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Analysis and modeling of linear and nonlinear microwave superconducting devices

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ANALYSIS AND MODELING OF LINEAR AND NONLINEAR
M ICROW AVE SUPERCONDUCTING DEVICES
by
Mohamed Abdel Fattah Megahed
A Dissertation Presented in Partial Fulfillment
o f the Requirements for the Degree
Doctor of Philosophy
ARIZONA STATE UNIVERSITY
August 1995
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DMI Number: 9538169
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ANALYSIS AND MODELING OF LINEAR AND NONLINEAR
MICROWAVE SUPERCONDUCTING DEVICES
by
Mohamed Abdel Fattah Megahed
has been approved
July 1995
APPROVED:
. Chairperson
3
d . (lJ u J o
A
iMJAa
9 -----------------Supervisory Committee
ACCEPTED:
Dentfmhent Chairperson
Dean, Graduate College
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ABSTRACT
High Temperature Superconducting (HTS) materials have potential applications
for microwave and millimeter-wave devices. The performance of superconductors is
substantially superior to normal conductors and semiconductors concerning low loss,
high sensitivity and low dispersion. However, new methodologies are needed for the
design and analysis o f such devices.
Field penetration effects must be taken into
consideration, especially for high power applications.
As w ith other fabrication
technologies, it is desirable to simulate these devices before they are built to save money
and time.
Similarly, to exploit the exciting characteristics o f these new materials,
accurate and flexible models have to be developed.
An accurate analysis for microwave and millimeter-wave devices, which include
high temperature superconductor materials, is presented in this dissertation. This study
covers both low linear and high nonlinear power applications. This analysis is based on
blending a fu ll electromagnetic wave model with phenomenological linear and nonlinear
superconductor model, and the two-fluid model. The linear model is based on the low
power London's model. On the other hand, the nonlinear model is developed using the
Ginzburg-Landau theory.
These models are capable o f fu lly characterizing HTS
microwave devices, including obtaining the current distributions inside the
superconducting material, the electromagnetic fields, and the power handling capability.
These solutions are obtained using the finite-differcnce scheme.
The
superconductor thickness is rigorously modeled. No approximations are made to the
superconductor thickness. The anisotropy associated with the superconductor is also
considered. The linear problem can be solved in either the frequency or the time domain.
iii
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However, the nonlinear solution must be performed in the time domain. This approach is
employed to investigate HTS transmission lines and filter. Results have shown that the
number o f superfluid electrons decreases near the edges o f transmission strips as the
applied power increases, indicating the breaking o f superfluid electrons pairs. The linear
model underestimates the magnetic field penetration inside the superconductor. The
change in the losses with the applied field is much larger than the change in the velocity
o f the wave propagating along the device. A variation in the frequency spectrum o f the
applied signal resulting from the nonlinearity is seen. Also, simulation o f HTS filte r has
revealed that dimension and layout o f HTS filters must be optimized in the design cycle
to avoid nonlinearity effects.
A novel nonlinear phenomenological tw o-fluid model for superconducting
materials has also been proposed, where the thermodynamics and electromagnetics
properties o f HTS are considered simultaneously.
This model is very useful for
computer-aided design applications.
iv
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This work is dedicated to the memory of my parents who have
brought me up and given me their love, to my wife Sawsan who
always comforts and consoles, never complains or interferes, and asks
nothing and endures all, and to my children Ahmed, Jylan, and Samer.
v
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ACKNOWLEDGMENTS
I would like to express my deepest indebtedness to Dr. Samir M. El-Ghazaly for
his consent to be my thesis advisor, for suggesting the area o f research, and his constant
support and patience. His valuable guidance, as well as the time and effort he spent, make
this dissertation finally come true. I thank him for sending me to the IEEE MTT-S, IEEE
AP-S, URSI, and PIERS Conferences, and giving me a place among other researchers. I
am indebted to him for his able assistance and interest in this academic research and
many other areas of my life.
I would also like to thank my committee members Dr. I. Kaufman, Dr. R.
Grondin, Dr. E. El-Sharawy, Dr. J. Aberle and Dr. K. Schmitt for their critical reading of
this dissertation and valuable comments.
Deep appreciation is due to my friends and colleagues at Arizona State University
for their constant help and useful discussions. In particular, special thanks to M. A lSunaidi for many stimulating conversations and T. El-Shafiey for his eternal friendship.
Finally, I acknowledge God who not only loaned me the talent and abilities
necessary to complete this degree, but put me in the right circumstances so that I am
where I am today. He is, at the root, responsible for all the acknowledgments above.
vi
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TABLE OF CONTENTS
Page
LIST OF TA BLE S .............................................................................................................. x
LIST OF FIGURES............................................................................................................ xi
CHAPTER
1
INTRODUCTION...............................................................................................1
1.1
Problem Definition..........................................................................................2
1.2
Motivation for the Selected Techniques.......................................................... 4
1.3
Thesis O utline................................................................................................ 7
2
LINEAR AND NONLINEAR SUPERCONDUCTOR M ODELS...................13
2.1
Material Models of Superconductors.............................................................15
2.2
The Two-Fluid M odel................................................................................... 16
2.3
Temperature Dependence of Superconductors..............................................17
2.4
London Phenomenological M odel.................................................................18
2.5
Ginzburg-Landau Phenomenological M odel.............................................. 20
2.5.1 Theory................................................................................................ 20
2.5.2 Limitations on GL Theory.................................................................. 26
2.5.3 GL for Microwave HTS Applications................................................ 27
2.6
Solution of GL Equations...............................................................................29
2.6.1 Numerical Scheme............................................................................. 29
2.6.2 Bulk Superconductor (Superconducting Half-Space)........................ 32
2.6.3 Thin Superconductor F ilm ................................................................. 35
2.7
3
Summary........................................................................................................ 40
FINITE-DIFFERENCE APPROACH.................................................... 42
3.1
Finite-Difference Frequency-Domain (FDFD)............................................. 45
3.1.1 Wave Equation for Superconducting Microwave Structure............. 45
vii
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CHAPTER
Page
3.1.2 Eigenvalue Problem............................................................................ 48
3.1.3 FDFD Application to Microstrip L in e ................................................ 52
3.2
Finite-Difference Time-Domain (FD TD )......................
63
3.2.1 Nonuniform Finite-Difference Mesh Generator................................. 65
3.2.2 Absorbing Boundary Conditions: Perfectly Matched Layers............ 69
3.2.3 Parallel Implementation of the FDTD on MASPAR Machine.......... 77
3.2.4 FDTD Application to Microstrip L in e ................................................ 86
3.3
Comparison between FDFD and FDTD Solutions........................................ 88
3.4
Summary
4
:................................................................................................92
ANISOTROPIC SUPERCONDUCOR ON ANISOTROPIC SUBSTRATES 93
4.1
Anisotropic High Temperature Superconductor Model................................. 95
4.2
Anisotropic Finite-Difference Time-Domain Approach................................ 97
4.3
Anisotropic Superconductor Microstripline on Anisotropic Sapphire
Substrate....................................................................................................... 100
4.4
Anisotropic Superconductor Coplanar Waveguide on Anisotropic Sapphire
Substrate....................................................................................................... 108
4.5
5
Summary........................................................................................................112
FULL-W AVE NONLINEAR ANALYSIS OF MICROWAVE
SUPERCONDUCTING DEVICES..................................................................113
5.1
Time-Domain versus Frequency-Domain Numerical Techniques................116
5.2
Nonlinear Full-Wave Superconductor M odel...............................................117
5.2
Nonlinear Full-Wave Superconductor Simulator..........................................121
5.4
Nonlinear Analysis o f Superconducting Microstrip Lines........................... 124
5.5
Nonlinear Analysis o f Superconducting Filters............................................138
5.5.1 Simulation of Microstrip Resonator Array HTS F ilte r.................... 139
viii
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CHAPTER
Page
5.5.2 High Power Design Consideration for HTSF ilters.......................... 147
5.6
6
Summary...................................................................................................... 148
NOVEL NONLINEAR PHENOMENOLOGICAL TW O FLU ID M O D E L. 150
6.1
Nonlinear Phenomenological Two fluid model........................................... 152
6.2
Macroscopic Model o f Nonlinear Constitutive relationsin H T S ................. 157
6.3
HTS Nonlinear Surface Impedance...............................................................159
6.4
Nonlinear Model Validation and Verification..............................................163
6.5
Summary....................................................................................................... 167
7
CONCLUSIONS.................................................................................:............ 168
7.1
Summary of Findinds and Conclusions........................................................ 169
7.2
Recommendations for Future Research........................................................ 172
REFERENCES..................................................................................................................174
ix
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LIST OF TABLES
Table
3.1
Page
Number o f Processors and Machine Size for available options........................82
x
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LIST OF FIGURES
Page
Figure
2.1
Superconducting Half Space (bulk superconductor)..........................................33
2.2
Normalized superfluid current density in one-dimensional bulk Y B A 2Cu3 0 7 .x
XA
T-ITO
i l l J ......................................................................................................................................................,....................................................................................................................._
>
*
+
2.3
Y B A 2CU3O7-X HTS strip (thin film superconductor).........................................35
2.4
Tangential magnetic field intensity to a YBa2Cu307-x HTS strip in microstrip
line configuration............................................................................................... 36
2.5
Normalized superfluid current density distribution in YBa2Cu307-x HTS strip
in microstrip line configuration at 0.2 Pcrf applied power................................ 37
2.6
Normalized superfluid current density distribution in YBa2Cu307-x HTS strip
in microstrip line configuration at 0.45 Pcrf applied power.............................. 38
2.7
Normalized superfluid current density distribution in YBa2Qi307-x HTS strip
in microstrip line configuration at 0.9 Pcrf applied power................................ 39
2.8
Normalized superfluid electron density distribution in YBa2Cu307-x HTS strip
in microstrip line configuration at 0.45 Pcrf applied power.............................. 40
3.1
Mesh indices ij and the corresponding xy axis..................................................49
3.2
Superconductor microstrip line geometry..........................................................52
3.3
Phase constant of superconducting microstrip line filled with air at different
temperatures...................................................................................................... 54
3.4
Attenuation constant of superconducting microstrip line filled with air at
different temperatures....................................................................................... 55
3.5
Propagation constant of superconducting microstrip line on lossless
dielectric............................................................................................................ 56
3.6
Phase constant of superconducting microstrip line on lossless substrate at
different temperatures........................................................................................57
3.7
Attenuation constant of superconducting microstrip line filled with air at
different temperatures....................................................................................... 58
3.8
Attenuation constant of superconducting microstrip line on a lossless substrate
at different penetration depth.............................................................................60
3.9
Relative phase constant of superconducting microstrip line on a lossless
substrate at different penetration depth..............................................................61
xi
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Figure
3.10
Attenuation constant o f superconducting microstrip line on a lossless and lossy
substrates at different temperatures....................................................................62
3.11
Nonuniform mesh for half microstrip line structure.................................
