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Modeling and Analysis of DefectiveGround Plane Microstrip Structures

Evi Van Nechel

Academic year2013-2014

Master thesis submitted under the supervision ofProf. dr. ir. Yves RolainProf. dr. ir. Gerd Vandersteen

The co-supervision ofir. Matthias Caenepeel

In order to be awarded the Master’s Degree inElectronics and Information Technology Engineering

Acknowledgements

First of all I would like to thank my promoter Yves for giving me the best guidance during the course

of the thesis and for the condence he has put in me. I also want to thank my adviser Matthias

for all his good advice and support and for the motivation he gave me. Next, I would like to thank

Gerd for exploring the edges of the EM-simulator with me and for coming up with new possibilities

every time. Furthermore, I want to thank the whole ELEC-department for creating a nice and

comfortable environment with special thanks to Johan Pattyn for always carefully fabricating my

PCB-designs and to Maral for taking my mind of the thesis when needed and who always kept me

motivated.

Very special thanks goes to my boyfriend Kenny for his support, his love and his enormous amount

of patience. I want to thank all my classmates, especially Piet and Steluta, for the joyful moments

and the memorable quotes of the week. I also want to thank Glenn and Glenn for the relaxing lunch

times during the year. And last but not least, I want to thank my family and friends for their care

and understanding.

Abstract

Title - Modeling and analysis of defective ground plane microstrip structures

Author - Evi Van Nechel

Master's degree - Electronics and Information Technology Engineering

Academic year - 2013-2014

Abstract - Coupled line lters are widely used in modern microwave communication systems.

The classical parallel-coupled transmission-line-resonator lter developed by Cohn [1] has two main

shortcomings when designed in microstrip technology: 1) the presence of spurious passbands at the

harmonic responses; 2) the diculty to design wideband lters. When the classical approach fails,

alternative solutions are needed. In this work, we investigate whether introducing slots in the ground

plane can result in wideband lters with a low insertion loss. The aim of this thesis is to examine

the eect of resonant apertures in the ground plane of microstrip structures and to investigate their

potential benets in lter design.

The work consists of ve main results:

The EM-simulator is shown to be capable of simulating a slotted ground plane accurately.

This is proven with actual designs and measurements.

A simple microstrip line with multiple slots is examined. It is shown that this structure acts as

a bandstop lter. Adding this structure to classical parallel-coupled line lters can attenuate

the harmonic responses, hereby solving the rst drawback.

A lumped-element equivalent circuit model is generated to predict the behavior of the slots.

The novelty in the proposed model is twofold: 1) it predicts anti-resonances and quality factor

accurately; 2) it is symmetric.

A MIMO-LTI-model-based comparison between the EM-simulations and the lumped circuit

model shows that the frequencies of the poles and zeros is accurately predicted, while the

quality factor needs some additional improvement.

A nal analysis of a simple coupled line structure with a slotted ground plane shows that

introducing slots in the ground plane indeed increases the coupling. This can be used to solve

the second drawback of the classical approach. Further research is needed to include this

properly in a lter design procedure.

Index Terms - Slotted ground plane, defective ground plane, microstrip technology, slotline, mod-

eling, coupled line lter design.

Samenvatting

Titel -Modeleren en analyseren van resonerende defecten in het grondvlak van microstrip structuren

Auteur - Evi Van Nechel

Academische graad - Ingenieurswetenschappen in elektronica en informatie-technologie

Academiejaar - 2013-2014

Abstract - Gekoppelde lijn lters worden vaak gebruikt in moderne microwave communicatie-

systemen. Voor de microstrip technologie heeft het klassieke parallel-gekoppelde transmissie-lijn-

resonator lter, ontwikkeld door Cohn [1], twee belangrijke tekortkomingen: 1) de aanwezigheid van

ongewenste harmonische doorlaatbanden; 2) de moeilijkheid om breedbandige lters te ontwerpen.

Waar de klassieke aanpak tekort schiet, zijn alternatieve oplossingen nodig. In dit werk onderzoe-

ken we of het introduceren van apertures in het grondvlak kan leiden tot betere breedbandige lters

met een lage insertion loss. Het doel van dit eindwerk is dan ook om het eect van resonerende

apertures in het grondvlak van microstrip structuren en hun potentiële voordelen in lterontwerp

te onderzoeken.

Het eindwerk bestaat uit vijf belangrijke resultaten:

Er wordt aangetoond dat de ADS EM-simulator in staat is om defecten in het grondvlak

accuraat te simuleren. Dit resultaat wordt gestaafd door middel van praktische designs en

metingen.

Een eenvoudige microstrip lijn met meerdere apertures wordt onderzocht. Er wordt aange-

toond dat deze structuur zich als een bandstop lter gedraagd. Door deze structuur toe te

voegen aan de klassieke parallel-gekoppelde lijnen lters kunnen de ongewenste harmonischen

onderdrukt worden, wat een oplossing biedt voor de eerste tekortkoming.

Een lumped-element equivalent circuit model wordt gegenereerd om het gedrag van de apertu-

res nauwkeurig te voorspellen. Er zijn twee verbeteringen aanwezig in het voorgestelde model

in vergelijking met de literatuur: 1) het model kan anti-resonanties en eindige kwaliteitsfacto-

ren nauwkeurig voorspellen; 2) het model is symmetrisch.

De EM-simulaties en het lumped circuit model worden vergeleken gebruik makend van een

geëstimeerd MIMO-LTI-model. Hierdoor wordt aangetoond dat de frequenties van de polen

en nullen nauwkeurig voorspeld worden. Het voorspellen van de kwaliteitsfactor is mogelijk

maar vereist een bijkomende optimalisatie.

Uit een laatste analyse van een eenvoudige gekoppelde lijn structuur met apertures in het

grondvlak blijkt dat de invoering van apertures de koppeling inderdaad verhoogt. Dit kan

gebruikt worden om de tweede tekortkoming van de klassieke ontwerp-aanpak op te lossen.

Verder onderzoek is nodig om tot een praktische ontwerp procedure te komen voor deze lters

met een onderbroken grondvlak.

Indextermen - Slotted ground plane, defectief grondvlak, microstrip technologie, slotlijn, modele-

ren, lter design.

Résumé

Titre - Modélisation et analyse d'ouvertures résonantes dans le plan de masse d'une structure

microstrip de ligne de transmission

Auteur - Evi Van Nechel

Grade universitaire - Ingénieur civil en électronique

Année universitaire - 2013-2014

Résumé - Les ltres à lignes couplées sont largement utilisés dans la conception de systèmes de

communication modernes opérant aux hyperfréquences. Le ltre classique composé de résonateurs

à lignes couplées développé par Cohn [1], a deux défauts principaux lorsqu'il est réalisé en lignes

microstrip: la présence de bandes passantes parasites à des fréquences harmoniques et la diculté à

réaliser des ltres à large bande passante. Lorsque la méthode classique échoue, il devient impératif

de trouver des solutions ou des techniques alternatives. Cette thèse est consacrée à l'introduction

d'ouvertures résonantes dans le plan de masse an d'obtenir des ltres à large bande passante ayant

peu de perte d'insertion. Le but nal de la thèse est de déterminer si les ouvertures résonantes

possèdent le potentiel nécessaire pour réaliser ces ltres très pointus.

La thèse contient cinq résultats principaux:

Le simulateur électromagnétique est capable de simuler les ouvertures d'une façon précise. Les

résultats simulés sont à cette n vériés en mesurant des structures réalisées et mesurées qui

sont semblables aux structures simulées.

Une ligne microstrip avec plusieurs ouvertures est étudiée. On démontre que cette structure se

comporte comme un ltre coupe-bande. Il résulte également que l'introduction d'ouvertures

dans le plan de masse d'un ltre classique peut atténuer ces bandes passantes parasites

présentes aux fréquences harmoniques. Cette propriété permet de résoudre le premier dé-

faut de la méthode classique.

Un nouveau circuit équivalent ne comportant que des éléments concentrés est généré an de

prédire le comportement de ces ouvertures. Le modèle proposé prédit aussi bien les antiréso-

nances que les facteurs d'amortissement avec précision. Le modèle reste symétrique comme

les lignes couplées.

L'identication de modèles MIMO-LTI à partir de la réponse en fréquence des simulations

électromagnétiques et du circuit équivalent permet de comparer la position des pôles et des

zéros. Cette comparaison démontre que les fréquences de résonance des pôles et des zéros

sont prédites avec une précision acceptable. La prédiction des facteurs de qualité doit être

améliorée.

L'analyse nale d'une paire des lignes couplées d'amortissement est bonne après l'emploi d'une

optimisation numérique avec plusieurs ouvertures dans le plan de masse démontre que leur

introduction augmente l'eet de couplage entre les lignes. Cette propriété permet de résoudre

le deuxième défaut de la méthode classique. Une recherche plus poussée reste néanmoins

nécessaire pour permettre l'usage des ouvertures lors de la conception de ltres.

Mots clés - Ouvertures dans le plane de masse, lignes microstrip, ouvertures résonantes, modéli-

sation, conception des ltres.

Contents

1 Introduction 1

2 Simulator settings 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Momentum setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Simulation mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.2 Mesh settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.3 Frequency plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.4 Substrate denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.5 Dening ports to interact with the design . . . . . . . . . . . . . . . . . . . . 6

2.3 Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Example: coupled line lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Momentum setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1.1 Simulation mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1.2 Mesh settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1.3 Frequency plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1.4 Substrate denition . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1.5 Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

i

ii CONTENTS

3 Modeling a microstrip line with slotted ground plane 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Single microstrip line with rectangular slots . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 A lumped-element circuit model for the general unit cell . . . . . . . . . . . . . . . . 17

3.4 Example: transmission line with slotted ground plane . . . . . . . . . . . . . . . . . 19

3.4.1 Geometry and settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.2 Lumped-element circuit model vs. EM-simulations . . . . . . . . . . . . . . . 21

3.4.3 Tuned lumped-element circuit model vs. EM-simulations . . . . . . . . . . . 28

3.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Simplied structure: a single slot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5.1 Geometry and settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5.2 Lumped-element circuit model vs. EM-simulations . . . . . . . . . . . . . . . 30

3.5.3 Tuned lumped-element circuit model vs. EM-simulations . . . . . . . . . . . 31

3.5.4 Symmetrizing the lumped-element circuit model . . . . . . . . . . . . . . . . 33

3.5.5 Introducing losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6 Modeling a single slot structure: the MIMO-model . . . . . . . . . . . . . . . . . . . 40

3.6.1 Geometry and settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6.2 Improved lumped-element circuit model vs. EM-simulations . . . . . . . . . . 40

3.6.3 MIMO-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Coupled line with slotted ground plane 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Geometry and settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Comparing EM-simulations to measurements . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Eect of the slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Conclusion and future perspectives 59

Nomenclature

ADS Advanced Design System

DUT device under test

EM electromagnetic

FRF frequency response function

LTI linear time-invariant

MIMO multiple-input-multiple-output

PP pole pair

RF radio frequency

RMSE root-mean-square error

S-parameters scattering parameters

SML Sampled Maximum Likelihood

TE transverse electric

TEM transverse electromagnetic

TML-calibration transmission line calibration

VNA vector network analyzer

ZP zero pair

iii

Chapter 1

Introduction

Coupled line lters are widely used in modern microwave communication systems. The classical and

widely used parallel-coupled transmission-line-resonator lter design method developed by Cohn [1]

has proven to be successful, but it has two main shortcomings when a design in microstrip technology

is aimed at: 1) spurious passbands are present at the harmonic response; 2) wideband lters (>

30 % [1]) are very hard to obtain.

Wideband lters require a strong coupling between the coupled lines. To achieve this strong coupling

in a classical coupled line, the gap between the lines needs to decrease accordingly. However,

decreasing the dimensions of the gap is not always possible since, at some point, mechanical tolerance

induces that the structure cannot be fabricated accurately anymore.

When the classical approach fails, alternative solutions are needed. In this work, we investigate

whether introducing slots in the ground plane can result in wideband lters with a low insertion

loss. The aim of this thesis is to examine the eect of resonant apertures in the ground plane of

microstrip structures and to investigate their potential benets for an application in lter design.

Chapter 2 veries whether the used EM-simulator [2] is capable of simulating a slotted ground

plane accurately and determines this accuracy. We give an overview of the dierent settings of the

EM-simulator that inuence the accuracy and based on this a set of valid settings is obtained to

simulate a nite ground plane structure. The choice of the settings is veried using an actual design

and its realization as we compare the simulated to the measured S-parameters.

