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iv
To
My Father, ZHANG Liangzhu
and
My Mother, NI Anqi
v
ACKNOWLEDGEMENTS Ac knowledgements
I would first like to thank my advisor, Prof. Kevin J. Chen, who guided me
through my Ph. D. study. Without his help and support, I could not have this
opportunity to come to the Hong Kong University of Science and Technology.
During the past three and a half years, I had learnt a great deal from him, especially
the relentless pursuit in understanding the most fundamental mechanisms and
providing intuitive explanations to seemingly complex problems. I also would like
to thank my thesis defense committee members, Prof. Kei May Lau, Prof. Andrew
W. O. Poon, Prof. Lilong Cai, and Prof. Quan Xue for their support and feedback.
Among all the members of Prof. Chen’s group that I have had the privilege to
work with, the first one that I would like to thank is Dr. Jinwen Zhang. She is the
one who guides me through the fabrication process of my first research project. I
enjoyed working with her and learned many hands-on skills about microfabrication
from her. Her dedication to the quality of the work is one thing that I hope I never
forget. Although all of the works presented in this dissertation have been done out of
the clean-rooms, I have to say that I learned how to do research during the time
spent on the microfabrications. Mr. Kwok Wai Chan is the next person that I am
indebted to. He helped me to get familiar with the microwave measurements, which
are not as simple as they look. For this matter, Dr. Lydia L. W. Leung is also to be
recognized for many valuable discussions about the measurement problems and
other questions. Her intuitions about the essentials of various complex problems
impress me. Mr. Cheong Wai Hon (Golo) and Mr. William Chun San Chu also
deserve to be mentioned here for those helpful discussions in the group meetings.
vi
Special thanks also go to Dr. Zhengchuan Yang. He is my source of information
about many things both technical and non-technical. Without his help about the
mask drawings, much of my time will be wasted in these tedious tasks.
Work is part of life and life is part of work. I am glad I had such a nice group of
colleagues around me. Mr. Kenneth K. P. Tsui, Mr. Kwong Fu Chan, Mr. Rongming
Chu (a man who is satisfied with engineering alone), Mr. Shuo Jia, Dr. Jie Liu, Dr.
Zhiqun Cheng, Dr. Yong Cai (a man with inexhaustible energy and my squash
partner), Mr. Ruonan Wang, Mr. King Yuen Wong, Mr. Di Song, Mr. Yichao Wu,
Ms. Song Tan, Dr. Congshun Wang, Dr. Wei Huang, Ms. Congwen Yi and Mr.
Xiaohua Wang.
Finally, I thank my father Liangzhu Zhang and my mother Anqi Ni, who were
my first teachers. Their unconditional love and encouragement inspired my passion
for learning. It is to commemorate their love that I dedicate this dissertation to them.
vii
TABLE OF CONTENTS
Table of Contents
Title Page i
Authorization ii
Signature iii
Acknowledgements v
Table of Contents vii
List of Figures ix
List of Tables xvii
Abstract xviii
CHAPTER 1 Introduction 1 1.1 History of Microwave Circuits 2 1.2 Planar Microwave Circuits 3
1.2.1 Applications of Planar Microwave Circuits 3 1.2.2 Structures of Planar Microwave Passive Circuits 8
1.3 Motivation and Overview of This Dissertation 13
CHAPTER 2 Synthesis of Microwave Filters 16 2.1 Introduction 16 2.2 Basic Principles for Generating the Rational Polynomials of the General Chebyshev Filters 17 2.3 Circuit Model of the Filter and the Coupling Matrix 19 2.4 Synthesis of General Chebyshev Filters Using GA 22
2.4.1 Basic Elements of GA 22 2.4.2 Synthesis of the Filters 27
2.5 Summary 34
CHAPTER 3 Design of Compact Microwave Bandpass Filters 35 3.1 Introduction 35 3.2 Topology of the Proposed Tri-Section Stepped-Impedance Resonator and Theoretical Analysis 36 3.3 A Microstrip Bandpass Filter Designed Using the Proposed Tri-Section SIR 45
3.3.1 Circuit Prototypes of the Third-Order Bandpass Filter 45 3.3.2 Experimental Results 48
3.4 Topology of the Proposed Slow-Wave CPW Stepped-Impedance Resonator 51 3.5 CPW Microwave Bandpass Filters Designed Using the Proposed Slow-Wave SIR 55
3. 5. 1 Circuit Prototypes of the Fourth-Order Bandpass Filter 55 3. 5. 2 Experimental Results 57
3.6 Summary 60
CHAPTER 4 Design of Microwave Bandpass Filters with Reconfigurable Transmission Zeros and Tunable Center Frequencies 62
viii
4.1 Introduction 62 4.2 Bandpass Filters with Reconfigurable Transmission Zeros and Tunable Center Frequencies 64
4.2.1 Bandpass Filters with Reconfigurable Transmission Zeros 64 4.2.2 Bandpass Filters with Reconfigurable Transmission Zeros and Tunable Center Frequencies 75
4.3 Bandpass Filters with Reconfigurable Transmission Zero 80 4.4 Summary 87
CHAPTER 5 Dual-Band Microwave Bandpass Filters, Couplers and Power Dividers 89 5.1 Introduction 89 5.2 Dual-Band Quarter-Wavelength Transmission Line 91 5.3 Dual-Band Filter Design 94 5.4 Applications to Other Dual-Band Passive Components 102
5.4.1 Branch-Line Coupler for Dual-Band Operations 102 5.4.2 Rat-Race Couplers for Dual-Band Operations 108 5.4.3 Wilkinson Power Divider for Dual-Band Operations 126
5.5 Summary 130
CHAPTER 6 Parameter Extractions for Tuning of the Microwave Bandpass Filters 132 6.1 Introduction 132 6.2 Parameter Extraction for Microwave Filter Tuning 134
6. 2. 1 Basic Equations for the Parameter Extractions of the Filters 134 6. 2. 2 Genetic Algorithm and Its Implementation for the Parameter Extractions 135 6. 2. 3 Coupling Coefficients Extractions of the Filters with Only Mistuned Inter-Resonator Couplings 138 6. 2. 4 Coupling Coefficients Extractions of the Filter with Both Mistuned Inter-Resonator Couplings and Mistuned Resonators 146
6.3 Summary 148
CHAPTER 7 Conclusion and Future Work 149 7.1 Conclusion 149 7.2 Future Work 150
REFERENCES 152
APPENDIX: PUBLICATION LIST 166
ix
LIST of FIGURES List of Figures
Figure 1.1.1: Band designations and applications of microwaves. 1
Figure 1.2.1: Photo of a microwave circuits using waveguide. 4
Figure 1.2.2: General structures of (a) microstrip line and (b) coplanar
waveguide line.
4
Figure 1.2.3: (a) Photo of an HTS planar microwave filter [1], (b) a
comparison between a conventional transceiver (left) and an
HTS transceiver (right) [2].
5
Figure 1.2.4: Die micrograph of (a) a 24-GHz power amplifier. Chip size:
0.7mm×1.8mm [4]. and (b) a 77-GHz power amplifier. Chip
size: 1.35mm×0.45mm [5].
7
Figure 1.2.5: The block diagram of a wireless transceiver. 8
Figure 1.2.6: The structure of the image rejection mixer circuit. 8
Figure 1.2.7: Structures of (a) combline filters and (b) interdigital filters. 10
Figure 1.2.8: Structures of (a) parallel-coupled filters and (b) hairpin-line
filters.
11
Figure 1.2.9: Topology of the planar branch-line coupler. 12
Figure 1.2.10: Topology of the planar rat-race coupler. 13
Figure 1.2.11: Topology of the Wilkinson power divider. 13
Figure 2.3.1: The equivalent circuit representing an general two-port
network.
19
Figure 2.3.2: The equivalent circuit of n-coupled resonators. 20
Figure 2.4.1: The flowchart of the proposed genetic algorithm (GA). 24
Figure 2.4.2: Three basic operators of GA. (a) Reproduction. (b) Crossover.
(c) Mutation.
26
x
Figure 2.4.3: The roulette wheel represents the reproduction process in the
GA.
27
Figure 2.4.4: The S-parameters represented by the synthesized coupling
matrix of the fourth-order filter.
30
Figure 2.4.5: The S-parameters represented by the synthesized coupling
matrix of the sixth-order filter.
32
Figure 2.4.6: The S-parameters represented by the synthesized coupling
matrix of the fifth-order filter.
34
Figure 3.2.1: General topology of the conventional stepped-impedance
resonator.
37
Figure 3.2.2: General topology of the proposed tri-section SIR. 38
Figure 3.2.3: The computed minimum electrical length with different values
of k and m for the tri-section SIR. (i) 1≤m≤k≤10 (ii) k≤m≤10
(iii) 0<m≤1.
41
Figure 3.2.4: The computed electrical length for case(iii) with different m
and θ1 under the condition (a) k=2 (b) k=4 (c) k=6.
42
Figure 3.2.5: The computed electrical length for case(ii) with different m
and θ1 under the conditions (a) k=2 (b) k=3 (c) k=4.
43
Figure 3.2.6: The structures of (a) conventional SIR (b) the new tri-section
SIR.
44
Figure 3.2.7: Simulated resonance frequencies of the conventional and the
new SIR.
44
Figure 3.3.1: Coupling schemes of the conventional microwave bandpass
filters. (a) inductive-coupled (b) capacitive-coupled (c) mixed-
coupled.
46
Figure 3.3.2: Coupling schemes of the third-order CT bandpass filter with
inductive cross-coupling.
47
xi
Figure 3.3.3: Layout of the third-order CT filter with a transmission zero in
the upper stopband.
49
Figure 3.3.4: Photo of the fabricated filter. 49
Figure 3.3.5: The measured results of the fabricated filter. 50
Figure 3.4.1: Layout of (a) conventional CPW SIR (b) proposed tri-section
CPW SIR (Type A) (c) proposed tri-section SIR (Type B).
52
Figure 3.4.2: Schematic illustrating the basic structure of the proposed slow-
wave SIR and the wave propagating path in the slow-wave
SIR.
53
Figure 3.4.3: Layout of (a) proposed slow-wave CPW SIR (Type C) (b)
proposed slow-wave CPW SIR (Type D).
53
Figure 3.4.4: Simulated results of the five SIRs shown in Fig. 3. 4. 1 & Fig.
3. 4. 3.
55
Figure 3.5.1: Coupling schemes of the fourth-order CQ bandpass filter with
capacitive cross coupling.
56
Figure 3.5.2: Layout of the second-order microwave bandpass filter based
on Type C slow-wave SIR.
57
Figure 3.5.3: Simulated and measured responses of the second-order
bandpass filter.
58
Figure 3.5.4: Layout of the fourth-order quasi-elliptic filter. 60
Figure 3.5.5: Measured and simulated results of the designed CQ filter. 60
Figure 4.2.1: Topology of the bandpass filter with one reconfigurable
transmission zero.
65
Figure 4.2.2: The equivalent circuit of a λ/2 resonator with a tapped stub
serving as a K-inverter.
65
Figure 4.2.3: The equivalent circuit for the stub working as a resonator with
the varactor tuning the resonant frequency.
65
xii
Figure 4.2.4: Circuit model of a varactor loaded transmission line. 66
Figure 4.2.5: Theoretical resonant frequency tuning range for the varactor-
loaded transmission line.
67
Figure 4.2.6: The simulation results of the filter with one reconfigurable
transmission zero.
68
Figure 4.2.7: The photo of the fabricated filter with one reconfigurable
transmission zero.
70
Figure 4.2.8: The measurement results of the filter. (a) Measured results
under different bias around the passband. (b) Measured wide
band characteristics.
71
Figure 4.2.9: The bandpass filter with two reconfigurable transmission
zeros. (a) The topology of the filter. (b) The equivalent circuit
of the filter.
73
Figure 4.2.10: Fabricated filter with two reconfigurable transmission zeros.
(a) The photo of the filter. (b) The measured results under
different biases.
74
Figure 4.2.11: Filter with tunable center frequency and one zero. (a) The
topology of the filter. (b) The photo of the filter.
75
Figure 4.2.12: Measured results for the filter with tunable center frequency
and one transmission zero. (a) The measured results under
different biases near the passband. (b) The wide band
measured results.
77
Figure 4.2.13: Filter with tunable center frequency and two transmission
zeros. (a) The topology of the filter. (b) The photo of the filter.
79
Figure 4.2.14: Measured results of the filter with tunable center frequency
and two transmission zeros.
80
Figure 4.3.1: The circuit prototype used for the proposed reconfigurable 81
xiii
bandpass filters.
Figure 4.3.2: The theoretical performances of the bandpass filter under two
different states.
82
Figure 4.3.3: The topology of the reconfigurable filter (topology I)
constructed on the Rogers RO3210 board.
84
Figure 4.3.4: Photo of the tested filter (topology I). 84
Figure 4.3.5: Measured results of the tested reconfigurable filter (topology I)
built on the Rogers RO3210 board, where state 1 represents
the state with transmission zero located at the upper band and
state 2 with the zero at the lower band.
85
Figure 4.3.6: The topology of the reconfigurable filter (topology II)
designed on FR4 board.
86
Figure 4.3.7: Measured results of reconfigurable filter (topology II) on the
FR4 PCB board.
86
Figure 5.2.1: The topology of the proposed dual-band quarter-wavelength
transmission line.
92
Figure 5.3.1: The topology of the dual-behavior resonator. 95
Figure 5.3.2: The equivalent circuit of the dual-band bandpass filter. 98
Figure 5.3.3: The topology of the dual-band bandpass filter. 98
Figure 5.3.4: The simulation and measurement results of the fabricated dual-
band bandpass filter.
99
Figure 5.3.5: The structure of the L-shape bandstop filter used to suppress
the spurious harmonics.
100
Figure 5.3.6: The simulation results of the bandstop filters. 101
Figure 5.3.7: The pattern of the dual-band filter with harmonic suppressions. 101
Figure 5.3.8: The simulation and measurement results of the fabricated dual-
band bandpass filter with harmonic suppression.
102
xiv
Figure 5.3.9: The measurement results of the dual-band bandpass filter
with/without harmonic suppression.
102
Figure 5.4.1: The topology of the proposed stub tapped dual-band branch-
line coupler.
103
Figure 5.4.2: Computed normalized branch-line impedances (Z0 =50 Ω)
used in the dual-band branch-line coupler under different
frequency ratios. (a) line impedances for the 2/50 Ω branch,
(b) line impedances for the 50 Ω branch.
104
Figure 5.4.3: Photo of the fabricated dual-band branch-line coupler. 105
Figure 5.4.4: Measurement results of the fabricated dual-band branch-line
coupler (a) the return loss(S11) and the isolation(S41), (b) the
insertion loss, (c) the phase responses at the two designed
ports.
107
Figure 5.4.5: General topology of the proposed dual-band rat-race coupler.
(a) The whole pattern, (b) the proposed unit cell acting as a
quarter-wavelength line at two working frequencies.
109
Figure 5.4.6: Normalized line impedances used in the type I rat-race coupler
under different frequency ratios. (a) Line impedances for
branch I, (b) line impedances for branch II.
111
Figure 5.4.7: Photo of the fabricated type I rat-race coupler. 112
Figure 5.4.8: Measured return loss and isolation of the type I dual-band rat-
race coupler.
113
Figure 5.4.9: Measured insertion losses and phase responses of the in-phase
outputs (S21 and S41) of the type I rat-race coupler. (a) Insertion
loss, (b) phase responses.
114
Figure 5.4.10: Measured insertion losses and phase responses of the anti-
phase outputs (S23 and S43) of the type I rat-race coupler. (a)
115
xv
Insertion loss, (b) phase responses.
Figure 5.4.11: General topology of the type II dual-band rat-race coupler. 116
Figure 5.4.12: (a)Even- and (b) odd- mode topologies of the proposed type II
dual-band rat-race coupler.
117
Figure 5.4.13: Normalized branch impedances used in the type II dual-band
rat-race coupler under different frequency ratios.
121
Figure 5.4.14: Photo of the fabricated type II rat-race coupler. 123
Figure 5.4.15: Measured return loss and port isolation of the type II rat-race
coupler.
123
Figure 5.4.16: Measured insertion losses and phase responses of the in-phase
outputs (S21 and S41) of type II dual-band rat-race coupler. (a)
Insertion losses, (b) phase responses.
124
Figure 5.4.17: Measured insertion losses and phase responses of the anti-
phase outputs (S23 and S43) of type II dual-band rat-race
coupler. (a) Insertion losses, (b) phase responses.
125
Figure 5.4.18: General topology of the proposed dual-band Wilkinson power
divider.
127
Figure 5.4.19: The computed design parameters for different frequency ratios
of the dual-band Wilkinson power divider.
127
Figure 5.4.20: The photo of the fabricated Wilkinson power divider. 128
Figure 5.4.21: The insertion losses of the tested dual-band Wilkinson power
divider.
128
Figure 5.4.22: The return losses and the isolation of the tested dual-band
Wilkinson power divider. (a) S11 and S23, (b) S22 and S33.
129
Figure 5.4.23: The phase responses ( 3121, SS ∠∠ ) of the tested dual-band
Wilkinson power divider.
130
Figure 6.2.1: The flowchart of the proposed algorithm. 136
xvi
Figure 6.2.2: Ideal response of the fourth-pole chebyshev filter. 139
Figure 6.2.3: Responses of the fourth-order chebyshev filter with slightly
mistuned inter-resonator couplings (the extracted ones are the
same as the assigned ones).
140
Figure 6.2.4: Comparisons between assigned and extracted responses of the
fourth-order chebyshev filter with highly mistuned inter-
resonator couplings. (a) S21. (b) S11 (the frequency range for
S11 is between -2 and 2 for the purpose of clarity).
141
Figure 6.2.5: Comparisons between assigned and extracted responses of the
eighth-order quasi-elliptical filter with slightly mistuned inter-
resonator couplings. (a) S21. (b) S11.
143
Figure 6.2.6: The flowchart of the improved GA simulation process. 144
Figure 6.2.7: Comparisons between assigned and extracted responses of the
eighth-order quasi-elliptical filter with highly mistuned inter-
resonator couplings. (a) S21. (b) S11.
145
Figure 6.2.8: Comparisons between assigned and extracted responses of the
fourth-order chebyshev filter with mistuned resonators and
inter-resonator couplings. (a) S21. (b) S11.
147
xvii
LIST OF TABLES
Table 3.3.1: Total phase shifts for the two paths in a third-order bandpass
filter with inductive cross-coupling.
48
Table 3.5.1: Total phase shifts for the two paths in a fourth-order bandpass
filter with capacitive cross-coupling.
56
List of Tables
xviii
Compact, Reconfigurable and Dual-Band Microwave
Circuits
by ZHANG Hualiang
Department of Electronic and Computer Engineering
The Hong Kong University of Science and Technology Abstract
ABSTRACT
Microwave systems have an enormous impact on modern society. Applications
are diverse, from entertainment via satellite television, to civil and military radar
systems. In particular, the recent trend of multi-frequency bands and multi-function
operations in wireless communication systems along with the explosion in wireless
portable devices are imposing more stringent requirements such as size reduction,
tunability or reconfigurability enhancement, and multi-band operations for the
microwave circuits.
In this dissertation, we intend to address the design issues related to microwave
passive circuits. Several novel design concepts for meeting the above-described
challenges in microwave bandpass filters are presented based on in-depth theoretical
analysis and practical implementation. For compact bandpass filters, a microstrip tri-
section stepped impedance resonator (SIR) and a CPW (coplanar waveguide) tri-
section slow-wave SIR are proposed. Compared with the conventional two-section
SIR, the size reduction of the new SIRs can be up to 40 percents. Filters based on the
new SIR structures are designed and implemented in low-cost PCB, with excellent
agreement between the designed and measured characteristics. To achieve
xix
reconfigurability, two types of filters with electronically reconfigurable transmission
zeros are proposed using varactor-tapped stubs. In addition, one of these proposed
bandpass filters features robust reconfigurability in both the transmission poles
(center frequency) and transmission zeros. To achieve multi-band operation, a dual-
band quarter-wavelength transmission line is proposed, which can acts as the dual-
band impedance inverter. A second-order dual-band filter is constructed based on a
dual-band resonator in conjunction with this dual-band impedance inverter. The
performance of this filter is verified by measurement results. The proposed dual-
band transmission line can be also applied to other microwave passive circuits for
dual-band operations. A branch-line coupler, a Wilkinson power divider and two
types of rat-race couplers featuring dual-band characteristics are designed and
fabricated. The desired dual-band performances are verified by measurement results.
The practical issue such as the realizable frequency ratio between the two working
frequencies is also discussed.
For theoretical analysis, we have developed a synthesis process based on the
genetic algorithm (GA). The direct searching property of the GA obviates the
computations of the gradients. To demonstrate the effect of the proposed method,
several general Chebyshev filters with different orders and different performances
are synthesized. This method is applied to get the prototype design parameters of the
filters presented in this dissertation. Besides, the genetic algorithm (GA) is applied
to the parameter extraction for the tuning and optimizing of filters. Not much apriori
knowledge is required for this method, facilitating an automated computer-aided
tuning and optimization platform. To demonstrate the feasibility of our method,
filters with both mistuned resonators and mistuned inter-resonator couplings have
xx
been studied. For all of these filters, the extracted coupling matrices fit the assigned
ones well.
1
CHAPTER 1
INTRODUCTION
Modern microwave technology is an exciting and dynamic field, due in large part
to the advances in modern electronic device technology and the explosion in demand
for voice, data and video communication capacity. Prior to this, microwave
technique was the nearly exclusive domain of the defense industry. The recent and
dramatic increase in demand for communication systems such as mobile phone,
satellite communications and broadcast video has transformed this field to the
commercial and consumer market. As a result, the diversity of applications and
operational environments has led, through the accompanying high production
volumes, to tremendous advances in cost-efficient manufacturing capabilities of
microwave products. This, in turn, has lowered the implementation cost of a new
wireless microwave service. Inexpensive handheld GPS navigational aids,
automotive collision-avoidance radar and widely available broadband digital service
access are among these. Microwave technology is naturally suited for these
emerging applications in communications and sensing, since the high operational
frequencies permit both large numbers of independent channels for uses as well as
significant available bandwidth per channel for high speed communication.
The current trend in microwave technology is toward circuit miniaturization,
high-level integration and cost reduction. To meet these requirements, both active
and passive microwave circuits featuring the properties such as compact size,
tunability and multi-band operations need to be designed. In this dissertation, we
will discuss the novel implementations of microwave passive circuits, with special
focus on the designs of microwave filters.
2
1.1 History of Microwave Circuits
Microwave technology has been developed for over seventy years. The most
fundamental characteristic that distinguishes microwaves with other terms is the
working frequency. Generally speaking, the microwave electronic systems operate
in the frequency range from 300 MHz to 100 GHz or even higher. Fig. 1.1.1 shows
graphically the most common frequency band designations and applications for
microwaves.
Fig. 1.1.1 Band designations and applications of microwaves.
