Frequency agile RF/microwave circuits using BST varactors

ABSTRACT

JIN, ZHANG. Frequency Agile RF/microwave Circuits using BST Varactors. (Under the direction of Amir Mortazawi)

The research has focused on characterizing barium strontium titanate or BST film at

RF/microwave frequencies, improving BST capacitor quality factors and designing

frequency agile RF/microwave circuits.

A simple and fast measurement technique is developed to extract BST loss tangent and

dielectric constant in a parallel plate BST capacitor. An error analysis is performed to

indicate the measurement accuracy. In addition, the BST capacitor layout is optimized to

achieve the best possible quality factor.

BST capacitor based tunable band-pass filters are designed. Two coupled

half-wavelength microstrip resonators are used in the filter. BST capacitors are placed at the end

of the resonators. Analysis shows that with the increase in BST capacitance, the filter skirt

sharpness increases, the circuit size decreases, and the tunability increases but the insertion

loss increases. Different filter specifications are achieved by using various BST capacitances.

In addition, a new topology is developed to maintain the 3-dB bandwidth of filters within the

tuning frequency range. The bandwidth change decreases from 54% to only 4% using the

new topology.

A tunable microstrip antenna is designed, fabricated and measured. Multiple varactors are

used to load a rectangular microstrip antenna. A figure of merit is defined to find the

optimum number of varactors. Good tunability of 25% and maximum gain of 7.8 dB are

obtained. The measurement results prove that tunable microstrip antennas using

multiple-varactor loading can achieve better performance than those using single-multiple-varactor loading.

A tunable high impedance surface is designed, fabricated and measured. The surface uses

lumped elements to form multiple parallel resonant circuits. Unlike other lumped-element

high impedance surfaces, the inductance in this surface is obtained from the grounded

substrate, which shows much higher quality factor and requires less substrate height than

those using vias to obtain the inductance. Measured tunability of 62% is achieved.

A novel digital phase shifter design at X-band is presented. The phase shifter operates

based on converting a microstrip line to a rectangular waveguide and thus achieving the

phase shift by changing the propagation constant of the medium. A 3-bit phase shifter has

been designed and constructed using PIN diodes as switches. An average insertion loss of

FREQUENCY AGILE RF/MICROWAVE CIRCUITS

USING BST VARACTORS

by

Zhang Jin

A dissertation submitted to the Graduate Faculty of

North Carolina State University in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

Electrical and Computer Engineering

Raleigh

2003

APPROVED BY:

_______________________

Prof. Amir Mortazawi (Chair)

_______________________ _____________________

Prof. Angus Kingon Prof. Jon-Paul Maria

BIOGRAPHICAL SUMMARY

Zhang JIN received the B.Sc degree in 1994 from the University of Electronic Science

and Technology of China, the M.Eng from National University of Singapore in 2000, and the

PhD from North Carolina State University in 2004. From 1994 to 1997, she worked as a

research engineer at Chengdu Institute of Technology. Her research interests include

ACKNOWLEDGEMENTS

I would like to express my gratitude to my husband, Liu Li, my parents, Jin Gongwang

and Zhang Manjing, and my sister and brother-in-law, Jin Jing and Liu Jin. Without their

encouragement, patience, and sacrifice this work would not be possible.

I would like to thank my advisor Prof. Amir Mortazawi for his guidance during my

graduate studies. I would also like to thank Prof. Angus Kingon, Prof. Jon-Paul Maria, Prof.

Gianluca Lazzi, and Prof. Griff Bilbro for their valuable suggestions and serving on my Ph.D

committee.

Many thanks go to my past and present graduate student colleagues, Sean Ortiz, Mete

Ozkar, Xin Jiang, Ayman Al-zayed, Ali Tombak, Navin Gupta, Lora Schulwitz, Jonghoon

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

LIST OF APPENDICES

CHAPTER 1 INTRODUCTION ... 1

1.1 Motivations ... 1

1.2 Overview of current technologies for continuous tunable circuits ... 2

1.3 Overview of BST thin film process for high frequency applications ... 3

1.3.1 BST composition ...4

1.3.2 BST deposition on various materials ...4

1.3.3 BST film thickness...5

1.3.4 BST process temperature ...6

1.4 Previous work in RF and microwave circuits using BST ... 6

1.4.1 BST tunable capacitors ...6

1.4.2 BST tunable filters ...7

1.4.3 BST phase shifters ...8

1.4.4 BST tunable antennas ...9

1.5 Research objectives and dissertation organization ... 9

1.5.1 Optimization of BST capacitor layout to achieve high Q factor ...10

1.5.2 Characterization of BST film at RF and microwave frequencies ...10

1.5.3 Tunable integrated filters ...10

2.1 Introduction... 13

2.2 Quality factors of BST capacitors... 14

2.2.1 Integrated BST capacitors...17

2.2.2 Discrete BST capacitors...22

2.3 BST thin film characterization... 26

2.3.1 Measurement technique for BST loss tangent and dielectric constant extraction...26

2.3.2 Error analysis on BST loss tangent measurement...30

2.4 Conclusion ... 31

CHAPTER 3 TUNABLE INTEGRATED FILTERS... 32

3.1 Introduction... 32

3.2 Capacitively loaded resonators ... 32

3.3 Tunable band pass filter design using capacitively loaded resonators ... 40

3.4 Design of tunable filters with constant bandwidth ... 44

3.5 Conclusion ... 45

CHAPTER 4 TUNABLE MICROSTRIP ANTENNA... 46

4.1 Introduction... 46

4.2 Advantages of using multi-varactor loading... 47

4.2.1 Radiation pattern improvement...49

4.2.2 Antenna gain improvement...50

4.3 Design of tunable antenna using multi-varactor loading ... 53

4.3.1 Tunability...53

4.3.2 Figure of merit ...56

4.4 Measurement results ... 57

4.5 Conclusions... 60

CHAPTER 5 TUNABLE ARTIFICIAL HIGH IMPEDANCE SURFACE ... 62

5.1 Introduction... 62

5.2 Theory of artificial high impedance surface ... 63

5.3 Tunable high impedance surface design ... 69

5.4 Fabrication and measurement ... 77

5.6 Conclusion ... 80

CHAPTER 6 DESIGN OF A NEW DIGITAL PHASE SHIFTER AT X-BAND... 81

6.1 Introduction... 81

6.2 Background... 81

6.3 Theory... 83

6.4 Experimental results... 88

6.5 Conclusion ... 93

CHAPTER 7 CONCLUSIONS... 94

APPENDIX A ... 96

LIST OF FIGURES

Figure 2.1 Skin depth of the thin film platinum... 14

Figure 2.2 Conductor (Qm), dielectric (Qd) and total quality factors (Qt) as a function of frequency... 15

Figure 2.3 (a) Integrated and (b) discrete BST capacitors ... 16

Figure 2.4 The current distributions of (a) the capacitor and (b) the equivalent electrode. 18 Figure 2.5 Simulated current distributions of BST capacitor ... 18