Chapter 3 examines the eect of rectangular slots in the ground plane on the frequency response

of a single transmission line. A lumped-element equivalent circuit model is proposed and validated

using an example that is realized and measured again. Since the original lumped model taken from

the literature is not complex enough to yield an accurate model, Section 3.5 proposes improvements

to the the lumped circuit model: symmetry is hereby imposed and the circuit model can predict

anti-resonances and quality factor more accurately.

1

2 CHAPTER 1. INTRODUCTION

Obtaining a model that can predict the eects of slotted ground plane structures accurately is a

very important step towards a proper design procedure. In Section 3.6 we discuss a parametric

MIMO-model to verify how good the improved model actually is. Can it predict the behavior of

the transmission line structure suciently well? This comparison is based on the comparison of the

position of the poles and zeros of the model.

Next we examine a simple coupled line structure with a defective ground plane in Chapter 4. We

verify whether the extra complication of the line coupling, next to the slot coupling, still leads to

accurate and correct simulation results. Three dierent coupled line structures of increasing com-

plexity are selected to gradually work from a coupled line to a complete defective ground lineset.

The dierent steps are again validated with measurements. The structures are physically realized

and measured with the Vector Network Analyzer. The eect of the coupling of two lines in combi-

nation with slots in the ground plane is assumed and analyzed and the benets for the use in lter

design are discussed.

Finally, some conclusions are drawn about the research on defective ground plane microstrip struc-

tures and their potential use in lter design. We also briey touch the future perspectives on this

subject.

Chapter 2

Simulator settings

2.1 Introduction

The goal of this thesis is to examine the eect of apertures in the ground plane of microstrip

structures. Hence, it is very important to be able to properly simulate these eects since the rst

stage of every design is based on simulations. We want these simulations to be as close as possible to

the actual measured results and therefore, the settings of the used simulator should be chosen with

care. Since simulating in a correct way is very important, this chapter is devoted to the choice of

the simulator settings and its inuences. In this case, the Momentum simulator [2] of the Advanced

Design System (ADS) [3] software of Agilent is used.

Section 2.2 gives an overview of the dierent possible settings of the Momentum simulator. The

inuence of the dierent settings is discussed based on an example. As an example, a coupled line

lter is used (Section 2.4). The eect of the introduction of a nite ground plane on this lters'

response is examined and compared to the simulation with innite ground plane. The design is

realized in hardware and the measurements are compared to the simulations.

2.2 Momentum setup

This section outlines the dierent settings of the Momentum setup. The pros and cons of some

important settings are discussed. This is necessary to justify the choice of the settings for the

considered application.

2.2.1 Simulation mode

Momentum has two simulation modes, the microwave mode and the radio frequency (RF) mode.

The main dierence between them is the way the Greens' functions are calculated.

3

4 CHAPTER 2. SIMULATOR SETTINGS

The microwave or full-wave mode uses full-wave Greens' functions, which are frequency dependent

and fully characterize the substrate. In the RF or quasi-static mode, however, the Greens' functions

are frequency independent and hence, some simplications to the Maxwell equations are made. This

results in real and frequency independent inductances and capacitances, compared to complex and

frequency dependent element values in case of the full-wave Greens' functions. Furthermore the

RF mode does not incorporate the high frequency losses in the ground plane, unlike the microwave

mode.

Thus, the microwave mode uses less approximations and the results are more accurate when selecting

this mode. The downside of using the microwave mode is that the simulation time is much longer

than in case of the RF mode. [4]

2.2.2 Mesh settings

It is impossible for the simulator to calculate the voltage and current distributions at every single

point of a transmission line. Therefore, some points need to be specied at which the calculations

are done. Hence, the transmission lines are divided in small cells. The mesh is the collection of

these cells. At the crossings of the cells, the necessary functions are calculated.

There are several settings related to the mesh that are very important. The mesh frequency is

the frequency that determines the wavelength that is used for the mesh density. Two options are

possible here. We can use the default value, which sets the mesh frequency to the highest simulation

frequency. This ensures that the simulation results are accurate over the whole range of simulated

frequencies. The other option is that the user denes the mesh frequency himself.

The mesh density determines the number of cells per wavelength. This wavelength is calculated

from the mesh frequency and the relative permittivity. On the one hand the mesh needs to be dense

enough in order to capture real world spatial variations of the current/voltage distribution. A rule

of thumb is to use 20 cells per wavelength. A denser mesh can be obtained by increasing either the

mesh frequency or the number of cells per wavelength. The downside of making the mesh denser

is that it increases the simulation time. On the other hand, making the mesh too dense can even

result in wrong solutions due to numerical inaccuracy.

To overcome problems in the neighborhood of the edge of a structure, an edge mesh can be used.

This results in a higher accuracy for the simulations, but again this comes at the cost of an increased

simulation time.

Furthermore, the mesh reduction can be disabled. The mesh reduction is a tool that removes

possible redundancies and thus lowers the complexity of the mesh. Disabling the mesh reduction

results in a higher accuracy as well. The drawback is that the rules used to prune the mesh are not

clear and not parametrized. [5]

2.2. MOMENTUM SETUP 5

2.2.3 Frequency plan

An important setting here is the type of the frequency plan, which can be chosen to be 'linear'

or 'adaptive'. For both types the start and stop frequency needs to be specied. When a linear

frequency plan is used, the frequencies to be simulated are chosen to be equidistant. The distance

is set by the user by specifying either the number of points or the frequency step. The user has full

control over the exact simulation frequencies.

This is not the case if an adaptive frequency plan is used. The frequencies are chosen by the simulator

itself. The rule is to use a larger density in the frequency bands where the scattering parameters

(S-parameters) vary more. If the simulator notices large variations of the S-parameters, the number

of frequency points will be increased in that region and decreased in zones where nothing happens.

Hence, the user has no control anymore over the exact frequencies. Only the maximum number of

frequency points can be specied in this case.

2.2.4 Substrate denition

There are two types of substrate denitions that are used for the remainder of the text: an innite

substrate and a nite one.

Figure 2.2.1 shows a general denition for the innite substrate. The bottom layer represents the

innite ground plane and is assumed to consist of an innitely thin, perfectly conducting material.

The layer is assumed to be lossless and to have an innite conductivity. The 'cond'-layer is the

conduction layer that represents the signal layer. This layer has a nite thickness and conductivity

and the losses are taken into account. The layer in between these conduction layers is the dielectric.

The parameters of the dielectric are discussed in Section 2.3.

Figure 2.2.1: Denition of a substrate with innite ground plane.

When a nite substrate is used, a second conduction layer ('cond2') is introduced. This layer

represents the ground plane (Figure 2.2.2). It has the same properties as the rst conduction layer.


Figure 2.2.2: Denition of a substrate with nite ground plane.

2.2.5 Dening ports to interact with the design

A port consists of two terminals, a negative and a positive one. The positive terminal is situated at

the signal layer, the negative terminal at the ground layer.

In case of an innite substrate, the ports are dened as shown in Figure 2.2.3. Here, pins P1 and

P2 are the positive terminals. The negative terminals are implicitly located in the innite ground

plane, as close to the positive terminal as possible.

Figure 2.2.3: Port denitions for a substrate with innite ground plane.

For a nite substrate, the ports are dened as shown in Figure 2.2.4. In this case, the negative

terminals need to be dened explicitly. Pins P3 and P4 are dened as the negative terminals. They

are located at the same position as the positive terminals, P1 and P2.

Figure 2.2.4: Port denitions for a substrate with nite ground plane.

2.3. SUBSTRATE 7

An important setting here is the calibration type. There are a few possibilities, but only two are

applicable in our case, namely a TML-calibration (transmission line calibration) or no calibration.

If a TML-calibration is selected, Momentum automatically adds a transmission line feed line to the

port [6]. If 'None' is selected as calibration type, no calibration is performed.

2.3 Substrate

There are a number of substrate parameters that need to be set. The parameters that are needed

for the conduction layers are the conductance and the thickness of the conduction layer. This was

already mentioned in Section 2.2.4. The parameters of the dielectric are the relative permittivity,

the relative permeability and the dielectric loss model.

The relative permittivity of the dielectric, εr(f), is a complex function of the frequency and can be

written as

εr(f) = ε′(f)(1− j.tanδ(f)) (2.3.1)

with ε′(f) the real part of εr(f) and δ(f) the angle between the real and the imaginary part. The

imaginary part represents the substrate losses.

The relative permeability of the dielectric, µr(f), is also a complex function of the frequency. The

relative permeability of the materials used as a substrate in microwave applications is very near

unity and the conductivity of the dielectric is very low. Hence, we can set the real part of the

relative permeability to 1 and the imaginary part to 0.

There are four parameters for the dielectric loss model, namely the model type, the frequency at

which εr has been measured (fM ) and the lower and upper bound frequency for the model (fL

and fH respectively). There are two possible model types: the frequency independent model or the

Svensson/Djordjevic model.

The frequency independent model assumes that the relative permittivity, εr, is frequency inde-

pendent. The Svensson/Djordjevic model takes the frequency dependency of the permittivity into

account as follows:

εr(f) = ε∞ + a.lnfH + j.f

fL + j.f(2.3.2)

where ε∞ is the value of the permittivity when the frequency approaches innity and a is a constant.

These two parameters are calculated from the substrate parameters ε′, tanδ, fM , fL and fH . fM

is the frequency at which Equations (2.3.1) and (2.3.2) are equivalent for a certain, constant ε′ and

tanδ.

Thus, both models require the user to specify constant values for ε′ and tanδ. When the rst model

is selected, these constant values are used over the whole frequency range. In case of the second


model, they are used to calculate the parameters ε∞ and a that are needed for the calculation of

ε′(f) and tanδ(f) at the dierent frequencies.

A third option is possible, however. The user can enter frequency dependent expressions for ε′(f)

and tanδ(f). In that case, ADS [3] automatically turns o the Svensson/Djordjevic model. [7]

The substrate that is used in all future designs is a Rogers RO4003 [8] with a thickness of 60 mils.

The conducting material is copper with a conductance of 5.8e7 Sm . The thickness of the conduction

layer is 35µm. The real part of the relative permittivity and the loss tangent parameter are εr = 3.55

and tanδ = 0.0021 at a frequency of fM = 2.5 GHz. The Svensson/Djordjevic model is selected.

2.4 Example: coupled line lter

The example for which the dierent Momentum settings are discussed is a coupled line lter of order

three with a center frequency of 2 GHz. Figure 2.4.1 shows the circuit schematically. The lter is

simulated with an innite and a nite ground plane. The simulator settings as well as the results

are compared. Finally, the simulations are compared to real measurements.

Figure 2.4.1: Schematic of a coupled line lter of order 3 and with a center frequency of 2 GHz.

The layout of the lter with an innite ground plane is shown in Figure 2.4.2. When simulating the

same lter with a nite ground plane, the ground plane needs to be drawn explicitly at a second

conduction layer, 'cond2' (yellow surface in Figure 2.4.3).

2.4. EXAMPLE: COUPLED LINE FILTER 9

Figure 2.4.2: Layout of the coupled line lter with an innite ground plane.

Figure 2.4.3: Layout of the coupled line lter with nite ground plane.

2.4.1 Momentum setup

As mentioned in Section 2.2, there are some dierences in the settings of the substrate and the

port denition depending on the type of the ground plane (nite or innite). In both cases, the

simulation mode, the frequency plan and the mesh settings are kept constant.

2.4.1.1 Simulation mode

It is our intention to replace the innite ground plane by a nite one in order to be able to simulate

apertures in the ground plane. Since the simulation results must be as close to the actually measured

results as possible, we want the simulator to make as few approximations as possible. Therefore,

the microwave mode is preferred above the RF mode.

2.4.1.2 Mesh settings

The mesh frequency is set to be the highest simulation frequency. This ensures that the simulation

results are accurate over the whole range of simulated frequencies. The mesh density is set to 20

cells per wavelength. More accurate simulations could be obtained if the mesh density would be

set to a larger number of cells per wavelength but this increases the simulation time too much once

the nite ground plane is introduced. Furthermore, an edge mesh is used and the mesh reduction

is disabled. In order to properly compare the dierent simulations, the mesh settings are xed to

be the same for all future simulations.

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2.4.1.3 Frequency plan

There exists a maximum frequency of operation that is related to the thickness of the substrate.

This is explained as follows. When the frequency is increased, the wavelength decreases. At a certain

point, the wavelength comes very close to the thickness of the substrate and the substrate becomes

non-operational. Besides this, we also know that the simulation time increases very rapidly if the

highest simulation frequency increases. For those reasons, we will never simulate above 5 GHz. This

means all our future designs are designed so that at least the second harmonic is still below this

maximum of 5 GHz.

