with the liquid crystal. Since there are holes only in the upper cover, this process is carried out only in that cover. In this way since the filter is assembled, it is possible to polarize the LC from the outside, since the soldering is connected to the little polarization squares and the metal outside the uncoupling square is connected to ground.

• Copper. Finally, the last step is to rub a thin ﬁlm of polymide over the upper and lower covers. This polymide acts as an alignment layer. Therefore, LC molecules are aligned parallel to the waveguide and when no voltage is applied, the LC permittivity is minimum. This process is carried out in University of Carlos III (Madrid).

On the other hand, in order to get the central body of the ﬁlter, the next stages must be 5.2 Manufacturing process 105

Figure 5.12: Polarization holes in the upper cover.

Figure 5.13: Removing the adhesive material with the milling machine. followed.

• Fiducials markers. As for the case of the covers, the fiducials markers are done to allow the milling machine to locate the substrate plate. Also, at the end of this step, it is necessary to sand the substrate to remove the metal burrs. • Isolating holes. These holes are done in both left and right sides of the transition in order to avoid that the signal that goes through the microstrip goes away by these zones. The substrate laminate could behave as a parallel plates guide. Therefore, for avoiding this behaviour, the zones are isolated and the power of the microstrip is totally delivered to the filter. The model of this step in CST Studio is shown in figure 5.20. • Adhesive material. In this step, an adhesive material is stuck to the substrate. This is necessary because only the isolating holes and sidewalls of the waveguide are metal- 106 Results and measurements

Figure 5.14: Metallization of the ﬁlter. (a) Metallization of sidewalls and holes. (b) ProCon- duct Paste.

Figure 5.15: Layer of the union holes modeled in CST Studio.

lized. Therefore, to avoid the metallization of the upper and lower sides, they must be covered by the adhesive material. If the sidewalls are not metallized, there will not be electric continuity inside the waveguide and, hence the filter will not conduct the electromagnetic field. As before, since the substrate is covered by an adhesive material and it is desired to metallized outline of the filter it is necessary to remove the adhesive material in this zone. This process makes the contour of the filter, removing the adhesive material in the holes done before as shown in the model in CST Studio in figure 5.21. The resulting process is shown in figure 5.22.

• Metallization. The material used for producing the ﬁlter is the laminate RT/duroid 5870 [101] of ROGERS, which is a PTFE composite reinforced with glass microﬁbers. This material has the lowest dielectric constant of all products (εr = 2.33), the most similar to the values of the permittivity of the liquid crystal. However, the PTFE mate- 5.2 Manufacturing process 107

Figure 5.16: Lower cover with the union holes.

Figure 5.17: Layer of the squares modeled in CST Studio. (a) Uncoupling square. (b) Small squares.

rial cannot be metallized by following an electrolysis process (i.e, depositing copper in a electrolytically way) In order to metallize, the LPKF ProConduct paste [99] is used as before (Figure 5.14(b)). This paste is applied in the holes and in the outline of the ﬁlter. After applying the ProConduct, in order to have it along the whole surface of the holes, the substrate is placed over the vacuum table and suctioned. This produces that the paste goes along the holes covering the whole walls. Last part of this step is to remove the adhesive material from the substrate and introduce the device in the oven (ProtoFlow S [100]) during 30 minutes to afﬁx the ProConduct paste to the substrate as shown in Figure 5.23.

• Union holes. Eight holes are done in every layer to join the whole ﬁlter with screws after manufacturing. This is the same process that the union holes in the covers. 108 Results and measurements

Figure 5.18: Squares. (a) Uncoupling square. (b) Small squares.

• Transition. The exponential taper is cut following the model in CST Studio shown in the ﬁgure 5.24. Next, the interior of the ESIW is detached and it becomes an empty waveguide. Afterward, to form the linear taper and the microstrip line, the copper around them is milled by using the milling machine. This step is modeled in CST Studio as shown in ﬁgure 5.25. The device is cut and separated from the substrate sheet. The resulting piece is shown 5.26.

Now, the three manufactured parts, the upper cover (figure 5.19), the lower cover (figure 5.16), the and central body (figure 5.26), are soldered and joint with 8 screws using the union holes. After that, the filter can be filled with the liquid crystal. It is introduced inside the central body of the filter, through one of the polarization holes. It is necessary to keep another hole without soldering, to allow the air inside the structure to flow. These holes are left open in the soldering process of the polarization holes, and they are soldered after the filling process. The filled with LC is carried out in University of Carlos III (Madrid). Finally, two SMA-microstrip transitions are soldered in order to feed the filter and be able to measure its frequency response. The filter once the manufacturing process has finished is shown in figure 5.27.

5.3 Measurements

As mentioned at the end of the previous section, once the filter is manufactured, it is filled with liquid crystal (LC). Before filling with LC, a simulation of the filter not filled with liquid 5.3 Measurements 109

Figure 5.19: Soldering the polarization holes.

Figure 5.20: Layer of the isolating holes modeled in CST Studio. crystal has been conducted. In figure 5.8 it can be observed that the frequency response of the empty structure has the shape of a bandpass filter. Hence, in order to verify the simulation results, the filter is measured before filling with LC. The results of these measurements will allow to validate the efficiency and accuracy of the tools developed in this master thesis. To measure the aforementioned structure, the Vector Network Analyzer (VNA) E8363A [102] from Keysight Technologies is used. It is a VNA with scattering parameters measurement capacity from 45 MHz to 50 GHz. The filter has been measured with the aforementioned VNA, obtaining the data in a .s2p file. This file has been read with Matlab and plotted with the same software. The frequency response is shown in 5.28. It can be observed that the filter keeps its response shape, centered at 9.4 GHz. The central frequency is shifted 620 MHz compared to the simulation results (5.8). This shift in the frequency response is due to tolerances in the manufacturing process, that 110 Results and measurements

Figure 5.21: Layer of the contour of the ﬁlter modeled in CST Studio.

Figure 5.22: Removing the adhesive material from the central body with the milling machine. is one of the reason because of the reconfiguration of a filter after manufacturing is necessary. It is expected that once the filter is filled with LC, the frequency response can be approximate to the desired one. The measured return losses and the simulated ones are the same, being around -5 dB. The transmission coefficient in the bandpass (parameter S21) presents high losses, but they are expected as observed in the simulation. 5.3 Measurements 111

Figure 5.23: Substrate in the oven to afﬁx the ProConduct paste.

Figure 5.24: Layer of the exponential transition modeled in CST Studio.

Figure 5.25: Layer of the linear transition modeled in CST Studio. 112 Results and measurements

Figure 5.26: Central body manufactured.

Figure 5.27: Complete manufactured ﬁlter.

