tolerance analysis of a multi-mode ceramic resonator658392/fulltext01.pdf · khawar naeem tolerance...

FACULTY OF ENGINEERING AND SUSTAINABLE DEVELOPMENT .

TOLERANCE ANALYSIS OF A MULTI-MODE

CERAMIC RESONATOR

Khawar Naeem

August 2013

Master’s Thesis in Electronics

Master’s Program in Electronics/Telecommunications

Examiner: Jose’ Chilo

Supervisor’s: Piotr Jedrzejewski & Reine Josefsson

Khawar Naeem TOLERANCE ANALYSIS OF A MULTI-MODE CERAMIC RESONATOR

i

Preface

While working on this thesis, I have been lucky enough to receive valuable help from many skilled

personnel. I would like to send my regards and thank each and every one of the many people who

were involved in the completion of this thesis work.

Firstly, I would like to thank my thesis supervisors Piotr Jedrzejewski, Patrik Lindell & Anders

Jansson for their invaluable assistance, encouragement, patience and guidance throughout the thesis

period. I would also like to extend my regards to my Manager Reine Josefsson for his extensive

cooperation and support during my time at Ericsson. They are unquestionably the best mentors a thesis

worker could ask for. I would also like to thank Qian Li for his friendship and support during my

research.

I would also like to thank the entire staff at Ericsson Kista for their much needed assistance during the

thesis period. In particular I would like to thank Hamid Jahja for his technical advice on ceramics and

all other members of the Filter Design team.


ii

Abstract

This thesis describes the design, test and measurement of a dual-mode ceramic cavity resonator for

cellular communications. Dual mode resonators present a challenging and exciting topic in the current

microwave filter industry. Dual mode dielectric resonators are identified by their characteristic

property of two resonances which can be used to construct RF filters capable of dual mode operation.

The motivation for this project has been derived from the novelty of multi-mode operation of a

resonator giving rise to new line of filters capable to deliver high performance, cost efficiency and

small size options to generate quality wireless communication systems. Multi-Mode Ceramic

Resonators provide an exciting solution to meet these challenging requirements.

Initially, the thesis discusses and quantifies the various developments in the modern microwave

industry on this topic by an extensive literature review.

The main thesis objective was to develop a routine to characterize a dual mode ceramic resonator

inside a test cavity with coupling bandwidth. Moreover, the resonator had to go through detailed

tolerance analysis so that its behavior can be specified in the form of a technical document.

Having this objective, various techniques and methods to develop the routine for dual mode ceramic

resonator inside a test cavity are discussed and evaluated in detail. The desired requirements were

achieved by simulations in different software environments according to the need. The 3-D finite

element method (FEM) electromagnetic simulation tool, HFSS from Ansys, was employed to

determine the ideal geometry of the resonator inside the test cavity.

Agilent’s Advanced Design System (ADS) environment was used to create equivalent circuit models

for both the test cavity and the resonator for various cases to establish correlation between geometry

and circuit parameters and to be able to characterize the dual-mode resonator inside a test cavity

along-with mutual and input-output couplings.

MATLAB by Mathworks was used to develop a measurement routine to define and simplify the dual

mode resonator measurement procedure.


iii

Abbreviations

ADS – Advanced Design System

DR – Dielectric Resonator

HFSS – High Frequency Structure Simulator

RF – Radio Frequency

SQP – Sequential Quadratic Programming

TE – Transverse Electric

TM – Transverse Magnetic


iv

Table of contents

Preface ...................................................................................................................................................... i

Abstract ................................................................................................................................................... ii

Abbreviations ......................................................................................................................................... iii

Table of contents .................................................................................................................................... iv

1 Introduction ..................................................................................................................................... 1

1.1 Technical Motivation .............................................................................................................. 1

1.2 Objective ................................................................................................................................. 2

1.3 Outline ..................................................................................................................................... 2

2 Theory ............................................................................................................................................. 3

2.1 Dielectric Resonators .............................................................................................................. 3

2.1.1 Resonators ....................................................................................................................... 3

2.1.2 Modes & Field Patterns of a Dielectric Resonator .......................................................... 3

2.1.3 Dielectric Permittivity ..................................................................................................... 5

2.1.4 Quality Factor .................................................................................................................. 6

2.1.5 Resonant Frequency ........................................................................................................ 7

2.1.6 Tuning ............................................................................................................................. 7

2.1.7 Coupling .......................................................................................................................... 8

2.1.8 Metallic Cavity ................................................................................................................ 8

2.1.9 Dielectric Resonators ...................................................................................................... 8

2.1.10 Multi-Mode Dielectric Resonators .................................................................................. 9

2.2 Filters & Their Types ............................................................................................................ 10

2.3 Two Port Network Theory .................................................................................................... 11

2.3.1 Y & Z-Parameters ......................................................................................................... 11

2.3.2 ABCD-Parameters ......................................................................................................... 12

2.3.3 S-Parameters .................................................................................................................. 13

3 Process & Results .......................................................................................................................... 15

3.1 Results ................................................................................................................................... 16


v

3.1.1 HFSS Simulations of TM Mode Dielectric Resonator .................................................. 16

3.1.2 Empty Cavity ................................................................................................................. 16

3.1.3 Single Mode DR ............................................................................................................ 17

3.1.4 Dual Mode DR .............................................................................................................. 20

3.2 Dual-Mode DR with diagonal cut ......................................................................................... 23

3.3 Circuit-Geometry Correlation ............................................................................................... 27

3.3.1 Coupling Probes ............................................................................................................ 27

3.3.2 Single Mode DR ............................................................................................................ 27

3.3.3 Dual Mode DR with Diagonal Cut ................................................................................ 29

3.4 Test Cavity ............................................................................................................................ 33

3.4.1 Reference Resonator ...................................................................................................... 33

3.4.2 Circuit Model & Correlated Response .......................................................................... 33

3.4.3 Tolerance Analysis ........................................................................................................ 35

3.4.4 Test Cavity with Reference Resonator & Losses .......................................................... 43

3.4.5 Tuning Screw Simulation & Quality Factor .................................................................. 44

4 MATLAB Routine ........................................................................................................................ 45

4.1 Nominal Resonator ................................................................................................................ 45

4.2 Test Cavity without losses ..................................................................................................... 46

4.3 Test Cavity with Losses ........................................................................................................ 49

4.4 Nominal Resonator Tolerance Analysis with MATLAB Routine ........................................ 51

4.4.1 Case: Width ................................................................................................................... 51

4.4.2 Case: Thickness ............................................................................................................. 52

4.4.3 Case: Cut Length ........................................................................................................... 53

4.4.4 Case: Cut Width ............................................................................................................ 54

4.4.5 Case: Geometry Variation ............................................................................................. 55

4.4.6 Comparison between HFSS, ADS & MATLAB Routines ............................................ 56

4.4.7 How to use MATLAB Routine ..................................................................................... 57

5 Discussions/Conclusions ............................................................................................................... 59

5.1 Suggested Future Work ......................................................................................................... 60


vi

References ............................................................................................................................................. 61

Appendix A .......................................................................................................................................... A1

Appendix B ........................................................................................................................................... B1

Appendix C ........................................................................................................................................... C1


1

1 Introduction

1.1 Technical Motivation

Dual Mode Dielectric Resonator filters were introduced in 1982 by Fedziuszko and Chapman [1] for

satellite communication systems but have currently been gaining increasing interest of the modern

microwave industry. This gives way to many design and development opportunities to be explored in

this field. The inspiration behind this thesis work has, therefore, been derived from the exciting new

development of using these dual mode resonators as components of new and efficient modern

telecommunication systems.

Ceramic filters offer smaller size compared to air cavity filters and better RF performance when the

size is kept constant. Compared to single mode filters, dual mode filters are considerably smaller in

size. Other key parameters depend on the mode type used in the dielectric resonators. For hybrid

modes, Q-factor is high while TM mode offers moderate Q values.

The idea for this project has been initiated by Ericsson, a global leader in provision of

telecommunications equipment and services to mobile and fixed network operators. A key area of

research at the Ericsson Filter Design group is to design RF filters for cellular base stations. The filters

in these base stations are set to select the bands for transmission and reception to deal with

interference issues. The quality and efficiency of the communication system is related to the

performance of these RF filters. Commercial focus in this research domain indicates that the

development of dual mode ceramic resonators is a challenging and significant problem to the cellular

filter design industry.


2

1.2 Objective

The ultimate aim of the project is to develop a routine to characterize and specify a dual mode ceramic

resonator inside a test cavity. Certain outcomes were set to achieve at the end of the project work. The

results generated during the project duration will act as a starting point for further innovation and

development. The outcomes to be derived from the project work are:

1. Develop a routine for characterization of a dual mode ceramic resonator inside a test cavity.

The focus of this project was to generate a method to identify the electrical parameters of a dual mode

ceramic resonator inside a test cavity.

1.3 Outline

This thesis describes the specification of a dual mode resonator inside a test cavity by developing a

robust and efficient routine for test and measurement.

Following the introduction of the thesis work in Chapter 1, detailed theoretical background of dual

mode dielectric resonator is given in Chapter 2 . It explains the dielectric resonator theory, modes

inside a resonator, resonator parameters as well as filter and two-port network theory.

Chapter 3 describes the detailed solution process developed to achieve the target as well as will

discuss the complete research results in detail. This include results for resonators such as single mode,

dual mode and dual mode with diagonal cut as well as detailed tolerance analysis or each case. Also

the test cavity development and tolerance analysis is also presented.

Chapter 4 discusses the MATLAB routine development in extensive detail. It starts with the

development of routine and then follows up with the testing of routine for various test cases developed

during the tolerance analysis process. At the end, the results are presented and also how to use the

MATLAB routine is also discussed.

Chapter 5 is the final chapter and deals with the conclusions drawn from the thesis work as well as the

future work that is possible to take the project further.


3

2 Theory

In this chapter, detailed theoretical background of dual mode dielectric resonator is presented. The

chapter starts with the basic resonator theory and modes inside a dielectric resonator. Later, all the

different resonator parameters are discussed in detail. This is followed by the theory involved with the

dual mode and other multi-mode resonators (triple, quadruple etc.). At the end, basic filter theory and

two port network theory is discussed in detail.

2.1 Dielectric Resonators

2.1.1 Resonators

A resonator is a circuit that resonates at its resonant frequency. Resonator circuits are usually made of

lumped or distributed elements. Resonator can be built using lumped elements that are useful in case

of lower frequencies. On the contrary, for higher frequencies lumped elements tend to be so small that

it is difficult to fabricate them. In order to resonate, the distributed realization is used at microwave

frequencies. Lumped element resonators are quite different since they have different elements to store

energy. There occurs an exchange of energy that happens every quarter cycle. In case of higher

frequencies losses occur, and if the loss is considered zero, then the exchange of energy is expected to

continue for infinite time. Distributed circuits are very similar to this one since same resonance

phenomena occurs here too, but it’s different from the previous one in this way that it uses same

region to store electric and magnetic energy. Distributed resonators are comparable to wavelength of

transmitted wave since they put the concept of forward and reverse travelling into practice [2]. Due to

high permittivity, materials like dielectric resonators confine the wave energy at their resonant

frequency.

2.1.2 Modes & Field Patterns of a Dielectric Resonator

Depending on the shape of the resonator different field patterns and modes can be found in dielectric

resonator. Despite of all those available pucks, the most commonly used is the cylindrical one

operating in the TE01 mode [1]. In 1968, Cohn initially described the TE01 mode as a simple case.

