### ESCUELA T´ ECNICA SUPERIOR DE INGENIER´IA DE TELECOMUNICACI ´ ON

UNIVERSIDAD POLIT´ECNICA DE CARTAGENA

En esta página se presenta el escudo institucional de la Universidad que muestra en su centro una estrella de ocho puntas en color dorado, símbolo de la Orden de Santa María del Mar. En el interior de la estrella aparece un círculo cuartelado con castillos y leones contrapuestos sobre fondo blanco y rojo Cartagena, tomados del escudo de la ciudad donde se ubica. Todo ello sobre fondo de masa azul y circundado exteriormente por la forma verbal

“UNIVERSIDAD POLITÉCNICA DE CARTAGENA” y la leyenda “fechos allend mar” sobre franja roja ribeteada de dorado. Este emblema de escudo se utilizará en protocolos, acciones y soportes oficiales.

## 1.01

Gabinete Echeverría Manual de Identidad Corporativa de la Universidad Politécnica de Cartagena

escudo institucional

### Proyecto Fin de Carrera

### Dise˜ no de Multiplexores en Configuraci´ on Manifold para Aplicaciones por Sat´ elite

### Manifold Multiplexers for Satellite Communications

### Autor: Alejandro Pons Abenza

### Director: Alejandro ´ Alvarez Melc´ on

### Co-Director: M´ onica Mart´ınez Mendoza

### September 8, 2014

I

En esta página se presenta el escudo institucional de la Universidad que muestra en su centro una estrella de ocho puntas en color dorado, símbolo de la Orden de Santa María del Mar. En el interior de la estrella aparece un círculo cuartelado con castillos y leones contrapuestos sobre fondo blanco y rojo Cartagena, tomados del escudo de la ciudad donde se ubica. Todo ello sobre fondo de masa azul y circundado exteriormente por la forma verbal

“UNIVERSIDAD POLITÉCNICA DE CARTAGENA” y la leyenda “fechos allend mar” sobre franja roja ribeteada de dorado. Este emblema de escudo se utilizará en protocolos, acciones y soportes oficiales.

### 1.01

Presentación

Gabinete Echeverría Manual de Identidad Corporativa de la Universidad Politécnica de Cartagena

escudo institucional

Autor Alejandro Pons Abenza

E-mail del Autor [email protected] Director Alejandro ´Alvarez Melc´on

E-mail del director [email protected] Co-director M´onica Mart´ınez Mendoza E-mail del co-director [email protected]

T´ıtulo del PFC Dise˜no de Multiplexores en Configuraci´on Manifold para Aplicaciones por Sat´elite

Descriptores Multiplexores Manifold, Modelos circuitales de Matrices de acoplo, Optimizaci´on CAD de Multiplexores

Titulaci´on Ingeniero de Telecomunicaci´on

Intensificaci´on Sistemas y Redes de Telecomunicaci´on

Departamento Tecnolog´ıa de la Informaci´on y la Comunicaci´on

Resumen

Este proyecto se ha centrado en el estudio de los circuitos de microondas utilizados para el dise˜o de multiplexores. Se repasan las principales aplicaciones de estos disposi- tivos, as´ı como las posibles configuraciones que se han ido utilizando a lo largo de los a˜nos.

Finalmente, centramos el trabajo en el dise˜no en configuraci´on manifold, estudiando el proceso de optimizaci´on completa en su modelo circuital y, posteriormente, aplic´andolo a un dise˜no que se ha llevado desde el modelo circuital hasta el circuito real en gu´ıa de onda.

II

En esta página se presenta el escudo institucional de la Universidad que muestra en su centro una estrella de ocho puntas en color dorado, símbolo de la Orden de Santa María del Mar. En el interior de la estrella aparece un círculo cuartelado con castillos y leones contrapuestos sobre fondo blanco y rojo Cartagena, tomados del escudo de la ciudad donde se ubica. Todo ello sobre fondo de masa azul y circundado exteriormente por la forma verbal

“UNIVERSIDAD POLITÉCNICA DE CARTAGENA” y la leyenda “fechos allend mar” sobre franja roja ribeteada de dorado. Este emblema de escudo se utilizará en protocolos, acciones y soportes oficiales.

### 1.01

Gabinete Echeverría Manual de Identidad Corporativa de la Universidad Politécnica de Cartagena

escudo institucional

Author Alejandro Pons Abenza

Author’s e-mail [email protected] Director Alejandro ´Alvarez Melc´on

Director’s e-mail [email protected] Co-director M´onica Mart´ınez Mendoza Co-director’s e-mail [email protected]

Project Title Manifold multiplexers for satellite communications Key words Manifold multiplexers, circuit model of a coupling

matrix, CAD multiplexers optimization Degree Telecommunications engineering

Intensification Telecommunication networks and systems Department Information and communication technology

Abstract

This project has been focused on microwave circuits used for multiplexing networks design. A general overview to its applications, and the most common multiplexer config- urations is done as the first part of the project. Finally, the main objective of the work is to develop a complete study of the manifold multiplexer, its circuit model optimization and a complete design (from the ideal circuit model to to the physical realization using waveguide technology) is realized.