Figure 5.28: Result of the measurement of the empty ﬁlter. Chapter 6

Conclusions and future research lines

Due to the growing interest of having reconfigurable microwave devices, in this master thesis, filters on H-plane implemented in ESIW technology filled with liquid crystal are used. They can adjust the frequency response once manufactured, since the dielectric material has a variable permittivity when applying a bias voltage of low frequency. Thus, when varying the dielectric permittivity of LC, the center frequency and the relative bandwidth of the filter also vary. Among the applications of the reconfigurable filters, this structure in ESIW technology filled with liquid crystal can be used in mobile base stations, in satellite base stations, and in any high frequency communications under certain thermal and mechanical conditions. This wide range of possibilities of use is due to the filter can be designed at any center frequency with the tools developed in this thesis. This project arises from the need of efficient and accurate analysis and design tools for the aforementioned filter topology. Normally, to carry out the analysis of these structures, commercial tools based on numerical methods are used. When these analysis tools have to be integrated in the process of a computer-aided design, CAD, this approach is not viable, since the computational cost is very high. Therefore, an efficient analysis tool has been developed with the aim of overcoming the aforementioned limitations for the topology under study. The efficiency of this tool is achieved by segmenting the structure in simple building blocks: waveguide sections filled with dielectric material, sections of discontinuities between waveguides, sections of change of medium, and waveguide sections with multiple dielectric discontinuities. Each one of these blocks is a canonical or sufficiently simple structure that can be analyzed using modal methods, much more efficient than numerical ones. Once the generalized scattering matrices of each block are obtained, they are efficiently linked, obtaining the global scattering matrix of the filter. In order to check the validity and efficiency of the analysis tool developed, a filter of two cavities and another one of four cavities has been analyzed. The obtained frequency response has been compared to three different responses: the ideal response of a Chebyshev filter, the response simulated by the commercial tool CST, and the response of applying a quick 114 Conclusions and future research lines frequency sweep to the designed tool. Furthermore, the computational cost of each of the tools is compared. As assessment of this study, it can be said that all the tools are equally accurate. The frequency responses are very similar, almost identical. On the other hand, it can be said that the tool developed with quick frequency sweep is the most efficient, since it is the one with less computational cost, being approximately 16 times faster than CST software in the case of two cavities, and 32 times faster than CST in the case of four cavities. Moreover, to see how the change of the dielectric permittivity affects the frequency response of the filters discussed above, an analysis of these filters is performed with the developed tool. The relative permittivity of the liquid crystal is varied in the different sections of the filter,i.e, resonant cavities and coupling windows. It can be observed that the permittivity in the cavities cannot be less than the permittivity in the windows, as the frequency response loses its shape. If the permittivity is varied in the windows, the bandwidth varies; whereas if it is varied in the cavities, the center frequency of filter changes. When the permittivity in both cavities and windows is changed, then, both the center frequency and bandwidth vary, while maintaining the relative bandwidth. According to design tools, the commercial ones available on the market are based on the analysis tools of general purpose. Since they use numerical methods, the design of microwave devices by using these tools requires a high computational cost and the ability of the designer to direct the design process. In order to solve the shortcomings of the aforementioned commercial tools, it has been developed an efficient design tool of H-plane reconfigurable filters. This tool is able to calculate the geometric parameters of the filters under design. The developed tool uses the concept of equivalent waveguides, which begins with the synthesis of a starting point and optimizes these initial dimensions in order to get a frequency response as much similar as possible to the ideal response. The efficiency of the tool is achieved by following a segmentation strategy, and using an hybrid optimization. Regarding to the segmentation, the filter is divided into simple building blocks, in order to transform the design process into a set of steps where a small number of parameters is calculated. Furthermore, the choice of a symmetrical structure further increases the efficiency of the design process, and the number of steps is reduced. Related to the hybrid optimization, a suitable combination of optimization algorithms is used instead of just one. The efficiency and robustness of the optimization process is achieved by combining several optimization strategies, using the most appropriate algorithm, which depends on the proximity of the starting point to the minimum. Moreover, to achieve the objective of reconfiguration, once the filter is designed, the liquid crystal must change its permittivity, thereby changing the frequency response of the filter. Therefore, a tool for calculating the permittivity of the liquid crystal has been developed. Then, the desired frequency response can be achieved. This ensures that the filters under study are reconfigurable. To check the validity and efficiency of the developed tools, two filters (two and four cavities) in ESIW technology filled with liquid crystal have been designed. Subsequently, the range of possible reconfiguration of the designed filters have been studied by calculating 115 new values of permittivity which meet the new required specifications, center frequency, and bandwidth. As a final assessment of the tools developed, it can be concluded that the results have been very satisfactory, achieving the proposed objectives.

• Firstly, the ﬁlters designed achieve a frequency response very close to the ideal one, being practically the same, that is, they have a frequency performance equal to the performance of the waveguide ﬁlters that had been using so far.

• Furthermore, the use of the liquid crystal as dielectric material reduces the size of the ﬁlter in its longitudinal dimension, compared to the conventional ﬁlters in empty waveguide. One compaction occurs also in the vertical dimension, because they are integrated in a planar substrate.

• Besides, one of the current need of high frequency communications is satisfied. It is the reconfiguration of the filters. This allows to achieve the performance of several filters in only one filter.

Before the manufacturing process, the technological considerations related to the manufacturing of filters in ESIW technology have been studied, including the problem of the need of designing the most appropriate transition for the topology under study, and the polarization of the liquid crystal. On the one hand, to complete the integration of the ESIW to the active circuits and measure the filter response, it is necessary to design a transition between the ESIW and the planar structures. The designed transition combines both a linear taper and an exponential one. After a tough design process, the transition has the desired frequency response, that is, low reflection (¡-20 dB) in the whole working band. On the other hand, the LC polarization needs two layers electrically separated. It has been proposed a solution for this problem, which consists on cutting the upper cover in small squares. This squares are decoupled from the rest of the structure. Thus, there is a grid where the field currents flow. It is important to mention that the number of squares have been optimized to obtain minimum losses. This design has been simulated for the variation range of the permittivity values of the liquid crystal. Thus, the reconfiguration of the filter is validated, since the different filter frequency responses keep the shape of a bandpass filter, changing their center frequency. As expected there is a small degradation of the frequency response when the value of the dielectric permittivity changes from the designing point. Finally, a filter has been manufactured and it has been measured before filling it with liquid crystal. The measured frequency response is similar to the simulated one, following the response of a bandpass prototype. Related to return and transmission losses, these results are worse than a filter designed at same center frequency. These losses arise because the filter response is too far from the designing point. This research line is going to be continued for further results since there are different projects in progress in the research group (GAM) about this topic. The next step is to fill the 116 Conclusions and future research lines

filter with the LC, and measure it to validate the simulations and the possibility to reconfigure the frequency response of the filter. Afterward, among the many future research lines, one possibility is the study of how to polarize the liquid crystal by using distributed capacitors, since they behave like an open circuit at bias voltage (low frequency), and they behave as a short circuit at microwave frequencies. Furthermore, other types of transitions can be studied to have better feed of the filter and other dielectric substrates that can be electrolytically metallized could be used. It would be also interested to investigate the use of other technologies for filters, such as Corrugated Substrate Integrated Waveguide (CSIW) [104] or Empty Substrate Integrated Coaxial Line (ESICL) [105], in order to know if they allow to polarize the LC in an easier way. Appendix A

Calculation of the Z matrices of multiple discontinuities

(1) (N) (1) (N) In order to get Zm, the equations Vm , Vm , Im and Im must be related. It is known that:

(1) +(1) −(1) Vm (z1 = 0) = Vm + Vm (A.1)