Cohn assumed the field as a perfect magnetic conductor covering the surfaces except the end caps.

The flaw in the structure allowed energy to leak out of the surface and the problem got reduced to

more of a waveguide problem [3]. The structure is illustrated below:


4

Fig. 1. Cylindrical resonator structure described by Cohn[1].

The mathematical analysis of dielectric resonators is not trivial and for this reason is explained in

references [4]. The theory states that there are 3 basic modes for dielectric waveguides. Those are

Transverse Electric, Transverse Magnetic and the Hybrid modes. In TE and TM modes electric and

magnetic field goes missing in the axial direction respectively. But for hybrid modes, both fields are

present and can propagate in all directions.

Most applications use and for single and dual mode operation respectively. When placed

within conducting boundaries, are the two lowest resonant modes inside a DR. Suitable

modes are chosen that best fit the respective application while designing. Field distribution is

illustrated in the following figure:

Fig. 2. Field patterns for (a) and (b) [5].

The dielectric resonator’s field dimensions and external boundary conditions determine if the mode

has the lowest resonant frequency [6].

Like many other 3D structures the DR has several modes of resonance. To be precise, there are infinite

modes satisfying all boundary conditions resonating inside a dielectric resonator. There have been

numerous conventions specifically geared towards naming those modes inside the DR [7], or [4]. But


5

the one proposed by Zaki et al., [8] is the one that is most widely used. According to Zaki et al., modes

can be named as , , , , . Here first two letters are indicative of

their origin. Those two letters indicate whether the aforementioned modes are of TE, TM or the

Hybrid variety. Whether the symmetry plane is an electric or magnetic wall can be known with the

third letter (E or H). That means if the E-field for the mode is tangential or perpendicular to the plane.

The subscript “n” indicates towards the order of the φ variation of the field and the second subscript

‘m’ indicates towards the resonant frequency. In the lowest resonant for particular mode ‘m’ is set at

zero. In this mode of designation, it is worth noting that for all TE and TM modes ‘n’ is set at “0”.

Radial and axial field variations are not indicated by this convention; rather it concentrates on the

resonant frequency of the modes [8].

The geometry and the size of the resonator play an important in determining the existence of various

modes in different resonance frequencies. Mode charts help to design DR cavities perfectly and

provide a better understanding of how modes behave inside a cavity. Useful references are Rebsch [9],

Kobayashi [10], and Zaki et al. [8] and [11].

2.1.3 Dielectric Permittivity

Microwave ceramic filters are made on the basis of dielectric resonators. The capability of the

dielectric resonator to store both electric and magnetic energy at resonance frequency is determined by

the permittivity of the resonator. In order to achieve higher Q at resonant frequency, materials with

high permittivity are used. Moreover, speed of the electromagnetic wave is dependent on the

permittivity of the dielectric material. The speed of the electromagnetic wave passing through the DR

is inversely proportional to the permittivity of the material [12]. Even though resonant frequency of

DR can be decreased with an increased permittivity, it also affects the bandwidth and reliability of the

DR. The following equation is a relation between the magnetic wave passing through the material and

its permittivity:

√ (1)

Where = Permittivity of the dielectric

= Free space wavelength

= Dielectric wavelength [13]


6

2.1.4 Quality Factor

The power loss of a microwave system is illustrated by its quality factor. The selectivity of microwave

filter, bandwidth and the pass band insertion loss are determined by the quality factor. Even though the

Q-factor is inversely proportional to bandwidth, it is also known that the loss of the microwave system

is inversely proportional to the Q-factor. It can be defined as:

(2)

Four types of losses have been identified that usually occur in the dielectric resonators. Those are

dielectric relaxation loss, radiation loss, conduction loss and external loss. It should be kept in mind

while coupling energy to the DR that higher dielectric constant would result in a closer coupling of the

resonator. Otherwise external loss could occur due to defect in external coupling. Two types of Q-

factor have been identified till date namely the loaded and unloaded Q-factor. Internal losses of the

resonator can be accounted as the unloaded Q-factor, whereas the external losses correspond to the

loaded Q-factor. Using the following equation, unloaded Q-factor can be calculated:

(3)

Where = Unloaded Quality Factor

= Quality Factor related to Dielectric Losses

= Quality Factor related to Conduction Losses

(4)

(5)

Where = Angular Resonant Frequency

= Total Stored Dielectric Energy in the Resonator at Resonance

= Power Dissipation related to Conduction Losses

= Power Dissipation related to dielectric Losses

In case of external losses, conduction and radiation losses can be ignored. Unloaded Q can then be

calculated as-


7

(6)

Where = Loss tangent of the resonator [14].

2.1.5 Resonant Frequency

Resonant frequency of a dielectric resonator is one where the stored magnetic and electric energies are

equal to each other. Minimum pass band insertion loss occurs at the instant when electromagnetic

energy is transferred to the load at resonant frequency. Both the resonant frequencies and spurious

modes of a dielectric resonator are a key factor when designing a DR filter. In order to design a DR

filter, fundamental and spurious mode resonant frequency has to be calculated beforehand [15].

The resonant frequencies of a dielectric resonator can be approximated by previously developed

models like Cohn’s and Itoh and Rudokah’s models [4]. All these models are subjected to limitations

in calculation accuracy of the resonant frequency. This presents the need to develop more robust and

accurate methods for the design of DR cavities and DR filters that are capable of including the effects

of surrounding environment of a DR also. For this reason, many iterative techniques have been

formulated to evaluate the exact solution by numerous approximations converging to the exact

solution. These techniques enable us to calculate the field strengths and resonant frequencies to a

certain accuracy level. Such techniques include mode matching method, finite element method and the

method of moments [16].

2.1.6 Tuning

In case of a dielectric resonator, operating frequency is dependent on resonator material permittivity

and its shape. One can tune the center frequency of the dielectric resonator in various ways. By design

or by putting an extra disk on top of the DR, one can fine tune the center frequency. Up to 15% change

can be achieved just by putting an extra tuning disk [17]. Tuning screws are used to mount the tuning

disk in the cavity. Frequency is affected in opposite direction when metallic disks are used instead of

the dielectric tuning disks. Dielectric tuning disks faces a reverse action when the DR fast approaches

the metallic disk resonant frequency. But additional tuning disks can have a negative impact on the Q

factor of the resonator, as it downgrades the Q value. Still, acceptable frequency response can be

achieved with this method. To overcome this issue, dielectric or metallic disks are mounted on screws

[3]. Certain frequencies can be obtained by turning the screw inwards or outwards. Mode of the

resonator and the desired tuning amount will define exactly which method to put into practice for this

instance [17].


8

2.1.7 Coupling

Energy is required to be coupled in or out of the resonators, in order for dielectric resonators to work

efficiently. There are several ways to do so such as waveguides transmission lines or electric and

magnetic coupling. The amount of coupling required or the resonator mode are determining factors for

coupling [5]. Magnetic coupling is the chosen solution for a cylindrical dielectric resonators operating

in mode. Apart from that, higher order filters can be achieved by coupling energy among

resonators. Direct coupling among resonators can be achieved by changing the resonator proximity.

Size and distance among resonators are defining factors for resonator coupling [5]. Several factors

such as the mode to be excited and transmission medium; should be taken into consideration while

choosing an appropriate structure to couple energy [17].

Coupling among the resonators is identified through strongest external field. Iris, an aperture within

the common wall of cavities is used to initiate coupling among resonators. Dimensions of iris are

changed frequently in order to attain varying coupling between resonators. Apart from iris, coupling

screws are used too.

2.1.8 Metallic Cavity

Radiation loss from a resonator can be overcome by adding a metallic cavity to the resonator. Cavity

dimension can also affect the response of the filter. Additionally, it can also affect the spurious

performance, temperature stability or the pass band insertion loss. Relationship between the cavity and

the resonator’s operating frequency can be described with the help of following equation:

√ (7)

Where = Volume of Metallic Cavity

= Dielectric Permittivity

A better Q-factor can be achieved by a large cavity size, but then again, it puts a negative impact on

the structure’s resonant frequency. Decreased x- dimension of the resonator cavity results in decreased

Q-factor. As a result, there is a growing insertion loss and decreased stop band rejection [15].

2.1.9 Dielectric Resonators

Due to high permittivity, dielectric materials can confine electromagnetic energy. A dielectric

resonator is defined as a material having high dielectric constant and usually having a cylindrical

shape. Typical shape is called as “puck” and is shown in the figure below. Due to difference in

permittivity of dielectric resonators, they can sustain resonance and energy inside at resonant

frequency [18]. The aim of enclosure is to stop the radiation from getting leaked outside, and the

conduction of enclosure is not dependent on the contact of puck. Resonant frequency is strongly


9

affected by dielectric constant and dimensions of a puck. A common dielectric resonator structure is

shown as under:

Fig.3. Conventional cross-section of a typical dielectric resonator structure [1].

The fields outside the dielectric resonator decay quickly as it moves at the opposite direction of the

puck. Dielectric resonators are mostly consisting of cylindrical pucks and have high Q-factor. The

spurious free response of the resonator can be achieved by introducing a hole inside the puck. The

fundamental TE01 mode of dielectric resonator and other higher order modes can be separated with

the help of this procedure. Adjustable metal plates above the resonator are used to fine tune the

dielectric resonators [19]. Center frequency of a resonator depends on physical dimensions and can be

adjusted by setting proper size. Low loss tangent and good temperature stability are the benchmark of

dielectric resonators [19]. Additionally, resonator modes are quite sensitive towards the diameter of

the dielectric resonator [6]. One more key characteristic of dielectric resonator is the High Q-factor to

volume ratio [17]. Dielectric resonator can also act as magnetic monopole in fundamental where

the E- field circulates in resonator and outward propagation of magnetic field is observed. Spurious

free bandwidth, Q-factor and the resonant frequencies are the most important properties of the

dielectric resonator [1]. But these properties are dependent on several factors such as the materials

used in the dielectric resonator, its shape and the mode used.

2.1.10 Multi-Mode Dielectric Resonators

Microwave resonators can be classified as single or multi-mode resonators. In a single mode resonator,

a singular field distribution is observed at the resonant frequency while in a dual mode resonator, we

have two field distributions at the desired resonant frequency and so on. The main reason for the usage

of multi-mode operation is size reduction as a single physical resonator is loaded with multiple

electrical resonators each having its own mode distribution. The multi-mode operation such as in dual

and triple mode resonators having multiple field distributions at the same frequency is generally due to

the degeneracy of the modes. These different field distributions in degenerate modes are mostly

orthogonal modes of a specific field distribution and arise due to symmetry of the resonator structure.

For this reason, 90-degree radial symmetry is generally used to realize dual mode resonators [20]


10

while triple mode operation is observed in cubic cavities or circular cavities with specific modes, [21]

,[22] and [23]. The following figure shows example of a dual, triple and quadruple-mode filters in

various technologies.

Fig.4. (left) Dual-Mode Micro-strip Filter [24], (middle) Waveguide Triple-Mode Filter [21], (right) Quadruple-

Mode Cavity in Circular waveguide [22]

The idea behind the multi-mode operation of dielectric resonators is the benefit in size reduction vital

for applications such as satellite communication systems but it also has some limitations. Even though

dielectric resonators can come in many shapes and forms, not all of them are easy to manufacture. The

main reason is the high pressure and temperature requirements needed while firing the ceramic.