### Contents

1 Introduction 3

1.1 Initial approach to the project . . . 3

1.2 Objectives . . . 3

1.3 Report structure . . . 4

1.4 Novel contributions . . . 4

1.5 Software involved . . . 5

2 Fundamentals of the project 7 2.1 An Introduction to Multiplexers . . . 7

2.2 Multiplexers typical configurations . . . 9

2.2.1 Hybrid coupled approach . . . 9

2.2.2 Circulator coupled approach . . . 10

2.2.3 Directional filter approach . . . 11

2.2.4 Manifold coupled approach . . . 12

2.2.5 Star-junction multiplexers . . . 13

2.3 Design of the manifold coupled multiplexer . . . 14

2.3.1 Design methodology . . . 14

2.3.2 Analysis of the CPRL and channel transfers . . . 15

2.3.3 Channel filter design . . . 16

2.3.4 Optimization strategy . . . 17

3 Manifold coupled multiplexer design 21 3.1 Setting up the filters . . . 22

3.1.1 Filter equivalent circuit synthesis . . . 22

3.1.2 Resonator elements . . . 26

3.1.3 The admittance inverter . . . 27

3.1.4 Frequency plan . . . 29

3.1.5 Channel filters simulated responses . . . 30

3.2 Setting up the multiplexer configuration . . . 31

3.2.1 Equivalent circuit of the rectangular waveguide . . . 33

3.2.2 Tee junctions of the multiplexer equivalent circuit . . . 36

3.3 Preparing the optimization . . . 38

3.3.1 Functions to optimize . . . 39

3.3.2 Optimization goals . . . 39

3.3.3 Choosing an appropriate optimizer . . . 42

3.4 Optimization process . . . 44

3.4.1 First Round . . . 45

3.4.2 Second Round . . . 50

3.4.3 Third Round . . . 52

3.4.4 Fourth Round . . . 56

3.4.5 Final Round . . . 58

3.4.6 Results . . . 62

3.5 Refinements to the design process . . . 63

3.5.1 First Round . . . 66

3.5.2 Second Round . . . 70

3.5.3 Third Round . . . 73

3.5.4 Fourth Round . . . 73

3.6 Conclusions . . . 76

4 Triplexer design 79 4.1 Setting up the multiplexer equivalent circuit . . . 80

4.1.1 Equivalent circuit correspondent to the new filter topology . . . 80

4.1.2 New manifold multiplexer equivalent circuit set up . . . 84

4.2 Optimization process . . . 85

4.2.1 First Round . . . 85

4.2.2 Second Round . . . 88

4.2.3 Third Round . . . 91

4.2.4 Fourth Round . . . 94

4.2.5 Fifth Round . . . 97

4.2.6 Final Round . . . 101

4.3 Results and conclusions . . . 101

5 Triplexer physical implementation 105 5.1 Filter physical implementation . . . 107

5.1.1 Basics of full-inductive rectangular waveguide filters . . . 107

5.1.2 Detailed FEST3D design procedure example . . . 109

5.1.3 Results of the final filter designs . . . 122

5.2 Manifold physical structure . . . 126

5.3 First multiplexer simulation . . . 129

5.4 Solution to the partial triplexer design . . . 131

5.4.1 Convergence study of the FEST3D projects . . . 131

5.4.2 Phase adjustment related to the input iris . . . 131

5.4.3 Final multiplexer refinement in FEST3D . . . 138

5.5 Results and conclusions . . . 146

6 Conclusions and future work 149 A Tee Junction S-Parameter Matrix from FEST3D 151 B Converting the output file from FEST3D to Touchstone Specification Format 155 B.1 Major refinements . . . 158 C Converting the Touchstone File output from MWO to FEST3D Data

File 163

D Matlab Script for Manifold-Structure Lengths Calculations with other

Reference Planes 167

E FEST3D Convergency Study 173

CONTENTS V

Bibliography 174

### List of Figures

2.1 Satellite payload diagram [2]. . . 8

2.2 Front-end base stations diagram [2]. . . 9

2.3 Hybrid coupled multiplexer [2]. . . 10

2.4 Circulator coupled multiplexer [2]. . . 11

2.5 Directional filter multiplexer [2]. . . 11

2.6 Directional filter implementations [2] . . . 12

2.7 A manifold-coupled multiplexer [2]. . . 13

2.8 Microstrip 3-channel multiplexer [2]. . . 13

2.9 General architecture of the star-junction multiplexer [7]. . . 14

2.10 Equivalent circuit of the manifold multiplexer [2]. . . 16

2.11 Junction characterizations [3]. . . 17

3.1 Filter response prototype. . . 23

3.2 4-pole filter topology. . . 23

3.3 Multi-coupled bandpass prototype network [2]. . . 24

3.4 Lowpass prototype equivalent [2]. . . 25

3.5 Equivalent circuit based on FIS elements. . . 26

3.6 Impedance and admittance inverters [10]. . . 27

3.7 Equivalent circuit of the admittance inverter. . . 28

3.8 Admittance inverter schematic. . . 29

3.9 MWO filter schematic. . . 30

3.10 MWO simulated responses. . . 32

3.11 Physical drawing of a rectangular waveguide section. . . 33

3.12 Transmission line piece diagram. . . 33

3.13 Some useful ABCD parameters. . . 35

3.14 Transmission Line Schematic. . . 36

3.15 E-plane T-junction model. . . 36

3.16 H-plane T-junction model. . . 36

3.17 Manifold multiplexer schematic. . . 37

3.18 Initial manifold multiplexer equivalent circuit response. . . 38

3.19 Window capture of the optimization goal for channel 1 for the S44 parameter. 40
3.20 Window capture of the optimization goal for channel 1 for the S_{14} param-
eter (upper rejection band). . . 40

