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EXPLORING MOORE SPACE-FILLING GEOMETRY FOR COMPACT
FILTER REALIZATION
EZRI MOHD
A project report submitted as partial
fulfilment of the requirements for the award of
Degree of Master of Electrical Engineering
Faculty of Electrical and Electronic Engineering
Universiti Tun Hussein Onn Malaysia
JANUARY 2018
iii
ACKNOWLEDGEMENT
, Praise to The Almighty Allah, finally with His limitless kindness for
granting me a healthy and strength to complete this work at last. For my beloved
mother and father, my wife and sons for supporting me during my study.
The author would like to express his sincere appreciation to his supervisor,
Prof. Madya Dr. Samsul Haimi bin Dahlan for the support given throughout the
duration for this project.
The cooperation given by the Center for Applied Electromagnetic
(Emcenter), Wireless and Radio Science Centre (WARAS) and Makmal Rekabentuk
Papan Litar Tercetak Lanjutan (MRPLT) are also highly appreciated. Appreciation
also goes to everyone involved directly or indirectly towards the compilation of this
report.
iv
ABSTRACT
A compact RF filter is desirable in modern wireless communication system. The
Moore curves fill the defined space and compact the effective geometry within the
same line length. The miniaturization rate of the planar microstrip filter can be
accomplished by applying the Moore curves with defined iteration and yet still
maintaining the acceptable performance. In this work, the 2nd iterated Moore curves
bandpass filter is designed to operate at center frequency, centerf = 2.45GHz on
RT/duroid®5880 substrate. The first development stage investigate the characteristics
of Moore resonator and consequently design Moore bandpass filter in next stage. The
resulted mathematical equations revealed the correlation between Moore geometry
with center frequency and bandwidth of the filter. The transmission response
exhibited symmetrical response with two finite transmission zeros around center
frequency to suggest a good selectivity and better rejection profile. Comparison
between parallel coupled line filter, the Moore filter compacted the size about 90%
while still presented reasonable performance. The fabricated Moore BPF has a good
agreement with simulated result, thus our works is validated through realization. As
a conclusion, the design of compact Moore BPF is successfully accomplished
according to the required specification and the possible enhancement is also proposed
in future work.
v
ABSTRAK
Penapis gelombang mikro padat diperlukan dalam sistem komunikasi moden tanpa
wayar. Lengkung Moore memenuhi ruangan tertentu dan memadatkan geometri
berkesan dengan panjang talian yang sama. Kadar pengecilan penapis mendatar
talian mikro boleh dicapai menggunakan lengkung Moore dengan pengulangan
semula yang tertentu dan masih mengekalkan hasilan yang boleh diterima. Dalam
kerja ini, penapis lulus jalur dengan pengulangan semula lengkung Moore sebanyak
dua kali direka untuk beroperasi pada frekuensi pertengahan, centerf = 2.45GHz di atas
bahan RT/duroid®5880. Pembangunan tahap pertama menyiasat ciri-ciri pengayun
Moore dan seterusnya merekabentuk penapis lulus jalur Moore dalam tahap
berikutnya. Penghasilan persamaan matematik mendedahkan perkaitan antara
geometri Moore dengan frekuensi pertengahan dan lebar jalur penapis. Reaksi
penghantaran mempamerkan tindakbalas bersimetri dengan dua penghantaran sifar
terhad di sekitar frekuensi pertengahan untuk mencadangkan pemilihan dan profil
penolakan yang baik. Perbandingan dengan penapis talian berkembar selari, penapis
Moore memadatkan saiz kira-kira 90% serta masih mempamerkan prestasi yang
munasabah. Penapis lulus jalur yang dibina mempunyai persetujuan yang baik
dengan hasil simulasi, maka kerja kami disahkan melalui pembuatan. Sebagai
kesimpulan, rekabentuk penapis lulus jalur Moore berjaya disempurnakan
sewajarnya dengan spesifikasi yang ditetapkan dan kemungkinan penambahbaikan
turut dicadangkan untuk kerja pada masa depan.