Historically, the development of microwave circuits has in many ways followed
that of the lower frequency electronics circuits, which is from tubes to solid state
devices and from large components to small and to the development of integrated
circuits. The fundamental concepts for the microwave propagating were developed
over 100 years ago. After that, most of the applications in the early 1900s occurred
primarily in the frequency band lower than 300 MHz, due to the lack of reliable
microwave sources and other components. It was not until the 1940s and the advent
of radar development during World War II that microwave theory and technology
L
S 2 GHz
C
XKu
K Ka
VW
4 GHz 8 GHz 12.4 GHz
18 GHz 26 GHz
40 GHz 110 GHz
75 GHz
Frequency (GHz) Band Designation 1 10 100
GSM
1 GHz
DCS PCS
DECT
2 GHz
WLAN Blue- tooth
WLAN Road- Price
5 GHz
SAT TV
10 GHz
Auto- motive Radar
77 GHz
Micro- wave Links
28 GHz 2.4 GHz
3
receive substantial interest. The waveguide components, microwave antennas, small
aperture coupling theory were developed during that time. The concept of planar
transmission line was also proposed at that time, resulting in the hybrid microwave
integrated circuits. The area of hybrid microwave integrated circuits grew rapidly in
the 1960s and was further developed over the 1960-1980 period accompanying the
exciting developments in the semiconductor technology. Many other significant
developments also occurred over that time, including the idea of monolithic
microwave integrated circuits (MMIC), where all microwave functions of analog
circuits could be incorporated on a single chip. But most of the applications of the
microwave circuits during that period were still in the area of military. After that,
especially since 1990s, the field of the microwave technology has experienced a
radical transformation and the developments of microwave circuits have been driven
mainly by the commercial and consumer market, due to the rapid developments in
the communication systems. The advantages offered by microwave systems,
including wide bandwidths and line-of-sight propagation, have proved to be critical
for both terrestrial and satellite communication systems and have thus provided an
impetus for the continuing development of low-cost miniaturized microwave circuits.
1.2 Planar Microwave Circuits
1.2.1 Applications of Planar Microwave Circuits
Generally speaking, microwave circuits can be divided into two categories,
planar and non-planar. The non-planar microwave circuits are mainly based on
waveguides. The photo of the circuits using waveguides is shown in Fig. 1. 2. 1.
This kind of circuits has good performance at high frequency, but its size is large
and it is of high cost. Due to these reasons, this kind of microwave circuits finds
4
Fig. 1. 2. 1 Photo of a microwave circuit using waveguide.
(a)
(b)
Fig. 1. 2. 2 General structure of (a) microstrip line and (b) coplanar waveguide line.
applications when performances are the primary considerations. The planar
microwave circuits are based on the planar transmission lines, among which the
microstrip line and the coplanar waveguide line are the most important ones. The
structures of these lines are given in Fig. 1. 2. 2. The planar microwave circuits have
the properties such as light weight, high-level integration and low cost, which make
them more suitable for the emerging applications in the wireless communication
systems. The works presented in this dissertation are based on these structures.
Substrate
Signal
Ground
E-Field
Microstrip Line
Coplanar Waveguide (CPW) line
Substrate
Ground
E-Field
Ground Signal
5
(a)
(b)
Fig. 1. 2. 3 (a) Photo of an HTS planar microwave filter [1], (b) a comparison between a
conventional transceiver (left) and an HTS transceiver (right) [2].
With the advances in the communication systems, the planar microwave circuits
have been applied to and substituted for the conventional form of microwave
circuitry such as the waveguide based circuit in virtually every application in the
fields of communications, radar and weapon systems. One application of these
circuits is for the wireless base-stations. In the past, due to the high requirement in
the interference rejection between the transmitting and receiving channels of the
base-stations, most of the transceivers used in these systems were constructed by
non-planar microwave circuits such as the coaxial cavities. As a result, the size of
these circuits was very large. To reduce the overall size, the planar microwave
circuits are applied in combination with the high temperature superconducting (HTS)
6
technique. The planar microwave circuits are constructed on the HTS materials to
achieve good performance such as the low noise level and the high selectivity.
Shown in Fig. 1. 2. 3 (a) is the photo of an HTS planar microwave filter [1], which is
an important component of the microwave circuits. The photos of the transceivers
using the conventional non-planar microwave circuits and the HTS planar circuits [2]
are shown in Fig. 1. 2. 3 (b), demonstrating large size reduction.
Another application of the planar microwave circuits is in the area of monolithic
microwave integrated circuits (MMIC). This concept was proposed in 1958 [3]. It
provides high-level integration between the passive and active parts of the
microwave circuits. In addition, it results in large size reduction. Thus, it is very
suitable for the modern wireless communication systems, where size and cost are the
primary concerns. In the past, most of the MMIC planar microwave circuits were
implemented on the GaAs substrate, since its semi-insulating property provides low
loss for the planar transmission lines in the circuits. With the developments in the
wireless communication systems, the working bands are moved to higher
frequencies and these planar microwave circuits begin to be implemented on the
silicon substrate. This is made possible, since the wavelength of the electromagnetic
wave is reverse proportional to the frequency, with the increase in the working
frequency, the size and the loss of the planar microwave circuits on the lossy silicon
substrate will be greatly reduced. Shown in Fig. 1. 2. 4 are two examples of these
kinds of circuits, where two power amplifiers for the systems working at 24 GHz
and 77 GHz are designed [4], [5]. In these two amplifiers, the planar microwave
circuits are implemented on the silicon substrate with good performances and the
areas of whole circuits are still small. The implementation of the MMIC circuits on
7
silicon substrate reduces the cost of the systems greatly and it is very promising for
the future applications.
In addition to these applications in the field of communications, the planar
microwave circuits are widely used in other areas such as the modern radar systems
and the microwave sensor systems.
(a)
(b)
Fig. 1. 2. 4 Die micrograph of (a) a 24-GHz power amplifier. Chip size: 0.7mm×1.8mm [4].
and (b) a 77-GHz power amplifier. Chip size: 1.35mm×0.45mm [5].
Passive Circuits RF In RF Out
Active Circuits
Passive Circuits
Active Circuits
RF In RF Out
8
Fig. 1. 2. 5 The block diagram of a wireless transceiver.
Fig. 1. 2. 6 The structure of the image rejection mixer circuit.
1.2.2 Structures of Planar Microwave Passive Circuits
Planar microwave circuits are composed of passive and active circuits. As
mentioned before, we will discuss the designs of microwave passive circuits in this
dissertation, with focus on the designs of bandpass filters.
Microwave bandpass filters play important roles in the microwave systems. They
are used to separate or combine different frequencies. The electromagnetic spectrum
is limited and has to be shared, filters are used to select or confine the RF /
microwave signals within assigned spectral limits. One application of microwave
LO Signal
LNA
Image Reject Filter
Mixer
IF Filter AGC
PA
Antenna 2 Antenna 1
Antenna Switch
T/R Switch
Bandpass Filter
Transmitter
Receiver
Z0
Wilkinson Power
Divider
Branch-Line coupler
RF Input
LO Input
Branch-Line coupler
IF Output
9
bandpass filters is in the transmitting and receiving systems to identify and transmit
the desired signals as shown in Fig. 1. 2. 5.
In addition to microwave bandpass filters, there are also other important
microwave passive circuits including the branch-line (90˚) coupler, the rat-race(180˚)
coupler and the Wilkinson power divider. These components are used in the circuits
to realize the appropriated magnitude and phase shifts. One application of these
circuits is illustrated in Fig. 1. 2. 6. Here, the power divider is used to generate two
equal-amplitude signals and the branch-line coupler will generate 90˚ phase
difference between the two output ports resulting in the cancellation of the image
signals.
In the following, we will give the general topologies of these microwave circuits.
1) Bandpass Filters
Work on microwave filters commenced in the 1930s. Since then, various kinds of
filters have been developed. As for the planar microwave bandpass filters, there are
normally four different types: the combline filters, the interdigital filters, the parallel
coupled filters and the hairpin-line filters. Their topologies are given in Fig. 1. 2. 7
and Fig. 1. 2. 8.
The combline filters are constructed by capacitor-loaded resonators, as shown in
Fig. 1. 2. 7(a). The resonators are oriented so that the short circuits are all on one
side of the filter (like a comb), and all of the capacitors at the other side. The
capacitive end-loading of the resonators gives a great size reduction compared with
the conventional quarter-wavelength resonators. This kind of filter was invented in
the Stanford Research Institute in the 1960’s. A theory about these filters can be
found in [6]. The drawback of combline filters is the asymmetry of their insertion
loss, which has weaker attenuation on the low-frequency side. Hence, attenuation
10
poles have to be introduced on the low-frequency side sometimes to compensate the
asymmetry.
The topology of the interdigital filters is shown in Fig. 1. 2. 7 (b). The design
theory about this filter has been given in [7], [8]. The symmetrical patterns of these
filters make the performances of them symmetrical. So it is simpler to design linear
phase filters using this kind of structure.
(a)
(b)
Fig. 1. 2. 7 Structures of (a) combline filters and (b) interdigital filters.
……
……
11
(a)
(b)
Fig. 1. 2. 8 Structures of (a) parallel-coupled filters and (b) hairpin-line filters.
The general structure of the parallel-coupled-line filters is shown in Fig. 1. 2. 8(a),
which is based on the half-wavelength resonators. A simple design procedure can be
found in [9].
The folded version of the parallel-coupled-line filters is known as hairpin-line
filter. Fig. 1. 2. 8 (b) gives the pattern of this filter. Compared with the unfolded
filters, the size for the hairpin-line filter has been greatly reduced. The design issues
related to this kind of filter have been discussed in [10].
2) Branch-Line Couplers
The branch-line coupler is also called 90˚-coupler or quadrature coupler [11],
[12]. The general structure of this coupler is given in Fig. 1. 2. 9. Four quarter-
……
……
12
wavelength transmission lines with suitable characteristic impedances comprise this
coupler. Referring to Fig. 1. 2. 9, when the input signal is from port 1 and port 4 is
terminated with 50Ω resistor, the output signals at port 2 and port 3 will be equal in
magnitude and with 90˚ difference in phase. This circuit has been widely used in the
systems with balanced structures to suppress unwanted signals.
Fig. 1. 2. 9 Topology of the planar branch-line coupler.
3) Rat-Race Couplers
The rat-race coupler is also called 180˚-coupler or ring hybrid [13]. The general
structure of this coupler is given in Fig. 1. 2. 10. It has four branches, three of which
are quarter-wavelength lines and the fourth line is a three-quarter-wavelength line.
When the input signals are from port 2 and port 4, they will be added at port 1 and
be subtracted at port 3. Besides, good isolation will be achieved between port 2 and
port 4. This circuit can be also used in the balanced structures to reject image signals.
4) Wilkinson Power Divider
The Wilkinson power divider was proposed in 1960 [14]. The general structure of
this coupler is given in Fig. 1. 2. 11. It is constructed by two quarter-wavelength
transmission lines. A resistor is connected between the two output ports for the
purpose of matching. When the signal is inputted from port 1 (as shown in Fig. 1. 2.
11), it will be split into two signals with equal phase and equal amplitude. This
circuit is widely used in the microwave mixers to combine and separate signals.
Port 1 Port 2
Port 3 Port 4
50Ω λ/4 line
50Ω λ/4 line
35.35Ω λ/4 line
13
Fig. 1. 2. 10 Topology of the planar rat-race coupler.
Fig. 1. 2. 11 Topology of the Wilkinson power divider.
1.3 Motivation and Overview of This Dissertation
Microwave technology play important roles in modern communication systems.
It naturally meets the requirement for the communication systems to transmit more
and more data at high speed. Meanwhile, advances in the communications especially
in the wireless communication systems continue to challenge microwave circuits
with ever more stringent requirements — compact size, tunability or
reconfigurability enhancement and multi-band operations. The property of compact
70.7Ω λ/4 line
Port 3 Port 4
Port 1 Port 2
70.7Ω 3λ/4 line
Port 1
Port 2
Port 3
70.7Ω λ/4 line
100Ω
14
size will lead to the reductions in both the volume and the weight of the systems,
making them portable. The tunablity makes the system more reliable and flexible. It
also reduces the size of the systems. The property of multi-band operations will
result in both size and cost reductions of the systems.
In this dissertation, we intend to make improvements in all these aspects of
microwave passive circuit designs with focus on filter designs. Other microwave
passive circuits such as couplers and power dividers with dual-band operations will
also be discussed. Besides, the genetic algorithm (GA) is used to do the synthesis
and the post-tuning of the performances of the filters.
Chapter 2 covers the theory for the synthesis of the general Chebyshev filters.
Most of the filters presented in this dissertation will be designed based on the
synthesized design parameters. The GA based optimization process is proposed to
do the synthesis of this kind of filters. Filters with different orders and different
performances are analyzed and the results are very close to the ideal ones.
Chapter 3 presents two kinds of compact resonators, one for the microstrip line
and the other for the CPW lines. Filters with high selectivities on the edges of the
passbands are designed using these new resonators. Great size reductions have been
achieved. These resonators can also be applied in the monolithic microwave
integrated circuits (MMIC) to shrink the size.
Chapter 4 gives two topologies suitable for the reconfigurations of the
transmission zeros, type I and type II. Varators are applied to provide the tunabilities.
In addition, the type I design is also used to realize the tunings of both the center
frequencies and the transmission zeros. All of the design concepts have been proved
by measurement results.
15
Chapter 5 proposed a dual-band quarter-wavelength transmission line. The
working principles of this novel structure are explained. It is used as the dual-band
impedance inverter for the design of a second-order filter working at 2GHz / 5GHz.
This structure is also applied to other microwave passive circuits including branch-
line coupler, rat-race coupler and Wilkinson power divider. Detailed design
equations are derived for all of these proposed circuits. Besides, their performances
are proved by measurement results.
Chapter 6 discusses the practical design issues related to the post-tunings of the
performances of the filters. Again, the genetic algorithm (GA) is applied to extract
the coupling coefficients according to the assigned results. The deviations of the
fabrication errors can be clarified by the comparisons between the extracted
coupling matrix and the ideal coupling matrix.
Finally, the conclusion is given in Chapter 7, where the future work for this
dissertation is also provided.
16
CHAPTER 2
Synthesis of Microwave Filters
2.1 Introduction
The modern microwave filters are based on the circuit prototypes featuring
certain desirable responses. In usual, these prototypes are categorized according to
their transfer functions. The most important filter types are Butterworth (maximally
flat) filter, Chebyshev filter, elliptic function filter, Gaussian (maximally flat group-
delay) filter and all-pass filter. Among them, the microwave filters incorporating the
Chebyshev class of filtering functions have found frequent applications within
microwave space and terrestrial communication systems. The general characteristics
of this kind of filter are the equiripple in-band amplitude, together with the sharp
cutoffs at the edge of the passband and high selectivity, which give an acceptable
compromise between lowest signal degradation and highest noise / interference
rejection. Besides, the ability to build in prescribed transmission zeros for improving
the close-to-band rejection slopes has enhanced its usefulness. Due to these reasons,
most of the filters presented in this thesis are this kind of filters. In this chapter, the
analysis of this kind of filters will be discussed, which is an important step for filter
design.
In the practical cases, to realize the filter with the desirable responses, we need to
synthesize the prototypes and extract the design coefficients. This kind of synthesis
process can be very complicated and time-consuming. Fortunately, it is found that
the transfer functions of the filters can be expressed in terms of a finite number of
complex zero and pole frequencies. Starting from this rational function, a general
theory of coupled-resonator filters has been developed [15] – [28]. Once the system
17
function is obtained, the following synthesis process is proceeded to extract the
element values of a coupling matrix. In the past, several different methods had been
used. In [29], Cameron proposed to use matrix transformation to reduce a potentially
full coupling matrix to a folded form as desired. In [30] [31], Gajaweera et al. used
the Newton-Raphson method to reduce the coupling matrix. In [32], Lamecki et al.
used the damped Levenberg-Marquardt (LM) method to extract the coupling
coefficients. In [33], Amari proposed to use the gradient-based optimization
technique to do the synthesis. However, this method requires the computations of
the gradients of the functions, which is not easy under some circumstances. In this
chapter, we use the Genetic Algorithms (GA), which is a direct optimization method,
to synthesize the coupling matrix. In this way, the computations of the gradients can
be avoided.
The first part of this chapter reviews an efficient technique for generating the
general Chebyshev transfer function, given the numbers and positions of the
transmission zeros required to realize. In the following part, we will explain the
concept about the coupling matrix of the filter. Then, the basic operations of the GA
will be presented. Finally we will use this optimization method to synthesize the
prototype of the general Chebyshev filters.
2.2 Basic Principles for Generating the Rational Polynomials of the General
Chebyshev Filters
In the synthesis of this chapter, we deal with the two-port low-pass prototype
with a normalized frequency (ω=1). The transfer and reflection functions can be
expressed as a ratio of two Nth degree polynomials:
18
)()()(11 ω
ωωN
N
EPS = (2.1a)
)()()(21 ωε
ωωN
N
EDS = (2.1b)
where ω is the real frequency variable related to the more familiar complex
frequency variable s by s = jω. For the Chebyshev filtering function, ε is a constant
normalizing S21 to the equiripple level at ω = ±1 and its value is given by:
1
10/ )()(
1101
=
⋅−
=ω
ωωε
N
NRL P
D (2.2)
where RL is the prescribed return loss level in decibels and it is assumed that all the
polynomials have been normalized such that their highest degree coefficients are
unity. S11(ω) and S21(ω) share a common denominator EN(ω) and the polynomial
DN(ω) contains the transfer function transmission zeros.
Using the conservation of energy formula for a lossless network
1221
211 =+ SS (2.3)
combined with (2.1), it is found:
))(1))((1(1
)(11)( 22
221
ωεωε
ωεω
NN
N
FjFj
FS
−+=
+=
(2.4)
where
)()()(
ωωω
N
NN D
PF =
FN(ω) is known as the filtering function of degree N and has a form for the
general Chebyshev characteristic:
19
])(coshcosh[)(1
1∑=
−=N
nnN xF ω (2.5)
where
n
nnx
ωωω
ω
−
−=
1
1
And jωn = sn is the position of the nth transmission zero in the complex s-plane. It
can be easily proved that when 1=ω , FN = 1, when 1<ω , FN < 1, and when
1>ω , FN >1, all of which are necessary conditions for a Chebyshev response.
Finally, the filtering function FN(ω) is computed using the well-known recursive
relations between the numerator and denominator of the polynomials.
2.3 Circuit Model of the Filter and the Coupling Matrix
I1R1
es
a1
b1
V1
I2 a2
b2
V2
Two-port circuit network
R2
Fig. 2.3.1 The equivalent circuit representing an general two-port network.
As described before, one key step in the filter synthesis is to convert the rational
polynomials to a coupling matrix. In this section, we will explain the origin of the
useful coupling matrix. Most microwave filters can be represented by a two-port
network as shown in Fig. 2. 3. 1, where V1, V2 and I1, I2 are the voltage and current
20
variables at the ports 1 and 2, R1, R2 are the terminal impedances, es is the source
voltage. To measure the signals’ intensity at microwave frequencies, the wave
variables a1, b1 and a2, b2 are introduced, with ‘a’ indicating the incident waves and
‘b’ the reflected waves.
The relationships between the wave variables and the voltage and current
variables are defined as:
21)(
21
)(21
andnIR
RVb
IRR
Va
nnn
nn
nnn
nn
=
−=
+=
(2.6)
The above definitions guarantee that the power at port n (n=1 and 2) is:
)(21)Re(
21 ∗∗∗ −=⋅= nnnnnnn bbaaIVP (2.7)
Fig. 2.3.2 The equivalent circuit of n-coupled resonators.
21
Next, we consider the equivalent circuit of n-coupled resonators, which is shown
in Fig. 2. 3. 2, where L, C, R denote the inductance, capacitance and resistance, Mij
represents the mutual inductance between resonators i and j. Using the well-known
Kirchhoff’s law, we can write down the loop equations for this circuit:
=+++⋅⋅⋅−−
⋅⋅⋅⋅
=−⋅⋅⋅++−
=−⋅⋅⋅−++
0)1(
0)1(
)1(
2211
222
2121
121211
11
nn
nnnn
nn
snn
iCj
LjRiMjiMj
iMjiCj
LjiMj
eiMjiMjiCj
LjR
ωωωω
ωω
ωω
ωωω
ω
(2.8)
The equations listed in equation (2.8) can be represented in matrix form:
⋅⋅⋅=
⋅⋅⋅
++⋅⋅⋅−−
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
−⋅⋅⋅+−
−⋅⋅⋅−++
0
0
1
1
1
2
1
21
22
221
1121
11s
n
nnnnn
n
n e
i
ii
CjLjRMjMj
MjCj
LjMj
MjMjCj
LjR
ωωωω
ωω
ωω
ωωω
ω
(2.9a)
or ][][][ eiZ =⋅ (2.9b)
where [Z] is an n×n impedance matrix.
By inspecting equations (2.6), (2.7), Fig. 2. 3. 1 and Fig. 2. 3. 2, it can be
identified that I1 = i1, I2 = -in, V1 = es – i1R1 and V2 = -inR2, which leads to:
22
2
1
111
11
02
22
Rib
aR
Rieb
Rea
n
s
s
=
=
−=
=
(2.10)
22
And the S-parameter can be given as:
s
n
a eiRR
abS 21
01
221
2
2
===
(2.11)
sa eiR
abS 11
01
111
212
−===
(2.12)
where i1 and in can be solved by the inversion of the impedance matrix [Z].
The impedance matrix [Z] can be further separated into three parts [Ω], [R] and
[M], where [Ω] is a n×n identity matrix, [R] is a n×n matrix with all elements zero
except R1,1 = Rn,n = 1 and [M] represents the coupling matrix. Referring to equations
(2.11) and (2.12), the S-parameters can then be expressed as:
[ ][ ][ ] [ ][ ] [ ]ejIZIMjR −==+Ω+− ω (2.13)
[ ] 112121 2 −−= nZRRjS (2.14)
[ ] 111111 21 −+= ZjRS (2.15)
In equations (2.13) – (2.15), we have made use of the coupling matrix to express
the S-parameters of the filters.
2.4 Synthesis of General Chebyshev Filters Using GA
2.4.1 Basic Elements of GA
Genetic Algorithm (GA) is a robust and stochastic search method based on the
principles and concepts of natural selection and evolution [34] – [36]. It is a direct
search optimizer, which makes it effective to find an approximate global maximum
in a multi-variable, multi-model function domain compared with the conventional
optimization methods (e.g. the gradient based method). In GA, a set of potential
solutions is caused to evolve toward a global optimal solution. Evolution toward a
23
global optimum occurs as a result of pressure exerted by a fitness-weighted selection
process and exploration of the solution space is accomplished by recombination (in
GA, it is often called ‘crossover’) and mutation of existing characteristics present in
the current population. The flowchart of the proposed GA in this chapter is
illustrated in Fig. 2. 4. 1. To make it clear, some key GA terminologies are
explained here.
Genes and Chromosomes: As in natural evolution, the gene is the basic building
block in the GA optimization. In this chapter, the gene is a binary number (0 or 1).
Within the GA paradigm, a string of genes is called a chromosome. Hence, a
chromosome in our work is a string of binary code.
Populations and Generations: In GA based optimizations a set of trial solutions
in the form of chromosomes is assembled as a population. The iterations in GA
optimization are called generations. The important GA operation, the ‘reproduction’,
is implemented to create a new generation to replace the original generation. In
theory, individuals (or chromosomes) with high fitness values produce more copies
of themselves in the subsequent generation, resulting in the general drift of the
population as a whole toward an optimal solution point. The whole process can be
terminated by the prescribed criteria.
24
Fig. 2.4.1 The flowchart of the proposed genetic algorithm (GA).
Fitness Function: It is the objective function defining the optimization goal. It
connects the physical problem with the GA optimization process. The fitness value
25
assigned to a chromosome gives the ‘goodness’ of the trial solution represented by
that chromosome.
Parents and Children: In the process of reproduction, the chromosomes selected
from the current generation are called ‘parents’ and the newly generated
chromosomes are called ‘children’.
The three conventional GA operators are reproduction, crossover and mutation.
Their basic structures are given in Fig. 2. 4. 2.
Reproduction: As shown in Fig. 2. 4. 2 (a), it is actually a chromosome selection
process. In a typical selection scheme, this process is modeled as a weighted roulette
wheel as shown in Fig. 2. 4. 3. Each element in the roulette wheel represents a
chromosome. The chromosome with larger fitness value occupies larger space in the
wheel, which makes it easier to be selected in the reproduction process. In this way,
we emulate the Darwinian concepts of natural selection and evolution.