+(1) −(1) (1) Vm Vm Im (zi = 0) = (1) − (1) (A.2) Z0m Z0m

(N) +(N) −(N) Vm (zN = 0) = Vm + Vm (A.3)

+(N) −(N) ! (N) Vm Vm Im (zN = 0) = − (1) − (1) (A.4) Z0m Z0m doing:

1 (1) 1 (1) +(1) (A.1) + Z0m · (A.2)−→Vm + Z0mIm = 2Vm

1 (1) 1 (1) −(1) (A.1) − Z0m · (A.2)−→Vm − Z0mIm = 2Vm

1 (N) 1 (N) +(N) (A.3) − Z0m · (A.4)−→Vm − Z0mIm = 2Vm

1 (N) 1 (N) −(N) (A.3) + Z0m · (A.4)−→Vm + Z0mIm = 2Vm hence,

1 1 1 V (N) − Z1 I(N) = A V (1) − Z1 I(1) + A V (1) − Z1 I(1) (A.5) 2 m 0m m 11 2 m 0m m 12 2 m 0m m 118 Calculation of the Z matrices of multiple discontinuities

1 1 1 V (N) + Z1 I(N) = A V (1) − Z1 I(1) + A V (1) − Z1 I(1) (A.6) 2 m 0m m 21 2 m 0m m 22 2 m 0m m Operating these equations, it is obtained:

(N) (1) 1 (1) 1 (N) Vm − (A11 + A12) Vm = Z0m (A11 − A12) Im + Z0mIm (A.7)

(N) (1) 1 (1) 1 (N) Vm − (A21 + A22) Vm = Z0m (A21 − A22) Im − Z0mIm (A.8) doing:

(1) (A.7) − (A.8) · (A21 + A22 − A11 − A12) Vm (A.9) (1) Vm can be calculated:

1 (1) 1 (N) Z0m (A11 + A22 − A12 − A21) Im + 2Z0mIm (A.10)

1 (1) 1 A11 + A22 − A12 − A21 (1) 2Z0m (N) Vm = Z0m Im + Im (A.11) A21 + A22 − A11 − A12 A21 + A22 − A11 − A12 | {z } | {z } (1,1) (1,N) Zm Zm similarly, doing: A + A (A.7) − 11 12 · (A.8) (A.12) A21 + A22 (N) Vm can be calculated:

(N) A11+A12 1 (A11+A12)(A21−A22) (1) Vm 1 − = Z A11 − A12 − Im + A21+A22 0m A21+A22 (A.13) 1 A11+A12 (N) Z 1 + Im 0m A21+A22

(N) 1 A11A22 − A12A21 1 1 A21 + A22 + A11 + A12 N Vm = 2Z0m Im + Z0m Im (A.14) A21 + A22 − A11 − A12 A21 + A22 − A11 − A12 | {z } | {z } (N,1) (N,N) Zm Zm

In equations (A.11) and (A.14) the values of Zm matrix have been highlighted. Thus, for each mode the matrix is expressed as follows: 119

  A11 + A22 − A12 − A21 2 1 Z0m   Zm =   A21 + A22 − A11 − A12   2 (A11A22 − A12A21) A21 + A22 + A11 + A12 (A.15) Therefore, the multimodal matrix Z is as follows:

 (1,1) (1,N)  Z1 0 ··· 0 Z1 0 ··· 0  (1,1) (1,N)   0 Z2 ··· 0 0 Z2 ··· 0   . . . . .   . .. . | .. .     0 ··· Z(1,1) 0 0 ··· Z(1,N) 0   M−1 M−1   (1,1) (1,N)   0 ··· 0 ZM 0 ··· 0 ZM      Z =   (A.16)      (N,1) (N,N)   Z1 0 ··· 0 Z1 0 ··· 0   (N,1) (N,N)   0 Z2 ··· 0 0 Z2 ··· 0   . . . . .   ......   . . | .   (N,1) (N,N)   0 ··· ZM−1 0 0 ··· ZM−1 0  (N,1) (N,N) 0 ··· 0 ZM 0 ··· 0 ZM

Appendix B

Resolution of the connection method for Generalized Scattering Matrices (GSM)

In order to obtain the global generalized scattering matrix (SG), the next equations are used as starting point.

b1 = SG,11a1 + SG,12aN+1 (B.1)

bN+1 = SG,21a1 + SG,22aN+1 (B.2) If the global network is symmetrical (as in most microwave devices), it is only needed to calculate SG,11 and SG,21, since SG,22 = SG,11 and SG,12 = SG,21. In order to obtain SG,11 and SG,21 it can be considered that there is no incidence on the output port, it means, aN+1 = 0. In this case:

b1 = SG,11a1 (B.3)

bN+1 = SG,21a1 (B.4) Based on the deﬁnition of the scattering parameters, and according to the ﬁgure 2.9, the following equations are met for each i network, i ∈ [1, ··· ,N − 1]:

bi = Si,11ai + Si,12bi+1 (B.5)

ai+1 = Si,21ai + Si,22bi+1 (B.6) For the last network, forward and backward waves in the output port (see ﬁgure 2.9) are deﬁned differently. Therefore the equations of the last network are:

bN = SN,11aN + SN,12aN+1 (B.7)

bN+1 = SN,21aN + SN,22aN+1 (B.8)

Replacing aN+1 = 0 in (B.7) and (B.8):

bN = SN,11aN (B.9)

bN+1 = SN,21aN (B.10) 122 Resolution of the connection method for Generalized Scattering Matrices (GSM)

Replacing (B.9) in (B.6) and i = N − 1, aN can be expressed as a function of aN−1:

−1 aN = (I − SN−1,22SN,11) SN−1,21 aN−1 (B.11) where I is the identity matrix of dimension M × M. Replacing (B.11) in (B.9), bN can be also expressed as a function of aN−1:

−1 bN = SN,11 (I − SN−1,22SN,11) SN−1,21aN−1 (B.12)

It is deﬁned:

−1 AN = (I − SN−1,22SN,11) SN−1,21 (B.13)

CN = SN,11 (B.14)

BN = CN AN (B.15) hence,

aN = AN aN−1 (B.16)

bN = CN aN = CN AN aN−1 = BN aN−1 (B.17)

Following aN−1 and bN−1 are expressed as a function of aN−2. For this purpose, ﬁrst (B.17) is replaced in (B.5) for i = N − 1, and it is obtained:

bN−1 = (SN−1,11 + SN−1,12 BN ) aN−1

= CN−1 aN−1 (B.18) where CN−1 = SN−1,11 + SN−1,12 BN (B.19) Replacing (B.18) in (B.6) for i = N − 2:

−1 aN−1 = (I − SN−2,22 CN−1) SN−2,21 aN−2

= AN−1 aN−2 (B.20) where −1 AN−1 = (I − SN−2,22 CN−1) SN−2,21 (B.21) Replacing (B.20) in (B.18):

bN−1 = CN−1 aN−1 = CN−1 AN−1 aN−2

= BN−1 aN−2 (B.22) where BN−1 = CN−1 AN−1 (B.23) 123

The same procedure can be followed to link bi and ai with ai−1 for i = [N − 2, ··· , 2]:

ai = Ai ai−1 (B.24)

bi = Bi ai−1 (B.25) where

Ci = Si,11 + Si,12 Bi+1 (B.26) −1 Ai = (I − Si−1,22 Ci) Si−1,21 (B.27)