3.12
Electric field calculated at the same physical positions using the uniform and
nonuniform discretizations with the same number of mesh points....................68
3.13
Perfectly matched layers absorbing boundary condition in three-dimension
cartesian coordinates..........................................................................................73
3.14
Perfectly matched layers absorbing boundary condition for waveguiding
structures that include different dielectric materials and metallic conductors.. 76
3.15
Virtual Layers in the processing elements memory........................................... 83
3.16
CPU time for MASAPR and DBM RS/6000 machines for problems that have
different sizes and equal number ot time steps.................................................. 85
3.17
Effective dielectric constant (er = 13). Comparison of the results obtained
fromthe FDTD with the empirical formula and the results presented in [52]... 87
3.18
Attenuation constant for copper and superconducting microstrip lines with
different strip thickness using the FDTD and FDFD approaches......................90
3.19
Effective dielectric constant for lossless, copper and superconducting
microstrip lines with different strip thickness using the FDTD and FDFD
approaches.......................................................................................................... 91
4.1
Anisotropic microstrip line on anisotropic sapphire substrate......................... 101
4.2
The propagation characteristics of anisotropic HTS on isotropic substrate using
the anisotropic FDTD....................................................................................... 102
4.3
Normalized normal-fluid, super-fluid, and total current densities at the bottom
surface o f the strip, for both the isotropic and anisotropic HTS cases, on
isotropic substrate............................................................................................. 103
4.4
Propagation characteristics for anisotropic HTS on different r-cut sapphire
substrates and on isotropic substrate with er = 10.03..................................... 106
4.5
Normal-fluid, super-fluid, and total current densities for anisotropic HTS on
different r-cut sapphire substrates and on isotropic substrate er = 10.03.......107
4.6
Anisotropic HTS coplanar waveguide on anoistropic substrate...................... 108
4.7
Propagation characteristics for anisotropic HTS coplanar waveguide on
different r-cut sapphire substrates and on isotropic substrate with er - 10.03110
Xll
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67
Figure
Page
4.8
Normal-fluid, super-fluid, and total current densities for anisotropic HTS
coplanar waveguide on different r-cut sapphire substrates and on isotropic
substrate sr = 10.03......................................................................................... I l l
5.1
Flow chart o f the nonlinear analysis algorithm................................................. 123
5.2
HTS microstrip line geometry........................................................................... 124
5.3
Normalized tangential magnetic field intensiy under the strip probed at 60 /im
and 150 } im .......................................................................................................126
5.4
Normalized tangential magnetic field intensiy under the strip probed at 150 fim
at different levels of applied power...................................................................127
5.5
Effective dielectric constant for the HTS microstrip line at different levels of
applied power................................................................................................... 128
5.6
Fractional change in the effective dielectric constant for the HTS microstrip
line with applied power w.r.t. the linear model................................................ 129
5.7
Attenuation constant for the HTS microstrip line at different levels of applied
power.................................................................................................................130
5.8
Fractional change in the attenuation constant for the HTS microstrip line with
applied power w.r.t. the linear model..............................................................131
5.9
Normalized longuitudenal super fluid current density at the bottom surface of
the HTS strip at different applied power levels................................................ 133
5.10
Normalized longuitudenal super fluid current density at the top surface o f the
HTS strip at different applied power levels..................................................... 134
5.11
Normalized longuitudenal super fluid current density at the side surface o f the
HTS strip at different applied power levels..................................................... 135
5.12
Normalized tangential magnetic field intensity at the top and bottom surface of
the HTS strip at different applied power levels................................................ 136
5.13
Fractional change in the amplitude of the frequency spectrum o f the output
pulse w.r.t the dc component at different applied power levels....................... 137
5.14
HTS microstrip resonator array filter structure................................................ 139
5.15
The calculated S21 parameter for the HTS microstrip staggered resonator array
filter................................................................................................................... 143
xiii
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Figure
Page
5.16
Comparison between the calculated and the measured S21 parameters for the
HTS microstrip staggered resonator array filter............................................... 144
5.17
Output input power relation o f the HTS microstrip resonator array bandpass
filter...................................................................................................................145
5.18
Electric field distribution in the substrate along the longitudinal direction of the
bandpass filter...................................................................................................146
6.1
Variation of the number of superfluid electrons and the magnetic field H near a
flux lin e ............................................................................................................ 153
6.2
The function ( l — )as function of h for different values of the
variable a ........................................................................................................ 162
6.3
Comparison between the calculated and the measured [106] critical magnetic
field for YBCO HTS as a function of temperature.......................................... 164
6.4
Comparison between the calculated and the measured [106] surface resistance
for YBCO HTS as a function of temperature at zero magnetic fie ld .............. 165
6.5
Comparison between the calculated and the measured [106] surface resistance
for YBCO HTS as a function of temperature and magnetic fie ld ..................166
xiv
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CHAPTER 1
INTRODUCTION
In late 1986, J.G. Bednorz and K.A. Muller o f IBM's Zurich Lab reported possible
superconductivity in La and Ba copper oxides at temperature o f 30°K. These materials
are known as High critical-Temperature Superconductors (HTS).
Following their
discovery, the HTS phenomena have been found in various ceramic oxides having critical
temperatures Tc as high as 123°K. The importance of these discoveries is based on the
fact that these materials can be cooled by using inexpensive liquid nitrogen, with a
77.4°K boiling point, rather than liquid helium so that superconductivity applications
become economically viable. This research opened a broad range o f applications in
electronic systems. The performance of superconductors is substantially superior to
normal conductors and semiconductors concerning low loss, dispersion, low noise, high
sensitivity, and highest frequency of operation.
Some HTS materials can be easily
deposited on a substrate in a thin film form. These films have critical fields and currents
that are well within the operating region of most microwave and microelectronics
circuits.
In addition, they provide lower resistance than either copper or bulk
superconductors and are able to carry much higher currents per cross section than other
metals. The low resistance o f superconducting materials is attractive for applications in
antennas, filters, delay lines, interconnects, microwave matching networks and other sub­
systems [1]. The workhorse HTS materials are YBa 2Cu 3 0 * and TIBaCaCuO. Good
quality HTS epitaxial films can be deposited on low loss dielectric substrates, such as
Silica, Sapphire, Lithium Niobate, MgO, or LaA 1 03 , to form planar microwave structures
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2
[2]. For these reasons, thin HTS films make a good choice for microwave applications,
and pave the way for extremely small conductor patterns and circuits.
Research and development performed on superconducting devices has accelerated
since the discovery of HTS materials [3]-[ 15]. One goal o f this research is to develop
devices that w ill have lower losses and better operating characteristics than normal metal
devices. The technology used to fabricate such devices is not mature yet. It is difficult to
build devices with consistently good results.
However, many devices have been
designed, built, and tested. As with other fabrication technologies, it is desirable to
simulate these devices before they are built to save time and money. To exploit the
exciting characteristics of HTS materials, accurate and flexible numerical models have to
be developed. Moreover, many fundamental electrical parameters of the superconductor
material, such as the surface resistance, conductivity, critical temperature, magnetic field
strength, and fie ld penetration are determined by measuring the propagation
characteristics and the quality factors of microwave devices [ 16]-[ 17]. Therefore, it is
also important to develop accurate numerical models to determine these electrical
parameters precisely from the measurements.
1.1 Problem Definition
The use of HTS in microwave and millimeter-wave devices presents new
challenges which are not relevant in the present design of normal metal devices. The first
major difference is that the currents in HTS strips are not limited to the surface o f the
conductors. The superconducting current flows along the cross section of the entire strip.
Hence, the field penetration effects on the device performance must be considered.
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3
Another important aspect of the physics of the HTS is the layered structure and
the associated large anisotropy [18]-[20].
It is generally believed that the two-
dimensional C u0 2 network is the most essential block of the HTS materials. The main
macroscopic parameters o f the material, the magnetic field penetration depth and the
normal conductivity, have been measured and found to be anisotropic according to the
layered crystal structure of the HTS. Experiments also reported that the critical current
density and the upper critical field are anisotropic [21 ]-[23].
Moreover, anisotropic
substrates seem to be appropriate substrate materials for HTS applications, such as
sapphire and boron nitride [24], Crystal lattice matching of the sapphire with the c-axis
oriented YBCO, small dielectric loss, and high thermal conductivity can be achieved
simultaneously [25]-[28]. The microwave device designer is now faced w ith several
choices, as the type o f material and film direction, to obtain the optimum configuration
that enhances the characteristics of the HTS material on anisotropic substrate. Hence, the
isotropy assumption for HTS materials is inappropriate for the accurate analysis o f HTS
microwave and millimeter-wave devices.
Passive HTS devices, such as filters, multiplexers, and delay lines, can provide
better performance than conventional thin-film technologies [29]-[38]. However, with
increasing input power, typical HTS devices become nonlinear and their losses increase.
Although, the high current value may not exceed the HTS critical current densities of
high quality YBCO films, they are high enough to drive the HTS into nonlinear behavior
[39]-[44], The nonlinear characteristics o f the HTS results in the generation of harmonics
and spurious products created by the mixing of multiple input signals [45]. To date, HTS
technology has only been able to address "receive" but not "transmit" applications due to
nonlinearity effects. In order to efficiently use HTS in microwave and millimeter-wave
applications, it is crucial to understand the dependence o f the field penetration depth, as
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4
well as the superconductor electron density, on the electromagnetic field inside the HTS.
Therefore, studying and simulating the nonlinearity associated with the new HTS
materials w ill open a new frontier for HTS applications.
Nonlinearity is a complicated issue in all areas o f engineering applications.
Typically, modeling and simulation of the nonlinearity is very involved and requires
extensive numerical processing. However, it is very important to include the nonlinearity
in the design process o f high power HTS applications. A simple nonlinear model, yet
accurate over a wide range o f material parameters, applied power, temperature, and
frequency, is needed for Computer-Aided-Design (CAD) o f microwave and millimeterwave devices. In addition to accuracy, speed is required so that the design process can
proceed at a reasonable rate. Until now, no nonlinear HTS models exist that are both fast
and accurate.
1.2 Motivation for the Selected Techniques
Most recent microwave and millimeter-wave applications problems are not
tractable to closed-form analytical expressions. Moreover, solutions involving the more
approximate variational and perturbadonal techniques are not suitable for these problems.
Thus, they can only be solved using numerical techniques. In some of the numerical
techniques such as the integral equation and spectral domain approaches, the methods are
known to be efficient but are restricted, in general, to structures that may involve
infinitely thin conductor patches [46], On the other hand, the Finite Difference (FD)
method is considered as one of the most flexible numerical methods used in describing awide range of structures, especially those including finite thickness conductor patches.
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5
Recently, this has become a very attractive method with the advance in speed and
memory size o f computer systems [47]-[48].
To account for the field penetration effects in HTS, the finite thickness o f the
strips is rigorously modeled using the finite difference approach. No approximations are
made to the strip thickness.
The mesh size is adjusted according to the physical
characteristics o f the HTS materials. If the mesh is uniform, it would lead to a very dense
mesh that requires a unrealistic memory storage. A graded nonuniform mesh is adopted
t
in all the work presented in the thesis [49].
An arbitrary three-dimensional structure can be embedded in a FD lattice simply
by assigning desired values o f electrical permittivity and conductivity to each lattice
electric field intensity component, and magnetic permeability and equivalent loss to each
magnetic field intensity component. The material parameters are interpreted by the FD
program as local coefficients [50]-[51]. Specification o f the media properties in this
component-by-component manner provides a convenient algorithm to. represent the
anisotropy in a media, and assures continuity of tangential fields at the interface of
dissimilar media with no need for special field matching.
Analysis and modeling the nonlinearity imposes some restrictions on the selected
numerical techniques. The frequency domain approach is based on analysis in the
Fourier transform domain. It provides an elegant tool for the reduction of the partial
differential equations of mathematical physics into ordinary ones, which in many cases
are amenable to further analytical processing. The time-dependent partial differential
equation is decoupled into a series of frequency-dependent ones. Hence, the solution is
separately carried on each frequency component. The time-domain solution can be
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6
obtained by the superposition of the results calculated at each frequency component. This
approach is widely used in problems containing linear materials. However, when a
nonlinear material is used, the partial differential equation can not be transformed to the
frequency domain. The equations for the various harmonics are no longer separable, and
the superposition technique is not allowed. Hence, the equations must be solved in time
domain.
One should notice that this is a fundamental issue.
It is not a matter of
approximation or simplification. Therefore, the Finite-Difference Time-Domain method
(FDTD) [52]-[58] is adopted in simulating the nonlinearity associated w ith the HTS
materials.
Recent advances in FDTD modeling concepts and software implementation,
combined with advances in parallel computers, have expanded the scope, accuracy, and
speed of the method. The field components at each point in the simulation domain of the
FDTD are calculated frorn their nearest points. The FDTD technique has an explicit or
semi-implicit scheme, which makes parallel computers an excellent environment to
execute such schemes. Thus, execution of a FDTD computer code on parallel machines
w ill decrease the required simulation time [59]-[61].
The Ginzburg-Landau (GL) theory is used to model the nonlinear mechanism in
the HTS materials [62]-[65], It is the only available approach where the macroscopic
parameters are field and temperature dependent simultaneously.
Thus, GL theory
provides an appropriate framework to describe nonlinear effects in the HTS materials.
However, the GL type scheme needs to be modified due to the large anisotropy
associated with the HTS materials [66].
