Treball de Fi de Grau

Grau en Enginyeria de Sistemes de Telecomunicacio

Advanced Filter Response Based OnReflectionless Concept

Daniel Ulinic

Director: Jordi Verdu

Departament de Telecommunicacio i Enginyeria de Sistemes

Universitat Autonoma de Barcelona

2

Bellaterra, Juliol 12, 2018

MTQ

El tribunal d’avaluació d’aquest Treball Fi de Grau, reunit el dia ___________________, ha acordat

concedir la següent qualificació:

President: ___________________________________

Vocal: ___________________________________

Secretari: ___________________________________

RES

Els sotasignants, Jordi Verdú Tirado, Professor de l’Escola d’Enginyeria de la Universitat Autònoma

de Barcelona (UAB).

CERTIFIQUEN:

Que el projecte presentat en aquesta memòria de Treball Fi de Grau ha estat realitzat sota la seva

direcció per l’alumne Daniel Ulinic.

I, perquè consti a tots els efectes, signen el present certificat.

Bellaterra, 25 de Juny de 2018.

Signatura:

ii

Abstract

Modern wireless communications systems require microwave circuits presenting high perfor-

mances in order to provide the end-user multiple services at ultralight data rates. In the case

of RF/Microwave filters, reflectionless configurations became an attractive solution since the

present good matching characteristic not only at the interest frequencies, but in all the frequency

spectrum which improves the overall linearity, efficiency, and reduces instability scenarios at the

system level.

Recently, extraction techniques of circuital coupling parameters from measured or simulated

scattering data have become a focus area of research for investigators of modern microwave

filter synthesis. In this study, the design methodology and equations of a reflectionless filter

are being reproduced while described for low-pass, high-pass and bandpass characteristics with

theoretically perfect input and output match at all frequencies. Often this kind of response is

useful in a wide variety of microwave systems where the poor out-of-band termination may cause

unwanted instability.

Several implementations and tunings were performed and their response was then compared

via digital tools intended for simulations and measuring. Furthermore, output data was analyzed

on all filter topologies and response presented illustrating the practical advantages, while showing

correlation between theoretical, implemented and simulated designs.

Although some contributions have been developed using planar technology or lumped com-

ponents, this configuration has not been applied to the case of acoustic wave resonators, allowing

a bandpass configuration with high selectivity taking advantage on the electrical performances

of such components.

The thesis concludes with the analysis of both the implementation and understanding of

reflectionless filters and shows that there are still technological boundaries to overcome. An

ideal prototype implementation of the bandpass reflectionless filter is proposed with microstrip

transmission lines as well as the first steps forwards introducing this concept with acoustic wave

(AW) resonators.

iii

Resum

Els sistemes de comunicacions inalambrics actuals requereixen de circuits de microones d’altes

prestacions per tal d’oferir a l’usuari final multiples serveis a altes velocitats de transmissio

de dades. En el cas dels filtres de microones/RF, les configuracions sense refleccio esdevenen

una solucio atractiva al presentar un bon acoplament, caracterıstica no nomes a les frequencies

d’interes, sino dins de tot l’espectre de frequencia millorant globalment la linealitat, eficacia, i

reduint escenaris d’inestabilitat a nivell de sistema.

Recentment, les tecniques d’extraccio de parametres circuitals d’acoplament a partir de

mesures o simulacions de parametres de dispersio, han esdevingut una area de recerca de molt

interes pels investigadors de sıntesi de filtres de microones moderns.

En aquest estudi, es reproduiran la metodologia de disseny i equacions d’un filtre sense

refleccio de senyal, a la vegada que s’estudia per a caracterıstiques de filtres pas baix, pas alt

i banda de pas presentant un acoplament perfecte de l’entrada i la sortida del sistema a totes

les frequencies. Sovint aquesta classe de resposta es util en una ampla varietat de sistemes de

microones on les terminacions fora de banda introdueixen inestabilitats indesitjades.

Es presenten diverses implementacions i afinaments i la resposta llavors es compara mit-

jancant eines digitals de simulacio i d’acotament. A mes, les dades de sortida s’analitzen per a

tot el conjunt de topologies de filtre presentades, donant a coneixer els avantatges practics tot

demostrant la correlacio teorica, d’implementacio i de simulacio dels dissenys.

Tot i el desplegament d’algunes contribucions utilitzant tecnologia planar o de components

encastats, s’estudiara aquesta configuracio no aplicada fins ara al cas de ressonadors acustics que

permeten una configuracio de banda de pas d’alta selectivitat tot aprofitant les caracterıstiques

electriques dels mateixos.

La tesi conclou amb l’analisi tant de la implementacio com de l’aprenetatge sobre els filtres

sense refleccio de senyal i dona a coneixer les fronteres tecnologiques a superar. Es proposa la

implementacio mitjancant lınies de transmissio microstrip d’un prototip ideal en la banda de

pas, aixı com els primers passos per tal d’introduir aquest concepte amb ressonadors acustics.

v

Resumen

Los sistemas de comunicaciones inalambricos actuales requieren de circuitos de microondas de

altas prestaciones para ofrecer al usuario final multiples servicios a altas velocidades de trans-

mision de datos. En el caso de los filtros de microondas/RF, las configuraciones sin reflexion

acontecen una solucion atractiva al presentar un buen acoplamiento, caracterıstica no solo a las

frecuencias de interes, sino dentro de todo el espectro de frecuencia mejorando globalmente la

linealidad, eficacia, y reduciendo escenarios de inestabilidad a nivel de sistema.

Recientemente, las tecnicas de extraccion de parametros circuitales de acoplamiento a partir

de medidas o simulaciones de parametros de dispersion, han acontecido una area de investigacion

de mucho interes por los investigadores de sıntesis de filtros de microondas modernos.

En este estudio, se reproduciran la metodologıa de diseno y ecuaciones de un filtro sin

reflexion de senal, a la vez que se estudia para caracterısticas de filtros paso bajo, paso alto y

banda de paso presentando un acoplamiento perfecto de la entrada y la salida del sistema a todas

las frecuencias. A menudo, esta clase de respuesta es util en una amplia variedad de sistemas

de microondas donde las terminaciones fuera de banda introducen inestabilidades indeseadas.

Se presentan varias implementaciones y afinamientos y la respuesta entonces se compara

mediante herramientas digitales de simulacion y de acotamiento. Ademas, los datos de salida se

analizan para todo el conjunto de topologıas de filtro presentadas, dando a conocer las ventajas

practicas demostrando la correlacion teorica, de implementacion y de simulacion de los disenos.

Dado el despliegue de algunas contribuciones utilizando tecnologıa planar o de componentes

encastados, se estudiara esta configuracion no aplicada hasta ahora en el caso de resonadores

acusticos que permiten una configuracion de banda de paso de alta selectividad aprovechando

las caracterısticas electricas de los mismos.

La tesis concluye con el analisis tanto de la implementacion como del aprendizaje sobre los

filtros sin reflexion de senal y da a conocer las fronteras tecnologicas aun por superar. Se propone

la implementacion mediante lıneas de transmision microstrip de un prototipo ideal en la banda

de paso, ası como los primeros pasos para introducir este concepto con resonadores acusticos.

vii

Acknowledgements

Thanks to Jordi Verdu Tirado, for supervising and guiding me through carrying out this project

and all the references and articles he suggested which were nevertheless helpful and essential to

get the work within this project done. Moreover, I would like to be thankful to all the teachers

I came across over this past years. It was a pleasure working with all of them and sure thing I

have learned a lot. Finally, I would like to definitely thank my parents for supporting me and

understanding the late hours and the skipped relevant appointments.