In this thesis work, we will be focusing mainly on the dual mode resonator with emphasis on rigorous

analysis of the resonator structure in order to characterize the dual mode response.

2.2 Filters & Their Types

Filters are tasked with shaping the amplitude of the transmitted signals in most applications. But, time

delay or the linearity of insertion phase is highly important in some applications [13].

Filters are classified in several groups based on their frequency selectivity. Some of them are listed

below-

1. Low-pass filters: These filters are tasked with overseeing the transmission of low frequencies

from DC to a specific frequency. Here attenuation level is kept to a minimum (Fig. 8.a).

2. High-pass filters: These filters are tasked with transmitting signals above the cutoff frequency.

Anything below the cutoff frequency gets rejected (Fig. 8.b).


11

3. Band-pass filters: These filters only transmit a specific band of frequency. Anything below or

higher gets rejected (Fig. 8.c).

4. Band-stop filters: These filters are tasked with eliminating a specific band of frequency out of

a frequency spectrum (Fig. 8.d).

Fig. 5. Filter Types w.r.t Frequency selectivity [25]

Here, , , , and the are considered as the corner frequency of both low-pass and high-pass

filters, upper corner frequencies of band-pass and band-stop filters and pass band and stop band center

frequency respectively.

2.3 Two Port Network Theory

2.3.1 Y & Z-Parameters

The voltages and currents are directly related to each other by virtue of impedances or admittances, as

depicted in the two port network illustrated in figure below. The voltages can be calculated by Z-

parameters using the following equation, where it is represented in terms of currents.


12

V1= Z11I1 + Z12I2 (8)

V2 = Z21I1 + Z22I2 (9)

Fig. 6. A two-port network

On the other hand, in the form of matrices, it can be best illustrated with the help of following

equation:

[

]= [

] [

] (10)

The Z- matrices come in handy when portraying systems where networks are connected in series

rather than in shunts. Like this, linkage between currents and voltages through Y- parameters can be

described with the help of following equation:

I1 = Y11V1+ Y12V2 (11)

I2 = Y21V1 + Y22V2 (12)

While describing it in matrices, following equation can be used:

[

] = [

] [

] (13)

Unlike the Z- matrices, Y-matrices are of great use when portraying systems where networks are

connected in shunt.

2.3.2 ABCD-Parameters

If the two port networks are connected in cascade then ABCD parameters could possibly have a telling

impact on the systems. In order to calculate the overall ABCD matrix for the whole system, one would

have to calculate the matrix for a single two- port network first, and then multiply the calculated

amount with the total number of two-port networks.


13

The ABCD parameters, while working with a two-port network, can be described using the following

equation:

V1= AV2 + BI2 (14)

I1= CV2 + DI2 (15)

Additionally, in the form of matrices it can illustrated as depicted below-

[

]= [

] [

]

2.3.3 S-Parameters

Even though, Y- and Z- parameters are enough to describe most networks; sometimes it’s hard to

analyze a network using these parameters. Y & Z- parameters are used for low frequency networks

mostly since direct measurement of those parameters at high frequency could be troublesome. Two

factors are held responsible for such behavior:

1. In case of non- TEM transmission lines, it might be quite difficult to define the voltages and

currents at high frequencies.

2. It is of utmost importance to use open and short circuits while formulating those parameters.

But, it may cause instability to the system while working at microwave frequencies, especially

applicable if active elements are involved there.

While defining parameters, it is thought to have a relation with power which much unlike, can be

measured at high frequencies. In case of a two port network, the S- parameters are illustrated with a

and b waves representing the forward and reverse travelling waves respectively as shown in the figure

below.

Fig. 7. A two-port network characterized by S-parameters.

B1= S11a1 + S12a2 (16)


14

B2= S21a1+ S22a2 (17)

In the form of matrices:

[

]= [

] [

] (18)

If b2 in port 1 is calculated as zero, then S11 can be considered as the reflection coefficient. But for this

to happen, a perfectly matched load and the absence of source at the port 2 are pre-requisites. Same

formula is applicable for S22 where the b1 is absent.


15

3 Process & Results

The problem statement given at hand was to develop a routine for characterization of a dual mode

ceramic resonator inside a test cavity. To solve this problem, the solution process was divided into

various steps to achieve the required target. Following steps were included in the solution process:

Optimized EM design in HFSS

Detailed tolerance analysis

Equivalent circuit model in ADS

Test cavity Setup

Establish circuit-geometry correlation

MATLAB routine

The first step was to achieve the optimized EM design model of the dual mode ceramic resonator in

HFSS. The design process was initiated with single mode dielectric resonator and was then progressed

to the dual mode dielectric resonator. In the last part, a perturbation in the form of a diagonal cut was

introduced to achieve the desired coupling bandwidth.

The next step was to perform the tolerance analysis of the resonator models developed in the HFSS

environment. Test cases were setup to model different geometric tolerance scenarios and results were

recorded. The tolerance analysis was performed over the entire design work done during our research

so all the resonator models (single mode, dual mode and dual mode with coupling bandwidth) as well

as the test cavity models ( with reference resonator both lossless and with losses) were subjected to

tolerance analysis to come up with specifications of our resonator designs.

The next step was to develop the equivalent circuit models in ADS that exhibit the same response as

our EM models from HFSS. Coupled resonators circuit design were implemented to emulate the EM

response from HFSS and key circuit parameters such as inductor and capacitor values as well as

coupling coefficients and losses were extracted.

The process carried on the with the design on test cavity and coupling probes as well as a test case

(band-pass filter) to test the resonator in the test cavity. Both lossless and loss bearing simulations

were performed with the reference resonator inside the test cavity and sniffing probes and tolerance

analysis values were recorded. Equivalent circuit models were developed and circuit-geometry

correlation was established.


16

In the last part, MATLAB routine was developed based on the circuit solution of the equivalent circuit

models created hence eliminating the need for the use of circuit simulator and a robust solution was

achieved to specify the dual mode resonator inside test cavity. The routine was thoroughly tested for

different cases of tolerance analysis and conclusions were drawn based on the performance of routine

in different scenarios.

In the next section, all the results during the design work are presented. The results breakdown is

given as under:

Empty cavity analysis

Single mode DR & tolerance analysis

Dual mode DR & tolerance analysis

Dual mode DR with diagonal cut & tolerance analysis

Circuit-geometry correlation (along with band-pass filter test case & coupling probes)

Test cavity with reference resonator (With & Without losses & tolerance analysis)

MATLAB routine

3.1 Results

3.1.1 HFSS Simulations of TM Mode Dielectric Resonator

In order to come up with the optimal dimensions of the dual-mode dielectric resonator, HFSS

simulations were performed. HFSS is a simulation tool that employs the Finite Element Method

(FEM) to obtain a complete three dimensional electromagnetic field inside a respective structure.

Using this electromagnetic field solution, HFSS can then calculate S-parameters. HFSS was employed

in this thesis project to determine the resonant frequencies of the dielectric resonator inside the test

cavity.

3.1.2 Empty Cavity

Simulations were undertaken for a cavity size of 30*30*30 (mm) without the addition of losses and

with the addition of losses to characterize the general behavior of the test cavity. The first two modes

above 1GHz were calculated for both cases. Losses were calculated for the cavity by assigning finite

conductivity of the value (S/m).


17

Fig.8. Empty cavity

The HFSS simulation results from the Eigen-mode solution are given as under:

Cavity

Mode 1

Frequency

(MHz)

Mode 2

Frequency

(MHz)

Mode 3

Frequency

(MHz)

Without Losses 7066.43 7066.69 7066.76

With Losses 7065.98 7066.23 7066.38

Table 1. Eigen-mode data from the empty cavity simulation

The Eigen-mode solution from HFSS with number of modes set to 3 returns us the values shown in the

table above. The little frequency shift observed in the three modes inside the empty cavity is due to

numerical inaccuracy of the simulation software which implies that in reality all these three modes are

resonating at the same frequency but due to error tolerance of HFSS Eigen-mode solver these three

frequencies are few kHz apart.

3.1.3 Single Mode DR

A single Mode Dielectric resonator puck was placed inside the test cavity having dimensions

8.3*8.35*30 mm and dielectric constant = 35 and no losses i.e. tan = 0 to obtain a resonant

frequency of 1960 MHz


18

Fig.9. Single mode DR inside test cavity

The following resonant frequency was obtained with the Eigen mode solution.

Case Mode 1 Frequency (MHz)

Single Mode DR 1960.19

Table 2. Eigen-mode data for single mode DR inside test cavity

3.1.3.1 Electric & Magnetic Field Distribution TM01 Mode

The electric field and the magnetic field distributions inside the single mode DR are given as under:

Fig.10. E-field & H-field for TM01 mode for a single mode DR

We can observe from the above figures that the electric field strength is concentrated inside the

dielectric puck whereas magnetic field is perpendicular to that of the electric field strength for the

single mode resonant frequency. To understand the TM01 mode operation, key references are Rebsch

[9], Kobayashi [10], and Zaki et al. [8] and [11].

3.1.3.2 Tolerance Analysis (Width & Thickness)

The next step was to run the geometry tolerance analysis for the single mode DR to observe its

behavior relative to the change in geometry. Two simulations were setup each for thickness and width

tolerance and its effect on the resonant frequencies was observed.

Fig.11. Tolerance Setup for single mode DR

The following resonant frequencies were obtained during the geometry tolerance analysis:


19

Width

(mm)

Thickness =8.35(mm)

Mode 1 Frequency (MHz)

5 2329.36

6 2182.37

7 2069.80

8 1980.52

9 1907.99

10 1848.28

11 1798.40

Table 3. Tolerance analysis of single mode DR w.r.t width while thickness is kept constant

Thickness

(mm)

Width =8.3(mm)

Mode 1 Frequency (MHz)

5 2333.52

6 2186.50

7 2074.05

8 1984.01

9 1911.65

10 1851.78

11 1801.89

Table 4. Tolerance analysis of single mode DR w.r.t thickness while width is kept constant

We can analyze the above measured tolerance data using the following plot:

Fig.12. Single mode DR tolerance analysis

We can see that due to symmetry in the structure, as width and thickness is varied; the resonant

frequencies follow each other in a same fashion for both the tolerance parameters. This is clearly

visible from the above graph as both the red and blue lines follow each other as the dimensions are

varied.

5 6 7 8 9 10 111700

1800

1900

2000

2100

2200

2300

2400

Dimensions (mm)

Fre

qu

en

cy (

MH

z)

Tolerance Analysis

Thickness

Width

Center Frequency 1960 MHz


20

3.1.4 Dual Mode DR

A cross-shaped dual-mode resonator puck was placed inside the test cavity with dimensions

8*7.92*30 mm having dielectric constant = 35 and no losses i.e. tan = 0 to obtain a resonant

frequency of 1960 MHz.

Fig.13. Dual-mode DR inside test cavity

The Eigen-Mode analysis generated the following resonant frequencies:

Case Mode 1 Frequency

(MHz)

Mode 2 Frequency

(MHz)

Dual- Mode DR 1960.52 1960.55

Table 5. Eigen-mode data for Dual mode DR inside test cavity

3.1.4.1 Electric & Magnetic Field Distribution for two orthogonal TM01 Modes

The electric field and magnetic field distributions inside the dual-mode DR for both the modes are

given as following:

(a) (b)


21

(c) (d)

Fig.14. (a) TM01 Mode1 E-field (b) TM01 Mode1 H-field (c) TM01 Mode 2 E-field (d) TM 01 Mode 2 H-field

The field plots of the two TM01 modes above indicate that we have two orthogonal modes inside the

test cavity and the two resonant frequencies are only few kHz apart which as explained earlier is due

to error tolerance inherent in the Eigen-mode solver of HFSS.