Figure 2.6 (a) the distributed and (b) one-stage equivalent circuit model of the capacitor 19 Figure 2.7 Normalized R’s at different capacitor dimensions. ... 21

Figure 2.8 Normalized Rbot at different sizes of bottom electrode... 22

Figure 2.9 Current distributions on a discrete capacitor ... 23

Figure 2.10 Figure 2.10 Normalized R’s at different capacitor dimensions... 24

Figure 2.11 Normalized Rbot at different sizes of bottom electrode... 24

Figure 2.12 Normalized R’s at different bonding positions... 25

Figure 2.13 Normalized R’s at different bonding diameters... 25

Figure 2.14 (a) the BST capacitor and standards (b) short #1 and (c) short #2 are fabricated on the same wafer ... 27

Figure 2.15 Equivalent circuits of the on wafer (a) capacitor, (b) short #1 and (c) short #2 28 Figure 2.16 BST capacitance and loss tangent versus frequency ... 29

Figure 2.17 Standard deviation of measured loss tangent versus frequency ... 31

Figure 3.1 Half wavelength resonator and its current and voltage distribution... 33

Figure 3.2 (a) Folded half wavelength resonator, and (b) folded half wavelength resonator with reduced size by using a tunable capacitor connecting at the two ends ... 34

Figure 3.3 (a) BST capacitor loaded resonator and (b) its equivalent circuit ... 34

Figure 3.4 Zeros and poles of resonators with and without loaded capacitor... 36

Figure 3.5 Relative distance between zero to pole as a function of Cbst... 37

Figure 3.6 Zero and pole of input impedance for capacitively loaded resonator as a function of loading capacitance ... 38

Figure 3.8 (a) a conventional two-pole hairpin filter consisting of two coupled half

wavelength resonators (b) hairpin filter with two coupled BST capacitor loaded

resonators ... 40

Figure 3.9 Filter response at over, under and critical coupled condition... 41

Figure 3.10 Simulated S-parameters of the band pass filters (a) design #1, Cbst=2.5 pF and (b) design #2, Cbst=1 pF ... 42

Figure 3.11 BST tuned hairpin filter with constant bandwidth design ... 44

Figure 3.12 Simulated S-parameters of the bandpass filter with constant bandwidth design ... 45

Figure 4.1 Microstrip antenna... 47

Figure 4.2 (a) single and (b) multiple varactor tuned microstrip antenna ... 48

Figure 4.3 The electrical field distribution along radiating edge and the radiation pattern at different bias voltage for (a) single varactor tuned and (b)multiple varactor tuned microstrip antenna... 49

Figure 4.4 (a) multiple varactor loaded radiating edge and its equivalent circuit (b) multiple varactors simplified as one effective varactor and its equivalent circuit ... 50

Figure 4.5 Antenna gain for a multiple and single varactor loaded antenna ... 53

Figure 4.6 (a) multiple varactor loaded antenna evenly divided by n cells where n is the number of varactors at each radiating edge (b) current distribution for each unit cell... 54

Figure 4.7 Antenna tunability as a function of loading capacitance C1... 55

Figure 4.8 Plot of figure of merit for the antenna as a function of number of varactors used ... 56

Figure 4.9 Picture of fabricated microstrip antenna with four varactors loading at each radiating edge... 57

Figure 4.10 Measured S-parameter of antenna ... 58

Figure 4.11 Setup for antenna radiation pattern and gain measurement... 59

Figure 4.12 Measured (a) E-plane and (b) H-plane of antenna radiation pattern ... 60

Figure 5.1 Conventional high impedance surface using corrugated metal slab whose

grooves are λ/4 from the bottom ground ... 63

Figure 5.2 Lumped-element based high impedance surface using vias to create inductance (a) top view (b) side view ... 64

Figure 5.3 (a) Origin of the capacitance and inductance in the high impedance surface (b) the equivalent circuit of the high impedance surface... 65

Figure 5.4 EM wave incident upon and reflected from a grounded dielectric substrate .... 66

Figure 5.5 Comparison of quality factors for a 5 nH inductor using grounded dielectric substrate and vias ... 68

Figure 5.6 Required height of via and grounded dielectric substrate for different inductance values ... 68

Figure 5.7 Modified high impedance surface structure using grounded dielectric substrate with no vias inside (a) top view (b) side view ... 69

Figure 5.8 (a) Tunable high impedance surface using varactors at the gap between two pads (b) equivalent circuit of the tunable high impedance surface... 70

Figure 5.9 (a) Equivalent circuit of high impedance surface with parasitics (b) simplified equivalent circuit... 71

Figure 5.10 (a) a shorted lossy transmission line and (b) its equivalent circuit using a same length lossless transmission line and a series impedance ... 72

Figure 5.11 Rc , RL and Rp as a function of Lsub /Cvar... 74

Figure 5.12 Bandwidth definition of high impedance surface... 74

Figure 5.13 Bandwidth of the high impedance surface as a function of Lsub/Cvar... 75

Figure 5.14 Bias network for the tunable high impedance surface... 76

Figure 5.15 Photograph of the high impedance surface ... 78

Figure 5.16 Measurement setup for the high impedance surface ... 78

Figure 5.17 Measured phase of reflection coefficient calibrated at the surface of high impedance structure ... 79

Figure 5.18 Rectangular waveguide phase shifter with tunable high impedance surfaces placed at the two side walls ... 80

Figure 6.2 Perspective and frontal views of the phase shifter with PIN diodes used as switches... 84 Figure 6.3 Phase shift as a function of operating frequency... 85 Figure 6.4 Electric field distribution for the (a) microstrip and (b) waveguide... 86 Figure 6.5 Waveguide impedance as a function of waveguide width a (b=15mil,εr =6

and f =10 GHz)... 87 Figure 6.6 Photograph of the 3-bit phase shifter... 89 Figure 6.7 The equivalent circuit of the PIN diode under (a) forward bias and (b) reverse

bias. ... 89 Figure 6.8 The cutouts near the diode to compensate the diode capacitance under reverse

bias. ... 90 Figure 6.9 (a) Simulated and (b) measured phase shift of the 3-bit phase shifter. ... 91 Figure 6.10 (a) Simulated and (b) measured return and insertion loss of the 3-bit phase

LIST OF TABLES

LIST OF APPENDICES

CHAPTER 1

INTRODUCTION

1.1

Motivations

With the rapid growth of communication systems including satellite, bluetooth, 3G

wireless phone, ultra-wide band (UWB) and optical network, it is desirable to be able to

design frequency agile RF front ends for operation in various frequency bands. Tunable RF

and microwave components are thus in largely demanded due to their frequency agile

characteristics. In addition, high-speed, small-size, and low-operation voltage components

are required in the current and next generation communication systems. These requirements

impose significant challenges on current tunable circuit technologies and illustrate the need

for new materials, technologies and designs. Barium strontium titanate ( (Bax,Sr1−x)TiO3) or

BST, a ferroelectric film whose dielectric constant can be controlled by the application of a

DC electric field, has shown great promise for the design of tunable RF and microwave

circuits such as phase shifters, filters, antennas, adaptive matching circuits, and

voltage-controlled oscillators. Therefore, the main objectives of this research work are to characterize

BST film and capacitors at RF and microwave frequencies and develop novel frequency agile

1.2

Overview of current technologies for continuous tunable circuits

There are currently five existing technologies for the design of continuous tunable

circuits based on mechanical tuning, ferrite, MEMs, semiconductor varactors and

ferroelectric materials. The earliest forms of tunable circuits were all mechanical, for

example, the rotary vane adjustable waveguide phase shifter first proposed by Fox in 1947

[1]. Mechanical circuits are cheap, simple to fabricate and have very low loss. However their

disadvantages include their large size and low tuning speed. In 1957 Reggia and Spencer

reported the first electronically variable ferrite phase shifter [2]. Ferrite can handle large

power levels and has faster switching time (few µs to tens of µs) than mechanical ones.