In applying GL approach to the HTS, the
structure discreteness becomes important and one expects a crossover from anisotropic
three-dimensional behavior to quasi-two-dimensional behavior.
The GL solution is
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7
conducted in two-dimensions, and the nonlinear macroscopic parameters are calculated
for the currents flowing in the longitudinal direction o f the strip. It is well known that the
currents in the transverse plane are small for most of the planar microwave and
millimeter-wave applications. Hence, the simple linear model is used to calculate the
currents in the transverse plane [67]-[69]. One must note that a microscopic theory
describing the physics of HTS is unavailable at the present time. Numerous issues are
still in part controversial, especially those dealing with phase transitions o f the vortex
lattice [66].
For applications such as CAD, a model is needed which incorporates the accuracy
o f the microscopic theory with the speed and intuitive nature o f the phenomenological
simple model. The expressions obtained from the linear two-fluid and London models
are elegant and simple in general. They are still in use today as a qualitative model.
However, they do not consider the bidirectional coupling between the thermodynamics
and electrodynamics in a superconducting system. Hence, developing a nonlinear twofluid model which incorporates the physics inherent in the nonlinear GL theory and the
speed of the linear models would be very useful for CAD o f microwave and millimeterwave HTS devices.
1.3 Thesis Outline
The main goal of the research is to conduct a nonlinear analysis o f microwave
HTS devices using a full-wave electromagnetic simulator, which incorporates the
anisotropic behavior of the superconducting strip and the substrate simultaneously. This
approach is not only useful to predict the nonlinearity effects on high power microwave
devices performance but also can be utilized in the characterization o f HTS materials.
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8
This work presents the first rigorous effort in modeling the nonlinearity o f HTS in the
time domain.
A novel nonlinear tw o-fluid model is developed based on a
phenomenological observation. This model w ill be very useful for CAD. It is the first
simple model that blends the electrodynamics and the thermodynamics o f the
superconducting material.
Chapter 2 introduces the linear and nonlinear models o f superconducting
materials. Material models, traditional two fluid model, and London's linear model are
briefly described, while the nonlinear GL theory is explained in details. It is used to
model the nonlinearity in HTS bulk and thin film forms. The GL equations are derived
by minimizing the total free energy with respect to the complex order parameter and the
magnetic vector potential near the critical temperature o f the superconductor.
The
resulting two coupled complex vector nonlinear differential equations govern the spatial
distribution o f the order parameter and the magnetic vector potential in equilibrium. The
anisotropic three-dimensional behavior is reduced to quasi-two-dimensional behavior for
the superconducting strip used in microwave devices operating in the low gigahertz
range. The current is assumed to flow mainly in the longitudinal direction. In this case,
it can be shown that the GL equations can be simplified to two coupled real scalar
nonlinear equations. First, the one-dimensional GL equations are solved numerically for
the superconductor half-space. The finite difference method is applied to approximate
the differential equations.
An iterative scheme is adopted for the solution o f GL
equations where the first nonlinear GL equation, which resolves the order parameter, is
solved using a Newton-SSOR iteration scheme. The second equation, which corresponds
to the superconducting current, is manipulated using a linearized scheme. Numerical
results for the super fluid current density and the order parameter at different applied
power levels are presented. It is observed that as the applied power increases, the super
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9
fluid electrons near the edges decreases, the magnetic field penetrates more into the HTS,
and the edge enhancement of the current density is suppressed. The HTS material can
even loose its superconductivity near the edges at moderate values of applied power.
This algorithm is used to estimate the rf power handling capability for HTS microstrip
lines.
Chapter 3 provides the implementation of the finite difference approach in the
thesis.
The finite-difference frequency-domain for structure containing nonuniform
dielectric material is presented in detail. The general wave equation governing the
longitudinal fields is derived. The boundary conditions are inherently satisfied inside the
wave equation, and the field discontinuity across the interface between different dielectric
materials is smoothly treated. The finite difference approximation is applied to the
general wave equation.. This leads to a finite difference scheme which is valid
everywhere in the simulated sU'ucture. The full wave analysis is carried on by solving the
eigen value problem for the propagation characteristics o f the transmission line. The
algorithm is applied to a linear HTS microstrip line. The attenuation and the phase
constants are calculated at different temperatures, magnetic field penetration depths, and
lossy substrates. The slow wave effects of HTS are obsen/ed along the microstrip line.
The losses increases with increasing the temperature or the penetration depth. The
attenuation associated with a lossy substrate dominates the losses in HTS structures.
Therefore, dielectrics with very low loss tangents are required for HTS applications. The
FDTD is briefly described since it was developed through the last 30 years.
Only
interesting features peculiar to its implementation are presented. These features are
necessary to successfully model the HTS microwave devices, where the field penetration
effects need to be taken into consideration.
They are the nonuniform graded mesh
generator, the Perfectly Matched Layer Absorbing Boundary conditions (PM L-ABC),
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10
and the execution of the computer code on a Massively Parallel Processors machine
(MPP).
Chapter 4 presents a technique based on the three dimensional finite-difference
time-domain method, to model transmission lines incorporating an anisotropic
superconducting material deposited on sapphire substrate. The anisotropy of both the
HTS material and the sapphire substrate are taken into account simultaneously. The
equations are derived directly from Maxwell's equations. The approach fits the needs for
accurate computation of the dispersion characteristics o f an anisotropic superconducting
transmission line. Also, effects of anisotropy on the field distribution inside the structure
and on the current distribution inside the HTS are investigated. Interesting comparisons
between isotropic and anisotropic structures, as well as a comparison between the
characteristics of the microstrip versus coplanar waveguide is presented
In chapter 5, a nonlinear full-wave solution, based on the GL theory is developed
using the Finite-Difference Time-Domain (FDTD) technique. The nonlinearity in the
HTS is modeled by the GL equations. The anisotropic three-dimensional behavior of
HTS superconductor is reduced to a quasi-two-dimensional one.
The physical
characteristics o f the HTS are blended with the electromagnetic model using the
phenomenological two flu id model.
Maxwell's and G L equations are solved
simultaneously in three-dimensions. This time-domain nonlinear model is successfully
used to predict the effects o f the nonlinearity on the performance of HTS transmission
lines and filters. This approach takes into account the field penetration effects. The
spatial distribution of the total electrons and the number of the super electrons compared
to the normal electrons vary with the applied power. A study o f the nonlinearity effects
on the propagation characteristics, current distributions, electromagnetic field
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11
distribution, and frequency spectrum is conducted. Numerical results show a change in
the phase velocity of the wave propagating with the applied power. The corresponding
increase in the attenuation is dramatic. The presented results show that the attenuation
constant is more nonlinear than the phase velocity. The effect on the electromagnetic
field distribution is also studied. It is more pronounced near the edge o f the HTS strip.
The superfluid current density distributions change dramatically with the applied field.
The change in the frequency spectrum is successfully depicted. The scattering parameter
o f microstrip resonator array bandpass HTS filte r is calculated and the results are
compared with experimental data. The output input power relation is also depicted. The
maximum operating power for the filter without nonlinearity effects is estimated. The
field distribution of the wave propagating along the array filter is studied. The field at the
input o f the filter near the connection of the feeding line with the microstrip resonator is
high. The dimension and the layout of resonator array filter must be optimized to reduce
the nonlinearity effects on HTS filters performance.
Chapter 6 introduces a novel nonlinear phenomenological two-fluid model for
superconducting materials.
The model is based on experimental observation for
superconductors. Both the temperature and field dependence is taken into consideration
simultaneously. The nonlinear main macroscopic parameters for superconductors are
derived. An empirical formula for the surface impedance of HTS that agrees very closely
with experimental measurements for YBCO superconductors is developed.
These
compact models are validated and verified by comparing the calculated results with data
obtained from experimental measurements. This model combines the physics associated
with the G L phenomenological model and the required sim plicity obtained from the
linear London's model.
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12
Chapter 7 summarizes the results and conclusions o f this dissertation. Possible
extensions o f the research are given. An extensive reference list is attached at the end.
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CHAPTER 2
LINEAR AND NONLINEAR SUPERCONDUCTOR MODELS
An effective use o f superconductor material in engineering applications
necessitates a better understanding o f its underlying physics.
In addition, practical
system design requires the development of tractable models o f the phenomenon. The
classical model was the first attempt at describing superconductivity.
This model
incorporates the fundamental superconducting properties of zero resistance and perfect
diamagnetism into electromagnetic constitutive relations known as the London equations
[68]. The mathematical expressions, proposed by Fritz and Heinz London in 1935, are
based on empirical observation rather than theoretical analysis. They are not deduced
from any microscopic mechanisms within the material. Nevertheless, these relations are
extremely useful. Just as it is possible to design a system using Ohm’s law without a
detailed knowledge of conduction processes, a superconducting system's relevant
parameters can be calculated and estimated using the London's equations.
Despite the power of the classical model, it is limited in several ways. First, it
does not provide a comprehensive understanding o f the superconducting phenomenon.
Second, many properties of superconductors can not be explained by the classical model.
In fact, superconductivity is a manifest quantum mechanical phenomenon. Fritz London
developed a macroscopic quantum model for superconducting materials in 1948. He
showed that the two London equations were a result of the quantum mechanical nature of
superconductivity. This description not only encompasses the results of the classical
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14
model but also self-consistently describes other properties of superconductors that are
important in many applications.
The macroscopic quantum model is related to the superconducting model
proposed by Vitaly Ginzburg and Lev Landau in 1950 [63]. The Ginzburg-Landau (GL)
phenomenological theory tied the thermodynamics and the electrodynamics o f the
superconductor intimately together. This theory contributed greatly to the understanding
o f superconductivity in general.
Despite their usefulness, all the models discussed thus far are phenomenological
in nature.
In other words, these models do not give any explanation as to how
superconductivity occurs. A microscopic theory as the one provided by John Bardeen,
Leon Cooper, and Robert Schreiffer in 1957 is required in this context [62]. However, it
may not be necessary for describing most applications. Indeed, it should be noted that in
1959, L. P. Gorkov showed that Bardeen-Cooper-Schreiffer (BCS) theory reduced to the
more tractable Ginzburg-Landau theory near the critical temperature of superconductors
[65], As a result, it is known that the important conclusions of the phenomenological
models are consistent with the microscopic theories.
In late 1986, J. G. Bednorz and K. A. M uller of IBM 's Zurich lab reported
possible superconductivity in La and Ba copper oxides at temperature o f 30° K. These
materials are known as High critical-Temperature Superconductors (HTS). However, it
appears that the BCS theory may not be adequate in explaining the phenomena of hightemperature superconductivity [66].
Despite the present lack o f a fu lly successful
microscopic explanation o f the HTS, the phenomenological theories seem to work
reasonably well. Moreover, there is experimental evidence that the superelectrons in the
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15
HTS materials also carry a charge that is twice that of an electron. Consequently, the
general ideas presented in the phenomenological models are just as applicable to these
newly discovered superconductors as they are to the conventional ones.
In this chapter, we shall adopt the two-fluid model as the basis for our analysis,
focusing on the macroscopic features of superconductors. The two-fluid model assumes
that the conduction current in a superconductor comprises two separate fluids, the normal
and the super electrons fluid. The temperature dependence of the superconductors is
presented. London phenomenological linear model is briefly described. A detailed
presentation o f the nonlinear GL theory is given. The limitations of the GL model for
microwave applications as applied to the new HTS materials, and the appropriate
approach overcoming them are explained. The solution algorithm for GL equations is
discussed. Application of GL two-dirpensional solution to bulk and thin film HTS is
performed. The spatial distribution o f the superfluid current density and the superfluid
electrons density is presented for different applied magnetic field intensity.
2.1 Material Models of Superconductors
The material parameters of superconductors can be derived from the MattisBardeen formula based on the microscopic BCS theory or a classical two-fluid model.
The Mattis-Bardeen formula predicts the sudden increase o f loss at or above the gap
frequency, but the two-fluid model doesn’t [62], Despite its failure at the gap frequency,
fc, and at temperatures close to the critical temperature, Tc, the two-fluid model provides
reasonable material parameters at frequencies significantly lower than the gap frequency.