3.1.4.2 Tolerance Analysis

The tolerance analysis is one of the key steps in specification of a resonator. The principle behind

tolerance analysis is to test and check as to how different a resonator can behave if its geometry is

subjected to manufacturing tolerances in terms of width, thickness, dielectric constant and other

geometric tolerances. These tolerances affect the resonating frequency as well as coupling coefficient

and other parameters related to the resonator and hence need to be tested and specified. The geometry

tolerance analysis was also performed on the dual-mode resonator to see the behavior of resonant

frequencies with changing physical dimensions. In the first step, two simulations (test cases) were

performed each for the thickness and width of the dielectric resonator while keeping the nominal value

intact for a given iteration.

Fig. 15. Tolerance Setup for Dual mode DR


22

Table 6. Tolerance analysis of Dual-mode DR w.r.t width while thickness is kept constant

Table 7. Tolerance analysis of Dual-mode DR w.r.t width while thickness is kept constant

The following trend was seen when the above recorded tolerance data was analyzed in the form of

respective dimensions versus frequency plot.

Fig.16. Dual-Mode DR tolerance analysis

5 6 7 8 9 10 111700

1800

1900

2000

2100

2200

2300

2400

Dimensions (mm)

Fre

qu

en

cy (

MH

z)

Dual Mode DR Tolerance Analysis

Thickness

Width

Center Frequency 1960 MHz

Width

(mm)

Thickness =7.92(mm)

Mode 1 Frequency

(MHz)

Mode 2 Frequency

(MHz)

5 2317.55 2317.70

6 2166.30 2166.63

7 2051.44 2051.72

8 1960.52 1960.55

9 1886.92 1887.01

10 1826.48 1826.53

11 1777.73 1777.85

Thickness

(mm)

Width =8(mm)

Mode 1 Frequency

(MHz)

Mode 2 Frequency

(MHz)

5 2301.98 2302.12

6 2156.28 2156.30

7 2042.20 2042.34

8 1954.24 1954.38

9 1882.52 1882.58

10 1823.47 1823.55

11 1774.51 1774.59


23

We can observe from the above plot the tolerance w.r.t both width and thickness. We can see that for

the smaller dimension size such as 5mm, the dual mode frequencies are significantly apart but as we

increase the dimension size, the dual-mode frequencies start to vary in a more synchronous manner for

both the tolerance parameters.

The next step was to check the tolerance against the dielectric constant of the dual-mode resonator.

The case for change in dielectric constant was setup as we have various materials provided by the

manufacturers with different dielectric constants, quality factor and cost. Based on the data collected

during this tolerance analysis, a dielectric material will be chosen for use in the dual mode case. HFSS

simulations were setup for the nominal values of thickness and width at center frequency of 1960

MHz and the following data was recorded.

Dielectric

Constant

Mode 1 Frequency

(MHz)

Mode 2 Frequency

(MHz)

Step Size

/

(MHz)

33 2018.07 2018.17 57.55/57.62

34 1989.66 1989.70 29.14/29.15

35 1960.52 1960.55 0

36 1934.25 1934.46 26.27/26.09

37 1909.09 1909.20 51.43/51.35

Table 8. Tolerance analysis of Dual mode DR w.r.t dielectric constant

3.2 Dual-Mode DR with diagonal cut

During the course of the design work in HFSS, several resonator models were created to intentionally

destroy the symmetry of the resonator and separate the two resonant frequencies and generate coupling

bandwidth between the two modes. Several geometries were designed and simulated but at the end the

single diagonal two-cut variant of the dual-mode DR was finalized. The dimensions of dual-mode DR

were adjusted to 8*8.2*30 mm with 1*1 cut area.

Fig.17. (a) Dual Mode DR with diagonal cut (b) Dual Mode DR with diagonal Cut Setup


24

Cut

Width

(mm)

Cut

Length

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

1 1 1943.83 1960.47 1977.11 0.016

Table 9. Eigen-mode data for Dual mode DR with coupling bandwidth

The electric and magnetic field distribution inside the cut variant of dual-mode DR can be seen as

under:

(a) (b)

(c) (d)

Fig.18. (a) Even Mode E-field (b) Even Mode H-field (c) Odd Mode E-field (d) Odd Mode H-field

We can see from the electric and magnetic field plots above that the symmetry of both the fields

within the test cavity has been destroyed and the two modes are not perfectly orthogonal to each other

as before giving rise to two different resonant frequencies and a coupling bandwidth of 33.28 MHz

was observed for a cut area of 1*1 These modes are termed as even and odd modes and have

been generated due to perturbation of the resonator by the diagonal cut. To read in detail about the

even and odd mode theory, reference Cameron [27] is highly useful.


25

The next step was to perform tolerance analysis for the cut variant of the dual-mode DR. The tolerance

analysis included thickness and width tolerance, dielectric tolerance, cut length, cut width and cut

offset tolerance. Simulations were setup for each case; for the eigen mode analysis of every parameter.

Following are the results for the Eigen-mode simulations for different tolerance parameters:

Table 10. Eigen-mode data for tolerance analysis of Dual mode DR with coupling bandwidth w.r.t thickness

while width is kept constant

Table 11. Eigen-mode data for tolerance analysis of Dual mode DR with coupling bandwidth w.r.t width while

thickness is kept constant

The next step was to check the tolerance against the dielectric constant of the dual-mode resonator.

HFSS simulations were setup for the nominal values of thickness and width at center frequency of

1960 MHz and the following data was recorded.

Thickness

(mm)

Width =8(mm)

Even Mode

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Center

Frequency

(MHz)

8 1959.47 1992.87 0.0169 1976.17 8.1 1951.33 1984.81 0.0170 1968.07 8.2 1943.83 1977.11 0.0169 1960.47 8.3 1936.17 1969.30 0.0169 1952.73 8.4 1928.64 1961.35 0.0168 1944.99

Width

Thickness =8.2(mm)

Even Mode

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Center

Frequency

(MHz)

7.8 1960.04 1989.45 0.0148 1974.74

7.9 1951.83 1983.15 0.0159 1967.49

8 1943.83 1977.11 0.0169 1960.47

8.1 1935.86 1970.91 0.0179 1953.38

8.2 1925.42 1962.63 0.0191 1944.02


26

Table 12. Eigen-mode data for tolerance analysis of Dual mode DR with coupling bandwidth w.r.t dielectric

constant

The next simulation that was carried out was the tolerance of the cut variant of the dual-mode DR with

cut width and cut length as the respective parameters. The following values were recorded:

Table 13. Eigen-mode data for tolerance analysis of Dual mode DR with coupling bandwidth w.r.t cut

width while cut length is kept constant

Table 14. Eigen-mode data for tolerance analysis of Dual mode DR with coupling bandwidth w.r.t cut width

while cut length is kept constant

Cut

Width

(mm)

Cut Length = 1(mm)

Even Mode

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Center

Frequency

(MHz)

0.8 1942.86 1973.47 0.0156 1958.16

0.9 1943.04 1975.26 0.0164 1959.15

1 1943.83 1977.11 0.0169 1960.47

1.1 1944.25 1978.75 0.0175 1961.50

1.2 1944.84 1980.32 0.0180 1962.58

Dielectric

Constant

Even Mode

Frequency

(MHz)

Odd Mode

Frequency

(MHz))

Coupling

Factor

k=[

]

Center

Frequency

(MHz)

Step Size

/

(MHz)

33 1999.48 2033.36 0.0168 2016.42 55.65/56.25

34 1971.08 2004.62 0.0168 1987.57 27.25/27.51

35 1943.83 1977.11 0.0169 1960.47 0

36 1917.71 1950.70 0.0170 1934.20 26.12/26.41

37 1892.59 1925.32 0.0171 1908.95 51.24/51.79

Cut Length

(mm)

Cut Width = 1(mm)

Even Mode

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Center

Frequency

(MHz)

0.8 1943.09 1969.70 0.0136 1956.39

0.9 1943.51 1973.48 0.0153 1958.49

1 1943.83 1977.11 0.0169 1960.47

1.1 1944.20 1980.94 0.0187 1962.57

1.2 1944.34 1984.67 0.020 1964.50


27

All the above data was recorded, the change in frequencies and coupling coefficient was observed for

the different cases and based on these values the manufacturing tolerances will be specified when

placing the order for commercial units of the resonator from ceramic manufacturers.

3.3 Circuit-Geometry Correlation

3.3.1 Coupling Probes

To couple electromagnetic energy to the modes inside the test cavity with a dielectric resonator,

coupling probes are required. For this purpose, two kinds of probes are generally used: electric and

magnetic loops. Different coupling probes were tested during the entire thesis period to couple the

energy efficiently to the two modes inside the test cavity and at the end, magnetic square loop was

finalized, because of its efficiency and ease of manufacturing in future.

Fig. 19. Magnetic Square Loop

3.3.2 Single Mode DR

To characterize the resonator, equivalent circuit models were created to establish the correlation

between circuit and geometry. For this purpose, ADS (Advanced Design System) software was

employed to create circuit models and simulate the circuit models. The following HFSS and circuit

(ADS) model was created for the single mode DR.

Fig. 20. Single mode DR inside test cavity with coupling probes

The lumped element equivalent circuit model for the single mode DR is given as under:


28

Fig. 21. Equivalent Circuit Model for Single mode DR inside test cavity with coupling probes

The responses from both HFSS and ADS environments were recorded as s2p files and were matched

with each other by tuning the circuit model until it gave the exact response as the physical model in

HFSS. Following is the graph of the responses from both HFSS & ADS simulations of single mode

DR plotted together giving the same response.

Fig. 22. Correlation between circuit & physical models for single mode DR.

Below is the key data from both the simulations for various essential parameters:

Table 15. HFSS parameters for single mode DR circuit-geometry correlation

Parameter Value

4r*3 )

Loop Diameter 1mm

35

8.35mm

8.30 mm

Cut Area 1*1( )


29

Table 16. ADS parameters for single mode DR circuit-geometry correlation (from equivalent circuit)

3.3.3 Dual Mode DR with Diagonal Cut

After completing the correlation between circuit and physical model for the single mode DR, the same

method was applied for dual mode DR as well. Coupling probes were used to couple the energy to the

two orthogonal modes inside the test cavity. Since filters are used to define the geometry of a

resonator, as a test case, two band-pass filters with 20 dB return loss specification were developed: one

using the inductive (magnetic) coupling probes and other with the capacitive (electric) coupling

probes.

3.3.3.1 Band-pass Filter Test Case (Inductive Coupled)

Since filters are used to define the geometry of a resonator, an inductive coupled band-pass filter was

developed as a part of characterizing the dual mode resonator inside the test cavity. Magnetic probes

were used to achieve strong coupling between input/output and the two modes inside the cavity and

band-pass response was generated.

Fig. 23. Band-pass filter test case (inductive coupled) for dual mode DR

The equivalent circuit model obtained for the dual mode inductive coupled case is given as under:

Parameter Value

0.0601

0.0601

8nH

0.8288pF


30

Fig. 24. Equivalent circuit model of band-pass filter test case (inductive coupled) for dual mode DR

Responses of both the HFSS and ADS models were recorded in the form of s2p files and plotted

together and the circuit model was tuned to get the exact same response as the physical mode for dual

mode DR. Following figure contains both HFSS & ADS responses plotted together:

Fig. 25. Correlation between circuit & physical models for dual mode DR w.r.t band-pass filter test

case (inductive coupled).