vi
TABLE OF CONTENTS
TITLE i
DECLARATION ii
ACKNOWLEDGEMENT iii
ABSTRACT iv
ABSTRAK v
TABLE OF CONTENTS vi
LIST OF TABLES ix
LIST OF FIGURES x
LIST OF SYMBOLS AND ABBREVIATIONS xiv
LIST OF APPENDICES xvi
LIST OF PUBLICATIONS xvii
CHAPTER 1 INTRODUCTION 1
1.1 Overview 1
1.2 Problem Statement 2
1.3 Objectives of the Project 2
1.4 Scope of the Project 3
1.5 Outline of the Report 4
CHAPTER 2 LITERATURE REVIEW 6
2.1 Introduction 6
2.2 Background of Fractal Geometry 6
2.3 RF Filter’s terminology 8
2.3.1 Bandpass definition 8
2.3.2 Selectivity 10
2.3.3 Quality factor, Q 11
2.4 Moore Fractal Resonator 12
2.5 Review of Fractal Resonator 13
2.5.1 Minkowski Resonator 13
vii
2.5.2 Hilbert Resonator 16
2.6 Coupling Coefficient of Coupled Line 17
2.6.1 Input and Output Coupling 17
2.6.2 Adjacent Resonator’s coupling 19
2.7 Class of Coupling 21
2.8 Review of Microwave Fractal Filter 22
2.9 Defected ground structure DGS 25
2.10 Conclusion 27
CHAPTER 3 METHODOLOGY 28
3.1 Introduction 28
3.2 Design and development study 28
3.3 Simulation 32
3.4 Fabrication 34
3.5 Measurement 36
CHAPTER 4 DEVELOPMENT OF MOORE FRACTAL
RESONATOR 38
4.1 Design of Moore Fractal Resonator 38
4.2 Resonator’s dimension 38
4.3 Performances of Moore Resonator 40
4.4 Parametric Studies of Moore Resonator 42
4.5 Measurement of Moore Resonator 44
4.6 Conclusion 47
CHAPTER 5 DEVELOPMENT OF MOORE BANDPASS
FILTER 48
5.1 Design Approach 48
5.1.1 Topology of Coupled Moore
Resonator 48
5.1.2 Technique of Feeding 50
5.2 Parametric Studies on geometry’s dimension 51
5.2.1 Length of Moore open-line
segment, x 51
viii
5.2.2 Distance gap between Moore
resonators elements, spacingg 55
5.3 Moore Filter design at center frequency,
centerf = 2.45GHz 60
5.3.1 Optimized dimension of designed
Moore fractal filter 60
5.3.2 Transmission Response 61
5.3.3 Surface current density 63
5.4 Comparison with Parallel-Coupled
Line Filter 64
5.5 Moore Filter with DGS 66
5.6 Realization of Moore bandpass filter 70
5.6.1 Fabrication 70
5.6.2 Measurement 70
CHAPTER 6 CONCLUSION AND RECOMMENDATION 73
6.1 Conclusion 73
6.2 Recommendation 75
REFERENCES 76
APPENDIX 79
ix
LIST OF TABLES
1.1 Scope of the proposed Moore filter 4
2.1 Fractal applications in different fields 8
2.2 Requirement of modern RF bandpass filter 8
4.1 Comparison of resonator’s size 40
4.2 Transmission characteristics of modelled
resonators 41
4.3 Relation between resonant frequency and
value of gap,g and width,w 43
4.4 Transmission characteristics of measured
resonators 45
5.1 Transmission characteristic associated to
variation of x and w 51
5.2 Validation of equation 5.1 between calculation
and simulation 53
5.3 Validation of equation 5.2 between calculation
and simulation 55
5.4 Summary of passband region attributes 62
5.5 Comparison of passband attributes 65
5.6 Simulated frequency suppression with different
length of DGS transverse slot, s 68
5.7 Summary of comparison between measurement
and simulation 72
6.1 Advantages and disadvantages of designed
Moore bandpass filter 75
x
LIST OF FIGURES
2.1 Examples of fractal geometries 7
2.2 Transmission coefficient, 21S of bandpass filter 9
2.3 Symmetrical response of bandpass filter 11
2.4 (a) Moore curves and (b) Hilbert curves with
2nd iteration level (n=2) 12
2.5 Square dual-mode resonator with Minkowski’s
iteration [8] 14
2.6 Relationship between L and g at 14GHz and
transmission characteristic for 1st iteration
square Minkowski resonators. [8] 14
2.7 Transmission characteristics of square, 1st and
2nd iteration Minkowski resonators. [8] 15
2.8 Transmission characteristics of 2nd square
Minkowski resonators. [8] 15
2.9 Open line λ/2 resonator to Hilbert fractal
curves. [8] 16
2.10 Transmission characteristics of Hilbert
Resonator. [8] 16
2.11 Types of coupling between resonators 17
2.12 Method used to determine the unloaded
quality factor 18
2.13 Reference plane added to RLC equivalent
circuit. [9] 19
2.14 ��� response with input/output are weakly
coupled [8] 20
2.15 Class of coupling between resonators. [8] 22
2.16 Transmission zero and selectivity 22
2.17 (a) 2nd iteration and (b) 3rd iteration of Hilbert
filters and its transmission response [10] 23
xi
2.18 3rd order meandered line with (a) magnetic
coupling and (b) electrical coupling and
their transmission responses [11] 24
2.19 3rd order Hilbert filter with (a) magnetic
coupling and (b) electrical coupling and
their transmission responses. [11] 24
2.20 (a) 2nd and (b) 3rd iteration of Moore fractal
and their transmission responses. [12] 25
2.21 (a) 3-Dimensional view of the DGS unit section.