The center frequency of the coupled line lter is approximately 2 GHz, with a 10 dB-bandwidth of

about 250 MHz. Since we are also interested in the out-of-band behavior and the second harmonic

response, we simulate from 1 GHz to 5 GHz . We use a linear frequency plan with a frequency step

of 50 MHz.

2.4.1.4 Substrate denition

The substrate denition for the innite ground plane is set as shown in Figure 2.2.1 of Section 2.2.4.

The denition for the nite ground plane is set as shown in Figure 2.2.2 of Section 2.2.4. The details

about the substrate properties are mentioned in Section 2.3.

2.4.1.5 Ports

The denition of the ports is shown in Figure 2.2.3 for the innite ground plane and in Figure 2.2.4

for the nite ground plane (Section 2.2.5).

What remains is to select the calibration type. It is eventually the intention to use the lter in

a larger circuit. In that case, transmission lines are connected at the ports of the lter. Hence,

selecting a TML-calibration is the most obvious choice.

2.4.2 Results

In Figure 2.4.4, the simulations obtained with the innite ground plane are compared to the mea-

surements. The simulations t the measurements quite well, except for a small frequency shift of

about 25 MHz. This frequency shift can be explained by the approximation that is used by ADS [3]

to calculate the eective permittivity and the uncertainty on the eective permittivity.


1 2 3 4 5

−60

−40

−20

0

f (GHz)

S21(dB)

1.8 1.9 2 2.1 2.2

−30

−20

−10

0

f (GHz)

S21(dB)

Figure 2.4.4: Comparison between EM-simulation results of a coupled line lter with innite groundplane (blue) and measurements (red). Left: S21 for a wide frequency band. Right: S21 for thefrequency band of interest. A frequency shift can be observed between the two results.

In Figure 2.4.5 the simulation results of the lter with nite ground plane (red curve) are compared

to the results obtained with the innite ground plane (blue curve). The results show two main

dierences:

Two peaks appear in the forward gain (S21-parameter) of the nite ground plane simulation

that are centered around the fundamental frequency. These peaks are not present in the

innite ground plane simulation, neither in the measurements (see Figure 2.4.4). They appear

due to the unwanted resonance mode of the ground plane.

The width of the ground plane is 42 mm. This corresponds to half of the wavelength at a

frequency of 1.28 GHz and this is in the very close neighborhood of the rst peak that appears

in the simulations.

The input reection factor (S11) of the nite ground plane simulation becomes larger than one

(i.e. larger than 0 dB) around a frequency of 4.6 GHz. This means that more energy is coming

out than is going in, or in other words, energy is created by the circuit. This is physically

impossible since we are dealing with a passive circuit here.


1 2 3 4 5−25

−15

−5

5

f (GHz)

S11(dB)

1 2 3 4 5

−60

−40

−20

0

f (GHz)

S21(dB)

Figure 2.4.5: Comparison of the EM-simulation results of a coupled line lter with nite (red) andinnite ground plane (blue). Left: the input reection factor S11. Right: the forward gain S21. Forthe case of a nite ground plane we see an unrealistic behavior of S11 around a frequency of 4.6 GHz.Also, two peaks appear in the S21-parameter.

There is denitely something wrong with the simulation. When the simulations are started, it

is noticed that the simulator takes a very long time for calibrating the ports. Furthermore, two

warnings appear:

1. The 2D port solver data is not available, using default values (50 Ohm) for the transmission

line parameters

2. S-parameter results show unphysical behavior for certain frequencies. Cause:

inaccurate (high frequency) calibration;

mesh density is too coarse.

The causes that are proposed by ADS [3] and are not applicable to our case are not mentioned

above. If the mesh density is too coarse, this can be easily adjusted but it strongly increases the

simulation time. Another possible cause is a problem with the calibration. The problem is explained

as follows: the addition of the transmission line feed line at the ports does not necessarily include

an additional extension of the nite ground plane at the ports, causing an incorrect calibration.

Since the TML-calibration leads to unrealistic results, we opt for an uncalibrated port (calibration

type: none). The results are shown in Figure 2.4.6.

Top left gure: The amplitude of S11 does not exceed 0 dB, thus the simulations do not show

unrealistic behavior this time.

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Top right gure: There is a quite good in-band match of the amplitude of the S21-parameter

between the nite and the innite ground plane simulation. However, the out-of-band behavior

shows some large dierences.

Middle left gure: The amplitude of the S21-parameter of the nite ground plane simulation

also matches the measurements quite well in-band, but again some dierences are visible in

the out-of-band behavior.

Middle right gure: There is again a frequency shift of about 25 GHz between the simulations

and the measurements, but this was expected. We can also see that the bandwidth of both

simulations is smaller than the bandwidth of the measured lter. This is possibly due to

the fact that the mesh density was still too coarse and hence, the simulation results are not

accurate enough, although no warnings appeared.

Bottom left gure: The phase of S11 of the measurements and the nite ground plane sim-

ulation match almost perfectly, if the small frequency shift of about 25 GHz is taken into

account.

Bottom right gure: For the phase of S21 of the measurements and the nite ground plane

simulation there is only a good match around the center frequency and around the second

harmonic, taking the small frequency shift into account again. Out-of-band the phase does

not match, but this is expected since the out-of-band amplitude of S21 does not match either

(middle left gure).

We can state that there are still some details that the simulator can not simulate correctly or

accurately enough. This problem is explained as follows. The dielectric loss model that is used,

namely the Svensson/Djordjevic model, does not t the real dielectric losses of the substrate that is

used. This is where the small dierences in simulations and measurements come from. Up to 5 GHz

these deviations are still acceptable. Since no better alternative dielectric loss model is found, the

Svensson/Djordjevic model is used for all future simulations.

2.4.3 Conclusion

We conclude that the ports must be uncalibrated in order to obtain realistic simulations when

simulating a nite ground plane. For the settings concerning the simulation mode, the mesh and

the substrate we refer to Sections 2.4.1.1, 2.4.1.2, 2.4.1.4 and 2.3.

These are the settings that are used for all the following examples in this thesis, unless it is explicitly

mentioned dierently.

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1 2 3 4 5−25

−20

−15

−10

−5

0

f (GHz)

S11(dB)

1 2 3 4 5

−60

−40

−20

0

f (GHz)

S21(dB)

1 2 3 4 5

−60

−40

−20

0

f (GHz)

S21(dB)

1.8 1.9 2 2.1 2.2

−30

−20

−10

0

f (GHz)

S21(dB)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S11(deg)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S21(deg)

Figure 2.4.6: Comparison of the EM-simulation results of the coupled line lter with the niteground plane with no calibration (red) to the EM-simulation with the innite ground plane (blue)and the measurements (green). Top left and right: respectively the amplitude of S11 and S21 ofthe innite ground versus nite ground. Middle left: the amplitude of S21 of the measurementsversus nite ground. Middle right: zoom on the frequency band of interest of the amplitude of S21.Bottom left: the phase of S11 of the measurements versus nite ground. Bottom right: the phaseof S21 of the measurements versus nite ground.

Chapter 3

Modeling a microstrip line with

slotted ground plane

3.1 Introduction

This chapter examines the eects of rectangular slots in the ground plane on the frequency response

of a single transmission line. Since the aim is to use these slotted ground plane structures in lter

designs, models are generated to predict these eects. Such models facilitate the design in two ways:

they dene an equivalent circuit that can be used in the synthesis; they provide a cheap alternative

to the numerically expensive EM-simulation (electromagnetic simulation) of Momentum [2] that can

be used to ne-tune the lter or determine the sensitivity of the frequency response function (FRF)

to geometric inaccuracies. The key idea is that this allows to predict the behavior of the design.

The model can then be used as a starting point for our future designs. This is a very important

step towards a proper design procedure since it allows us to make a design that is based on some

fundamental synthesis rather than merely using expensive overnight EM-based optimizations.

Section 3.2 describes the geometry for a single transmission line with slots in the ground plane

underneath it. Section 3.3 discusses a lumped-element equivalent circuit model, introduced in [9].

The model is studied by use of an example (Section 3.4). The EM-simulations, the lumped model

and the measurements of the realized design are compared next. The obtained result is not very

satisfactory since the lumped circuit model is not complex enough to model the structure.

The design is reduced to a transmission line with only one slot in the ground plane in order to restrict

the complexity of the structure (Section 3.5). After the improvements proposed in Sections 3.5.3,

3.5.4 and 3.5.5, the lumped circuit model predicts anti-resonances and quality factors accurately and

is symmetric, which is not the case for the original lumped circuit model. Section 3.6 discusses a

parametric multiple-input-multiple-output (MIMO) model for this one-slot structure to verify how

15

16 CHAPTER 3. MODELING A MICROSTRIP LINE WITH SLOTTED GROUND PLANE

good the improved lumped model represents the transmission line structure by comparing the poles

and zeros.

3.2 Single microstrip line with rectangular slots

The geometry of the structure used in Chapter 3 consists of a single microstrip line with rectangular

slots in the ground plane underneath it (Figure 3.2.1a). The width of the microstrip line is W . The

slots have a length d and a width Ws. The distance between the center of the slots is p. To model

this structure, we consider that it consists of a periodic repetition of an elementary cell as shown in

Figure 3.2.1b. The distance p is also used as the length of the cell. The number of slots is chosen

equal to N .

(a) (b) (c)

Figure 3.2.1: (a): Symmetrical slotted ground plane structure with N = 11. (b): Symmetrical unitcell of length p with slot width Ws, slot length d and microstrip line width W . (c): General unitcell of length p with slot width Ws, slot length d = d1 + d2 and microstrip line width W .

Figure 3.2.1c shows a slot that is coupled asymmetrically with respect to the line, i.e. the slotline

is not fed in its center. In that case, the length of the slot, d, should be split into two parts: d1 and

d2. These lengths are, respectively, the lengths of the slot that are situated 'above' and 'below' the

microstrip line.

This asymmetrical unit cell (Figure 3.2.1c) is more general than the previous one (Figure 3.2.1b)

since it contains the symmetrical case when d1 = d2. Therefore, this unit cell is used as the general

one for the remainder of the text. The inuence of the feedpoint of the slotline is discussed in

Section 3.4.2, when a lumped-element equivalent circuit model is proposed for the general unit cell.

3.3. A LUMPED-ELEMENT CIRCUIT MODEL FOR THE GENERAL UNIT CELL 17

3.3 A lumped-element circuit model for the general unit cell

Figure 3.3.1 shows a lumped-element equivalent circuit model for the general geometry of the unit

cell as introduced in Figure 3.2.1c. The model is proposed in [9]. The equations that link the

geometrical parameters to the electrical parameters are introduced there without any rationale. To

get a better understanding of this model, we study it more in-depth to check its qualities and its

deciencies, as this information is not present in the paper.

To obtain a model for a slotted ground plane structure that consists of N slots, N unit cells should

be cascaded.

Figure 3.3.1: Lumped-element circuit model for a general unit cell, proposed in [9].

The model is built up around a transmission line model for the microstrip line and a slotline model

for the slot in the ground plane. The coupling of the slotline to the microstrip line is represented

by an ideal transformer.

The shunt capacitance C and the series inductance L represent the lumped-element transmission line

model for a transmission line of length p. They are dened using the classical per unit capacitance,

Cpu, and per unit inductance, Lpu, as used in the transmission line model of the telegrapher's

equations [10]. They are given by

ZTL0 =

√LpuCpu

(3.3.1)

⇓

Lpu = Cpu (ZTL0 )2(3.3.2)

Cpu ≈ p

√εTLeff

c0 ZTL0

(3.3.3)

with C = pCpu and L = pLpu. Herein, εTLeff represents the eective permittivity of the microstrip

line, ZTL0 its characteristic impedance and c0 the speed of light in vacuum.

Note that transverse electromagnetic (TEM) propagation of the microstrip line is hereby imposed,

while a microstrip structure is actually built on an inhomogeneous medium (two guided-wave media:


dielectric and air). Hence, a microstrip line does not support a true TEM-wave. The propagation

of the elds along a microstrip line can be approximated by TEM-propagation, which is called

the quasi-TEM approximation. The inhomogeneous medium is then replaced by a homogeneous

medium with an eective permittivity εeff (as is done in Equation (3.3.3)). Note that the eective

permittivity is calculated dierently for a microstrip line (Equation (3.3.4)) and a slotline ([11]).

The microstrip line is also dispersive, which means that its characteristic impedance varies with the

frequency. This implies that Equation (3.3.1) is an approximation for√

Rpu+jωLpu

Gpu+jωCpu, which is only

valid if we assume that the line is lossless. Since the losses of the microstrip line are suciently low,

this approximation holds.

The quasi-static characteristic impedance of the line can be calculated by using the Wheeler closed-

form approximations [12]. The characteristic impedance is determined by the width of the microstrip

line, W , the thickness of the substrate, h, and the eective permittivity of the microstrip line, εTLeff .