June 2018,

Dani

ix

Contents

Abstract iii

Resumen v

List of Figures xvii

List of Tables xix

1 Introduction 1

1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 State-of-the art 5

2.1 Conventional Filter Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Transversal Filter Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Filters based on Bulk Acoustic Wave Resonators . . . . . . . . . . . . . . . . . . 8

3 Passive Microwave Networks Concepts 11

3.1 N-port Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Filter Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.1 Impedance Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.2 Frequency Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

xi

xii Contents

4 Lumped Reflectionless Filter 15

4.1 Reflectionless Principles Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Lumped Element Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.4 Cascade Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4.1 Cascading High- and Low-Order Low-Pass Filters . . . . . . . . . . . . . 22

4.4.2 Cascading Equal Order Low-Pass Filters . . . . . . . . . . . . . . . . . . . 23

4.5 Design of the band-pass reflectionless filter . . . . . . . . . . . . . . . . . . . . . . 24

5 Distributed Reflectionless Filter 27

5.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.2 Coupled Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Application for the Reflectionless Filters . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4 Cascade Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4.1 Cascading Equal Order Filters . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4.2 The Multi-band bandpass filter topology . . . . . . . . . . . . . . . . . . . 35

6 The Acoustic Wave (AW) Reflectionless Filter 39

6.1 The Low-pass prototype based on AW resonators . . . . . . . . . . . . . . . . . . 39

6.2 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.3 Cascade Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Conclusions And Future Work 45

Bibliography

Annex

Contents xiii

A iii

A.1 Determining a Filter’s Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

B v

B.1 Frequency Scaling Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . v

xiv Contents

List of Figures

2.1 Low-pass laddernetwork topology. Elements are labeled with normalized values. . 5

2.2 (a) Coupling scheme transversal filter. [Feh76] (b) Proposed dual-passband

transversal filtering section. [GG09] . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 (a) Illustrative example of the synthesized power transmission response for the

dual-passband transversal filtering section. (b) Power transmission and reflection

responses of an ideal synthesized dual-passband filter from [GG09] . . . . . . . . 7

2.4 (a) Schematic view of the Surface Acoustic Wave Resonator (b) Electrical input

impedance of an acoustic wave resonator from [Ver10] . . . . . . . . . . . . . . . 8

2.5 BAW Ladder-type filter configuration and its respective transmission response

from [Ver10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 N-port network schematic[Pac15, chap. 11.1]. . . . . . . . . . . . . . . . . . . . . 11

3.2 Transfer functions for all four basic third-order filter types. (a) Low-pass. (b)

High-pass. (c) Band-pass. (d) Band-stop. (∆ = 0.5)[Mor15] . . . . . . . . . . . . 13

4.1 Two-port symmetric network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Dual high-pass Cauer topology filters, circuits used in derivation of a reflectionless

low-pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Dual high-pass circuits modification to satisfy symmetry condition. . . . . . . . . 17

4.4 (a) Complete reflectionless low-pass filter of arbitrary order. (b) Final reflection-

less topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 Low-pass 5th order reflectionless filter circuital implementation. . . . . . . . . . 19

4.6 Low-pass 5th order reflectionless filter response. . . . . . . . . . . . . . . . . . . 20

4.7 Low-pass 7th order reflectionless filter. . . . . . . . . . . . . . . . . . . . . . . . 20

xv

xvi List of figures

4.8 Low-pass 7th order reflectionless filter response. . . . . . . . . . . . . . . . . . . 21

4.9 Comparison to previous lower-order filter. . . . . . . . . . . . . . . . . . . . . . 21

4.10 Cascade of high- and low-order low-pass filters. . . . . . . . . . . . . . . . . . . 22

4.11 Cascade of high- and low-order low-pass filters response. . . . . . . . . . . . . . 22

4.12 Cascade of equal order low-pass filters. . . . . . . . . . . . . . . . . . . . . . . . 23

4.13 High frequency selective filter response result of cascading equal order structures. 23

4.14 Cascade of same order low-pass filters. . . . . . . . . . . . . . . . . . . . . . . . . 24

4.15 Bandpass reflectionless filter response. . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Richard’s transformation (a) Inductor to short-circuited stub. (b) Capacitor to

open-circuited stub. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 The four Kuroda Identities (n2 = 1 + Z2Z1) . . . . . . . . . . . . . . . . . . . . . . 29

5.3 Coupled Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4 (a) Dual terminated low-pass filters as even- and odd-mode equivalent circuits. (b)

Application of Richard’s transformations converting reactive elements to trans-

mission line stubs and the resistive terminations matched by a quarter-wave trans-

mission line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.5 (a) Application of Kuroda’s identity on both sides to separate the stubs and con-

vert most of the series stubs into parallel stubs. (b) Order of the series stub at the

bottom of the even-mode is exchanged with the termination resistor, grounding

the series stub, making it a shunt stub instead. The first parallel stub on the

odd-mode is also virtually grounded turning it into a series connected stub. . . . 31

5.6 Three-port network transmission-line identity used in derivation for maintaining

the reflectionless concept from [Mor15] . . . . . . . . . . . . . . . . . . . . . . . . 31

5.7 (a) Addition of pertinent elements in order to maintain symmetry. (b) Final

reflectionless topology after application of the coupled-line identity in Figure 5.6. 32

5.8 Ideal transmission lines bandpass filter implementation. . . . . . . . . . . . . . . 33

5.9 Ideal transmission lines bandpass filter response. . . . . . . . . . . . . . . . . . . 33

5.10 Cascading of ideal transmission lines bandpass filter implementation. . . . . . . . 34

5.11 Cascading of ideal transmission lines bandpass filter response. . . . . . . . . . . . 34

List of figures xvii

5.12 Multiband filter response, cascading high- and low-frequency ideal transmission

lines bandpass filterss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.13 Microstrip transmission lines bandpass filter implementation. . . . . . . . . . . . 36

5.14 Microstrip transmission lines bandpass filter implementation response. . . . . . . 37

5.15 Parameters values where Z1 = 1146.843Ω and Z2 = 2.180Ω . . . . . . . . . . . . . 37

5.16 ADS microstrip coupled lines schematics illustration. . . . . . . . . . . . . . . . . 37

6.1 Nodal representation (left), Equivalent circuit schema (center) and lowpass BVD

model (right) for the series AW resonator. . . . . . . . . . . . . . . . . . . . . . . 40

6.2 Basic configuration for the lowpass reflectionless prototype based on AW resonators. 40

6.3 Transmission response S21 for Figure 6.2 and electrical input impedance of the

AW resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.4 Complementary configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.5 Transmission response S21 for Figure 6.4 and electrical input impedance of the

AW resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.6 Transmission response S21 for the concatenated sections. Dashed line corresponds

to the configuration in Figure 6.2 whereas the solid line to the complementary

implementation, Figure 6.4, and gray symbol line with the concatenated structure. 43

6.7 Transmission response S21 of the bandpass filter and input impedance of the

fabricated AW resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.8 Effect of the phase unbalance on the return losses for φi =0.5, 1, 2, 5 and 10. 44

B.1 Summary of Prototype filter Transformations (∆ = w2−w1w0

) . . . . . . . . . . . . v

xviii List of figures

List of Tables

A.1 Element Values for Equal-Ripple Low-Pass Filter Prototypes (g0=1, wc=1, N=1

to 10, 0.5 dB ripple) [Poz05] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

xx List of tables

Chapter 1

Introduction

1.1 Motivation and Objectives

The use of microwave resonators can be extended to a large number of devices such as filters,

oscillators, frequency meters and tuned amplifiers. The common characteristic among most

microwave resonators is that the excitation is by means of an electromagnetic wave. Therefore,

the size of the devices is directly related to the wavelength and propagation velocity of the

electromagnetic wave at a certain frequency.

Filters have been for the better part of the century a great area of research. They are impor-

tant components in virtually all electronic systems, from communications to radio astronomy.

Moreover, practical filters design and implementation has remained one of the most active fields

of study in the electronics community today. Therefore, the innovative discovery of the reflec-

tionless or absorptive filters capable of adapting the signal out-of-band rather than reflecting it

back to the source, with port impedances theoretically constant at all frequencies, has gained

much interest.

Conventional filters are well-known to provide a highly reactive out-of-band termination

which is often undesirable. However, in the rejection region the input impedance is usually

reactive, reflecting the signal while negatively affecting the performance of active devices such

as amplifiers or frequency-conversion mixers in terms of instability, linearity or efficiency.