(b) The newly proposed equivalent circuit. [14] 26
3.1 Process flow of the project 39
3.2 Mesh solver and adaptive mesh refinement setting 33
3.3 Boundary conditions setting 33
3.4 Flow chart of fabrication process using
photolithography technique 34
3.5 Laminator machine 35
3.6 UV exposure machine 35
3.7 The measurement outcome of network analyzer 37
3.8 Measurement setup to excite both ports of
Moore resonator 37
4.1 Evolution of Moore fractal 38
4.2 Geometry of designed resonators,
(a) half-wavelength resonator, (b) 1st iteration
Moore, (c) 2nd iteration Moore. 39
4.3 Transmission characteristic of modelled
resonators 40
4.4 Gap and width of Moore resonator 42
4.5 Transmission response with gap variation
[w=2.5mm] 43
4.6 Resonant frequency with gap variation
[w=2.5mm] 43
4.7 Relation between gap variation and
resonant frequency 44
xii
4.8 Realization of the resonators,
(a) half-wavelength resonator, (b) 1st iteration
Moore, (c) 2nd iteration Moore 45
4.9 Comparison between simulation and
measurement of λ/2 line resonator 46
4.10 Comparison between simulation and
measurement of 1st iteration Moore resonator 46
4.11 Comparison between simulation and
measurement of 2nd iteration Moore resonator 47
5.1 Optimized dimension of Moore resonator
element 48
5.2 Proposed topologies of the filter 49
5.3 Transmission characteristic 21S for electric and
magnetic coupling 49
5.4 Input/output feeding technique 50
5.5 Transmission response of Moore filter with
tapping and mutual coupling feedlines 50
5.6 Relation between center frequency,
centerf (GHz) and length of x (mm) 52
5.7 S21 response with different width of fractal at
center frequency, centerf =2.45 GHz 54
5.8 Distance spacing, spacingg between Moore
resonators with magnetic coupling 56
5.9 Coupling coefficient, k associated with
distance spacing, spacingg 56
5.10 Transmission response, 21S associated with
variation of distance spacing, spacingg 57
5.11 Extremely over-coupled with distance gap,
spacingg = 0.1mm 58
5.12 Over-coupled with distance gap,
spacingg = 0.3mm and 0.4mm 59
xiii
5.13 Under-coupled with distance gap,
spacingg = 2.0mm and 4.0mm 59
5.14 Optimized dimension of designed Moore filter 60
5.15 Transmission response of designed Moore filter 61
5.16 Bandpass response of designed Moore filter 61
5.17 Simulated surface current at 2.45GHz
(passband region) 63
5.18 Simulated surface current at 2.8GHz
(reject band region) 64
5.19 Conventional 6th order parallel coupled line
bandpass filter 64
5.20 Comparison of passband characteristic
between Moore filter and parallel coupled
line filter 65
5.21 The dumbbell DGS at the Moore I/O feedlines 67
5.22 Transmission response with different length of
DGS transverse slot, s 67
5.23 Relation between frequency suppression and
length of DGS transverse slot, s 68
5.24 Moore bandpass filter with dumbbell DGS 69
5.25 Transmission response of Moore bandpass
filter with DGS 69
5.26 The fabricated Moore bandpass filter 70
5.27 Comparison between simulated and
measured result of the proposed filter
(full observed range) 71
5.28 Comparison between simulated and
measured result of the proposed filter
(passband range) 71
xiv
LIST OF SYMBOLS AND ABBREVIATIONS
rε - Dielectric constant
effε - Effective dielectric constant
ARL - Maximum passband ripple
λ - Wavelength of the radio wave
cf - Resonant frequency
centerf - Center frequency of the filter
Hf - Upper frequency at -3dB below from insertion loss
Lf - Lower frequency at -3dB below from insertion loss
f∆ - Different between two split peak frequency
outP - Output power from filter
inP - Input power to filter
Q - Quality factor
eQ - External quality factor
LQ - Loaded quality factor
UQ - Unloaded quality factor
n - Level of Moore iteration
nS - Total perimeter length of Moore fractal with n-iteration
L - Length of outer area side Moore fractal
w - Width of fractal
g - Gap spacing between Moore segments
spacingg - Gap spacing between two coupled Moore resonators
x - Length of Moore open line segment
k - Coupling coefficient
11S - Reflection coefficient
21S - Transmission coefficient
xv
oZ - Characteristic impedance
BW - Bandwidth of bandpass filter
I/O - Input and output
UV - Ultra-violet
RF - Radio frequency
CST - Computer simulation technology
BPF - Band pass filter
xvi
LIST OF APPENDICES
APPENDIX TITLE PAGE
A Published Paper 79
B Moore BPF with upper cavity: A brief study 87
C Datasheet: RT/duroid®5870/5880 by Rogers
Corporation 88
D Datasheet: Bandpass Cavity Filter SF24251 by
Fairview Microwave [29] 91
xvii
LIST OF PUBLICATIONS
Journal:
(i) Mohd, E. & Dahlan, S. H. (2017). Miniaturization of Resonator based on
Moore Fractal. TELKOMNIKA (Telecommunication, Computing,
Electronics and Control), Universitas Ahmad Dahlan (UAD), Vol.13,
No.3, pp. 1433-1439.
CHAPTER 1
INTRODUCTION
1.1 Overview
In a front-end receiver of microwave communication systems, RF filter is an essential
component. The filters are designed to pass specific frequency within a particular
passband and reject the other frequency outside the operating frequency band.
Nowadays, the bandpass filter with high performance, compact size and low cost is
desired to enhance the overall performance of communication system. The well-
known planar structure filter such as microstrip is already in small size but there are
needs to reduce the size of a particular application especially in portable
communication devices.
Currently, the miniaturization of microwave filter focuses on a planar filter
structure. Open line resonator filter is a famous filter structure that had been widely
studied and commercialized. Parallel coupled line and hairpin resonator are preferred
for passband application and can frequently be found inside communication devices.
These structures are simple to design and the bandwidth can be controlled with ease.
As further development, they are benefited from its flexibility to fold the resonator
as a result of more compact form.
A good filter should have a higher return loss and a lower insertion loss in
passband region. Passband ripple relates to return loss. In other words, the Chebyshev
response with 0.1dB ripple will give the better return loss compared to Butterworth
response. Order of the filter determines the selectivity of the filter. Lower order is
expected to have poor selectivity while higher order filter is directly proportional to
the physical size of the filter. Hence, to compact a filter while maintaining or
improving the performance could be achieved by Moore fractal filter configuration.
Theoretically, the Moore fractal has space-filling properties characteristic that the
2
higher order of filter could be designed by iterative construction while the overall
physical realization is miniaturized.
The high dielectric constant substrate is used to miniaturize microstrip filter
and a variable geometry of filters is required. Hence, numbers of new filter
configuration can be modelled possibly. However, dissipation losses correlated with
the size of the filter and it affect the performance of the filter overall. The parametric
study will be conducted to define ‘trade-off’ between the technical performance and
miniaturization based on application.