3.21 Window capture of the optimization goal for channel 1 for the S_{14} param-
eter (lower rejection band). . . 40

3.22 Microwave Office window showing var folder. . . 46

3.23 Optimization round 1 step 1.a. . . 46

3.24 Optimization round 1 step 1.b. . . 47

3.25 Optimization round 1 step 2.a. . . 48

3.26 Optimization round 1 step 2.b. . . 49

3.27 Optimization round 1 step 2.c. . . 49

3.28 Optimization round 2 step 1.a. . . 51

3.29 Optimization round 2 step 1.b. . . 51

3.30 Optimization round 2 step 2.a. . . 52

3.31 Optimization round 2 step 2.b. . . 53

3.32 Optimization round 2 step 2.c. . . 53

3.33 Optimization round 3 step 1.a. . . 54

3.34 Optimization round 3 step 1.b. . . 55

3.35 Optimization round 3 step 2.a. . . 55

3.36 Optimization round 3 step 2.b. . . 56

3.37 Optimization round 3 step 2.c. . . 57

3.38 Optimization round 4 step 1.a. . . 57

3.39 Optimization round 4 step 1.b. . . 58

3.40 Optimization round 4 step 2.a. . . 59

3.41 Optimization round 4 step 2.b. . . 59

3.42 Optimization round 4 step 2.c. . . 60

3.43 Final spacings and stubs optimization. . . 60

3.44 Final Optimization. . . 61

3.45 S-parameters of the optimized multiplexer. . . 61

3.46 New manifold structure distribution. . . 63

3.47 Initial simulation with the new junction. . . 66

3.48 Second optimization, round 1 step 1.a. . . 67

3.49 Second optimization, round 1 step 1.b. . . 68

3.50 Second optimization, round 1 step 2.a. . . 68

3.51 Second optimization, round 1 step 2.b. . . 69

3.52 Second optimization, round 1 step 2.c. . . 69

3.53 Second optimization, round 2 step 1.a. . . 70

3.54 Second optimization, round 2 step 1.b. . . 71

3.55 Second optimization, round 2 step 2.a. . . 71

3.56 Second optimization, round 2 step 2.b. . . 72

3.57 Second optimization, round 2 step 2.c. . . 72

3.58 Second optimization, round 3 step 1.a. . . 73

3.59 Second optimization, round 3 step 1.b. . . 74

3.60 Second optimization, round 3 step 2.a. . . 74

3.61 Second optimization, round 3 step 2.b. . . 75

3.62 Second optimization, round 3 step 2.c. . . 75

3.63 Second optimization, round 4 step 1.a. . . 76

3.64 Second optimization, round 4 step 1.b. . . 77

3.65 Second optimization, final multiplexer refinement. . . 77

3.66 Multiplexer response corresponding to the first optimization. . . 78

3.67 Multiplexer response corresponding to the second optimization. . . 78

4.1 Structure of an individual circular-waveguide dual-mode filter [1]. . . 79

4.2 Central Filter response. . . 81

4.3 6-pole filter topology. . . 81

4.4 Filter equivalent circuit schematic for the figure 4.3 topology. . . 82

4.5 Ideal individual triplexer filters. . . 83

4.6 Manifold multiplexer MWO equivalent circuit schematic . . . 84

LIST OF FIGURES IX

4.7 Initial multiplexer response to optimize. . . 86

4.8 Multiplexer Response after spacings optimization. . . 86

4.9 Multiplexer Response after stubs optimization. . . 87

4.10 Channel 1 Optimization. . . 88

4.11 Channel 2 Optimization. . . 89

4.12 Channel 3 Optimization. . . 89

4.13 Multiplexer Response after spacings optimization. . . 90

4.14 Multiplexer Response after stubs optimization. . . 90

4.15 Channel 1 Optimization. . . 91

4.16 Channel 2 Optimization. . . 92

4.17 Channel 3 Optimization. . . 92

4.18 Multiplexer Response after spacings optimization. . . 93

4.19 Multiplexer Response after stubs optimization. . . 93

4.20 Channel 1 Optimization. . . 94

4.21 Channel 2 Optimization. . . 95

4.22 Channel 3 Optimization. . . 95

4.23 Multiplexer Response after spacings optimization. . . 96

4.24 Multiplexer Response after stubs optimization. . . 96

4.25 Channel 1 Optimization. . . 97

4.26 Channel 2 Optimization. . . 98

4.27 Channel 3 Optimization. . . 98

4.28 Multiplexer Response after spacings optimization. . . 99

4.29 Multiplexer Response after stubs optimization. . . 99

4.30 Channel 1 Optimization. . . 100

4.31 Channel 2 Optimization. . . 100

4.32 Channel 3 Optimization. . . 101

4.33 Final Refinement of the Triplexer. . . 102

4.34 Final Response of the Multiplexer . . . 103

5.1 Full-inductive rectangular waveguide filter used in this work. . . 105

5.2 An example of a fourth-degree filter with two full-inductive dual-mode res- onator cavities. . . 107

5.3 Dual-mode rectangular waveguide resonant cavity. . . 108

5.4 3D Model of the physical filter implementation. . . 110

5.5 Module response of the channel 1 filter. . . 110

5.6 Phase (S11 response) of the channel 1 filter. . . 110

5.7 Window from the Matlab application for getting the targets [9]. . . 111

5.8 Collection of segments considered in each optimization stage (forward and backward). . . 113

5.9 Input iris optimization. . . 113

5.10 3D model of the first optimized segment. The input iris is the yellow block. 113 5.11 Input iris optimization. Black points denotes goal function, while red line is the optimized S21 value. . . 114

5.12 3D model of the second filter segment. . . 115

5.13 Second forward step optimization schematic. . . 115

5.14 Second forward step optimization response. . . 116

5.15 Output iris optimization. . . 117

5.16 Output iris and second single-mode cavity optimization. . . 117

5.17 3D model corresponding to the third segment. . . 118

5.18 First transmission zero. . . 118

5.19 Second transmission zero. . . 118

5.20 Backward design step 3 optimization. . . 119

5.21 Forward design step 3 optimization. . . 120

5.22 Final optimization of the first channel filter. . . 121

5.23 Full-inductive rectangular-waveguide filter dimensions. . . 122

5.24 Final optimization of the second channel filter. . . 123

5.25 Final optimization of the third channel filter. . . 125

5.26 Complete 3D view of the manifold structure. . . 126

5.27 3D models for each of the three considered pieces of the manifold structure. 127 5.28 Manifold structure optimized modules in FEST3D. . . 128

5.29 Manifold structure optimized phases in FEST3D. . . 128

5.30 FEST3D schematic of the 3-channel multiplexer. . . 129

5.31 First Simulation of the Triplexer in FEST3D. . . 130

5.32 Theoretical vs Simulated CPRL of the triplexer. . . 130

5.33 Theoretical vs Simulated TX channels of the triplexer. . . 130

5.34 Solution proposed related to the stub length. . . 132

5.35 S_{11} phase adjustment of the first channel filter. . . 134

5.36 S_{11} phase adjustment of the second channel filter. . . 135

5.37 S_{11} phase adjustment of the third channel filter. . . 135

5.38 Example of FEST3D optimization with channels bandwidth only. . . 136

5.39 Multiplexer result after manifold lengths optimization. . . 137

5.40 Separated filter schematic with a new division of the filter stub. . . 137

5.41 Re-optimized module response of the channel 1 filter. . . 139

5.42 Re-optimized phase (S_{11} response) of the channel 1 filter. . . 139

5.43 Re-optimized module response of the channel 2 filter. . . 140

5.44 Re-optimized phase (S_{11} response) of the channel 2 filter. . . 140

5.45 Re-optimized module response of the channel 3 filter. . . 141

5.46 Re-optimized phase (S11 response) of the channel 3 filter. . . 141

5.47 Initial point for the multiplexer with re-optimized channel filters. . . 142

5.48 Multiplexer response after manifold lengths optimization. . . 143

5.49 Multiplexer response while optimizing channel 2 filter. . . 144

5.50 Multiplexer response after manifold lengths optimization, second round. . . 145

5.51 Multiplexer response while optimizing channel 2 filter, second round. . . . 145

5.52 Final triplexer response. . . 146

5.53 Final response of the triplexer. . . 147

A.1 Detail of the Cubic Junctions . . . 151

A.2 Names corresponding to each face of the cubic junction. . . 152

A.3 Example of a Tee Junction schematic, as seen in the official help content. . 153

A.4 Configuration of the Tee Junction port positions. . . 153

B.1 Detail of the application window . . . 158

B.2 Window caption of the new GUI implemented . . . 158

### List of Tables

3.1 Frequency plan of the 3-channel multiplexer. . . 30

3.2 Manifold structure length values for the 4-pole channel multiplexer. . . 62

3.3 Channel filters coupling matrix elements for the 4-pole channel multiplexer (1). . . 62

3.4 Channel filters coupling matrix elements for the 4-pole channel multiplexer (2). . . 63

4.1 Manifold structure length values for the 6-pole channel multiplexer. . . 101

4.2 Channel filters coupling matrix elements for the 6-pole channel multiplexer (1). . . 102

4.3 Channel filters coupling matrix elements for the 6-pole channel multiplexer (2). . . 102

5.1 Filter 1 Physical Parameters. . . 121

5.2 Filter 2 Physical Parameters. . . 124

5.3 Filter 3 Physical Parameters. . . 125

5.4 Comparative between the optimized and unoptimized manifold lengths in FEST3D. . . 143

5.5 Final manifold structure lengths. . . 147

5.6 Channel filters final lengths (1). . . 147

5.7 Channel filters final lengths (2). . . 148

### Listings

B.1 Function convierte() from FEST3d to Touchstone Application . . . 155

B.2 Function convierte() for 2-port parameter files . . . 158

B.3 Function convierte() for 4-port parameter files . . . 160

C.1 Function convierte() from Touchstone to FEST3D Application . . . 163

C.2 Function finalFile() from Touchstone to FEST3D Application . . . 165

D.1 Function getPropagationConstant.m which obtains β_{10} . . . 167

D.2 Function getPhysicalLength.m to obtain the corresponding physical length from an electrical length . . . 167

D.3 Function getElectricalLength.m to obtain the corresponding electrical length from a physical length . . . 168

D.4 Function parentScript.m obtaining the needed calculations for the refine- ment of the manifold multiplexer . . . 168

### Chapter 1 Introduction

The first chapter of the project is a general overview of the contents that can be found, relative to the developed work.

### 1.1 Initial approach to the project

Multiplexing networks synthesis and design have got a huge importance in modern satellite and mobile communication systems over the last decades. Inside the design process, the CAD has become a fundamental part of the work, gaining more weight as the computational power increases with the latest electronic achievements.