However, ferrite based circuits also have large size and mass, and need tunable magnetic

fields to operate. In the 1960s, semiconductor diodes were introduced in tunable circuits and

are still the dominant devices for making tunable circuits [3][4]. They are very small (in

µms), very fast (<1µs for pin diode and <1 ns for FET), and have large tunability (3:1~10:11

or 200%~900%). In addition, they can be easily integrated with other circuits for example in

monolithic microwave integrated circuits (MMICs). However, semiconductor based varactors

suffer from the junction noise and have poor power handling capability. Also, they require

reversed bias to keep them capacitive. This is a potential problem of semiconductor varactors

operating under large RF signals because the varactor diodes might be turned on (act as a

resistor) if the signal voltage amplitude is larger than the reverse bias. In early 1990s, MEMs

were started to use for tunable circuits [5]. MEMs have very little loss at RF and microwave

frequencies and can handle higher power levels. However, they have some disadvantages

including low tunability (<1.5:1, or 50%), slow switching speed (2-100 µs), and high bias

voltage (50-100 V). In addition, they require hermetically sealed packaging, which is

expensive and hard to integrate with other circuits. Another alternative approach for making

tunable devices is to employ ferroelectric based varactors. They are fast, low loss at RF and

microwave frequencies, and can handle more power than semiconductor varactors. Their C-V

curve is symmetric with respect to the bias voltage, thus there is no requirement for reverse

bias like semiconductor varactors. They can be used in bulk form so that planar circuits like

coplanar waveguide and microstrip lines can be directly fabricated on them. In addition, they

can be used in parallel plate or interdigital capacitors, which can be integrated in other

circuits. The recent results obtained from ferroelectric varactors indicate their potential for

making tunable RF front ends.

Strontium titanate, SrTiO3 (STO) and BST are two of the most popular ferroelectric films

currently being studied for the design of tunable RF circuits [6]-[16]. Tunable STO

characteristics can be obtained only at low temperatures thus allowing the use of high

temperature superconductors (HTS) to achieve lower loss. However, since STO exhibit very

little tunability at room temperature, they cannot be employed in systems operating at room

temperature. BST films can overcome these difficulties. Depending on the specific

composition, BST can exhibit paraelectric behavior and making it tunable at room

temperature (typically 2:1 or 100% tunabilities are reported). With the recent advances in

BST thin film deposition and processing, low loss tunable capacitors, filters, phase shifters

and antennas requiring lower drive voltages can be fabricated.

The electrical characteristics of BST thin films greatly depend on the BST composition,

the bottom electrode materials, the film thickness, and the process temperature. Researchers

from different groups have studied these properties.

1.3.1 BST composition

The BST properties are varied by the composition of Ti and Ba/Sr ratio. The Ti content is

studied from 51.5% to 55.0% by our group [13]. The result showed that the loss tangent and

tunability decreased when increasing the excess of Ti content. For 55.0% Ti and thickness

≤700 Å, BST thin films reached loss tangent as low as 0.003. The figure of merit, which

combines the information of loss tangent and tunability, showed that BST film with 53.3% of

Ti content gave a good compromise between a low loss tangent and a relatively high

tunability.

The dielectric constant can be tuned by adjusting the Sr content of the base composition.

For compositions with ≥35 mol% Sr, BST is cubic at room temperature and hence exhibits

paraelectric behavior [17]. Studies showed that the BST film has a higher dielectric constant

with a lower Sr content.

1.3.2 BST deposition on various materials

The electrodes deposition in parallel plate capacitor structures are challenging, especially

for the bottom electrode. It must provide good growth surface for BST and be stable at high

growth temperatures in an oxidizing atmosphere yet have high conductivity and be

compatible with substrates. BST thin films have been tried to deposit on different metals

capacitors. However, the BST growth prevents the use of Pt thickness over 0.1-0.2 µm [19],

which is smaller than the skin depth up to 2000 GHz2. This turns out to be the main factor that limits the total quality factor of the BST capacitors. Multi-layer bottom electrode

structure, which includes adhesion and interleaved barrier layers were investigated,

fabricated and measured by our group [19]. We successfully increased the pure Pt electrode

up to 0.5 µm, and thus increased the total quality factor of the BST capacitors.

BST thin films have been also deposited directly on dielectric materials including MgO

[20]-[23], LaAlO3 [21] [22] and sapphire [15]. Silicon is also used as the substrate for BST

[24]. The BST showed tunability with the applied DC voltage and reasonable loss tangent on

these materials. However, due to the lack of one metal layer, capacitors must be in the

interdigital form, which require high bias voltage (200-300V), and have less tunability since

the electrical field goes partially to the air.

1.3.3 BST film thickness

BST characteristics are strongly depended on the film thickness [13]. Our group reported

that the dielectric constant rises with film thickness. We have also observed that the loss

tangent increases with the film thickness up to 4000-5000 Å, and then drops slightly with

further increase in film thickness. However, the tunability decreases with film thickness up to

3000-4000 Å, and then slightly rises with further increase in film thickness (BST thin films

were deposited by a liquid-delivery-source CVD technique at approximately 640°C,

Ba/Sr=70/30) [13].

2 _{Skin depth}

ωµσ

1.3.4 BST process temperature

The substrate temperature during growth has a very strong influence on electrical and

physical properties of BST films [14]. A high growth temperature promotes formation of

more highly crystalline material, while at low temperatures microcrystalline or amorphous

material is formed [14]. The measurement results reported by our group indicates that the

dielectric constant increases with the rise of growth temperature.

1.4

Previous work in RF and microwave circuits using BST

1.4.1 BST tunable capacitors

There are basically two types of capacitors, interdigital and parallel plate. Interdigital

capacitors have smaller tuning range but are easy to fabricate (only have one metal layer) and

capable of high power operation due to their large area and higher bias voltage requirement.

Parallel plate capacitors can achieve maximum tunability at low power levels but have one

more metal layer than the interdigital capacitors. For both types of capcitors, the total quality

(Q) factor is dominated by the thin electrodes of the capacitors

[11][12][15][16][19][25]-[27]. Various approaches for increasing the thickness of the electrodes are being developed.