Bi = Ci Ai (B.28) The equations (B.24) and (B.28) are valid for all the values of i (i = [2, ··· ,N]), except from CN , since the correct value is CN = SN,11. Once all ai and bi are related to ai−1, it can be determined the scattering parameters of global matrix. Using equation (B.5) with i = 1 :

b1 = S1,11 a1 + S1,12 b2 = S1,11 a1 + S1,12 B2 a1

b1 = (S1,11 + S1,12 B2) a1 (B.29) and comparing to equation (B.3):

SG,11 = S1,11 + S1,12 B2 (B.30) On the other hand, by using equation (B.10):

bN+1 = SN,21 aN = SN,21 AN aN−1

bN+1 = SN,21 AN AN−1 aN−2

bN+1 = SN,21 AN AN−1 AN−2 aN−3 . . 2 ! Y bN+1 = SN,21 Ai a1 (B.31) i=N Comparing to equation (B.4): 2 ! Y SG,21 = SN,21 Ai (B.32) i=N If the global network is symmetrical, and using equations (B.30) and (B.32), the generalized scattering matrix is:

 2 !  Y  S1,11 + S1,12 B2 SN,21 Ai     i=N    SG =   (B.33)  2 !   Y   SN,21 Ai S1,11 + S1,12 B2  i=N

Appendix C

Routines

There are different routines implemented in Matlab that have been used in the development of this thesis. Among all the routines used, some of them had been implemented in the research group where the thesis has been carried out, the Microwave Applications Group (GAM). The rest of them have implemented in the thesis itself. The heading of each routine is shown below. Together with the heading, the use of the function is explained and the input and output parameters are detailed. The functions are divided in analysis and design routines indicating which ones are developed inside the thesis and which ones are used from the GAM routines. The code of the routines is not included by conﬁdentiality reasons.

C.1 Analysis routines

Ideal [S11,S21, f] = Ideal(N, Ncav, Ripple, fci, fcs, epsilonr, mur, bW g, aW g, fini, ffin, NF P o) This function is responsible for generating the ideal response of a Chebychev bandpass ﬁlter. This response is associated with the equivalent ﬁlter of distributed elements. This function is used from the GAM routines.

• Input parameters:

– N: Number of cavities in the structure. – Ncav: Number of cavities in the structure to be analyzed. – Ripple: Ripple of the filter in the passband in dB. – fci: Lower cutoff frequency of the passband in GHz. – fcs: Upper cutoff frequency of the passband in GHz. – epsilonr: Relative dielectric permittivity in each section of the filter. – mur: Relative magnetic permeability in each section of the filter. 126 Routines

– bW g: Height of the structure in mm. – aW g: Width of the cavities (N + 2) given in mm. – fini: Initial analysis frequency in GHz. – ffin: Final analysis frequency in GHz. – NF P o: Number of frequency points to be analyzed.

• Output parameters:

– S11: Reflection parameter of the filter in the analyzed frequency range. – S21: Transmission parameter of the filter in the analyzed frequency range.

Indfilt [S11,S21, f] = Indfilt(N, Ncav, aW gIN, aW gOUT, aW g, bW g, aW in, lW in, lW gRes, lW gIN, lW gOUT, epsilonr, fini, ffin, NF P o, M) This function calculates the scattering parameters of a resonant cavities filter given the dimensions of the filter already designed. This function is modified from one implemented by GAM. The previous function allowed the analysis of empty filters, but this one allows the analysis of filter filled with a dielectric material whose value of permittivity can vary.

• Input parameters:

– N: Number of cavities in the structure. – Ncav: Number of cavities in the structure to be analyzed. – aW gIN: Width of the input waveguide in mm. – aW gOUT : Width of the output waveguide in mm. – aW g: Width of the cavities (N) given in mm. – bW g: Height of the structure in mm. – aW in: Width of the coupling windows (N + 1) in mm – lW in: Length of the coupling windows (N + 1) in mm. – lW gIN: Length of the input waveguide in mm. – lW gOUT : Length of the output waveguide in mm. – lW gRes: Length of the resonant cavities given in mm. – epsilonr: Relative dielectric permittivity in each section of the ﬁlter. – fini: Initial analysis frequency in GHz. – ffin: Final analysis frequency in GHz. – NF P o: Number of frequency points to be analyzed. – M: Number of analyzed modes. C.1 Analysis routines 127

• Output parameters:

– S11: Reflection parameter of the filter in the analyzed frequency range (complex number). – S21: Transmission parameter of the filter in the analyzed frequency range (complex number). – f: Vector with the analyzed frequency points given in GHz

The aforementioned function (Indfilt) implements the segmentation of the filter under analysis. Therefore, modal functions are designed for the analysis of each building block of the whole filter. These functions are explained below.

FilledGuide S = F illedGuide(aW g, l, epsilonr, M, f) This function calculates the multimodal scattering matrix of a rectangular waveguide ﬁlled with a dielectric material of relative permittivity εr. It calculates only the TEm0 modes. This function has been implemented in the thesis.

• Input parameters:

– aW g: Width of the waveguide in mm. – lW g: Length of the waveguide in mm. – epsilonr: Relative dielectric permittivity. – M: Number of guided modes. – f: Single frequency point to be analyzed in GHz.

• Output parameters:

– S: Generalized Scattering Matrix (GSM) of the ﬁlled waveguide.

MultipleDiscontinuities S = MultipleDiscontinuities(aW g, lW g, epsilonr, M, f) This function calculates the multimodal scattering matrix of a waveguide with multiple discontinuities. It calculates only the TEm0 modes. This function has been implemented in this thesis.

• Input parameters:

– aW g: Width of the waveguide in mm. – lW g: Length of the waveguide in mm. – epsilonr: Relative dielectric permittivity. 128 Routines

– M: Number of guided modes. – f: Single frequency point to be analyzed in GHz. • Output parameters: – S: Generalized Scattering Matrix (GSM) of the waveguide section with multiple discontinuities.

Step S = Step(ae, ng, de, np, y0, f) This function calculates the scattering parameters of a step between two waveguide sections of different widths. This function had been implemented by the GAM. • Input parameters: – ae: Width of the largest waveguide in mm. – ng: Number of guided modes in the largest waveguide. – de: Width of the smallest waveguide in mm. – np: Number of guided modes in the smallest waveguide. – y0: Position of the bottom part of the largest waveguide respect to the bottom part of the smallest waveguide. – f: Single frequency point to be analyzed in GHz. • Output parameters: – S: Generalized Scattering Matrix (GSM) of the step.