1.1 Problem Statement
With rapid demand for smaller and reliable wireless communication system, RF
devices miniaturization become the research trend. A new filter circuit design which
suits the manufacturing ability and cost yet still maintaining the required
performance is developed.
The parallel coupled line resonator is popular for designing the narrowband
filter and widely used for many years since 1958. However, there are disadvantages
of this filter although it requires simple design and supports a wide range of fractional
bandwidth from 5% to 50%. The length of coupled line resonator is too long and
increases directly proportional to the order of the filter. The hairpin line filter is
developed based on parallel coupled line filter with folding the λ/2 resonator
structure. Recently, the microstrip fractal geometry had become a trend in designing
planar structure on RF devices such as antenna and filter. The Moore fractal has
several unique criteria such as essential dual properties, space curve filling and self-
similarity which is the iteration can be done in a defined space. Theoretically, this
feature will enable the miniaturization of the filter. In this thesis, by applying Moore
fractal construction, a compact bandpass filter applicable for wireless application in
ISM band is modelled, designed and validated.
1.1 Objectives of the Project
The primary aim of this work is to miniaturize a conventional microstrip bandpass
filter as small as possible while keeping its performance in good or acceptable
3
agreement. In this work, the miniaturization is expected to be achieved by using high
permittivity material, RT/duroid®5880 and fractal configuration with its variation of
resonant structure. The configuration of parallel coupled line filter is designed and
its physical realization are observed as well as filter’s performance to be compared
with designed Moore fractal filter.
The challenging task is to design a bandpass filter with lowest possible
insertion loss while miniaturizing the filter and keeping the low coupling between
resonators. Some parameters highly correlated to each other and achieve one
requirement may reduce the other one. Hence, analysis and parametric study of the
trend is presented in this work. To achieve the aim, the objectives of the study are:
(i) To design a compact bandpass filter using Moore fractal at ISM band.
(ii) To analyze the performance of Moore fractal filter in term of geometry and
technical aspect.
(iii) To explore the correlation between variations of Moore geometry with
electrical performance of the Moore BPF.
(iv) To prototype a Moore BPF and validate experimentally.
1.2 Scope of the Project
A compact resonator filter is achieved by using a Moore fractal structure. A reduction
in the form factor is made by folding the resonator line while maintaining its length
approximately and is represented in term of iteration. The iterative construction of
the fractal Moore filter geometry is limited up to the second iteration. This designed
filter is compared with conventional parallel coupled line. The scope of this work
are:
(i) Design and fabricate a microstrip 2nd order of bandpass Moore fractal filter at
center frequency of 2.45GHz with a narrowband response based on electrical
and material specifications as shown in Table 1.1.
(ii) Approximation of the microstrip parallel coupled line filter and Moore BPF’s
electrical properties using Computer Simulation Technology (CST)
Microwave Studio (MWS) software. CST is used as well to analyze and
compare the performances in term of miniaturization and transmission
characteristics.
4
(iii) The optimized design of the Moore BPF should consider the accuracy of in-
house fabrication which is set up to 0.5mm precision to guarantee the desired
geometry.
(iv) The resulted correlation between geometry’s dimensions and the transmission
characteristic of the designed Moore BPF are valid for a defined range of
width of fractal, w and center frequency, ������only.
(v) The simulated results are verified experimentally in a typical laboratory
environment without any special shielding and setup by using a calibrated
network analyzer that able to sweep for a range of frequency from 500MHz
to 10GHz.
Table 1.1: Specification of the proposed Moore filter
Specification Value
Targeted electrical properties at passband region
1. Insertion loss, IL ≤ 1.0dB
2. Magnitude of ripples, ARL ≤ 1.0dB
3. Bandwidth (narrowband respond) ≤ 490MHz or FBW ≤ 20%
Material substrate, RT/duroid®5880
1. Permittivity, rε 2.2
2. Thickness 1.57mm
3. Copper thickness 35µm
The maximum insertion loss, IL in the passband region is aimed to lower than
1dB to emulate electrical specification of commercialized BPF [29]. In order to
preserve the consistency of filtered-in RF signal, the maximum allowable ripples in
the passband region is limited to 1dB only. While the 3dB bandwidth is targeted at
below 490MHz to justify the narrowband specification with maximum fractional
bandwidth of 20%.
5
1.5 Outline of the Report
This report consists of six chapters. Chapter 1 presented the overview of the project
and its objectives. The scope limitations of the works are also described briefly in
this chapter.
Chapter 2 presented the literature review by discussing general knowledge of
microstrip fractal and its associated characteristics. The evolution of conventional
microstrip transmission line to Moore resonator fractals are explained. The main
filter characteristics that define the performance of the filter such as selectivity and
coupling technique are discussed. Then, the advantages and limitations of other
fractal resonators and filters by other researchers are reviewed.
Chapter 3 explained the methodology of the project to accomplish the
objectives with a scope above. The approach starts with the development of single
element of Moore resonator through simulation and realization to validate the
potential as a filter. With a verified result from Moore resonator, the Moore bandpass
filter is designed. This chapter also describes the simulation procedure and
fabrication technique.
Chapter 4 described the first stage of this project by presenting a resulted
preliminary study of Moore resonator by simulation and experiment. This study also
involved the comparison between different iteration of Moore fractals and their
performances. The correlation between Moore geometries and transmission
characteristics is reviewed.
Chapter 5 focused on developing the ultimate goal of this project which is
designing a Moore bandpass filter. In this chapter, the variation of technique to excite
the filter through different class coupling and feedline are explained. The outcomes
of parametric studies and the resulted correlation between different parameters with
transmission characteristic are discussed. This chapter also revealed the control
parameters of the Moore fractal geometry to achieve specific filtering criteria.