One of the biggest challenges for depositing thicker electrodes is to find suitable conductor

stacks for the parallel plate capacitor bottom electrodes that will survive the high growth

temperatures of BST, and maintain good adhesion during the subsequent processing. We

have tested several samples using multi-Pt layers with TiAlN or IrO2 as the adhesion or the

interleaved barrier [19]. Results showed that while TiAlN worked well as a sticking layer for

adhesion and an interleaved barrier layer, allowing successful formation of BST capacitors

on bottom electrodes as thick as 2 µm. It was also found that as little as one barrier layer

placed near the top of the Pt structure, it provided adequate protection for the multi-layer

bottom electrode. BST dielectric constants ranging from 150-400 (depending on film

thickness) and tunability of approximately 2:1 (100%) were achieved on these thick bottom

electrodes. The loss tangent of BST film was found to be less than 0.006 (±0.002) between

45 and 200 MHz [19]. Using this process, the total Q factor for a 310 pF parallel plate

capacitor with 1 µm multi-layer bottom electrode and 0.7 µm top electrode (Pt) was 77 at 50

MHz. In addition, the tunability of 2.4:1 (138%) at 5 V was obtained. To the best of our

knowledge, this is the highest quality factor for a 300+ pF tunable capacitor available in the

market.

York’s group has reported Pt on sapphire substrate [15], and Ti/Pt/Au electrode stacks

[16] on glass substrates for interdigital capacitors. They reported an interdigital capacitor of 7

pF with the tunability of 1.75:1 (75%) and Q factor of larger than 20 from 0 to 24 GHz at 90

V [15]. They also reported a 0.15-2 pF parallel plate capacitor with tunability of 2:1 (100%)

and Q factor of 30 at 20 GHz [26]. So far, these are the highest quality factors reported for pF

size BST capacitors at 20 GHz.

1.4.2 BST tunable filters

BST-based low pass and band pass filters were reported by our group [12] in 2001. The

circuits used lumped inductors and tunable BST capacitors forming a 3rd and 5th order 0.5 dB ripple Chebychev prototype low pass filter and a 3th order band pass filter at VHF frequencies. The parallel plate BST capacitors were fabricated on 500 µm thick silicon wafer

technique was used to grow the 3000 Å thick (Ba0.7Sr0.3)TiO3 thin films. The Q factor of the

32 pF BST capacitors was 50 at 160 MHz. For the 3rd _{order low pass filter, the maximum}

measured insertion loss in the pass-band was 0.8 dB and return loss better than 10 dB. The 3

dB cut-off frequency was tuned from 160 MHz to 210 MHz (30% tunability) with 9 V bias.

For the 5th order low pass filter, the tunability reached 40%. The insertion loss of the band pass filter was 7 dB at 0 V and was reduced to 5.1 dB at 10 V. A 45% tunability was

obtained. Note that most of the insertion loss (4.5 dB of 7 dB) was found from the low Q

factor of the inductors.

In the mean time, Paratek Microwave Inc. has commercialized two types of BST-based

band pass filters, including the hybrid microstrip line resonator filter at 2 GHz [28], and the

finline waveguide resonator filters at 22.5 and 38.5 GHz [29]. The first one is a 4-pole

microstrip combline bandpass filter with tunable BST capacitors. The insertion loss at the

pass band was 7.7 dB at 200 V, the center frequency can be tuned from 2.16 to 2.36 GHz

(9.3% tunability). The two-pole finline filter has the insertion loss of 3 dB at 38.5 GHz and a

small tunability of 1% at 200 V. The three-pole finline filter operates at 22.5 GHz, and the

maximum insertion loss was only 2 dB. The tunability of 2.2% is achieved with 300 V bias

voltage. The BST tunable capacitors used in these filters were thick film (8 µm thick BST

film) interdigital capacitors, which required high DC bias voltage. The thick films were

deposited on the 20-mil thick MgO, and the gold was used as the electrodes. There was no

report on the dielectric constant and loss tangent of these BST films.

1.4.3 BST phase shifters

gel technique. A phase shift of 165° was obtained at 2.4 GHz with insertion loss of below 3

dB by using a bias voltage of 250 V. In 1999, Van Keuls reported a thirteen- section Ku-band

coupled microstrip phase shifter [23], which used BST interdigital capacitors as the series

coupling components. A phase shift of 200° was obtained at 14 GHz with insertion loss of

below 4.7 dB by using a bias voltage of 400 V. York has reported several phase shifters

using parallel plate and interdigital BST capacitors [15][16][26][30]. The BST capacitors

were used in a as the periodically loaded CPW line. The best performance for the phase shift

of 240° was obtained at 10 GHz with insertion loss of below 3 dB at a bias voltage of 17.5 V

[31].

1.4.4 BST tunable antennas

A microstrip antenna consisting multi-dielectric layer structure with thin BST

sandwiched between two dielectric slabs was investigated in [32]. In this paper, the BST has

the dielectric constant varied from 1800 to 9000, which caused the null of H-plane radiation

pattern to move from 16º to 85º. The author did not report the change of the resonant

frequency.

1.5

Research objectives and dissertation organization

The research on BST has been going on for more than a decade. However, there are still

many challenges to overcome. For example, the properties of BST thin film alone (the

material properties excluding electrode) in the capacitor form have not been accurately

obtained at microwave frequencies, and the electrical means of improving the total Q have

not been implemented. Furthermore, the designs of microwave tunable circuits using BST

capacitor layout to obtain best Q to accurately characterize BST film at RF and microwave

frequencies and to develop novel frequency agile circuits using BST. This work includes the

following five areas:

1.5.1 Optimization of BST capacitor layout to achieve high Q factor

In order to obtain high Q BST capacitors, material scientists have put much effort to

increase the electrode thickness, which is still the most critical and challenging task in the

fabrication of high Q BST capacitors. Besides from improvement in electrode properties, the

quality factor can also be improved by choosing the optimum layout for the capacitors, which

can substantially affect the Q factor. One of the goals in this work is to study the current

distribution in parallel plate capacitors, and to propose suitable layout for achieving highest

possible Q.

1.5.2 Characterization of BST film at RF and microwave frequencies

Accurate and fast characterization of BST thin film can help material scientists to

optimize the film growth condition and also provide important information about key devices

in a tunable circuit. However, due to the high loss of the electrodes, it is very hard to

accurately extract the BST characteristics. The goal in this work is to develop a new and

simple measurement technique to accurately determine the dielectric constant and loss

tangent of BST thin film at RF and microwave frequencies.

1.5.3 Tunable integrated filters

Electronically tunable filters with fast tuning speed and narrow bandwidth are required

microstrip line resonators are investigated. Filters with low loss, high tunability, and small

size are designed. In addition, a common drawback of tunable filters is their bandwidth

change with frequency, which is normally not desired. In this work, new filter topology is

developed to achieve constant bandwidth.