The quasi-static characteristic impedance is calculated dierently for wide and for narrow microstrip

lines with respect to the thickness of the substrate using an empirical law.

For narrow microstrip lines (Wh < 1) ZTL0 is calculated by

ZTL0 =60√εTLeff

ln

(8h

W+ 0.25

W

h

)

and for wide strips (Wh ≥ 1) by

ZTL0 =120π√

εTLeff[Wh + 1.393 + 2

3 ln(Wh + 1.44

)]with

εTLeff =εr + 1

2+εr − 1

2

(1 +

10h

W

)−0.5(3.3.4)

The slotline in this circuit is represented by an ideal transmission line, or delay line, that is connected

to ground on both sides and tapped along the line to a transformer as in Figure 3.3.1. The position

of the tap in the circuit model corresponds to the position of the feedpoint of the slotline in the

transmission line structure. The tapped delay line has an electrical length θ = θ1 +θ2, where θ1 and

θ2 dene the position of the tap. They are expressed in radians and dened at a reference frequency

fref , that corresponds to the rst resonance frequency of the slotline.

The electrical length is dened as the product of the propagation constant, βslotm , and the length of

the slot, d1 or d2:

θ1,2 = βslotm d1,2 (3.3.5)

3.4. EXAMPLE: TRANSMISSION LINE WITH SLOTTED GROUND PLANE 19

with

βslotm =ωmvϕ

=2π fmc0√

εsloteff µr

(3.3.6)

fm ≈ mc0

2 d√εsloteff

(3.3.7)

Herein, vϕ represents the phase velocity, d = d1 + d2, m the mth resonance mode of the slotline,

fm the mth resonance frequency of the slotline, εsloteff the eective permittivity of the slotline and µr

the relative permeability of the substrate, which can be considered equal to 1. Note that Equation

(3.3.7) is dened in [9] as an approximation.

An ideal transformer is used to represent the coupling between the slotline and the microstrip line.

The turns ratio of the transformer, n , is dened as

n =

√ZsourceZload

It is approximated in [9] by

n ≈

√ZTL0

Zslot0

(3.3.8)

with Zslot0 the characteristic impedance of the slotline.

Note that TEM-propagation of the slotline is hereby imposed, while the mode of propagation in a

slotline is actually non-TEM as it is almost transverse electric (TE). Hence, the slotline is strongly

dispersive which means Zslot0 strongly varies with the frequency: Zslot0 (f). As mentioned before,

the microstrip line is also dispersive. As a result, n also strongly varies with the frequency: n(f).

Therefore, Equation (3.3.8) is a rather weak approximation. Paper [9] includes a graph showing the

large variety of Zslot0 (f) and n(f). It is suggested to use the values of Zslot0 (f) and n(f) at the rst

resonance frequency to limit the eect of the dispersion.

The characteristic impedance of the slotline Zslot0 and its eective permittivity εsloteff are calculated

by using [11]. In this paper, dierent cases for calculating Zslot0 and εsloteff are proposed, depending

on the relative permittivity of the substrate, εr, the width of the slotline, Ws, and the free space

wavelength at the rst resonance frequency, λ0. We are in the case

2.22 ≤ εr ≤ 3.8 and 0.0015 ≤ Ws

λ0≤ 0.075

3.4 Example: transmission line with slotted ground plane

This section discusses the use and the properties of the previously described model using an example.

The response of the EM-simulations and the corresponding circuit model are compared. The results


of this study allow to draw some conclusions about the usefulness and the correctness of the model.

3.4.1 Geometry and settings

The geometry of the example that is studied is shown in Figure 3.4.1. The used substrate is again

a Rogers R4003 of thickness 1.5 mm (see Section 2.3).

Figure 3.4.1: Layout of a microstrip line with N = 11 slots in the ground plane. The slots aresymmetrically coupled to the microstrip line.

Chapter 2 showed that the simulated S-parameters deviate from the expected real behavior from

a certain frequency on. These deviations remain acceptable up to a frequency of 5 GHz. The

dimensions of the structure are therefore selected to avoid that the frequency band of interest would

exceed 5 GHz. Equation (3.3.7) shows that the rst resonance frequency, f1, is directly related to the

length of the slots, d. In order to make the second and third harmonic repetitions to remain below

5 GHz, d is calculated using Equation (3.3.7) to have a the fundamental frequency f1 ≈ 1.25 GHz.

The range of simulated frequencies is chosen to be from 0.1 GHz to 5 GHz. The other simulator

settings are identical to those used in Chapter 2.

The microstrip line is a 50 Ω-line with a corresponding width of W = 3.4 mm, which was calculated

using the LineCalc Toolbox of ADS [3]. The N = 11 slots in the ground plane are laid out symmet-

rically with respect to the transmission line. All slots have a width Ws of 5 mm. The corresponding

value for Zslot0 is found in Table 3.1. The length of the slots, d, is chosen equal to 80 mm so that

the fundamental frequency f1 is about 1.25 GHz.

The distance between the slots is chosen to be p = 7 mm. This is about 3 % of the wavelength at

the center frequency, where λ ≈ 240 mm. In that case, p is small enough with respect to λ for the

transmission line model of the telegraphers equation to hold.

To fabricate the design, extra delay lines are added to the microstrip line to be able to solder the

connectors without overlapping the slots. The length of these delay lines is set to 10 mm for practical

reasons.

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dit is niet de guided wavelength!! hier gewoon c/f


The corresponding lumped model circuit is shown in Figure 3.4.2. The lumped model-block repre-

sents the unit cell of Section 3.3 and is repeated N = 11 times. The model parameters are given

in Table 3.1. Parameters C, L, f1 and n are calculated from Equations (3.3.3), (3.3.2), (3.3.7) and

(3.3.8) respectively. For the equation for parameter Zslot0 we refer to [11].

Table 3.1: Values of the model parameters.

Parameter Value

C 0.78 pFL 2.0 nHf1 1.24 GHzZTL0 50 OhmZslot0 18e1 Ohmn 0.54

The extra delay lines that were added to the distributed design are also included in the model (see

Figure 3.4.2) in order to be able to compare the two simulations. These lines have a characteristic

impedance Z0 of 50 Ω and an electrical length θ that corresponds to a physical length of 10 mm.

Figure 3.4.2: The complete lumped circuit model that corresponds to the distributed design inFigure 3.4.1.

3.4.2 Lumped-element circuit model vs. EM-simulations

Figure 3.4.3 compares the reection and the transmission properties of the EM-simulations (red)

and the lumped model (blue). The resonance frequency of the transmission line circuit is about 4 %

higher than the one that is calculated for the model via Equation (3.3.7), relative to the expected

resonance frequency of the lumped circuit model. For a slotline with a length d = 80 mm, the


resonance frequency that is calculated from Equation (3.3.7) is f1 = 1.24 GHz. However, the actual

resonance frequency of the structure with slots that have a length d = 80 mm is f1 = 1.29 GHz.

Thus, the resonance frequency that is implemented in the model deviates from the actually realized

one. Although this deviation is not signicant (4 %), this means that the formula for the resonance

frequency (Equation (3.3.7)) is not accurate enough.

The structure behaves as a bandstop lter. The rst stopband starts at a frequency f1. The

bandstop behavior comes from the fact that energy is stored and dissipated in the slots along the

line because of the coupling between the slots and the microstrip line. Since the slots are oriented

perpendicularly to the microstrip line, the coupling between the line and the slots is very tight [13].

The more slots are present beneath the line, the stronger the total coupling will be. Hence, the

more energy is stored and the more conspicuous the bandstop behavior is.

1 2 3 4 5

−30

−20

−10

0

f (GHz)

S11(dB)

1 2 3 4 5−80

−60

−40

−20

0

f (GHz)

S21(dB)

Figure 3.4.3: Comparison of the EM-simulation of the distributed design in Figure 3.4.1 (red) andthe corresponding lumped circuit model (blue). Left: the amplitude of S11. Right: the amplitudeof S21. A frequency shift is visible between the two simulation.

The absence of the second harmonic response is due to the fact that the second order mode of the

resonating slot has a knot that is situated exactly below the center of the microstrip line (see Figure

3.4.4). Since there is almost no eld under the microstrip line, no energy coupling can occur between

the line and the slots. Hence, the slot mode is not excited and this results in the absence of the

corresponding harmonic.


Figure 3.4.4: Transmission line with slots in the ground plane, showing the rst three resonancemodes of the slots and their knots, where the electrical eld is zero.

More generally, this means that we can choose, up to some level, which harmonics we want to excite

and which ones we want to avoid. This is a very nice feature that can be used to lter some harmonic

responses. The third harmonic, for example, is not excited if we position the slots asymmetrically

with respect to line. More specically if the feedpoint of the slotline lies at one third instead of in

its center (see Figure 3.4.5).

Figure 3.4.5: Transmission line with a slot in the ground plane that is coupled at one third of theline. Hence, the feedpoint coincides with a knot of the third mode. This implies that the third modewill not be excited.

Theoretically, the electrical eld underneath the transmission line then becomes zero. In practice

however, the eld will not be exactly zero because the smallest inaccuracy during the fabrication


process can cause that the slot is not perfectly aligned. Hence in practice, the mode will be slightly

hit.

To compensate for the frequency shift that was shown earlier we can manually apply a rst order

frequency correction to the model elements. This correction only aects the reference frequency

for the electrical lengths of the ideal transmission lines in the model (Figure 3.3.1). The results

obtained after the frequency correction are shown in Figure 3.4.6.

From 0.1 GHz up to the rst resonance frequency, f1 ≈ 1.3 GHz, the model ts the results quite

well. This is shown by the error (gray -.-) that is plotted on the amplitude plots of S11 and S21 (top

gures of Figure 3.4.6). The peaks in the error show that the resonance frequencies of the circuit

model and the EM-simulations are still a little bit dierent in this frequency band.

From f1 = 1.3 GHz to about 2 GHz the rst stopband is visible. After the frequency compensa-

tion, the resonance frequency of the slotline in the EM-simulation is now equal to the one of the

corresponding delay line in the circuit model. The slopes at the edge of the stopband of the two

simulations however, still dier signicantly. The slope of the model is much steeper than the

slope of the EM-simulation. This means the quality factor of the resonator is much higher in the

model compared to the EM-simulation. The 3 dB-bandwidth of the two simulations also diers

signicantly.

In the stopband, the attenuation of the circuit model is several orders of magnitude higher than

the attenuation of the EM-simulation. This indicates that the model does not take any losses into

account. This is also clearly visible in the plot of the S11-parameter where almost everything is

reected in the stopband.

The match of the ripple in front of the second resonance is not as good as for the ripple before the

rst resonance. However, this can be improved by tuning the parameters of the model. The quality

of the t of the second resonance is similar to the t of the rst resonance.

A root-mean-square error (RMSE), as dened in Equation (3.4.1), of about −11 dB and −9.5 dB is

obtained over the frequency band of interest for S11 and S21 respectively, which is still high for a

good quality model. This indicates that the overall t is not that good.

RMSESij=

√√√√ 1

N

N∑f=1

[Sij, sim(f)− Sij,mod(f)]2

(3.4.1)

When looking at the phase of S11 and S21 (middle gures), the same observations can be made

as for the amplitudes. The frequency bands right before the resonances show a very good match.

However, from the rst resonance frequency on, the phase of the S-parameters of the circuit model

does not match the EM-simulations anymore. This is also clearly visible when the S-parameters are

plotted on a Smith Chart (bottom gures). On the Smith Charts, only the rst resonance (from

0.5 GHz to 2 GHz) is shown to prevent that the plot would become unreadable.


1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S11(dB)

1 2 3 4 5−80

−60

−40

−20

0

f (GHz)

S21(dB)

1 2 3 4 5−300

−200

−100

0

100

f (GHz)

6S11(deg)

1 2 3 4 5−300

−200

−100

0

100

f (GHz)

6S21(deg)

0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 ∞ 0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 ∞

Figure 3.4.6: Comparison of the EM-simulations of the distributed design in Figure 3.4.1 (red) andthe corresponding lumped model with corrected resonance frequency (blue). Top left: the amplitudeof S11 and the error (gray -.-). Top right: the amplitude of S21 and the error (gray -.-). Middle left:the phase of S11. Middle right: the phase of S21. Bottom left: S11-parameter plotted on the SmithChart. Bottom right: S21-parameter plotted on the Smith Chart. The ripple is matched very well.The bad match of the stopband shows that the model does not capture the stopband behavior.