Crossover: It is implemented when two chromosomes (parents) are selected. In
our case, the one-point crossover method is used, as shown in Fig. 2. 4. 2 (b). A
crossover point is set randomly and the binary numbers after this point are
exchanged, forming two new chromosomes (or so called ‘children’). The children
inherit advantages (i.e. good fitness) of the parents, but with new features.
Mutation: This process is done on the new chromosomes (‘children’) as shown in
Fig. 2. 4. 2 (c). Some mutation points are picked randomly and the binary numbers
at these positions of the chromosomes are flipped. The ‘crossover’ and the
‘mutation’ processes insure the divergence of the solutions generated by the GA
method.
In the next, we will use the GA to find the optimal solutions for the synthesis of
the general Chebyshev filters.
26
(a)
(b)
(c)
Fig. 2. 4. 2 Three basic operators of GA. (a) Reproduction. (b) Crossover. (c) Mutation.
27
8 % 17 %
10 %
5 %
11 %30 %
5 %
4 %
6 %4 %
Fig. 2. 4. 3 The roulette wheel represents the reproduction process in the GA.
2.4.2 Synthesis of the Filters
To synthesize the general Chebyshev filters using the GA, we need to define an
appropriate fitness function. In this work, we use the positions of the transmission
zeros, reflection zeros and reflection maxima to construct the fitness functions. In
detail, the positions of the transmission zeros are prescribed for the general
Chebyshev filters. The positions of the reflection zeros and reflection maxima can be
computed from the polynomial PN(ω). The object in the synthesis process is to find
an optimal set of parameters to minimize the value of the defined cost functions,
which is the inverse of the fitness functions. We will apply this method to the
general Chebyshev filters with different orders and different specifications. And all
the synthesized filter prototypes are normalized to the frequency ω = 1.
As the first example, a fourth-order general Chebyshev filter with a pair of finite
transmission zeros is considered. This kind of filter is also called the cascaded-
quadruplet (CQ) filter. For this example, the positions of the two transmission zeros
28
are at Ω1,2 = ±1.8 and the in-band return loss is -20dB. Using the transformation
formula as follows:
[ ] 2/110/ 110 −−= RLε (2.16)
where RL represents the in-band return loss level, ε represents the in-band insertion
loss level. The in-band insertion loss level is 0.1005.
Since it is a fourth-order filter, it has four transmission zeros in theory. In this
case, with two transmission zeros assigned to the finite frequencies, the other two
transmission zeros are at Ω3,4 = ± ∞. With the positions of the four transmission
determined, the polynomial PN(ω) can be computed as:
8315.05639.64238.6)( 244 +−= ωωωP (2.17)
From equation (2.17), the four reflection zeros are at ωz1,z2 = ±0.9347 and ωz3,z4 =
±0.3849, the three reflection maxima are at 0 and ±0.7376.
Based on these reflection zeros, reflection maxima and transmission zeros, the
cost function for this filter is defined as:
211211211
211211
4
1
2
121
211
1)7376.0(
1)7376.0(
1)0(
1)1(
1)1(
)()(
εεω
εεω
εεω
εεω
εεω
ω
+−=+
+−−=+
+−=+
+−=+
+−−=+
Ω+= ∑ ∑= =
SSS
SS
SSKi i
izi
(2.18)
where S11 and S21 can be computed via equations (2.14) and (2.15).
According to equations (2.14) and (2.15), the S-parameters are related to the
coupling matrix. Hence, we need to find the optimal set of coupling coefficients in
the coupling matrix to minimize the value of cost function K as defined in equation
(2.18). For this fourth-order filter, there are four unknowns to be optimized, which
29
are m12, m23, m14 and R1. In our GA optimizations, the population size is chosen as
200, which means there are 200 chromosomes. Each chromosome is constructed by
four 14-bit binary strings, which means the length of each chromosome is 56. The
fitness function of each chromosome is defined as:
i
i KF 1
= (2.19)
where Ki is the cost function value of the chromosome i and Fi is the fitness function
value of the chromosome i.
The roulette wheel for the process of ‘reproduction’ is constructed according to
the fitness values of the chromosomes. Let the fitness of chromosome j be Fj and the
percentage for chromosome j in the roulette wheel is defined by:
∑
=
= n
ii
jj
F
FP
1
(2.20)
Using equations (2.18) – (2.20) combined with the flowchart illustrated in Fig. 2.
4. 1, the optimization based on GA is implemented, where the crossover rate is set as
0.6 and the mutation rate is set as 0.0333. After 100 iterations (100 generations), the
fitness value of the best chromosome generated is 79.7922. Converting this
chromosome to real numbers results in the synthesized coupling matrix as given in
equation (2.21) and the input/output impedance is 1.054. Fig. 2. 4. 4 shows the
frequency responses of the synthesized coupling matrix. It is found that the desired
transmission zeros at Ω1,2 = ±1.8 and the -20dB in-band return loss have been
reached.
30
−
−
08608.002233.08608.007889.0007889.008608.02233.008608.00
(2.21)
-4 -3 -2 -1 0 1 2 3 4-80
-60
-40
-20
0
S21
& S1
1 (d
B)
Normalized Frequency
Fig. 2. 4. 4 The S-parameters represented by the synthesized coupling matrix of the
fourth-order filter.
The second example is a sixth-order general chebyshev filter with four
transmission zeros at finite frequencies. The four finite transmission zeros are at Ω1,2
= ±1.5 and Ω3,4 = ±2.1. The in-band return loss level is again -20dB. The
corresponding P6(ω) is:
1449.145377.345184.21)( 2466 −+−= ωωωωP (2.22)
31
From equation (2.22), the six reflection zeros are at ωz1,z2 = ±0.9743, ωz3,z4 =
±0.7549 and ωz5,z6 = ±0.2931, the five reflection maxima are at 0, ±0.5521 and
±0.8945. The cost function for the synthesis of this filter is defined as:
2
211
2
211
2
211
2
211
2
211
2
211
2
211
4
321
6
1
2
121
211
1)8945.0(
1)8945.0(
1)5521.0(
1)5521.0(
1)0(
1)1(
1)1(
))((10)()(
εεω
εεω
εεω
εεω
εεω
εεω
εεω
ω
+−=+
+−−=+
+−=+
+−−=+
+−=+
+−=+
+−−=+
Ω×+Ω+= ∑∑ ∑== =
SS
SS
SSS
SSSKi
ii i
izi
(2.23)
There are six unknowns to be optimized, which are m12, m23, m34, m25, m16 and
R1. Applying the same GA method as for the fourth-order filter, the fitness value of
the best chromosome generated after 100 generation is 334.7305. Converting this
chromosome to real numbers results in the synthesized coupling matrix as given in
equation (2.24) and the input/output impedance is 0.993. Fig. 2. 4. 5 shows the
frequency responses of the synthesized coupling matrix. It is found that the desired
transmission zeros at Ω3,4 = ±2.1 and the -20dB in-band return loss have been
reached, but the transmission zeros at Ω1,2 = ±1.5 have been shifted to ±1.48. This
shift can be compensated by changing the coefficients in equation (2.23), which are
related to the transmission zeros at Ω1,2.
−
−
08294.00000245.08294.005716.001923.0005716.007243.000007243.005716.0001923.005716.008294.0
0245.00008294.00
(2.24)
32
-4 -3 -2 -1 0 1 2 3 4-100
-80
-60
-40
-20
0
S21
& S
11 (d
B)
Normalized Frequency
Fig. 2. 4. 5 The S-parameters represented by the synthesized coupling matrix of the sixth-
order filter.
The final example is a fifth-order filter with two asymmetrically located
transmission zeros. The two finite transmission zeros are at Ω1 = -1.43 and Ω2 = -
2.45. The in-band return loss level is -20dB. The corresponding P5(ω) is:
9302.0729.29735.75644.141508.81208.13)( 23455 ++−−+= ωωωωωωP
(2.25)
From equation (2.25), the five reflection zeros are at ωz1 = -0.9739, ωz2 = -0.7472,
ωz3 = -0.249, ωz4 = 0.4222 and ωz5 = 0.9267, the four reflection maxima are at -
0.8924, -0.5318, 0.0817 and 0.7212. The cost function for the synthesis of this filter
is defined as:
33
2
211
2
211
2
211
2
211
2
211
2
211
221
5
1121
211
1)8924.0(
1)5318.0(
1)0817.0(
1)7212.0(
1)1(
1)1(
)(10)(5)(
εεω
εεω
εεω
εεω
εεω
εεω
ω
+−−=+
+−−=+
+−=+
+−=+
+−=+
+−−=+
Ω×+Ω×+= ∑=
SS
SS
SS
SSSKi
zi
(2.26)
There are twelve unknowns to be optimized, which are m11, m22, m33, m44, m55,
m12, m23, m34, m45, m13, m35 and R1. For this example, the population size is 100 and
the generation number is 60. The fitness value of the best chromosome generated is
207.9148. Converting this chromosome to real numbers results in the synthesized
coupling matrix as given in equation (2.27) and the input/output impedance is 1.035.
Fig. 2. 4. 6 gives the frequency responses of the synthesized coupling matrix. It is
found that the desired transmission zeros at Ω1 = -1.43 and Ω2 = -2.45 have been
satisfied, but the in-band return loss level is a little bit higher than -20 dB. This
difference can be compensated by setting a lower return loss level (e.g. -22dB) in the
cost function (2.26).
−−
−−−
−−
0374.07413.04498.0007413.05834.0514.0004498.0514.01428.06081.02344.0006081.02686.08401.0002344.08401.00623.0
(2.27)
34
-4 -2 0 2 4-100
-80
-60
-40
-20
0
S21
& S1
1 (d
B)
Normalized Frequency
Fig. 2.4.6 The S-parameters represented by the synthesized coupling matrix of the fifth-
order filter.
2.5 Summary
In this chapter, we have applied the genetic algorithm (GA) to synthesize the
general Chebyshev filters. In the synthesis, the performance of the filter is expressed
by the well-known coupling matrix. The rational polynomial is used to find the
positions of the reflection zeros and maxima, which are the necessary conditions for
the success of the synthesis process. The cost function is defined based on these
conditions. Finally, the GA is applied to find the optimal coupling coefficients to
minimize the value of the cost function. To prove the performance of the proposed
algorithm, three filters with different orders and different characteristics have been
synthesized. The results are very close to the specifications. In addition, the results
can be further improved by increasing the population size (more chromosomes) and
the generation number in the simulation.
35
CHAPTER 3
Design of Compact Microwave Bandpass Filters
3.1 Introduction
Microwave bandpass filters are the key components for the modern
RF/microwave systems. They provide the isolations between the receiver and the
transmitter circuits, which enables the integrations of these two systems in one chip.
In general, there are two types of filters, one is composed of lumped elements and
the other is composed of distributed elements. The most important feature of the
lumped elements filters is its compact size. However, at high frequency (e.g. at the
radio frequency band and the microwave frequency band), the distributed effect will
be dominant, which degrades the performance of the lumped elements filters. Due to
this reason, most of the microwave bandpass filters are based on distributed
elements (e. g. waveguides, microstrip lines, coplanar waveguide lines). Compared
with the waveguide filters, microstrip lines and coplanar waveguide lines based
filters are easy to integrate with active circuits and low in cost. These characteristics
make them the main candidates for microwave filter designs. However, the size of
the planar microwave filters is still the problem, especially when these filters are
applied in the monolithic microwave integrated circuits (MMIC). To alleviate this
problem, in this chapter, we try to design microwave filters with more compact size.
As for the miniaturization of the microwave filters, numerous methods were
proposed in the past [37] – [39]. Among them, the using of high dielectric constant
substrate, the slow-wave effect and the stepped-impedance resonator (SIR) structure
are the most effective techniques to achieve compact size. In our work, we employ
the last two ones to do the size reductions. First, a compact resonator prototype
36
based on stepped-impedance techniques is proposed. This resonator is then applied
to the microstrip lines to construct a compact third-order microstrip bandpass filter.
Based on this newly proposed stepped-impedance resonator, combining the slow-
wave effect, we develop another compact CPW resonator. This CPW resonator has
been applied to a fourth-order CPW bandpass filter to demonstrate the performance
of the proposed structure.
3.2 Topology of the Proposed Tri-Section Stepped-Impedance Resonator
and Theoretical Analysis
As mentioned in the introduction, this resonator is based on the stepped-
impedance techniques. The general structure of a conventional stepped-impedance
resonator is shown in Fig. 3. 2. 1. This structure was first proposed by Dr. Makimoto
[40], [41] and was successfully applied in various types of bandpass filters
(Butterworth, Chebyshev, quasi-elliptical etc.) to shrink the overall size [42] – [44].
As shown in the figure, this kind of resonator is composed of two sections with
different impedance (Za ≠ Zb), but with the same electrical length (θa = θb). By the
introduction of the impedance discontinuity, the overall electrical length (2θa +2θb)
at resonance can be greatly reduced compared to the conventional uniform half-
wavelength resonator. Physically speaking, the presence of the impedance obstacle
makes some of the wave reflect. The reflection wave plus the incident wave make
the resonance condition satisfied at a shorter electrical length. The rigorous
theoretical analysis will be given in the following when we derive the design
equations of the proposed tri-section stepped-impedance resonator. It should be
pointed out here that, to reduce the size of the resonator, the impedance of the center
section Za should be larger than Zb.
37
Fig. 3.2.1 General topology of the conventional stepped-impedance resonator.
Although the size reduction in the conventional stepped-impedance resonator is
very large, it is still desirable to further shrink its size. After a carefully checking, we
find it possible to realize this kind of miniaturization by introducing another section
besides the original two sections, so called tri-section SIR. The basic structure of the
newly proposed resonator is given in Fig. 3. 2. 2. The additional section is inserted
between the other two sections with the characteristic impedance of Z3. As shown in
the figure, starting with the open-terminated left end, the admittance Y looking into
the right end of the SIR is given by:
∆
−−−= 32
232131312132 tantantantantantan θθθθθθ ZZZZZZZY (3.1)
Where 321
221
323223
221321
tantantan
tantantan
θθθ
θθθ
ZjZ
ZjZZjZZZjZ
−
++=∆
and Z1, Z2, Z3 are the characteristic impedances of the three cascaded sections and θ1,
θ2, θ3 are the corresponding electrical lengths. At resonance, Y=0, which results in
the condition for resonance:
1tantantantantantan 313
121
2
132
2
3 =++ θθθθθθZZ
ZZ
ZZ
(3.2)
Za , 2θa
Zb , θb Zb , θb
38
Fig. 3.2.2 General topology of the proposed tri-section SIR.
Using equation (3.2), we can easily get the design formula for the conventional
SIR. By setting θ3=0, we have a conventional two-section SIR, and equ. (3.2)
becomes:
1tantan 212
1 =θθZZ
(3.3)
To obtain the minimum size for the two-section SIR, it is required to have
tanθ1=tanθ2= (Z2/ Z1)1/2.
For the simplicity of analysis of tri-section SIRs, we keep the condition of θ1= θ2,
and we set: k = Z1/Z2 and m = Z1/Z3. Equation (3.2) can be rewritten as:
1tantantan)( 12
31 =++ θθθ kmkm (3.4)
which can be further expressed as:
1
12
3
tan)(
tan1tan
θ
θθ
mkm
k
+
−= (3.5)
Since the overall electrical length (θtotal) of the resonator is 2θ1+ θ3, using equation
(3.5) to substitute θ3 with θ1, θtotal can be expressed as:
)tan)(
tan1tan2(*21
12
11
+
−+= −
θ
θθθ
mkm
ktotal (3.6)
From equation (3.6), we can see the overall electrical length (θtotal) is mainly
determined by k, m, and θ1. When m and k are fixed, θtotal becomes a function of θ1
Z1, 2θ1
Z3 ,θ3 Z3 ,θ3
Z2,θ2 Z2 ,θ2
Y
39
and its minimum value (θtotal,min) could be found. By scanning through the realizable
values of k and m, we aim at finding the range of the values in k and m that yield
smaller θtotal,min compared to the conventional two-section SIR, achieving the goal of
further size reduction by the tri-section SIR structure. Such an analyzing process can
be carried out using computer simulation.
Before the analysis, there are several conditions for the values of k, m, and θ1
should be mentioned. One is that since θ3 can not be negative so from equation (3.5)
we get that 0≤ θ1≤tan-1(k-0.5). The other is that since the resonators will be based on
microstrip lines and the possible largest impedance for the microstrip line is limited
by the resolution of the etching and the achievable smallest impedance is limited by
the fact that the width of the stub should be smaller than a quarter wavelength to
avoid the excitation of higher order modes [45]. Also this limit will vary with the
different property of the PCB board used. In our case we use the PCB board with
dielectric constant of 4.5 and substrate thickness of 1.6mm. Since in our process the
minimum gap we can get is 0.2mm, at the frequency of 2GHz, the impedance of the
microstrip line is limited between 12Ω and 129Ω. So it is reasonable to set
0<k,m≤10 in our analysis. Another thing important is that since we want to find the
condition to further decrease the overall electrical length of the conventional SIR, so
all the results are normalized to the shortest electrical length of the conventional
two-section SIR, which is equals to 4tan-1(k-0.5). Finally, k is always chosen to be
larger than 1, as the case in conventional two-section SIRs.
The analysis will be divided into three parts due to the fact that there are three
kinds of cases for the relation of the k and m, which are (i) 1≤m≤k, (ii) k≤m, and (iii)
0≤m≤1. For case (i) we compute the minimum value of the θtotal and the results are
given in Fig. 3. 2. 3 (i). From the results we can see that the minimum value of the
40
tri-section SIR occurs when m=1 or m=k and the normalized value is 1. These facts
mean the two-section SIR is already the best choice under this condition. We can not
decrease θtotal further using the tri-section SIR under this circumstance. For case (ii)
the results are given in Fig. 3. 2. 3 (ii), it is clear we can decrease θtotal very much
under this condition. And the larger the difference between k and m the more we can
decrease the total electrical length. For case (iii), actually it is equivalent to the
case(ii), so it is also possible to decrease θtotal under this condition and the results are
given in Fig. 3. 2. 3 (iii). From the figure, it seems that under this situation we can
decrease the size most. But actually there are some problems in the realization of
this condition, which will be explained later.
0 2 4 6 8 100.998
1.000
1.002
1.004
1.006
1.008
1.010
1.012
1.014
1.016
1.018
1.020
k=10 k=8 k=6 k=4 k=2
Nor
mal
ized
to th
e sh
orte
st e
lect
rical
leng
th o
f
the
conv
entio
nal S
IR
m (1<= m <=k)
(i)
2 4 6 8 10
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
Nor
mal
ized
to th
e sh
orte
st e
lect
rical
leng
th o
f
the
conv
entio
nal S
IR
m (k<= m <=10)
k=8 k=6 k=4 k=2
(ii)
41
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Nor
mal
ized
to th
e sh
orte
st e
lect
rical
leng
th o
f
the
conv
entio
nal S
IRm (0< m <=1)
k=10 k=6 k=2
(iii)
Fig. 3. 2. 3 The computed minimum electrical length with different values of k and m for the
tri-section SIR. (i) 1≤m≤k≤10 (ii) k≤m≤10 (iii) 0<m≤1.
Based on the analysis above, now we want to realize the new resonator using
microstrip line. As we know both case (ii) and case (iii) are effective in decreasing
θtotal. Now we will look into the detail to see which one is the best choice. In Fig. 3.
2. 4 we plot the normalized electrical length versus θ1 with the different values of k
and m. It is obvious from these figures, with all these different k values, to decrease
the size, θ1 can not be too small. Another thing should be noted is that, under all
these situations, only when m is smaller than 0.5 the decrease tends to be evident.
Since θ1 and θ2 is not small, to occupy small area, Z2 can not be too small. This fact
will cause some problem in the realization of the case (iii) in the real circuit. For
example in the microstrip line considered by us, under the condition k=2 and m=0.5,
at 2GHz if Z3 equals to the upper limit impedance (130Ω), then Z2 equals to 32.5Ω.
The width of the stub for that impedance is about 5.8mm. In fact, it is too large. And
in this example the value of k and m have been selected very carefully to release the
worry about the size of the stub. For a larger k and a smaller m, the condition will be
too difficult to be satisfied. Clearly, it is not a good choice.
42
(a)
(b)
(c) Fig. 3. 2. 4 The computed electrical length for case (iii) with different m and θ1 under the
condition (a) k=2 (b) k=4 (c) k=6.
43
(a)
(b)
(c)
Fig. 3. 2. 5 The computed electrical length for case (ii) with different m and θ1 under the conditions (a) k=2 (b) k=3 (c) k=4.
As for case (ii), in Fig. 3. 2. 5 we show the computed results of normalized θtotal
with different k, m and θ1. Same with case (iii) we can not choose θ1 too small. The
44
larger the value of m the more we can decrease the overall length. It seems that to
get a larger m we have to make the stub of impedance Z3 wider, which causes the
same problem in case (iii). But remember this time since θ1 is not small, from
equation (3.5) we can choose a small θ3. And the very short stub gives us the
possibility to fold it to reduce the size. So finally we select Z3<<Z1, Z2 and θ3<< θ1,
θ2 to construct the tri-section SIR.
(a)
(b)
Fig. 3. 2. 6 The structure of (a) conventional SIR (b) the new tri-section SIR.
Fig. 3. 2. 7 Simulated resonance frequency of the conventional and the new SIR.
Wb3
Lb2
Lb1
Wb2
Wb1
Wa1 Wa2
La
45
To prove our prediction, considering two SIR structures, one is the conventional
SIR and the other is the proposed SIR. Their structures are shown in Fig. 3. 2. 6.
And their performance are simulated by the full-wave EM simulator IE3D [46]. The
simulated result is given in Fig. 3. 2. 7.
In the simulation, we set the substrate same with the normal FR4 board having
εr=4.5, tanδ=0.019, and h=1.6mm. The dimensions of the SIR are
Wa1=Wb1=2.5mm, Wb3=0.2mm, Wa2=Wb2=0.5mm, La=Lb1=29mm,
Lb2=7.5mm as labeled in Fig. 3. 2. 6. And the two SIR structures have the same
length and almost the same step width. The only difference is that the new SIR has a
narrow central step with a very small electrical length and a very small characteristic
impedance comparing with the other two steps. As we predict, the simulation result
in Fig. 3. 2. 7 shows that the resonant frequency for the conventional SIR is
2.13GHz and the new SIR is 1.96GHz. This means the proposed SIR can really
shorten the total length of the conventional SIR.
3.3 A Microstrip Bandpass Filter Designed Using the Proposed Tri-Section SIR
3.3.1 Circuit Prototypes of the Third-Order Bandpass Filter To prove the performance of the proposed compact resonator, in this section we
will implemented it to a third-order microwave filter. As for the filter, besides the
size, another important property is its response in the center frequency. With the
development of modern wireless communication systems, there have been
increasing demands for filter with high selectivity near the passband compared with
the conventional Chebyshev filters. It is well-known that the quasi-elliptical filter
(general Chebyshev filter with transmission zeros) which has transmission zeros
near the upper or lower side of the passband is suitable for this kind of applications.
46
In the following, we will first discuss the general principles of this kind of filter and
then implemented it using the proposed resonator.
The conventional microwave filters are composed of several resonators. The
energy is transmitted through these resonators via the capacitive or inductive
couplings as shown in Fig. 3. 3. 1. According to the combinations of the different
types of couplings, it can be divided into three categories (capacitive-coupled,
inductive-coupled, mixed-coupled). The conventional Chebyshev and Butterworth
filters are all of this kind.