The important parameters of both the HFSS model and ADS model are listed as under:

Table 17. HFSS parameters for dual mode DR circuit-geometry correlation w.r.t band-pass filter test

case (inductive coupled)

Parameter Value

9.5r*9.2 )

Loop Diameter 1mm

35

8.7mm

8.6 mm

Cut Area 1*1( )


31

Table 18. ADS parameters for dual mode DR circuit-geometry correlation w.r.t band-pass filter test

case (inductive coupled)

3.3.3.2 Band-pass Filter Test Case (Capacitive Coupled)

Similar method was used to generate the band-pass filter test case with capacitive (electric) probes.

Electric probes were created to couple energy efficiently to the two resonant modes inside the test

cavity to give rise to the band-pass filter response.

Fig. 26. Band-pass filter test case (capacitive coupled) for dual mode DR

The equivalent circuit model was created in ADS which is given as under:

Fig. 27. Equivalent circuit model of Band-pass filter test case (capacitive coupled) for dual mode DR

Parameter Value

0.0247

0.2400

0.2400

8nH

0.864 pF


32

Both models were simulated and there results were exported as s2p files which were then used to

correlate both of them by tuning the circuit model. Here are the results for the correlated responses of

both the models:

Fig. 28. Correlation between circuit & physical models for dual mode DR w.r.t band-pass filter test case

(capacitive coupled).

Important parameters of both the models for this test case are tabulated as:

Table 19. ADS parameters for dual mode DR circuit-geometry correlation w.r.t band-pass filter test case

(capacitive coupled)

Table 20. HFSS parameters for dual mode DR circuit-geometry correlation w.r.t band-pass filter test

case (capacitive coupled)

Parameter Value

0.015

& 0.138pF

8nH

0.687 pF

Parameter Value

12.5mm

Loop Diameter 1mm

35

8 mm

8 mm

Cut Area 1*1( )


33

3.4 Test Cavity

The next step in the process was to characterize the test cavity for the dual mode DR. For this purpose,

the reference resonator from the inductive coupled band-pass filter test case was put inside the test

cavity and weak coupling probes (sniffers) were used to measure the test cavity response. A circuit

model was created for the test cavity as the first step and then a detailed tolerance analysis was

performed with several parameters on the reference resonator inside the test cavity.

Fig. 29. Test cavity with reference resonator & sniffing probes

3.4.1 Reference Resonator

The reference resonator has been picked up from the inductive coupled two pole bandpass filter test

case and has been placed inside the test cavity to analyze its behavior. The reference resonator

parameters are given as under:

Table 21. Reference Resonator Parameters

3.4.2 Circuit Model & Correlated Response

Just like the previous cases, an equivalent circuit model was developed in ADS to specify the test

cavity response with weak coupling probes and the reference resonator. Following figure shows the

equivalent circuit model developed in ADS for the test cavity.

Parameter Value

35

8.7 mm

8.6 mm

Cut Area 1*1( )


34

Fig. 30. Equivalent Circuit model for Test cavity with reference resonator & sniffing probes

The correlated response for both the HFSS and ADS models for the test cavity case is given by:

Fig. 31. Correlation between circuit & physical models for test cavity with reference resonator & coupling

probes

Several key parameters from both HFSS and ADS models are given in the tables below:

Table 22. HFSS parameters for test cavity circuit-geometry correlation

Parameter Value

2*1.5r )

Loop Diameter 1mm

Cavity Size 30*30*30 (mm)


35

Table 23. ADS parameters for test cavity circuit-geometry correlation

3.4.3 Tolerance Analysis

Detailed tolerance analysis was performed for the reference resonator inside test cavity w.r.t all the

different tolerance parameters such as thickness, width, dielectric constant variation, cut length, cut

width and cut offset.

The first simulation was tolerance analysis w.r.t thickness of the reference resonator. The limits for the

thickness tolerance were specified to be ±0.2mm. Here are the results recorded for the tolerance

simulation:

Fig. 32. Test cavity tolerance analysis setup w.r.t thickness

Cross

Thickness

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

8.5 1873.41 1894.70 1915.99 0.0225

8.6 1866.09 1887.26 1908.43 0.0224

8.7 1859.41 1880.62 1901.83 0.0225

8.8 1852.77 1873.92 1895.08 0.0225

8.9 1846.17 1867.14 1888.12 0.0224

Table 24. Driven-modal data for test cavity tolerance analysis w.r.t thickness

Parameter Value

0.0225

0.0036

0.0036

8nH

0.8956 pF


36

Responses for each simulation was saved as an s2p file and were later on plotted together to see the

difference in test cavity response due to change in thickness of the reference resonator. Following

figure depicts the thickness tolerance clearly.

Fig. 33. Test cavity tolerance analysis w.r.t thickness

The next simulation was to do the tolerance analysis of the reference resonator w.r.t its width. The

limits specified for the width tolerance were same as thickness tolerance and following data was

recorded:

Fig. 34. Test cavity tolerance analysis setup w.r.t width


37

Cross

Width

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

8.4 1873.27 1893.03 1912.79 0.0208

8.5 1865.88 1886.54 1907.20 0.0218

8.6 1859.41 1880.62 1901.83 0.0225

8.7 1852.28 1874.20 1896.12 0.0233

8.8 1846.27 1868.87 1891.47 0.0241

Table 25.Driven-modal data for test cavity tolerance analysis w.r.t width

The above data was plotted together in the form of s2p files and following plot was observed:

Fig. 35. Test cavity tolerance analysis w.r.t width

The next iteration was performed for dielectric constant variation tolerance analysis and following

data was recorded:

Dielectric

Constant

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

33 1914.03 1936.11 1958.19 0.0228

34 1886.54 1908.48 1930.43 0.0229

35 1859.41 1880.62 1901.83 0.0225

36 1835.32 1856.70 1878.08 0.0230

37 1811.30 1832.52 1853.74 0.0231

Table 26. Driven-modal data for test cavity tolerance analysis w.r.t dielectric constant


38

The above recorded data was plotted together in one plot resulting in the following figure:

Fig. 36. Test cavity tolerance analysis w.r.t dielectric constant

The next simulation sequence was the cut length tolerance. Limits were specified as same as the

thickness tolerance i.e. ±0.2mm and following results were obtained:

Fig. 37. Test cavity tolerance analysis w.r.t cut length


39

Cut

Length

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

0.8 1859.03 1877.48 1895.93 0.0196

0.9 1859.34 1879.21 1899.08 0.0211

1 1859.41 1880.62 1901.83 0.0225

1.1 1859.65 1882.35 1905.06 0.0241

1.2 1859.63 1884.29 1908.96 0.0261

Table 27. Driven-modal data for test cavity tolerance analysis w.r.t cut length

Just like previous cases, all the relevant s2p files for the above data were plotted together as under:

Fig. 38. Test cavity tolerance analysis w.r.t cut length

The next simulation was done in the same fashion for cut width given as follows:

Fig. 39. Test cavity tolerance analysis w.r.t cut width


40

Cut

Width

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

0.8 1858.87 1878.50 1898.13 0.0208

0.9 1858.63 1879.12 1899.61 0.0218

1 1859.41 1880.62 1901.83 0.0225

1.1 1860.13 1882.10 1904.07 0.0233

1.2 1860.65 1883.40 1906.16 0.0241

Table 28. Driven-modal data for test cavity tolerance analysis w.r.t cut width

The data shown in the above table was plotted in a single plot as under:

Fig. 40. Test cavity tolerance analysis w.r.t cut width

The last set of simulations in the tolerance analysis was the cut offset (w.r.t cut width) tolerance

analysis. A limit of ±0.3mm was assigned to cut offset tolerance. Two cases were formulated: one

where the cut offset was in the similar direction on both diagonals and the other where it was opposite

on respective diagonals. The following results were recorded:


41

Fig. 41. Test cavity tolerance analysis setup w.r.t cut offset (similar direction)

Cut Offset

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

-0.3(up) 1860.50 1882.06 1903.63 0.0229

Normal 1859.41 1880.62 1901.83 0.0225

+0.3(down) 1860.04 1881.07 1902.11 0.0223

Table 29. Driven-modal data for test cavity tolerance analysis w.r.t cut offset (similar direction)

The above data was plotted in the following figure:

Fig. 42. Test cavity tolerance analysis w.r.t cut offset (similar direction)

For cut offset in the opposite direction, the following behavior was recorded:


42

Fig. 43. Test cavity tolerance analysis w.r.t cut offset (opposite direction)

Cut Offset

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Case 1 1859.47 1881.22 1902.97 0.0231

Normal 1859.41 1880.62 1901.83 0.0225

Case 2 1860.00 1881.95 1903.91 0.0233

Table 30. Driven-modal data for test cavity tolerance analysis w.r.t cut offset (opposite direction)

The following plot was observed when the data in the above table was plotted together:

Fig. 44. Test cavity tolerance analysis w.r.t cut offset (opposite direction)


43

3.4.4 Test Cavity with Reference Resonator & Losses

In the next simulation task, losses were added to the test cavity and resonator to see how the test cavity

response changes in the presence of losses. Three cases were developed in which first only the cavity

was changed to be having losses, then losses were added to resonator only and in the last case, both

cavity and resonator were modified to be as entities with losses.

For the first case, when cavity was modified to be having finite conductivity whereas the dielectric

material was ideal, we observed the following results:

Table 31. Parameters for the test cavity with losses

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Mode 1

Q

Mode 2

Q

1863.43 1885.48 1907.53 0.0233 9946 8273

Table 32. Eigen-mode data for test cavity with losses

For the second case, reference resonator was modified to be the one with the losses and cavity was

assigned to be lossless. Following results came out as a result of that simulation:

Parameter Value

35

3.33 e-5

30000

Table 33. Parameters for the reference resonator with losses

Parameter Value

Silver

41000000(S/m)

35

8.7 mm

8.6 mm

Cut Area 1*1( )

Cavity Size 30*30*30( mm)


44

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Mode 1

Q

Mode 2

Q

1863.12 1885.20 1907.28 0.0234 31143 31474

Table 34. Eigen-mode data for reference resonator with losses

The last case was setup such that both the cavity and resonator included losses. This case resulted in

the following results:

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Mode 1

Q

Mode 2

Q

1863.52 1885.60 1907.69 0.0234 7811 6351

Table 35. Eigen-mode data for test cavity & reference resonator both with losses

3.4.5 Tuning Screw Simulation & Quality Factor

The tuning screw test case was setup in order to check if we can compensate the effect of geometric

(manufacturing tolerances) without compromising our quality factor too much. These values will give

us an idea of how much tuning range we have without destroying the quality factor. In the next

simulation sequence, a brass tuning screw of radius 1 mm was inserted from the top of the test cavity

while the cavity and resonator were both having losses as in the previous case. The quality factor Q

was observed for both the modes and is given as under:

Tuning

Distance

(mm)

Even Mode

Frequency

(MHz)

Center

Frequency

(MHz)

Odd Mode

Frequency

(MHz)

Coupling

Factor

k=[

]

Mode 1

Q

Mode 2

Q

1 1863.42 1886.06 1908.70 0.0240 7089 7246

2 1863.07 1886.59 1910.11 0.0249 7083 7150

3 1862.81 1887.56 1912.31 0.0262 7080 7049

4 1861.84 1887.90 1913.96 0.0276 7072 6931

5 1860.88 1888.26 1915.65 0.0289 7069 6846

Table 36. Eigen-mode data for tuning screw simulation for test cavity & reference resonator both with losses


45

4 MATLAB Routine

In the last part of the thesis work, a MATLAB routine was developed based on the ABCD-parameters

two port network solution of the equivalent circuit of the test cavity. Non-linear equations were

developed describing the equivalent circuit and ABCD to S-parameter conversion was performed to

get the response in terms of S-parameters to correlate with the circuit model. The complete set of non-

linear equations is given in Appendix A under the MATLAB code.