In fact, it is believed also to be a good approximate model for HTS. On the other hand,
the applicability o f the Mattis-Bardeen formula to HTS is debatable, since it only
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16
describes the extreme anomalous lim it where the coherence length, £, is large compared
with the penetration depth, A . This lim it is not realized, for example, in the YBaCuO
ceramic superconductor. Measurements show a typical value o f A(0) = 1500 A 0 fo r the
penetration depth for current flow w ithin the copper-oxygen planes but reveal
considerably larger values for current flow perpendicular to the planes. Estimates o f the
coherence length are£ = 5 - 2 0
A 0, depending on the crystalorientation [66]. Therefore,
the two-fluid model may be considered as one of the most appropriatechoices, currently
available for modeling HTS materials.
2.2 The Two-Fluid Model
The two-fluid model postulates that the conduction current in a superconductor
consists o f two separate fluids, the normal and the super electrons fluids,
J = Jn + J s
(2.1)
with
7, = n„q{vm)
(2.2)
J, = n,qv,
(2.3)
where Jn and Js are the normal and super currents, respectively; nn and ns are the
densities of the super and normal electrons, respectively; v, is the velocity o f the super
fluid electrons, and (v„) is the average velocity of the normal electrons. The total number
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17
of electrons n in the superconducting material is constant, according to the conservation
of particles law, and equals to
(2.4)
n = n„+n,
The superconducting fraction o f the conduction electrons is in the lowest energy state,
while the normal fraction is in the excited state. Under the influence o f external electric
fields, the motion o f normal electrons includes the effects o f both resistance and inertia.
:
The movement o f superconducting electrons is inertial only. This phenomenological
model was originally used by London to explain the first microwave experiment with
superconductors and it is still the framework in which many physicists and engineers
visualize many superconductive phenomena [68].
This model is satisfactory for
microwave engineers who wish to develop a simple, yet intuitive, formulation for the
superconductor phenomena without having to delve too deeply into the underlying theory
of superconductivity, which does not exist for HTS.
2.3 Temperature Dependence of Superconductors
The most successful temperature dependence of superconductors was developed
by Gorter and Casimir in 1934 [67]. They assumed that the fraction o f the conduction
electrons in the superfluid state ns varies from unity at T = 0 to zero at the temperature of
transition to the completely normal state Tc. They found that the best agreement with the
thermal properties o f superconductors was obtained when the fraction o f the super­
electrons and normal-electrons was chosen to have the form,
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IE
4
n
=
n
(2.5)
1-
It is known that application of a magnetic field stimulates the flow o f a current in
the superconductor, and energy is added to the superconductor. This can be represented
by an equivalent magnetization. A t some value o f magnetic field, the energy o f the
magnetization is larger than the condensation energy, which is the difference o f energies
between the superconducting and normal states.
So, it is more favorable for the
superconductor to be in the normal state. Since the energy o f condensation into the
superconducting state depends on the temperature, the critical value of the magnetic field
H c does also. The relation between H c and T is known experimentally to follow to within
a few percent o f the relation
(2.6)
which is consistent w ith the tw o-fluid model o f Gorter-Casimir developed for
conventional superconductor.
2.4 London Phenomenological Model
London equations are derived by combining the hydrodynamic o f a
superconductor with one o f Maxwell's equations. The hydrodynamic equations of the
superfluid, which is assumed to be collision free, is expressed
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19
dt
(2.7)
and the normalfluid, which is governed by the momentum conservation equation, is
written as
(2 .8)
where vs is the velocity of the super fluid electrons, (vn) is the average velocity o f the
normal electrons, which have an average momentum relaxation time t„ , and m and e are
the mass and charge of single electron, respectively. The first London equation is derived
by combining Eqs. (2.3) and (2.7), yielding
where A is the field penetration depth in the superconducting material, and equals to
A = /n/nJe2. The second London equation is obtained by substituting Eq. (2.7) into
Faraday's Law, which yields
(2.10)
These equations when coupled with the two-fluid model represent the fu ll constitutive
model that describes the total current in terms of the electric and magnetic fields. It is the
role o f Maxwell's equations to predict the electric and magnetic field in terms of these
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20
currents. The complete solution is achieved when the outputs o f London's equations, the
two-fluid model and Maxwell's equations are consistent.
2.5 Ginzburg-Landau Phenomenological Model
Ginzburg and Landau proposed a phenomenological extension o f the London
theory to take account of spatial variation of the macroscopic parameters o f the
superconductor. The construction of the GL theory is independent o f the microscopic
mechanism and is purely based on the ideas o f the second order phase transition. In the
G L theory, the macroscopic parameters o f the superconductor are field-dependent, which
provides an appropriate way to describe non-linear-effects.
2.5.1 Theory
The macroscopic electromagnetic London's equations are able to satisfactorily
account for the current persistence and the magnetic flux exclusion (Meissner) effect
[67], However, they do not give a completely satisfactory macroscopic picture o f all
superconducting phenomena in a magnetic field, because they regard the superconducting
material as being entirely superconducting or entirely normal. These deficiencies were
overcome in 1950 by Ginzburg and Landau, who proposed a phenomenological set o f
equations, allowing for spatial variations in the superconducting order due to the presence
of a magnetic field. The accuracy of GL equations have been validated by Gorkov based
on the microscopic theory for conventional superconductor materials [65], and extended
beyond their region of validity by other researchers [70]. However, an exact theory
describing the new HTS has not yet been developed.
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21
The construction of GL theory is independent of the microscopic mechanism and
is purely based on the ideas o f the second order phase transition only. G L theory begins
by introducing a quantity to characterize the degree of superconductivity at various points
in the material. This quantity is called the "order parameter" and denoted by y/(r). The
local order parameter is defined to be zero for a normal region and unity for a fu lly
superconductive region at zero temperature with zero magnetic field. Clearly, y / { f ) must
be closely related to the superfluid fraction in a two-fluid model, but the two quantities
are not identical. Rather, to allow for supercurrent flow, ijf( r) is taken as a complex
function and interpreted as analogous to a "wave function" for superconductivity, so that
its magnitude square can be identified with the superfluid density Ns(r),
(2.11)
It should be noted that y/(r) is not the system wave function for the electrons in the
material, since it is defined to be zero in the normal state.
However, pursuing the
interpretation o f the order parameter as a wave function, it is reasonable to write the
expression for the supercurrent J3, in the absence of a magnetic field, as
J. =
2 in
(2. 12)
where e* and m’ are the charge and mass of the entities whose wave function is y/(r), h
is the reduced Plank's constant, and i is V - l . It is equally natural to include the magnetic
field via the vector potential A :
J. =
1
2m
< /( p)| y V - e ‘A ( r ) j i/ ( r ) + ^ ( r ) f y V - e ’ A ( r ) \ v ' ( r )
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(2.13)
From the point o f view o f the two-fluid hydrodynamic model, the superfluid component
has been regarded as a "quantum fluid" and a quantum wave function is associated with
it. Reinforcing this interpretation, a more suggestive form o f Eq. (2.3) is
J,(r) = e
N s( r ) V s( r )
(2.14)
where the superfluid velocity v3(r) can be related to the phase o f the order parameter by
(2.15)
Furthermore, i f there is no position dependence to the superconductivity and the order
parameter yr(r) is independent of r , then Eq. (2.14) reduces to the London's relation
Js(r) = ^ N sA(r)
(2.16)
m
Next, a relation determining y/ must be constructed.
Ginzburg and Landau
focused on the free energy of the material Fs, which they assume as a functional of y/
and y/*. The equation determining yr is obtained by requiring that Fs, be a minimum
with respect to variations o f yr *. The condition that Fs be a minimum can be shown to
be equivalent to requiring that the superfluid and normal components o f the two-fluid
model be in stable equilibrium with respect to each other. Thus, the functional Fs plays a
role analogous to a Lagrangian o f Schrodinger wave mechanics, while its minimum value
with respect to y/ and y/* is just the free energy o f the superconducting phase in
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thermodynamic equilibrium with the magnetic field. Ginzburg and Landau chose the
following form for Fs,
2
(r )|! ) + |H (r
[ y V - e'A (r )1 ^ ( r )
(2.17)
The term F„ is the free energy, at the same temperature, of the underlying normal phase,
which could be obtained by increasing the magnetic field above its critical value. The
function F > 0 represents the free energy lowering of the system due to having formed
the superconducting correlation. For temperatures just below the zero field transition
temperature, Tc - T « T C, it should be sufficient to expand F in a power series and keep
just the first two non vanishing terms:
(2.18)
The next term in Eq. (2.16), the magnetic field energy, represents the increase in
superconducting free energy due to the expulsion of magnetic flux. The final term in Eq.
(2.16) represents the increase in superconducting free energy coming from the spatial
variations in the order parameter and from the current flow. It could also be written,
using Eq. (2.15), as
(2.19)
which illustrates more clearly that the term is in the form of a supercurrent kinetic energy
plus a stiffness against rapid changes in the superfluid density. It is the latter contribution
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24
which tends to prevent the formation o f superconducting-conducting-normal domain
boundaries. A further argument for the plausibility of the form of these terms is that Eq.
(2.13) for the supercurrent follows from a variation o f the free energy functional with
respect to A . Thus minimizing Fs, with respect to A is equivalent to solving Maxwell's
equation,
V x / / = y,
(2.20)
GL phenomenological theory results in a set o f two equations relating the order
parameter yr and the magnetic vector potential A . These equations can be reduced to a
dimensionless form by taking all the lengths in units o f the weak-field penetration depth
X, measuring the magnetic field in terms of the thermodynamic critical magnetic field H c,
and introducing a reduced order parameter normalized by its zero-field positionindependent value. Then, the dimensionless GL equations can be expressed as follows,
(2.21)
(2.22)
The subscript N denotes normalized quantities. The boundary conditions for AN and y/N
at the superconductor-insulator interface are given by
(2.23)
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25
n x ( V i r x A N) = n x n 0H ft
(2.24)
Eq. (2.23) forces the normal component o f the conduction current to vanish at
superconducting-insulator boundary, resulting in the boundary condition for the order
parameter yr. Eq. (2.24) indicates that the boundary conditions imposed on the magnetic
vector potential A correspond to the magnetic field tangential to the superconductor
surface. The dimensionless parameter k , known as GL parameter is defined as
:
K = 4 l [ e l t i ) n 0H cX \
(2.25)
where
XL = ^ m l n 0e \ l
\v~\2=fh
Hn = H /H c
an
=
a / x lh ,h c
V „ = A ,V
V'v = V / Y ~
k l and nL are the low field London penetration depth and the low field superconducting
electron density. The subscript N w ill be omitted in the rest of the thesis for simplicity.
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26
Even though the Ginzburg-Landau theory relies on the two experimental
parameters H c and X , in fact it is just a one parameter theory. Only the differences o f
k
from one material to another prevent the equations from scaling perfectly into a single
law of corresponding states valid for all superconductors. It can be seen that whereas the
magnetic field varies spatially over characteristic length o f order X , the reduced order
parameter y/ has spatial variations with a quite different characteristic length, called the
coherence length
which equals X / k . The GL coherence length is expressed as
*
(2.26)
which is the characteristic decay length for a disturbance o f y/ from its value in the
absence of currents and magnetic fields, yr„. The ratio of the penetration depth and
coherence length, and thus
k
itself, determines the relative balance between rapidity of
the variation o f the magnetic field and the order parameter in the final solution.
2.5.2 Limitations on GL Theory
The ordinary superconductors prior to the new revolution which began 1986 had
Tc < 25K and coherence lengths at zero temperature much larger than the interatomic or
interelectronic distance (of the order o f 500 A 0 or more). It is quite accurate to use near
Te mean field theory such as the GL macroscopic theory for this reason. In the new high
Tc systems, in contrast, the coherence length is small (of the order o f 15 A 0 in x-y
parallel plane and 2-3 A° along z-perpendicular axis) and the GL type schemes need to be
modified. This means that we must take account o f the strong anisotropy o f the new
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27
systems in developing the G L type scheme. In the original scheme one obtains a local
differential equations for the order parameter whereas in the new systems, besides
introducing the strong anisotropy, one may want to develop a differential equations in x-y
plane different than the z-plane.