1.5.4 Tunable antennas

Microstrip antennas are widely used in wireless communication systems because they are

lightweight, compact, conformable to planar and non-planar surfaces, simple and inexpensive

to manufacture using modern printed-circuit technology. However, the main disadvantage of

microstrip antennas is their narrow operation bandwidth. In situations when there is no need

for high instantaneous bandwidth, like frequency hopping in cell phone systems, one can

improve the operating frequency range of antennas by making them tunable. Microstrip

antennas with multiple-varactor loading at the radiating edges are designed, fabricated and

measured. This type of tuning can achieve more tunability than using multi-dielectric layers

with BST in between. In addition, constant radiation patterns and improved 2-3 dB gain are

achieved by using multiple varactors.

1.5.5 Tunable high impedance surfaces

A high impedance surface has properties that are opposite to normal metal surface which

has low surface impedance. It does not allow ac current to flow on the surface and thus does

not support surface waves. In addition, the image currents due to high impedance surface are

not phase reversed, therefore they can be used as antenna reflectors without affecting the

radiation efficiency. The surface can be described using photonic band gap (PBG) concepts

developed, which uses the basic non-tunable design in [34] with further modification to

improve loss and simplify the fabrication.

In the dissertation, chapter 2 reports the characterization of BST film which includes the

quality factor improvement and measurement techniques to obtain loss tangent. Tunable

filters, antennas and high impedance surfaces are described in chapters 3, 4 and 5

respectively. In chapter 6, a new phase shifter design is presented. The dissertation is

CHAPTER 2

CHARACTERIZATION OF BST

CAPACITORS

2.1

Introduction

One of the most important parameters for a capacitor, which indicates the loss is its

quality factor defined as

CR

Q=1/ω (2.1) where, ω is the angular frequency, C is the capacitance and R is the equivalent series resistance. R represents the total loss in a capacitor, which includes metal and dielectric

losses. In this chapter, major loss sources in a parallel plate form are examined and various

capacitor layouts are simulated and optimized to achieve the best possible Q. In addition, it is

important to be able to determine the loss tangent of the dielectric material (here BST thin

film) in a capacitor in order to optimize the material growth process. A quick and accurate

measurement technique is developed to extract the loss tangent of BST film in a parallel plate

2.2

Quality factors of BST capacitors

The total quality factor of the BST capacitors is determined by two different loss

mechanisms, the conductor loss due to the electrodes and the dielectric loss from the BST

thin film. It can be expressed as:

d m

t Q Q

Q

1 1

1 _{= +}

(2.2)

where, Q_t is the total quality factor,

m m

CR Q

ω

1

= is the conductor (electrode) quality factor

(Rm: equivalent resistance of the electrodes), and

δ

tan 1 = d

Q is the dielectric (BST) quality

factor. Since the electrode thickness (normally 0.1~0.2 µm) is much less than the skin depth

(up to 100 GHz), as shown in Figure 2.1, Rm remains almost constant (same as its DC value)

at microwave frequencies.

Figure 2.1 Skin depth of the thin film platinum

(σ=3e+6 S/m)

205.57

65.01

20.56

6.50 _{2.06 0.65} 0.00

50.00 100.00 150.00 200.00 250.00

1 10 100 1000 10000 100000

Frequency (MHz)

S

ki

n

de

pt

h

(u

m

Figure 2.2 Conductor (Qm), dielectric (Qd) and total quality factors (Qt) as a function of frequency

(C=5 pF, tanδ=0.005)

Figure 2.2 shows the contribution of the conductor and dielectric quality factors to the

total Q for a 5 pF capacitor. It can be seen that the conductor Q dominates at high frequencies

(>1 GHz). Therefore, focuses must be on Rm in order to increase the total Q of BST

capacitors. To reduce Rm, the most obvious way is to increase the electrode thickness, d, as

Rm is proportional to 1/d. However, due to the difficulties associated with increasing the

electrode thickness [19], other solutions that do not rely on thick electrodes need to be

investigated. Among these is the effect of capacitor layout on Rm. The capacitor layout is

investigated on two different implemented capacitors, integrated and discrete capacitors as

shown in Figure 2.3. Both types of capacitors can be used in tunable circuits. Integrated

capacitors can be fabricated together with other parts of circuit while discrete capacitors are

fabricated on a separate wafer and must be diced before wire bonded. In integrated

capacitors, there are possible BST cracks along the edges of the bottom electrode as shown in 0.10

1.00 10.00 100.00 1000.00 10000.00 100000.00

10 100 1000 10000 100000

Frequency (MHz)

Q

u

al

it

y f

act

o

r

Figure 2.3 (a), which might cause the capacitor failure. On the other hand, discrete capacitors

are easier to fabricate and are more reliable. Integrated capacitors can be much smaller than

discrete capacitors (usually larger than 7 pF based on BST thickness=700 Å, εr=200,

size=17×17 µm2_{), which may not be suitable for RF/microwave applications, but can be used}

for IF (inter-medium frequency, 50-200 MHz) applications. In the following section, the

current distributions of both types of capacitors are studied.

Figure 2.3 (a) Integrated and (b) discrete BST capacitors (a)

(b)

Top electrode

BST Bottom electrode

Interconnecting line

Wirebonds

Possible BST cracks here

2.2.1 Integrated BST capacitors

Figure 2.4 shows the current distribution of the integrated BST capacitor. The total RF

current flowing into the capacitor consists of the conduction and displacement current

(Figure 2.4a). Conduction current flows on the surface of the electrode and displacement

current flows into the dielectric with direction vertical to that of the conduction current. As

conduction current flows, part of it is converted to displacement current. When displacement

current reaches the bottom electrode, it changes back to conduction current and flows along

the bottom electrode. Therefore, the density of conduction current along the capacitor

electrodes varies. On the top electrode, the conduction current density is maximum at the

“RF IN” terminal, and decreases to zero (or very small value if fringe fields are not

neglected) at the end of top electrode. While on the bottom electrode, it is zero at the

beginning and increases to its maximum at the “RF OUT” terminal. In addition, as the

dimension of the capacitor is much smaller than the operation wavelength (<0.01λ up to 100

GHz) the voltage along the electrode can be regarded as a constant. Therefore electrical field

density is uniform in the dielectric, which implies that the displacement current density is

also uniform since

E j

J_d = ωεr (2.3)

where Jd is the displacement current density, ε is the permittivity and Er is the electric

field in the dielectric. The capacitor is simulated using a commercial 3-D EM simulator

Figure 2.4b shows an electrode having the same dimensions as the capacitor electrode

(equivalent electrode). The conduction current on the equivalent electrode flows uniformly

since there is no displacement current involved.

Figure 2.4 The current distributions of (a) the capacitor and (b) the equivalent electrode

Figure 2.5 Simulated current distributions of BST capacitor

Conduction current

Displacement current

L

W

RF OUT RF IN

(b)

Unit cells

(a)

Conduction current

1 2 n-1 n RF IN

RF OUT Top electrode

Bottom electrode BST Displacement current

Thus the resistance of the capacitor electrode is different from a same shape equivalent

electrode. To find the relation of the resistance and inductance between these two, a

distributed model as shown in Figure 2.6 (a) is used. The one-stage lumped-element

equivalent circuit is shown in Figure 2.6 (b). As indicated in Figure 2.4 (a), the capacitor is

divided into n small cells along the electrode.