(a)
(b)
(c)
Fig. 3.3.1 Coupling schemes of the conventional microwave bandpass filters. (a) inductive-
coupled (b) capacitive-coupled (c) mixed-coupled.
47
In the quasi-elliptical filter, the transmission zeros are produced by the cross-
couplings between non-adjacent resonators. When these transmission zeros approach
the center frequency, the selectivity of the filter can be greatly improved. To provide
a general understanding of the principles of these cross-coupling techniques, a
simplified topology is considered as given in Fig. 3. 3. 2. Every resonator in this
figure is constructed by a parallel LC tank and the inductors and the capacitors stand
for inductive coupling and capacitive coupling respectively. It is known that, at high
frequency, the phase change of the inductor will be about -90˚, the phase change of
the capacitor will be about +90˚ and the phase change of the resonator will be +90˚
below resonance and -90˚ above resonance. Applying these rules to the circuit, the
total phase change in the filter can be computed as shown in Table 3. 3. 1.
According to the data listed in the table, there is an out of phase frequency point
above resonance, which corresponds to the transmission zero. Therefore, for the
topology shown in Fig. 3. 3. 2, the inductive cross-coupling will introduce an upper
band transmission zero [47]. Normally, this kind of structure is called cascaded
triplet (CT). We will design a CT bandpass filter using the tri-section SIR.
Fig. 3.3.2 Coupling schemes of the third-order CT bandpass filter with inductive cross-
coupling.
Resonator 1
Resonator 2 +/- 90°
Resonator 3 - 90°
+ 90° + 90°
48
Table 3.3.1: Total phase shifts for the two paths in a third-order bandpass filter with
inductive cross-coupling
Below Resonance Above Resonance
Path 1-2-3 +90˚ + 90˚ + 90˚ = +270˚ +90˚ - 90˚ + 90˚ = +90˚
Path 1-3 -90˚ -90˚
Results In phase Out of phase
As mentioned before, the cross coupling is the key point to achieve transmission
zero in the CT filter. However, for the conventional SIR, it is not easy to realize the
coupling between the non-adjacent resonators. For the new tri-section SIR as shown
in Fig. 3. 2. 6 (b), it is obvious that the added central section is long comparing with
the other two stubs. So when we folded the half-wavelength SIR into a hairpin type,
it is easy to use the new SIR to introduce cross coupling. This feature combined with
the compact size of the resonator make it attractive in the design of advanced
microwave bandpass filters.
3.3.2 Experimental Results
To demonstrate this property, a third-order CT filter centered at f0=2.05GHz
with a FBW of 5.85% was designed and measured. The minimum in-band return
loss of the filter is less than -15dB. The other design parameters are K12=K23=0.051,
K13=-0.019, where Kij refers to the coupling coefficients between the different
resonators (i,j are the numbers of the resonators) and external quality factor
Qe=15.34. All of these coupling parameters are synthesized using the GA method
proposed in Chapter 2. An upper band transmission zero is located at fzero=1.06f0.
All the physical parameters are then obtained by adjusting the spaces between the
49
resonators and the ports to get the desired coupling coefficients and external quality
factors [48], [49].
Fig. 3. 3. 3 Layout of the third-order CT filter with a transmission zero in the upper
stopband.
Fig. 3. 3. 4 Photo of the fabricated filter.
The final pattern of the tested filter [50] is shown in Fig. 3. 3. 3. It was designed
and fabricated on a FR4 PCB board of dielectric constant εr=4.5 and thickness
L6
Port 2
W 0
W 1
Port 1
S2
S1
W 7
W 4
Resonator 1
W 2 W 3
Resonator 2 S3
L2 L1
L9 W 5
L4
L3
L5
W 6
L8 L7
L10 L11
L12
L13
Resonator 3
50
h=1.6mm. Its photo is shown in Fig. 3. 3. 4. The dimensions of the filter are W0=3.1,
W1=2.5, W2=0.5, W3=W4=0.2, W5=2.5, W6=0.5, W7=0.5, S1=0.2, S2=0.4, S3=0.2,
L1=7.25, L2=2.3, L3=4.65, L4=4.9, L5=5.55, L6=4.45, L7=6.95, L8=4.95, L9=3.7,
L10=5.75, L11=7.75, L12=2.5, L13=8.35. All these numbers are in the unit of mm.
From Fig. 3. 3. 3 and Fig. 3. 3. 4, we can see that the pattern is symmetrical which
means that the resonator1 is same with the resonator3. The width of the feed line is
3.1mm to give the 50 Ω impedance. It should be mentioned that in the design we use
a long narrow line (L12 +L13) to implement the strong input and output couplings. If
you want to give stronger couplings which mean a smaller external quality factor,
you only need to lengthen the line further along the edge of resonator1 and
resonator3. Another thing is that we fold the central line of resonator1, 3 to a right
angle to give the cross coupling. This change will not shift the resonant frequency of
the two SIR. And it is clear that by using this kind of structure we can easily adjust
the spaces between different resonators to give appropriate direct and cross
couplings.
Fig. 3. 3. 5 The measured results of the fabricated filter.
The measured and simulated results of the filter are given in Fig. 3. 3. 5. A
transmission zero is observed at the frequency of 2.27GHz as we expected.
51
Comparing with the simulation performance, there is a frequency shift of about
100MHz for both the position of the transmission zero and the center of the filter’s
passband. The reason of this shift lies in the accuracy of the line spacing in our PCB
fabrication.
3.4 Topology of the Proposed Slow-Wave CPW Stepped-Impedance
Resonator
In the past, there are several designs proposed to reduce the size of the CPW
microwave bandpass filters, which employed the concept of lumped-element or the
conventional SIR [51] – [53]. It is noted that the size of the resonator can be further
shrank using the multi-section SIR. The topology of the conventional CPW SIR is
shown in Fig. 3. 4. 1 (a). The application of the tri-section SIR proposed in last two
sections to the coplanar waveguide(CPW) lines results in two kinds of tri-section
CPW SIR. The patterns of these resonators are given in Fig. 3. 4. 1 (b) and (c). As
shown in Fig. 3. 4. 1 (c), the wide folded midsection in the tri-section SIR results in
size reduction. These folded arms can also be extended into the ground lines and
play the role of capacitive shunt stubs. The capacitive shunt stubs can introduce
slow-wave effects, which can further reduce the size of the SIR [54], [55]. In the
slow-wave tri-section SIR, the midsection is very wide and is extended into the
ground plane of the CPW line. This midsection can be treated as a shunt open stub
with an enlarged capacitance to the ground. As a result, the resonant frequency of
the resonator is reduced, enabling further size reduction. The slow-wave effect can
be explained in the way of wave guiding. Considering the pattern in Fig. 3. 4. 2, the
signal is propagating between the signal line and the ground. Comparing with the
conventional SIR, the propagating path of the signal for the slow-wave SIR has been
52
increased by the midsection. So by using the normally unused ground plane of the
CPW structure, the size of the slow-wave SIR is smaller than that of the traditional
SIR. In the practical designs given in this paper, the shunt stub is folded in the
ground plane to reduce transverse size.
(a)
(b)
(c)
Fig. 3.4.1 Layout of (a) conventional CPW SIR (b) proposed tri-section CPW SIR (Type A)
(c) proposed tri-section SIR (Type B).
Wb2 Wb1
Wb3
Lb3
Wb0
Lb0
La0
Wa1
La3
Wa3
Wa2
Wa0
L0
W0
W1 W2
53
Fig. 3. 4. 2 Schematic illustrating the basic structure of the proposed slow-wave SIR and the
wave propagating path in the slow-wave SIR.
(a)
(b)
Fig. 3. 4. 3 Layout of (a) proposed slow-wave CPW SIR (Type C) (b) proposed slow-wave
CPW SIR (Type D).
To prove our proposed miniaturized CPW tri-section SIRs, the full-wave EM
simulator IE3D is used to simulate the performance of these resonators. For the
Ld0
Ld3
Wd0
Wd1 Wd2
Lc0
Lc3
Wc0
Wc1
Wc2
Sc0
54
slow-wave CPW SIR, two types of structures have been proposed as shown in Fig. 3.
4. 3. The simulation results are shown in Fig. 3. 4. 4. For the ease of comparing
these resonators’ size, all of the SIRs in Fig. 3. 4. 1 and Fig. 3. 4. 3 are tuned to
resonate at the same frequency, 2.4 GHz. In the simulation, we set the substrate the
same as the normal FR4 board having εr = 4.5, tanδ = 0.019, and h = 1.6 mm. The
dimensions of these SIRs are W0 = Wa0 = Wb0 = Wc0 = Wd0 = 6.8 mm (i.e. the
distance between the ground lines is kept the same), W1 = Wa1 = Wb1 = Wc1 = Wd1 =
0.4 mm (i.e. the line width of the first section is kept the same), W2 = Wa2 = Wb2 =
Wc2 = Wd2 = 4.8 mm (i.e. the line width of the third section is kept the same), La3 =
1.4 mm, Wa3 = 0.2 mm, Lb3 = 1.4 mm, Wb3 = 11.6 mm, Lc3 = 0.4 mm, Sc0 (the gap
between the shunt stub and the ground) = 0.2 mm, Ld3 = 0.4 mm. The overall lengths
of the various structures are: L0 = 11.4 mm, La0 = 10.6 mm, Lb0 = 10.0 mm, Lc0 = 7
mm and Ld0 = 8.6mm.
The dimensions of the proposed four tri-section SIRs are all smaller than that of
the conventional two-section SIR. Type A SIR and type B SIR are the tri-section
SIRs having a very wide or narrow central section. At the resonant frequency of
2.4GHz, the length reduction for type A and B resonators is 7% and 12%,
respectively. Type C SIR and type D SIR are the proposed slow-wave SIR. As
mentioned before, the stubs have been folded into the ground. The length of the
shunt stub is about 7.2mm. Type C has two arms and type D has one arm. From the
analysis presented above, type C will be more effective in reducing the size since it
has two shunt stubs. But type D will be more convenient in the design of the quasi-
elliptic filters (cascaded triplet filter, cascaded quadruplet filter, etc.) which normally
do not feature wide ground lines on both sides of the resonators. Comparing with the
conventional two-section SIR, the longitudinal size reduction for type C SIR is
55
about 39% and for type D SIR is about 25%. It is evident that the slow-wave SIRs
are more effective in size reduction as compared with the tri-section CPW SIR.
Hence, we will design two CPW bandpass filters based on this kind of resonator.
Fig. 3. 4. 4 Simulated results of the five SIRs shown in Fig. 3. 4. 1 & Fig. 3. 4. 3.
3.5 CPW Microwave Bandpass Filters Designed Using the Proposed Slow-
Wave SIR
3. 5. 1 Circuit Prototypes of the Fourth-Order Bandpass Filter
In this section, the slow-wave resonators will be implemented in two microwave
bandpass filters. One is a second-order direct-coupled filter, the other is a fourth-
order quasi-elliptical filter with cross couplings. The general topology of the second-
order filter is as that given in Fig. 3. 3. 1 and the detail of this filter has been
thoroughly discussed [56], [57]. The working principle of the fourth-order filter is
more complicated and will be briefly discussed here. As shown in Fig. 3. 5. 1, there
is a capacitive cross coupling between resonator 1 and resonator 4. This additional
coupling introduces another propagating path (1-4) besides the conventional path (1-
56
2-3-4). Thus, the signals transmit through these two paths will be canceled at the
designed frequency, resulting in the transmission zeros. The total phase changes of
this filter are summarized in Table 3. 5. 1. Below resonance, the phase change of
path 1-4 is -90˚ and the phase change of path 1-2-3-4 is +90˚. They are out of phase
and a transmission zero will appear below resonance. Due to the same reason,
another transmission zero will present above resonance. Hence, there are two
transmission zeros for this kind of filter, one above center frequency and the other
below center frequency. The roll-off of this filter will be sharp on both sides of the
passband. In general, this fourth-order filter is called cascaded quadruplet (CQ) filter.
Fig. 3. 5. 1 Coupling scheme of the fourth-order CQ bandpass filter with capacitive cross
coupling.
Table 3. 5. 1: Total phase shifts for the two paths in a fourth-order bandpass filter with
capacitive cross-coupling
Below Resonance Above Resonance
Path 1-2-3-4 -90˚ + 90˚ - 90˚ + 90˚ - 90˚ = -90˚ -90˚ - 90˚ - 90˚ - 90˚- 90˚= -450˚
Path 1-4 +90˚ +90˚
Results Out of phase Out of phase
Resonator 1
Resonator 2 +/- 90°
Resonator 4
- 90° - 90°
Resonator 3 +/- 90°
+ 90°
- 90°
57
3. 5. 2 Experimental Results
Both of these two filters [56], [57] are constructed on the FR4 PCB board with
dielectric constant εr = 4.5, loss tangent tanδ = 0.019 and substrate thickness h =
1.6mm. The Agilent 8720ES network analyzer is used to measure the performance
of the filters.
The two-pole filter is constructed using type C SIR as shown in Fig. 3. 4. 3 (a).
Narrow meander lines connected to the ground lines are used at the input and output
ports to realize the external coupling. The EM simulator IE3D is used to optimize
the design. The final layout of the filter is given in Fig. 3. 5. 2, showing an overall
size of 12.8 mm×17.8mm. The gap (g) between the two resonators is 2.2mm.
The simulated and the measured responses of the filter are shown in Fig. 3. 5. 3.
The measured results show an insertion loss of approximately 3.3dB and a return
loss of better than 12dB in the passband. The center frequency (f0) is about 2.29GHz.
Comparing with the simulated response, the center frequency shifts down 140MHz.
This shift is caused by the limitation in the control of line width and spacing during
PCB fabrication (in our fabrication, the minimum feature size is 0.2mm and the
fabrication tolerances are +/- 0.05mm).
Fig. 3. 5. 2 Layout of the second-order microwave bandpass filter based on Type C slow-
wave SIR.
Inductive arms for external coupling
17.8 mm
g
12.8 mm
58
Fig. 3. 5. 3 Simulated and measured responses of the second-order bandpass filter.
The fourth-order CQ filter is designed and measured based on the type D slow-
wave SIR and the layout is shown in Fig. 3. 5. 4. As mentioned before, type D SIR
has only one shunt stub and is less effective in reducing the size comparing with
type C slow-wave SIR. But for the design of quasi-elliptic filter, this kind of SIR is
more convenient because we can easily adjust the spacing between adjacent
resonators to obtain desired direct and cross coupling coefficients.
In our design, the filter has a fractional bandwidth (FBW) of 10% at 2.41GHz.
The attenuation poles are at Ωa = ±1.5, where Ωa is the frequency variable
normalized to the passband cutoff frequency. The coupling coefficients used here
are listed below:
K12 = K34 = 0.08044,
K23 = 0.08132,
K14 = -0.03214,
Qe = 10
59
where Qe is the external quality factor for the input and output ports and the K’s are
the coupling coefficients between various resonators. K23 and K14 are of opposite
signs so that two attenuation poles can be generated. The synthesis method presented
in last chapter is used to get these coupling coefficients. The EM simulation tool
IE3D [46] is used to satisfy the given parameters. A practical design procedure can
be summarized as follows: Firstly, the gap between resonators 1 and 4 as shown in
Fig. 3. 5. 4 is adjusted to get the desired coupling coefficient K14. The similar
procedure is applied to resonators 2 and 3 to satisfy K23. After these two steps, the
relative position of resonators 1 and 2(also the relative position of resonators 3 and 4)
at the longitudinal direction is fixed. Then the width of the ground plane between
resonators 1 and 2 is tuned to meet the desired coupling coefficient K12. Finally, the
lengths of the inductive arms at the input and output ports are adjusted to satisfy the
specified external quality factor. The overall size of the finalized designed filter is
20.1mm×20.4mm.
The simulated and measured responses of the filter are shown in Fig. 3. 5. 5. The
two attenuation poles, one at the upper band and one at the lower band, can be
observed from the results. These two poles are due to the cross coupling between the
resonator 1 and resonator 4. The insertion loss is 5.8dB and the return loss is 24dB
in the passband. The relatively large insertion loss is due to the conduction loss and
the possibly excited radiation loss with the asymmetrical structure of the resonators
used. There is a frequency shift of 80MHz between the measurement and simulation
results. This difference is due to the variation in the fabrication of the filter.
60
Fig. 3. 5. 4 Layout of the fourth-order quasi-elliptic filter.
Fig. 3. 5. 5 Measured and simulated results of the designed CQ filter.
3.6 Summary
In this chapter, we have proposed a novel tri-section stepped-impedance
resonator structure that enables the shrinkage of the overall electrical length. Most
importantly, the tri-section SIR structure also enables the introduction of cross-
20.1 mm
20.4 mm
Resonator 2 Resonator 1 Resonator 4 Resonator 3
61
coupling for any circuits that feature multiple SIRs, such as cascaded triplet
bandpass filters. To prove this prediction, we successfully designed a microstrip CT
bandpass filter. The measurement results match with the theoretical predictions.
Meanwhile, we developed another more compact slow-wave tri-section SIR on
the CPW lines, which provides additional flexibility in designing miniaturized CPW
bandpass filters. Compared to the conventional two-section stepped-impedance
resonators, the inserted midsection of the slow-wave SIR is embedded into the
ground lines to introduce significant slow-wave effect. And this effect, in turn, leads
to great size reduction of the resonators. Using the slow-wave tri-section SIRs, a
second-order end coupled filter and a fourth-order quasi-elliptic bandpass filter are
demonstrated with compact size. The measured results agree with the theoretical
results. The design concepts can also be implemented on semiconductor substrates
for monolithic microwave integrated circuit applications.
62
CHAPTER 4
Design of Microwave Bandpass Filters with
Reconfigurable Transmission Zeros and Tunable Center
Frequencies
4.1 Introduction
Advances in modern wireless communication applications impose new
requirements for multi-frequency bands and multi-functions. This trend has led to
the development of various types of reconfigurable or tunable filters. According to
the different tuning properties, these reconfigurable filters can be summarized as
filters with variable center frequency, bandwidth and skirt selectivity. Up to now, the
tuning of the center frequency has been frequently mentioned and many methods
have been proposed [58] - [62]. Such a tuning capability enables one filter working
at several different frequency bands, and hence, saves the overall system size and
cost. Recently, filters with tunable bandwidth or constant bandwidth under variable
center frequencies are another topic of interest [63] - [65], since the flexibility in
both center frequency and bandwidth is required in practical applications. Compared
with filters with tunable passband, the ones with reconfigurable transmission zero
(or skirt selectivity) are seldom mentioned in the literature. Due to the sophisticated
network prototypes for the filters with highly selective skirt shapes, it is challenging
to design filters with tunable skirt selectivity. On the other hand, there is strong
demand for filters with reconfigurable transmission zeros. The flexibility in
achieving high selectivity in upper band, lower band, or both is desirable for the
communication systems where multiple frequency bands exist. For example, filters
63
with reconfigurable transmission zeros are favored in applications such as
transmit/receive diplexers, which requires high inter-band isolation between
the transmitting and receiving bands. Recently, a new network prototype suitable for
this kind of transmission zero reconfiguration was proposed and demonstrated with a
combline structure and varactor diodes [66], using a filter with fixed center
frequency.
It is beneficial if we can develop other filter structures, in which simpler tuning
can achieve the same goal. In this chapter, we will show that it is possible to achieve
the reconfigurations of the transmission zeros using other topologies. Two different
types of filters have been designed and tested. In type I design, the transmission
zeros which enable sharp out-of-band roll-off are introduced by tapped quarter-
wavelength stubs [67], [68]. The transmission zeros become tunable when varactors
are loaded at the open ends of the stubs. Bandpass filters with one transmission zero
configured to appear at either the upper stop band or the lower stop band will be
demonstrated. We will also demonstrate filters with two reconfigurable transmission
zeros, one locating at the upper band and one locating at the lower band. Meanwhile,
it is found that this type of filter can be used to realize the reconfigurations of both
the transmission zeros and the passband. The tuning of the passband was achieved
by varactors in series with a half-wavelength resonator. The type II design is a
second-order filter which has a zero-shifting effect. In this kind of filters,
transmission zero is determined by the resonant frequencies of the resonators and is
independent of the coupling coefficients. Using two designs with different
resonators but the same coupling scheme, zero-shifting behavior has been
demonstrated in pseudo-elliptic microstrip line filters [69], [70]. We implement this
zero-shifting concept in a single filter with transmission zero electronically
64
reconfigured by varactors that are parts of the resonators. Two filters are devised in
this way. All the designed filters were fabricated on the PCB board. The design
concepts were demonstrated by measured results [71], [72].
4.2 Bandpass Filters with Reconfigurable Transmission Zeros and Tunable
Center Frequencies
4.2.1 Bandpass Filters with Reconfigurable Transmission Zeros
As mentioned above, the main task for this chapter is to design bandpass filters
with reconfigurable transmission zeros. There are several methods to realize the
transmission zeros. One approach is to utilize multi-path propagation as explained in
last chapter. Another method is to use the tapped stubs which will introduce the
desired transmission zero without affecting the performance of the resonator. This
kind of structure will be employed in this section.
The topology of the proposed filter with one transmission zero is shown in Fig. 4.
2. 1. A λ/2 resonator is tapped with a λ/4 shunt stub at the center. The shunt stub is
loaded with a varactor. This filter structure has several important features. First, it
produces a RF short at the center of the λ/2 resonator. Thus, two λ/4 resonators are
formed. The order of the bandpass filter changes from 1 to 2. Second, tapped stub
also works as a K-inverter between two λ/4 resonators [68]. An equivalent circuit is
given in Fig. 4. 2. 2. The coupling between the two λ/4 resonators is dependent on
the electrical length and the impedance of the shunt stubs. Third, without
considering the effect of the loaded varactor, the open-end λ/4 shunt stub is
equivalent to a series LC resonator and produces a transmission zero. The varactor’s
variable capacitance, in series with the LC tank, can effectively change the
65
frequency at which the transmission zero occurs. The equivalent circuit model of the
stub plus the loaded varactor is given in Fig. 4. 2. 3.
Fig. 4.2.1 Topology of the bandpass filter with one reconfigurable transmission zero.
Fig. 4. 2. 2 The equivalent circuit of a λ/2 resonator with a tapped stub serving as a K-
inverter.
Fig. 4. 2. 3 The equivalent circuit for the stub working as a resonator with the varactor
tuning the resonant frequency.
K- inverter
Z , λ/4
Port 1 50Ω
Lstub
J-inverter
J-inverter
Port 2 50Ω
Z , λ/4
Cstub
Cvaractor
Port 1 Port 2 Vbias
L1
L2 W2
W1 S1
L3
66
Fig. 4. 2. 4 Circuit model of a varactor loaded transmission line.
From the analysis presented above, it is clear that the transmission zero is
dependent on the shunt stub and the varactor. The design equation can be found by
examining the impedance looking into the stub loaded with the varactor, as shown in
Fig. 4. 2. 4. Using the well known impedance transforming equation, Zin can be
given by:
0var00
0var000
tan)tan1(
ZCjCZZ
Z in ωθθω
+−
= (4.1)
where ω0 = 2πf0 (f0 is the frequency at which the transmission zero occurs), θ0 is the
electrical length of the stub at f0. At resonance, Zin equals to zero and a virtual RF
ground appears at the tapping point. Signals propagating in the resonator are shorted
to ground via this path, resulting in the transmission zero of the filter. With Zin = 0,
from equation (4.1), we get the relation between the frequency of transmission zero
and the circuit elements’ parameters:
0var0
0 tan21
θπ CZf = (4.2)
This equation gives a theoretical guideline of choosing the filters’ physical
dimensions. In order to achieve large tuning range of the transmission zero, the
characteristic impedance of the stub (Z0) should be relatively small. In the practical
Zin
Z0 , θ0 Cvar
67
designs, an impedance of 25 Ω was chosen. The filters are designed and fabricated
on a low-cost FR4 board with a substrate thickness of 1.6 mm and a dielectric
constant of 4.5. The center frequency is 900 MHz. The varactor1 used has a typical
capacitance tuning range from 0.8 pF to 9.3 pF. To show the tuning range of the
transmission zero, the frequency of the transmission zero under different stub length
and different capacitance are computed using equation (4.2) and plotted in Fig. 4. 2.