Therefore, this study is intended to reproduce, for a better understanding, the analysis,

design, implementation and simulation of a reflectionless bandpass prototype filter leading to

further complex Acoustic Wave (AW) resonators filters’ investigation. Advantages of meeting

the reflectionless concept, in terms of the system point of view, are those such as not having

loaded amplifiers thereby, increasing their power or the absence of components by means of

isolation preceding active elements.

1

2 Chapter 1. Introduction

Usually, filters design process and other microwave circuit imply a high percentage of the

total time dedicated to implementation. Since equations in this field are all approximations,

once the filter is designed, often the response is optimized by tuning different parameters or

layout dimensions manually. Therefore, tools such as Advanced Design System (ADS) or Matlab

used upon this study, automate most of the parametrization process. Hence the simulated and

theoretical prototype response are as optimal and matching as possible.

In order to achieve the means of this thesis, some previous requirements and knowledge are

to be taken into account:

• Understanding of microwave filter structures, specially those the study is theoretically

based on.

• Understanding of the reflectionless concept and its principles, designs, implementation and

applications.

Finally, the main goals are the following:

• To implement reflectionless filter topology based on different derived principles and the

diverse solutions proposed for research purposes such as lumped-elements, ideals and mi-

crostrip transmission lines networks.

• To verify if a more sophisticated frequency response is achieved while retaining the reflec-

tionless concept.

• To investigate new applications of the reflectionless concept onto filters based on acoustic

waves (AW) resonators.

• To physically implement the prototype of a reflectionless filter using microstrip transmis-

sion lines.

1.2 Thesis outline

In this chapter, the motivation and objectives of the study are presented as well as the main

applications given nowadays to microwave filters. The advantages of reaching implementations

based on acoustic wave resonators are described in terms of fabrication processes given the

singular properties of its resonance wave and propagation velocity.

Chapter 2 includes the discussion over other current microwave filters implementations and

usages. A variety of state-of-the-art solutions are described such as conventional or traditional

topologies, transversal or based on acoustic waves resonators’ designs. Hence, this will be leading

to the correlation and research within this study.

1.2. Thesis outline 3

In Chapter 3, the main concepts of passive microwave filters are reviewed in order to give a

better understanding and context for further analysis.

Chapter 4 covers the design and solutions procedure for a bandpass response by taking

advantage on the properties of reflectionless structures. Then, for validation purposes, low

pass and bandpass filters are being analyzed using implementations and simulations with the

correspondent responses.

Chapter 5, prototypes of the conversion from structures based on lumped-elements realiza-

tions to transmission-lines designs are introduced and explained.

Chapter 6, the prototype of a reflectionless bandpass filter based on AW resonators is pro-

posed while describing the application of the reflectionless concept upon derivate topologies.

Advantages of such designs and the obtained simulation results are outlined as well.

Finally, the conclusions and future line of work are presented.

Chapter 2

State-of-the art

2.1 Conventional Filter Topologies

The most common topology used for lumped-element filters is the laddernetwork, or Cauer

topology, consisting in alternating series and shunt elements as shown in Figure 2.1

Port 1 Port 2

g1

g2

g3

g4

g5

g6

g7

Figure 2.1: Low-pass laddernetwork topology. Elements are labeled with normalized values.

where gk are the normalized values of the elements from left to right. In order to have a

filter with a cutoff frequency wc, designed to be matched in the pass-band to a characteristic

impedance of Z0, each capacitor and inductor has to be set to:

Ck =gkY0wc

, Lk =gkZ0

wc. (2.1)

Table A.1 from Appendix A, gives the normalized element values for a number of filters

having different numbers of poles and ripple factors thus determining filter’s order. This table

parameters relate to filters of special significance when it comes to the reflectionless topology

analyzed within this study. Notice that the load impedance gN+1 6= 1 for even N.

Butterworth filters can also be implemented using the Cauer topology, where different ele-

ment values are assigned so as to move the poles further away from the jw axis.

In order to create zeros at finite frequencies, capacitors in parallel with the series inductors

need to be added, or vice-versa.

5

6 Chapter 2. State-of-the-art

2.2 Transversal Filter Designs

The minimization of the co-channel, the adjacent channel and the intersymbol interference (ISI)

is often essential in data communication. Common contribution to these system weaknesses is

the out-of-band energy of the signal ergo at frequencies above and below the specified signal

bandwidth given in terms of theoretical minimum Nyquist restriction. Hence the need of an

efficient use of the electromagnetic spectrum in modern communication systems, therefore it is

required the development of high selective frequencies components.

Working with analog passive or active filter designs, a minimum bandwidth channel with a

low ISI is hardly achievable. Furthermore, analog filters are not easily adaptable for fluctuating

data rates, a clear drawback in terms of flexibility when required. This need of rejecting certain

unwanted frequencies, leads to the development of microwave filters whose insertion loss response

have zeros at finite frequencies. The main difference with respect to conventional filter topologies,

is that in transversal filters the input signal is coupled at the same time to multiple resonators,

as shown in Figure 2.2 (a). The solid lines represent coupling between the input/output ports

and resonators, while dashed the line represents coupling between the ports.

Figure 2.2: (a) Coupling scheme transversal filter. [Feh76] (b) Proposed dual-passband transver-

sal filtering section. [GG09]

This kind of filter network, by means of transversal signal-interference concepts, allow dual-

band bandpass filtering actions to be generated as well. Despite the amount of solutions currently

available, there are some limitations in this field which have not been entirely overcome. For

instance, filters which designs are based on coupled-resonators circuits make implementations

difficult when it comes to feasibility of the minimum inter-coupled-line space separation through

the circuit manufacturing process. Also, designs methodology of filters presenting transmission

zeros at both sides of each passband have not achieved full functional implementation.

In [GG09] a new signal-interference technique consisting of transversal filtering sections

shaped by two transmission lines connected in parallel is described. The principle, on which this

transversal combination of two signal components derived from the input signal, aims to achieve

2.2. Transversal Filter Designs 7

a dual-passband filtering behavior, once propagated by its two constitutive transmission lines.

This proposed dual-passband transversal filtering is shown in Figure 2.2 (b) where Z1, Z2 are

the characteristic impedances and θ1, θ2 represent the electrical lengths of the line segments.

Furthermore, these characteristic impedance variables allow to control the transfer function

performances by shaping the dual-passband filtering responses. An illustrative example of the

power transmission curve of this kind of transversal filtering sections is next presented in Figure

2.3 (a). It corresponds to a Type I filter solution for its design parameters and the produced

dual-band center frequencies derivation represent it.

The concept of cascading a number of identical sections is also described by [GG09] in order

to achieve high selective frequencies components. Thereby, some features are worth mention-

ing, first, an increase of the out-of-band power rejection level and second, an insertion loss

minimization. Figure 2.3 (b) shows a representation of such facts.

Figure 2.3: (a) Illustrative example of the synthesized power transmission response for the dual-

passband transversal filtering section. (b) Power transmission and reflection responses of an

ideal synthesized dual-passband filter from [GG09]

8 Chapter 2. State-of-the-art

2.3 Filters based on Bulk Acoustic Wave Resonators

The exponential growth in wireless communications systems recently has led to requirements

of small high performance microwave devices. Actual implementations based on traditional

technologies are limited in terms of manufacturing process of standard integrated circuits (IC).

Hence, microwave devices based on acoustic resonators overcome this boundary by being com-

patible with standard IC technologies. Since they are excited by means of an acoustic wave with

propagation velocity four or five orders lower than the propagation velocity of electromagnetic

waves, this allows lowering device’s size.

The most basic configuration of a Bulk Acoustic Wave Resonator (BAW) consists of a piezo-

electric layer interleaved between two metal electrodes, as in a parallel plate capacitor, where

the electric field is generated in the direction of thickness, thereby exciting the acoustic wave.

Figure 2.4 (a) represents it.