To find the three unknown of the system of non-linear equations, first the equations were made non-

dimensional. Constants like L, C, which are of order of nH are all divided by 1 ; frequency is

multiplied by 1 . By scaling these variables, unknowns will be of order, O(1). After getting non-

dimensional set of equations, parameters are solved by optimization.

The nonlinear equations which are to be solved are made into a single objective function of norm2

form. For the correct set of parameters, the objective function becomes zero. By this, the problem of

finding the unknown parameters is formulated into an optimization problem. For an initial guess of

parameters, objective function can be reduced iteratively by nonlinear programming algorithms like

sequential quadratic programming, active-set etc. But, it is found that the solution is highly sensitive to

initial guess, and is resulting in a local minima. To avoid convergence to local minima, a hybrid

approach is followed. First, the problem is solved by heuristics algorithm like genetic algorithm,

which can bring the solution close to global minima. After going through some iterations, best solution

obtained from genetic algorithm is taken as initial guess to sequential quadratic programming

algorithm. The SQP algorithm quickly reduces the objective function and gives the required

parameters.

4.1 Nominal Resonator

The reference resonator has been picked up from the inductive coupled two pole band-pass filter test

case and has been placed inside the test cavity to analyze its behavior. The reference resonator

parameters are given as under:

Parameter Value

35

8.7 mm

8.6 mm

Cut Area 1*1( )


46


The loss parameters added to both the cavity and the reference resonator, are listed in the tables below:

Table 38. Reference Resonator Loss Parameters

Table 39. Test cavity Loss Parameters

4.2 Test Cavity without losses

For the first case, both the test cavity and reference resonator were considered ideal and having no

losses.

Fig. 45. Test cavity without losses

Equivalent Circuit model for the test cavity with weak coupling was created in ADS and is given as

under:

Parameter Value

3.33 e-5

3000

Parameter Value

Material Silver

41000000(S/m)

Cavity Size 30*30*30 ( )


47

Fig. 46. Equivalent Circuit Model for Test cavity without losses

The next step was to correlate the responses from both HFSS and ADS and following plot was created

after tuning the circuit model to match the HFSS response:

Fig. 47. Correlation between physical and circuit model for Test cavity without losses

Now to get a manual solution for the equivalent circuit for the test cavity, a T-network transformation

was applied on the original equivalent circuit given in Appendix B. The following equivalent circuit

was created as a result:


48

Fig. 48. T-Equivalent circuit model for Test cavity without losses

The T-equivalent circuit response was generated and was found to be only a few kHz apart from the

original equivalent circuit. Here is the T-equivalent circuit response:

Fig. 49. T-Equivalent circuit model response for Test cavity without losses

The manual solution was written for the T-equivalent circuit in the form of ABCD-parameters which

were then converted to S-parameters according to the conversion shown in Appendix C. Non-linear

equations were created and necessary approximations were made to write a MATLAB code that gave

the same response as the original circuit for the test cavity. The MATLAB code is listed in Appendix

A. Following MATLAB response was observed for the test cavity without losses scenario:

1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency(GHz)

dB(S

12)

MATLAB RESPONSE OF TEST CAVITY WIHTOUT LOSSES

dB (S12)

Original Frequency Points


49

Fig. 50. MATLAB response for Test cavity without losses

4.3 Test Cavity with Losses

The last case was to add losses to both the test cavity and the reference resonator and to model the

same response in MATLAB as the previous case.

Fig. 51. Test cavity & reference resonator with losses

The equivalent circuit model was created for the above losses modeled into the circuit cavity and

resonator and is given as following:

Fig. 52. Equivalent circuit model for test cavity & reference resonator with losses

The frequency response from HFSS and ADS simulations was plotted together in similar manner as in

the previous case and is given as below:


50

Fig. 53. Correlation between physical and circuit models for test cavity & reference resonator with losses

T-equivalent circuit model was then created for the test cavity and reference resonator having losses

cases in ADS.

Fig. 54. T-Equivalent circuit model for test cavity & reference resonator with losses

The T-equivalent circuit given in the previous figure gave out the following frequency response:

Fig. 55. T-Equivalent circuit model response for test cavity & reference resonator with losses

The newly added losses to the test cavity and reference resonator appeared in the form of a resistor in

the circuit models. These resistive losses were then modeled into the manual solution from the


51

previous case and MATLAB routine was modified to account for the losses. The following MATLAB

response was observed for the test cavity and reference resonator with losses case:

Fig. 56. MATLAB response for test cavity & reference resonator with losses

The following data for the circuit parameters was extracted. Inductances across the whole design work

were kept constant at 8nH:

Case

(MHz)

(MHz)

(pF)

(pF)

(Ohm)

Without Losses 1859.47 1901.79 0.8956 0.8956 0.0036 0.0036 0.02250 0

With Losses 1859.13 1901.93 0.8957 0.8957 0.0036 0.0036 0.02276 0.045

Table 40. Circuit Parameters for Nominal Resonator

4.4 Nominal Resonator Tolerance Analysis with MATLAB Routine

4.4.1 Case: Width

The next case was to test the nominal resonator with the change in its dimensions. The simulations

were setup such that the width of one of the two orthogonal resonators merged to form the cross

resonator was reduced by a nominal value whereas the width of the other resonator was increased w.r.t

same nominal value. Same method was followed as in the previous cases and the following results

were obtained:

1.85 1.86 1.87 1.88 1.89 1.9 1.91 1.92-75

-70

-65

-60

-55

-50

-45

-40

-35

FREQUENCY (GHz)

S12(d

B)MATLAB RESPONSE OF TEST CAVITY WITH LOSSES

S12 dB

Original Frequency Points

Parameter Value

35

8.7 mm

8.6± nominal value mm

Cut Area 1*1( )


52


The nominal value for this case was set to be nominal width ± 0.2 mm. The figure below explains

clearly how this case was setup:

Fig. 57. Test Case setup for nominal resonator

For constant inductance value of 8nH, we have the following circuit parameters for this case:

Case

(MHz)

(MHz)

(pF)

(pF)

(Ω)

Without Losses 1857.98 1903.48 0.8955 0.8955 0.0036 0.0036 0.02420 0

With Losses 1857.40 1903.90 0.8957 0.8957 0.0036 0.0036 0.02476 0.045

Table 42. Circuit Parameters for Nominal Resonator with change in dimensions

4.4.2 Case: Thickness

The new case was to test the nominal resonator with the change in its dimensions w.r.t thickness. The

simulations were setup such that the thickness of one of the two orthogonal resonators combined to

generate the cross resonator was reduced by a nominal value whereas the width of the other resonator

was increased w.r.t same nominal value. Same method was followed as in the previous cases and the

following results were obtained:

Parameter Value

35

8.7 mm ± nominal value mm

8.6 mm

Cut Area 1*1( )


53


The nominal value for this case was set to be nominal thickness ± 0.2 mm. The figure below explains




Case

(MHz)

(MHz)

(pF)

(pF)

(Ω)

Without Losses 1856.18 1901.84 0.8972 0.8972 0.0036 0.0036 0.02430 0

With Losses 1855.35 1900.34 0.8983 0.8983 0.0036 0.0036 0.02396 0.045


4.4.3 Case: Cut Length


simulations were setup such that the cut length of one of the diagonal cuts to the cross resonator was

reduced by a nominal value whereas the cut length of the other diagonal cut was increased w.r.t same

nominal value. Same method was followed as in the previous cases and the following results were

obtained:


The nominal value for this case was set to be nominal cut length ± 0.2 mm. The figure below explains


Parameter Value

35

8.7 mm

8.6 mm

Cut Length 1mm ± nominal value mm

Cut Width 1 mm


54



Case

(MHz)

(MHz)

(pF)

(pF)

(Ω)

Without Losses 1859.97 1902.11 0.8952 0.8952 0.0036 0.0036 0.02240 0

With Losses 1858.77 1900.80 0.8964 0.8964 0.0036 0.0036 0.02236 0.045


4.4.4 Case: Cut Width


simulations were setup such that the cut width of one of the diagonal cuts to the cross resonator was

reduced by a nominal value whereas the cut width of the other diagonal cut was increased w.r.t same

nominal value. Same method was followed as in the previous cases and the following results were

obtained:


The nominal value for this case was set to be nominal cut width ± 0.2 mm. The figure below explains


Parameter Value

35

8.7 mm

8.6 mm

Cut Length 1mm

Cut Width 1 mm ± nominal value mm


55



Case

(MHz)

(MHz)

(pF)

(pF)

(Ω)

Without Losses 1859.05 1901.36 0.8960 0.8960 0.0036 0.0036 0.02250 0

With Losses 1859.20 1901.43 0.8959 0.8959 0.0036 0.0036 0.02246 0.045


4.4.5 Case: Geometry Variation

The new case was to test the nominal resonator with the change in its geometry. The simulations were

setup in HFSS environment. Same characterization method was followed as in the previous cases and

the following results were obtained:


The nominal value for this case was set to be nominal geometry ± 0.3 mm. The figure below explains


Parameter Value

35

8.7 mm

8.6 mm

Cut Area 1*1( )

Case Nominal Geometry ± 0.3 mm


56



Case

(MHz)

(MHz)

(pF)

(pF)

(Ω)

Without Losses 1859.26 1901.58 0.8958 0.8958 0.0036 0.0036 0.02250 0

With Losses 1858.88 1901.11 0.8962 0.8962 0.0036 0.0036 0.02246 0.01


4.4.6 Comparison between HFSS, ADS & MATLAB Routines

In the following table, different mutual coupling coefficient K values have been compared for various

simulations:

Case

Nominal

Resontaor

Without Losses 0.02250 0.02250 0.02250

With Losses 0.02275 0.02276 0.02276

Dimension

Change-Width

Without Losses 0.02418 0.02420 0.02420

With Losses 0.02472 0.02476 0.02475

Dimension

Change-

Thickness

Without Losses 0.02429 0.02340 0.02340

With Losses 0.02395 0.02396 0.02395

Dimension

Change-Cut

Length

Without Losses 0.02239 0.02240 0.02240

With Losses 0.02235 0.02236 0.02236

Dimension

Change-Cut

Width

Without Losses 0.02250 0.02250 0.02250

With Losses 0.02245 0.02246 0.02245

Geometry

Variation

Without Losses 0.02250 0.02250 0.02250

With Losses 0.02245 0.02246 0.02244

Table 51. Comparison of coupling coefficient values from different solutions.


57

The above table shows that MATLAB routine has successfully characterized the dual mode resonator

response with very little error. The values found were in agreement with the values of electromagnetic

model from HFSS and circuit model of ADS.