5. It is observed that, with the change of the varactor’s capacitance, the transmission
zero can be reconfigured from about 1.1 GHz to 0.6 GHz.
0 2 4 6 8 10
0.6
0.8
1.0
1.2
Freq
uenc
y (G
Hz)
Capacitance (pF)
Stub length=37mm Stub length=36mm Stub length=35mm Stub length=34mm
Fig. 4. 2. 5 Theoretical resonant frequency tuning range for the varactor-loaded transmission
line.
1 Silicon tuning diodes, BB833, Infineon Technology, 2004
68
There is another challenging issue in designing filters that include lumped
components (i.e. varactors). Filters solely using transmission lines can be designed
and optimized with EM simulators. However, the varactors can not be incorporated
in EM simulators and a hybrid design procedure needs to be developed. To solve
this problem, we developed a two-step simulation procedure. First, a full wave EM
simulation based on MoM method (using IE3D from Zeland, Inc.) is carried out on
the three-port network including the tapped stub, without considering the effect of
the varactors. The simulation provides S-parameters of the three-port network. Then
the S-parameter results are imported into Agilent’s Advanced Design System (ADS).
The capacitor is then connected to the three-port network in ADS and simulation of
the overall filter can be performed.
0.8 1.0-50
-40
-30
-20
-10
0
S21
& S1
1(dB
)
Frequency (GHz)
S11&S21 for C=1.5pF S11&S21 for C=2pF S11&S21 for C=4pF
Fig. 4. 2. 6 The simulation results of the filter with one reconfigurable transmission zero.
69
Using the hybrid design procedure, the filter is designed on FR4 board. The
structure is the same as that shown in Fig. 4. 2. 1. The dimensions are: L1 = 92 mm,
L2 = 35 mm, L3 = 43mm, W1 = 3 mm, W2 = 8.2 mm, S1 = 0.4 mm. The simulation
results with three different values for the varactor’s capacitance are given in Fig. 4. 2.
6. As has been predicted, the positions of the transmission zeros can be shifted by
the value of the varactor’s capacitance. With the capacitance changing from 1.5 pF
to 2 pF and then 4pF, the transmission zero change from 1.0 GHz to 0.95 GHz and
0.82 GHz. The transmission zero has changed from the upper side to the lower side
of the filter’s passband. It is also found that the center frequency and the bandwidth
have experienced certain variation as the transmission zero is tuned. This is due to
the slight change in the coupling coefficient between the two λ/4 resonators.
To prove the designs obtained from the simulation, the filter was fabricated and
measured as shown in Fig. 4. 2. 7. The DC bias point is directly connected to the
center of the λ/2 stub without any RF chokes. This is made possible since, at
resonance, the center point is an ideal RF short and no RF signal can leak into the
DC source and degrade the filter’s performance. A shorting pin connects the
varactor to the ground. The experimental results under two different biases are given
in Fig. 4. 2. 8 (a) and (b). The measured insertion loss of the filter at center
frequency is about 3.1 dB. The return loss in the passband is below 10 dB. By
applying a bias of 10 V or 5 V to the loaded varactor, the transmission zero can be
tuned to 0.99 GHz or 0.83 GHz, respectively. Furthermore, a direct comparison can
be done between the measured and the simulated results. In Fig. 4. 2. 8 (a), the
measured transmission zero is at 0.99 GHz when the DC bias across the varactor is
10 V and the corresponding varactor’s capacitance value is 1.4 pF. From Fig. 4. 2. 6,
the simulated transmission zero is at 1.0 GHz when the capacitance is 1.5 pF. Thus,
70
the measurement agrees with simulation very well. In Fig. 4. 2. 8 (b), the wide band
performances of the filters are given. Several features are worth of notice. First, in
addition to the passband at 900 MHz, another transmission peak appears at near 1.8
GHz and its position shifts as a function of the bias voltage. A closer examination
reveals that this transmission peak occurs at a frequency that doubles the frequency
of transmission zero, and can be attributed to the path that includes the tapped stub
together with the shunting varactor. At the fundamental passband, this path operates
as a bandstop filter. As the frequency is doubled, however, this path works as a half-
wavelength resonator and operates as a bandpass filter. Second, it is observed that
the transmission peaks at 0.9 GHz and 2.7 GHz do not vary with the varactor’s bias
voltage. These two peaks are produced by the unloaded λ/2 stub. However, the
second transmission peak has been changed from 1.8 GHz (as would occur for a λ/2
resonator) to 2.7 GHz. Such a property belongs to λ/4 resonators. Thus, it can be
concluded that the tapped λ/4 stub has transformed the half-wavelength resonator
into two quarter-wavelength resonators.
Fig. 4. 2. 7 The photo of the fabricated filter with one reconfigurable transmission zero.
DC Bias Point
Ground Varactor
Port 1 Port 2
71
0.8 1.0-40
-30
-20
-10
0
S21
& S1
1(dB
)
Frequency (GHz)
S11&S21 for Bias=10V S11&S21 for Bias=5V
(a)
0 1 2 3-50
-40
-30
-20
-10
0
S11
& S2
1(dB
)
Frequency (GHz)
S11&S21 for Bias=10V S11&S21 for Bias=5V
(b)
Fig. 4. 2. 8 The measurement results of the filter. (a) Measured results under different bias
around the passband. (b) Measured wide band characteristics.
To show the flexibility of the shunt stubs in filter designs, another bandpass filter
with two shunt arms and two transmission zeros is devised. This kind of filter is
72
suitable for the case where both the upper and the lower sides of the passband
require sharp roll-off. The two transmission zeros can also be placed on the same
side of the passband to further improve the skirt slope.
The basic structure of this filter is shown in Fig. 4. 2. 9 (a). The equivalent circuit
of the structure is given in Fig. 4. 2. 9 (b), where two parallel LC-tanks represent the
two shunt stubs. These LC-tanks together can be perceived as a cascade of one LC
resonator and two equivalent K-inverters, and the order of the filter increases from 2
to 3, as explained in [68]. To control the positions of the two zeros independently,
two separate DC bias circuits are needed. Thus, DC block capacitors are applied to
enable the isolation between the two biasing networks. Two short stubs (as shown in
Fig. 4. 2. 9 (a) with length L6) are added to provide interconnections for the
additional discrete components. For the purpose of minimizing the effects of the
additional DC block capacitors on the RF properties, relatively large capacitance
value of 6.8 nF has been used for Cblock. Finally, the same simulation procedure
presented before is applied here to optimize the design. The stub length used in the
EM simulation is set to L6+L5.
The final dimensions of the filter are: L4 = 92 mm, L5 = 30 mm, L6 = 5 mm, L7 =
37 mm, W3 = 3 mm, W4 = 8.2 mm, S2 = 0.3 mm. FR4 board is used and the photo of
the fabricated filter is shown in Fig. 4. 2. 10 (a). The two varactors are biased with
various bias combinations and the measured results are plotted in Fig. 4. 2. 10 (b).
When the two bias voltages are all 4.5 V, the transmission zeros appear at the lower
side of the passband with a sharp roll-off. With one varactor biased at 5.9 V and the
other biased at 25 V, the two transmission zeros are tuned to appear on both sides of
the passband. When the two varactors were biased at 9.5 V and 25 V, the zeros all
73
appear at the upper side of the passband and the skirt selectivity is improved
consequently.
(a)
(b)
Fig. 4. 2. 9 The bandpass filter with two reconfigurable transmission zeros. (a) The topology
of the filter. (b) The equivalent circuit of the filter.
Vbias 1
Vbias 2
Cblock
Cblock
Port 1
Port 2
Varactor
Varactor
L5
L4
W3
W4
L6
L7 S2
Z , λ/4
Port 1 50Ω Shunt
Stub 1
J-inverter
J-inverter
Port 2 50Ω
Z , λ/4
Shunt Stub 2
74
(a)
0.8 1.2 1.6 2.0-50
-40
-30
-20
-10
0
S11
& S
21(d
B)
Frequency (GHz)
S11&S21 for Bias1=4.5V & Bias2=4.5V S11&S21 for Bias1=5.9V & Bias2=25V S11&S21 for Bias1=9.5V & Bias2=25V
(b)
Fig. 4. 2. 10 Fabricated filter with two reconfigurable transmission zeros. (a) The photo of
the filter. (b) The measured results under different biases.
Ground
DC Block Capacitor
Varactor
Vbias 1
Vbias 2
Port 1 Port 2
75
(a)
(b)
Fig. 4. 2. 11 Filter with tunable center frequency and one zero. (a) The topology of
the filter. (b) The photo of the filter.
4.2.2 Bandpass Filters with Reconfigurable Transmission Zeros and Tunable
Center Frequencies
The ultimate reconfigurable bandpass filters require tunability in both the center
frequency (passband) and transmission zeros. As described in section 4. 2. 1, the
Port 1 Port 2 Vbias for shifting the center frequency
L8
L9
W6 W5
S3
Vbias for tuning zero’s position
Cblock
L10
L11
Ground
DC Bias Point
Varactor
Port 1 Port 2
76
passbands of the presented filters are determined by the λ/2 resonator. If we can
introduce tuning capability to alter the overall electrical length from port 1 to port 2,
the passband will be tuned. In this section, two such kinds of filters are designed,
one with one transmission zero and one with two transmission zeros. The topology
of the filter with one transmission zero is given in Fig. 4. 2. 11(a) and the
photograph of the fabricated filter is shown in Fig. 4. 2. 11(b). Two varactors are
added to the open ends of the half-wavelength stub in the filter to shift the center
frequency. The center of this stub is chosen as the biasing point for these two
varactors. This biasing circuit does not require RF chokes and saves the space. For
the biasing of the varactor controlling the transmission zero, a DC block capacitor
(6.8 nF) is used. Since the shunt arm does not resonate at the center frequency, no
RF choke is needed for this biasing circuit.
The same FR4 board was used in the fabrication of the filter. All the design
parameters are determined by the hybrid simulation procedure described in section II.
The final dimensions are: L8 = 68 mm, L9 = 26 mm, L10 = 3 mm, L11=27.5mm, W5 =
3 mm, W6 = 8.2 mm, S3 = 0.3 mm.
The measured results are given in Fig. 4. 2. 12. Four different bias combinations
are presented. The center frequency can shift from 890 MHz to 680 MHz and the
insertion loss changes from 2.6 dB to 5.8 dB. Under each center frequency, the
transmission zero can be set at either the lower side or the upper side of the passband
by tuning the varactor’s bias. In Fig. 4. 2. 12 (a), with the DC bias for tuning the
center frequency at 7.5 V and the bias for the transmission zero at 25 V, the
transmission zero appears at the upper side of the passband. For the other three cases,
the zeros appear at the lower side. The wide band performance of the filter is shown
in Fig. 4. 2. 12 (b). It is observed that the third order harmonic of the filter (near 2.7
77
GHz) is suppressed. This is due to that this frequency is near the self-resonant
frequency of the varactor used. So the signal at this frequency is shorted to ground
via the varactors loaded to the λ/2 stub and no wave is transmitted.
0.4 0.6 0.8 1.0 1.2 1.4-50
-40
-30
-20
-10
0
S21(
dB)
F requency (G H z)
S 21 fo r center_bias=7.5V , zero_b ias=25V S 21 fo r center_bias=9.5V , zero_b ias=4.5V S 21 fo r center_bias=4.5V , zero_b ias=2.2V S 21 fo r center_bias=5.5V , zero_b ias=3.5V
(a)
0 1 2 3 4-50
-40
-30
-20
-10
0
S11
& S2
1(dB
)
Frequency (GHz)
S11&S21 for center_bias=7.5V, zero_bias=25V S11&S21 for center_bias=9.5V, zero_bias=4.5V S11&S21 for center_bias=4.5V, zero_bias=2.2V S11&S21 for center_bias=5.5V, zero_bias=3.5V
(b)
Fig. 4. 2. 12 Measured results for the filter with tunable center frequency and one
transmission zero. (a) The measured results under different biases near the passband. (b)
The wide band measured results.
78
The topology of the filter with tunable center frequency and two reconfigurable
zeros is shown in Fig. 4. 2. 13 (a). The same biasing networks are used as that for
the filter with one zero. Compared with the structure shown in Fig. 4. 2. 11, the only
difference is that this structure has two tapped stubs, which can supply sharp roll-off
at both the lower and the upper side of the passband. The dimensions of the final
design are: L12 = 68 mm, L13 = 27.5 mm, L14 = 26 mm, L15 = 3 mm, W7 = 3 mm, W8
= 8.2 mm, S4 = 0.2 mm. The photo and the measured results of the fabricated filter
are given in Fig. 4. 2. 13 (b) and Fig. 4. 2. 14. With the bias voltage changing from
20 V to 4.5 V, the resonant frequency shifts from 1020 MHz to 680 MHz. The
insertion loss varies from 3.2 dB to 6.1 dB. The four varactors allow us to tune the
filter’s performance in the passband and skirt selectivity (transmission zeros) at both
band edges simultaneously. In Fig. 4. 2. 14, six different bias combinations are
presented illustrating the filter tuned to three different center frequencies. At each
center frequency, two configurations of transmission zeros are obtained by varying
the bias controlling the transmission zeros (see Fig. 4. 2. 13 (a)): one representing a
filter with two transmission zeros on both sides of the passband and the other
representing a filter with two zeros on single side of the passband. For example,
when the DC bias for tuning the center frequency was at 4.5 V and the bias voltages
for the two transmission zeros were at 1.4 V and 4.6 V, the designed filter worked at
680 MHz with the two transmission zeros on both sides of the passband. After
changing the bias voltage for the low stopband zero from 1.4 V to 4.6 V, both of the
two zeros were located at the upper stopband with the center frequency unchanged.
79
(a)
(b)
Fig. 4. 2. 13 Filter with tunable center frequency and two transmission zeros. (a) The
topology of the filter. (b) The photo of the filter.
Port 1 Port 2 Vbias for center frequency
L12
L14 W8 W7
S4
Vbias for controlling Zero 2
Cblock
L15
Vbias for controlling Zero 1
L13
Ground DC Bias
Point
Varactor
Port 1 Port 2
80
0.4 0.6 0.8 1.0 1.2 1.4-70
-60
-50
-40
-30
-20
-10
0
S21(
dB)
Frequency (GHz)
center_bias=4.5V, zero_bias1=4.6V, zero_bias2=1.4V center_bias=4.5V, zero_bias1=4.6V, zero_bias2=4.6V center_bias=6V, zero_bias1=1.5V, zero_bias2=2V center_bias=6V, zero_bias1=5V, zero_bias2=2.4V center_bias=20V, zero_bias1=4.5V, zero_bias2=4.5V center_bias=20V, zero_bias1=5.8V, zero_bias2=25V
Fig. 4. 2. 14 Measured results of the filter with tunable center frequency and two
transmission zeros.
4.3 Bandpass Filters with Reconfigurable Transmission Zero
As for the type II design, we employ the topology suggested in [69], but applying
varactors to realize the reconfiguration of the position of the transmission zero. A
two-pole bandpass filter is used to demonstrate our design concept. Fig. 4. 3. 1
shows the circuit prototype of the designed filters. A detailed theoretical analysis can
be found in [69]. The loop currents, which are grouped in a vector [I], can be given
by the matrix equation:
[ ][ ][ ] [ ][ ] [ ]EjIAIMjR −==+Ω+− ω (4.3)
where [R] is a (n+2)×(n+2) matrix having R1,1=Rn+2,n+2=1 with the other elements
equal to zero, [Ω] is similar to the (n+2)×(n+2) identity matrix, except that Ω1,1=
81
Ωn+2,n+2= 0, and [M] is the matrix illustrating the coupling coefficients. [E] is the
excitation, which is [1, 0, 0, …, 0]t. ω is the normalized frequency, which equals to
f0/∆f(f/f0 – f0/f). n is the order of the filter analyzed.
Fig. 4. 3. 1 The circuit prototype used for the proposed reconfigurable bandpass filters.
The transmission and reflection coefficients are given as follows:
[ ] 1,21
21 2 +−−= nAjS (4.4)
[ ] 1,11
11 21 −+= AjS (4.5)
In this way, the S21 and S11 can be computed. The synthesis of the coupling
coefficients of the structure can be done with the gradient-based method or the
genetic algorithms developed in Chapter 2.
Based on the analysis method presented above, it is noted that the most attractive
property of the circuit in Fig. 4. 3. 1 is the reconfiguration of the position of the zero.
It is done by changing the signs of the diagonal elements in the matrix [M]. Two sets
of coupling coefficients which have this kind of duality will be used to design the
filter. Equation (4.6) and equation (4.7) show the parameters utilized, with equation
(4.6) providing the coefficients for upper band zero (State 1) and equation (4.7)
providing the parameters for lower band zero (State 2). The diagonal elements in
these two matrices have been underlined. The theoretical results plotted using (4.4)
and (4.5) are illustrated in Fig. 4. 3. 2. As desired, the transmission zero’s position
Port 1 Port 2
Resonator 1
Resonator 2
82
has been interchanged, which means that we can realize the reconfiguration by
shifting the resonant frequency of the two resonators analytically. Such a change in
the resonant frequency can be realized by varactors that are part of the resonators.
−−
−
00743.16544.000743.14991.100743.16544.006450.16544.000743.16544.00
(4.6)
−−
−
00743.16544.000743.14991.100743.16544.006450.16544.000743.16544.00
(4.7)
-10 -5 0 5 10-60
-40
-20
0
-60
-40
-20
0
S11,
S21
(dB)
Normalized Frequency
S11&S21 of State 1 S11&S21 of State 2
Fig. 4. 3. 2 The theoretical performances of the bandpass filter under two different states.
To apply the varactors to this circuit, the physical dimensions of the original
patterns need to be adjusted. Equation (4.1) and (4.2) given in last section can be
83
used here. Besides, the transformation of the coupling coefficients to the real design
parameters is done using the equations listed below:
jiji MFBWK ,, ×= (4.8)
0)1( fMFBWf iii ×+≈ (4.9)
FBWMQ Sse /1,, = (4.10)
FBWMQ Lnle /,, = (4.11)
where MS,1 and Mn,L stand for the coupling coefficients from the input (source) and
output (load) ports to the resonators. fi is the resonant frequency of the i’th resonator.
Ki,j is the coupling coefficient between resonators ‘i’ and ‘j’.
The circuit prototypes, combined with varactors (determined by the design
equations (4.1) and (4.2)) and equations (4.8) – (4.11), form the proposed filters. The
design procedures are summarized as follows:
1) Determine the coupling coefficients to be used in combinations with the
center frequency and the fraction bandwidth of the bandpass filters.
2) Convert these coefficients to the real parameters using (4.8) – (4.11).
3) Obtain the appropriate resonant frequencies for different resonators at
different states.
4) Choose the final dimensions of the resonators after adjustment using
equations (4.1) and (4.2).
Following the procedures presented above, a microstrip filter (Topology I) is
fabricated on the Rogers RO3010 board with dielectric constant εr = 10.2, and
substrate thickness h = 1.27mm. The pattern of the designed filter is given in Fig. 4.
3. 3, where L1 = 5.4mm, L2 = 4.4mm, L3 = 9mm, L4 = 7mm, L5 = 1.8mm, W1 = W2 =
W3 = 1.2mm, S1 = 0.3mm, S2 = 0.2mm. The resonator on the upper half as shown in
84
Fig. 4. 3. 3 is the one with electrical length equating to λ at the center frequency, the
other resonator in the lower half is the λ/2 resonator. The center frequency of this
filter is 1.9 GHz and the fraction bandwidth is 3%. The high-Q GaAs varactors
MV31020 are used to tune the resonant frequencies of the resonators.
Fig. 4. 3. 3 The topology of the reconfigurable filter (topology I) constructed on the Rogers
RO3210 board.
Fig. 4. 3. 4 Photo of the tested filter (topology I).
Port 1 Port 2
L3 L2 L1
W1 W2
W3
S1
S2
L4
L5
Ground
VaractorDC Bias
Point
85
Fig. 4. 3. 4 shows the photograph of the tested filter. The biasing points have
been marked in the figure. For the λ/2 resonator, the bias point is at the center of the
line, and for the λ resonator, the bias point is at the position of quarter wavelength
from the end of line. The bias point is chosen at the position where an ideal RF short
occurs.
Measurement results performed by the Agilent 8720ES are displayed in Fig. 4. 3.
5. From the measurements, the center frequency of the filter is about 1.92 GHz and
the insertion loss in the passband is about 1.2dB. As predicted, after changing the
capacitance values of the loaded varactors properly, the transmission zero has been
reconfigured from 2.03 GHz (State 1) to 1.81 GHz (State 2). The biasing voltages
for state 1 are 15V and 10V for the half-wavelength resonator and the whole-
wavelength resonator, where 20V and 7V are used respectively for state 2.
1.6 2.4 3.2-30
-20
-10
0
-30
-20
-10
0
upper zerolower zero
S21
(dB)
Frequency (GHz)
State 1 State 2
Fig. 4. 3. 5 Measured results of the tested reconfigurable filter (topology I) built on the
Rogers RO3210 board, where state 1 represents the state with transmission zero located at
the upper band and state 2 with the zero at the lower band.
86
Fig. 4. 3. 6 The topology of the reconfigurable filter (topology II) designed on FR4 board.
0.5 1.0 1.5-40
-30
-20
-10
0
-40
-30
-20
-10
0
lower zeroupper zero
S21
(dB)
Frequency (GHz)
State 1 State 2
Fig. 4. 3. 7 Measured results of reconfigurable filter (topology II) on the FR4 PCB board.
To provide more design choices for the reconfigurable filter, another filter
(topology II) working at 0.9 GHz is designed. The layout of this filter is given in Fig.
4. 3. 6. It is constructed on the low-cost FR4 PCB board with dielectric constant εr =
Port 1 Port 2
Biasing Point
L3
L2
L1
W1 W2
W3
S1
S2
L4
Biasing Point
S3
87
4.5, and substrate thickness h = 1.6mm. The varactors used in this design are the
Infineon’s BB833. As labeled in the figure, the dimensions of this structure are L1 =
125.4mm, L2 = 16.8mm, L3 = 69.8mm, L4 = 4.6mm, W1 = W2 = W3 = 3mm, S1 = S2 =
0.3mm, S3 = 5.4mm.
The same coupling coefficients are used for this design, but with the fractional
bandwidth changed to 5%. Compared with the pattern shown in Fig. 4. 3. 3, the
topology for this filter has been adjusted to provide the stronger couplings between
the ports and the resonators.
Measurement results are given in Fig. 4. 3. 7. Two reconfigurable states have
been observed. For state 1, the bias voltages applied are 20V for both the λ/2
resonator and the λ resonator respectively, which have been changed to 25V and
6.9V for state 2. There is a lower band zero observed in state 1. This additional
transmission zero is produced by the coupling between the source and the load ports.
In state 2, this zero merges with the designed transmission zero.
4.4 Summary
In this chapter, we designed two types of microwave bandpass filters with
reconfigurable transmission zeros. The type I design can be reconfigured on both the
positions of the transmission zeros and the center frequencies. In details, tapped
stubs combined with varactors are used to reconfigure the positions of the
transmission zeros. And varactors connected to a λ/2 resonator are used to tune the
center frequency. To account for the effects of the lumped components in the
circuits (e.g. varactors), a design procedure based on the mixed mode simulations
including EM simulation and circuit simulation is developed. Using this design
method, several types of reconfigurable bandpass filters are designed and fabricated.