As for the electrical performance of the acoustic wave resonators, it is characterized by the

presence of two resonances. The first one is the frequency, fr, where the magnitude of the elec-

trical impedance tends to its minimum as well as its leading the resonator to behave as an open

circuit at the center of the structure. On the other hand, there is the antiresonance frequency

fa where the magnitude of the electrical impedance tends to infinity as well as representing a

short circuit behavior. In Figure 2.4 (b) fr and fa can be appreciated in more detail.

Figure 2.4: (a) Schematic view of the Surface Acoustic Wave Resonator (b) Electrical input

impedance of an acoustic wave resonator from [Ver10]

Lateral standing waves in the electrical behavior produce on the acoustic wave (AW) res-

onators spurious or unwanted resonances. The degradation due to these spurious modes is widely

discussed and out of the scope of this study. Finally, as for filters based on AW resonators the

conventional laddernetwork is composed of consecutive series and shunt AW resonators. Its

response is of high selectivity due to the presence of two notches, although presenting a poor

2.3. Filters based on Bulk Acoustic Wave Resonators 9

out-of-band (OoB) rejection which can be improved by increasing the order of the filter. A rep-

resentation of the laddernetwork and its transmission response can be appreciated in the left-

and right-hand section, respectively, of Figure 2.5.

Figure 2.5: BAW Ladder-type filter configuration and its respective transmission response from

[Ver10]

Chapter 3

Passive Microwave Networks

Concepts

Before intending to derive to a reflectionless filter, it would be useful to go over some of the fun-

damental principles of microwave networks. This chapter is a brief overview of terms regarding

scattering parameters and their mathematical relationship to impedance. Finally, transmission-

line concepts will be described.

3.1 N-port Networks

Usually the point of view of a N-port network is represented by N terminals to which the

input and output transmission lines are connected as shown in Figure 3.1.

N-Port

Network

Figure 3.1: N-port network schematic[Pac15, chap. 11.1].

Therefore, the relationships of each network access is as follows:

[b] = [S][a] (3.1)

11

12 Chapter 3. Methodology

Where [S] is the matrix of the scattering parameters, generally complex, defining the linear

circuit while [b],[a] are linear combinations of normalized voltage and current. The characteristic

impedance is illustrated as Z0.

Furthermore, the scattering parameters, sij , are the ratio of the outgoing wave amplitude at

port i divided by the ingoing wave amplitude at port j. For instance,

s11 =b1

a1. (3.2)

In order for s11 to not be complex both a1 and b1 need to have the same phase or opposite

one. Since the input impedance Zin of the network is given by the ratio of total voltage and

current at port 1, solving for parameter s11 in terms of impedance is as follows

s11 =Zin − Z0

Zin + Z0. (3.3)

This relationship between the impedance and complex reflection coefficient is usually repre-

sented in a Smith chart [Nik05]. Meanwhile the reflected and transmitted powers are typically

expressed in logarithmic units, decibels (dB).

3.2 Duality

The concept of duality is based on the symmetrical relationship between the electric and

magnetic field in Maxwell’s equations, and duals result from interchanging the two. Therefore,

for almost every quantity or concept in electromagnetic theory and electronic circuits, there

exists a dual. Moreover, voltage is the dual of current, inductance is the dual of capacitance,

resistance is the dual of conductance and so on. It is worth mentioning that this ensures both

modes will have equal amplitude reflection coefficients but with the opposite sign.

3.3 Filter Transformations

Filters are normally categorized as being one of the following response types: low-pass, high-

pass, band-pass and band-stop as show in Figure 3.2. Take into account networks were designed

such as all the prototype parameters having unit value.

In order to convert any of the given low-pass filters to high-pass, band-pass or band-stop one

needs to employ well-known impedance and frequency scaling.

3.3. Filter Transformations 13

Figure 3.2: Transfer functions for all four basic third-order filter types. (a) Low-pass. (b)

High-pass. (c) Band-pass. (d) Band-stop. (∆ = 0.5)[Mor15]

3.3.1 Impedance Scaling

Impedance scaling needs to be applied in the prototype design. The source and load resistances

are unity (except for equal ripple filters with even N, which have no unity load resistance).

Therefore a source resistance of R0 can be obtained by multiplying all the impedances of the

prototype design by R0. Hence, the new filter component values would be as follows:

L′ = R0L, C ′ =C

R0(3.4)

where L, C are the component values for the original prototype.

3.3.2 Frequency Scaling

On the other hand, in order to change the cutoff frequency of a low-pass prototype from unity

to wc, it is required that the frequency dependence of the filter is scaled by the factor 1wc

, which

is accomplished by:

w ← w

wc(3.5)

Targeting a low-pass transfer function Figure B.1 from Annex B, provides appropriate transfor-

mations in order to create high-pass, band-pass and band-stop implementations where w1 and

w2 denote the edges of the passband and w0 the resonant frequency.

Chapter 4

Lumped Reflectionless Filter

In this chapter, the reflectionless filters derived from principles under the assumption that the

structure would be symmetric following [Mor11] and prototypes are presented alongside the

simulations and results obtained in order to verify the theory.

4.1 Reflectionless Principles Derivation

Introducing the reflectionless concept, if based on an arbitrary symmetric two-port network

Figure 4.1 illustrates it. Thus both ports are excited simultaneously with equal signal amplitudes

and matching phase, there can be no currents crossing from one side of the symmetry plane to

the other. This is called the even mode. Otherwise, with equal amplitude but 180 out-of-

phase, then all nodes on the respective symmetry plane must have zero potential with respect

to ground. This is called the odd mode. Moreover, filter’s response can be fully described to

these two standard excitations, respectively.

Port 1 Port 2

Figure 4.1: Two-port symmetric network.

Symmetry allows the network to be divided in half, assigning either one an opened or shorted

circuited behavior. In this case, there would be an even-mode equivalent circuit and an odd one,

respectively.

15

16 Chapter 4. State-of-the-art

The scattering parameters for this two-port structure then would be given as the superpo-

sition of the reflection coefficients of the even and odd mode equivalent circuits, whereas the

transmission coefficient will be proportional to the difference. This is as follows:

s11 = s22 =1

2(Γeven+ Γodd) (4.1)

s21 = s12 =1

2(Γeven− Γodd) (4.2)

Moreover, for the circuit to be reflectionless (s11 = 0), the condition for the perfect in-

put match is derived from ensuring the even- and odd-mode reflection coefficients are equal in

amplitude and opposite in sign.

Γeven = −Γodd (4.3)

Finally, the transfer function of the two-port network is given directly by the even-mode

reflection coefficient if (4.1) and (4.2) are combined.

s21 = Γeven (4.4)

In terms of the reflectionless concept, if the even- and odd- mode discussed are duals to each

other, it can be said that normalized even-mode input impedance and odd-mode admittance,

zodd = yeven, or vice versa, are equivalent.

4.2 Lumped Element Realization

The original reflectionless filter is derived from first principles under the even-odd mode analysis.

Meaning, the structure would be a two-port network which can be fully described by its response

to two standard excitations.

In order for the circuit to be reflectionless (s11 = 0), the even- and odd-mode reflection

coefficients need to be equal in amplitude and opposite sign. The symmetry plane therefore is

either an open or short circuit, respectively. Thereby, fulfilling equation 4.4, the low-pass transfer

characteristic is obtained by using a high-pass filter with one end terminated in the even-mode

circuit. Moreover, the odd-mode equivalent circuit attending to the duality condition, where

impedances are inverses of each other, is also illustrated in the right-hand side of Figure 4.2.

The circuit dual is therefore the exchange of inductors with capacitors, or vice-versa.

Although the identically zero reflection coefficient is provided, these two circuits need to

accomplish as well the symmetry condition in order for equations 4.1 and 4.2 to be valid. The

steps onto doing so are next described.

4.2. Lumped Element Realization 17

.

.

.

.

.

.

Even-Mode

Equivalent Circuit

Odd-Mode

Equivalent Circuit

Figure 4.2: Dual high-pass Cauer topology filters, circuits used in derivation of a reflectionless

low-pass filter.