Figure 3.4.7 compares the simulations to the measurements. Besides some unwanted extra ripples

that are present between 0.1 GHz and 0.8 GHz, the EM-simulations predict the real behavior (mea-

surements) very well. This is veried when looking at the error that is plotted in gray (-.-) (top

gures). Again, the high peaks in the error are explained by a small dierence in the resonance

frequencies of the circuit model and the EM-simulations. An RMSE of about −18 dB and −19 dB is

obtained over the whole frequency band for S11 and S21 respectively, which is denitely acceptable.

The stray ripples present in the measurements can be due to a lower accuracy of the calibration

of the vector network analyzer (VNA) or to the presence of more than one EM-mode, as this can

indeed be present in the real circuit but is not simulated by the EM-simulator.

Both the amplitude as well as the phase show a very good match. This is conrmed when comparing

the S-parameters on a Smith Chart. Again, only the rst resonance (from 0.5 GHz to 2 GHz) is shown

to prevent that the plot would become too complex.

Since we are not satised with the quality we obtain with the current model, we will try to improve

it by tuning the model parameters (Sections 3.4.3).


1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S11(dB)

1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S21(dB)

1 2 3 4 5−300

−200

−100

0

100

f (GHz)

6S11(deg)

1 2 3 4 5−300

−200

−100

0

100

f (GHz)

6S21(deg)

0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 ∞ 0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 ∞

Figure 3.4.7: Comparison of the EM-simulations of the distributed design in Figure 3.4.1 (red) andthe measurements (green). Top left: the amplitude of S11 and the error (gray -.-). Top right: theamplitude of S21 and the error (gray -.-). Middle left: the phase of S11. Middle right: the phase ofS21. Bottom left: S11-parameter plotted on the Smith Chart. Bottom right: S21-parameter plottedon the Smith Chart. The simulations match the measurements very well. Unwanted ripples arepresent between 0.1 GHz and 0.8 GHz, probably due to inaccurate calibration.


3.4.3 Tuned lumped-element circuit model vs. EM-simulations

In an attempt to improve the model, the parameters of the lumped-element circuit model are tuned.

Our goal is to decrease the slope that sets the selectivity and decrease the attenuation of the circuit

model in the rst stopband. At the same time, we want to keep the description of the ripple as it

is.

Figure 3.4.8 shows that the slope can be decreased signicantly by decreasing the capacitor and

inductor value and the value of the turns ratio. After tuning, the slope matches the EM-simulation

quite well. However, the match of the ripple is gone. This is because the ripple is mainly due

to the LC-circuit. Changing the capacitor and inductor values therefore changes the ripple. The

disadvantage is that the slope could not be decreased without changing L or C. Furthermore, the

attenuation in the stopband is not changed. Hence, improving one part of our goal, deteriorates the

quality of the other parts.

1 2 3 4 5

−30

−20

−10

0

f (GHz)

S11(dB)

1 2 3 4 5−80

−60

−40

−20

0

f (GHz)

S21(dB)

Figure 3.4.8: Comparison of the EM-simulations of the distributed design in Figure 3.4.1 (red) andthe corresponding tuned, lumped model (blue). Left: the amplitude of S11. Right: the amplitudeof S21. The match of the slope is better, at the cost of the match of the ripple.

The parameter values that are achieved for the tuned model are given in Table 3.2.

Table 3.2: Parameter values after tuning.

Parameter Value

C 0.44 pFL 0.45 nHf1 1.29 GHzZTL0 50 OhmZslot0 18e1 Ohmn 0.31

3.5. SIMPLIFIED STRUCTURE: A SINGLE SLOT 29

3.4.4 Conclusion

We can conclude that there are some fundamental issues with the proposed circuit model. The

advantage is that the ripple is modeled very well. However, the model does not properly predict the

real in-band behavior of the system. The two main issues are the poor modeling performance of the

losses and the quality factor that the model proposes. The fact that the quality factor of the model

is too high can be attributed to the fact that the transformer is ideal while the coupling between

the slots and the line is not.

The proposed circuit model has also some other shortcomings. As was already mentioned in Section

3.3, the transmission line and the slotline are dispersive because the microstrip structure is built

on an inhomogeneous dielectric medium (two guided-wave media: dielectric and air). The value of

the turns ratio, n, proposed for the transformer by an empirical law, dened by Equation (3.3.8), is

therefore a very weak approximation.

Secondly, the representation of the transmission line by the series inductance and shunt capacitance

is actually only valid for an innitesimal slice of the transmission line or for a transmission line

used below the rst resonance. Here, the LC-equivalent must represent a transmission line slice, p,

of 7 mm, which corresponds to about λ/35 at the center frequency. Since p is about 1.5 orders of

magnitude smaller than λ, this approximation is neither strong nor weak.

Furthermore, there is the fact that we needed to carry out a frequency correction in order to get

the frequency right. All these factors play a signicant role in the quality of the model.

We cannot reach our goal only by tuning the model parameters. Therefore, we will try a dierent

approach. In order to get more insight in the behavior of the structure, we reduce the complexity

of the model structure. To keep the structure as simple as possible, a transmission line with only

one slot in the ground plane is studied. Section 3.5 compares the EM-simulation of this one-slot

structure to the lumped-element circuit model from paper [9].

3.5 Simplied structure: a single slot

Since the previous attempt to t the lumped-element circuit model to the EM-simulations was not

very successful, we take one step back. Instead of using a design with multiple slots, we restrict

ourselves to a single slot structure.


The geometry of the single slot structure is shown in Figure 3.5.1. The same dimensions are used as

in Section 3.4.1 (see Table 3.1). The only model parameter that is changed is the number of slots,

N , which is set to 1 here. In the transmission line structure, the length of the extra delay-lines is

increased to 50 mm so that the length of the transmission line is larger than the length of the slot.


Figure 3.5.1: Layout of a microstrip line with one slot in the ground plane underneath it.

3.5.2 Lumped-element circuit model vs. EM-simulations

Figure 3.5.2 shows the EM-simulation (red) and the lumped model (blue) from 0.5 GHz to 5.5 GHz.

Surprisingly, the resonance frequency of the EM-simulated system is shifted to f1 = 1.6 GHz, while

it was expected to be at 1.3 GHz. The actual resonance frequency is now about 20 % higher than

expected, which is a big deviation. It is obvious that the resonance frequency f1, depends on the

number of slots, N , since this is the only parameter that is changed. In paper [9], however, in the

model there is no dependence of the resonance frequency on N (Equation (3.3.7)). This proves

again that the model is not accurate enough to be used for a practical design.

To properly compare the EM-simulation and the lumped model, a frequency correction was carried

out to compensate for the frequency shift in the same way as in Section 3.4.2. The results of the

lumped model without this correction are not useful. Hence, Figure 3.5.2 immediately shows the

lumped model with the correction applied.

Again, the results are not very satisfactory. Keeping in mind that a frequency correction was

needed beforehand, we can conclude that the model does not t the EM-simulations well. Both

the amplitude and the phase show a large mismatch. An RMSE of about −12 dB and −10 dB is

obtained over the whole frequency band for S11 and S21 respectively, which is again too high. The

error is plotted in gray (-.-) on the amplitude gures. We will again try to improve the t by tuning

the model parameters.


1 2 3 4 5−30

−25

−20

−15

−10

−5

0

f (GHz)

S11(dB)

1 2 3 4 5−30

−25

−20

−15

−10

−5

0

f (GHz)

S21(dB)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S11(deg)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S21(deg)

Figure 3.5.2: Comparison of the EM-simulations of the distributed design in Figure 3.5.1 (red) andthe corresponding lumped model (blue) of a transmission line with one slot in the ground planeunderneath it. Top left: the amplitude of S11 and the error (gray -.-). Top right: the amplitude ofS21 and the error (gray -.-). Bottom left: the phase of S11. Bottom right: the phase of S21. Themodel does not t the EM-simulations.

3.5.3 Tuned lumped-element circuit model vs. EM-simulations

Figure 3.5.3 shows the results for a tuned model. Now, the circuit model response tracks the EM-

simulation better, both for the amplitude and the phase of S11 and S21. Similar to the example with

multiple slots, the in-band attenuation of the model is still much higher than for the EM-simulation.

However the overall envelope of the model matches the EM-simulation quite well.

Remarkable is the anti-resonance that is present in the EM-simulation at a frequency of about

4.8 GHz. This is most probably a spurious mode that is attributed to the nite ground plane.


1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S11(dB)

1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S21(dB)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S11(deg)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S21(deg)

Figure 3.5.3: Comparison of the EM-simulations of the distributed design in Figure 3.5.1 (red) andthe corresponding tuned lumped model (blue). Top left: the amplitude of S11 and the error (gray-.-). Top right: the amplitude of S21 and the error (gray -.-). Bottom left: the phase of S11. Bottomright: the phase of S21. The model t is now more acceptable.

The error is plotted in gray (-.-) on the amplitude gures. An RMSE of about −14 dB and −17 dB

is now obtained for S11 and S21 respectively. This is an improvement, but still not low enough.

Although we can improve the t by tuning the model, the main issues remain. The modeling

performance of the losses and the quality factor is still very poor. Since we are not satised with

the quality of the current model, we will try to improve it by introducing losses in the circuit model

in Section 3.5.5.

Before we do this, another important thing should be remarked. Figure 3.5.4 compares the real and

the imaginary part of S11 and S22. These plots show that the circuit model is not symmetric, while

the transmission line structure clearly is. This is a big disadvantage of the lumped-element circuit

model as suggested in [9]. Therefore, the circuit model is adjusted in Section 3.5.4 in order to get

rid of this disadvantage.


1 2 3 4 5−1

−0.5

0

0.5

1

f (GHz)

Re(Sii)

1 2 3 4 5−1

−0.5

0

0.5

1

Figure 3.5.4: Comparison of the S11- (green) and S22-parameter (blue) of the lumped-element circuitmodel. Top left: the real part of S11 (green) and S22 (blue). Top right: the imaginary part of S11

(green) and S22 (blue).

3.5.4 Symmetrizing the lumped-element circuit model

A big disadvantage of the original lumped-element circuit model is that it is asymmetric. Since the

transmission line structure is a passive structure with a symmetric geometry, it is clearly symmet-

rical. Therefore, an important step in improving the circuit model is to make it symmetrical too.

In order to do so, the LC-circuit, that represents the transmission line, is split in two equal parts

distributed around the slot, as is shown in Figure 3.5.5.

Figure 3.5.5: Lumped-element circuit model for a general unit cell that is made symmetrical.

Hereby, the value of the total capacitance and inductance should remain the same as before (Equa-

tions (3.3.3) and (3.3.2)). To fulll that, the values of the capacitors and the inductors, as dened

in Figure 3.5.5, should be halved:

C =pCpu

2≈ p

√εTLeff

2 c0 ZTL0

L =pLpu

2=pCpu (ZTL0 )2

2

evi.vannechel

Callout

voordeel: 1 LC structure only corresponds to p/2 anymore so assumption of Telegraphers equation that an LC piece represents infinitesimal small piece of TL is more fulfilled


In Figure 3.5.6 the absolute error between S11 and S22 of the real and the imaginary part of the

symmetric lumped model are shown. They perfectly overlap, as we expect. The errors have a

magnitude that is approximately equal to the accuracy of Matlab.

1 2 3 4 50

0.4

0.8

1.2

1.6x 10

−15

f (GHz)1 2 3 4 5

0

0.3

0.6

0.9

1.2x 10

−15

f (GHz)

Figure 3.5.6: The absolute error between S11 and S22 of the symmetrical lumped-element circuitmodel. Top left: the absolute error of the real part. Top right: the absolute error of the imaginarypart. The S11- and S22-parameters are perfectly overlapping as expected.

Making the circuit model symmetrical is however not enough to fulll all our initial goals. Therefore,

we will make an attempt to do so in the next section.

3.5.5 Introducing losses

In an attempt to improve the modeling performance, we introduce losses in the equivalent circuit

by including resistors at dierent positions in the symmetric circuit model. Hereby, care is taken to

keep the symmetry of the circuit model. The goal is still to decrease the slope and the attenuation

in the stopband, while keeping the match of the ripple as much as possible.

In order to check the eects separately, the following losses are introduced one by one:

1. The ideal inductor is replaced by an inductor with an equivalent series resistance to model

the skin eect (see Figure 3.5.7). Modeling the skin eect requires a resistance that varies

with the frequency, R(f). We use a simplied model for the skin eect by using a constant

resistance at a frequency that is equal to the rst resonance frequency, R(f1).


Figure 3.5.7: Symmetrical lumped-element circuit model with a non-ideal inductor.

2. The ideal capacitor is replaced by a capacitor with an equivalent parallel resistance to model

the dielectric material losses (see Figure 3.5.8).