The final Chapter 6 outlined the conclusions of the project and recommended
the future works for enhancing the performance and flexibility of Moore bandpass
filter.
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
Chapter 2 discussed the general background of microwave fractal filter based on
planar microstrip technology. The suitability of Moore fractal curve to be used as a
microstrip microwave bandpass filter is explored. Studies are done based on
knowledge on planar coupled mode microstrip since the fractal curve will have many
adjacent microstrip structures on completed filter’s geometry and induced a coupling
effect. Coupling between microstrip coupled line is one of the elements that made a
microwave filtering’s effect. Besides, reviews about other previous studies on
microwave resonator or filter based on fractal curves are discussed. Some fractal
design consideration and transmission characteristics are compared. At the end of
this chapter, the advantages and limitation of the microwave fractal filters which are
done by previous researchers will be summarized.
2.2 Background of Fractal Geometry
The fractal curves had become interest to several mathematicians in the late of 19th
century. The fractal structure is continuous curves with an infinite length that
occupied finite space. This unique property is called space-filling curve (SFC). An
infinite limit of iterative construction would determine the fractal structure [1]. It
involves the complex mathematical functions within continuous but
nondifferentiable curves with self-similar sets and sets with fractional dimensions
[2]. The infinite iteration construction process defines the fractal curves and makes
the miniaturization is possible.
7
There are many types of fractal iteration that had been introduced as early as
1870s. The simplest structure to describe the fractal iteration is Cantor. The Cantor
set give us an early idea about fractal iteration and consists a set points lying on a
single line segment. The Cantor iteration has infinite numbers and self-similarity
properties. Others types of fractal geometry are Peano, Hilbert, Sierpinski, Von Koch
and Moore in Figure 2.1. From visual observation, one should notify that these curves
are actually repeating themselves in fractional dimension with finite occupied area.
Figure 2.1: Examples of fractal geometries
Nowadays, fractals are implemented in many fields due to their main
properties; self-similarity and space filling. In microwave field, fractal have been
used to develop compact RF devices, most notably antennas and filters. Some of
these applications are summarized in Table 2.1.
(a) Cantor line (b) Peano curve (c) Hilbert curve
(d) Sierpinski triangle (e) Von Koch snowflake (f) Moore curve
8
Table 2.1: Fractal applications in different fields.
Field Application
Statistical Probability theory called statistical fractal to predict extreme/catastrophic
events.
Chemistry Research structural design and optimized design of microfluidic and
nanofluidic devices for mixing, reaction and heat transfer.
Acoustics Invention of a revolutionary type of acoustic barrier of soundproof walls.
Cosmology Understanding the distribution of luminous matter in the universe.
Computer Science Apply degree of recursion and self-similarity at different scales in creating
an animation or to compress lossy image.
Electromagnetic Miniaturization of RF devices and performance enhancement such as
selective frequency and harmonic suppression. [3]
2.3 RF Filter’s terminology
In order to produce high performance of the filter, there are main criteria that the
filter must be able to fulfill. Table 2.2 concludes the general need and the proposed
realization.
Table 2.2: Requirement of modern RF bandpass filter
Criteria Proposed realization
High selectivity Additional of transmission zeros near band edges [4][5]
High rejection High filter order and reduce parasitic coupling between elements
[5][6]
Low insertion loss High quality factor of resonator and reduce filter order [6]
Compactness Using different material and compact resonator such as fractal,
reduce filter order [6]
2.3.1 Bandpass definition
This section reviewed on main specification of the RF filter that need to be
considered during development of bandpass filter in general. As term of bandpass,
the filter must be able to reject a certain region of frequency while allowing other
region to pass through. The allowed region can be more than one region to create
9
dual-band of bandpass filter. The characteristic response of passband filter also
should be considered prior to its application.
When designing a bandpass filter, we must determine the level of rejection or
attenuation that must be achieved by the filter at frequencies outside of its passband
bandwidth. The bandwidth is associated with cut-off frequencies upper and lower of
filter’s center frequency at -3dB from maximum insertion loss in passband region.
Figure 2.2 illustrated the terminology to express the bandwidth of passband filter.
Figure 2.2: Transmission coefficient, 21S of bandpass filter
The bandwidth is calculated by subtracting the lower frequency, Lf where the
response attenuates by 3dB from the higher frequency, Hf with same attenuation
level as equation 2.1.
LH ffBW −= (2.1)
The bandwidth of the filter can be categorized as narrowband or wideband. It depend
on the range of the passband region and the suitability based on application.
Generally, the category of the filter is determined by fractional bandwidth, FBW in
percentage.
Fractional bandwidth, FBW = 100*centerf
BW (2.2)
The 3dB drop associates with half-power reduction as the filter attenuates the
received power by half. The relation of this understanding can be shown in equation
2.3. 21S parameter is a transmission coefficient and normally is represented in
10
magnitude decibel (dB). Insertion loss is an indicator of forward transmission gain
of 2-port network system.
Insertion loss, dBIL = in
out
P
P10log10 (2.3)
To realize the definition of filtering characteristics for RF system, the device
must be able to attenuate 50% of the input power in stopband region. Calculate using
the formulation of insertion loss, dBIL would be approximately -3.01dB.
Theoretically, the low insertion loss in passband region is desired due to zero loss in
allowing all the input power and high insertion loss as possible as can be is desired
in stopband region.
2.3.2 Selectivity
Selection of proper RF filter for a certain application is determined by its selectivity.