In the analysis of symmetric networks, both parts are typically simplified by eliminating

inactive components such as open-ended or shorted-out elements and by alterations which would

have no effect on the circuit behavior.

First, open-circuited elements are added to the even-mode side. An inductor is placed

between the input node of this mode and the symmetry plane, whereas representing an open

circuit would not be affecting the response provided. Second, exchanging positions of the series

resistor and capacitor while directly connecting the node between them to the symmetry plane

will turn into an open circuit.

On the other hand, in the odd-mode equivalent circuit there would be changes from absolute

to virtual ground of some elements. The first inductor and the termination resistor are connected

this way, while a capacitor is added from the symmetry plane to ground which would be shorted

at both ends (virtual and absolute ground) and therefore inactive.

.

.

.

.

.

.

Even-Mode

Equivalent Circuit

Odd-Mode

Equivalent Circuit

1

4

2

3

5 6

Figure 4.3: Dual high-pass circuits modification to satisfy symmetry condition.

18 Chapter 4. State-of-the-art

All of the above alterations, as mentioned, would not affect the behavior of either equivalent

circuits. The final symmetric reflectionless topology either is of an arbitrary order or two-port

circuit are shown in Figure 4.4 (a) and Figure 4.4 (b), respectively.

.

.

.

.

.

.

⇒

La1=2 La1=2

Lb2 Lb2

C1

R

Cn2

C1

R

Cn2

Figure 4.4: (a) Complete reflectionless low-pass filter of arbitrary order. (b) Final reflectionless

topology.

Since symmetry and duality constrain the circuit elements values, the first element on the

even-mode must be complementary to the first one on the odd-mode and so on. As for the

termination resistors R, since they have to be their own dual, R = 1R = 1 = Z0, in terms of

simplest normalized form (w2c = LC) where elements have unit value and wc is the frequency of

attenuation zero. The values constrained to duality and symmetry condition are as follows:

C1 =1

Z20

· La12

; Lb2 = Z20 · C1; Cn2 =

1

Z20

· 2Lb2; R = Z0 (4.5)

Additional flexibility can be obtained by adding inductors between the circuit nodes and

the symmetry plane. Similarly, elements of any type may be added in series with the final

capacitor while conveniently balancing those with components just before the dual end resistor

terminations.

Ck =1

Z20

· La12

; Lb,k = Z20 · Ck−1; Cn+1 =

1

Z20

· 2Lb,k+1 (4.6)

4.3. Design Examples 19

4.3 Design Examples

In order to achieve and experiment with reflectionless filter topologies, low-pass distinct network

structures are presented. It is worth mentioning that the values of each element should be

coherent with those from Table A.1. Thereby, if there was to be an implementation of a third

order reflectionless low-pass filter, the third row of values ought to be selected. Attending to

the constraint of the values previously described, following reflectionless designs is not based on

this criteria.

A first approach to a circuit implementation for a low-pass filter, in this case of order N = 5,

is shown in Figure 4.5. The simulated frequency responses, describing a stopband peak of -14.47

dB, are illustrated in Figure 4.8.

Figure 4.5: Low-pass 5th order reflectionless filter circuital implementation.

20 Chapter 4. State-of-the-art

Figure 4.6: Low-pass 5th order reflectionless filter response.

Furthermore, by increasing the order of the filter to N = 7, a new former topology is

presented with its correspondent transmission response, S21. This is illustrated in Figure 4.7

and 4.8, respectively.

Figure 4.7: Low-pass 7th order reflectionless filter.

From the obtained different order reflectionless structure responses, it can be stated that

4.4. Cascade Combinations 21

Figure 4.8: Low-pass 7th order reflectionless filter response.

the higher the order of the filter, the sharper the cutoff characteristics are obtained. Moreover,

higher out-of-band peaks are developed. This two topologies’ comparison is illustrated in Figure

4.9 for a better visualization. Also as a result, it seems unlikely to obtain better out-of-band

rejection by increasing the order of the filter.

Figure 4.9: Comparison to previous lower-order filter.

4.4 Cascade Combinations

For the low-pass filter structures, it has been seen that by increasing the order, responses de-

rived often manifest as a degradation of out-of-band rejection at high-frequency. As a solution,

by cascading reflectionless filters, the combination is actually the sum of its parts since the

interaction of stop-bands does not prevent cascades from behaving as predicted. Therefore, the

well-matched stop-band could extend to very high-frequencies.

22 Chapter 4. State-of-the-art

4.4.1 Cascading High- and Low-Order Low-Pass Filters

Cascading a sharp cutoff high-order design (N = 7) with a lower fourth order filter, allows a

well-matched stop-band to extend to higher frequencies, as it can be appreciated from Figure

4.11.

Figure 4.10: Cascade of high- and low-order low-pass filters.

Figure 4.11: Cascade of high- and low-order low-pass filters response.

4.4. Cascade Combinations 23

4.4.2 Cascading Equal Order Low-Pass Filters

By cascading three sixth order reflectionless filters, sharper cutoff characteristics are obtained

as well as higher out-of-band peaks since each cascaded section will provide an additional value

of its individual return losses of rejection in the stop-band.

Figure 4.12: Cascade of equal order low-pass filters.

Figure 4.13: High frequency selective filter response result of cascading equal order structures.

24 Chapter 4. State-of-the-art

4.5 Design of the band-pass reflectionless filter

Similar statements as described previously can also be applied for bandpass filters, which can be

derived in a similar manner. The signal in this case takes the path through the filter illustrated

in Figure 4.14.

Figure 4.14: Cascade of same order low-pass filters.

4.5. Design of the band-pass reflectionless filter 25

When a signal is in the filter’s passband, it passes directly through from one port to the

other, whereas if present in the stopband, it is blocked from following this path, and instead is

routed directly to the dual end absorbing resistor terminations. The response of this bandpass

implementation is shown in Figure 4.15 with a central cutoff frequency of 2.4 GHz.

Figure 4.15: Bandpass reflectionless filter response.

Chapter 5

Distributed Reflectionless Filter

This chapter expands on the previously explained reflectionless filters, having zero reflection

coefficient at all frequencies, by introducing the derivation of lumped-element to include trans-

mission line prototypes as in [Mor15]. Therefore, this is achieved under the assumption that

both left and right out-of-band terminations are coupled to each other using the same previously

described two-port network.

5.1 Derivation

5.1.1 Transmission Lines

Transmission lines are a fundamental aspect of a microwave engineer’s perspective of electronic

circuits, hence the scattering parameters are assumed to work when those are connected to the

ports. Basic concepts are to be explained in this subsection.

Waves in transmission lines, which are characterized by voltage and current amplitudes on

the conductors, are propagating from one end to the other and reflecting off of the terminating

impedances at either end. The current is to be defined as positive in the direction of the

propagation. The length is therefore defined by the exponential solutions of the Helmholtz

equation. If there were to be port 1 located at x = 0, and port 2 at x = l, l is the length of

the line. In this study, all transmission-line sections and stubs will be assumed to be a quarter-

wavelength long at the designed center frequency (w0 = 2.4GHz). Since the load impedance is

finite (Z = 50Ω), but the line λ4 at w0, then in all design cases the length is, βl = π

2 .

The lumped-element components allow filters to be designed and work well generally at

low frequencies. Thus, two problems arise at higher RF and microwave frequencies. First,

components such as inductors and capacitors are only available for a limited range of values.

27

28 Chapter 5. State-of-the-art

Secondly, open- or short-circuited transmission line stubs are used to approximate these ideal

lumped elements.

Richard’s Transformation solves the first problem by transforming the reactive elements

to transmission-line stubs or sections. The transformation

Ω = tanβl = tan(wlνp

) (5.1)

maps the w plane to the Ω plane, repeating periodically bywp

νp= 2π. Therefore, by replacing

the frequency, the reactance of an inductor can be defined as

jXL = jΩL = jLtanβl (5.2)

whereas the susceptance of a capacitor as

jBC = jΩC = jCtanβl. (5.3)

These clarifies the replacement of an inductor with a short-circuited stub while the capacitor

is replaced with an open-circuited one. Both inductors and capacitors having a characteristic

impedance of L and 1C , respectively. This can be seen in Figure 5.1

Figure 5.1: Richard’s transformation (a) Inductor to short-circuited stub. (b) Capacitor to

open-circuited stub.