As a result, the role of the engineer as a microwave designer is focused more in controlling and programming the Computer Aided Design applications, like full-wave simulators (e.g. Ansoft HFSS, FEST3D), mathematics software (e.g. Matlab as the most representative) and other electronic and modelling suites (e.g. OrCAD spice, Microwave Office). However, the fundamentals of the circuit design must not be forgotten, since this type of software has just appeared to make life easier but does not replace the accumulated knowledge over the years.

The aim of this project is to join both theoretical and computational knowledge into the design of one of the most important microwave circuits applied to satellite commu- nications: the manifold multiplexer, whose use has been extended with the advanced optimization techniques we were talking about.

### 1.2 Objectives

The main objective of this project is the study and application of the complete design process of microwave multiplexers based in Manifold Configuration.

After the first analysis of the complete conceptual technique of manifold optimization, we will go through the complete design from the first circuit model until the last real dimensions using a practical implementation of a duplexer in waveguide technology. In this project we will cover the design of 3-channel multiplexer, but it is also possible to ex- tend this technique to an N-channel multiplexer design, only by increasing the design time.

A complete E-plane 3-channel multiplexer based on rectangular-waveguide filters with inductive irises is proposed as the main result of the project. Using this design example, all Microwave Office, Matlab and FEST3D design steps are explained in detail throughout the project.

### 1.3 Report structure

This report has been subdivided in a total of 5 chapters.

In this first chapter an initial approach of the contents of the report is covered, as well as the main objectives of the multiplexing networks study and the design approach to be followed.

In the second chapter of this report we will take a look at the fundamentals of the multiplexing networks, types of multiplexers and which advantages and disadvantages are relevant in order to choose between one or another design. Moreover, the theoretical basis of the manifold multiplexer design is also covered. Also, in the final subsection of this chapter, the complete sequential optimization process is described, linking with the optimization process developed in Microwave Office (chapter 3.4).

The third chapter will lead us to the full optimization process of the manifold multiplexer using a simple filter network. We will start dealing with the equivalent circuit of the multiplexer used to implement a simple model in Microwave Office and then we will learn how to use the available optimizers, and how the optimization must be performed.

In the fourth chapter we will begin a different approach, starting with the design of a different kind of filter. The objective here is to build an inductive filter 3-channel manifold multiplexer, applying the general knowledge obtained from the study of the chapter number two. Main differences and similarities between these two designs will be described, emphasizing on the final response characteristics.

Then, we will move to the fifth and last chapter of the report where we will describe the design techniques needed to achieve the final manifold multiplexer in rectangular- waveguide configuration. In this chapter i will also explain some common problems related to the design of manifold multiplexers, together with a description on the possible solutions.

Finally, in the appendix section, I cover some specific points of the report which are needed but are not essential to the understanding of the project. This includes additional software to convert format files exchanged between the used programs and some other technical questions.

### 1.4 Novel contributions

The main novel contribution to the state of the art is the application of the already known techniques of Manifold optimization to full inductive dual-mode rectangular-waveguide

1.5. SOFTWARE INVOLVED 5

filters instead of the circular-waveguide filters used in [1].

This approach is as advantageous as novel. Design of dual-mode circular cavities involves a lot of tuning and optimization, making the whole process long and tedious.

However, even when the circuit-model of both approaches are the same, the design of the all-inductive rectangular-waveguide filters is simpler and it gives, in general, more accurate results as no tuning screws are involved in the process. Additionally this is not the only advantage; fabrication cost is also a major point. This filter technology is easier to manufacture than its circular analogous.

There is one more advantage compared to the circular-waveguide design: in this project the presented technique does not use the distributed model to transform the circuit model of the filter into the real dimensions filter.

This is a major advantage which is traduced into a slight reduction of the design time.

However it is necessary to notice that this technique will only work with small bandwidth systems. In addition, because of the exclusion of the distributed model in the whole design process, spurious effects are not taken into account.

### 1.5 Software involved

This project is mostly optimization-based, thereby there are not many software imple- mentations. Instead, commercial software such as Microwave Office (MWO) and FEST3D have been used. Besides, some Matlab scripts have been developed to allow the file con- version between MWO and FEST3D. This is because MWO uses a standard file format for microwave device parameters representation, whereas FEST3D only accepts plain data format to import files into FEST.

### Chapter 2

### Fundamentals of the project

This chapter presents a theoretical point of view of the multiplexing networks studied in this project.

In the first part of the chapter a brief historical introduction can be found, additionally with the most typical multiplexer configurations. These configurations are described by a block diagram and a little description of their key advantages and disadvantages. The manifold multiplexer configuration is also introduced here, making a comparison between this configuration and the other configurations which have not been developed in this work.

The second part of the chapter begins with the design procedure of the manifold multiplexer, the main subject of this project.

### 2.1 An Introduction to Multiplexers

Multiplexers (MUXs) are mostly used in communication system applications, where there is a need to separate a wideband signal into a particular number of narrowband signals.

Channelization of the allocated frequency band is the key to a good flexibility for the flow of communication traffic in a multi-user environment. Amplification of individual channels also eases the requirements on the high-power amplifiers (HPAs), allowing them to operate at relatively high efficiency with an acceptable degree of nonlinearity.

Multiplexers are also employed to provide the opposite function: combining several narrowband channels into a single wideband composite signal for transmission via common antenna. For this reason, multiplexers are also known as channelizers or combiners. Due to the reciprocity characteristic of filter networks, it is possible to configure a MUX to separate the transmit and receive frequency bands within the same device, referred to as a duplexer or diplexer.

Many applications such as in satellite payloads, wireless systems, and electronic warfare systems have taken advantage of this kind of microwave technology.

Despite the fact that the principles of combining or separating frequency-diverse signals (channels) for interfacing with a single port of an antenna system have been known for many years, the advent of satellite communication systems motivated major advances in this field. Figure 2.1 illustrates a simplified block diagram of a conventional

Uplink Antenna

Low-noise Receiver

Input Multiplexer Output Multiplexer

High-Power Amplifiers

Downlink Antenna

Figure 2.1: A simplified block diagram of a satellite payload [2].

satellite payload system. It consists of a receive and transmit antenna, a low-noise receiver, input and output multiplexers, and HPAs for each channel. The payload behaves as an orbiting repeater that receives, amplifies, and transmits signals in the allocated frequency bands. The practical constraints on HPAs require the channelization of the received signal into a number of RF channels by a multiplexer (IMUX). After narrowband channels are amplified separately by the HPAs, the channels are recombined by an output multiplexer (OMUX) for transmission back to the ground via a common antenna. In satellite payloads, the input and output multiplexers have an important role on the RF channels, substantially impacting on the performance of the overall payload.

For example, required number of filters for input and output multiplexer ranges from 50 to over 100 typically.

Telephony systems are another important application of the microwave multiplexers.

Transmit/receive diplexers are commonly used in base stations of mobile telephone systems, thanks to their build-standard requirements accommodating high- and low- power signals within one housing. Usually these diplexers are located at the top of transmitter masts exposed to the worst of climatic extremes. Figure 2.2 shows a simplified block diagram of the front-end of a cellular base station. The purpose of the receive filter is to reject the out-of-band interference prior to the low-noise amplification and down-conversion. The transmit filter is used primarily to limit the out-of-band signals generated by the transmit portion of the base station. The transmit filter must also have a very high level of rejection in the receive band in order to eliminate the possibility of intermodulation products being fed into the receiver through the common antenna.