Figure 2.6 (a) the distributed and (b) one-stage equivalent circuit model of the capacitor

The equivalent circuit of each cell consists of a small series resistance (∆R) and inductance (∆L) for the top and bottom electrodes, and a small shunt capacitance (∆Cbst) and

resistance (∆Rbst), which are given by:

n R R₌ eq_elec

∆ (2.4)

n L L₌ eq_elec

∆ (2.5) (b)

Lm

Rbst

Cbst Rm

(a)

RF IN _{∆R ∆L}

∆R ∆L

∆Cbst ∆Rbst

∆R ∆L

∆R ∆L

∆Cbst ∆Rbst ∆Cbst _{RF OUT}

1

+ = ∆

n C

C bst

bst (2.6)

bst

bst n R

R =( +1)

∆ (2.7)

where, Req_elec and Leq_elec represent the total resistance and inductance of the equivalent

electrode (Figure 2.4 (b)), respectively. This distributed circuit model should be equal to the

one-stage equivalent circuit model of the capacitor shown in Figure 2.6 (a) and (b). The two

circuits are simulated in a circuit simulator (Agilent ADSTM ). The equivalent series resistance and inductance are:

Rm=2/3×Req_elec (2.8)

Lm=2/3×Leq_elec (2.9)

Lm is different from results obtained by Pucel [35].

To separate the top and bottom electrode resistance, Rm can be expressed as:

bot top

m R R

R = + (2.10)

where, Rtop and Rbot represent the equivalent resistance for the top and bottom electrodes

respectively.

Based on equation (2.8), Rtop and Rbot can be expressed as:

ρ

× × =

W L

Rtop 1

2 3 1

(2.11)

ρ ρ+ × ×

× =

W L W

L

R_bot 2 1 2

3 1

(2.12)

where, ρ is the DC sheet resistance of the conductor, L1 is the length of the top and bottom electrodes, L2 is the connecting line length at the bottom electrode, and W is the width of

Rm’s as a function of L1, L2 and W are calculated based on the above analysis. To verify

its validity, the same structure is also simulated using a two and half dimensional commercial

EM solver (HP-MomemtumTM). The simulation results agree very well with the calculation, as shown in Figure 2.7. According to the current distribution on the integrated BST

capacitors, lower Rm is obtained by shortening L1 and L2 and widening W.

Figure 2.7 Normalized R’s at different capacitor dimensions.

In addition, it is found that the size of the bottom electrode has an impact on Rbot.From

Figure 2.3(a), it can be seen that the width of the bottom electrode, Wb1, can be increased

without changing the capacitance. The increase of Wb1 can reduce Rbot because it widens the

current path on the bottom electrodes. Figure 2.8 shows the simulated Rbot at different Wb1. It

can be seen that Rbot drops almost to half of its original value when Wb1 doubles. With further

increase of Wb1, Rbot reduces slightly.

R₀ is R when

L₁=W=50 µm L2=20 µm, D=25 µm

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

L/W

R/R

0

R-calculation R-simulation Rbot-calculation Rbot-simulation Rtop-calculation

Rtop-simulation L1 L2

Figure 2.8 Normalized Rbot at different sizes of bottom electrode.

2.2.2 Discrete BST capacitors

Discrete BST capacitors can be used in circuits where wirebonding is allowed. Figure 2.9

shows the total current through bond wires. The current then spreads on the top electrode,

flows through the BST as displacement current and ultimately flows through the bottom

electrode.

From Figure 2.9, it can be seen that the current distribution on the bottom electrode is the

same as that of the integrated capacitor. This is because that although the current distribution

changes on the top electrode, it can not be seen by the bottom electrode because of the

uniform displacement current in the dielectric. This is verified by the simulation results from

Agilent MomentumTM, as shown in Figure 2.10. It can also be seen from Figure 2.10 that the Rtop is much smaller than Rbot. This is because part of the top electrode is covered by the

wirebond, which effectively increases the thickness of the top electrode, thus reducing Rtop.

0 0.2 0.4 0.6 0.8 1 1.2

1 1.4 1.8 2.2 2.6 3 3.4 3.8

W_b1/W Rbot

/Rbot

0

Wb1

W

Figure 2.9 Current distributions on a discrete capacitor

In discrete capacitors, not only Wb1 but also Wb2 of the bottom electrode can be increased

without changing the capacitance, as shown in Figure 2.11. It can be easily predicted that Wb1

of the bottom electrode has the same effect on Rbot as that of the integrated capacitors, which

is verified by the simulation as shown in Figure 2.11. However, the extended size on Wb2 has

little effect on Rbot.

Top electrode

Bottom electrode BST

Figure 2.10 Figure 2.10 Normalized R’s at different capacitor dimensions

Figure 2.11 Normalized Rbot at different sizes of bottom electrode.

Results for different wirebonding positions and diameters are shown in Figures 2.12 and

2.13 respectively. The conclusion is that wirebonds should be made in the center of the

electrodes and the diameter of the wirebonds needs to be as large as possible to reduce Rtop.

As R dominates the loss in discrete capacitors, the effect of wirebond position and size is

0 0.2 0.4 0.6 0.8 1 1.2

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

L/W

R/R

0

R-simulation Rbot-simulation Rbot-calculation

Rtop-simulation L1 L2 W

D

R₀ =R when

L₁=W=50 µm L2=20 µm, D=25 µm)

0 0.2 0.4 0.6 0.8 1 1.2

1 1.4 1.8 2.2 2.6 3 3.4 3.8

Wb/W

Rbot /Rbot

0

Wb1 and Wb2 increase together

Wb1 increases only _Wb1

Wb2

W

Rbot0 =Rbotwhen L1=Wb1=Wb2=W

Figure 2.12 Normalized R’s at different bonding positions.

Figure 2.13 Normalized R’s at different bonding diameters.

In summary, the design rules for making BST capacitors are:

™ For both types of capacitors, choose the minimum length for L1 and L2, and let W be

determined by the required capacitance.

™ For both types of capacitors, increase bottom electrode width by three times.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.5 0.6 0.7 0.8 0.9 1 1.1

D/D₀

R/R

0

R-simulation Rbot-simulation Rbot-calculation Rtop-simulation

D

R₀=R when

L₁=W=50 µm D0=25 µm

0 0.2 0.4 0.6 0.8 1 1.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8

L/L₁

R/R

0

R-simulation Rbot-simulation Rbot-calculation Rtop-simulation

L1

L

R₀ =R when L=25µm L1=50

™ For discrete capacitors, choose the largest possible wirebond diameter and bond to the

center of the top electrode

2.3

BST thin film characterization

2.3.1 Measurement technique for BST loss tangent and dielectric constant extraction

At microwave frequencies, usually the conductor loss due to the interconnecting lines and

electrodes is a main contributor to the overall loss of thin film BST capacitors [11][26], thus

must be carefully extracted for an accurate determination of BST loss tangent. Furthermore,

the probe contact resistance increases the inaccuracy of loss tangent extraction [35]. A simple

approach is developed here to reduce these effects. In this approach, parallel plate capacitors

as well as two “short” standards are fabricated on the same wafer (Figure 2.14). Reflection

coefficient (one-port S-parameter) measurements are performed on the capacitor and the two

“short” standards individually using a vector network analyzer (VNA).