2.5.3 G L fo r Microwave HTS Applications
The GL theory discussed so far has concentrated on a time-independent thermal
equilibrium situation, where the superconductor is in a static magnetic field. However, in
a non equilibrium situation where an electric field is also present in the superconductor,
the time-dependent G L has to be considered.
The order parameter relaxes to its
equilibrium value in the temperature dependent relaxation time xo which is given by [62]
T = ----- ^ — 7
0 8ArB|r e- T |
(2.27)
I f the characteristic time scale of the electromagnetic field ( 1 // where / is the
frequency) is much larger than x0, we can neglect the time dependence of yr and use the
time-independent GL equations. In this case, the order parameter ys responses to A so
rapidly that we can assume that it adjusts itself instantaneously to A . For example, for
Tc - T = 13K, x0 ~ 10"13s and for operating frequency in the order of tens gigahertz 1 //
~10-10.y which means that x0 is about three orders o f magnitude less than 1 //.
A common feature of the family of HTS, including YBaCuO and TIBaCaCuO, is
that they all have layered crystal structures.
It is generally believed that the two-
dimensional C u02 network is the most essential building block of the HTS materials [21].
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28
The metallic Cu-0 layers are separated by other layers of atoms. This structure causes a
pronounced anisotropy o f the electronics properties of HTS. The anisotropy in HTS
materials is a complicated issue, which is analyzed fully in a chapter 4. A model taking
this into account is the Lawrence-Doiach model. It describes the system as a stack of
superconducting layers with spacing s, each treated within a 2D Ginzburg-Landau theory,
coupled by interlayer Josephson tunneling [71].
However, th in -film microwave
transmission lines w ill favor films in which conducting sheets lie in the plane of the film.
To overcome the lim itation in applying the GL theory to the new HTS materials, the
solution for GL equations is only performed for the longitudinal superfluid z-current
component in the HTS strip. The transverse x- and y-current components are calculated
using London low field model since the current in the transverse plane is relatively small
for most microwave and millimeter-wave applications working in the low gigahertz band.
Using the London gauge V.A = 0 and assuming y/ = |y/|exp(id), the normalized GL
equations fo r the superfluid z-component current density can be sim plified to the
following expressions,
V p L = |v /|2A;
(2.28)
(2.29)
where r stands for the transverse x-y direction and V,0 = 0. The required boundary
conditions become
n x ( V t x z A . ) = MoH,
(2.30)
n- V , \y \ = 0 .
(2.31)
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29
2.6 Solution of GL Equations
The G L coupled nonlinear differential equations are solved simultaneously to
obtain the superconducting current and the order parameter. The first nonlinear G L
equation, which resolves the order parameter, is solved using a Newton-SSOR iteration
scheme. The second GL equation, which corresponds to the superconducting current, is
manipulated using a linearized scheme. These two equations are solved iteratively until
convergence, starting with
initial conditions
= 1 and ,4. = 0 .
The described
procedure is rapid and robust, and is successfully applied to the solution of G L equations
both in one- and two-dimensions. The solution converges in a few number o f iterations,
which depends on the applied magnetic field intensity.
2.6.1 Numerical Scheme
The Newton-SSOR iteration scheme for solving a system of nonlinear equation is
based on the following Newton algorithm
Xw
given
f o r k = 1, 2
r (i_1) = - F ( ^ (i_))
x^ = x ^
+ d^ -»
end (/')
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30
which is modulated by the successive over relaxation method for faster convergence and
robust solution. The Newton-SSOR algorithm can be summarized as follows
x w given
f o r k = 0,1
r ik) = - F ( x ik))
dw = 0
f o r j = 1,...... m (*)
ii
\
l<>
(><
y. = Q){k'j)y. + ( l - CQ^-’ ^ d j( / - I )
end (/)
d: = -
/(*)
ri
\
(<i
l> i
d\j) =CO^i)di + ( l - ( D a 'i ) )yi
end (/)
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31
end O')
x iM) = x (k) + d imi>))
end ( k )
where the Jacobi J is defined as follows
r?L
dxl dx2
dxm
(2.32)
J - F'(x) =
d fn
d fn
Kdxy dx2
d fn
dx.n J
The k iteration is the outer, or primary iteration, and the j iteration is the inner, or
secondary iteration. The values for m(t) depends in general on k. It could be chosen
independent of k and equals m, since the system y (t).va+1) = b w does not need to be
solved exactly for lower values o f k.
G L coupled nonlinear equations are discretized by using a centered difference
scheme for the second order derivatives :
d 2f
fi+i ~ 2 f i + f i -
dx
A.v
2
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(2-33)
32
where / stands for y/ and A . The boundary conditions are applied to the system of
equadons. The solution procedure starts by the iniual conditions y/ = 1 and A = 0. Eq.
(2.28) is manipulated using a linear solver to calculate the magnetic vector potential. The
updated A is substituted into Eq. (2.29) which is solved using the Newton-SSOR
numerical scheme with F(x(i)) < lO ^F ^x10*) as a stopping criteria. The calculated value
for y/ is then substituted back into Eq. (2.18), and the whole process is repeated until A
and y/ converge.
2.6.2 Bulk Superconductor (Superconducting Half-Space)
In this section, GL equations are solved for the bulk superconductor, shown in
Fig. 2.1 in one dimension. The applied field is parallel to the plane surface of the
superconducting half space, which is of infinite extent in this dimension. The 1-D GL
equations are
(2.34)
(2.35)
with boundary conditions
(2.36)
dx
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(2.37)
33
L
y
►
Hy
Fig. 2.1 Superconducting Half Space (bulk superconductor)
By imposing the boundary condition on the tangential field Hy at the boundary o f the half
space, G L equations are solved numerically fo r
y/(x) and A.(;t)
inside
the
superconductor. The superconducting current is calculated using the following relation
(2.38)
Js{ x ) = r ( x ) * A :{x)
In our analysis, the HTS macroscopic parameters are those measured for
YBa 2Cu3C>7-x atT = U K with Tc equals to 90K. The GL parameter
k
equals to 44.8.
The corresponding penetration depth A (T) and critical magnetic flux density n 0H c(T)
equals to .323 fim and 0.1 T, respectively.
Fig. 2.2 presents the variations o f the
superfluid current density with distance for one-dimensional superconducting slab at
different magnetic fields. These results are in excellent agreement with those in Lam et
al. [72], The peak values observed in the plots of the normalized current density are
about the same and approximately equal to 0.544. This value is very close to the GL
depairing critical current density [67]. As a matter of fact, the critical current density for
type II superconductor is about one order of magnitude less than the GL depairing critical
current density. This is explained by the vortex pinning which is not present in the
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34
conventional superconductor [18]. The nonlinearity in the superconducting material is
clear, especially near the edge of the material. The superconducting current near the edge
is suppressed in favor of the normalfluid current, which explains the increase in the losses
as the applied field increases. Also, the predicted penetration depth using G L theory is
higher than the low-field value calculated from the linear London model, which can
explain the more pronounced slow wave effects associated with the superconducting
material as the magnetic field intensity increases. Thus, the field penetration effects on
the superconducting material is represented more rigorously in the GL theory compared
to the London model. These results confirm the success of the phenomenological GL
model to evaluate a field and position dependent macroscopic parameters for the
superconducting material.
•
Lam etal.
- - - H = 1.0H,
H = 0.9Hc
a
- - H = 0.7h '
H = 0.5H
u 0.6
3
2
*3
*3
o
§•0.4
t/5
-o
io.2
o
z
0
0.5
1
2
1.5
Normalized distance (x^J
2.5
3
Fig. 2.2 Normalized superfluid current density in one-dimensional bulk
YBa2Cu30 7_x HTS.
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35
2.6.3 Thin Superconductor Film
The solution for the thin HTS film is performed in the transverse two-dimensional
plane. It is reasonable to assume that the local order parameter y/ depends on the total
magnetic field at coordinates (x,y), and denoted by y/(x,y).
To demonstrate the
versatility of our numerical scheme, two-dimensional solution for a typical HTS strip,
shown in Fig. 2.3, used in microwave and millimeter-wave devices is presented. The
strip width, W, and thickness, t, are 7.5 fim and 1.0 \±m, respectively. The strip is
divided into numerical grid cells. The generated mesh is uniform in the vertical direction
and nonuniform in the horizontal direction. The horizontal mesh size decreases near the
edges o f the strip where mere rapid change in the macroscopic parameters of the
superconductor is expected.
The mesh size is chosen smaller than the low -field
penetration depth. The HTS parameters are the same as previously described.
Fig. 2.3 YBa 2Cu3 0 7 -x HTS strip (thin film superconductor).
The typical tangential magnetic field shown in Fig. 2.4 is applied to the strip.
This Field distribution is obtained for the HTS microstrip line using a full-wave
electromagnetic simulator, which w ill be explained later. The corresponding applied
power is calculated. GL equations are solved at different levels o f applied power. The
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36
maximum r f power P crf , where the HTS m icrostrip com pletely looses its
superconductivity, is predicted. Its value equals to 920 W / cm2.
strip side
strip thickness
strip bottom
0.5
- half-strip width
-
-0.5
strip top
0
0.75
1.5
2.25
3.0
3.75
Distance (pm)
Fig. 2.4 Tangential magnetic field intensity to a YBa 2Cu3 0 7 _x HTS strip
in microstrip line configuration.
The normalized superconducting current distributions are shown in Figs. 2.5-2.7 for
different applied power levels : 834 W / cm2, 410 W / cm1, and 181.8 W I cm2 denoted
by 0.9 Pcrf, 0.45 Pcr/, and 0.2 P crf. respectively. Fig. 2.5 shows that the HTS may be
considered linear when the applied magnetic field is low, i.e. approximately less than 0.2
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37
Pcrf- As the applied power increases, the dependence of the macroscopic parameter of
the superconducting material on the magnetic field becomes nonlinear.
Applied power = 0.2 Pcrf
Jsz
0.6
0.4
0.2
—■
—
0.6
0.4
0.75
half-strip width (um)
Fig. 2.5
2.25
0.2
strip thickness (um)
Normalized superfluid current density distribution in
YBa2Cu307-x HTS strip in microstrip line configuration at 0.2 Pcrf applied
power.
A typical distribution for the normalized superconducting current density in this
nonlinear region at 0.45 Pcrf is presented in Fig. 2.6. London's equation fails to predict
the superconductor behavior in this nonlinear region, even the type II superconducting
material is still in the mixed state, and possesses a relatively good superconductor nature.
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38
Applied power = 0.45 Pcrf
Jsz
y-
0.8
0.6
0.4
s
-/
0.2
0.6
0.4
0.75
half-strip width (um)
Fig. 2.6
2.25
0.2
strip thickness (um)
Normalized superfluid current density distribution in
YBa2Cu307-x HTS strip in microstrip line configuration at 0.45 Pcrf
applied power.
Fig. 2.7 demonstrates that the superconducting material partially looses its
superconductivity at high rf power 0.9 Pcrf.
It is obvious that the material lost its
superconductivity near the edge of the strip where the singularity in the field is expected.
On the other hand, the material behaves as a good superconductor at the center of the
strip. This behavior w ill not only introduce nonlinearity effects but it w ill also increase
the noise.
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39
Applied power = 0.9 Pcrf
strip thickness (um)
half-strip width (um)
Fig. 2.7
Normalized superfluid current density distribution in
YBa2Cu307-x HTS strip in microstrip line configuration at 0.9 PCTf applied
power.
Fig. 2.8 presents the normalized superfluid electron density at 0.45 Pcrf. It is clear that
the bottom part of the strip looses its superconductivity much faster than the top section.
This can be explained by the effect of the dielectric substrate underneath the strip, which
increases the field intensity at the bottom side. Thus, superconducting applications w ill
favor low dielectric substrate to decrease the nonlinearity effects in microwave and
millimeter-wave applications.
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40
Applied power = 0.45 Pcrf
y<
Ns(x,y)
0.8
0.6
0.4
0.2
0.6
0.4
0.75
half-strip width (um)
Fig. 2.8
2.25
0.2
strip thickness (um)
Normalized superfluid electron density distribution in
Y B a 2Cu307-x HTS strip in microstrip line configuration at 0.45 Pcrf
applied power.
2.7 Summary
In this chapter, the linear and nonlinear models of superconductor are presented.