Multiplexers are also used in wireless applications, where the base station may need to transmit various frequency channels in different directions by using directive antennas.

A multiplexer is needed to separate the overall band into different channels, radiated in various directions. Another possible application of multiplexers in wireless base stations is in cases where the base station provides service to a number of independent operators that are licensed to operate only in specific channels within the frequency band covered by the base station. Multiplexers have also been used in electronic warfare systems being

2.2. MULTIPLEXERS TYPICAL CONFIGURATIONS 9

LNA

HPA

Down Converter

Up Converter Receive Filter

Transmit Filter Diplexer Antenna

Figure 2.2: A simplified block diagram of the front end in base stations [2].

part of switched filter banks, essential building blocks in wideband receivers. Their most important feature is the ability to operate in a hostile signal-dense environment; switches are integrated into the multiplexer to allow the selection of a particular channel, effectively allowing the realization of a tunable filter with a variable bandwidth and a variable center frequency.

### 2.2 Multiplexers typical configurations

Over the past four decades (i.e., since the mid-1970s) there have been many advances in the design and implementation of multiplexing networks. The most commonly used con- figurations are hybrid-coupled multiplexers, circulator-coupled multiplexers, directional filter multiplexers and Manifold-coupled multiplexers, whose study is undertaken in this master project. Their main features are going to be described, as well as some common ways of implementing these types of multiplexers.

### 2.2.1 Hybrid coupled approach

The hybrid coupled multiplexer was one of the main multiplexer configurations used in the early days of the microwave multiplexer. Figure 2.3 shows a block diagram for this configuration generalized to n channels.

Each channel is composed by two identical filters and two identical 90° hybrids.

The main advantage of the hybrid-coupled approach is its directional property, which minimizes the interaction among the channel filters. As a consequence, the hybrid coupled multiplexer is amenable to a modular concept. It also allows the integration of additional channels at a later date without disrupting the existing multiplexer design (required in some systems), and it is probably the most important feature of this configuration. Additionally, only half of the input power goes through each filter, making this solution suitable for high-power applications.

f1 f1 f1

f2 f2 f2

fn fn fn

f1,f2, ..., fn

Figure 2.3: Layout of a Hybrid-coupled multiplexer [2].

On the other hand, its large size is a disadvantage, since every channel needs two filters and two hybrid couplers. Moreover, multiplexer designers have to consider that such multiplexers have a phase deviation between the two filter paths within each channel, because there are two signals undergoing before they add constructively at the end of the channel. As a consequence, phase minimization is a critical design criteria, leading to a delicate manufacturing of the channel structure (filters and hybrids). A tolerance study is often needed to minimize this measure particularly in planar circuit applications, where it is difficult to use tuning elements to balance the two paths.

### 2.2.2 Circulator coupled approach

In this solution hybrids are replaced by circulators as isolator elements. Each channel consists of a channel-dropping circulator and one filter (instead of two filters like the hybrid multiplexer), as shown in figure 2.4. This configuration was born to solve the main disadvantages of the hybrid coupled multiplexer: with only one filter per channel, phase deviation is not appearing here.

The unidirectional property of the circulator provides the same advantages as the hybrid-coupled approach does in terms of amenability to modular integration and ease of design and assembly.

However, the insertion loss of each circulator becomes a problem. First channel in- sertion loss is the sum of the insertion loss of the channel filter and the insertion loss of the circulator. The subsequent channels exhibit a relatively higher loss due to the insertion loss incurred during each trip through the channel dropping circulators. The main consequence of this fact is that it will not be possible to build multiplexer with a high number of channels, since with every channel we connect to the system will suffer a higher insertion loss, leading to a malfunction if the number of channels is too large.

Nevertheless, it is a good option with a few channels in the design.

2.2. MULTIPLEXERS TYPICAL CONFIGURATIONS 11

f1 f2 fn

f1, f2, ..., fn

Figure 2.4: A circulator-coupled Multiplexer [2].

### 2.2.3 Directional filter approach

Another good configuration for a multiplexing network can be achieved by cascading 4- port devices which make the work of isolating and filtering in every channel. This 4-port device is known as a directional filter.

f1 Z1

a

Input Signal f1, f2, ..., fn

f2 Z2

b

f3 Z3

c

f4

Figure 2.5: A directional filter multiplexer [2].

Figure 2.5 illustrates a layout of a multiplexer Directional filters are terminated in a load at one port. The other three ports of the directional filter essentially act as a circulator connected to a bandpass filter. The input power emerges at the second port with a bandpass frequency response, while the reflected power from the filter emerges at the third port. Conceptually it is a simple way of implementing a multiplexer, but there is difficulty which becomes its main disadvantage: the directional filter implementation.

Figure 2.6 illustrates two possible approaches for realizing directional filters in waveg-
uide and microstrip, respectively. The waveguide directional filter is realized by coupling
rectangular waveguides, operating in T E_{10} mode, to a circular waveguide filter operating
in T E_{11} modes. In the microstrip version, each 360^{◦}wavelength ring resonator is coupled
to another wavelength ring resonator and to two transmission lines. This multiplexing ap-

Figure 2.6: (a) Waveguide directional filter [2]. (b) Microstrip directional filter [2].

proach has the same advantages as the hybrid-coupled and circulator-coupled approaches.

It is, however, limited to narrowband applications [2].

### 2.2.4 Manifold coupled approach

Finally we arrive at the main studied configuration in the project. The manifold-coupled multiplexer is nowadays the optimum choice as far as miniaturization and absolute insertion loss are concerned. This type of multiplexer requires the presence of all the channel filters at the same time so that the effect of channel interactions can be compensated in the design process. Figure 2.7 shows a typical manifold multiplexer block diagram, where channel filters are connected directly by junctions and transmission lines between these junctions.

Channel interaction compensation implies that the manifold multiplexer is not amenable to a flexible frequency plan. Any change in the allocation of the channels requires a new multiplexer design. Additionally, as the number of channels increases, this approach becomes more difficult to implement. The manifold-coupled multiplexer shown in Figure 2.7 acts as a channelizer. The same configuration can be used as a combiner.

The configuration of the manifold multiplexer can be implemented as well in planar circuits as shown in Figure 2.8. Here, three microstrip filters are integrated with a microstrip manifold. In this particular case, one of the channels is connected directly to the manifold. Other practical implementations of manifold multiplexers in microstrip technology can be found in [11].

There are three distinct categories of multiplexing networks required by communica- tion systems, namely, RF channelizers, RF combiners, and transmit-receive diplexers.

The design of each type of multiplexer is dictated by its application and system constraints.

2.2. MULTIPLEXERS TYPICAL CONFIGURATIONS 13

f1 f2 fn

f1, f2, ..., fn

El1 El2 Eln

Figure 2.7: A manifold-coupled multiplexer [2].

Figure 2.8: A three channel manifold multiplexer [2] based on microstrip technology.

### 2.2.5 Star-junction multiplexers

In addition to the different multiplexer configurations described in the previous sections, there is another important multiplexer family based on a different connection topology.

These multiplexers are called star-junction multiplexers.