The input impedance is then obtained using:

11 11 0

1 1

S S Z Z

− +

= (2.13)

where Z0 is the reference impedance (50 Ω) of the VNA, and S11 is the reflection coefficient.

Zc, Zs1 and Zs2 represent the input impedance of the capacitor, short #1 and short #2

respectively. Zs1is used to extract the parasitics of the pads, interconnecting lines and the

discontinuities. By subtracting Zs1 from Zs2, characteristics of an equivalent electrode are

Figure 2.14 (a) the BST capacitor and standards (b) short #1 and (c) short #2 are fabricated on the same wafer

The equivalent circuits of the capacitor and two shorts are shown in Figure 2.15. Their

impedances are given by:

bst bst bst m

s c

C R j

R Z

Z Z

ω

+ + + =

1

1 (2.14)

1 1

1 s s

s R j L

Z = + ω (2.15) 2

2

2 s s

s R j L

Z = + ω (2.16) )

( _{2 1}

3 2

s s m

m

m R j L Z Z

Z = +

ω

W1

W

2L1+ 2L2 +L3

W1

Electrodes BST

W1 L1 L2 L3

W

Pads

W1

W1

W

2L1+L3

Lines

(a)

(b)

where, Rs1, Ls1, Rs2 , and Ls2 are the series resistance and inductance of short #1 and short #2,

respectively. Zm is the impedance of the electrodes in the capacitor. Rbstis the shunt resistance

of thin film BST, which is given by:

δ ω tan 1 bst bst C

R = (2.18)

where Cbst is the capacitance of the two parallel plate capacitors in series, and tanδ is the

loss tangent of BST.

Figure 2.15 Equivalent circuits of the on wafer (a) capacitor, (b) short #1 and (c) short #2

BST loss tangent can be determined by removing the interconnect and electrode loss. The

error due to probe contact resistance is also reduced due to the impedance subtraction in the

loss tangent calculation. By solving equations (2.13)-(2.18) the BST loss tangent and the

device capacitance are obtained:

[

]

[

]

s s s c Z Z Z Z Z Z Z Z − − × + − × − − = ) ( Im ) ( Re tan 1 2 3 2 1 1 2 3 2 1

δ (2.19)

) tan 1 ]( ) ( Im[ 1 2 1 2 3 2 1 δ ω + × − − + = c s s s bst Z Z Z Z

C (2.20)

Rs1 Ls1 Rm Lm

Rbst

Cbst

(a)

(b) (c)

No special BST processing is required to use this measurement technique. The

measurement is performed on the devices that will be used in the actual circuits, like tunable

filters [12].

Figure 2.16 BST capacitance and loss tangent versus frequency

Figure 2.16 shows the capacitance and BST loss tangent obtained for a 0.4 pF parallel

plate capacitor. The capacitance is almost frequency independent up to 10 GHz. Thus the

permittivity of the BST film can be treated as frequency independent up to this frequency. In

addition, the BST loss tangent is also almost frequency independent. The average value of

the loss tangent is approximately 0.006 up to 10 GHz, which proves BST has potential

applications for the design of tunable RF and microwave circuits such as phase shifters,

filters and antennas.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

0 2 4 6 8 10

Frequency (GHz)

Loss tangent

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

2.3.2 Error analysis on BST loss tangent measurement

In order to evaluate the accuracy of the measurement technique, an error analysis is

performed on the measured loss tangent. The goal is to calculate the standard deviation of the

measured loss tangent. It is assumed that the error is due to the uncertainty of VNA only. The

measured S parameter can be expressed as:

11 0 _ 11 11

~ S S

S = + (2.21)

where, S11_0 is the true value of S parameter and S~₁₁ is the measurement error.

It can be assumed that the mean of S~₁₁ is zero. Also the standard deviation of S~₁₁ is the

uncertainty provided by the VNA datasheet [37]. The detailed derivation for the standard

deviation of measured loss tangent is in Appendix A.

A Matlab code is written to calculate the standard deviation of the measured BST loss

tangent in the 0.4 pF capacitor, and the result is shown in Figure 2.17. It can be seen that the

error is frequency dependent, and reaches its minimum when 1/ωC≅50 Ω. This is because that VNA’s impedance is 50 Ω and therefore has the minimum error when DUT is close to

50 Ω. Based on this, depending on the frequency, at which BST loss tangent should be

determined, one need to choose a capacitance value which provides close to 50 Ω reactance.

For example one can use a larger capacitor to get accurate low frequency loss tangent data

and a smaller capacitor for accurate high frequency loss tangent. Figure 2.17 also shows that

from 4.5 to 8 GHz the measurement error for loss tangent error is around 0.003. Therefore

the true value of loss tangent is around 0.006±0.003 from 4.5 to 8 GHz. The discontinuity in

Figure 2.17 Standard deviation of measured loss tangent versus frequency

2.4

Conclusion

In this chapter, the current and field distributions for integrated and discrete capacitors

are investigated. Design rules for choosing the best capacitor geometry are developed to

achieve the best possible quality factor. In addition, a measurement technique is developed to

obtain the capacitance and loss tangent of BST. An error analysis is performed to evaluate

CHAPTER 3

TUNABLE INTEGRATED FILTERS

3.1

Introduction

Electronically tunable filters with fast tuning speed and narrow bandwidth are required

for many applications including electronic surveillance and agile wireless communication

systems. They have been successfully implemented by using varactors [38]-[42], FETs

[44]-[46], MEMs [47]-[49], PZT [50][51], STO [52][53], and BST [28][29][52]-[56]. In this

chapter, tunable microstrip resonators using BST capacitors are used to design tunable

integrated elliptical filters. Design tradeoffs between insertion loss, tunability, filter skirt

sharpness and circuit size are analyzed. A new circuit topology is developed to maintain the

bandwidth during the whole tuning frequency range.

3.2

Capacitively loaded resonators

Figure 3.1 shows the structure of a half wavelength microstrip resonator and its current

l j

Z Z_in

β

tan

0

= (3.1)

where, Z0 and β are the characteristic impedance and propagation constant of the microstrip

line, respectively. Obviously, Zin is capacitive since the resonator length is λ/2. Thus, a

capacitor can be used to replace the two segments of line l at the ends of the resonator. This

reduces the filter size, as shown in Figure 3.2. If the capacitor is tunable like BST capacitors,

the equivalent length of the resonator can be changed, therefore the resonant frequency can

be tuned.

Figure 3.1 Half wavelength resonator and its current and voltage distribution

In general, to accurately model this resonator over a wide bandwidth, a distributed circuit

model is needed. To simplify our task, a lumped element equivalent circuit is used as shown

in Figure 3.3. Although it can not represent the resonator in the wide-band frequency range, it

is valid on a narrow-band basis, namely, near the resonance, which is the frequency of

interest.