88
Measurement results demonstrate the tunabilities of the new filters, which are
suitable for the applications in the multi-functional and multi-band systems.
In type II design, a circuit prototype featuring zero-shifting properties has been
implemented to the designs of the reconfigurable bandpass filters. The desired
transmission zero is related to the couplings between the ports and the resonators
and the reconfiguration of the transmission zero can be carried out by changing the
resonant frequency of the resonators without changing the coupling coefficients. To
validate the theoretical predictions, two filters with different topologies have been
fabricated and measured. The experimental results verify the reconfigurations of the
positions of the zeros.
Compared with the topology proposed previously, the new reconfigurable
bandpass filters given in this chapter is more straightforward for practical
implementation and easier to design. Thus, they provide more choices for the
applications in the modern communication systems.
89
CHAPTER 5
Dual-Band Microwave Bandpass Filters, Couplers and
Power Dividers
5.1 Introduction
With the development in the wireless mobile communications, systems with
multi-band operations have become quite popular. This property is very attractive
since both the size and cost of the whole system can be reduced in this way. To
satisfy this kind of multi-band behavior, each constitutive element (e.g. antennas,
filters, couplers, amplifiers) of such a system needs to be redesigned. In this chapter,
we will address the issues about the designs of dual-band passive components
including filters, branch-line couplers, rat-race couplers and Wilkinson power
dividers.
For the designs of microwave passive components, the quarter-wavelength
transmission lines are basic building blocks. Hence, it is important to develop this
kind of structure with dual-band operations. For this purpose, we proposed a tapped
stub structure, which behaves as quarter-wavelength transmission line at two
frequencies. This structure is then applied to filters, couplers and power dividers for
dual-band operations.
For the design of dual-band filters, there are several different methods. In [73],
the Zolotarev function is used, which resulted in the so-called Zolotarev dual-band
filter. In [74], two different filters are set in parallel to operate in the two desired
frequencies. However, for this design, the size is usually very large. In [75] – [77],
the dual-band filters are designed based on the complicated frequency
transformations. In [78], the dual-mode resonator is used for the dual-band filter. In
90
[79], the stepped-impedance resonator (SIR) is used. The impedances of the two
sections of the SIR are tuned to resonate at the two assigned frequencies. In addition,
the coupling coefficients in this kind of filter need to be adjusted to work at the two
frequencies, which is not easy under some circumstance. In [80], Quendo et al.
propose to design the dual-band filters using the dual-behavior resonator constructed
by three open stubs. In [81], Tsai et al. show that only two open stubs in parallel are
enough to behave as a dual-band resonator, which can simplifies the designs in [80].
Also in this paper, several kinds of dual-band impedance inverters are proposed to
connect the dual-band resonators. However, the impedance inverters suggested have
two open stubs loaded in the two ends, which makes the connections between the
inverters and the resonators difficult.
In this chapter, we use the two section dual-behavior resonators as that in [81] to
devise the dual-band filter. The developed dual-band transmission line, which
behaves as the dual-band impedance inverter, is combined with the dual-band
resonator to construct a second-order dual-band filter. The measurement results
prove the dual-band operations. However, an unwanted resonant peak is observed
between the two working frequencies, which degrades the whole performance. To
suppress this resonance, bandstop filters are integrated with the original bandpass
filter and good suppression has been achieved.
The tapped line structure is also applied to other microwave components. As for
the design of dual-band branch-line coupler, only several designs were proposed in
the past [82] – [86]. Since the conventional branch-line coupler is constructed by
quarter-wavelength transmission lines with different characteristic impedances, the
proposed tapped stub structure can be easily implemented for the design of dual-
band coupler. To verify the theoretical prediction, a dual-band branch-line coupler
91
working at 0.9 GHz / 2 GHz is designed and fabricated [87]. The measurement
results prove the dual-band operations.
Compared with the design of dual-band branch-line coupler, fewer designs of rat-
race coupler with dual-band operations can be found in the literature [88]. In our
works, two types of dual-band rat-race couplers are proposed, type I and type II [89].
The type I design is similar with that of the branch-line coupler and the type II
design is totally different with them. The design equations are derived for both of
these two designs. Two experimental dual-band couplers are fabricated, one for type
I working at 2 GHz / 5 GHz and the other for type II working at 1 GHz / 3.5 GHz.
Measurement results prove the theoretical analysis.
Finally, this kind of dual-band transmission line is used for the design of dual-
band Wilkinson power divider [90], [91]. An experimental dual-band power divider
working at 1 GHz and 2.5 GHz is designed and fabricated. The desired dual-band
operations are proved by the measurement results.
The arrangement of this chapter is as follows. At first, the structure and design
equations of the new dual-band quarter-wavelength transmission line are presented.
Then this dual-band transmission line will be used for the design of dual-band
microwave bandpass filter. Finally, the designs of other microwave passive
components (branch-line couplers, rat-race couplers etc.) are discussed.
5.2 Dual-Band Quarter-Wavelength Transmission Line
The structure of the proposed dual-band quarter-wavelength transmission line is
shown in Fig. 5. 2. 1. A shunt stub is tapped to the center of a conventional line
forming a T-shaped, where Za, Zb, θa and θb represent the characteristic impedances
and the electrical lengths of series and shunt sections.
92
Fig. 5. 2. 1 The topology of the proposed dual-band quarter-wavelength transmission line.
The ABCD-matrix is applied to derive the design equations. By cascading the
matrix of the three different sections, the ABCD-matrix of the T-shaped pattern can
be written as:
=
TT
TT
DCBA
aa
a
aaa
b
ba
a
a
aaa
Zj
jZ
Zj
Zj
jZ
θθ
θθθ
θθ
θθ
cossin
sincos
1tan
01
cossin
sincos (5.1)
With each element of the ABCD-matrix given by:
b
baaaaaTT Z
ZDA
θθθθθ
tancossinsincos 22 −−== (5.1a)
b
baaaaaT Z
ZjjZB
θθθθ
tansincossin2
22
−= (5.1b)
b
ba
a
aaT Z
jZ
jC θθθθ tancoscossin2 2
+= (5.1c)
Since the proposed structure should be equivalent to a quarter-wavelength
transmission line, the ABCD matrix of the structure should be equal to that of the
conventional λ/4 line, yielding:
±
±=
01
0
c
c
TT
TT
Zj
jZ
DCBA
(5.2)
Za , θa Za , θa
Zb , θb
93
where Zc is the characteristic impedance of the conventional line. By setting
AT=DT=0, we obtain:
aaa
aabb Z
Zθθ
θθθ
cossin)sin(cos
tan22 −
= (5.3)
Substituting (5.3) into (5.1b) and (5.1c), we get:
aaT jZB θtan= (5.4a)
aa
T ZjC
θtan1
= (5.4b)
Thus, under the condition (5.3), the ABCD-matrix of the T-shaped line is:
±
±=
010
0tan1
tan0
c
c
aa
aa
Zj
jZ
Zj
jZ
θ
θ (5.5)
For the purpose of dual-band operation, the necessary conditions implied by (5.5)
are:
cafa ZZ ±=1tan θ (5.6a)
cafa ZZ ±=2tan θ (5.6b)
where θaf1 and θaf2 are electrical lengths of the lines at the two desired operating
frequencies (θaf1< θaf2). The solution of (5.6) is:
12 afaf n θπθ += (5.7)
where n = 1, 2, 3,…, and with the relation of
2
1
2
1
ff
f
f =θθ
(5.8)
It can be deduced that:
πθ naf =0 (5.9)
where f0 = f2-f1 and θaf0 = θaf2 - θaf1.
94
As a result, once the two operating frequencies are determined, the electrical
lengths of the series section at these two frequencies (θaf1 and θaf2) will be
determined. During the above analysis, (5.3) has been assumed to be valid. Since we
have θaf2 = θaf1 + nπ, combined with (5.3) yields:
21 tantan bfbf θθ = (5.10)
where θbf1 and θbf2 are electrical lengths of the shunt section (θb) at the two desired
operating frequencies (θbf1< θbf2). So the length of the line (θb) is that:
πθ mbf =0 (5.11)
where m = 1, 2, 3,…, and f0 = f2-f1.
Following the same procedures as for the series stubs, the electrical lengths for
the shunt section (θbf1 and θbf2) can be computed.
5.3 Dual-Band Filter Design
For the application in filter design, the proposed dual-band transmission line will
behave as the impedance inverter transforming the impedance level between
different resonators. The value of the impedance inverter is determined by the
effective characteristic impedance of the dual-band transmission line. There are
normally two kinds of impedance inverters, J-inverter and K-inverter. As a
conventional J-inverter, we have:
cZ
J 1= (5.12)
And as a conventional K-inverter, we have:
cZK = (5.13)
where Zc is defined in equations (5.5) and (5.6).
95
In our work, a so-called dual-behavior resonator is used as the dual-band
resonator. The general structure of the dual-behavior resonator used is given in Fig.
5. 3. 1. It is composed by two parallel open stubs.
Fig. 5. 3. 1 The topology of the dual-behavior resonator.
The total admittance of the resonator is
)tan()tan()(1
221
11 ffjY
ffjYfYt θθ += (5.14)
where f1 is the center frequency of the first passband.
The susceptance slope at the resonant frequency fr can be obtained as:
)](sec)(sec[
21
)Im(2
21
22
121
1
21
11 θθθθ
ff
ffY
ff
ffY
fYfb
rrrr
ff
tr
r
+=
∂∂
== (5.15)
Assume that f2 is the center frequency of the second passband, and r is the ratio of
f2 to f1. The needed susceptance slopes are b1 and b2 at f1 and f2, respectively. Hence,
the resonant conditions and susceptance slopes at the two resonant frequencies
should satisfy the following four equations:
Y1 , θ1
Y2 , θ2
Port 1 Port 2
96
0tantan 2211 =+ θθ YY (5.16)
122
2212
11 2secsec bYY =+ θθθθ (5.17)
0)tan()tan( 2211 =+ θθ rYrY (5.18)
222
2212
11 2)(sec)(sec brrYrrY =+ θθθθ (5.19)
The four parameters, Y1, Y2, θ1 and θ2 can be computed by solving equations
(5.16) – (5.19). And the susceptance slopes b1 and b2 in these equations are
determined from the bandwidths of the filter. Let ∆1 and ∆2 be the fractional
bandwidths at f1 and f2. From the classical filter synthesis method, we have:
1
101 ∆
=ggGb (5.20a)
2
102 ∆
=gg
Gb (5.20b)
for the resonator at the input, and
1
11 ∆
= +nn ggGb (5.21a)
2
12 ∆
= +nn ggGb (5.21b)
for the resonator at the output, where G is the termination conductance (usually is
1/50 S). For the conventional Butterworth and Chebyshev filters, equations (5.20)
and (5.21) are the same. The admittance inverters between resonators are determined
by:
1
10
1
22
21
21
11,+++
+ =∆=∆=iiiiii
ii ggggG
ggb
ggbJ (5.22)
It is found that equations (5.16) – (5.19) can be further reduced to:
)tan(tan)tan(tan 1221 θθθθ rr = (5.23)
97
]tansectansec[]tan)(sectan)(sec[ 122
2212
11122
2212
12 θθθθθθθθθθθθ −∆=−∆ rrr
(5.24)
The solving of equations (5.23) and (5.24) can be very complicated. Since the
focus of our work is to show the performance of the proposed dual-band impedance
inverter, we will consider a simple case in this chapter, where f1∆1= f2∆2. Under this
assumption, the solutions of θ1 and θ2 are:
11 +
=rπθ (5.25)
1
22 +
=r
πθ (5.26)
To prove the theoretical analysis, a second-order dual-band Chebyshev bandpass
filter is designed. It works at 2GHz / 5GHz, which means that r = 2.5. For a
passband ripple of 0.04321dB, the design parameters are g0 = 1, g1 = 0.6648, g2 =
0.5545 and g3 = 1.2210. And the fractional bandwidths are ∆1 = 5 % and ∆2 = 2 %.
Substituting these parameters into equations (5.20) – (5.26), we get:
=
=
=
=
=
022.0
2@74
012.0
2@72
0419.0
2,1
2
2
1
1
J
GHz
SY
GHz
SY
πθ
πθ
(5.27)
To realize J1,2 using the proposed impedance inverter, we need to solve equations
(5.6) – (5.11) with Zc = 1/ J1,2 = 1/0.022 = 45.455. The results are as follows:
98
=
Ω=
=
Ω=
GHz
Z
GHz
Z
J
J
J
J
2@76
8.38
2@72
25.36
2
2
1
1
πθ
πθ (5.28)
where ZJ1, ZJ2, θJ1, θJ2 are as labeled in Fig. 5. 3. 3.
Fig. 5. 3. 2 The equivalent circuit of the dual-band bandpass filter.
Fig. 5. 3. 3 The topology of the dual-band bandpass filter.
ZJ1 , θJ1 ZJ1 , θJ1
ZJ2 , θJ2
Z1 , θ1
Z2 , θ2
Z1 , θ1
Z2 , θ2
Port 1 Port 2
Port 1 50Ω
Resonator 1 J12
Port 2 50Ω
Resonator 2
L1 L2 C2 C1
99
2 3 4 5-70
-60
-50
-40
-30
-20
-10
0
S11_sim
S11_mea
S21_sim
S21_meaS_
para
met
ers
(dB)
Frequency (GHz)
Fig. 5. 3. 4 The simulation and measurement results of the fabricated dual-band bandpass
filter.
The equivalent circuit of the designed filter is given in Fig. 5. 3. 2. The final
pattern of the filter is given in Fig. 5. 3. 3, where Z1 = 1/Y1, Z2 = 1/Y2 and other
parameters are the same with that in (5.27) and (5.28). It was constructed on the
Rogers RO3006 board with dielectric constant = 6.15, substrate thickness = 1.27mm
and loss tangent = 0.0025. The simulation and the measurement results are shown in
Fig. 5. 3. 4. It is observed that the simulation results match with the measured
results. The measured center frequencies are 2.05 GHz and 5.01 GHz. The return
losses are below -11 dB at the two operating frequencies. The insertion losses are
about 0.9 dB at 2.05 GHz and 2.2 dB at 5.01 GHz. There is also another peak at
about 3.08 GHz, which is related to the impedance inverter.
To suppress this unwanted resonant peak, we use the L-shape lines as the
bandstop filters. The structure of this kind of bandstop filter is shown in Fig. 5. 3. 5.
100
Two L-shape lines are used in this filter. Both of them perform as the bandstop filter.
The combination of these two L-shape lines makes the resonant tank deep. As
shown in the figure, each L-shape line is constructed by two quarter-wavelength
lines. The total length is half-wavelength. It behaves as a shunt series LC tank. The
simulation results of this bandstop filter is shown in Fig. 5. 3. 6. The center
frequency of the stopband is about 3.08 GHz, which is the frequency of the
unwanted resonant peak.
The bandstop filter is then connected to the input and output ports of the dual-
band filter to suppress the harmonic. The pattern of the filter is shown in Fig. 5. 3. 7.
The simulation and measurements results are given in Fig. 5. 3. 8. The desired
harmonic suppression is observed in the measurement results. To further show the
suppression of the harmonic, the measured responses with or without the harmonic
suppressions are plotted in Fig. 5. 3. 9. According to this figure, up to 20 dB
spurious suppression is achieved.
Fig. 5. 3. 5 The structure of the L-shape bandstop filter used to suppress the spurious
harmonics.
Port 1 Port 2
λ/4
λ/4
101
1 2 3 4 5-40
-30
-20
-10
0
S-pa
ram
eter
(dB)
Frequency (GHz)
S11 S21
Fig. 5. 3. 6 The simulation results of the bandstop filters.
Fig. 5. 3. 7 The pattern of the dual-band filter with harmonic suppressions.
ZJ1 , θJ1 ZJ1 , θJ1
ZJ2 , θJ2
Z1 , θ1
Z2 , θ2
Z1 , θ1
Z2 , θ2
Port 1 Port 2
Band-stop filter
Band-stop filter
Original filter
102
2 3 4 5-60
-50
-40
-30
-20
-10
0
S21_mea
S21_sim
S11_mea
S11_simS_
para
met
ers(
dB)
Frequency(GHz)
Fig. 5. 3. 8 The simulation and measurement results of the fabricated dual-band bandpass
filter with harmonic suppression.
2 3 4 5-60
-50
-40
-30
-20
-10
0 unwanted harmonic
S_pa
ram
eter
s(dB
)
Frequency(GHz)
S21(without suppression) S21(with suppression)
Fig. 5. 3. 9 The measurement results of the dual-band bandpass filter with/without harmonic
suppression.
5.4 Applications to Other Dual-Band Passive Components
5.4.1 Branch-Line Coupler for Dual-Band Operations
The dual-band quarter-wavelength transmission line as shown in Fig. 5. 2. 1 can
be further applied to the passive components such as coupler and power combiner
103
for dual-band operations. The first example is the design of dual-band branch-line
coupler.
Fig. 5. 4. 1 The topology of the proposed stub tapped dual-band branch-line coupler.
The basic structure of the proposed coupler is shown in Fig. 5. 4. 1. The four
branches of the conventional coupler are replaced respectively by the proposed
tapped-line structures. Two of the branches are with the characteristic impedance of
50Ω (as shown in Fig. 5. 4. 1 the branch of Z3, Z4, L3 and L4) and the other two with
the impedance of 36.35Ω (the branch of Z1, Z2, L1 and L2). The design procedures of
this coupler can be summarized as follows:
1) Using (5.9) and (5.11) combined with (5.7), (5.8) (as given in Section 5.2) to get
the values of electrical lengths at the given two operating frequencies (f1 and f2).
The values of n and m always start with 1 for compactness.
2) Using (5.6) and the desired characteristic impedance (Zc) to compute the value
of Za.
3) Computing the value of Zb using (5.3) combined with the parameters obtained in
the former two steps.
4) Determining whether the values of Za and Zb can be realized in practice. If not
realizable, go back to step 1 and increase the values of n and m.
Port 1 Port 2
Port 3 Port 4
Z1 , 2L1
Z2 , L2
Z3 , 2L3
Z4 , L4
104
5) Computing the physical lengths of the two stubs under the different
characteristic impedances (Za and Zb).
1.25 1.50 1.75 2.00 2.25 2.50 2.750.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
Nor
mal
ized
Impe
danc
e
Frequency Ratio (f2 / f1)
Z1 / Z0 Z2 / Z0
(a)
1.25 1.50 1.75 2.00 2.25 2.50 2.750.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
Nor
mal
ized
Impe
danc
e
Frequency Ratio (f2 / f1)
Z3 / Z0 Z
4 / Z
0
(b)
Fig. 5. 4. 2 Computed normalized branch-line impedances (Z0 =50 Ω) used in the dual-band
branch-line coupler under different frequency ratios. (a) Line impedances for the 2/50 Ω
branch, (b) line impedances for the 50 Ω branch.
105
In practice, the construction of the dual-band branch-line coupler is constrained
by the range of realizable impedance, between 20 Ω and 120 Ω for PCB based
microstrip lines. It is found that, by changing the stub lengths (which corresponds to
changing the values of n and m in (5.7), (5.9) and (5.11)), the frequency ratios (f2/f1)
from 1.25 to 2.85 can be realized, where n is confined between 1 and 2 and m is
between 1 and 6. This range of the achievable frequency ratio is comparable to what
can be achieved in the dual-band coupler described in [84]. A simple numerical
searching program has been developed to find these values. The results are shown in
Fig. 5. 4. 2., where Z1, Z2 are used for the 2/50 Ω branch and Z3, Z4 are used for
the 50 Ω branch of the coupler, as shown in Fig. 5. 4. 1. All of these values have
been normalized to Z0 = 50 Ω.
Fig. 5. 4. 3. Photo of the fabricated dual-band branch-line coupler.
A microstrip coupler is devised in this way to validate the theoretical analysis.
The initial parameters of the coupler are given based on the equations derived in
section 5.2. The working frequencies of the coupler are selected as 0.9GHz and
Port 1 Port 2
Port 3 Port 4
106
2GHz. It is constructed on the Rogers’ board RO4003 with dielectric constant of
3.38, substrate thickness of 0.81mm and loss tangent of 0.0027. The resulting
theoretical parameters for the 2/50 Ω branch are Z1 = 24Ω, L1 = 30.3mm, Z2 =
75.5Ω, L2 = 65mm, the parameters for the 50Ω branch are Z3 = 33.9Ω, L3 = 30.9mm,
Z4 = 106.8Ω, L4 = 66.4mm, where the symbols are as labeled in Fig. 5. 4. 1. The
final pattern of the tested coupler is optimized using the full-wave EM simulator
IE3D and fabricated on the Rogers’ board. All the tapped stubs are pointing inward
to the center of the branch-line coupler to achieve minimum size. The photo of the
fabricated coupler is shown in Fig. 5. 4. 3.
Measurement results are given in Fig. 5. 4. 4. It is found that the two measured
center frequencies are 0.92 GHz and 2.03GHz. The return loss and the isolations are
below -24dB at 0.92GHz and below -19dB at 2.03GHz. The magnitudes of the
insertion loss are given in Fig. 5. 4. 4 (b), where S21 = -3.17dB, S31 = -3.50dB at 0.92
GHz and S21 = -3.76dB, S31 = -3.83dB at 2.03 GHz. Fig. 5. 4. 4 (c) gives the phase
response of the proposed coupler. The phase differences between port2 and port3
( 2131 SS ∠−∠ ) are -90.57˚ at 0.92GHz and 90.92˚ at 2.03GHz. Finally, the bandwidths
of the designed coupler are examined under the conditions of equal amplitude and
quadrature phase difference. With the mismatches in amplitude and quadrature
phase below 0.5dB and 5˚, the bandwidths of the coupler are 80MHz at both of the
two operating bands.
107
0 1 2 3 4-40
-30
-20
-10
0
f2f1
S11
& S
41 (d
B)
F requency (GHz)
s41 s11
(a)
0 1 2 3 4-40
-30
-20
-10
0
f2f1
S21
& S3
1 (d
B)
Frequency (GHz)
s31 s21
(b)
0 1 2 3 4-200
-100
0
100
200
f1
f2
S31
S21
Phas
e of
S21
& S
31 (d
egre
e)
Frequency (GHz)
(c)
Fig. 5. 4. 4. Measurement results of the fabricated dual-band branch-line coupler (a) the
return loss (S11) and the isolation(S41), (b) the insertion loss, (c) the phase responses at the
two designed ports.
108
5.4.2 Rat-Race Couplers for Dual-Band Operations
We have also designed two types of dual-band rat-race couplers, type I and type
II. The type I rat-race coupler is similar with the dual-band branch-line coupler
presented in the last section. Four branches of the conventional rat-race coupler are
replaced by the tapped-line structure. The desired dual-band performances are
realized by setting these stubs to have +90˚ / -90˚ phase shift at the two different
design frequencies.
The general topology of the type I dual-band rat-race coupler is given in Fig. 5. 4.
5 (a). In this design, the tapped open stub is used to construct an equivalent quarter-
wavelength line at the two working frequencies as shown in Fig. 5. 4. 5 (b). The
theoretical formulae for this structure are as follows:
x
cx
ZZ
θtan±= (5.29)
x
yxxy
ZZ
θθθ
2tan1tantan
−= (5.30)
where Zx, Zy, θx, θy are the impedances and electrical lengths of the branches and
stubs, as shown in Fig. 5. 4. 5 (b). Zc is the equivalent characteristic impedance of
the stub-tapped quarter-wavelength line.