Figure 3.5.8: Symmetrical lumped-element circuit model with a non-ideal capacitance.

3. The two ideal transmission lines are replaced by equivalent microstrip lines with dielectric and

skin losses.

4. The LC-circuit is replaced by an equivalent microstrip line with dielectric and skin losses.

5. A resistor is added in series with the ideal transmission lines (see Figure 3.5.9) to model the

losses in the dielectric.

Figure 3.5.9: Symmetrical lumped-element circuit model with an additional resistor in series withthe ideal transmission lines.

6. A resistor is added in parallel with the whole circuit model (see Figure 3.5.10). This resistor

is added for the following reason. In the original, symmetrical circuit model the signal is com-

pletely reected at the resonance frequency. In the transmission line structure however this is


not the case. By adding a resistor from the input to the output port, the total reection as

predicted by the circuit model disappears and the reection of the transmission line structure

can be modeled well.

Figure 3.5.10: Symmetrical lumped-element circuit model with an additional resistor in parallelwith the whole circuit model.

Trial (1) results in a decrease of the level of S11 and S21 over the whole frequency range, as shown

in Figure 3.5.11. Note that the result of the simulations with multiple slots is shown here because

the eect was more clear than in the simulations with only one slot. The slope of the stopband

transition is a bit less steep than before. This corresponds to an S11-parameter that decreases more

smoothly in the rst stopband (see rectangle in left gure). The match of the ripple is slightly better

(see rectangle in right gure). An RMSE of about −11 dB and −10 dB is obtained for S11 and S21

respectively, which is a small improvement compared to Figure 3.4.6.

1 2 3 4 5

−30

−20

−10

0

f (GHz)

S11(d

B)

1 2 3 4 5−80

−60

−40

−20

0

f (GHz)

S21(d

B)

Figure 3.5.11: Comparison of the EM-simulations of the distributed design in Figure 3.4.1 (red) andthe corresponding symmetrical lumped model with introduction of an equivalent series resistanceat the inductor (blue). Left: the amplitude of S11. Right: the amplitude of S21. The rectanglesindicate the improvements with respect to the lossless model from Figure 3.4.6.


Unfortunately, trials (2), (3) and (4) do not show a considerable improvement. Therefore, these

results are not shown and the adaptation of the circuit is not performed.

By introducing resistors (5) and (6), however, a considerable improvement is noticed. Adding a

resistor in series with the ideal transmission lines allows us to set the attenuation (see Figure 3.5.12,

top right), and hence also the reection (see Figure 3.5.12, top left), of the frequency band around

the resonance frequencies that are not excited (black rectangles). Adding a resistor in parallel with

the whole circuit model allows us to set the attenuation (see Figure 3.5.12, top right), and hence

also the reection (see Figure 3.5.12, top left), of the frequency bands around the excited resonance

frequencies (green rectangles).

Figure 3.5.12 shows the results after adding these two resistors and tuning the parameters. The

overall shape of the amplitude of S11 and S21 matches very well. Also the phase of both S11 and

S21 shows a very good match. Spurious modes are present at about 3.4 GHz and around 5 GHz.

These are attributed to the nite ground plane.

The slope of the circuit model now matches the slope of the EM-simulations. Also the attenuation

of the circuit model is decreased at the resonance frequencies. This means our two goals are now

reached. An RMSE of about −18 dB and −20 dB is obtained for S11 and S21 respectively, which

is a very big improvement compared to Figure 3.5.2 and an acceptable level. The S-parameters

are plotted on a Smith Chart in the bottom gures of Figure 3.5.12 over a range of 0.5 GHz to

2.5 GHz. Only the rst resonance is shown to prevent that the plot becomes too complex. The good

match of the improved lumped model and the EM-simulations is clearly visible. There are still some

unwanted spurious modes present in the EM-simulations, but since we believe they are due to the

nite ground plane eect, we will not try to model them.


1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S11(d

B)

1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S21(d

B)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S11(deg)

1 2 3 4 5−200

−100

0

100

200

f (GHz)

6S21(deg)

0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 \infty 0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 \infty

Figure 3.5.12: Comparison of the EM-simulations of the distributed design in Figure 3.5.1 (red) andthe corresponding symmetrical lumped model (blue) after improving the model by introducing theresistors (1), (5) and (6). Top left: the amplitude of S11 and the error (gray -.-). Top right: theamplitude of S21 and the error (gray -.-). Middle left: the phase of S11. Middle right: the phase ofS21. Bottom left: S11-parameter plotted on the Smith Chart. Bottom right: S21-parameter plottedon the Smith Chart. The model t is now acceptable. There are still some unwanted spurious modespresent in the EM-simulations, but we are not interested in them so they are not modeled.


To investigate whether or not the material losses are the reason for the dierence in behavior, the

reverse test is carried out. Instead of adding losses to the model, the substrate of the transmission

line simulation is made completely lossless. The results are shown in Figure 3.5.13.

1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S11(dB)

1 2 3 4 5−40

−30

−20

−10

0

f (GHz)

S21(dB)

Figure 3.5.13: Comparison of the EM-simulations of the lossy distributed design in Figure 3.4.1(red) and the lossless distributed design (blue). Left: the amplitude of S11. Right: the amplitudeof S21.

Unfortunately, this result shows that the material losses do not inuence the S-parameters very

much at all. The substrate is so good that leaving out the losses does not change the simulations

signicantly.

3.5.6 Conclusion

A rst step in improving the model is to symmetrize it. This is a logical step since it is known

that the transmission line circuit that is to be modeled is symmetrical. The circuit model is made

symmetrical by splitting the equivalent series inductance and parallel capacitance of a transmission

line into two equal parts.

As a second step, losses were introduced into the model. The equivalent series resistance at the

inductor results in a small improvement (Figure 3.5.11). The additional resistor in series with the

transmission lines and the resistor in parallel with the whole circuit allow us to improve the circuit

model considerably (Figure 3.5.12). By including these two resistors we were able to reach our

two goals: 1) to decrease the slope of the circuit model such that it matches the slope of the EM-

simulation; 2) to decrease the attenuation of the circuit model at the resonance frequencies in order

to match the attenuation that is obtained via the EM-simulation.

The other trials however did not improve the match between the transmission line simulations and

the model simulations signicantly. Hence, they are deleted again and are not considered for further

use.


Because we want to see how good the obtained, improved circuit model represents the transmission

line structure, we will t a model of a rational form on both the circuit model and the EM-simulations

in Section 3.6.

3.6 Modeling a single slot structure: the MIMO-model

In this section, a parametric MIMO-model is t to both the lumped-element circuit model and the

EM-simulations of a single slot structure. The model is a parametric MIMO-model with common

denominator. The selected MIMO-models for the lumped circuit model and the EM-simulations are

chosen to have the same order and their poles and zeros are compared.


The one-slot-structure that was used in Section 3.5 has a resonance frequency at 1.6 GHz. Hence, its

third resonance mode is already close to 5 GHz. Since we want to restrict the maximum simulation

frequency to 5 GHz at the most, we would like to decrease the rst resonance frequency of the

structure. We have seen before that increasing that length of the slotline(s) results in a decrease of

the resonance frequency. Therefore, a structure with a longer slotline is used in this section.

The geometry of the structure remains the same as in Section 3.5 (see Figure 3.5.1). The transmission

line has a characteristic impedance of 50 Ohm and a corresponding width of 3.4 mm. The width

of the slotline remains 5 mm, but the length is increased to 160 mm so that the rst resonance

frequency, calculated from Equation (3.3.7), is about 600 MHz. We select a frequency simulation

range from 500 MHz to 3 GHz with a frequency step of 20 MHz.

3.6.2 Improved lumped-element circuit model vs. EM-simulations

Figure 3.6.1 compares the improved circuit model (blue) to the EM-simulations (red). The real

resonance frequency is about 800 MHz instead of the expected frequency of 600 MHz. Note that this

frequency shift is already compensated in Figure 3.6.1.

An RMSE of about −22 dB and −23 dB was obtained for S11 and S21 respectively. That shows that

the improved model represents the transmission line structure very well. The match is good for the

amplitude as well as for the phase over the whole frequency band of interest. This is shown in the

top gures, where the error is plotted in gray (-.-), and the middle gures of Figure 3.6.1. The good

quality of the t is also clearly visible when the S-parameters are plotted on a Smith Chart (bottom

gures). Some spurious modes are present around 1.8 GHz in the EM-simulation. It is our belief

that they are devoted to the nite size of the ground plane.

3.6. MODELING A SINGLE SLOT STRUCTURE: THE MIMO-MODEL 41

0.5 1 1.5 2 2.5 3−40

−30

−20

−10

0

f (GHz)

S11(dB)

0.5 1 1.5 2 2.5 3−40

−30

−20

−10

0

f (GHz)

S21(dB)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S11(deg)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S21(deg)

0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 \infty 0.2

0.5

1.0

2.0

5.0

+j0.2

−j0.2

+j0.5

−j0.5

+j1.0

−j1.0

+j2.0

−j2.0

+j5.0

−j5.0

0.0 \infty

Figure 3.6.1: Comparison of the EM-simulations of the distributed design in Figure 3.5.1 (red)and the corresponding improved lumped model (blue) of a transmission line with one slot in theground plane underneath it. Top left: the amplitude of S11 and the error (gray -.-). Top right: theamplitude of S21 and the error (gray -.-). Middle left: the phase of S11. Middle right: the phase ofS21. Bottom left: S11-parameter plotted on the Smith Chart. Bottom right: S21-parameter plottedon the Smith Chart. The model t is now acceptable. Spurious modes are present around 1.8 GHzdue to the nite ground plane.


3.6.3 MIMO-model

In this section, we want to verify how good the improved lumped circuit model represents the EM-

simulations. To this end, we compare the position of the poles and zeros of both tted linear time-

invariant (LTI) models. We estimate a 2x2 parametric MIMO-model with a common denominator

starting from both the EM-simulation data and the lumped circuit model data. The Frequency

Domain Toolbox [14] is used for this purpose.

We start with the estimation of a common denominator MIMO-model using the EM-simulation data.

The parametric MIMO-model is estimated for the complete S-matrix of the lter. The available

simulation data represent the sampled S-parameter data of the lter, evaluated on a pre-specied

frequency grid. Only the simulation data that describe the rst resonance (range [0.5 GHz,1.5 GHz]

with a frequency step of 20 MHz) is used to estimate the MIMO-model, both in order to restrict the

model complexity and because the simulation data at higher frequencies may not prove to be very

accurate.

The simulated S-parameters, for which we want to estimate a MIMO-model, are evaluated on the

simulation frequency grid fi, 1 < i < N :

V ec(STEM (fi)) =

S11,EM (fi)

S12,EM (fi)

S21,EM (fi)

S22,EM (fi)

(3.6.1)

with SEM (fi) the S-matrix of the EM-simulation data.

The estimator requires an input signal to be useable, and this is set to a constant value to indicate

that an FRF is to be estimated.

The response of the system is then set to

V ec(STMIMO(fi)) =

S11,MIMO(fi)

S12,MIMO(fi)

S21,MIMO(fi)

S22,MIMO(fi)

with 1 < i < N and where SMIMO(fi) represents the parametric 2x2 MIMO-model of a rational

form of the S-matrix of the device under test (DUT).

The covariance matrix of these simulation data is chosen as in Equation (3.6.2) to impose a relative

error criterion on the model t.


Cov(SEM (fi)) =

1% |S11,EM (fi)|2 0 0 0

0 1% |S12,EM (fi)|2 0 0

0 0 1% |S21,EM (fi)|2 0

0 0 0 1% |S22,EM (fi)|2

(3.6.2)

In the Sampled Maximum Likelihood (SML) framework, Cov(SEM (fi)) represents the covariance

matrix of SEM (fi) with 1 < i < N . Note that the covariance matrix represents the simulation

error of ADS [3] since we are dealing with simulations and not with measurements. Therefore, this

quantity is extremely hard to get as it is almost impossible to force a dierent realization of the

noise. Since we do not have any information available about the correlation, we choose it to be zero.

The cost function is then dened as:

LK =

N∑i=1

eT (fi)Cov(SEM (fi)) e(fi)

with e(fi) = |V ec(SEM (fi))− V ec(SMIMO(fi))| the error to be minimized.

The SML estimator is used to estimate the parametric MIMO-model. The estimation and validation

process is not discussed in detail here since this is not our main focus and the measurement based

criteria do not really hold in this simulation context. For the full theoretical framework we refer to

[15].