MediumChange S = MediumChange(aW g, epsilonr, M, f) This function calculates the multimodal scattering matrix of a change of medium. It calculates only the TEm0 modes. The change of medium is produced from ε0 (vacuum) to εr. • Input parameters: – aW g: Width of the waveguide in mm. – epsilonr: Relative dielectric permittivity. – M: Number of guided modes. – f: Single frequency point to be analyzed in GHz. • Output parameters: – S: Generalized Scattering Matrix (GSM) of the change of medium. C.2 Design routines 129

Swindow [S11,S21] = Swindow(aIN, aOUT, b, aW in, lW in, lIN, lOUT, epsilonr, M, f) This function calculates the scattering parameters of the section shown in Figure 3.8. This section is formed by two waveguides ﬁlled with a dielectric material, two changes of medium, two impedance changes, and a section with multiple discontinuities. This function is implemented in this thesis. • Input parameters: – aIN: Width of the input waveguide of the section in mm. – aOUT : Width of the output waveguide of the section in mm. – b: Height of the waveguide. – aW in: Width of the coupling window in mm. – lW in: Length of the coupling window in mm. – lIN: Length of the input waveguide of the section in mm. – lOUT : Length of the output waveguide of the section in mm. – epsilonr: Relative dielectric permittivity. – M: Number of guided modes. – f: Single frequency point to be analyzed in GHz. • Output parameters: – S: Generalized Scattering Matrix (GSM) of the window section.

C.2 Design routines

PtoIniCavRellena [lW gRes, aW in, lW in] = P toIniCavRellena(P aramIP I, Ripple, fci, fcs, SimulaResponse) This function is responsible for determining the initial dimensions of an inductive filter using the equivalent circuit based on lumped elements. This follows the process described in section 3.2. This function is modified from one implemented by GAM. The previous function allowed the design of empty filters, but this one allows the design of filter filled with a dielectric material whose value of permittivity can vary. • Input parameters: – P aramIP I: File where data is stored for calculating the starting point. The stored parameters are the number of cavities in the structure (N), the width of the input/output waveguides (a), height of the structure (b), length of each coupling window (ti), minimum and maximum values of the parameter to calculate, number of input and output modes, relative dielectric permittivity in each section of the filter, relative magnetic permeability in each section of the filter. 130 Routines

– Ripple: Ripple of the ﬁlter in the passband in dB. – fci: Lower cutoff frequency of the passband in GHz. – fcs: Upper cutoff frequency of the passband in GHz. – SimulaResponse: Parameter to know if the ﬁlter response after optimization is simulated (0 → notshown, 1 → shown).

• Output parameters:

– aW in: Vector with the width of the inductive windows given in mm. – lW in: Vector with the length of the inductive windows given in mm. This vector is not calculated since the inductive windows are given in the input parameters. However, it is needed as a output for using it as the input of other functions. – lW gRes: Vector with the length of resonant cavities given in mm.

Optimization [fallo, Xos] = Optimization(Xini, log, rutaIN, rutaOUT ) This function implements the optimization strategy described in section 3.3 by using the following techniques: segmentation and hybrid optimization algorithms. This function is a modification of the function developed in the research group GAM. The previous function allowed the optimization of empty filters, but this one allows the optimization of filter filled with a dielectric material whose value of permittivity can vary.

• Input parameters:

– Xini: Starting point of the strategy calculated with the function described above, P toIniCavRellena. This is a vector with the initial values of the length of the resonant cavities (lW gRes), and the width of the coupling windows (aW in). – log: Output file where the evolution of the optimization process will be stored. Data as the different optimization algorithms used, the obtained accuracy, and the spent time are stored in this file. – rutaIN: Directory where the input files are stored. The stored files are the stages to follow in the optimization process and the goals to meet in each stage of the optimization. – rutaOUT : Directory where the result of the optimization process will be stored.

• Output parameters:

– fallo: An integer whose value is equal to zero if the operation is correct and it is nonzero if something went wrong. – Xos: Vector with the resulting optimized values of the width of the coupling windows and the length of the resonant cavities. These values are given in mm. C.2 Design routines 131

Reconfiguration [fallo, NewEr] = Reconfiguration(Xos, log, rutaIN, rutaOUT ) This function implements the reconfiguration strategy described in section 3.5. It calculates the new values of the relative permittivity of the dielectric material needed to obtain the desired frequency response. This allows the reconfiguration of the filter, i.e, the change of the frequency response as the central frequency and bandwidth. This function has been entirely developed in the thesis.

• Input parameters:

– Xos: Dimensions of the filter calculated with the function described above, Optimization. This is a vector with the values of the length of the resonant cavities (lW gRes), and the width of the coupling windows (aW in). – log: Output file where the evolution of the reconfiguration process will be stored. Data as the different algorithms used, the obtained parameters in each iteration (function error, and accuracy), and the spent time are stored in this file. – rutaIN: Directory where the input files are stored. The stored files are the scattering paremeters of the goal frequency response (new central frequency, and bandwidth) and the parameters of the search function of Matlab. These parameters are the termination tolerance on the error function value (T olF un), the termination tolerance on the solution (T olX), the maximum number of error function evaluations allowed (MaxF unEvals), and the maximum number of iterations allowed (MaxIter) – rutaOUT : Directory where the result of the reconfiguration process will be stored. The resulting dielectric permittivity values of each section will be stored and the frequency response of the filter with the new values of permittivity (scattering parameters, S11 and S21). • Output parameters:

– fallo: An integer whose value is equal to zero if the operation is correct and it is nonzero if something went wrong, for example, if the goal cannot be obtained. – NewEr: Vector with the resulting values of the relative permittivity of the dielectric material, which the ﬁlter is ﬁlled.

Bibliography

[1] A. Belenguer, H. Esteban, and V. E. Boria, “Novel empty substrate integrated waveguide for high performance microwave integrated circuits,” IEEE Trans. Microwave Theory and Techniques, vol. 62, no. 4, pp. 832–839, April 2014.

[2] C. Bachiller, “Metodos´ h´ıbridos para el analisis´ y diseno˜ eﬁciente de ﬁltros avan- zados para sistemas de comunicaciones espaciales,” Ph.D. dissertation, Universidad Politecnica´ de Valencia, Valencia, 2010.

[3] H. Esteban, “Analisis´ de problemas arbitrarios de dispersion´ electromagnetica´ mediante metodos´ h´ıbridos,” Ph.D. dissertation, Universidad Politecnica´ de Valencia, Va- lencia, 2002.

[4] S. Cogollos, “Caracterizacion´ modal de gu´ıas arbitrarias y aplicacion´ a dispositivos de microondas,” Ph.D. dissertation, Universidad Politecnica´ de Valencia, Valencia, 2002.

[5] J. V. Morro, “Nuevas estrategias de diseno˜ automatizado de componentes pasivos para sistemas de comunicaciones de alta frecuencia,” Ph.D. dissertation, Universidad Politecnica´ de Valencia, Valencia, 2011.

[6] P.J.Collings and M.Hird, Introduction to Liquid Crystals Chemistry and Physics. 1st Edition. Taylor I& Francis, London, 1997.

[7] J.Torrecilla, E.Avila-Navarro,´ C.Marcos, V.Urruchi, J. Sanchez-Pena,´ J.Arias, and M. Sanchez,´ “Resonador microstrip spurline en espiral para ﬁltros sintonizables con tecnolog´ıa cristal l´ıquido,” XXVII Simposium Nacional Union´ Cient´ıﬁca Internacional de Radio (URSI), September 2012.