For the purpose of dual-band operation at f1 and f2, the necessary conditions
for the electrical lengths of the stubs are:
πθ mxf =0 (5.31)
πθ nyf =0 (5.32)
where m = 1, 2, 3,…, f0 = f2 ± f1, θxf0 is the electrical length of the stub(Zx) at f0 and
n = 1, 2, 3,…, f0 = f2 ± f1, θyf0 is the electrical length of the stub(Zy) at f0.
109
(a)
(b)
Fig. 5. 4. 5. General topology of the proposed dual-band rat-race coupler. (a) The whole
pattern, (b) the proposed unit cell acting as a quarter-wavelength line at two working
frequencies.
For the application to the rat-race coupler, we have Zc = 1.414Z0 = 70.7 Ω. Both
+90˚ and -90˚ phase shifts can be realized by this proposed structure with the sign
selections of ‘+’ / ‘-’ in equation (5.29). Hence, by setting the top three branches
(connected to port 1 and 4 as shown in Fig. 5. 4. 5 (a)) with the same stub-tapped
line and the bottom branch with an additional 180˚ phase change compared with the
other three branches, a rat-race coupler can be formed.
To reveal the relations of the four branches in the proposed rat-race coupler, we
define the top three branches as branch I (as shown in Fig. 5. 4. 5 (a) with
parameters of Z1, Z2, θ1, θ2) and the fourth branch as branch II (as shown in Fig. 5. 4.
5 (a) with parameters of Z3, Z4, θ3, θ4). To achieve dual-band rat-race coupler
operations, there are four phase combinations of branch I and branch II. Due to the
Zx , θx Zx , θx
Zy , θy
Identical Branches
Port 1 (Σ port)
Port 4
Port 2 Port 3 (∆ port)
Z1 , θ1 Z1 , θ1
Z2 , θ2
Z3 , θ3 Z3 , θ3
Z4 , θ4
110
symmetry of the whole pattern, these four combinations can be summarized as two
cases, for the other two cases, we can find the values by simply exchanging the
designs of branch I and branch II described in the following cases:
+−
=
−+
=
2
1
2
1
@90@90
@90@90
)(
ff
IIbranchofshiftPhase
ff
IbranchofshiftPhase
icase
o
o
o
o
(5.33)
++
=
−−
=
2
1
2
1
@90@90
@90@90
)(
ff
IIbranchofshiftPhase
ff
IbranchofshiftPhase
iicase
o
o
o
o
(5.34)
Finally, the design procedures of the proposed rat-race coupler are as follows:
1) Using equations (5.33) and (5.34) to determine the phase shifts to be used in the
relative branches.
2) Using equations (5.29)-(5.32) and the desired characteristic impedance (Zc =
70.7 Ω) to compute parameters of the four branches (in our case the parameters
are Z1, Z2, Z3, Z4, θ1, θ2, θ3, θ4 as shown in Fig. 5. 4. 5 (a)).
3) Computing the physical dimensions based on the parameter values obtained in
step 2).
As mentioned before, in practice, the frequency ratios realizable in the type I rat-
race coupler are constrained by the values of the available transmission-lines’
characteristic impedances. A simple numerical searching procedure is applied here
to find the appropriate parameters. In the analysis, we limit the range of the
frequency ratios (f2/f1) to be between 1.7 and 2.8. Within this range, only the
frequency ratios of 2.23, 2.41, 2.45, 2.59, 2.6 and 2.61 can not be realized when the
branch lines’ impedance is limited between 20Ω and 120Ω. For other frequencies
111
within the specified range of frequency ratio, the computed parameters are given in
Fig. 5. 4. 6, where we have combined the results obtained for case (i) and case (ii).
As for the frequencies beyond this range (e. g. f2/f1 < 1.7), it is found that most of
the frequency ratios can be realized using this structure. The complicated relation
between the branch impedance and the frequency ratio shown in Fig. 5. 4. 6 is a
reflection of the constrain we added to the impedance values and stub lengths.
1.7 1.9 2.1 2.3 2.5 2.7
0.6
1.2
1.8
2.4
Z2 / Z0
Z1 / Z0
Nor
mal
ized
impe
danc
e
Frequency ratio (f2 / f1)
(a)
1.7 1.9 2.1 2.3 2.5 2.7
0.6
1.2
1.8
2.4
Nor
mal
ized
impe
danc
e
Frequency ratio (f2 / f1)
Z3 / Z0 Z4 / Z0
(b)
Fig. 5. 4. 6. Normalized line impedances used in the type I rat-race coupler under different
frequency ratios. (a) Line impedances for branch I, (b) line impedances for branch II.
112
To validate the analysis of the type I rat-race coupler, an experimental prototype
was designed and fabricated on the Rogers’ RO3006 PCB with a dielectric constant
of 6.15, the board thickness of 1.27mm and the loss tangent of 0.0025.
To design the prototype of the dual-band rat-race coupler, the two working
frequencies are selected to be 2 GHz and 5 GHz, yielding a frequency ratio of 2.5.
Following the design procedures presented above, the theoretical characteristic
impedances and electrical lengths of the four different branches are:
=Ω=
=Ω=
=Ω=
=Ω=
GHzZ
GHzZ
GHzZ
GHzZ
2@32,23.61
2@34,82.40
2@34,23.61
2@32,82.40
44
33
22
11
πθ
πθ
πθ
πθ
(5.35)
where Z1, Z2, Z3, Z4, θ1, θ2, θ3, θ4 are as labeled in Fig. 5. 4. 5 (a).
Fig. 5. 4. 7. Photo of the fabricated type I rat-race coupler.
By converting these parameters into physical dimensions, the experimental dual-
band rat-race coupler is constructed as shown in Fig. 5. 4. 7. The measurement
results of the coupler are plotted in Fig. 5. 4. 8 - Fig. 5. 4. 10. The measured working
Port 1 (Σ port)
Port 4
Port 3 (∆ port) Port 2
113
frequencies for this coupler are at 2.1 GHz and 5.11 GHz. This kind of shift in the
center frequency is due to the process variations in the circuit fabrications. The
return loss (S11) is below -16 dB and the isolation (S31) is better than -15 dB as
shown in Fig. 5. 4. 8. Fig. 5. 4. 9 gives the phase and magnitude performances of the
in-phase outputs (S21, S41). The magnitude responses are S21 = -3.23 dB, S41 = -3.66
dB at 2.1GHz, S21 = -3.95 dB, S41 = -4.27 dB at 5.11 GHz, and the phase responses
are °=∠−∠ 0.54121 SS at 2.1 GHz, °=∠−∠ 3.44121 SS at 5.11 GHz. Fig. 5. 4. 10 gives the
anti-phase outputs results (S23, S43). The magnitudes results are S23 = -3.58 dB, S43 =
-3.12 dB at 2.1 GHz, S23 = -4.74 dB, S43 = -3.99 dB at 5.11 GHz, and the phase
responses are °=∠−∠ 76.1754323 SS at 2.1 GHz, °=∠−∠ 51.1834323 SS at 5.11 GHz. The
operating bandwidth of the coupler is about 30MHz at the lower band and 50MHz at
the upper band. The relatively large insertion loss of S21, S41, S23, S43 at 5.11 GHz is
due to that the loss introduced by the PCB board used will be larger with a higher
working frequency.
2 3 4 5 6-50
-40
-30
-20
-10
0
S11,
S31
(dB)
Frequency (GHz)
S11 S31
Fig. 5. 4. 8. Measured return loss and isolation of the type I dual-band rat-race coupler.
114
2 3 4 5 6-50
-40
-30
-20
-10
0
S21,
S41
(dB)
Frequency (GHz)
S21 S41
(a)
2 3 4 5 6-200
-100
0
100
200
Phas
e of
S21
, S41
(deg
ree)
Frequency (GHz)
S21 S41
(b)
Fig. 5. 4. 9. Measured insertion losses and phase responses of the in-phase outputs (S21 and
S41) of the type I rat-race coupler. (a) Insertion loss, (b) phase responses.
115
2 3 4 5 6-40
-30
-20
-10
0
S23,
S43
(dB)
Frequency (GHz)
S23 S43
(a)
2 3 4 5 6-200
-100
0
100
200
S43
S23
Phas
e of
S23
, S43
(deg
ree)
Frequency (GHz)
(b)
Fig. 5. 4. 10. Measured insertion losses and phase responses of the anti-phase outputs (S23
and S43) of the type I rat-race coupler. (a) Insertion loss, (b) phase responses.
116
Fig. 5. 4. 11. General topology of the type II dual-band rat-race coupler.
The design of type II dual-band rat-race coupler is totally different with that of
the type I coupler. It needs only one additional tapped stub for the dual-band
operations. The even-odd mode analysis is applied to derive the design equations.
The topology of the type II coupler is given in Fig. 5. 4. 11. The scattering matrix
of this coupler is:
S =
00
00
434241
433231
423221
413121
SSSSSSSSSSSS
(5.36)
For the coupler to perform as an ideal rat-race coupler, the following conditions
need to be satisfied:
22
43413221 ==== SSSS (5.37)
04231 == SS (5.38)
°=∠−∠ 04143 SS (5.39)
°=∠−∠ 04121 SS (5.40)
°=∠−∠ 1802132 SS (5.41)
Port 1 (Σ port)
Port 2 Port 3 (∆ port)
Port 4
φ,BZ
φ,BZφ,BZ
22 ,θZ11,θZ
117
(a)
(b)
Fig. 5. 4. 12. (a)Even- and (b) odd- mode topologies of the proposed type II dual-band rat-
race coupler.
The even-odd mode analysis is then applied to study the properties of the
proposed structure. Fig. 5. 4. 12 gives the equivalent topologies under even- and
odd- mode excitations. The ABCD-matrices of these circuits are used to find their
transmission (Τe, Τo) and reflection coefficients (Γe, Γo). The resulting equations have
been listed in the following:
( )oeS Γ+Γ=21
11 (5.42)
( )oeS Τ+Τ=21
41 (5.43)
( )oeS Γ−Γ=21
21 (5.44)
( )oeS Τ−Τ=21
32 (5.45)
Port 1 Port 2
2, φ
BZ 11,θZ
φ,BZ
2, φ
BZ 11,θZ
Port 1 Port 2 φ,BZ
22 ,2 θZ
118
ee
ee
eee
e
eDZCZ
BA
DZCZBA
+++
−−+=Γ
00
00 (5.46)
ee
ee
eDZCZ
BA +++=Τ
00
2 (5.47)
oo
oo
ooo
o
oDZCZ
BA
DZCZBA
+++
−−+=Γ
00
00 (5.48)
oo
oo
oDZCZ
BA +++=Τ
00
2 (5.49)
11 tan
sincos
θφ
φZZ
A Bo += (5.50)
φsinBo jZB = (5.51)
11
11 tan2
tan
sintan
cos
2tan
cossin
θφφ
θφ
φφφ
Zj
Zj
Zj
ZjC
BB
o −−−= (5.52)
2tan
sincosφφφ +=oD (5.53)
21
2121
1221
tantan2sintan2tansin
cosθθ
φθθφφ
ZZZZZZZ
A BBe −
+−= (5.54)
φsinBe jZB = (5.55)
212
121
1212121
tantan2
tan2
tansin2tancos22
tansintancossincos2
tan
θθ
θφφθφφφθφφφφ
ZZZ
ZZZZj
Zj
ZjC
BBe −
−+−++=
(5.56)
119
2
tansincos φφφ −=eD (5.57)
Besides, under the assumption that the coupler is lossless, we have the relations:
122 =Τ+Γ ee (5.58)
122 =Τ+Γ oo (5.59)
where Γe, Γo are even- and odd- mode reflection coefficients and Τe, Τo are even- and
odd- mode transmission coefficients respectively.
Combining equations (5.37)-(5.41), (5.58) and (5.59), it is found that the
sufficient and necessary conditions for ideal rat-race coupler are:
ee Τ=Γ (5.60)
oo Τ−=Γ (5.61)
As for the dual-band operations, equations (5.60) and (5.61) have to be satisfied
at both of the desired frequencies (f1: lower band, f2: upper band). To facilitate the
analysis at these two frequencies, we define that:
+=
−=
2
1
@2
3
@2
f
f
επφ
επφ (5.62)
+−=
−=
211
11
@2
@2
fn
f
δππθ
δπθ (5.63)
−−=
+=
222
12
@2
@2
fn
f
ψππθ
ψπθ (5.64)
120
where n1, n2 are integers. As labeled in Fig. 5. 4. 11 and Fig. 5. 4. 12, φ, θ1 and θ2 are
the electrical lengths associated with the major branches and the tapped stubs.
Combining equations (5.60)-(5.64), the solutions for the proposed dual-band rat-
race coupler can be given as:
12 / ff=β (5.65)
β
ππε+
−=1
22
(5.66)
β
ππδ+
−=12
1n (5.67)
21
2 πβ
πψ −
+=
n (5.68)
0cos2cos2 ZZ B ε
ε= (5.69)
01 cot)1(sin2cos2 ZZ
δεε
−= (5.70)
022 ]1sin)1(sin)[cotsin1(cot2cos2 ZZ
++−−=
εεδεψε
(5.71)
From equations (5.65)-(5.71), the characteristic impedances of the branch lines
and the tapped stubs (ZB, Z1, Z2) are determined by the frequency ratio β (β = f2/f1).
In practice, the realizable impedance values are constrained. Therefore, the
realizable frequency ratio using this rat-race coupler is also limited. However, with
the changing of the stub lengths (which corresponds to changing the values of n1 and
n2 in (5.67) and (5.68)), the range of frequency ratios (f2/f1 from 3.1 to 4.9) can be
realized. A simple numerical searching program has been developed to find these
values. In our case, we have limited the values of n1 and n2 between 1 and 9. The
realizable characteristic impedance is set between 20Ω and 120Ω. For the purpose of
121
demonstration, Fig. 5. 4. 13 shows the computed normalized line impedances of ZB,
Z1, Z2 (referred to Z0=50Ω) with the frequency ratio β changing from 3.1 to 4.9.
Since the results shown in Fig. 5. 4. 13 are the combinations of computed
impedances allowed by the impedance limits by selecting different sets of n1 and n2
values, the curves are not smooth. But it should be pointed out that this kind of non-
smooth change in the curves does not indicate any stability problem. In practice, for
a given frequency ratio, the values of n1 and n2 are fixed and the effects of small
changes in impedances (or line widths) on the corresponding parameters of the
coupler will be smooth and continuous. Hence, the small changes in the design
parameters will not affect the performances of the coupler greatly.
3.1 3.4 3.7 4.0 4.3 4.6 4.90.3
0.9
1.5
2.1
2.7
Nor
mal
ized
Impe
danc
e
Frequency Ratio (f2/f1)
ZB/Z0 Z1/Z0 Z2/Z0
Fig. 5. 4. 13. Normalized branch impedances used in the type II dual-band rat-race coupler
under different frequency ratios.
122
To validate the analysis of the proposed type II dual-band rat-race coupler, an
experimental prototype was designed and fabricated on the Rogers’ RO3006 PCB
with a dielectric constant of 6.15, the board thickness of 1.27mm and the loss
tangent of 0.0025. The structure of the coupler is shown in Fig. 5. 4. 11. It is
designed to work at the frequencies of 1GHz and 3.5GHz. Referring to Fig. 5. 4. 13
and the equations (5.62) – (5.71), the characteristic impedances and electrical
lengths of the three branches are:
=Ω=
=Ω=
=Ω=
GHzZ
GHzZ
GHzZ B
1@92,7.53
1@32,2.48
1@94,6.69
22
11
πθ
πθ
πφ
(5.72)
The prototype coupler is then designed by converting the parameters listed in
(5.72) into physical dimensions. The full-wave EM simulator IE3D was used to
optimize the complete structure to account for the junction effect and substrate loss.
The photo of the fabricated coupler is shown in Fig. 5. 4. 14. The measured
performances of this coupler are plotted in Fig. 5. 4. 15-Fig. 5. 4. 17. The measured
center frequencies of this coupler are found to be 1.02GHz and 3.55GHz. The shift
in center frequency is due to the process variations in circuit fabrication. The return
loss (S11) is below – 29 dB and the isolation (S31) is below -40 dB at the two
operating frequencies, as shown in Fig. 5. 4. 15. The phase and the magnitude of the
in-phase outputs (S21, S41) are given in Fig. 5. 4. 16, where the magnitudes are S21 =
-3.08 dB, S41 = -3.15 dB at 1.02 GHz, S21 =-3.50 dB, S41 = -3.10 dB at 3.55 GHz,
and the phase responses are °=∠−∠ 5.24121 SS at 1.02 GHz, °=∠−∠ 8.54121 SS at
3.55GHz. The anti-phase outputs results (S23, S43) are given in Fig. 5. 4. 17. The
magnitudes are S23 = -3.18 dB, S43 = -3.09 dB at 1.02 GHz, S23 = -3.25 dB, S43 = -
123
3.52 dB at 3.55 GHz, and the phase responses are °=∠−∠ 76.1824323 SS at 1.02 GHz,
°=∠−∠ 20.1754323 SS at 3.55GHz. The operating bandwidth of the coupler is about
80MHz at the two working frequencies. The measurement results match well with
the simulation results.
Fig. 5. 4. 14. Photo of the fabricated type II rat-race coupler.
1 2 3 4-50
-40
-30
-20
-10
0
S11,
S31
(dB)
Frequency (GHz)
S11 S31
Fig. 5. 4. 15. Measured return loss and port isolation of the type II rat-race coupler.
Port 1 (Σ port)
Port 4
Port 3 (∆ port)
Port 2 1—λ @ f1 3
2—λ @ f1 9
1—λ @ f1 9
124
1 2 3 4-40
-30
-20
-10
0
S21,
S41
(dB)
Frequency (GHz)
S21 S41
(a)
1 2 3 4-200
-100
0
100
200
Phas
e of
S21
, S41
(deg
ree)
Frequency (GHz)
S21 S41
(b)
Fig. 5. 4. 16. Measured insertion losses and phase responses of the in-phase outputs (S21 and
S41) of type II dual-band rat-race coupler. (a) Insertion losses, (b) phase responses.
125
1 2 3 4-40
-30
-20
-10
0
S23,
S43
(dB)
Frequency (GHz)
S23 S43
(a)
1 2 3 4-200
-100
0
100
200S23 S43
Phas
e of
S23
, S43
(deg
ree)
Frequency (GHz)
(b)
Fig. 5. 4. 17. Measured insertion losses and phase responses of the anti-phase outputs (S23
and S43) of type II dual-band rat-race coupler. (a) Insertion losses, (b) phase responses.
126
5.4.3 Wilkinson Power Divider for Dual-Band Operations
The same design concept has been implemented to a dual-band Wilkinson power
divider. The basic structure of this power divider is given in Fig. 5. 4. 18. It is
constructed by simply replacing the quarter-wavelength line of the Wilkinson power
divider with the proposed dual-band transmission line. The characteristic impedance
of the dual-band transmission line is 02Z (Z0 = 50Ω). As mentioned before, the
realizable frequency ratio (f2/f1) is constrained by the practical impedance values for
the microstrip line. Since the desired characteristic impedance in the Wilkinson
power divider is 02Z , the design parameters for this divider will be different with
the dual-band branch-line coupler and rat-race coupler mentioned before. A
numerical searching program is applied here to get the design parameters for
different frequency ratios. In this calculation, the realizable impedance value is
confined between 20Ω and 120Ω. The results are given in Fig. 5. 4. 19. It is found
that a wide frequency ratio with f2/f1 from 1.1 to 2.9 can be realized.
To verify the design concept, we have designed and tested a dual-band Wilkinson
power divider on Rogers’ board RO3006. The working frequencies are selected to
be 1GHz / 2.5GHz. The full-wave EM simulator IE3D was again used to optimize
the complete structure to account for the junction effect and substrate loss. The
photo of the fabricated divider is shown in Fig. 5. 4. 20. The measured performances
of this coupler are plotted in Fig. 5. 4. 21-Fig. 5. 4. 23. The measured center
frequencies of this power divider are found to be 1.04GHz and 2.59GHz. The shift
in center frequency is due to the process variations in PCB fabrication. As shown in
Fig. 5. 4. 21, the insertion losses at the two output ports (S21 and S31) are S21 = -3.161
dB, S31 = -3.13 dB at 1.04 GHz, S21 =-3.294 dB, S31 = -3.412 dB at 2.59 GHz. The
measured return losses and port isolations are given in Fig. 5. 4. 22. The return loss
127
of port 1 (S11) is below -29 dB and the isolation (S23) is below -25 dB at the two
operating frequencies as shown in Fig. 5. 4. 22 (a). The return loss of port 2 (S22) is
below -28dB and port 3 (S33) is below -23 dB at the two working frequencies as
shown in Fig. 5. 4. 22 (b). The phase responses of the power divider are shown in
Fig. 5. 4. 23, where the phase differences between port 2 and port 3 are
°=∠−∠ 8.03121 SS at 1.04 GHz and °=∠−∠ 1.23121 SS at 2.59 GHz.
Fig. 5. 4. 18. General topology of the proposed dual-band Wilkinson power divider.
1 .1 1 .7 2 .3 2 .9
0 .6
1 .2
1 .8
2 .4
Z 2 /Z 0Z
1/Z
0
Nor
mal
ized
Impe
danc
e
F re q u e n c y R a tio ( f 2 / f 1 )
Fig. 5. 4. 19 The computed design parameters for different frequency ratios of the dual-band
Wilkinson power divider.
Port 1
Z1 , θ1 Port 2
Port 3
Z1 , θ1
Z2 , θ2
210 &@Z2 ff 2Z0
128
Fig. 5. 4. 20 The photo of the fabricated Wilkinson power divider.
1 2 3-50
-40
-30
-20
-10
0
S21,
S31
(dB)
Frequency (GHz)
S21 S31
Fig. 5. 4. 21 The insertion losses of the tested dual-band Wilkinson power divider.
Port 2
Port 3
Port 1
129
1 2 3-50
-40
-30
-20
-10
0
S11,
S23
(dB)
Frequency (GHz)
S11 S23
(a)
1 2 3
-30
-20
-10
0
S22,
S33
(dB)
Frequency (GHz)
S22 S33
(b)
Fig. 5. 4. 22 The return losses and the isolations of the tested dual-band Wilkinson power
divider. (a) S11 and S23, (b) S22 and S33.
130
1 2 3-200
-100
0
100
200
Phas
e of
S21
, S31
(deg
ree)
Frequency (GHz)
Θ21 Θ31
Fig. 5. 4. 23 The phase responses ( 3121, SS ∠∠ ) of the tested dual-band Wilkinson power
divider.
5.5 Summary
In this chapter, we have proposed a novel dual-band quarter-wavelength
transmission line, which is constructed by a transmission line tapped with stubs.
First, this dual-band line is used in the filter for dual-band operations. It behaves as
the dual-band impedance inverter, and a second-order chebyshev bandpass filter is
designed in this way. A test filter is fabricated on the Rogers’ RO3006 board. The
measurement results match with the theoretical predictions. However, there is an
unwanted resonant peak appearing between the two working frequencies, degrading
the performance of the designed dual-band filter. To suppress this kind of resonant
peak, band-stop filters using L-shape lines are added to the input and output ports of
the filters. Up to 20dB harmonic suppression has been achieved using this structure.
131
Then this kind of structure is applied to other dual-band microwave passive
components. A dual-band branch-line coupler working at 0.9 GHz / 2 GHz is
designed, fabricated and measured. For the rat-race coupler, two types of prototypes
are proposed using the tapped line structure but with different implementation
schemes. The experimental rat-race coupler for type I design working at 2 GHz / 5
GHz and the experimental rat-race coupler for type II design working at 1 GHz / 3.5
GHz are fabricated and measured. Finally, a Wilkinson power divider operating at 1
GHz / 2.5 GHz is fabricated and tested. All of the measurement results prove the
desired dual-band operations.