Selectivity is described as the level of attenuation at some frequencies that is rejected
from passband region of the filter. It usually is represented by a slope which indicate
how many decibel drop from cut-off frequency in both lower and upper side. For an
example, the cavity resonator at 155MHz has a selectivity which can be described as
15dB drop at interval of 1MHz in both lower and upper cut-off frequencies.
Normally, the transmission coefficient of RF bandpass filter is not
symmetrical. The response may be described of having 5dB drop at interval 1MHz
at lower side of center frequency and 10dB at upper side. This condition indicated
that the filter has a better selectivity at upper side of center frequency. The Chebyshev
response has a better selectivity than Butterworth response which is indicated by a
sharp slope, while quasi-elliptic response usually has an equal selectivity in both side
from center frequency.
The selectivity of the filter can be improved by using a quasi-elliptic response
[4]. Quasi-elliptic response has transmission zeros in both side of center frequency.
Figure 2.17 illustrates the importance of having symmetrical response with
occurrences of finite transmission zeros. The writers proposed a source load coupling
to realize quadrupled in second order filter in order to create two finite transmission
zeros for improving the selectivity of the filter.
11
Figure 2.3: Symmetrical response of bandpass filter [4]
2.3.3 Quality factor, Q
High performance of the filter generally achieve high value of quality factor. The Q
factor can described as a unit less ratio of the total energy stored in the lumped
element of the resonator (inductor and capacitor) to the dissipated power in the
resistor. As we may recalled, the microstrip microwave resonator can be represented
by an equivalent lumped elements circuit. However the extraction of the accurate
lumped elements of the filter always become a great challenges to the researchers
especially when dealing with complex structural like fractal geometry.
There are two ways to extract the quality factor of resonator; equivalent of
lumped elements circuit and transmission type measurement [7]. Since in this work
implement the Moore fractal, the latter method of extraction is reviewed in section
2.5.1 later on.
2.4 Moore Fractal Resonator
Microwave resonators that are designed using planar technologies such as microstrip
can be compacted by adopting the fractal construction. Recently, researchers are
exploring the fractal construction in order to meet the needs of smaller antennas and
filters with reduced cost.
The Moore fractal curve is a continual line that is folded several times in term
of iteration. Moore fractal begin with a single line and has a special recursive
procedure so that the length of the line remains but the fractal is filling the space.
12
SFC make a miniaturization on RF devices is possible. Hilbert and Moore curve are
outlined in Figure 2.4. The main different is the endpoints of these fractal, Moore has
its endpoint coincides with each other and create a spacing gap. From microstrip
technology’s perspective, a gap between microstip lines creates a capacitance which
influences the coupling characteristic.
Figure 2.4: (a) Moore curves and (b) Hilbert curves with 2nd iteration level(� = 2).
The total perimeter length, S of Moore fractal is determined by with n is the iteration
level.
( ) LSn
n
n *12
112
−++= (2.4)
L is the length of outer area side Moore fractal and determined the area
occupied by fractal curve. From study, the total length of line segment is longer from
its origin but the physical area is miniaturized. To develop a resonator using Moore
fractal, the single resonator with certain length is folded to some iteration.
Theoretically, the total length is maintained to resonate it to the same frequency for
different iteration. However, the length need to be optimized in physical realization
to achieve same resonant frequency for a half wavelength resonator, first iteration
and second iteration respectively.
Ideally, the designer would prefer to fit resonator in the small area as much
as possible by increasing the iteration level of Moore curve. As for microstrip, the
width, w and the gap space between, g are the parameters needed to be defined to
model a Moore curve. It depends on the iteration level, n. A higher iteration Moore
curve would decrease the resonator width, w and spacing between lines, g to fit the
(a) (b)
13
total perimeter length, S curves in the smaller area. The relation between dimensions
as depicted by:
( ) ggwL n−+= *2 (2.5)
The external side, L could be controlled by defining the width of the
resonator, w and spacing between lines, g. However, from the study the other external
side, x would be varying to resonate the structure to a specific frequency for second
iteration (� = 2) and is defined as an optimization technique.
A smaller width of resonator, w would increase the dissipation losses, thus
the quality factor, Q should diminishes. So, the designer should consider this
parameter if the application emphasize on higher quality factor, Q and at the same
time, miniaturization is required. The trade-off between miniaturization and
dissipation losses should be defined well according to priority.
2.5 Review of Fractal Resonator
This section discussed about studies that had been done on microwave resonators.
The studies reveals the suitability and potential of two types of resonators for
compacted RF devices and filters based on Minkowski and Hilbert fractal curves. [8]
2.5.1 Minkowski Resonator
The square dual-mode resonator is designed to be operated at center frequency,
14GHz. As depicted in Figure 2.5, the iteration of Minkowski fractal can be done by
removing several small square or rectangular repeatedly. The iteration techniques can
be done in two methods; square iteration or rectangular iteration. The designer should
be able to determine the value of g and l based on his preference of iteration’s
technique.
14
Figure 2.5: Square dual-mode resonator with Minkowski’s iteration [8]
We should notice that the value of L is preserved while the value of g
decreases after each iteration. After an infinite number of iterations, we should expect
that there is no more left as the iteration has consumed the space square.
For a dual mode resonator to be operated at a specific frequency is determined
by the value of L and g. The writers concluded the relationship between L and g to
maintain a resonant frequency at 14GHz and some cases to validate the transmission
characteristics in Figure 2.6.