The best model t is obtained for a model order [NB + 1 , NA + 1] of [5, 5], with NB the number of

zeros and NA the number of poles. Figure 3.6.2 and 3.6.3 respectively show the t of the amplitude

and phase of the S-parameters of the MIMO-model (full blue line) to the EM-simulation data (full

red line). The user selected standard deviation of the EM-simulated FRFs (1 % relative error)

(dashed blue line) is compared to the obtained model error (dashed gray line (-.-)) to assess the

quality of the estimated model.

Even if the simulation of the transmission line structure were perfect, the presence of this model

error is expected. We t a model of a rational form to the transcendental simulation data and

clearly these functions do not behave equally. However, the simulation data actually represent a

meromorphic function. Since the simulation frequency band is nite, the meromorphic transcen-

dental function can be approximated with an arbitrary precision by a rational form of nite order

[16]. This approximation error will decrease if the model order of the rational function is increased.

However, increasing the model order will also increase the model complexity and the model might

become unstable. Hence, a good balance between the model variability (model order) and the model

complexity is needed and is obtained here for an order of [5, 5].

To get a better idea of the average level of the model error, the RMS-value of the model error is

calculated for each S-parameter:


RMSE =

[RMSES11 RMSES12

RMSES21 RMSES22

]=

[−29 −31

−31 −28

]dB

The RMSE-values are similar to the level of the user selected standard deviation of the to be modeled

FRFs (Equation (3.6.2)). Based on this knowledge we may assume there is a good balance between

the model complexity and the model variability.

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S11(dB)

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S12(dB)

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S21(dB)

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S22(dB)

Figure 3.6.2: Comparison of the amplitudes of the parametric MIMO-model (blue) and the EM-simulation data (red) and comparison of the user selected standard deviation (Equation (3.6.2)) ofthe EM-simulation data (dashed blue) and the model error (gray -.-). Top left: S11. Top right: S12.Bottom left: S21. Bottom right: S22.

Note that the S11- and the S22-parameter of the EM-simulations are not perfectly equal (this is

clearly noticeable when looking at the RMSE values). This implies that the structure is asymmetri-

cal. However, we know that the transmission line structure modeled here is perfectly symmetrical.


The small deviation between S11 and S22 can therefore only be attributed to simulation inaccuracies

of ADS [3].

0.4 0.6 0.8 1 1.2−500

−400

−300

−200

−100

f (GHz)

6S11(deg)

0.4 0.6 0.8 1 1.2−200

−100

0

100

200

f (GHz)

6S12(deg)

0.4 0.6 0.8 1 1.2−200

−100

0

100

200

f (GHz)

6S21(deg)

0.4 0.6 0.8 1 1.2−500

−400

−300

−200

−100

f (GHz)

6S22(deg)

Figure 3.6.3: Comparison of the phases of the parametric MIMO-model (blue) and the EM-simulation data (red). Top left: S11. Top right: S12. Bottom left: S21. Bottom right: S22.

In a second step, a common denominator MIMO-model is t to S-matrix of the lumped-element

circuit model data evaluated on the same frequency grid. The input and output matrices and their

sample noise covariances are again dened as in Equations (3.6.1) and (3.6.2).

We now want to compare the S-matrix response of the estimated MIMO-model to the one that was

t on the EM-simulations. Therefore, the model order is chosen to be the same as for the EM-based

MIMO-model, namely [5, 5].

Figure 3.6.4 and 3.6.5 respectively show the t of the amplitude and the phase of the S-parameters

of the MIMO-model (full blue line) to the lumped model data (full red line).


The user selected standard deviation of the FRFs of the lumped circuit model (dashed blue line)

is again compared to the model error (dashed gray line (-.-)) to assess the quality of the estimated

model. The RMS-values of the model error are similar to the level of the user selected standard

deviation (Equation (3.6.2)):

RMSE =

[RMSES11

RMSES12

RMSES21RMSES22

]=

[−28 −32

−32 −28

]dB

Again, a good balance exists between the model complexity and the model variability.

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S11(dB)

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S12(dB)

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S21(dB)

0.4 0.6 0.8 1 1.2−60

−45

−30

−15

0

f (GHz)

S22(dB)

Figure 3.6.4: Comparison of the amplitudes of the parametric MIMO-model (blue) and the lumpedmodel data (red) and comparison of the user selected standard deviation of the FRFs of the lumpedmodel (dashed blue) and the model error (gray -.-). Top left: S11. Top right: S12. Bottom left:S21. Bottom right: S22.

Note that the S11- and the S22-parameter of the lumped model are indeed perfectly equal. This is


as expected since the lumped model was made symmetrical in Section 3.5.4.

0.4 0.6 0.8 1 1.2−500

−400

−300

−200

−100

f (GHz)

6S11(deg)

0.4 0.6 0.8 1 1.2−200

−100

0

100

200

f (GHz)

6S12(deg)

0.4 0.6 0.8 1 1.2−200

−100

0

100

200

f (GHz)

6S21(deg)

0.4 0.6 0.8 1 1.2−500

−400

−300

−200

−100

f (GHz)

6S22(deg)

Figure 3.6.5: Comparison of the phases of the parametric MIMO-model (blue) and the lumpedmodel data (red). Top left: S11. Top right: S12. Bottom left: S21. Bottom right: S22.

Let us rst introduce some names for simplicity.

EM MIMO-model: the parametric MIMO-model that is estimated on the EM-simulation data

circuit MIMO-model: the parametric MIMO-model that is estimated on the lumped circuit

model data

Figure 3.6.6 compares the poles and zeros of EM MIMO-model (red) and circuit MIMO-model

(blue). All S-parameters have the same poles, since the MIMO-model is a common denominator

model. Note that the imaginary axis is scaled by 2π to convert the angular frequency ω = 2πf into

frequency f . Remark that the MIMO-models are both stable. The dashed blue lines indicate the

frequency range for which the model is extracted.


0 5 10 15

x 109

−5

0

5

x 109

−1 0 1 2

x 109

−1

−0.5

0

0.5

1

x 109

−1 0 1 2

x 109

−1

−0.5

0

0.5

1

x 109

0 5 10 15

x 109

−5

0

5

x 109

Figure 3.6.6: Comparison of the the poles and zeros of the parametric MIMO-model estimated onthe EM-simulation data (red) and the parametric MIMO-model estimated on the lumped modeldata (blue). The dashed blue lines indicate the frequency range of operation. Top left: S11. Topright: S12. Bottom left: S21. Bottom right: S22.

Tables 3.3 and 3.4 give the relative deviations, ξ, between the poles and zeroes of the two MIMO-

models respectively. The errors of the real and imaginary part are evaluated separately. The relative

error for e.g. the real part of a pole is calculated as follows:∣∣∣∣Re(pole of the EM MIMO model)− Re(pole the circuit MIMO model)

Re(pole of the EM MIMO model)

∣∣∣∣


Table 3.3: Relative errors, ξ, of the real, Re, and imaginary part, Im, of the poles of EM MIMO-model and circuit MIMO-model. Since it is a common denominator model, all S-parameters havethe same pole pairs (PP).

PP 1 PP 2ξ(Re) (%) 9.6 0.25ξ(Im) (%) 0.89 3.0

The real part of the pole pair (PP) 1 shows a deviation of about 10 %, which is very high, and

explains the dierence in damping present in the FRFs. The PP 1 of EM MIMO-model is located

at a larger distance from the imaginary axis than PP 1 of circuit MIMO-model. This means that

the quality factor of circuit MIMO-model is still 10 % higher than the one of EM MIMO-model.

We can conclude that the quality factor of the poles in the EM-simulation is still not modeled well

enough by the lumped-element circuit model.

The deviation of the real part of the zero pair (ZP) 1 of the S12 and S21-parameters is quite high.

Note that the match of ZP 1 of S22 is much better than the match of ZP 1 of S11. We expect

a similar match would for both since the structure is symmetric, but this is not the case due to

simulations inaccuracies of ADS [3].

The other PPs and ZPs show a very good match. Remark that the errors of the imaginary parts

of all PPs and ZPs are acceptable. This is a good thing since the imaginary part is related to the

frequency of the poles and zeros. We can conclude that the frequencies at which the poles and zeros

are located show a very good match between the two MIMO-models.

Table 3.4: Relative errors of the real and imaginary part of the poles of EM MIMO-model andcircuit MIMO-model. Parameter S12 and S21 have the same zero pairs (ZP) because the circuit issymmetric. The ZPs of S11 and S22 of EMMIMO-model slightly dier due to simulation inaccuraciesof ADS [3].

S11 ZP 1 S12 ZP 1 S12 ZP 2 S22 ZP 1ξ(Re) (%) 6.5 6.2 2.1 4.5ξ(Im) (%) 0.28 2.1 0.57 3.2

Note that the remaining zeros of S11 and S22 are not discussed yet. We see that these zeros are

completely dierent for the two MIMO-models. This is not a very big problem since we know

that positive real zeros, called minimum phase zeros, are necessary to make the phase delay, that is

present in Figures 3.6.3 and 3.6.5, possible. The fact that these zeros dier for the two MIMO-models

is not a problem since we could make these zeros disappear by doing a phase correction.

3.6.4 Conclusion

We conclude that the improved lumped-element circuit model matches the EM-simulations quite

well. The level of agreement is shown by estimating a parametric MIMO-model using the EM-


simulation data and the lumped circuit model data and comparing the obtained poles and zeros.

The poles of the lumped-element circuit model do indeed match the poles of the EM-simulation data.

Hence, the circuit model captures the essence of the behavior of the transmission line structure.

An important disadvantage is that the quality factor of the poles and zeros is still not modeled very

well. To improve on the modeling of the quality factor, the element(s) of the circuit that create

this damping should be located. Next, we should check whether it is possible to adjust the element

values of the losses to improve the modeling of the quality factor.

3.7 Conclusion

It was shown in this chapter that the transmission line structure with slotted ground plane behaves

as a bandstop lter. Hence, this structure can be used to attenuate the harmonic responses in lter

design. An extra benet is that we can choose up to some level which harmonics we want to excite

simply by the selecting the position of the feedpoint of the slot(s).

A lumped element circuit model was generated that is symmetric and able to accurately predict the

quality factors and anti-resonances of the transmission line structure. Hence, this lumped model is

a very important step towards a proper lter design with slotted ground plane.

Chapter 4

Coupled line with slotted ground

plane

4.1 Introduction

In Chapter 3, we examined a transmission line with a periodic repetition of slots in the ground

plane underneath it. The next step is to examine the use of this slotted ground plane structure in

combination with a coupled line in an attempt to use it in lter design later on. Since introducing

a coupled line results in the introduction of an extra coupling next to the coupling to the slots

and changes the eld pattern quite extensively, there is an extra complication in the simulation.

It is possible that the EM-simulator of ADS [3] is not really capable anymore to simulate the real

behavior of this more complex structure correctly and accurately enough for our modeling purposes.

Therefore this chapter is devoted to the verication of the correctness of the EM-simulations for

coupled line structures with slotted ground plane.

Section 4.2 describes the geometry of the coupled line structure itself. Section 4.3 compares the

simulations and the measurements of three dierent coupled line structures. The three structures

are selected to show dierent steps in the validation process. First, a structure without any slots is

realized as a reference element. Next, a coupled line with one slot is used to see and understand the

inuence of the slot coupling on the line coupling. Finally, a structure with three slots is realized

to represent a practically useable lter. Section 4.4 discusses the eects of the slots in the ground

plane underneath the coupled line.

51

52 CHAPTER 4. COUPLED LINE WITH SLOTTED GROUND PLANE

4.2 Geometry and settings

A simple coupled line structure is designed as shown in Figure 4.2.1. The microstrip lines have a

width W = 3.4 mm corresponding to a characteristic impedance of 50 Ohm. The lines are coupled

over λ/2 and the resonance wavelength, λ, is selected to correspond to a resonance frequency of

1 GHz. This yields λ ≈ 176 mm for the considered substrate (see Section 2.3). The spacing between

the lines, S1, is set to 1.5 mm. Hereby, the spacing is small enough so that coupling between the

lines occurs, but not too small to make sure the design can still be fabricated accurately and a

potential increase in the coupling between the lines can easily be noticed.

Three dierent designs of similar structures are realized: one without any slots, one with only one

slot and one with three slots in the ground plane. Figure 4.2.1 only shows the structure with one

slot. The other two are similar and the slots are always arranged symmetrically with respect to

the coupled line section.. The approach explained in Chapter 3 is used to design the dimensions.

The slots have a width Ws = 5 mm, which corresponds to a characteristic impedance of the slotline

of Zslot0 ≈ 14e1 Ohm. The length of the slotlines is d = 60 mm, which corresponds to a resonance

frequency of f1 ≈ 1.7 GHz. The length of a unit cell is p = 8 mm. The slots have two feedpoints

that are positioned symmetrically with respect to the lines in both directions.