4.4.7 How to use MATLAB Routine

The following steps need to be followed for using the developed MATLAB routine:

1. Put the reference resonator in test cavity

2. Read s2p file

3. Read 3 Points from s2p file.

4. Input three Frequency Points into the MATLAB routine

5. Execute and get the circuit parameters.

4.4.7.1 Steps 1 & 2:

Steps 1 & 2 have been followed under the test cavity section for each and every case.

4.4.7.2 Steps 3& 4: Reading & Selection of Three Points:

Pass on the following three points i.e. two peaks and the center frequency taken from the dual mode

response to the MATLAB functions shown as follows:

Fig.62. Selection of 3 Points from the dual mode response

1.84 1.86 1.88 1.90 1.92 1.94 1.96 1.98 2.00Freq [GHz]

-70.00

-60.00

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

Y1

HFSSDesign1XY Plot 1 ANSOFT

m3

m2m1

Curve Info

dB(S(1,2))Setup1 : Sweep

dB(S(1,1))Setup1 : Sweep

Name X Y

m1 1.89338668 -29.07315433

m2 1.93681936 -28.57380040

m3 1.91467093 -42.91719935


58

Fig.65. Appending 3 Points into the MATLAB Routine

4.4.7.3 Step 5: Execute & Get Circuit Parameters:

Execute the main Function provided in the MATLAB routine. The result will be displayed as follows:

Fig.63. Circuit Parameters calculated via MATLAB routine


59

5 Discussions/Conclusions

Based on the research work, single mode and dual mode DR inside a test cavity have been

characterized by Eigen-mode and two port simulations in HFSS, equivalent circuit models were

created in ADS to establish the correlation between geometry and circuit parameters and at the end, a

MATLAB routine was generated to characterize the response of the test cavity with a dual mode

reference resonator inside by using ABCD & S-parameters and non-linear equations.

In simulation sequences, a 30*30*30 cavity was used for all cases whereas the resonators were

designed to resonate at a center frequency of 1960 MHz. The field distributions of both single and dual

mode DR were studied closely to understand the modes present inside the test cavity. For the dual

mode DR, a novel method of introducing a diagonal cut was introduced to couple the two resonant

modes inside the test cavity. Different electric and magnetic probes were created to couple the energy

efficiently to the two modes inside the test cavity. Single mode and dual mode dielectric resonators,

both, were subjected to rigorous tolerance analysis w.r.t physical dimensions and other parameters to

record their behavior inside a test cavity. Equivalent circuit models were developed in each case to

establish the relationship between physical and circuit models for the resonators. Two 2-pole 20 dB

RL band-pass filters were developed as test cases for the dual mode DR by coupling via electric and

magnetic probes. These filters are used to define the geometry of the reference resonator inside test

cavity. Test Cavity model was developed with weak coupling probes and having the two pole band-

pass filter inside as reference resonator. Equivalent circuit model and correlation was achieved and

extensive tolerance analysis was performed. A tuning screw simulation was setup to measure the Q

value which showed the resonator has a good quality factor inside the test cavity.

A MATLAB routine was developed in the end to make the measurement of a dual mode resonator

inside a test cavity robust and to eliminate the use of a circuit simulator. Successful characterization of

the dual mode response was achieve in MATLAB by implementing the manual two port network

solution and the resulting non-linear equations in the MATLAB environment. Hence, it can be inferred

that a dual mode resonator has been developed in HFSS environment, its variant was generated for the

coupling bandwidth purpose, equivalent circuit models were developed to specify the relationship

between circuit and physical parameters and resonator response has been characterized by the detailed

test cavity model and tolerance analysis. The resonator measurement routine has been established in

MATLAB to specify the dual mode response.


60

5.1 Suggested Future Work

The Future work can include the development of a single and simplified equation that denotes the full

response of a dual mode resonator by rigorous manual solution of the non-linear equations of the two

port network. The development of physical model of the test cavity for the dual mode resonator is also

a pivotal task. The ideal placement of probes in the test cavity as well as development of detailed

specifications of the dual mode resonator tolerances should be included in the future goals.The

fabrication and lab measurement of the dual mode resonator will then follow.


61

REFERENCES

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volume 48, The institution of electrical engineers, London, UK.

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edition, John Willey & sons, August 1993.

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transactions on microwave theory and techniques, vol. MTT-16, NO. 4, April 1968.

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[6] C. Wang, K. A. Zaki, A. E. Atia, and T. G. Dolan, “Dielectric Combline Resonators and

Filters”, IEEE transactions on microwave theory and techniques, vol. 46, No. 12,December

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[7] Y. Kobayashi, N. Fukuoka, and S. Yoshida, “ Resonant modes for a shielded-dielectric rod

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Theory Tech., vol. MTT-34, pp. 815-824, July 1986.

[9] D. L. Rebsch, D. C.Webb, R. A. Moore, and J. D. Cowlishaw, “A mode chart for accurate

design of cylindrical dielectric resonators”, IEEE Trans. Microwave Theory Tech., vol. MTT-

13, pp. 468–469, May 1965.

[10] Y. Kobayashi and S. Tanaka, “ Resonant modes of a dielectric rod resonator short-circuited at

both ends by parallel conducting plates”, IEEE Trans. Microw. Theory Tech., vol. MTT-28,

pp. 1077–1085, Oct. 1980.

[11] K. A. Zaki and A. E. Atia, “Modes in dielectric-loaded waveguides and resonators”, IEEE

Trans. Microw. Theory Tech., vol. MTT-31, pp. 1039–1045, Dec. 1983.

[12] P. S. Neelakanta, “Handbook of electromagnetic materials Monolithic and composite versions

and their applications”, CRC press New York. January 1995.

[13] K. Mehmet, “Network Transformations for Realization of Lumped and Distributed Filters”,

M.S. Thesis in Electrical and Electronics Engineering Dept., Middle East Technical

University, Ankara, Turkey, September 1994.

[14] C. Wang and K. A. Zaki, “Dielectric Resonators and Filters”, IEEE microwave magazine, Oct

2007.

[15] M. T. Sebastian, “Dielectric materials for wireless communication”, First Edition, Elsevier

Linacre House, Jordan Hill, Oxford OX2 8DP, UK, 2008.

[16] M. Sadiku, “Numerical Techniques in Electromagnetics”, CRC Press, Florida, U.S.A., 2000.


62

[17] R. R. Mansour, “High Q tunable dielectric resonator filters”, IEEE microwave magazine,

October 2009.

[18] M. Memarian, and R. R. Mansour, “Quad-Mode and Dual-Mode Dielectric Resonator Filters”,

IEEE transactions on microwave theory and techniques, vol.57, No.12, December 2009.

[19] I. C. Hunter, J. D. Rhodes and V. Dassonville, “Dual-Mode Filters with Conductor-Loaded

Dielectric Resonators”, IEEE transactions on microwave theory and techniques, vol. 47, NO.

12, December 1999.

[20] A.E. Atia and A. E. Williams, “Narrow-Bandpass Waveguide Filters”,IEEE Trans. Microw.

Theory Tech., vol. MTT-20, No.4, pp. 258-265, April 1972.

[21] S. Amari, and U. Rosenberg, “New in-line dual- and triple-mode cavity filters with non-

resonating nodes,” IEEE Trans. Microw. Theory Tech., vol. MTT-53, pp. 1272–1279, April

2005.

[22] R.R. Bonetti, and A.E. Williams, “Application of Dual TM Modes to Triple- and Quadruple-

Mode Filters”, IEEE Trans. Microw. Theory Tech., vol. MTT-35, pp. 1143- 1149, Dec 1987.

[23] W. Tang and S.K Chaudhuri, “A True Elliptic-Function Filter Using Triple-Mode Degenerate

Cavities,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, pp. 1449 – 1454, Nov. 1984.

[24] S. Fok et al., “A Novel Microstrip Square-Loop Dual-Mode Bandpass Filter with

Simultaneous Size Reduction and Spurious Response Suppression”, IEEE Trans. Microwave

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[25] I. Oksar , “Design and realization of mixed element broadband bandpass filters”, M.S. Thesis

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Systems”, New Jersey: J. Wiley & Sons, 2007.


A1

Appendix A

% The MAIN executable function

clc

clear all

% x0 = [0.0288

% 1.7914

% 0.3641];

x0 = ones(3,1);

options_nlp = optimset;

options_nlp = optimset(options_nlp,'Display','iter');

options_nlp = optimset(options_nlp,'TolX',1e-15);

options_nlp = optimset(options_nlp,'TolCon',1e-15);

options_nlp = optimset(options_nlp,'MaxFunEvals',1e5);

options_nlp = optimset(options_nlp,'MaxIter',1e4);

options_nlp = optimset(options_nlp,'Algorithm','sqp');

options_ga = gaoptimset;

options_ga = gaoptimset(options_ga,'CrossoverFraction',0.8);

options_ga = gaoptimset(options_ga,'Display','iter');

options_ga = gaoptimset(options_ga,'EliteCount',2);

options_ga = gaoptimset(options_ga,'Display','iter');

options_ga = gaoptimset(options_ga,'Generations',10000);

options_ga = gaoptimset(options_ga,'HybridFcn',@fmincon,options_nlp);

options_ga = gaoptimset(options_ga,'PopulationSize',1000);

options_ga = gaoptimset(options_ga,'StallGenLimit',50);

options_ga = gaoptimset(options_ga,'TimeLimit',10);

options_ga = gaoptimset(options_ga,'TolCon',1e-15);

options_ga = gaoptimset(options_ga,'TolFun',1e-15);

% lb = 0.01*x0;

% ub = 100*x0;

lb = 0.001*x0;

ub = 1000*x0;

% lb = 0.0001*x0;


A2

% ub = 10000*x0;

x =

ga(@double_coupled_eqns_reverse_nondimensional,3,[],[],[],[],lb,ub,[],optio

ns_ga);

%Solution

fprintf('Lc = %g\n',x(1)*1e-9);

fprintf('Ct = %g\n',x(2)*1e-12);

fprintf('Lc_ = %g\n',x(3)*1e-9);

Kin=x(1)/(sqrt(8)^2);

Kout = x(1)/(sqrt(8)^2);

Kmut= x(3)/(2*sqrt(8)^2);

Cap= x(2)/2;

fprintf('K_input = %g\n',Kin);

fprintf('K_output = %g\n',Kout);

fprintf('K_mutual = %g\n',Kmut);

fprintf('C1 = %g\n',Cap*1e-12);

fprintf('C2 = %g\n',Cap*1e-12);

double_coupled_eqns_nondimensional(x);

% The Function with the solution of non-linear system of equations

function [w_out,out_value ] = double_coupled_eqns_nondimensional(in)

%function [ out ] = double_coupled_eqns( in )

%Lc=in(1);

%Ct=in(2);

%Lc_=in(3);

out=[];

L=double(8);

C=double(0.8764);

% R=0; % Lossless Case

% R = 0.045; %(Lossy Case)

R = 0.01; %(Geometry1_lossy case)


A3

%[Lc,Ct,Lc_]=find_indcap_double(end_coup,inter_coup,L,C);

% x1 = 0.0288;

% x2 = 2*C;

% x3 = 0.3472;

Lc = in(1);

Ct = in(2)/1000;

Lc_ = in(3);

La=L-Lc;

La_=L-Lc_;

Lb=La;

Lb_=La_;

%%%%%%%%%%%%%%% Plot Frequency %%%%%%%%%%%%%%

f= linspace(1.840,1.920, 10000);

w=2*pi*f;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for q=1:length(w)