[8] J.Torrecilla, V.Urruchi, J.M.Sanchez-Pena,´ N.Bennis, A.Garc´ıa, and D.Segovia, “Im- proving the pass-band return loss in liquid crystal dual-mode bandpass ﬁlters by microstrip patch reshaping,” Materials, vol. 7, pp. 4524–4535, June 2014.

[9] S.Missaoui and M.Kaddour, “Liquid crystal-reconﬁgurable conformal monopole antennas structures for microwave applications,” Journal of Engineering and Technology Research, vol. 43, no. 3, pp. 57–64, February 2012. 134 BIBLIOGRAPHY

[10] J.Torrecilla, C.Marcos, V.Urruchi, and J. Sanchez-Pena,´ “Tunable dual-mode bandpass ﬁlter based on liquid crystal technology,” 43rd European Microwave Conference (EuMC), pp. 806–809, October 2013.

[11] H. Patzelt and F. Arndt, “Double-plane steps in rectangular waveguides and their applications for transformers, irises and ﬁlters,” IEEE Trans. Microwave Theory and Techniques, vol. 30, no. 5, pp. 771–776, May 1982.

[12] Advanced Modal Analysis - CAD Techniques for Waveguide Components and Filters. John Wiley I& Sons, Ltd., Chichester, 2000.

[13] D. M. Pozar, Microwave Engineering. 4th Edition. John Wiley & Sons Inc., 2012.

[14] C. Balanis, Advanced Engineering Electromagnetics. John Wiley & Sons, 1989.

[15] C. Bachiller, H. Esteban, V. E. Boria, A. Belenguer, and J. V. Morro, “Efﬁcient technique for the cascade connection of multiple two port scattering matrices,” IEEE Trans. Microwave Theory and Techniques, vol. 55, no. 9, pp. 1880–1886, September 2007.

[16] A. Tasic, W. Serdijn, and G. Setti, Circuits and Systems for Future Generations of Wireless Communications. Springer Netherlands, 2009.

[17] K. Rawat, M. S. Hashmi, and F. M. Ghannouchi, “Dual-band RF circuits and components for multi-standard software deﬁned radios,” IEEE Circuits Systems, vol. 12, no. 1, pp. 12–32, January 2012.

[18] J. Uher and J. Bornemann, Waveguide Components for Antenna Feed Systems: Theory and CAD. Artech Print on Demand, December, 1993.

[19] V.E.Boria, D.Camilleri, A.Coves, H.Esteban, B.Gimeno, M.Guglielmi, and L.Polini, “Contributions to the analysis and design of all-inductive ﬁlters with dielectric resonators,” 33rd European Microwave Conference (EuMC), vol. 3, pp. 1247–1250, Oc- tober 2003.

[20] M.Capurso, M.Piloni, and M.Guglielmi, “Resonant aperure ﬁlters: Improved out-of- band rejection and size reduction,” 31st European Microwave Conference (EuMC), pp. 1–4, September 2001.

[21] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microwave and Wireless Components Letters, vol. 11, no. 2, pp. 68–70, February 2001.

[22] ——, “Accurate modeling, wave mechanisms, and design considerations of a substrate integrated waveguide,” IEEE Trans. Microwave Theory and Techniques, vol. 54, no. 6, pp. 2516–2526, June 2006. BIBLIOGRAPHY 135

[23] M. Almalkawi, L. Zhu, and V. Devabhaktuni, “Magnetically tunable substrate integrated waveguide bandpass ﬁlters employing ferrites,” in 36th International Confer- ence on Infrared, Millimeter and Terahertz Waves (IRMMW-THz), October 2011, pp. 1–2.

[24] S. Adhikari, Y.-J. Ban, and K. Wu, “Magnetically tunable ferrite loaded substrate integrated waveguide cavity resonator,” IEEE Microwave and Wireless Components Let- ters, vol. 21, no. 3, pp. 139–141, April 2011.

[25] A. Ghiotto, S. Adhikari, and K. Wu, “Ferrite-loaded substrate integrated waveguide switch,” IEEE Microwave and Wireless Components Letters, vol. 22, no. 3, 2012.

[26] M. Armendariz, V. Sekar, and K. Entesari, “Tunable siw bandpass ﬁlters with pin diodes,” in European Microwave Conference (EuMC), September 2010, pp. 830–833.

[27] J. C. Bohorquez, B. Potelon, C. Person, E. Rius, C. Quendo, G. Tanne, and E. Fourn, “Reconﬁgurable planar siw cavity resonator and ﬁlter,” IEEE MTT-S International Mi- crowave Symposium Digest, pp. 947–950, June 2006.

[28] F. He, K. Wu, and H. Wei, “Electrically tunable half mode substrate integrated waveguide resonator,” Asia Paciﬁc Microwave Conference, pp. 1166–1169, December 2009.

[29] Y. Ding and K. Wu, “Varactor-tuned substrate integrated waveguide phase shifter,” IEEE MTT-S International Microwave Symposium, IMS, pp. 1–4, June 2011.

[30] S. Adhikari, A. Ghiotto, and K. Wu, “Simultaneous electric and magnetic two- dimensional tuning of substrate integrated waveguide cavity resonator,” IEEE MTT-S International Microwave Symposium, IMS, pp. 1–3, June 2012.

[31] ——, “Simultaneous electric and magnetic two-dimensionally tuned parameter-agile siw devices,” IEEE Trans. Microwave Theory and Techniques, pp. 423–435, January 2013.

[32] J. Rayas-Sanchez´ and V. Gutierrez-Ayala,´ “A general EM -based design procedure for single-layer substrate integrated waveguide interconnects with microstrip transitions,” IEEE MTT-S International Microwave Symposium Digest, pp. 983–986, June 2008.

[33] I. Hunter, Theory and Design of Microwave Filters. London: The Institution of Electrical Engineers, 2001.

[34] M. Koshiba and M. Suzuki, “Application of the boundary-element method to waveguide discontinuities,” IEEE Trans. Microwave Theory and Techniques., vol. 34, pp. 301–307, February 1986.

[35] J. P. Webb and S. Porihar, “Finite element analysis of H-plane rectangular waveguide problems,” Proc. of the Institute of Electrical Engineering, vol. 133, no. 2, pp. 91–94, April 1986. 136 BIBLIOGRAPHY

[36] A. Taﬂove, Computational Electromagnetics: The Finite-Difference Time-Domain Method. Artech House, 1995.

[37] A. Alvarez, G. Connor, and M. Guglielmi, “New simple procedure for the computation of the multimode admittance or impedance matrix of planar waveguide junctions,” IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 3, pp. 413–418, March 1996.

[38] Y. Leviatan., P. Li, A. Adams, and J. Perini, “Single-post inductive obstacle in rectangular waveguide,” IEEE Trans. Microwave Theory and Techniques, vol. 31, no. 10, pp. 806–811, October 1983.

[39] A. Valero and M. Ferrando, “Full-wave equivalent network representation for multiple arbitrarily shaped posts in H-plane waveguide,” IEEE Trans. Microwave Theory and Techniques, vol. 47, no. 10, pp. 1997–2002, Oct. 1999.