Figure 4.2.1: Coupled line structure with slots in the ground plane underneath it. The slots havetwo feedpoints that are arranged symmetrically with respect to the lines. The spacing between thelines is S1. The length of the coupled lines is λ

4 . The slots have a width Ws = 5 mm and a lengthd = 60 mm. The length of a unit cell is p = 8 mm.

The structures are simulated over a frequency range of 100 MHz to 3 GHz with a frequency step of

20 MHz. For the settings of the EM-simulator is referred to Chapter 2.

4.3. COMPARING EM-SIMULATIONS TO MEASUREMENTS 53

4.3 Comparing EM-simulations to measurements

Figure 4.3.1 compares the EM-simulations (red) to the measurements (green) of the structure with-

out slots in the ground plane, as described in Section 4.2. Since no slots are present, we expect that

the simulations predict the real behavior very well because this type of line is one of the classical

structures simulated with ADS [3].

0.5 1 1.5 2 2.5 3−1

−0.8

−0.6

−0.4

−0.2

0

f (GHz)

S11(dB)

0.5 1 1.5 2 2.5 3−60

−50

−40

−30

−20

−10

f (GHz)

S21(dB)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S11(deg)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S21(deg)

Figure 4.3.1: Comparison of the EM-simulations (red) and the measurements (green) of the coupledline structure, introduced in Figure 4.2.1, without slots. Top left: the amplitude of S11. Top right:the amplitude of S21 and the error (gray -.-). Bottom left: the phase of S11. Bottom right: thephase of S21. The EM-simulations match the measurements very well.

As expected, the EM-simulations predict the real behavior of the structure very well. This is proven

by looking at the value of the RMSE. An RMSE of about −32 dB and −39 dB is obtained over the

frequency band of interest for S11 and S21 respectively. The magnitude of the squared complex

dierence (gray -.-) between the simulation and the measurement of S21 is shown in Figure 4.3.1


(top right). For S11 the error is not shown since the overall error is about 30 % lower than S11,

relative to the amplitude of S11 (see RMSE-value of S11 given previously). The ripples in the

error are explained as follows. The measured data have an irrational form. The EM-simulation is

an approximation for this irrational form. Therefore, the simulated data are wrapped around the

measured data, resulting in ripples.

A small dierence in time delay is present between the simulations and the measurements. This can

be explained by the approximation that is used by ADS [3] to calculate the eective permittivity

and the position of the reference plane of the calibration of the VNA. An intermediate connection

piece with a length of about 0.5 mm was present during the measurements that is not accounted for

in the measurement data that are shown. At 3 GHz this length corresponds to a delay of about 30°,

which is consistent with the delay that is present in Figure 4.3.1 .

Figure 4.3.2 and Figure 4.3.3 compare the EM-simulations (red) to the measurements (green) of the

coupled line structure with respectively one slot and three slots in the ground plane.

The error (gray -.-) is shown on the amplitude plots of S11 (top left) and S21 (top right) on Figures

4.3.2 and 4.3.3. The ripples in the error are again explained by the fact that the simulated data are

wrapped around the measured data.

4.3. COMPARING EM-SIMULATIONS TO MEASUREMENTS 55

0.5 1 1.5 2 2.5 3−60

−45

−30

−15

0

f (GHz)

S11(dB)

0.5 1 1.5 2 2.5 3−60

−45

−30

−15

0

f (GHz)

S21(dB)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S11(deg)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S21(deg)

Figure 4.3.2: Comparison of the EM-simulations (red) and the measurements (green) of the coupledline structure, introduced in Figure 4.2.1, with one slot. Top left: the amplitude of S11 and theerror (gray -.-). Top right: the amplitude of S21 and the error (gray -.-). Bottom left: the phase ofS11. Bottom right: the phase of S21. The EM-simulations match the measurements very well.

The obtained RMS-values for the two structures are given in Table 4.1. For the structure with three

slots, the RMS-values for S11 and S21 are higher than for the one-slot structure. This is due to the

high error peak that is present in the three-slot structure (Figure 4.3.3) around 1.5 GHz. This error

peak is present due to a small shift in the resonance frequency between the EM-simulations and

the measurements around 1.5 GHz. This peak pushes the RMSE to higher values as what can be

expected.

Table 4.1: RMSE values for the coupled line structures with 1 and 3 slots.

1 slot 3 slotsS11 −28 dB −27 dBS21 −32 dB −25 dB


0.5 1 1.5 2 2.5 3−60

−45

−30

−15

0

f (GHz)

S11(dB)

0.5 1 1.5 2 2.5 3−60

−45

−30

−15

0

f (GHz)

S21(dB)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S11(deg)

0.5 1 1.5 2 2.5 3−200

−100

0

100

200

f (GHz)

6S21(deg)

Figure 4.3.3: Comparison of the EM-simulations (red) and the measurements (green) of the coupledline structure, introduced in Figure 4.2.1, with three slots. Top left: the amplitude of S11 and theerror (gray -.-). Top right: the amplitude of S21 and the error (gray -.-). Bottom left: the phase ofS11. Bottom right: the phase of S21. The EM-simulations match the measurements very well.

Both EM-simulations show a good prediction of the real behavior of the structure. The simulated

resonances are clearly present in the measurements and all the measured resonances are predicted by

the EM-simulator. This clearly shows that the EM-simulator may be used in a real design context.

The match of the phases is also very good, except for a small dierence in time delay due to

the calibration plane of the VNA and the approximation of the eective permittivity in ADS [3].

However, this dierence was also present in the design without slots. This means this delay dierence

is not related to the slots but has the same origin as in the design with the full ground plane.

4.4. EFFECT OF THE SLOTS 57

4.4 Eect of the slots

Comparing Figures 4.3.1, 4.3.2 and 4.3.3 shows the eect of introducing one or more slots in the

ground plane underneath a coupled line.

First of all, the maximum forward transmission, S21, of the structure without slots (Figure 4.3.1)

reaches more or less −10 dB. This is much lower than the maximum of S21 of the structure with

one or three slots, where S21 arrives at almost 0 dB. This means that introducing slots underneath

the coupled line increases the coupling between the lines signicantly.

This is an advantage that we can try to exploit for lter design. In wideband lters a strong coupling

is required between the coupled lines. To achieve a strong coupling in a classical coupled line setting,

the gap between the lines is decreased accordingly. However, decreasing the dimensions is limited

by the accuracy of the production process. The strength of the coupling of the lines is therefore also

limited. This is where introducing slots in the ground plane can bring a solution. As they allow to

increase the coupling of the lines without having to push the dimensions of the structure to their

lower limit, they can pave the way to broadband designs with low insertion losses.

Explaining the eect of the slots on the shape of the amplitudes of S11 and S21 is not straight-

forward. Further research on the modeling of these eects is necessary to come up with a design

procedure.

4.5 Conclusion

We can conclude that the EM-simulation of a coupled line structure with slots in the ground plane

is very reliable. This is a big advantage if we want to use these kind of structures for lter design

procedures.

An advantage of introducing slots in the ground plane of a coupled line is that the slots can increase

the coupling between the line. In lter design, this can be of great benet to increase the bandwidth

of the lters keeping the insertion losses low and without having to decrease the dimensions of the

lter signicantly. The interpretation of the overall eects of the slots on the coupled line is not

straight-forward and requires further research.

Chapter 5

Conclusion and future perspectives

The aim of this thesis is to examine the use of slots in the ground plane of coupled microstrip line

lters and to investigate their use in practical lter design procedures, hereby working towards the

design of wideband lters with low insertion losses. This research yields ve main results.

In the rst part of the thesis we examine and adapt the settings of the EM-simulator that are required

to accurately simulate a nite ground plane. The validity of the settings that are required to properly

simulate a slotted ground plane is veried by comparing the EM-simulations to the measurements

of the realized designs. We show that the EM-simulator is indeed capable of simulating defective

ground plane structures accurately.

Secondly, we examine a simple microstrip line with a slotted ground plane. Simulations and mea-

surements that this structure behaves as a bandstop lter. This eect can be used in lter design

to attenuate the spurious harmonic responses that are present in the classical parallel-coupled line

lters. Moreover, the position of the feedpoint of the slots can be used to select, up to some level,

which harmonics are attenuated and which are not.

Next, we investigate an existing lumped-element equivalent circuit model that should predict the

behavior of the slots. The original lumped circuit model is not accurate enough to be used in

practical design purposes. We propose several improvements to the original lumped circuit model

that remediate its shortcomings. First of all the improved lumped model is made symmetric. This

is necessary because the transmission line structure is also clearly symmetric by nature. Secondly,

the model is adapted to better predict the quality factor and the anti-resonances.

A parametric MIMO-LTI-model is estimated next starting from the EM-simulations and the lumped

circuit model response to verify how accurately the improved model can predict the behavior of the

transmission line structure. Comparing the poles and zeros shows a very good agreement indeed. It

is shown that the resonance frequency of the poles and zeros is predicted accurately. Some additional

improvement is still needed to predict the quality factor of the transmission line structure with very

high precision.

59

60 CHAPTER 5. CONCLUSION AND FUTURE PERSPECTIVES

Finally, a simple coupled line structure with slotted ground plane is analyzed. It is shown that

introducing slots in the ground plane indeed increases the coupling between the lines. This can be

of great benet when used in broadband lter design, since it allows to increase the bandwidth of

the lters while keeping the insertion loss low and without having to increase the dimensions of the

lters signicantly.

We can conclude that there is a lot of potential in the use of defective ground planes in lter design.

It is shown that introducing slots in the ground plane can solve the two main shortcomings of the

classical parallel-coupled transmission-line-resonator lters developed by Cohn [1]. Further research

is needed to include this properly in a lter design procedure.

Although this thesis is especially targeted to the use of a slotted ground plane in classical coupled-

line lter design. Defective ground planes can also be of great use in the design of hairpin resonator

lters, patch antennas and possible for dierent applications. There is still a lot of research potential

in dierent aspects of this subject.

Bibliography

[1] S. B. Cohn, Parallel-coupled transmission-line-resonator lters, Microwave Theory and Tech-

niques, IRE Transactions on, vol. 6, no. 2, pp. 223231, April 1958.

[2] Agilent Technologies, Momentum - the leading 3D planar EM simulator, 2013. [Online].

Available: http://cp.literature.agilent.com/litweb/pdf/5990-3633EN.pdf

[3] , Advanced design system, 2013. [Online]. Available: http://cp.literature.agilent.com/

litweb/pdf/5988-3326EN.pdf

[4] , Theory of operation for momentum, ADS 2011 Help.

[5] , Dening mesh settings for momentum, ADS 2011 Help.

[6] , Selecting the calibration type, ADS 2011 Help.

[7] , About dielectric loss models, ADS 2011 Help.

[8] Rogers Corporation, RO4000 series - High frequency circuit materials, 2013. [Online]. Avail-

able: http://www.rogerscorp.com/documents/726/acm/RO4000-Laminates---Data-sheet.pdf

[9] C. Caloz, H. Okabe, T. Iwai, and T. Itoh, A simple and accurate model for microstrip structures

with slotted ground plane, Microwave and Wireless Components Letters, IEEE, vol. 14, no. 4,

pp. 133135, April 2004.

[10] R. E. Colline, Foundation for microwave engineering, 1992.

[11] R. Janaswamy and D. Schaubert, Characteristic impedance of a wide slotline on low-

permittivity substrates (short paper), Microwave Theory and Techniques, IEEE Transactions

on, vol. 34, no. 8, pp. 900902, Aug 1986.

[12] H. Wheeler, Transmission-line properties of parallel strips separated by a dielectric sheet,

Microwave Theory and Techniques, IEEE Transactions on, vol. 13, no. 2, pp. 172185, Mar

1965.

[13] S. B. Cohn, Slot line on a dielectric substrate, Microwave Theory and Techniques, IEEE

Transactions on, vol. 17, no. 10, pp. 768778, 1969.

61

http://cp.literature.agilent.com/litweb/pdf/5990-3633EN.pdf



http://www.rogerscorp.com/documents/726/acm/RO4000-Laminates---Data-sheet.pdf

62 BIBLIOGRAPHY

[14] R. Pintelon, Frequency Domain Toolbox, Wiley. [Online]. Available: http://wiley.

mpstechnologies.com/wiley/BOBContent/searchLPBobContent.do

[15] R. Pintelon and J. Schoukens, System identication: A frequency domain approach, 2012.

[16] P. Henrici, Applied and computational complex analysis: Power series integration conformal

mapping location of zeros, 1974.

http://wiley.mpstechnologies.com/wiley/BOBContent/searchLPBobContent.do

http://wiley.mpstechnologies.com/wiley/BOBContent/searchLPBobContent.do

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