Figure 2.6: Relationship between L and g at 14GHz and transmission characteristic
for 1st iteration square Minkowski resonators. [8]
With same resonant frequency at 14GHz, the rate of miniaturization can be
controlled by the value of g. This is a definite conclusion that the Minkowski iteration
is capable of producing a smaller square dual-mode resonator. In term of transmission
characteristics, the first spurious frequency which is twice resonant frequency occurs
for all cases. The spurious frequency for the smallest Minkowski resonator occurs at
15
the highest frequency and the writers concluded the miniaturization also could
improve the spurious response.
Further investigation, Minkowski with 2nd iteration resonator had been
developed to observe the transmission characteristics. Figure 2.7 shows the
comparison of the square resonator with 1st and 2nd Minkowski’s iteration. The first
spurious frequency is attenuated significantly in higher iteration and in addition to
that, also produce the transmission zero characteristics. The writers also designed
four different sizes of 2nd Minkowski’s iteration to examine the transmission
characteristics and the result is depicted in Figure 2.8. The smaller size of the
resonator with same Minkowski’s iteration would also provide a better spurious
response.
Figure 2.7: Transmission characteristics of square, 1st and 2nd iteration Minkowski
resonators.[8]
Figure 2.8: Transmission characteristics of 2nd square Minkowski resonators.[8]
16
2.5.2 Hilbert Resonator
The Hilbert fractal curve is constructed by folding a single open line resonator to ‘U’
shape and is iterated by a special curves. The iteration can be done by repeated
folding again and again into finite space with infinite length. A λ/2 open line
resonator is folded according to Hilbert’s curve up to 2nd Hilbert’s iteration as in
Figure 2.9. All the resonators are designed with respective dimension to have same
resonant frequency at 9.78GHz.
From transmission characteristics as illustrated in Figure 2.9, we noticed that
open line resonator and ‘U’ shape resonator have a spurious response at twice the
resonant frequency. The 1st iteration Hilbert had a spurious frequency slightly above
twice the resonant frequency and produced transmission zeros response. The 2nd
iteration Hilbert also had a spurious frequency above twice the resonant frequency
and the transmission zero is apparently moved closer to the resonant frequency. This
response suggested an improvement in selectivity.
Figure 2.9: Open line λ/2 resonator to Hilbert fractal curves. [8]
Figure 2.10: Transmission characteristics of Hilbert Resonator. [8]
17
As a conclusion, the fractal resonator especially Minkowski and Hilbert tend
to suggest that the higher iteration number, the miniaturization factor should be
increased. The performances with better selectivity due to the occurrence of
transmission zero and harmonic suppression is possible.
2.6 Coupling Coefficient of Coupled Line
Proximity between two or more transmission line introduces coupling effect through
their fringing fields respectively. This coupling effect could be desirable or
undesirable. In printed circuit board, the coupling effect is undesirable as it cause a
crosstalk, but in microwave devices such as directional coupler, it is desirable. The
coupling between transmission lines enable power transfers between lines at the
resonant frequency.
For a microstrip transmission lines, there are two types of coupling, input or
output coupling and coupling between adjacent resonators as illustrated in Figure
2.11. Both characterize the excitation between the resonators.
Figure 2.11: Types of coupling between resonators
2.6.1 Input and Output Coupling
The input coupling characteristic depends on the spacing gap, g between the first
resonator and input feedline and output coupling depends on the spacing between last
Wave propagation
Substrate
Input port
feedline Output port
feedline
g g g
coupling between
adjacent resonators
output coupling input coupling
18
resonator and output feedline. From reflection coefficient transmission characteristic,
���, we could calculate the input coupling by measuring its resonant frequency, ��
and bandwidth corresponding to 180�phase shift from resonant frequency. Then, the
input coupling is represented by external quality, �� factor given by:
f
fQe
∆= 0 (2.6)
Since the performance of filter is referred to unloaded quality factor, ��,
spacing gap between input or output feedlines can be manipulated. To eliminate the
effect of input and output coupling, the spacing between feedlines and resonators is
adjusted until attenuation, ��� transmission characteristic at least 20dB from its
maximum value at resonant frequency as depicted in Figure 2.11. The 3dB bandwidth
is measured to calculate unloaded quality factor, �� by extrapolating external quality
factor, �� given by
)(1 21 o
e
fS
QQu
−= (2.7)
Higher quality factor suggests more power is delivered between resonator lines due
to strong coupling between resonators.
Figure 2.12: Method used to determine the unloaded quality factor.
2.6.2 Adjacent Resonator’s coupling
Coupling between adjacent resonators determine the maximum power transfer at
certain resonant frequency. A good filter resonate at designed resonant frequency.
Variation in spacing gap between resonator elements shift the resonant frequency.
When the spacing between resonators is shorter, the resonant frequency will be
Output port
feedline
Input port
feedline
Gap is adjusted to minimize
the I/O coupling
19
decreases. These assumption has been used to optimize the modelled design in order
to achieve desired operating frequency of planar resonator.
As suggested by Y. Hongyu [9], the calculation methods of coupling
coefficient between resonators are presented. When two resonators are located in
close proximity, two modes of couplings are existed; electric coupling and magnetic
coupling. The electric coupling creates capacitance effect while the magnetic
coupling introduces inductance effect as showed in equation (1) in this paper. From
the general coupling coefficient equation, the researchers propose three methods in
order to calculate the coupling coefficient more accurately based on certain
conditions.
One of the methods is to add an electric and magnetic wall between coupled
resonators. This is done by adding a reference plane in the middle of RLC equivalent
circuit as illustrated in Figure 2.13.