Kuroda’s Identities are a total of four identities all allowing the use of transmission line

sections to achieve more practical microwave filter implementations. The idea stands on the

conversion of series-connected short-circuited stubs to shunt-connected open-circuited stubs,

and vice versa as shown in Figure 5.2. Each box represents a unit element, or transmission line.

There are the so called unit elements lines which are no more than equivalent with the stubs

used to implement the inductors and capacitors of the prototype design.

5.1. Derivation 29

Figure 5.2: The four Kuroda Identities (n2 = 1 + Z2Z1)

5.1.2 Coupled Transmission Lines

If two microstrip lines are brought very close together in parallel (Figure 5.3), the wave propa-

gating on one line can induce waves propagating in the other. When this happens it is said that

both lines are coupled. This results from mutual inductance and capacitance between the two

of them. Theoretically, these lines can be characterized via an even- and odd-mode analysis.

There exists a symmetry plane down the middle of the gap between the two lines which acts

as an electric or magnetic boundary condition very near both equivalents circuits forming. It

alters the characteristic impedance and more often the propagation constant.

In order to find the full four-port scattering parameters of the coupled lines, superposition

must be applied to the response of the even - and odd-mode equivalent circuits, resulting in

equations similar to 4.1 and 4.2.

Port 1

Port 3

Port 2

Port 4

Z0o; Z0e

Figure 5.3: Coupled Transmission Lines

30 Chapter 5. State-of-the-art

5.2 Application for the Reflectionless Filters

In order to enhance circuit dimensions, as the frequency increases, the lumped-element model

results in narrow band and low quality factor behaviors. Therefore, a solution with transmission

lines is presented according to the concepts described in the previous chapter. In order to

keep the already implemented and studied model results, Richard’s Transformation comes in

handy while it synthesizes the equivalence in impedance of an LC network using open- and

short-circuited transmission line stubs on different frequency domains.

This derivation starts with the previous analyzed even- and odd-mode equivalent circuits,

duals to each other. Thereby, applying Richard’s transformations, the reactive elements are

converted to transmission lines, where the center frequency of the band is that for which the

transmission line is a quarter wavelength long. For this reason, the resistances at both ends of

each equivalent circuit are also offset by a quarter-wavelength matched transmission line. These

steps so far are illustrated in Figure 5.4, respectively.

R

L

C

R

L

C

Even-Mode

Equivalent Circuit

Odd-Mode

Equivalent Circuit

⇒

R

R

Even-Mode

Equivalent Circuit

Odd-Mode

Equivalent Circuit

unitelement

unitelement

Figure 5.4: (a) Dual terminated low-pass filters as even- and odd-mode equivalent circuits. (b)

Application of Richard’s transformations converting reactive elements to transmission line stubs

and the resistive terminations matched by a quarter-wave transmission line.

From Richard’s transformation direct application, most of the lumped-element circuits are

now either series- or parallel-connected stubs. This can raise several issues. First, it becomes

difficult to realize the circuit with any transmission line geometries like microstrip, for instance.

Second, since the stubs terminations coincide with the connection points of a lumped-element

structure, coupling of adjacent stubs becomes challenging. In order to convert series-connected

short-circuited stubs to shunt-connected open-circuited stubs, and vice versa, at the other end of

an equal length transmission line, the most common way to do this is applied by substitution of

transmission line identities. Thereby, Kuroda’s identity is applied to do so while also swapping

5.2. Application for the Reflectionless Filters 31

the series stub and termination resistor on the even-mode side, effectively achieving a shunt

stub. By also changing the grounding on the odd-mode from absolute to virtual, a series stub

is achieved as well. Figure 5.5 is a snapshot of the steps just explained.

R

R

Even-Mode

Equivalent Circuit

Odd-Mode

Equivalent Circuit

⇒

R R

Even-Mode

Equivalent Circuit

Odd-Mode

Equivalent Circuit

Figure 5.5: (a) Application of Kuroda’s identity on both sides to separate the stubs and convert

most of the series stubs into parallel stubs. (b) Order of the series stub at the bottom of the

even-mode is exchanged with the termination resistor, grounding the series stub, making it a

shunt stub instead. The first parallel stub on the odd-mode is also virtually grounded turning

it into a series connected stub.

Furthermore, in order to restore symmetry, in Figure 5.7 (a) matching changes are made

and thus achieving a fully symmetric circuit. In terms of the reflectionless filter structure

presented, the application of Richard’s transformation leaves a combination of series and parallel

stubs closely connected at multiple nodes in an electrically small cluster. Mostly, the series

stubs directly between the two ports at the top of the circuit represent an inconvenience in

manufacturing process. Eliminating them requires a further three-port transmission-line identity

illustrated in Figure 5.6,

Figure 5.6: Three-port network transmission-line identity used in derivation for maintaining the

reflectionless concept from [Mor15]

32 Chapter 5. State-of-the-art

where the coupled-line section on the right-hand side of the representation, its parameters

are those of a general four-port coupled-line section where the fourth port is left open. The full

three-port network parameters are identical and all the lines have the same electrical length,

with the following element values equations 5.4:

Zodd = Z1(1−1

n)

Zeven = Z1(1 +1

n)

n =

√Z2

Z1+ 1

(5.4)

The final transmission-line reflectionless filter after application of the above mentioned iden-

tity is shown in Figure 5.7 (b).

R R

Even-Mode

Equivalent Circuit

Odd-Mode

Equivalent Circuit

⇒

Zeven; Zodd Zeven; Zodd

Z1 Z1

Z0 Z0

Z2 Z2

Final Reflectionless Topology

Figure 5.7: (a) Addition of pertinent elements in order to maintain symmetry. (b) Final reflec-

tionless topology after application of the coupled-line identity in Figure 5.6.

Moreover, for ρ being the coupling factor of the coupled lines while the symmetry and duality

required for reflectionless operation constrains the element values, we have equations 5.5:

Zeven = Z0√ρ

Zodd =Z0

ρ

Z1 = 2Z0√ρρ+ 1

(ρ− 1)2

Z2 =Z20

Z1

(5.5)

5.3. Design Example 33

5.3 Design Example

By applying the concepts of this chapter within the transmission-line framework the imple-

mentation and its correspondent transmission response, S21, are shown in Figure 5.8 and 5.9,

respectively. As seen in Chapter 4 higher-order networks have higher stop-band peaks or less

stop-band rejection, here applying as well.

Figure 5.8: Ideal transmission lines bandpass filter implementation.

Figure 5.9: Ideal transmission lines bandpass filter response.

34 Chapter 5. State-of-the-art

5.4 Cascade Combination

With the same purpose as in previous chapters, transmission lines implementation of the reflec-

tionless filter has been cascaded. In so doing, the results seem to be favorable as well as next

described.

5.4.1 Cascading Equal Order Filters

By cascading equal-order filters expanded to transmission lines, better out-of-band rejection

results are obtained as shown in Figure 5.11.

Figure 5.10: Cascading of ideal transmission lines bandpass filter implementation.

Figure 5.11: Cascading of ideal transmission lines bandpass filter response.

5.4. Cascade Combination 35

5.4.2 The Multi-band bandpass filter topology

By cascading high and low frequency sections, a multi-band filter topology is achieved, as shown

in Figure 5.12. It is worth mentioning that this implementation is important since great effort has

been put into developing multifunction radiofrequency (RF) front-ends, capable of supporting

multiple standards at the same time with multi-band operation capabilities in the recent years.

This has high relevance in fields such of wireless communications where urges the need to allocate

different frequencies bands for each service. Moreover, basing the design in the reflectionless

concept structures, advantages such as low cost, small size and high performance could be met.