[40] J. Reiter and F. Arndt, “Rigorous analysis of arbitrarily shaped H- and E-plane discontinuities in rectangular waveguides by a full-wave boundary contour mode-matching method,” IEEE Trans. Microwave Theory and Techniques, vol. 43, no. 4, pp. 796–801, April 1995.

[41] R. Collin, Foundations for Microwave Engineering. Hoboken: John Wiley & Sons Inc., 2001.

[42] T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. New York: John Wiley & Sons, 1989.

[43] S. Ahmed and P. Daly, “Finite element method for inhomogeneous waveguides,” Proc. Inst. Electrical Engineering, vol. 116, no. 10, pp. 1661–1664, Oct 1969.

[44] P. B. Johns, “The solution of inhomogeneous waveguide problems using a transmission-line matrix,” IEEE Trans. Microwave Theory and Techniques., vol. 22, no. 3, pp. 209 – 215, March 1974.

[45] R. Harrington, Field Computation by Moment Methods. New York: The Mac Millan Company, 1968.

[46] A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics. IEEE Press, 1998.

[47] T. Okoshi, Y. Uehara, and T. Takeuchi, “The segmentation method: An approach to the analysis of microwave planar circuits (short papers),” IEEE Trans. Microwave Theory and Techniques., vol. 24, pp. 662– 668, October 1976.

[48] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., vol. 73, no. 2, pp. 325–348, 1987. BIBLIOGRAPHY 137

[49] P. Arcioni, V. E. Boria, M. Bozzi, G. Conciauro, and B. Gimeno, “Analysis of H-plane waveguide components with dielectric obstacles by the BI-RME method,” in 32nd Eureopean Microwave Conference (EuMC), Milano, 2002, pp. 1–3. [50] Ansoft Corporation, “HFSS: 3D high-frequency electromagnetic simulation,” http://www.ansoft.com/products/hf/hfss/index.cfm. [Online]. Available: http://www. ansoft.com/products/hf/hfss/index.cfm [51] CST Computer Simulation Technology AG, “CST Microwave Studio,” https://www.cst.com/Products/CSTMWS. [Online]. Available: https://www.cst.com/ Products/CSTMWS [52] A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Trans. Microwave Theory and Techniques, vol. 15, no. 9, pp. 508–517, September 1967. [53] C. Bachiller, H. Esteban, V. E. Boria, H. Mata, M. Valdes,´ A. Belenguer, and J. V. Morro, “Hybrid mode matching method for the efficient analysis of metal and dielectric rods in h-plane rectangular waveguide devices,” IEEE Trans. Microwave Theory and Techniques, vol. 58, no. 12, pp. 3634–3644, December 2010. [54] O. P. Franza and W. C. Chew, “Recursive mode matching method for multiple waveguide junction modelling,” IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 1, pp. 87–92, January 1996. [55] R. C. Booton, Computational Methods for Electromagnetics and Microwaves. Wiley- Interscience, January 2001. [56] V. E. Boria, “Analisis´ de problemas electromagneticos´ mediante metodos´ modales y matrices generalizadas, y aplicaciones,” Ph.D. dissertation, Universidad Politecnica´ de Valencia, Valencia, 1997. [57] MATLAB, R2014a. Natick, Massachusetts: The MathWorks Inc., 2014. [58] J. V. Morro, H. Esteban, C. Bachiller, and V. E. Boria, “Automated design of complex waveguide filters for space systems: A case study,” Int. J. of RF and Microwave Computer-Aided Engineering, vol. 17, no. 1, pp. 84–89, January 2007. [59] N. Marcuwitz, Waveguide Handbook. New York: McGraw-Hill, 1951. [60] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Artech House Inc., 1980. [61] R. Levy, R. Snyder, and G. Matthaei, “Design of microwave filters,” IEEE Trans. Microwave Theory and Techniques, vol. 50, no. 3, pp. 783–793, 2002. [62] R. Levy and S. B. Cohn, “A history of microwave filter research, design, and development,” IEEE Trans. Microwave Theory and Techniques, vol. 32, no. 9, pp. 1055–1067, 1984. 138 BIBLIOGRAPHY

[63] M. Guglielmi, “Simple CAD procedure for microwave ﬁlters and multiplexers,” IEEE Trans. Microwave Theory and Techniques, vol. 42, no. 7, pp. 1347–1352, July 1994.

[64] J. Alos and M. Guglielmi, “Simple and effective EM-based optimization prodecure for microwave ﬁlters,” IEEE Trans. Microwave Theory and Techniques, vol. 45, no. 5, pp. 856–858, May 1997.

[65] E. Diaz, J. V. Morro, H. Esteban, V. E. Boria, C. Bachiller, and A. Belenguer, Simulation-Driven Design Optimization and Modeling for Microwave Engineering. Singapore: Imperial College Press, 2013, ch. Simulation-Driven Design of Microwave Filters for Space Applications.

[66] G. Temes and D. Zai, “Least pth approximation,” Circuit Theory, IEEE Transactions, vol. 16, no. 2, pp. 235–237, May 1969.

[67] J. W. Bandler, S. H. Chen, R. M. Biernacki, L. Gao, K. Madsen, and H. Yu, “Hu- ber optimization of circuits: a robust approach,” IEEE Trans. Microwave Theory and Techniques, vol. 41, no. 12, pp. 2279–2287, December 1993.

[68] R. Fletcher, Practical Methods of Optimization. 2nd Edition. John Wiley & Sons, 1987.

[69] W. H. Press, W. T. V. S. A. Teukolsky, and B. P. Flannery, Numerical Recipes in C. Cambridge University Press, 1992.

[70] J. V. Morro, P. Soto, H. Esteban, V. E. Boria, C. Bachiller, M. Taroncher, S. Cogollos, and B. Gimeno, “Fast automated design of waveguide ﬁlters using aggressive space mapping with a new segmentation strategy and a hybrid optimization algorithm,” IEEE Trans. Microwave Theory and Techniques, vol. 53, no. 4, pp. 1130–1142, April 2005.

[71] J. W. Bandler, “Optimization methods for computer-aided design,” IEEE Trans. Mi- crowave Theory and Techniques, vol. 17, no. 8, pp. 533–552, August 1969.

[72] H. H. Rosenbrock, “An automatic method for ﬁnding the greatest or least value of a function,” The Computer Journal, vol. 3, no. 3, pp. 175–184, 1960.

[73] J. Nelder and R. Mead, “A simple method for function minimitazion,” The Computer Journal, vol. 7, no. 4, pp. 308–313, 1965.

[74] W. T. V. W. H. Press, S. A. Teukolsky and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientiﬁc Computing. Cambridge University Press., September 2007.