The London model is briefly reviewed. Detailed analysis of Ginzburg-Landau theory is
illustrated.
The application of GL theory to the nonlinear modeling o f HTS
superconducting bulk and thin films is discussed. The anisotropic three-dimensional
behavior of HTS superconductor is reduced to a quasi two-dimensional one. The solution
of GL equations is performed under the assumption of negligible transverse current. The
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41
complex vector GL nonlinear differential equations are simplified to a coupled set of real
scalar differential equations governing the spatial variations o f the order parameter and
the magnetic vector potential.
The simplified equations are normalized fo r the
convenience o f the numerical calculations.
The GL coupled nonlinear differential equations are solved simultaneously to
obtain the superconducting current and the order parameter. The first nonlinear GL
equation, which resolves the order parameter, is solved using a Newton-SSOR iteration
scheme. The second GL equation, which corresponds to the superconducting current, is
manipulated using a linearized scheme. These two equations are solved iteratively until
convergence. The described procedure is rapid and robust, and is successfully applied to
the solution o f GL equations both in one- and two-dimensions. The required boundary
conditions for the thin film solution is obtained from the full-wave simulator described in
chapter 5.
Numerical results show that as the magnetic field at the boundary increases, the
order parameter near the edges decreases, indicating the breaking of superfiuid electron
pairs. The HTS material can even loose its superconductivity near the edges at moderate
value of applied power. The rf critical power density for HTS strip used in microwave
applications depends not only on the physical characteristics o f the HTS but also on the
structure configuration. Special attention has to be placed in predicting the efficient
operating region of the HTS materials used in microwave and m illimeter-wave
applications.
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CHAPTER 3
FINITE-DIFFERENCE APPROACH
Most partial differential equations describing recent physical phenomena of
science and engineering are not amenable to closed-form solution. Solutions involving
the more approximate variational and perturbational techniques are not suitable for these
problems. Thus, they can only be solved using numerical techniques. In some numerical
techniques such as the integral equation and spectral domain approaches, the methods are
known to be efficient but are restricted, in general, to structures that may involve
infinitely thin conductors.
On the other hand, the finite difference (FD) method is
considered as one o f the most flexible numerical method used in describing a wide range
of structures, especially those including finite thickness conductors. Moreover, the FD
solution of Maxwell's equations is one of the most suitable numerical modeling
approaches for the electromagnetic analysis of volumes containing arbitrary shaped
dielectric and metal objects.
FD is relatively simple in its concept and execution.
However, it is remarkably robust, and provides highly accurate modeling predictions for
a wide variety of electromagnetic wave interaction problems [50]-[58].
The FD
technique is based on approximations which permit replacing partial differential
equations by finite difference equations. The formulation is algebraic in form, relating
the value o f the dependent variable at a point in the solution region to the values at the
neighboring points. In this analysis, both the finite difference frequency domain (FDFD)
[49], [80] and the finite difference time domain (FDTD) [74]-[84] are employed to study
the linear and nonlinear electromagnetic characteristics of microstrip lines, coplanar
waveguides, and microstrip multiarray resonators filters.
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43
In previous research work, the transmission line characteristics are described in
terms of their integrated quantities, and strip thickness is mostly treated by some
approximation. Also, some studies of transmission line properties of strip and microstrip
lines have been made with the assumption that TEM modes are supported by both
structures. This assumption may be valid for the strip line structures w ith uniform
dielectric filling. However, it is not tenable when an air-dielectric interface is present, as
in microstrip lines, for example. Moreover, these calculations may be valid at low
frequencies, but not at higher frequencies. Modeling superconducting microwave and
millimeter-wave devices imposes new restrictions on the selected numerical approach.
The strip thickness can not be approximated by infmitesimally thin perfect conductor.
The field penetration inside the strip must be considered.
The problem o f calculating the propagation characteristics o f superconducting
microstrip lines are tackled by different approaches. The microstrip line is modeled
using a modified spectral-domain immitance approach, based on the transverse resonance
method, which takes into account the complex resistive boundary conditions [4]-[5], The
spectral domain formulation is also used in conjunction with the method o f moments [ 6 ].
An equivalent single strip, which has the same internal impedance as the original
superconducting line, is utilized in the phenomenological loss equivalence method [7].
The superconducting strip is replaced by a frequency-dependent surface impedance
boundaries in the characterization of a thin film line [ 8 ], Monte-Carlo method is used to
calculate the propagation characteristics of superconducting interconnects [33]. Despite
the usefulness of the above mentioned approaches, the field penetration effects are not
taken into account. This effect must be considered, especially when evaluating losses for
the propagating wave inside the superconducting microstrip line [ 10 ]-[ 11 ].
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44
In this chapter, technique based on the FDFD method is presented to model a
microstrip line incorporating a superconducting material. The equations are derived
directly from Maxwell's equations. The approach fits the needs for accurate computation
o f the dispersion characteristics of a superconducting transmission line. The model is
flexible, and can be used for any planar transmission line structures containing HTS
materials.
It incorporates all the physical aspects o f the HTS through London's
equations. The electromagnetic characteristics and the required boundary conditions in
the structure are represented using Maxwell's equations. The physical characteristics o f
t
the HTS are blended with the electromagnetic model by using the phenomenological two
fluid model. The complex propagation constant is calculated. Hence, the losses inside
the superconductor material are considered. The technique is also used to analyze
microstrip line structures at different temperatures and different frequencies. The effect
o f the losses associated with a lossy dielectric substrate on the performance of the
superconducting microwave transmission line is also investigated. The FDTD is briefly
described in this chapter.
presented.
Only interesting features peculiar to our approach are
They are the nonuniform graded mesh generator, the Perfectly Matched
Layer Absorbing Boundary Conditions (PML-ABC), and the execution o f the computer
code on Massively Parallel Processors machine (MPP) [83]. These features are necessary
to successfully model the HTS microwave devices, when the field penetration effects
must be considered. Finally, a comparison between the FDFD and FDTD solutions is
conducted.
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45
3.1 Finite-Difference Frequency-Domain (FDFD)
3.1.1 Wave Equation for Superconducting Microwave Structure
The relation between the HTS current density and the fields inside the
superconducting material is described by London's equation [ 6 8 ]. The dependence of the
effective penetration depth (XjJ, as well as the ratio of the densities o f superconducting
and normal electrons, on the temperature are expressed using Gorter-Casimir expressions
t
[67]. According to this model, the superconductor has almost constant penetration depth
at a temperature well below the critical temperature Tc and the field penetration remains
almost unchanged with frequency. In general, the effective penetration depth is greater
than the one depicted in the two fluid model [18].
Combining Ampere's Law with the two fluid model, and using London's equation,
we get the follow ing relation between the electric and magnetic fields inside HTS
materials [19]
V x H = j( 0 £0£sE
(3.1)
where the complex relative dielectric constant ej is
£. =
j ~
CO£„
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(3.2)
46
Here, the electrical properties of the superconductor are assumed isotropic. The negative
dielectric constant for the superconducting strip may be explained by the stored electric
kinetic energy associated with the superconductor electrons pair motion. The negative
real part of the dielectric constant means that the electromagnetic wave is expelled from
the HTS material without attenuation. The negative imaginary part o f the dielectric
constant shows that the structure expels the wave from its inside with attenuation due to
the presence of normal electrons.
Consider an electromagnetic wave propagating in the +z direction with a
propagation constant y = a + //}, that is yet to be determined. The 3-D problem is
transformed into a 2-D one by assuming that the transmission line is infinite in the +zdirection. A ll the fields have an e~r' dependence on z, so that d/dz can be replaced by
—yz.
The problem may be formulated for either T M Z or TEZ. In this section, the
equations are derived for the T M Z mode, for simplicity only. The TE Z derivation is
analogous to the T M Z case.
The wave equation governing the longitudinal electric field E . in the case of
uniform dielectric constant at the points inside air, dielectric, or superconductor regions
may be written as,
i r
+ 1r
+,^ =
0
where hp is the transverse wave number, and is expressed as follows;
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(33)
47
h; = y 2 + co2p o£p
(3.4)
and the subscript "p" designates air, dielectric, or superconductor regions. The dielectric
constant, ep, equals e0er in case of lossless dielectric, equals £0er( l- y 'ta n 5 ) in case of
lossy dielectric,
and equals
£o£r[ ( l - l / t u 2/z0£o£rA ^ ) - 7'(crn/G)£(,£r )j in sid e
the
superconductor.
At the interface points of the FD simulation domain, the dielectric constant, £p, is
equal to the average of the dielectric constants of the materials across the interface. The
boundary conditions are inherently satisfied inside the wave equation, and the field
discontinuity across the interface is smoothly treated. When the interface between two
different materials lies parallel to the x-axis, the wave equation has the following
expressions,
d 'E .
d 2E.
Bx"
CO~jJ.0£p
■) d ' 1
b e
By2 ~ r ~fy h i dy
'
+ h2
pE: = 0
(3.5)
and when the interface is parallel to the y-axis, the corresponding wave equation can be
written as,
t d
d 2E.
03 I10£
op
dx2
~ T
1Tx
1 BE,
yh2
x dx
+^J±
2 + h2
'P„ E. = 0
By
(3.6)
The general wave equation for a nonuniform dielectric constant can be obtained by
combining (3.5) and (3.6). It is expressed as follows,
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48
d 2E.
co^o
0£p
dx2
d 2E,
*— r
dy'
1 d E .]
d / 1 dE ^
dx h: dx
+ h lE = 0
(3.7)
\ hyi dy / /
where subscripts "x" and "y" designate x and y directions, respectively. These equations
are derived in a form convenient for formulating an eigenvalue problem using the finite
difference approach. Also, it successfully describes the points lying on the interface
between the different material constituting the microwave structures.
The fu ll wave analysis can be carried out by applying the same procedure to the
TEz mode, then combining both modes together. We may conclude that this approach,
presented above, results in a consistent formulation for the wave equation throughout the
structure, including the interface points. The technique is general, and can be applied to
any superconducting microwave structure.
3.1.2 Eigenvalue Problem
In general the wave equations governing linear transmission line structure result
in an eigenvalue problem when discretized using the finite difference approximations.
The eigenvalues are the propagation constants of the structure while the corresponding
eigenvectors are the possible modes in the structure. This section explains how to
calculate the complex propagation constant at a given frequency.
The finite difference method is used to approximate the wave equations for Ez
presented in the previous section. Employing the notation shown in Fig. 3.1, Ez becomes
Ezij= E : {iAx + jA y)
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(3.8)
49
Substituting in the general wave equation shown in (3.7), and after some
manipulation, we get the following general finite difference equation,
1
Ax 2
1
,2
(
£ ‘+ l.j
hi.j
X
p
1
^3+ 1,j
(e
Ay 2
1
ftw j
u
Ax 2
1
f t ^P
£.
.
*'
ft 1
f £u -l f t 1
£ M ■; h j j
1
Ax"
£ i.j
h i 1,;.
^£fc i . / + l fti2. j
Ay2 V
P
+
ft.J
‘ Ay 2 I £.v f t - J
f t +I,
l2
f£
h2
i,;+ l
^ £;
E - j . j -1 +
/l; j
£ , ;.
/I* ,.,.
£
h*
^
fci . ; - l
n i.j
P
i.j
h2
i,j-l y
i.j -1
i+1, j
i- l.j
i, j +1
Fig. 3.1 Mesh indices ij and the corresponding xy axis.
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(3.9)
50
The finite difference equation is consistent with the partial differential equation
representing the wave equation. The coefficients inserted are only used at the interface
between the different materials constituting the structure. The discontinuities at the
interface are carefully handled to satisfy the matching conditions without generating
undesired numerical singularities.
The discontinuity in the field in the direction
perpendicular to the interface, as a result of different dielectric values at both sides, is
treated using the electric field-dielectric constant product. Similarly, the finite difference
equations for the homogeneous media, the interface parallel to the x-axis, and the
interface parallel to the y-axis can be easily derived. This results in a finite difference
scheme which is valid everywhere in the transmission line structure. There is no need to
impose unnecessary boundary conditions anywhere inside the structure.