The main idea of this configuration is to reduce the number of junction pieces of the multiplexer to one, reducing considerably the size of the final multiplexer. However, its main disadvantage is related to the resonant junction, which is very complex to design

Figure 2.9: General architecture of the star-junction multiplexer [7].

when the number of connected channels is too high. In practice, these multiplexers are limited to 3 or 4 channels only. In [7], the author G. Machiarella implements a 3-channel star-junction multiplexer from the equivalent circuit to the final manufactured device, presenting measured results from the device.

However there are recent design techniques to improve the performance of these kind of multiplexer. For example, in [12], the author implements a 4-channel star-junction multiplexer achieving good performance responses, using novel techniques which allowed to increase the number of multiplexing channels while reducing the required connections to the resonant node.

### 2.3 Design of the manifold coupled multiplexer

### 2.3.1 Design methodology

Since there are no directional or isolating elements in the manifold multiplexer (circu- lators, hybrids), all channel filters are electrically connected to each other through the near lossless manifold (see figure 2.7). Therefore the entire system must be considered in the design process. Consequently, the design of any channel individually inside the manifold multiplexer will never work, since interactions between each filter would not be considered and the process would never reach a solution.

In the early days, a number of ingenious techniques were invented, to design the individual filters such that they would properly interact with the other filters on the same manifold and function virtually as if they were operating into a matched termination.

However, the huge increase in computer power that has become available in recent years has made practical multiplexer design procedure more optimization-based to achieve the final result, instead of the more limited analytic techniques.

2.3. DESIGN OF THE MANIFOLD COUPLED MULTIPLEXER 15

### 2.3.2 Analysis of the common port return loss and channel transfer characteristics

Inside any circuit optimizer there is an efficient analysis routine. During the optimization, the analysis routine will be called many thousands of times at different frequencies as it progresses optimizing the various parameters. In general, the two parameters from which the overall cost function is built up are the multiplexer Common-Port Return Loss (CPRL) and the individual channel transfer characteristics.

During the optimization process, these subroutines will have to calculate the CPRL and the individual channel transfer characteristics over the specified frequency range.

There are two ways of dealing with the process: it would be possible to construct an admittance matrix for the whole multiplexer circuit and analyze it at each frequency point to obtain the desired transfer and reflection data. However, this matrix will be quite large for the multiplexer with a large number of channel filters and will take a significant amount of CPU time to invert it. Moreover, a lot of data will be obtained that is not used for the optimization.

On the other hand, the strategy shown in this project involves optimizing the channel filters one after one in a repeated cycle. As the optimization parameters of each filter in turn are being optimized, it is only necessary to calculate the transfer characteristic of this channel filter individually; the others will be relatively unaffected by the changes (assumed small) being made to the filter that is the object of the optimizer’s attentions at this stage in the cycle.

To speed up the overall optimization, it becomes more efficient to analyze each filter’s input-to-common-port transfer characteristic individually, in addition to the CPRL.

The manifold of the multiplexer may be most conveniently represented as an open-wire circuit, with a short circuit at one end and three-port junctions spaced along its length (see figure 2.10). Every channel filter will be connected to the third port of each junction, and transmission lines (stubs) will be separating them from its correspondent junction as seen in figure 2.10.

In the particular case of the waveguide realization of the manifold, the junctions may be E-plane or H-plane, and since their intrinsic parameters do not change during the optimization, they are best characterized with three-port S-parameters pre-calculated by a full-wave routine and stored over an adequate range of frequencies (for example, covering the bandwidths of all the channel filters). Typically, the junctions are symmetric about ports 1 and 2, leaving the possibility for port 3 to be a different size as described in figure 2.11 [3]. However, in this project we will always work with equal port lengths for each manifold junction.

The complete theoretical process can be found at [3]. We are not going through all of it because the major part of the calculations shown there are not really needed to perform the optimization procedure described in this project as we will see later on the next chapter.

Figure 2.10: Open-wire model of waveguide manifold multiplexer (three-channel) [3].

### 2.3.3 Channel filter design

The best way of implementing the channel filters is using purely reactive components (i.e., no resistive elements between the channel inputs and the common-port output).

This means that channel filters will only interact with each other through the manifold itself. If the channels are widely spaced, the channel interactions are quite low, because over one filter’s passband the other filters are well into their reject regions and will be presenting short circuits at their ports nearest to the manifold.

The channel filters may be designed as double terminated networks separate from the entire system. The key point is, when the individual filter gets connected onto the manifold, all that is needed are adjustments to the along-manifold spacings and stub lengths and minor adjustments to the first three or four elements of the filter to recover a good CPRL. These elements are the most sensible to the global multiplexer response, and they will determine

However, as the guard bands between channel filters decrease towards contiguity, they begin to interact strongly along the manifold. Now significant adjustments to the filter parameters are needed in addition to the manifold and stub phase lengths to reach an acceptable CPRL. Although it is possible to optimize double terminated filters to operate in a contiguous channel environment, a starting point much closer to the final optimal result is obtained if single terminated filter prototypes are used in these conditions.

Despite of this fact, doubly terminated filters have been only used in this project.

In this project we are mainly going to work with two types of filters; the first one is not going to be physically implemented because it is only proposed for showing a simple example to perform an initial optimization over the manifold model we will use in the following chapter, and the second one is a full-inductive filter implemented in rectangular waveguide technology.

2.3. DESIGN OF THE MANIFOLD COUPLED MULTIPLEXER 17

Figure 2.11: E-plane and H-plane waveguide junctions and S-parameter matrix represen- tation [3].

The realization of this work has been based on two different filter topologies.

Both filters will implement two transmission zeros, allowing The first topology will be represented by a simple filter with 4 poles which is not physically implemented in the project as it will only serve to present the procedure and the equivalent circuit of a particular topology with its coupling matrix. Finally, there is a 6-pole filter whose design is covered from the equivalent circuit to the physical model. Waveguide technology has been chosen to implement the multiplexers designs shown in the project.

Summarizing the channel filter design process: we have to obtain a suitable coupling matrix to implement the chosen topology. Then, it is necessary to synthesize the equiva- lent circuit of the filter, to start with the multiplexer design in the whole equivalent circuit.

The resultant coupling matrix for each equivalent filter circuit will be transformed into rectangular-waveguide technology following the technique from [9], whose details will be described later, instead of its equivalent circuit. Afterwards it will be possible to join all channel filters with their correspondent junctions already implemented. Minor refine- ments over the first three or four parameters of the filter will be needed. Additionally we will have to discuss some considerations about how the equivalent circuit and the real dimensions circuit are linked, in order to achieve the final result.

### 2.3.4 Optimization strategy

The networks that model even a moderate-sized waveguide manifold multiplexer tend to be quite complex. An open-wire equivalent circuit of a six-channel manifold multiplexer with sixth-degree quasi-elliptic dual-mode filters has in the order of 90 frequency sampling points and 100 electrical elements of varying sensitivities and different constraints that all need to be correctly valued before the overall multiplexer will operate to specification.

If all the channels parameters are optimized simultaneously, not only will the amount of CPU time be enormous, but there is little likelihood that the global optimum will

be attained, there being a myriad of shallow local optimum solutions. With manifold multiplexer routinely incorporating 20 channel, and perhaps up to 30 in the future, global optimization is clearly unsuitable directly from the start.