%%%%%%%%%%%%%%%%%% Stage1%%%%%%%%%%%%%%%%%%%%

Za(q)=1i*w(q)*La;

Zac(q)=La/Lc;

X(q)=(2*Za(q))+(Zac(q)*Za(q));

Za_(q)=R+1i*w(q)*La_; %%% Loss addition

Zac_(q)=(R+1i*w(q)*La_)/(1i*w(q)*Lc_);

X_(q)=2*Za_(q)+Zac_(q)*Za_(q);

Zb(q)=1i*w(q)*Lb;

Zb_(q)=R+1i*w(q)*Lb_; %%% Loss addition

Zbc(q)=Lb/Lc;

Zbc_(q)=(R+1i*w(q)*Lb_)/(1i*w(q)*Lc_);

Yc1(q)=1i*w(q)*Ct;

Yc(q)=(-1i)/(w(q)*Lc);


A4

Yc_(q)=(-1i)/(w(q)*Lc_);

%%%%%%%%%%%%%%%%%%%%%% Stage2%%%%%%%%%%%%%%%%

Z1ac(q)=1+Zac(q);

Z1bc(q)=1+Zbc(q);

Z1ac_(q)=1+Zac_(q);

Z1bc_(q)=1+Zbc_(q);

%%%%%%%%%%%%%%%%%%%%% Stage3%%%%%%%%%%%%%%%%

A1(q)=Z1ac(q)+Yc1(q)*X(q);

A2(q)=Z1ac_(q)+Yc1(q)*X_(q);

C1(q)=Yc(q)+Yc1(q)*Z1bc(q);

C2(q)=Yc_(q)+Yc1(q)*Z1bc_(q);

%%%%%%%%%%%%%%%%%%%%% Stage4 %%%%%%%%%%%%%%%%

A1_(q)=A1(q)*A2(q)+X(q)*C2(q);

B1_(q)=A1(q)*X_(q)+X(q)*Z1bc_(q);

C1_(q)=C1(q)*A2(q)+Z1bc(q)*C2(q);

D1_(q)=C1(q)*X_(q)+Z1bc(q)*Z1bc_(q);

%%%%%%%%%%%%%%%%%%%%% Stage5 %%%%%%%%%%%%%%%%

A(q)=A1_(q)*Z1ac(q)+B1_(q)*Yc(q);

B(q)=A1_(q)*X(q)+B1_(q)*Z1bc(q);

C(q)=C1_(q)*Z1ac(q)+D1_(q)*Yc(q);

D(q)=C1_(q)*X(q)+D1_(q)*Z1bc(q);

%%%%%%%%%%%%%%% ABCD to S12 Transformation %%%%%%%%%%%%

Zo=50;

S12(q)=(2*Zo)/abs(A(q)*Zo+B(q)+(C(q)*Zo^2)+D(q)*Zo);

out_value(q)=20*log10(S12(q));

%%%%%%%%%%%%%%%%%% Building 3 equations %%%%%%%%%%%%%%%%

%%%S12_lin(q)=10^(S12_log(q)/20);

%%%magnitude(q)=abs(A(q)*Zo+B(q)+(C(q)*Zo^2)+D(q)*Zo);

%%%out(q)=S12_lin(q)*magnitude(q)-2*Zo;

end


A5

%out=out';

w_out = w/(2*pi);

plot(w/(2*pi),out_value,'blue');

hold on

%%%%%%%%%%%%%%% Data From Graph %%%%%%%%%%%%%

% w_true=[1.859910 1.881100 1.902290]; %test cavity_lossless (nominal res)

% S12_log_true=[-4.438 -72.484 -5.159 ];

% w_true=[1.859590 1.881010 1.902440]; %test cavity_lossy (nominal res)

% S12_log_true=[-38.351 -72.584 -39.829];

% w_true=[1.858540 1.880790 1.904130];

% S12_log_true=[-1.802 -73.111 -0.003]; % Width_lossless

% w_true=[1.857850 1.880600 1.904490];

% S12_log_true=[-38.290 -73.309 -39.898]; % Width_lossy

% w_true=[1.856690 1.879010 1.902430];

% S12_log_true=[-5.031 -73.142 -2.705]; % Thickness_Lossless

% w_true=[1.855850 1.878390 1.900910];

% S12_log_true=[-38.321 -73.025 -39.876]; %Thickness_lossy

% w_true=[1.860420 1.881040 1.902610];

% S12_log_true=[-5.414 -72.439 -0.313]; %cutlength_lossless

% w_true=[1.859200 1.879780 1.901290];

% S12_log_true=[-38.364 -72.422 -39.817]; %cutlength_lossy

% w_true=[1.859500 1.8802000 1.901860];

% S12_log_true=[-1.727 -72.475 -1.520]; %cutwidth_lossless

% w_true=[1.859640 1.880320 1.901920];

% S12_log_true=[-38.360 -72.460 -39.820]; %cutwidth_lossy

%

% w_true=[1.859710 1.880420 1.902070];

% S12_log_true=[-5.701 -72.476 -5.421]; %geometry1_lossless


A6

w_true=[1.859330 1.880730 1.901600];

S12_log_true=[-38.361 -72.467 -39.821]; %geometry1_lossy(R = 0.01)

plot(w_true,S12_log_true,'*');

hold off

end

% The Function For getting circuit parameters from Frequency Response

%function [ ] = double_coupled_eqns( end_coup,inter_coup )

function J = double_coupled_eqns_reverse_nondimensional( in )

out=[];

L=double(8);

C=double(0.8764);

% R=0; % Lossless Case

% R = 0.045; %(Lossy Case)

R = 0.01; %(Geometry1_lossy case)

%[Lc,Ct,Lc_]=find_indcap_double(end_coup,inter_coup,L,C);

x1 = in(1);

x2 = in(2);

x3 = in(3);

Lc = x1;

Ct = x2/1000;

Lc_ = x3;

La=L-Lc;

La_=L-Lc_;

Lb=La;

Lb_=La_;

%%%%%%%%%%%%%%% Data From Graph %%%%%%%%%%%%%


A7

% w=2*pi*[1.859910 1.881100 1.902290]; %test cavity_lossless (nominal res)

% S12_log=[-4.438 -72.484 -5.159 ];

% w=2*pi*[1.859590 1.881010 1.902440]; %test cavity_lossy (nominal res)

% S12_log=[-38.351 -72.584 -39.829];

% w=2*pi*[1.858540 1.880790 1.904130];

% S12_log=[-1.802 -73.111 -0.003]; % Width_lossless

% w=2*pi*[1.857850 1.880600 1.904490];

% S12_log=[-38.290 -73.309 -39.898]; % Width_lossy

% w=2*pi*[1.856690 1.879010 1.902430];

% S12_log=[-5.031 -73.142 -2.705]; % Thickness_Lossless

%

% w=2*pi*[1.855850 1.878390 1.900910];

% S12_log=[-38.321 -73.025 -39.876]; %Thickness_lossy

% w=2*pi*[1.860420 1.881040 1.902610];

% S12_log=[-5.414 -72.439 -0.313]; %cutlength_lossless

% w=2*pi*[1.859200 1.879780 1.901290];

% S12_log=[-38.364 -72.422 -39.817]; %cutlength_lossy

% w=2*pi*[1.859500 1.8802000 1.901860];

% S12_log=[-1.727 -72.475 -1.520]; %cutwidth_lossless

% w=2*pi*[1.859640 1.880320 1.901920];

% S12_log=[-38.360 -72.460 -39.820]; %cutwidth_lossy

% w=2*pi*[1.859710 1.880420 1.902070];

% S12_log=[-5.701 -72.476 -5.421]; %geometry1_lossless

w=2*pi*[1.859330 1.880730 1.901600];

S12_log=[-38.361 -72.467 -39.821]; %geometry1_lossy

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for q=1:length(w)


A8

%%%%%%%%%%%%%%%%%% Stage1%%%%%%%%%%%%%%%%%%%%

Za(q)=1i*w(q)*La;

Zac(q)=La/Lc;

X(q)=(2*Za(q))+(Zac(q)*Za(q));

Za_(q)=R+1i*w(q)*La_; %%% Loss addition

Zac_(q)=(R+1i*w(q)*La_)/(1i*w(q)*Lc_);

X_(q)=2*Za_(q)+Zac_(q)*Za_(q);

Zb(q)=1i*w(q)*Lb;

Zb_(q)=R+1i*w(q)*Lb_; %%% Loss addition

Zbc(q)=Lb/Lc;

Zbc_(q)=(R+1i*w(q)*Lb_)/(1i*w(q)*Lc_);

Yc1(q)=1i*w(q)*Ct;

Yc(q)=(-1i)/(w(q)*Lc);

Yc_(q)=(-1i)/(w(q)*Lc_);

%%%%%%%%%%%%%%%%%%%%%% Stage2%%%%%%%%%%%%%%%%

Z1ac(q)=1+Zac(q);

Z1bc(q)=1+Zbc(q);

Z1ac_(q)=1+Zac_(q);

Z1bc_(q)=1+Zbc_(q);

%%%%%%%%%%%%%%%%%%%%% Stage3%%%%%%%%%%%%%%%%

A1(q)=Z1ac(q)+Yc1(q)*X(q);

A2(q)=Z1ac_(q)+Yc1(q)*X_(q);

C1(q)=Yc(q)+Yc1(q)*Z1bc(q);

C2(q)=Yc_(q)+Yc1(q)*Z1bc_(q);

%%%%%%%%%%%%%%%%%%%%% Stage4 %%%%%%%%%%%%%%%%

A1_(q)=A1(q)*A2(q)+X(q)*C2(q);

B1_(q)=A1(q)*X_(q)+X(q)*Z1bc_(q);

C1_(q)=C1(q)*A2(q)+Z1bc(q)*C2(q);

D1_(q)=C1(q)*X_(q)+Z1bc(q)*Z1bc_(q);


A9

%%%%%%%%%%%%%%%%%%%%% Stage5 %%%%%%%%%%%%%%%%

A(q)=A1_(q)*Z1ac(q)+B1_(q)*Yc(q);

B(q)=A1_(q)*X(q)+B1_(q)*Z1bc(q);

C(q)=C1_(q)*Z1ac(q)+D1_(q)*Yc(q);

D(q)=C1_(q)*X(q)+D1_(q)*Z1bc(q);

%%%%%%%%%%%%%%% ABCD to S12 Transformation %%%%%%%%%%%%

Zo=50;

%%%S12(q)=(2*Zo)/abs(A(q)*Zo+B(q)+(C(q)*Zo^2)+D(q)*Zo);

%%%out_value(q)=20*log10(S12(q));

%%%%%%%%%%%%%%%%%% Building 3 equations %%%%%%%%%%%%%%%%

S12_lin(q)=10^(S12_log(q)/20);

magnitude(q)=abs(A(q)*Zo+B(q)+(C(q)*Zo^2)+D(q)*Zo);

out(q)=S12_lin(q)*magnitude(q)/(2*Zo) - 1;

end

out=out';

J = 0;

for i = 1:length(w)

J = J + out(i)*out(i);

end

%J = sqrt(J);

%plot(w/(2*pi),out_value);

end


B1

Appendix B

Fig.63: Equivalent T-Circuit of a Linear Transformer [26]


C1

Appendix C

[ ] [ ]

Where

tolerance analysis of a multi-mode ceramic resonator658392/fulltext01.pdf · khawar naeem tolerance...

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