132
CHAPTER 6
Parameter Extractions for Tuning of the Microwave
Bandpass Filters
6.1 Introduction
In Chapter 2, we have studied the synthesis of the filter with various topologies.
Based on the filter prototypes synthesized, the designs of microwave filters featuring
different characteristics (compact size, reconfigurability and dual-band operation)
have been thoroughly discussed in Chapter 3 – 5. In this chapter, we will address the
parameter extractions of the filters, which is another important issue for the filter
designs.
In the practical implementation and fabrication of the filters, it is commonly
found that the measured performances differ from the designed frequency responses.
As a result, microwave filters usually go through the tuning and optimization
process that becomes more time-consuming and expensive as the number of
resonators increases in high-order filters. In the tuning process of the filters, the key
step is to extract the coupling coefficients and resonant frequencies of individual
resonators from the measured frequency responses. Careful comparison between the
extracted parameters and the synthesized (designed) values will then lead to pin-
pointing the resonators that need to be tuned and optimized. As a result, an efficient
computer-aided approach to the parameter extraction for tuning coupled-resonator
microwave filter is desirable.
Recently, several techniques have been employed for the parameter extractions of
microwave filters. A time-domain technique is developed by Dunsmore [92].
133
However, this method may have difficulties in dealing with cross-coupled resonator
filters. Several frequency-domain techniques are also developed [93] - [95], where
closed-form recursive formulas and a sequential computer-aided tuning procedure
are proposed respectively. Different from the time-domain and frequency-domain
techniques, a parameter extraction algorithm based on fuzzy logic is also developed
by Miraftab and Mansour for tuning microwave filters [96], [97]. This approach
requires appropriate choices of fuzzy sets and fuzzy rules to extract the desired
parameters.
Meanwhile, the genetic algorithm (GA) as an evolutionary optimization method
has been applied successfully in many areas since its invention by Holland [98].
Compared with other conventional techniques such as quasi-Newton method and
conjugate-gradient method, the most distinctive feature of GA is the concept of
implicit parallelism [34]. The GA searches many different regions of objective
surfaces simultaneously, avoiding the optimization being trapped by a local
maximum or minimum point. Therefore, this method is mostly suitable for the
problems with multi-functions and multi-variables [35], [36]. In the area of
microwave engineering, the applications of GA are quite extensive. For example,
GA has been used to optimize the physical dimensions of various types of antennas
[99] - [103]. Genetic algorithm is also being used in the design optimization of the
microwave filters and other more complicated 2D or 3D microwave components
[104] - [108]. As for the circuit application, Araneo [109] used this method to
extract the parameters for the equivalent circuits of the microwave discontinuities.
Chen et al. [110] used GA to extract the model parameters of the RF on-chip
inductors. More applications of GA in the microwave optimization can be found in
[111].
134
In this chapter, we again employ the GA to the parameter extractions of the
coupled-resonator microwave filters, for both slightly mistuned and highly mistuned
cases. A conventional binary genetic algorithm has been used for extracting the
coupling coefficients and resonant frequencies from filters’ performance generated
by a set of presumed parameters. The extracted parameters match the presumed ones
with high accuracy, demonstrating the feasibility of the GA-based method. The
organization of this chapter is as follows. First, we define the problem to be
discussed. Then, the basic theory and the data structure of the chromosome used in
the GA searching are presented. In the third part, a fourth-order chebyshev filter and
an eighth-order general chebyshev (quasi-elliptical) filter with only mistuned inter-
resonator couplings are studied. Finally, a fourth-order chebyshev filter with both
mistuned resonators and inter-resonator couplings is studied.
6.2 Parameter Extraction for Microwave Filter Tuning
6. 2. 1 Basic Equations for the Parameter Extractions of the Filters
The performance of a typical coupled-resonator filter can be represented in a
matrix form. As described in Chapter 2 and Chapter 4, the loop currents of a filter
can be grouped in a vector [I] and described by a matrix equation:
[ ] [ ][ ][ ] [ ][ ] [ ]EjIAIMRj −==+Ω+− ω (6.1) where [R] is a n×n matrix having diagonal elements R1,1 = Rn,n= r with the other
elements equal to zero, [Ω] is a n×n identity matrix, and [M] is the coupling
coefficient matrix (M-matrix). [E] is the excitation, which is [1, 0, 0, …, 0]t. n is the
order of the filter analyzed. ω is the normalized frequency, which is normalized
against the center frequency of the passband by:
135
)( 0
0
0
ff
ff
ff
−∆
=ω (6.2)
In equation (6.2), f0 is the desired center frequency, f is the variable representing
the frequency and ∆f is the bandwidth of the filter. The transmission and reflection
coefficients of the filters can be calculated by equations (2.14) and (2.15).
Normally, the performances of the coupled-resonator filters are determined by the
coupling coefficients in the matrix [M]. In practical implementation and fabrication
of the microwave filters, some of the coupling coefficients deviate from the designed
values due to various reasons such as process variation, design mistakes, etc. These
deviations are reflected in the measured characteristics (i.e. S-parameters), which
also deviate from design specifications. To determine the root cause of the
undesirable filter performance, the coupling coefficients need to be extracted
accurately from the measured S-parameters. Then, the comparisons between the
extracted ones and the ideal ones can help us identify the mistuned parts, which can
then be tuned toward the designed values. The object of this paper is to make use of
genetic algorithm to carry out the parameter extraction efficiently and accurately.
For simplicity, all of the frequencies used in the following sections are normalized
frequencies as given by equ. (6.2). The basic principle of our method is to sample
the given responses of the filter and apply genetic algorithm to find the optimal set
of coupling coefficients that fit these sampled data.
6. 2. 2 Genetic Algorithm and Its Implementation for the Parameter Extractions
As mentioned in Chapter 2, GA’s are iterative optimization procedures that begin
with a set of randomly initialized chromosomes that represent potential solutions.
The chromosomes gradually evolve toward better solutions according to certain
reproduction rules. For each reproduction, fitness functions are checked. The
136
reproduction cycle is repeated until some terminating conditions (usually when the
solutions are fit enough) are met. The flowchart of the proposed GA in this work is
the same with that given in Chapter 2 and illustrated again in Fig. 6. 2. 1. To make it
clear, some fundamental GA elements are explained here.
Fig. 6. 2. 1 The flowchart of the proposed algorithm.
137
‘Population’ is a group of randomly initialized individuals (represented by
chromosomes). Each chromosome consists of the genes. In this chapter, the genes
are the coupling coefficients (mij) in the [M]-matrix. And a 16-bit binary code will
be used to represent one coupling coefficient. Hence, a chromosome is a chain of
16-bit binary codes. The chain size depends on the number of coefficients to be
extracted. The population size (the number of chromosomes) significantly affects the
speed of the simulation. In our study, the population size of 50 or 100 has been used.
‘Fitness function’ evaluates a chromosome’s fitness value. Normally, the fitness
value is directly related to the error function. In our method, the error function is the
difference between the filter’s response from the extracted parameters and the
presumed response. To find the value of this error function, the response of the
extracted parameters will be evaluated at several discrete points (sampling points).
Let these points be represented as (p1, p2, p3, p4, …. pn). The S21 values of the
response to be fitted at these points are (X1, X2, X3, X4, … Xn,), and the S21 values of
the response generated by the GA at these points are (Y1, Y2, Y3, Y4, … Yn,). The
fitness value is then defined by:
ii
n
iXY
F−Σ
=
=1
1
α (6.3)
where α is a scaling factor between 0 and 1.
The three conventional GA operators are reproduction, crossover and mutation.
Their basic structures have been explained in Chapter 2, as given in Fig. 2. 4. 2.
Here, we will only mention the necessary variables to be used for the parameter
extractions.
138
‘Reproduction’ is actually a chromosome selection process. Let the fitness of
chromosome j be Fj and the probability for chromosome j to be chosen in the
‘reproduction’ process of this work is defined as:
∑
=
= n
ii
jj
F
FP
1
(6.4)
where n is the total number of the chromosomes. In our work, the selection is based
on probability ranking. In other words, the chromosome with the largest fitness
value is selected first. The population size in our simulation is selected to be 50 or
100. The ‘crossover’ in this work is implemented through the one-point crossover
method [36]. The ‘mutation’ is done based on the prescribed mutation rate.
Based on the steps given in Fig. 6. 2. 1, we can search for the optimal parameters
using the genetic algorithms. In our simulations, the crossover and mutation rates are
set as 0.6 and 0.35, respectively.
6. 2. 3 Coupling Coefficients Extractions of the Filters with Only Mistuned Inter-
Resonator Couplings
To demonstrate the performance of the GA method in the parameter extraction,
several types of filters with mistuned inter-resonator couplings are tested. The first
example is the fourth-order chebyshev filter with 20 dB passband return loss. The
ideal coupling matrix of this filter is given in (6.5) and the performance of the filter
is shown in Fig. 6. 2. 2. Here the parameters for input and output couplings are R1,1 =
R4,4 = 1.0274.
139
09106.0009106.007000.0007000.009106.0009106.00
(6.5)
-4 -2 0 2 4
-60
-40
-20
0
S1
1 &
S21
(dB)
Normalized Frequency
S21 S11
Fig. 6. 2. 2 Ideal response of the fourth-pole chebyshev filter.
If the coupling coefficients deviate from the ideal values, the performance of the
filter will degrade. We will use the GA to clarify this kind of deviations. Both
slightly mistuned and highly mistuned filters are tested. For the case of slightly
mistuned coupling matrix as assigned in (6.6), nine sampling points in the GA
simulations are used, where the points in the passband are at 0.9, 0.6, 0.3, 0, -0.15, -
0.45, -0.75 and the points outside of the passband are at 2, -3. The extracted matrix
is the same as the assigned one, when five significant digits are used for the coupling
coefficients. The number of the chromosomes (the population size in the following
parts) used in the GA is 50, the scaling factor α is 1 and the number of the
generations is 40. The fitness value of the best chromosome is larger than 300. Since
140
the extracted responses (S-parameters) are the same as the assigned ones, we show
them in the same figure (Fig. 6. 2. 3).
The coupling matrix of the highly mistuned filter is assigned in (6.7). The same
nine sampling points are used for this example and the scaling factor α for the fitness
function is still 1. To improve the feasibility of the extraction, a larger population
size of 100 and generation number of 60 are applied here. The fitness value of the
best chromosome generated is 17.0017. Converting these chromosomes to real
numbers results in the extracted coupling matrix as given in equation (6.8). Fig. 6. 2.
4 shows the responses of the extracted and the assigned coupling matrices. The
agreement is excellent in the passband and stopband except the ones near the two
peaks, where small shifts have been observed. Further improvement of the accuracy
can be achieved by increasing the number of sampling points.
07106.0007106.008000.0008000.000106.1000106.10
(6.6)
-4 -2 0 2 4-50
-40
-30
-20
-10
0
S11
& S
21 (d
B)
Normalized Frequency
S21 S11
Fig. 6. 2. 3 Responses of the fourth-order chebyshev filter with slightly mistuned inter-
resonator couplings (the extracted ones are the same as the assigned ones).
141
Mhighly_assigned =
05000.0005000.002000.1002000.104000.0004000.00
(6.7)
Mhighly_extracted =
05199.0005199.001976.1001976.103839.0003839.00
(6.8)
-4 -2 0 2 4-60
-40
-20
0
S21
(dB)
Normalized Frequency
Extracted Assigned
(a)
-2 -1 0 1 2-14
-12
-10
-8
-6
-4
-2
0
S11
(dB)
Normalized Frequency
Extracted Assigned
(b)
Fig. 6. 2. 4 Comparisons between assigned and extracted responses of the fourth-order
chebyshev filter with highly mistuned inter-resonator couplings. (a) S21. (b) S11 (the
frequency range for S11 is between -2 and 2 for the purpose of clarity).
142
To show the flexibility of the proposed method in the filter tuning, an eighth-
order quasi-elliptical filter with transmission zero near the passband is also tested.
The ideal coupling matrix of this filter is given in (6.9). The slightly mistuned
coupling matrix is given in (6.10). In the GA simulations, we have assumed that the
matrix is symmetrical and there are six parameters to be extracted (m12, m23, m34, m45,
m36, m27). When the population size is 100, the scaling factor is 1 and the number of
generation is 50, the fitness of the best chromosome is 35.6065. The same nine
sampling points are used in this simulation. The extracted matrix is given in
equation (6.11). Comparisons between the assigned and the extracted responses are
shown in Fig. 6. 2. 5. Good matches have been achieved.
Mideal =
−
−
08231.00000008231.005917.00000251.0005917.005516.000781.000005516.004925.00000004925.005516.000000781.005516.005917.0000251.00005917.008231.00000008231.00
(6.9)
Mslight,assigned =
09417.00000009417.007349.00001534.0007349.009179.001336.000009179.004283.00000004283.009179.000001336.009179.007349.0001534.00007349.009417.00000009417.00
(6.10)
143
Mslight,extracted =
09372.00000009372.007306.00001598.0007306.009194.001382.000009194.004216.00000004216.009194.000001382.009194.007306.0001598.00007306.009372.00000009372.00
(6.11)
-4 -2 0 2 4
-60
-40
-20
0
-2 0 2-3-2-10
S21
(dB
)
Normalized Frequency
Assigned Extracted
(a)
-2 0 2-40
-30
-20
-10
0
S11
(dB)
Normalized Frequency
Assigned Extracted
(b)
Fig. 6. 2. 5 Comparisons between assigned and extracted responses of the eighth-order
quasi-elliptical filter with slightly mistuned inter-resonator couplings. (a) S21. (b) S11.
144
Fig. 6. 2. 6 The flowchart of the improved GA simulation process.
Mhigh,assigned =
05959.00000005959.007152.00000585.0007152.000260.100752.000000260.108982.00000008982.000260.100000752.000260.107152.0000585.00007152.005959.00000005959.00
(6.12)
145
Mhigh,extracted =
06016.00000006016.007121.00000548.0007121.009732.000194.000009732.008695.00000008695.009732.000000194.009732.007121.0000548.00007121.006016.00000006016.00
(6.13)
-4 -2 0 2 4-80
-60
-40
-20
0
S21
(dB)
Normalized Frequency
Assigned Extracted
(a)
-2 0 2-60
-40
-20
0
S11
(dB)
Normalized Frequency
Assigned Extracted
(b)
Fig. 6. 2. 7 Comparisons between assigned and extracted responses of the eighth-order
quasi-elliptical filter with highly mistuned inter-resonator couplings. (a) S21. (b) S11.
146
As for the highly mistuned eighth-order filter, the same generation number,
population number and the scaling factor as those for the slightly detuned case are
applied. The positions of the sampling points are not changed. Since the problem
under this situation is complex than that in the slightly mistuned one, we modify the
implementation of our GA method to increase the accuracy. The flowchart of this
process is given in Fig. 6. 2. 6. In this new process, we do several iterations of the
GA simulation as that in Fig. 6. 2. 1. The best chromosome during each iteration is
recorded. In the final simulation, all of these best-chromosomes are part of the initial
populations, the others are still randomly generated. In this way, the convergence
and the accuracy of the proposed algorithm can be improved. Inevitably, the time
used in the simulations has been increased. Using this procedure, the coupling
coefficients of the highly mistuned filter are extracted. Ten iterations have been used
to obtain the initial chromosomes. The fitness value of the final best chromosome is
41.5719. The assigned and the extracted coupling matrices are given in (6.12) and
(6.13). The corresponding S-parameters of these matrices are shown in Fig. 6. 2. 7.
The differences between them are very small and are mainly caused by the small
numbers of sampling points.
6. 2. 4 Coupling Coefficients Extractions of the Filter with Both Mistuned Inter-
Resonator Couplings and Mistuned Resonators
Mmistunedresonator_assigned =
3000.29000.0009000.05000.19000.0009000.01000.25000.0005000.07000.1
(6.14)
147
Mmistunedresonator_extracted =
2649.29281.0009281.04473.18580.0008580.01856.25164.0005164.06688.1
(6.15)
-6 -4 -2 0 2 4 6-80
-60
-40
-20
0
-4 -2 0
-4
-2
0
S21
(dB)
Normalized Frequency
Assigned Extracted
(a)
-6 -4 -2 0 2 4 6-8
-6
-4
-2
0
S11
(dB)
Normalized Frequency
Assigned Extracted
(b)
Fig. 6. 2. 8 Comparisons between assigned and extracted responses of the fourth-order chebyshev filter with mistuned resonators and inter-resonator couplings. (a) S21. (b) S11.
148
In this section, we will discuss another filter tuning problem, where both the
resonators (resonant frequencies) and the inter-resonator couplings are mistuned. A
fourth-order chebyshev filter with the same ideal performance as that given in Fig. 6.
2. 2 is considered. With all the resonators mistuned, the number of the parameters to
be extracted in this case is seven (m11, m22, m33, m44, m12, m23, m34). Equation (6.14)
gives the tested matrix with all these parameters mistuned. The main effect of the
mistuned resonators is the shift in the center frequency. Therefore, the positions of
the sampling points have been changed for this example. Eleven normalized
frequency points are chosen, which are at -4, -2.95, -2.6, -2.25, -1.9, -1.55, -1.2, -
0.85, -0.5, 2, 5. The same simulation process as described in Fig. 6. 2. 6 is used,
where the number of iterations is 10, the generation number is 50 and the population
size is 100. In addition, the scaling factor used here is 0.3 and the linear fitness-
scaling algorithm [36] is applied to improve the performance. The fitness value of
the best chromosome for this case is 21.5674. The exacted coupling matrix is given
in (6.15). Fig. 6. 2. 8 shows the S-parameters of the extracted and the assigned
characteristics. These responses are very close, which demonstrates the good
performance of the proposed GA method in this kind of filter tuning problem.
6.3 Summary
In this chapter, the genetic algorithm (GA) has been applied to extracting the
coupling matrices of the assigned filters’ responses. Both the filters with mistuned
resonators and mistuned inter-resonator couplings have been studied. For all these
filters, the extracted coupling matrices fit the assigned ones well. Besides, all of
these good performances are achieved with small number of sampling points, which
demonstrate the efficiency of the proposed method.
149
CHAPTER 7
Conclusion and Future Work
7.1 Conclusion
In this dissertation, we have discussed thoroughly the design issues related to the
microwave passive circuits especially the microwave bandpass filters. In the first
part, we employ the genetic algorithm (GA) to synthesize the filter with prescribed
performance. In this way, we avoid the computations of the derivatives of the
functions, which are necessary for the gradients-based optimization method. Filters
with different performances and different orders are synthesized for the purpose of
demonstration. The results are very close to the rigorous solutions, which prove the
effectiveness of the proposed method. As for the modern microwave bandpass filters
designs, the size, the tunability and the multi-band operations are the mostly
concerned topics. To explain the basic principles related to these problems, we have
discussed them respectively. In Chapter 3, a compact microstrip tri-section SIR and
a compact CPW tri-section slow-wave SIR are proposed. Compared with the
conventional two-section SIR, the size reduction of the new SIRs can be up to 40
percents. To prove the performance of the new resonators, filters with different
characteristics are designed based on them. The properties of both compact size and
sharp roll-off near the passband have been achieved. In Chapter 4, two types of
filters with reconfigurable transmission zeros are proposed. Besides, it is found that
the type I filter can be used to realize the tunabilities of the positions of both the
zeros and the center frequencies. Both the theory and the simulation details are
addressed for these two kinds of filters. Experimental prototypes are also fabricated
and measured. The results validate the desired reconfigurations. In Chapter 5, a new
150
dual-band quarter-wavelength transmission line is proposed. This dual-band
transmission line acts as the impedance inverter in the filter. Combining this new
structure and the so-called dual-behavior resonator, a second-order dual-band filter
working at 2 GHz / 5 GHz is designed. Besides, bandstop filters are connected with
the devised dual-band filter to suppress the spurious harmonic between the two
working bands. The measurement results prove the desired dual-band characteristics.
This kind of dual-band transmission lines can be also used to design other dual-band
microwave passive circuits. A dual-band branch-line coupler, a dual-band Wilkinson
power divider and two types of dual-band rat-race couplers are designed based on
the proposed structure. The design equations for all of these dual-band circuits are
derived based on the ABCD-matrix and the even-odd mode analysis. The
performances of these circuits are verified by measurement results. Another practical
issue for the filter design is the post-tuning of the filters. This is due to that there is
usually difference between the measurement and the simulation results. To clarify
where these differences happen, we need to extract the coupling matrix based on the
experimental data and compare the data with the theoretical ones. In Chapter 6, we
again use the GA to do this kind of parameter extractions. Filters with both slightly
and highly mistuned cased have been studied. The extraction parameters match with
the desired ones, which demonstrate the efficiency of this method.
7.2 Future Work
In Chapter 3, we proposed a compact slow-wave CPW SIRs. This resonator can
be applied in many aspects especially in the MMIC circuits at high frequencies (20
GHz or even higher) to reduce the size of the whole die. In the future, the
151
applications of this kind of resonators will be studied. It is suggested to combine this
resonator with the active devices to form some circuit building blocks.
In Chapter 6, we proposed to use the genetic algorithm (GA) to do the parameter
extractions. This method can be applied for the tuning of the planar filters from the
mistuned state to the tuned state based on the measurement results.
152
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with a broadside-cooupled multiconductor transmission-line model,” IEEE
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APPENDIX: PUBLICATION LIST
JOURNAL:
[1] H. Zhang and K. J. Chen, "A Tri-Section Stepped-Impedance Resonator for
Cross-Coupled Bandpass Filters," IEEE Microwave and Wireless Components
Letters, vol. 15, no. 6, pp. 401 - 403, June 2005.
[2] H. Zhang and K. J. Chen, "Miniaturized Coplanar Waveguide Bandpass
Filters Using Multi-Section Stepped Impedance Resonators," IEEE Trans.
Microwave Theory and Techniques, vol. 54, no. 3, pp. 1090 - 1095, March
2006.
[3] H. Zhang and K. J. Chen, "Bandpass Filters with Reconfigurable
Transmission Zeros Using Varactor-Tuned Tapped Stubs," IEEE Microwave
and Wireless Components Letters, vol. 16, no. 5, pp. 249 - 251, May 2006.
[4] H. Zhang and K. J. Chen, "A Stub Tapped Branch-Line Coupler for Dual-
Band Operations," to appear in IEEE Microwave and Wireless Components
Letters, Feb. 2007.
[5] H. Zhang and K. J. Chen, "A Dual-Band Rat-Race Coupler with a Single
Tapped Stub," submitted to IEEE Trans. Microwave Theory and Techniques.
CONFERENCE:
[1] H. Zhang, J. W. Zhang, L. L. W. Leung, and K. J. Chen, "Bandpass and
Bandstop Filters Using CMOS-Compatible Micromachined Edge-Suspended
Coplanar Waveguides," Proceedings of 2005 Asia-Pacific Microwave
Conference (APMC 2005), Suzhou, China, Dec. 4 - 7, 2005.
[2] H. Zhang and K. J. Chen, "Compact Bandpass Filters Using Slow-Wave
Coplanar Waveguide Tri-Section Stepped Impedance Resonators,"
167
Proceedings of 2005 Asia-Pacific Microwave Conference (APMC 2005),
Suzhou, China, Dec. 4 - 7, 2005.
[3] H. Zhang and K. J. Chen, "A Microstrip Bandpass Filter with an
Electronically Reconfigurable Transmission Zero," 2006 European Microwave
Conference (EuMW 2006), Manchester, U. K., Sep. 10 – 15, 2006.
[4] J. Zhang, H. Zhang, K. J. Chen, S. G. Lu and Z. Xu, "Microwave
Performance Dependence of BST Thin Film Planar Interdigitated Vatactors on
Different Substrates," IEEE International Conference on Nano/Micro
Engineering and Molecular Systems (NEMS 2007), Bangkok, Thailand, Jan.
16 - 19, 2007.