, It is well known for the case considered in this analysis for the microstrip line that
the modes may be subdivided into the TEz and TM Z modes. It is apparent that the lowest
T M Z mode is of great interest. Therefore, we w ill focus our analysis to this mode. Using
the even symmetry o f this fundamental mode, the numerical model is simulated over half
of the structure. A perfect magnetic wall is inserted at the plane o f symmetry. The
ground plane is represented by a perfect electric conductor, for simplicity. The open
boundaries can be closed with perfect conducting walls, either electric or magnetic. The
effect of these walls on the final solution is minimized by placing them relatively far
away from the strip.
A nonuniform mesh is generated over the transmission line cross-section. The
numbers of patches increase in the area where rapid field variations are expected. The
electric parameters are averaged over the patches lying at the interface between different
materials, as explained before. The mesh is constructed such that all the interfaces and
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51
the boundaries lie exactly on one side o f a patch. The nonuniform mesh generator is
explained later.
The finite difference approach presented above results in a finite set o f algebraic
equations, which have the form o f a matrix eigenvalue problem,
A E .= X E .
(3.10)
where A is the eigenvalue of the matrix. In our case, the eigenvalue value A equals to the
negative of the square of the propagation constant
Some elements of the matrix A are functions o f y2. These elements represent the
points on the interface between the different materials in the structure. Physically, they
couple the solution of the wave propagating in the different regions of the structure.
The eigenvalue problem is solved using a direct method. First, a reasonable value
for the propagation constant y is estimated and supplied to the numerical solver, which
produces a more accurate estimate for y. The updated propagation constant y value is
supplied again to the numerical solver. This iterative process continues until the solution
converges. Thus, an accurate propagation constant is obtained iteratively.
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52
3.1.3 FDFD Application to Microstrip Line
Numerical results fo r the finite difference method are generated fo r two
configurations. One configuration is a superconductor microstrip line w ith a lossless
dielectric substrate o f £d = 23, as shown in Fig. 3.2. The other structure is completely
fille d w ith air.
The latter structure is relatively simple.
It is used to validate the
generated results, since it is a pure TEM structure. Therefore, it provides an accurate tool
for comparing the results. Both configurations have the same dimensions; strip width W
= 500 pm, substrate thickness d = 450 pm, and superconducting strip t = 1.0 pm. The
characteristics of the superconducting material are as follows, penetration depth at T = 0
K is X(0) = 0.18 pm, the conductivity for the normal electron gas at T = Tc equals to
<Jn = 104 S/cm, and Tc = 100 K. The results are calculated at two different temperatures,
the liquid nitrogen boiling point T = 77 K, and T = 89 K.
Air
W/2
Dielectric
x
Fig. 3.2 Superconductor microstrip line geometry.
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53
The phase constant and relative phase constant shift for the air-filled structure are
shown in Fig. 3.3. The propagating mode is a quasi-TEM mode, since E. must exist due
to the losses. The percentage change in the phase constant is expressed as follows,
M
P
= W O - / ? ( I r . o ) . x ioo
(3.U )
P(T=0)
The slow-wave effect o f the superconducting material can also be observed. Fig. 3.4
shows the corresponding attenuation constant. The attenuation increases with frequency
and temperature as expected. As the critical temperature for the superconductor material
is approached, the attenuation increases dramatically.
The phase constant o f the superconducting microstrip line as function o f
frequency, at T = 0 K, is shown in Fig. 3.5. The results are verified by comparing them
with the phase constant of a microstrip line with perfect conductors [85]. As expected,
the two curves are very similar. The phase constant of the superconducting structure is
slightly greater than that of the perfectly conducting structure due the internal inductance
of the superconducting material.
The attenuation characteristics for the superconducting microstrip line on a
lossless dielectric substrate o f Ed = 23 at different temperatures, are illustrated in Fig. 3.6.
The attenuation increases with frequency and temperature as shown before.
The
corresponding phase constant and relative phase constant shift are shown in Fig. 3.7. The
propagating mode is T M Z. The low dispersion of the superconducting transmission line
is clear. The percentage changes in the phase constant at T = 77K and T = 89K are also
depicted in Fig. 3.7. It should be noted that the slope of the phase constant /3-curves is
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54
always positive in Fig. 3.3 and Fig. 3.7. The fact that A (3J[3 curves have negative slope
in Fig. 3.3 does not imply that the phase constant /}-curve itself has a negative slope. As
the temperature increases, the phase constant slightly increases. The increase in the
attenuation constant with the temperature is due to the increase of the normal electron
current penetration in the material. The increase in the phase constant with temperature
may be explained by the slow wave effect resulting from the increase in the internal
inductance associated with the superconducting material. Although, the phase constant
shift is small, it is predicted by our calculations.
0.01
1400
73
Phase constant
1200
0.008
2sT
cr.
<
o
5
1000
0.006
3
o
o
o
t/5
o
U
o
T =0K
800
eu
T= 89K
T=77K
600
0.004
0.002
>
"GO
400
10
20
30
40
50
60
70
Frequency (GHz)
Fig. 3.3 Phase constant of superconducting microstrip line filled with air
at different temperatures.
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0.0004
0.0003
s
To
§
T=89 K
0.0002
3
C
3
3
C
O
<
5
T=77K
0.0001
10
20
30
40
50
60
70
Frequency (GHz)
Fig. 3.4 Attenuation constant of superconducting microstrip line Filled
with air at different temperatures.
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6000
5000
4000
3000
perfect conductor
2000
1000
10
20
30
40
50
60
70
Frequency (GHz)
Fig. 3.5
Propagation constant of superconducting microstrip line
lossless dielectric.
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0.004
/H
0.003
T=89 K
aC*5
o
u
0.002
s
o
•3
3
5
<
T=77 K
0.001
10
20
30
40
50
60
70
Frequency (GHz)
Fig. 3.6 Attenuation constant of superconducting microstrip line filled
with dielectric at different temperatures.
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58
6000
0.01
Phase constant (rad/m)
0.008
Phase constant
4000
0.006
T=0K
3000
0.004
T= 89K
T=77K
2000
0.002
Relative Phase constant shift (Ap/p %)
5000
1000
10
20
30
40
50
60
70
Frequency (Ghz)
Fig. 3.7 Phase constant o f superconducting microstrip line on lossless
substrate at different temperatures.
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59
To study the effect o f different HTS materials on the line performance, structures
with different London's penetration depths are investigated. A ll other parameters are
unchanged. The attenuation and phase constants at T = 77K are shown in Fig. 3.8 and
Fig. 3.9, respectively. The increase in the attenuation constant with the penetration depth
is due to the increase of the normal electron current penetration in the material, and the
corresponding decrease in the super electron current in the HTS material. The increase in
the phase constant can be explained by the slow wave effect resulting from the increase in
the internal inductance associated with the superconducting material, as previously stated.
The former results are for HTS microstrip lines on a lossless dielectric substrate,
to emphasize the effect of a superconducting material on a microwave structure (e.g., the
slow wave effect and the low losses of the superconducting material). The characteristics
o f a microstrip line, deposited on LaGa0 3 /LaAlC >3 substrate, are shown in Fig. 10. The
substrate loss tangent equals 5.x 10‘5. The microstrip line has the same dimensions and
parameters as previously described. The attenuation constants o f the superconducting
microstrip line, with both lossless and lossy substrates, at two different temperatures, T =
77K and 89K, are shown in Fig. 3.10. The attenuation is dominated by the dielectric
substrate losses, since the attenuation constant for the lossy substrate case at the two
different temperatures of interest is almost the same.
Also, one may note that the
attenuation constant is drawn on a logarithmic scale. The effect o f the losses, due to the
lossy dielectric substrate, on the phase constant is extremely small.
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0.02
T = 77K
s 0.015
A. ( (Jm ) =
,558g -
i
1
CJ
0.01
c
0
ars
3
1
<
Am
0.005
.3553
.2235
10
20
30
40
50
60
7 0
Frequency (GHz)
Fig. 3.8 Attenuation constant of superconducting microstrip line on
lossless substrate at different penetration depth.
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0.01
T = 77K
0.008
k (pm) = .5588
c/5
|
0.006
c/5
a
0
u
1
.4471
0.004
s
>
I
"o
.3353
0 .0 0 2
10
20
30
40
50
60
_
70
Frequency (GHz)
Fig. 3.9 Relative phase constant o f superconducting microstrip line on
lossless substrate at different penetration depth.
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ILossy dielectric]
T =89kl
0.01
I Lossless Dielectric!
1
<
0.001
0.0001
10
20
30
40
50
Frequency (GHz)
60
70
Fig. 3.10 Attenuation constant of superconducting microstrip line on
lossless and lossy substrates at different temperatures.
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63
3.2 Finite-Difference Time-Domain (FDTD)
The FDTD applies second-order accurate central-difference approximations for
the space and time derivatives of the electric (E) and magnetic (H) field intensities
directly to the differential operators of the curl equations. It is a marching-in-time
procedure which simulates the continuous actual waves by sampled-data numerical
analogs propagation in a computer data space. An arbitrary 3-D structures can be
embedded in an FDTD lattice simply by assigning desired values o f electrical
permittivity and conductivity to each lattice electric field intensity (E) component, and
magnetic permeability and equivalent loss to each magnetic field intensity (H)
component. These are interpreted by the FDTD program as local coefficients for the
time-stepping algorithm [51]. Specification of the media properties in this componentby-component manner assures continuity of tangential fields at the interface of dissimilar
media with no need for special field matching. Moreover, recent advances in FDTD
modeling concepts and software implementation, combined with advances in computers,
have expanded the scope, accuracy, and speed of FDTD modeling to the point where it
may be one of the best choice for large electromagnetic wave interaction problems. Also,
FDTD technique has an explicit or semi-implicit scheme, where parallel computers
provide a good environment to run such schemes [61].
The FDTD is well known [50]-[58],
Hence, it is only briefly discussed.
Specifically, only the features peculiar to its implementation in the thesis are described.
These features are necessary to successfully model the HTS microwave devices, where
the field penetration effects need to be taken into consideration. They are the nonuniform
graded mesh generator, the perfectly matched layer absorbing boundary conditions, and
the execution of the computer code on massively parallel processors machine.
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64
The follow ing finite difference equation for Ez is obtained by combining
Ampere's law with the two fluid model,
A/
e : +1(/, j , k
+ 1/ 2 ) = —
e : (/, j , k + 1/ 2 ) + £,z O 'M -+ i/ 2 )
;
1 , <fz{U.k + l/2) A ’ J’
' } 1( az{i,j,k + \l2)
2e~(i,j,k + 1/ 2 )
2ez(i,j,k + l/2)
(H n
y+'/2{i +1/2 J , k + 1/2) - / / ; +V2(/ - 1/ 2 , 7 , k + 1/2))
z i r ( / - l / 2 ,;,A: + l/ 2 )
+1/2,/: +1/2) - H n
x+V2{ i , j -1 /2 , k + 1/2))
(3.12)
A y (/,7 -l/2 ,* + l/2)
J”+i/2( i\ M + 1/2 )]
where the superconducting current density Js; is obtained from the discretized form of
London equation [67], which can be written as,
• C V2( i . M + 1/2) =
+ 1/2) +
At
VoK{iJ,k + 1/ 2 )
En
z { i , j , k + 1/2)
(3.13)
It is obvious that the physical parameters of the different materials are defined at each
point in the three dimensional simulation domain.
Similarly, the finite difference
equations required for Ex and Ey can be obtained. The finite difference equations for the
H field components obey the modified Yee's algorithm [51].
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65
3.2.1 Nonuniform Finite-Difference Mesh Generator
The developed three-dimensional finite-difference scheme is capable o f modeling
the finite thickness of the HTS strip. No approximations are made to the strip thickness.
This leads to a very dense uniform mesh that requires unrealistic memory storage. To
alleviate this problem, a graded nonuniform mesh is implemented along the cross section
o f the waveguiding structure. This nonuniform discretization imposes some restrictions
which do not exist with the uniform discretization.
The computational domain is
discretized according to the following expression
(3.14)
where Ax is the mesh size, W distance to be discretized, n number of points, and p is the
mesh resolution factor. The smallest mesh size is chosen inside and around the HTS
strip.