Furthermore, with the same topology of the cascaded filters, only by making them resonate

at different frequencies, there is no need in combining more complex filter design topologies in

order to obtain equivalent results.

Figure 5.12: Multiband filter response, cascading high- and low-frequency ideal transmission

lines bandpass filterss.

36 Chapter 5. State-of-the-art

Finally, a possible implementation is presented in Figure 5.13. The proposed structure has

been designed using parameters values of a Rogers RO4003 substrate [Cor18], with a thickness

of 0.508 mm, a relative dielectric constant εr = 3.55, a dielectric loss tangent TanD = 0.0021

and a conductor thickness T = 35µm at a cutoff frequency f0 = 2.4e9 GHz.

Figure 5.13: Microstrip transmission lines bandpass filter implementation.

Values and response of the expansion to microstrip lines of the reflectionless filter can be

found in Figure 5.14, where the value table represents parameters from equation 5.5 while CdB

and ρ, the coupling factor, relate to each other as follows:

c = 10CdB20

ρ =1− c1 + c

(5.6)

5.4. Cascade Combination 37

Figure 5.14: Microstrip transmission lines bandpass filter implementation response.

Due to high coupling factor value, the implementation has been tunned in order to achieve

a structure to be manufactured. Thereby, the obtained results of impedances Z1 and Z2 by

lowering this parameter up to ρ = −13.7 dB, get bigger and out of technological limit range of

120 and 20 Ω, since fulfilling conditions in equation 5.5. These values and the the transmission

response of a lower coupling factor are therefore shown in Figure 5.15.

Figure 5.15: Parameters values where Z1 = 1146.843Ω and Z2 = 2.180Ω .

Finally, the obtained space between the coupled line is one that could have been manufac-

tured (S ∼= 250µm, Figure 5.16) but due to the value of the impedances previously mentioned,

this design represents a technological limitation at the moment that has not been overpassed.

Figure 5.16: ADS microstrip coupled lines schematics illustration.

Chapter 6

The Acoustic Wave (AW)

Reflectionless Filter

6.1 The Low-pass prototype based on AW resonators

At this point, the electrical behavior of the used resonator is key in obtaining a desired frequency

response. In this sense, acoustic wave (AW) resonators have shown a particular frequency be-

havior, characterized by two resonant frequencies [Ver17], [Tri17], at which the input impedance

is zero and infinite. Taking advantage on the extracted pole nature of such resonators, very high-

selective bandpass responses can be achieved by the presence of transmission zeros below and

above the passband. As a first approach, the low-pass prototype is described taking advantage

of the properties of reflectionless structures explained along previous chapters.

Moreover, the same principles’ derivation is taken into account, and in order to satisfy

symmetry and duality conditions, the value of the elements remains constrained such as,

ZL =1

ZC(6.1)

The electrical behavior of the AW resonator is usually modeled in the bandpass domain by

the Butterworth-Van Dyke (BVD) model, formed by a series LAW -CAW tank, modeling the

acoustic behavior, in parallel with a static capacitance C0 modeling the electric behavior as in

[Lar00]. On the other hand, the dangling resonator is an attractive and useful representation of

the lowpass model [Ver17] which is at the same time related with the lowpass BVD model as

shown in Fig. 6.1. The impedance of the series AW resonator is defined for the lowpass BVD

equivalent circuit as,

ZAW (Ω) =jXSE−0 (ΩLSE−m +XSE−m)

XSE−0 + ΩLSE−m +XSE−m(6.2)

39

40 Chapter 6. State-of-the-art

B

JR

b

JML -JML

JR

-JML

JML

jB

jb

s

LSE-m jXSE-m

jXSE-0

Figure 6.1: Nodal representation (left), Equivalent circuit schema (center) and lowpass BVD

model (right) for the series AW resonator.

6.2 Design Examples

Taking into consideration the symmetry conditions mentioned above but also the technological

restrictions which force the specific BVD model, the configuration in Figure 4.4 can be modified

to the case of using AW resonators as shown in Figure 6.2. The inversion of the impedance,

as stated in equation (6.1), is achieved by means of admittance inverters, being J = 1 in the

lowpass prototype, and R = 1 Ω. It is important to state that the AW resonator is the same for

all the structure.

J-J

J-J

J-J

J-J

Figure 6.2: Basic configuration for the lowpass reflectionless prototype based on AW resonators.

The obtained transmission response where at fr1 the AW resonator behaves as a short-

circuit in the passband signal path, and open-circuit in the stopband signal path, while at fa1,

the passband signal path is blocked, which entails a transmission zero at such frequency. The

6.2. Design Examples 41

transmission zero observed close to the bandpass occurs approximately at the central frequency

of the AW resonator, where a destructive interference is given. Hence for this particular ideal

prototype, the return losses have been avoided since, due to symmetry conditions, they are neg-

ligible unless the odd- and even-mode branches are not perfectly complementary. The response

presents resonant frequencies at fr1 = 1962 MHz and fa1 = 2019 MHz, as represented in Figure

6.3.

Frequency (GHz)-20 -15 -10 -5 0 5 10 15 20

Tran

smis

sion

Res

pons

e S 21

(dB

)

-60

-50

-40

-30

-20

-10

0

AW

Res

onat

or In

put I

mpe

danc

e (d

B+

)

-20

0

20

40

60

80

100

fr

fa

Figure 6.3: Transmission response S21 for Figure 6.2 and electrical input impedance of the AW

resonator.

Based on this basic configuration, also the complementary structure can be considered as

shown in Figure 6.4. Unlike the previous case, the signal is blocked in the passband signal path

at fr2 since the impedance of the AW resonator is 0Ω given between admittance inverters.

J -J J -J

J -J J-J

Figure 6.4: Complementary configuration.

The frequency at the bandpass is now found at fa2. Again, the return losses have been

avoided in the figure since they are negligible. Moreover, resonant frequencies in this case are

found at fr2 = 1908 MHz and fa2 = 1962 MHz, respectively, both shown in Figure 6.5.

42 Chapter 6. State-of-the-art

Frequency (GHz)-20 -15 -10 -5 0 5 10 15 20

Tran

smis

sion

Res

pons

e S 21

(dB

)

-60

-50

-40

-30

-20

-10

0

AW

Res

onat

or In

put I

mpe

danc

e (d

B+

)

-20

0

20

40

60

80

100

fr

fa

Figure 6.5: Transmission response S21 for Figure 6.4 and electrical input impedance of the AW

resonator.

6.3 Cascade Combination

In order to obtain high selectivity below and above the passband, the previous configurations can

be concatenated without loading effects of one on the other since perfect matching is achieved at

all frequencies. The transmission response of the concatenated sections configuration is shown in

Figure 6.6. To achieve a bandpass response, the AW resonator from Figure 6.2 must be detuned

with respect the one in Figure 6.4. The achievable bandwidth of the bandpass response depends

on how detuned are both resonators. The trade-off is that the higher the detuning, the higher

ripple in-band.

In order to obtain a bandpass response, it is mandatory that the AW resonator in Figure

6.2 must be resonating at higher frequencies than the AW resonator in Figure 6.4. Otherwise,

an attenuated band is obtained.

Therefore, the design procedure can be summarize in the next steps:

• Design of fr1, and fa2 of the AW resonators at the central frequency of the bandpass.

• Design of the fa1 and fr2 of the AW resonator at the position of the transmission zeros

below and above the passband respectively.

Thereby, the obtained transmission response and the input electrical impedance of each

resonator are illustrated in Figure 6.7 where the proposed structure has been designed using

manufactured Bulk Acoustic Wave - Solid Mounted Resonator (BAW-SMR). Furthermore, the

position of the transmission zeros and the center of the passband are consistent with the pre-

6.3. Cascade Combination 43

+ (rad/s)-20 -10 0 10 20

Tran

smiss

ion

Resp

onse

S21

(dB)

-60

-50

-40

-30

-20

-10

0

Figure 6.6: Transmission response S21 for the concatenated sections. Dashed line corresponds

to the configuration in Figure 6.2 whereas the solid line to the complementary implementation,

Figure 6.4, and gray symbol line with the concatenated structure.

viously described design methodology. Transmissions zero close to the bandpass and in the

out-of-band region are solutions for S21 = 0, when zeven = 0Ω and yodd = 0 S.