For these reasons, most of the major satellite output-multiplexer designers and man- ufacturers have developed more efficient methods for manifold multiplexer optimization.

Among these is the piecewise approach, optimizing parts of the multiplexer separately in repeated cycles while converging upon an optimal solution. The parts or parameter groups being referred to here might include the first five elements of each channel filter (narrow- band domain), or all the manifold interjunction or stub lengths (wideband domain). It is usual practice to commence the optimization process with the wideband sections first (parameters relating to the manifold and stubs), followed by a shift in emphasis to the narrow-band sections (filter parameters) as the CPRL begins to take shape. A typical design optimization project might proceed as follows:

• Design

1. Design the channel filter transfer/reflection functions to meet the individual in-band and rejection specifications.

2. Synthesize the corresponding coupling matrices, doubly terminated if the chan- nel filter design bandwidths (DBWs) are separated by guard bands greater than about 25% of the DBWs, singly terminated if otherwise. If singly terminated prototypes are to be used, a more practical design results if the initial proto- type is generated with a low return loss to bring the value of the termination at the end opposite to the manifold as close to unity as possible.

3. Set initial manifold spacings between E- or H-plane junctions at mλ_{g}/2, where
m is as low as possible for a convenient mechanical layout. Set the initial
manifold short/first junction spacing at λg/4 (H-plane) or λg/2 (E-plane). λg

is the wavelength in the manifold waveguide at the center frequency of the nearest filter to the length of waveguide in the direction of the common port.

4. Set initial manifold junction-filter stub lengths at nλg/2. Again, n should be as small as possible.

• Optimization

1. Design wideband components. Optimize spacings between the junctions and between the short and first junction and the stubs lengths. This often has the most dramatic effect in terms of improvement in CPRL.

2. Optimize stubs and the first three or four parameters of channel filter 1 [M_{s1}
(filter manifold coupling), M11 (first resonance tuning), and M12 (resonance 1
to resonance 2 coupling)].

3. Repeat the cycle for all the channels, possibly omitting the stub length since this does not change much after the first cycle of optimization, until the im- provements in the cost function begin to become negligible.

• Refinement

1. Repeat optimization of manifold and stub lengths.

2.3. DESIGN OF THE MANIFOLD COUPLED MULTIPLEXER 19
2. Reoptimization with a fine step on all of each channel filter’s parameters, most
lightly on the elements furthest away from the manifold, and not at all on the
final coupling M_{LN} (i.e., the input coupling to the multiplexer filter from the
channel high-power amplifier).

### Chapter 3

### Manifold coupled multiplexer design

After describing the general theory of the multiplexing networks and its main configu- rations, this chapter illustrates a complete design of the manifold coupled multiplexer.

We are going to use 4th order channel filters, with a Bandwidth of 36 MHz, 20 dB Return Loss and two transmission zeros that produce side lobes in the rejection band under 40 dB. The multiplexer is going to be centred at 11 GHz frequency and it is a non-contiguous, 3 channel multiplexer. To give a general idea about the complexity of the optimization process, we will just work with the lumped model of the multiplexer.

However, this approach will be useful for us to develop a multiplexer using different filter designs in a Manifold structure.

As it was said in the previous chapter, the manifold multiplexer has no isolating or directional elements. In terms of design strategy, this means that channel interactions between each other have to be considered in the optimization algorithm. As the project is optimization-based, these interactions will be compensated in the equivalent circuit optimization. In this project we will not use any analytic technique to design the manifold multiplexer.

First of all we need to define the general stages of the optimization procedure followed in this chapter. The first sections of this chapter explains how to build the equivalent circuit of the manifold multiplexer. In general we will refer to each of the equivalent circuit parts as sub-circuits. By looking at figure 2.10 we can see that we will need to define equivalent circuits for the channel filters, junctions an transmission lines connected between them.

In the present project we are working with rectangular waveguide technology, so that will be the theoretical model to synthesize the equivalent circuits for the transmission lines and junctions. Additionally, in the last design of the project we will describe a procedure to work with filters built with rectangular waveguides too, leading to a full rectangular waveguide device.

Once all these sub-circuits get synthesized, the set up for the optimization algorithm will be described. An iterative process is followed with this equivalent circuit [3], optimizing each sub-circuit of the manifold multiplexer individually while considering the global response of the whole system.

With this iterative algorithm, in each subsequent iteration we will worsen the

responses individually less and less until we reach a solution. If we do not get a better result after any following iteration, this might be an indication that something is wrong in the process. If this situation occurs, it is recommended to take the values from the previous iteration, and then review the list of variables to be optimized, as well as that the correct goal functions are used in the optimization.

Due to its power and simplicity, AWR Microwave Office 2002 (MWO) software has been chosen to optimize the equivalent circuit of the manifold coupled multiplexer we are going to design. Additional software like FEST3D will be needed in some steps during the setting up of the multiplexer equivalent circuit. This will be needed in order to increase the accuracy in the characterization of the three port junctions shown in figure 2.10. FEST3D will also be used during the final design of the physical multiplexer in waveguide technology.

Since the response of the multiplexer is independent on the source and load impedance values, the design of all its parts will be performed using normalized termination impedance of 1 Ω.

### 3.1 Setting up the filters

The main part of the manifold multiplexer is the channel filter. For each channel of the multiplexer, there must be a suitable filter at the center frequency of the corresponding channel bandwidth. As we said in the introduction, an equivalent circuit model of the filter will be needed to perform the optimization.

This section covers the set up of the equivalent circuit of the filters. These filters are the heart of every multiplexer channel, and need to be synthesized to obtain a suitable response individually, before connecting them onto the manifold multiplexer. As a result of the procedure described here, the equivalent circuit will be defined as a sub-circuit in our MWO project.

The first step is the synthesis of the ideal filter with two transmission zeros located in such a way that side lobes lower than 40 dB are obtained in the rejection band. Also, the reflection level is fixed to -22 dB.

Figure 3.1 illustrates the general response of the filters we are going to use. Here we
can see the side lobes described earlier: two transmission zeros are limiting the passband,
allowing the appearance of secondary transmission lobes in the rejection band, caused by
these abrupt falls in the S_{21} parameter. Reflection response shows 4 poles, leading to a
4-resonator filter.

### 3.1.1 Filter equivalent circuit synthesis

The filter topology used to implement the response shown in figure 3.1 follows the diagram of figure 3.2.

This filter topology is known as the quartet topology, also known as the extended box in some books [2]. This topology is characterized by 4 resonators connected in series

3.1. SETTING UP THE FILTERS 23

10.9 10.92 10.94 10.96 10.98 11 11.02 11.04 11.06 11.08 11.1

−70

−60

−50

−40

−30

−20

−10 0

Frequency (GHz)

|S ij| dB

Central Channel Filter

|S11|

|S21|

Figure 3.1: Filter response prototype.

2 ^{//} 3

S ^{//} 1

OO // 4 ^{//} L

Figure 3.2: 4-pole filter topology.

configuration, whose first and last resonators are also coupled (there is a cross-coupling between resonators 1 and 4).