Figure 2.13: Reference plane added to RLC equivalent circuit. [9]
In this method, the coupling coefficient is calculated in term of electrically or
magnetic coupling only. The electric coupling is calculated by short-circuiting the
reference plane (T-T’ plane) to create an electric wall while the magnetic wall is
created by open-circuiting the plane. Then the resonant frequency based on additional
of electric/magnetic wall is calculated respectively. Coupling coefficient for both
capacitance and inductance coupling can be found by knowing the value of resonant
frequency in both modes. Below equation summarize the method:
(i) Coupling coefficient for capacitance coupling (electric coupling),
22
22
em
emm
ff
ff
C
Ck
+
−== (2.8)
20
(ii) Coupling coefficient for inductance coupling (magnetic coupling),
22
22
me
mem
ff
ff
L
Lk
+
−== (2.9)
From this equation, we should expect that if coupling coefficient polarity is negative,
then we can conclude that the resonators are magnetically coupled and otherwise.
Coupling coefficient also can determined with S-parameter method
introduced by P. Jarry and J. Beneat [8]. This method require the gap adjustment
between input/output feedline and the resonators. To define the resonator coupling
coefficient, the input/output couplings need to be coupled weakly in order to prevent
any interference with coupling between resonators. Attenuation of at least 10dB
between two weakly coupled resonant frequencies ��and ��is assumed to make sure
the interference do not occurred as depicted in Figure 2.14.
Figure 2.14: ��� response with input/output are weakly coupled. [8]
Then, the coupling coefficient, k between resonators can be calculated by:
2
1
2
2
2
1
2
2
ff
ffk
+
−= (2.10)
This method applies to both electric and magnetic coupling. Farid Ghanem, Tayeb
A. Denidni and Renato-G Bosisio presented the same method to calculate the
coupling coefficient, k for planar cross-coupled resonator filters with electric or
magnetic coupling and mixed coupling [13]. The result revealed that the dimension
of the planar structure could be determined based on coupling coefficient required to
21
exhibit desired response. They also explained the application of equation (2.10) in
comprehensive graphical representation.
2.7 Class of Coupling
For a periodic structure, the arrangement of planar resonator determines the class of
coupling either electrical or magnetic coupling. In electrical coupling, an electrical
field is concentrated inside coupling area or sometimes is referred as positive
coupling. In other hand, a magnetic coupling has its electrical field is concentrated
outside the coupling area. Hence, it is opposite with electrical coupling and known
as negative coupling. The term positive and negative can be seen in phase
characteristic as it is opposite to each other. We can observe that with same distance
of the gap between resonators but a different class of coupling, magnetic coupling
provide stronger coupling. The orientation of the resonators may introduce mixed
coupling too. This method can be used as a mechanism to control the bandwidth of
the filter (narrowband or wideband).
Depending on the application, transmission zero also sometimes is desirable
in filter design. Transmission zero often relates to the ability of the filter to reject the
unwanted frequency in the stop-band region. Interestingly, we can control the
occurrence of transmission zero by electrical or magnetic coupling. Electrical
coupling creates transmission zero at low pass section whereas magnetic coupling
influences transmission zero at high pass section. Besides that, it also helps to
attenuate or eliminate the spurious frequency and enhance band-stop rejection as well
as selectivity.
22
Figure 2.15: Class of coupling between resonators. [8]
Figure 2.16: Transmission zero and selectivity
2.8 Review of Microwave Fractal Filter
There are many research on microwave filter based on fractal geometry. This section
will review some techniques and result. As fractal geometries have two common
properties, self-similarity and space-filling, the designer has an option to investigate
different fractal geometries.
Microstrip bandpass filter based on Hilbert fractal geometry of 2nd and 3rd
iteration are designed by Y.S. Mezaal [10]. The Hilbert fractal geometry is designed
from conventional half-wavelength resonator filter and considerable miniaturization
is achieved. Two structures of Hilbert fractal are coupled electrically to create a
bandpass region which operated at 2.4GHz with input and output tapping feedline.
23
In term of performance, the filter produces transmission zero at both low and high
pass region at least -50dB (��� response). The 3rd iteration of Hilbert fractal produce
better transmission response with symmetrical transmission zero around resonant
frequency. This result suggest better band-stop rejection and improve selectivity.
However, the width trace become smaller (w = 0.22mm) and need precision value
during fabrication. With smaller trace width, dissipation losses is higher and the
tradeoff between miniaturization rate and performance should be a concern. Hence,
this technique might not suitable to be used if fabrication is done in general printing
circuit board (university) which is not specialized in microwave.
Figure 2.17: (a) 2nd iteration, (b) 3rd iteration of Hilbert filters and its transmission
response [10]
To further improve the miniaturization, P. Jarry, J. Beneat and E. Kerherve
implemented the pseudo-elliptic response [11]. For the same selectivity, this
approach allows further reduction in size and the construction will be simplified by
using lower order of the filter. Meandered line, Hilbert and dual-mode Minkowski
bandpass filter are designed and compared. 3rd order meandered line and Hilbert
fractal are fabricated and experimental results are presented. Interestingly, they also
re-arranged the structures to demonstrate magnetic and electrical coupling. From
both experimental results, it suggests that class of coupling will determine the
location of transmission zero either in left (electrical coupling) or right (magnetic
coupling). This information is useful for the designer to decide where they would
require the transmission zero to occur based on application in term of band rejection
24
and selectivity. In this paper, all the input and output feedline is excited by mutual
coupling.
Figure 2.18: 3rd order meandered line with (a) magnetic coupling, (b)
electrical coupling and their transmission responses [11]
Figure 2.19: 3rd order Hilbert filter with (a) magnetic coupling, (b) electrical
coupling and their transmission responses [11]
(a)
(b)
(b)
(a)
76
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