Frequency (GHz)1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Tran

smis

sion

Res

pons

e S 21

(dB

)

-60

-50

-40

-30

-20

-10

0

AW

Res

onat

or In

put I

mpe

danc

e (d

B+

)

-20

0

20

40

60

80

100

-20

0

20

40

60

80

100

1.93 1.94 1.95 1.96 1.97 1.98 1.99 2-3

-2.5

-2

-1.5

-1

-0.5

0

Figure 6.7: Transmission response S21 of the bandpass filter and input impedance of the fabri-

cated AW resonator.

44 Chapter 6. State-of-the-art

Finally, aiming to study the effect of the unbalance on the return losses, the co-simulation

has been carried out including a phase shifter in one of the branches of the two configurations.

Moreover, the effect being considered resulted in obtaining better than 20dB return losses for

an unbalance of φi=5, without any alteration on the insertion losses. Figure 6.8 illustrates just

that.

Frequency (GHz)1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

Ret

urn

Loss

(dB

)

-100

-80

-60

-40

-20

0

fi

Figure 6.8: Effect of the phase unbalance on the return losses for φi =0.5, 1, 2, 5 and 10.

Chapter 7

Conclusions And Future Work

Although several well-known techniques like for instance the insertion of fixed attenuators on

heterodyne applications have resulted in fixing common issues of conventional filters, those still

remain an adjacent to other components since their stop-band has adverse effects on system

performance. Moreover, when intended to overpass these setbacks it usually ends up resulting

in more complex procedures. Therefore, although requirements may vary from one system to

another within this study, a recent new class of reflectionless low-pass, high-pass and band-pass

filter prototype has been researched, studied, electrically implemented, simulated and measured.

Graphical responses have been presented showing the reflection absence of signal back to the

source. Basic concepts helpful to understand the theory behind the reflectionless concept have

been discussed as well as comparisons have been made with the simulations results obtained.

Moreover, responses resulting from cascading different reflectionless topologies have been dis-

cussed as well as the advantages the new networks obtained represent.

Furthermore, the first reflectionless acoustic wave resonator filter’s electrical behavior has

been presented to allow designing sections describing transmissions zero below the bandpass, and

also the complementary whereas allocated above it. While both sections are perfectly matched

in-band and stop-band, different sections were concatenated to achieve a bandpass response with

transmissions zero below and above the passband. On the other hand losses better than 20dB

have also been considered while validating the proposed topology.

For future work some modification could be applied to the topology presented in this study,

thus, resulting onto new structures and networks achieving presumably sharper cutoff charac-

teristics. Additionally, designs with new responses would appear maybe comparable with actual

conventional optimized Chebyshev filters or others. Also, expansion to transmission lines frame-

work of these designs was considered although the challenge it represents the manufacturing

process of prototypes suitable for higher frequencies reaches technological boundaries.

45

46 Chapter 7. State-of-the-art

Finally, fabrication is a further step since the layout for such filter derived from the reflec-

tionless principles has been generated. It remains an open issue finding ways to overcome the

technological boundaries and have a prototype electromagnetically modeled and manufactured.

Also it is intended to continue with the investigation on the proposed implementation based on

acoustic wave (AW) resonators in the same prototyping direction in order to validate all results

shown in this work.

In terms of acoustic wave resonators, since the proportionality between electromagnetic and

acoustic waves has been mentioned, the research line of this study could lead to consolidated

lower-in-size microwave devices and systems, while describing reflectionless properties and char-

acteristics.

Bibliography

[Cor18] Rogers Corporation, “RO4000 R© Series High Frequency Circuit Materials”, pag. 4, 2018.

[Feh76] K. Feher, R. De Cristofaro, “Transversal Filter Design and Application in SatelliteCommunications”, IEEE Transactions on Communications, Vol. 24, no 11, pags. 1262–1268, nov 1976.

[GG09] Roberto Gomez-Garcia, Manuel Sanchez-Renedo, Bernard Jarry, Julien Lintignat,Bruno Barelaud, “A class of microwave transversal signal-interference dual-passbandplanar filters”, IEEE Microwave and Wireless Components Letters, Vol. 19, no 3,pags. 158–160, mar 2009.

[Lar00] John D. Larson, Paul D. Bradley, Scott Wartenberg, Richard C. Ruby, “ModifiedButterworth-Van Dyke circuit for FBAR resonators and automated measurement sys-tem”, Proceedings of the IEEE Ultrasonics Symposium, Vol. 1, pags. 863–868, 2000.

[Mor11] Matthew A. Morgan, Tod A. Boyd, “Theoretical and Experimental Study of a NewClass of Reflectionless Filter”, IEEE Transactions on Microwave Theory and Tech-niques, Vol. 59, no 5, pags. 1214–1221, may 2011.

[Mor15] Matthew A. Morgan, Tod A. Boyd, “Reflectionless filter structures”, IEEE Transac-tions on Microwave Theory and Techniques, Vol. 63, no 4, pags. 1263–1271, 2015.

[Nik05] Pavel V. Nikitin, K. V.Sephagiri Rao, Sander F. Lam, Vijay Pillai, Rene Martinez,Harley Heinrich, “Power reflection coefficient analysis for complex impedances in RFIDtag design”, IEEE Transactions on Microwave Theory and Techniques, Vol. 53, pags.2721–2725, sep 2005.

[Pac15] Pedro de Paco, Transmision por soporte fısico, Departament de Telecomunicacio id’Enginyeria de Sistemes, 2015.

[Poz05] D. Pozar, Microwave Engineering Fourth Edition, 2005.

[Tri17] Angel Triano, Jordi Verdu, Pedro de Paco, Thomas Bauer, Karl Wagner, “Relationbetween electromagnetic coupling effects and network synthesis for AW ladder typefilters”, 2017 IEEE International Ultrasonics Symposium (IUS), pags. 1–4, IEEE, sep2017.

[Ver10] Jordi Verdu Tirado Advisors, Pedro de Paco Sanchez Oscar Menendez Nadal, “BulkAcoustic Wave Resonators and their Application to Microwave Devices”, 2010.

[Ver17] Jordi Verdu, Iuliia Evdokimova, Pedro De Paco, Thomas Bauer, Karl Wagner, “Syn-thesis methodology for the design of acoustic wave stand-alone ladder filters, duplexers

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ii Bibliography

and multiplexers”, IEEE International Ultrasonics Symposium, IUS , pags. 1–4, IEEE,sep 2017.

Appendix A

A.1 Determining a Filter’s Order

Table A.1: Element Values for Equal-Ripple Low-Pass Filter Prototypes (g0=1, wc=1, N=1 to10, 0.5 dB ripple) [Poz05]

N g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

1 0.6986 1.0000

2 1.4029 0.7071 1.9841

3 1.5963 1.0967 1.5963 1.0000

4 1.6703 1.1926 2.3661 0.8419 1.9841

5 1.7058 1.2296 2.5408 1.2296 1.7058 1.0000

6 1.7258 1.2479 2.6064 1.3137 2.4758 0.8696 1.9841

7 1.7372 1.2583 2.6381 1.3444 2.6381 1.2583 1.7972 1.0000

8 1.7451 1.2647 2.6564 1.3590 2.6964 1.3389 2.5093 0.8796 1.9841

9 1.7504 1.2690 2.6678 1.3673 2.7239 1.3673 2.6678 1.2690 1.7504 1.0000

10 1.7543 1.2721 2.6754 1.3725 2.7392 1.3806 2.7231 1.3485 2.5239 0.8842 1.9841

iii

iv Bibliography

Appendix B

B.1 Frequency Scaling Transformations

Figure B.1: Summary of Prototype filter Transformations (∆ = w2−w1w0

)

v