The input impedance of the resonator is derived: Voltage

distribution

l l

A

A'

Current distribution

Half wavelength resonator

)] 2 ( 2 [ ) ( 1 2 2 _ s bst s s bst s s c in C C L C j C C L Z + − + − = ω ω ω (3.2)

where, Ls and Cs are the equivalent series inductance and capacitance of the microstrip line,

respectively.

Figure 3.2 (a) Folded half wavelength resonator, and (b) folded half wavelength resonator with reduced size by using a tunable capacitor connecting at the two ends

Figure 3.3 (a) BST capacitor loaded resonator and (b) its equivalent circuit

It can be seen from equation (3.2) that Zin_c has two poles and one zero.

The Zin_c’s zero is given by:

) ( 1 _c z C C L + =

0

_ 1 c = p

ω (3.4)

) 2 ( 1 _ 2 s bst s c p C C L + =

ω (3.5)

For comparison, the input impedance of a half wavelength resonator (without any

capacitor loading) is given by:

] 2 [ 1 2 2 s s s s s in C L C j C L Z ω ω ω − −

= (3.6)

Its zero and poles are:

s s z C L 1 =

ω (3.6)

ωp1 =0 (3.7)

2 1 2 s s p C L =

ω (3.8)

For both resonators, the first poles (ωp1 and ωp1_c) are at zero frequency, which are

normally not interesting to us since they are out of operation frequency band. ωp2 and ωp2_c

are important in filters as they determine the filter pass band frequency. ωz and ωz_c form

transmission zeros in the filter stop band. These transmission zeros can sharpen the filter

skirt. This can be employed in wireless systems that require small order filter with sharp filter

Figure 3.4 Zeros and poles of resonators with and without loaded capacitor

(substrate: εr=6, thickness=15 mil, Microstrip line width=23 mil, Resonator without loaded capacitor: line length=2800 mil, Resonator with loaded capacitor, Cbst=2 pF, line length=1200 mil.)

The poles and zeros for the two resonators are marked in Figure 3.4. The locations of

poles for both resonators are the same for best comparison. Note that the imaginary part of

the impedances close to poles is finite. This is due to the circuit losses including microstrip

conductor and dielectric, as well as radiation and BST capacitor losses.ωp1and ωp1_c are out

of the frequency of interest and are not shown in the figure. It can be seen that for the

resonator with loaded capacitor, the zero is much closer to the pole than that of the half

wavelength resonator. Therefore, it is expected that filters using resonators with loaded

capacitor have much sharper filter response than half wavelength filters. In addition, the

impedance at pole for resonators with loaded capacitor is smaller than that of the half

wavelength resonator. This indicates that the quality factor for the resonator with loaded -600

-400 -200 0 200 400 600 800

0 0.4 0.8 1.2 1.6

Frequency (GHz)

Im

(Z

) (o

h

m

)

without loaded capacitor with loaded capacitor

ωp=ωp_c

ωz

capacitor loss is added in the resonator. Thus filters using this type of resonator will have a

larger insertion loss.

In addition, based on equations (3.3) and (3.5), the distance between zero and pole can be

controlled by the Cbst and Cs. The relative distance (RD) between zero and pole is defined as:

      + − − = − = 1 2 1 1 1 _ 2 _ _ 2 s bst c p c z c p C C RD ω ω ω (3.9)

Figure 3.5 shows a plot of RD as a function of Cbst. Cs is assumed to be constant and it is

determined by the microstrip line. The curve in Figure 3.5 indicates that RD decreases with

the increase in Cbst, which means the filter skirt will be sharper for larger Cbst.

Figure 3.5 Relative distance between zero to pole as a function of Cbst

(Cs=1 pF)

Furthermore, ωp2 and ωz can also be tuned by changing Cbst and lms (length of microstrip

line). Shown in Figure 3.6 are ωp2and ωz as a function of Cbst while keeping lms unchanged. It

can be seen that as C increases, both ω and ω decrease. This means that the length of 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0 1 2 3 4 5 6

microstrip can be reduced by using larger Cbst, which decreases the circuit size. For example,

as can be seen from Figure 3.6, ωp2 is reduced to less than half of its value when Cbst=3 pF in

stead of Cbst=0.2 pF, which is equivalent to a half circuit size reduction.

The tunability of the resonator is defined as:

% 100 2 _ 2 2 _ 2 1 _

2 − _×

= c p c p c p Tunability ω ω ω (3.10)

where ωp2_c1 and ωp2_c1 are the poles of input impedances when the BST capacitance is

tuned at its minimum and maximum values, respectively. The resonator tunability as a

function of Cbst is shown in Figure 3.6. It can be seen that the tunability increases with Cbst3.

Figure 3.6 Zero and pole of input impedance for capacitively loaded resonator as a function of loading capacitance

(line length=1200 mil)

3

For a detailed analytical derivation of tunability as a function of C , please refer to Chapter 4, section

0 5 10 15 20 25 30 35 40

0 0.5 1 1.5 2 2.5 3

It is also desirable to maintain the position of ωp2 as the filter is fixed. To maintain ωp2 in

position, according to equation (3.5), one should keep Ls(Cbst+ Cs/2) constant. For microstrip

lines:

0

L l

L_s = _ms× (3.11)

0

C l

C_s = _ms× (3.12)

where, L0 and C0 are the line inductance and capacitance per unit length, respectively.

They are constants for a specific microstrip line. By substituting (3.11) and (3.12) in to (3.5),

we get:

) 2 (

1

0 0

_ 2

ms bst

ms c

p

l C C l

L +

=

ω (3.12)

Figure 3.7 Imaginary part of input impedance of resonator with the same pole but different capacitance and line length

There are two unknown parameters Cbst and lms in equation (3.12). Figure 3.7 shows the

imaginary part of impedances with the same pole but different lms and Cbst. It can be seen that

with the increase of Cbst, lms decreases, and thus the size of the resonator is reduced. Also, the

-500 -250 0 250 500

0.5 0.7 0.9 1.1 1.3 1.5

Frequency (GHz)

Im

(Z

in

) (

o

h

m

)

zero moves closer to the pole as Cbst increases, which will result in a sharper filter skirt.

However, the advantages are gained at the cost of higher filter insertion loss since the

resonator quality factor decreases with the increase in Cbst.

In summery, tunable filters using a larger Cbst can achieve smaller circuit size, more

tunability, sharper filter skirt but higher insertion loss.

3.3

Tunable band pass filter design using capacitively loaded resonators

Figure 3.8 (a) a conventional two-pole hairpin filter consisting of two coupled half wavelength resonators (b) hairpin filter with two coupled BST capacitor loaded resonators

A band pass filter can be formed using two or more coupled resonators, as shown in

Figure 3.8 (a). The detailed design procedure can be found in [57][58]. The critical part in the

design is to obtain the optimum coupling coefficient between two resonators to achieve

critical coupled condition. The coupling coefficient is defined as [57][58]:

2 2

2 2

m e

m e

f f

f f K

+ − =

m m

m m

C L LC

LC CL

+ +

= (3.13) Cbst

Cbst

<λ/2

(b) Half wavelength resonators

λ/2