The corresponding coupling matrix to implement the filter topology of 3.2 can be synthesized to obtain the prototype response from figure 3.1 leading to the following matrix [2]:

M =

0 1.0742 0 0 0 0

1.0742 0 0.9357 0 −0.1079 0

0 0.9357 0 0.7670 0 0

0 0 0.7670 0 0.9357 0

0 −0.1079 0 0.9357 0 1.0742

0 0 0 0 1.0742 0

(3.1)

As we can see from the filter topology shown in figure 3.2 our coupling matrix is squared and its dimensions are 6 × 6.

We will implement the equivalent circuit of the filter in MWO using inductive and capacitive lumped elements and constant susceptances. The equivalent circuit of a general cross coupled filter equivalent circuit can be represented by the generalized coupling matrix, and it will be similar to the one shown in figure 3.3.

Figure 3.3 shows an equivalent circuit composed of a cascade of lumped-element series resonators intercoupled by transformers. It can be observed in the prototype additional elements that are constant susceptances in the form of FIR elements. These elements enable the circuit to represent frequency asymmetric characteristics.

Figure 3.3: Multi-coupled series-resonator bandpass prototype network, modified to in- clude FIR elements and separate self-inductor [2].

3.1. SETTING UP THE FILTERS 25 The positive frequency characteristics of the bandpass prototype (BPP) circuit can be related to the lowpass prototype (LPP) characteristics through the lumped-element bandpass-to-lowpass mapping as follows:

s = j ω_{0}
ω_{2}− ω_{1}

ω_{B}
ω_{0} − ω_{0}

ω_{B}

(3.2)

Here, ω_{0} = √

ω_{1}ω_{2} is the center frequency of the bandpass prototype, ω_{2} and ω_{1}
are the upper and lower band-edge frequencies, respectively, and ω_{B} is the bandpass
frequency variable.

Because the coupling elements are frequency invariant, the series resonator circuit itself can be transformed to the lowpass domain by the following steps:

1. Replace all the mutual inductive couplings, provided by transformers, with inverters with the same values as the mutual couplings of the transformers. The inverters then provides the same amount of coupling energy betweeen the resonator nodes as the transformers, and with the same 90° phase change (Figure 3.4).

2. Transform the bandpass network to a lowpass prototype network with the band edges at ω = ±1 by letting the value of the series capacitance go to infinity (zero series impedance).

Figure 3.4: Lowpass prototype equivalent of the bandpass network, as shown in figure 3.3, with inverter coupling elements [2].

Figure 3.4 shows the lowpass filter equivalent circuit built with impedance inverters and FIR (Frequency Invariant Reactive) elements, with series inductances together with inverter couplings. However in our design we will use the dual equivalent circuit to the one shown in Figure 3.4. The corresponding lowpass prototype will be composed of shunt

capacitances, shunt frequency invariant susceptances (FIS) and admittance inverters. In the particular case of the quartet topology explained in figure 3.2. the final bandpass equivalent circuit is represented in Fig. 3.5. In the following sections I will explain how the different element values of this circuit are computed from the synthesized coupling matrix shown in eq. (3.1). Equivalent circuit of the filter from figure 3.5 will have the frequency response shown in figure 3.1.

B=M11 B=M22 B=M33 B=M44

Port 2 1 Ohm Port 1

1 Ohm

Inverter M12

Inverter M23

Inverter M34

Inverter M4L Inverter

Ms1

Inverter M14

Figure 3.5: Circuit model of the filter with parallel resonators and FIS elements, based on figure 3.3 circuit.

In addition, in the process I will show how the complete equivalent circuit can be built using MWO software tool.

### 3.1.2 Resonator elements

First I will describe how the inductance and capacitance of each resonator in the equiv- alent circuit is computed. With the low-pass capacitors (which are 1 in our normalized lowpass prototype filters), we can obtain the bandpass resonator elements by applying the bandpass transformation [2]:

C_{BP} = C_{LP}

BW_{ωBP} = 1

2πBW_{f BP} (3.3)

LBP = BWωBP

ω_{c}^{2}C_{LP} = BWf BP

2πf_{c}^{2} (3.4)

Equations 3.3 and 3.4 can be entered in the filter MWO schematic, using the option in the menu Schematic - Add equation.

In the MWO window we can also find the resonator components in the elem tab, placed in Lumped elements - Resonator (PLC).

Additionally, there is one more component of the resonator node which has not been described yet: the constant susceptance. This component will be modelled as a frequency-invariant admittance, and its susceptance value will be directly the auto- coupling value of the coupling matrix from eq. (3.1).

Individually, each resonant node without the FIS element will resonate at the same
frequency (since C_{1} = C_{2} = C_{3} = C_{4} = C_{BP} and L_{1} = L_{2} = L_{3} = L_{4} = L_{BP}). The auto-
coupling value will make these resonators shift their peak value to their corresponding

3.1. SETTING UP THE FILTERS 27 frequency. Therefore, this component is very important as a part of the equivalent circuit of the resonant node synthesis.

### 3.1.3 The admittance inverter

The next needed component is the admittance inverter. Figure 3.8 shows some general realizations of impedance and admittance inverters. These circuits essentially form in the input port the inverse of the load impedance or admittance.

Figure 3.6: Impedance and admittance inverters [10]. (a) Operation of impedance and admittance inverters [10]. (b) Implementation as quarter-wave transformers [10]. (c) Implementation using transmission lines and reactive elements [10].

The equivalent circuit we are going to use for the admittance inverter is based on three admittance components arranged in π configuration, illustrated in figure 3.7

As we can see from figure 3.6 the input admittance for the admittance inverter is
J^{2}/Y_{L}(where Y_{L} is the load admittance connected at the output of the figure 3.7 circuit).

In the equivalent circuit shown in figure 3.7 it is possible to calculate the admittance values to equal input admittance of the ideal inverter with the input admittance of the

Figure 3.7: Equivalent circuit of the admittance inverter.

circuit shown in figure 3.7.

With a Y_{L} load connected at the end of the figure 3.7 circuit, the analysis of the circuit
applying the classical analysis theory leads to the expression for the input admittance
(Y_{in}):

Y_{in} = Y_{1}+ (Y_{2}+ Y_{L})Y_{3}

Y_{2} + Y_{3}+ Y_{L} (3.5)

Each admittance has the form Y = G + jB, but since we need a lossless network, we
must set G = 0. It is possible to obtain the admittance inverter input by letting Y_{2} = Y_{3}
and Y_{1} = −jB, Y_{3} = +jB. Making these transformations the expression 3.5 results:

Y_{in} = Y_{1}+ Y_{1}Y_{3}+ Y_{L}Y_{3}

Y_{1}+ Y_{3}+ Y_{L} (3.6a)

Yin = −jB + −jB(+jB) + jYLB

−jB + jB + Y_{L} (3.6b)

Y_{in} = −jB +B^{2}+ jYLB

Y_{L} (3.6c)

Y_{in} = −jB +B^{2}

Y_{L} + jB (3.6d)

Y_{in} = B^{2}

Y_{L} (3.6e)

Therefore we can see that B = J leads to the input admittance from figure 3.6.

Figure 3.8 shows the final equivalent circuit of the admittance inverter in MWO.

The admittances can be found in General - Passive - Other - Admittance, in the Elem tab. Figure 3.8 shows that there are two ports in the circuit. The use of schematic and sub-circuits is an efficient way of working with MWO. By declaring the circuit with an input and output port in a separate schematic, MWO hierarchy will allow us to