[75] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimizations and Nonlinear Equations. Siam, 1996. BIBLIOGRAPHY 139

[76] B. N. Das, K. V. S. V. R. Prasad, and K. V. S. Rao, “Excitation of waveguide by stripline and microstrip-line-fed slots,” IEEE Trans. Microwave Theory and Tech- niques, vol. 34, no. 3, pp. 321–327, March 1986. [77] W. Grabherr, B. Huder, and W. Menzel, “Microstrip to waveguide transition compatible with mm-wave integrated circuits,” IEEE Trans. Microwave Theory and Tech- niques, vol. 42, no. 9, pp. 1842–1843, September 1994. [78] L. J. Lavedan, “Design of waveguide-to-microstrip transitions specially suited to milimiter-wave applications,” Electronics Letter, vol. 13, no. 20, pp. 604–605, Septem- ber 1977. [79] S. Moochalla, “Ridge waveguide used in microstrip transition,” Microwaves & RF, vol. 23, pp. 149–152, March 1984. [80] D. Deslandes and K. Wu, “Integrated transition of coplanar to rectangular waveguides,” IEEE MTT-S International, vol. 2, pp. 619–622, May 2001. [81] H. Nam, T. Un, K.B.Kim, K.C.Yoon, and J.C.Lee, “Ku-band transition between microstrip and substrate integrated waveguide (SIW),” Asia Pacific Microwave Confer- ence (APMC’05), vol. 1, December 2005. [82] F. Miranda, G. Subramanyam, F. V. Keuls, R. Romanofski, J. Warner, and C. Mueller, “Design and development of ferroelectric tunable microwave components for ku-band and k-band satellite communication systems,” IEEE Trans. Microwave Theory and Techniques, vol. 48, no. 7, pp. 1181–1189, July 2000. [83] J. R. White, C. J. White, and A. H. Slocum, “Octave-tunable miniature rf resonators,” IEEE Microwave and Wireless Components Letters, vol. 15, no. 11, pp. 793–795, November 2005. [84] A. Brown and G. Rebeiz, “A varactor-tuned rf filter,” IEEE Trans. Microwave Theory and Techniques, vol. 48, no. 7, pp. 1157–1160, July 2000. [85] Q.Xiang, Q.Feng, X.Huang, and D.Jia, “A novel tri-mode bandwidth tunable filter with harmonic suppression,” Progr. Electromagnet. Res. C, vol. 34, pp. 183–184, 2013. [86] G.Yang, X.Xing, A.Daigle, M.Liu, O.Obi, K.Naishadham, and N.Sun, “Tunable miniaturized patch antennas with self-biased multilayer magnetic films,” IEEE Trans. Antennas and Propagation, vol. 57, no. 7, pp. 2190–2193, July 2009. [87] J. A. Yeh, C. A. Chang, C. Cheng, J. Huang, and S. Hsu, “Microwave characteristics of liquid-crystal tunable capacitors,” IEEE Electronic Device Letters, vol. 26, no. 7, pp. 451–453, July 2005. [88] S. Muller, P. Scheele, C. Weil, M. Wittek, C. Hock, and R. Jakoby, “Tunable passive phase shifter for microwave applications using highly anisotropic liquid crystals,” IEEE MTT-S, vol. 2, June 2004. 140 BIBLIOGRAPHY

[89] N. Martin, P. Laurent, C. Person, P. Gelin, and F. Huret, “Size reduction of a liquid crystal-based, frequency-adjustable patch antenna,” 34th European Microwave Con- ference (EuMC), vol. 2, pp. 825–828, October 2004.

[90] F.Goelden, A.Gaebler, O.Karabey, M.Goebel, A.Manabe, and R.Jakoby, “Tunable band-pass ﬁlter based on liquid crystal,” 5th German Microwave Conference, pp. 98– 101, March 2010.

[91] T. Matsumoto and A. Katsuki, “Input impedance characteristics of an electronic choke - inﬂuence by inductor with different winding structure,” in IEEE Industrial Electron- ics, IECON 2006 - 32nd Annual Conference on, Nov 2006, pp. 1633–1638.

[92] Murata Inductors, “RF Inductor For Power Lines Wire Wound Type - LQW Series -,” http://psearch.en.murata.com/inductor/lineup/lqw/index.html. [Online]. Available: http://psearch.en.murata.com/inductor/lineup/lqw/index.html

[93] ——, “RF Inductor For Power Lines Wire Wound Type - LQW18ANR10G0Z - ,” http://psearch.en.murata.com/inductor/product/LQW18ANR10G0Z[Online]. Avail- able: http://psearch.en.murata.com/inductor/product/LQW18ANR10G0Z%23.html

[94] ——, “RF Inductor For Power Lines Wire Wound Type - LQW18ANR16G0Z - ,” http://psearch.en.murata.com/inductor/product/LQW18ANR16G0Z[Online]. Avail- able: http://psearch.en.murata.com/inductor/product/LQW18ANR16G0Z%23.html

[95] National Instruments AWR Design Environment, “AWR Microwave Office,” http://www.awrcorp.com/products/microwave-office. [Online]. Available: http:// www.awrcorp.com/products/microwave-office

[96] LPKF Laser Electronics, “LPKF Milling Machine ProtoMat S103: The Premium System,” http://www.lpkfusa.com/protomat/s103.html. [Online]. Available: http://www.lpkfusa.com/protomat/s103.html

[97] ——, “LPKF CircuitPro PM Software Suite for All Circuit Board Plotters of the S- and D-Line,” http://www.lpkf.com/products/rapid-pcb-prototyping/software/circuitpro- pm.htm. [Online]. Available: http://www.lpkf.com/products/rapid-pcb-prototyping/ software/circuitpro-pm.htm

[98] Rogers Corporation, “R04003C High Frequency Circuit Materi- als,” https://www.rogerscorp.com/documents/726/acm/RO4000-Laminates—Data- sheet.pdf. [Online]. Available: https://www.rogerscorp.com/documents/726/acm/ RO4000-Laminates---Data-sheet.pdf

[99] LPKF Laser Electronics, “LPKF ProConduct: In- house PCB through-hole plating without chemicals,” http://www.lpkfusa.com/RapidPCB/ThroughHolePlating/proconduct.htm. [Online]. Available: http://www.lpkfusa.com/RapidPCB/ThroughHolePlating/proconduct.htm BIBLIOGRAPHY 141

[100] ——, “LPKF ProtoFlow S: Reflow Ovens for Lead-Free Solder- ing,” http://www.lpkfusa.com/SMTAssembly/protoflow.htm. [Online]. Available: http://www.lpkfusa.com/SMTAssembly/protoflow.htm

[101] Rogers Corporation, “RT/duroid 5870 High Frequency Laminates,” http://www.rogerscorp.com/acs/products/31/RT-duroid-5870-Laminates.aspx. [Online]. Available: http://www.rogerscorp.com/acs/products/31/ RT-duroid-5870-Laminates.aspx

[102] Keysight Technologies, “E8363A PNA Series Network Analyzer, 45 MHz to 40 GHz,” http://www.keysight.com/en/pd-1000004608[Online]. Avail- able: http://www.keysight.com/en/pd-1000004608%3Aepsg%3Apro-pn-E8363A/ pna-series-network-analyzer-45-mhz-to-40-ghz?cc=ES&lc=eng

[103] D. Chen and K. Eccleston, “Substrate integrated waveguide with corrugated wall,” in Microwave Conference, 2008. APMC 2008. Asia-Paciﬁc, December 2008, pp. 1–4.

[104] A. Belenguer, A. L. Borja, H. Esteban, and V. E. Boria, “High-performance coplanar waveguide to empty substrate integrated coaxial line transition,” Not yet published.