It is equal to a fraction of the magnetic field penetration depth for the HTS
material, or the skin depth for normal lossy conductors. The mesh resolution factor p
must be optimized to minimize the dispersion introduced by the nonuniform
discretization. In general, the ratio between two consecutive mesh step distances must
not exceed 2. The Courant stability condition, which determines the time increment, At,
for the FDTD algorithm, equals to
where
is the maximum wave phase velocity within the structure under investigation,
8 ^ the smallest mesh size, and n is the number of space dimensions. For practical
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66
reasons, it is best to choose the ratio of the time increment to spatial increment as large as
possible yet satisfying Eq. (3.14). I f the physical mesh points are aligned with the
electric field components, the mesh spacing for the magnetic field components could be
calculated according to the following expressions
+
(3.16,
To ensure the accuracy o f the-computed results, the spatial increment must be small
compared to the wavelength. As a rule of thumb and to reduce the truncation and grid
dispersion errors, the smallest wavelength A ^ , i.e. at the highest frequency, existing in
the computational domain must be at least 2 0 times greater than the maximum step size
(5max in the discretized mesh,
^•min - 20<5max
(3.17)
One must note that the mesh is uniform ly discretized along the direction o f wave
propagation for most of the guided wave problems considered in this thesis.
Fig. 3.11 shows the nonuniform finite-difference mesh for half microstrip line
structure along its cross-section. It is seen that the conducting strip is approximated by an
adequate number of points, which corresponds to the skin depth for the normal conductor
and the magnetic field penetration depth for the superconductor. In our case for strip of
dimension of 15 pm x 1 pm, the number o f nodes are 24 x 10. The mesh is dense around
the strip edges where the variation in the fields is expected to be large.
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67
Fig. 3 .11 Nonuniform mesh for half microstrip line structure
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68
The calculated electric field using the FDTD approach for a microstrip line is
presented in Fig. 3.12. The substrate height is 10 pm, and the conducting strip o f width
15 jim is assumed to be perfectly conducting w ith zero thickness.
The results are
obtained for the uniform and nonuniform discretization meshes. The mesh size for the
uniform case is 0.625 pm, while the smallest mesh size in the nonuniform discretization
case equals to 0.125 pm. The number of points used are 127 x 64 x 127 for both cases.
Both cases have the same simulation time with equal time step.
0.8
Nonuniform mesh
— — Uniform mesh
S
0.575
tS
c_>
•s
b
3
0.35
■N
3
Z
0.125
-
0.1
0
0.2
0.4
0.6
0.8
1
Time (ps)
Fig. 3.12 Electric field calculated at the same physical positions using the
uniform and nonuniform discretizations with the same number of mesh
points.
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69
The electric fields are probed underneath the center o f the conducting strip at different
positions along the direction of propagation. It can be observed that the nonuniform mesh
discretization gives better convergence than the uniform case. There is no need to
decrease the mesh size in an uniform scheme, which in turn increases the number nodes
required to represent the same structure. This w ill lead to unrealistic memory allocation.
3.2.2 Absorbing Boundary Conditions: Perfectly Matched Layers
The FDTD has been applied to various electromagnetic problems such as
scattering, radiation, and integrated-circuit component modeling [50]-[58],
Many
applications involve modeling electromagnetic fields in an unbounded open space. Due
to the limited storage space o f computers, numerical computation domains must be finite.
A certain type of boundary condition, the so-called the Absorbing Boundary Condition
(ABC), must be applied on the outer boundaries of the computation domain to simulate
the unbounded physical space. Practical absorbing boundary conditions usually cannot
absorb outgoing waves completely, and generate some error in numerical solutions,
especially the frequency domain parameters. With better absorbing boundary conditions,
not only is the numerical solutions more accurate, but the outer boundaries may be
brought closer to the modeled targets, resulting in considerable savings in computer
memory space and computation time. Special attention must be given to unshielded
microwave devices. The side and top walls must be placed relatively far from the device
to include all the physical waves propagating in the device. It is well known that the
Fourier transform of the time domain results is very sensitive to numerical errors, notably
those resulting from the imperfect treatment of the absorbing boundary conditions used to
truncate the numerical computations of an open domain [52], The available absorbing
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70
boundary conditions for the discretized wave equations are either not accurate enough or
require impractical large computer memory to model microwave devices. Also, they are
originally developed for lossless uniform dielectric configuration, where no metal strips
exist. The effects of these ABC on the evanescent waves which may be present in some
structures are not studied, and in general do not work for these cases. A dispersive ABC
is implemented in [ 86 ], but it effects are not perfect. Most o f the implemented ABC are
based on the one-way wave equations suggested by Mur [87]. Actually, the stability o f
the absorbing boundary conditions can not be achieved exactly; there w ill always be
some non-physical reflected wave returning to the computation domain due to the
boundary treatment.
Most of the accurate numerical computation performed for
microwave devices are obtained by making the computation domain sufficiently large
and stopping the computation after the useful information has been obtained [56],
Recently, a new type o f absorbing boundary condition algorithm has been
developed, which greatly improves the accuracy of the local absorbing boundary
conditions. The new technique, called Perfectly Matched Layer (PML), was developed
by Berenger in 1994 [ 8 8 ], It was validated and extended by Taflove et al. [89]-[90]. This
approach is based on the use of an absorbing layer especially designed to absorb without
reflection the electromagnetic waves. The theoretical reflection factor of a plane wave
striking a vacuum-layer interface is null at any frequency and at any incidence angle. So,
the layer surrounding the computational domain can theoretically absorb without
reflection any kind of wave traveling towards boundaries, and it can be regarded as a
perfectly matched layer.
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71
3.2.2.1 Theory of the PML Medium
A reflectionless transmission of a plane wave propagating normally across the
interface between free space and an outer boundary layer is achieved by satisfying the
following condition
a
o
(3.18)
where cr and cr* denotes the electric and magnetic conductivities, respectively, o f the
outer boundary medium. In other words, impedance o f the outer boundary medium is
matched to the impedance o f the air. Layers o f this type have been previously used to
terminate FDTD grids. However, the absorption is thought at best to be in the order of
the analytical ABC's because o f increasing reflection at oblique incident angles [ 8 8 ], The
PML technique introduces a new degree of freedom in specifying loss and impedance
matching by splitting the fields components into sub-components. The magnetic field
component H . can be splitted into H .x and H 0 , as example. In three-dimensions, all six
Cartesian field vector components are split, and the resulting PM L modification of
Maxwell's equations yields 12 equations, as follows:
(3.19)
(3.20)
(3.21)
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72
dE„
£
iL + ( jE =
£° dt
° y‘b»
£ — h .+ a E = 0 dt
z yz
dt
+* a
E' - r
w r -“
d iH ^ + H )
v ^
dy
’
dE
d (H yx + H r )
£ ^ + £ 7 . ^ = - - L2L__2i2
dt
dz
dz
° dt
dx
f
(3.22)
(3.23)
( 3 .2 4 )
dx
The wave in the PML layers propagates with exactly the same velocity as the interior
medium, but decays exponentially with distance. Also, the wave impedance of the PML
medium exactly matches that of the interior region regardless of the angle of propagation
or frequency. This makes the PML approach an excellent boundary truncation for
microwave and millimeter-wave devices.
Berenger proposes a lossless free-space FDTD computational zone surrounded by
a PM L backed by perfectly conducting (PEC) walls, as shown in Fig. 3.13. A t both the
upper and lower sides of the grid, each PML has a t and ax matched according to Eq.
(3.18) along with a y = cr* = cr. = o ’. = 0 to permit reflectionless transmission across the
interior region-PML interface. A t both the left and right sides o f the grid, each PML has
a y and cr* matched according to Eq. (3.18) along with a x = cr'x = cr. = cr[ = 0. A t both
the front and back sides of the grid, each PML has cr. and cr* matched according to Eq.
(3.18) along with csx - cr* = crv = <j* = 0. A t the four normal edges o f the grid (along xaxis), where there is overlap of two PML's, each PML has o \, cr*, cr. and cr*, which are
set equal to those of the adjacent PML's, along with a x = a x = 0. A t the four transverse
edges of the grid (along y-axis), each PML has crx, cr*, cr. and cr‘ , which are set equal
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73
to those o f the adjacent PML's, along with a y = cr* = 0. A t the four longitudinal edges of
the grid (along z-axis), each PML has a x, cr*, a y and cr*, which are set equal to those of
the adjacent PML's, along with a. = cr* = 0 . A t the eight comers o f the grid , where
there is overlap of three PM L’s, all eight losses, a x, cr‘ , ay, cr*, a , and cr*, are present
and set equal to those o f the adjacent PML's.
Fig. 3.13 Perfectly matched layers absorbing boundary condition in threedimension cartesian coordinates.
Beringer suggests that the loss should increase gracefully with depth, p , within each
PML. The electric loss rises from zero at p = 0 , which is the interface between the
interior region and the PML load, to a maximum value of crmax at p = 8, the location of
the conducting wall backing the PML, as following
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74
n
(3.25)
*(P ) = < * » * ( ! )
where S is the PM L thickness and a is either o x, a y or a. ., and n determines the rate
o f increase of the losses in the PML.
Beringer assumed that the loss increases
quadratically, i.e. n - 2. The corresponding magnetic loss is expressed as
*v
(3.26)
where p* is shifted half spatial step size with respect to p to account for the physical
location of the fields in the FDTD lattice. Then, crmajt can be chosen to bound the
apparent reflection coefficient
2 0 ^ 5 cosfl
R(9) = e
n+1
cc
(3.27)
where 9 is the incidence angle defined with respect to the interface between the interior
region and the PML's, and equals zero for normal incidence. Numerical experiments
confirm that the reflection does not depend on the incidence angle [ 8 8 ]. The reflection
coefficient R reduces to a key user-defined parameter. In fact, sharp variations o f
conductivity create numerical reflection. Numerical studies have shown that the optimum
value of R depends on the number of PML's, the permittivity of the interior medium, the
factor n , and the types of waves incident on the PML.
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75
The attenuation to outgoing waves afforded by a PML medium is so rapid that the
standard Yee time-stepping algorithm cannot be used. The field components may be
advanced by using the explicit exponentially difference equations [91]. W ith the usual
FDTD notation, the following equations are presented as examples:
1/2{ i J + 1 / 2 , k + 1 / 2 ) =
; + 1 / 2 ,Jfc+ 1 / 2 ) +
az{ i , j + 1/2,k + l/2 ) A z
(3.28)
[£ ;( /,; +1 / 2,* +1 / 2) - En
y ( i J +1 / 2,k - 1 / 2)]
- a z(i+ \ll,j,k )& i/c p n
E ”« ( i + l / 2 J , k ) = e
e ; ( ; + i / 2,m )+
cr. (/' + ! / 2 ,j,k)A z
(3.29)
3.2.2.2 PML ABC for Waveguiding Structures
The PM L ABC is applied to terminate all the microwave devices simulated in the
thesis. A ll o f these devices consist of different dielectric medium, and include metallic
strips. The FDTD lattice is terminated by extending the dielectric layers into its matching
PML [89]. The conductors are extended without PML region. This configuration, shown
in Fig. 3.14, simulates extremely lossy waveguiding structure. The propagation wave is
highly attenuated in the PML without reflection at the interface between the PM L and the
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76
actual structure. This approach fits the need for an appropriate ABC for dispersive
multimodal propagation, which was not available. It is essential for calculating the
dispersion characteristics o f a transmission line, especially the losses. The enhancement
applied to the PML-ABC eliminates any possible discontinuity effects at the end of the
line.
Air
PML-Air
Conductor
Dielectric
; P M L-D iel;
Fig. 3.14 Perfectly matched layers absorbing boundary condition for
waveguiding structures that include different dielectric materials and
metallic conductors.
It is obvious that the reflection coefficient calculated using the PM L is very small
compared to the one-way wave equation [90]. Although the PML is successfully applied
at the back of the FDTD lattice, special precautions have to be taken into considerations
when applying the PML to the top and sides walls. It is essential to place the PML's far
enough from the device to prevent absorbing the physical power propagating along the
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77
device. The simulation domain can also be closed at the front surface using PML. The
source plane is inserted next to the PML layers.
3.2.3 Parallel Implementation of the FDTD on MASPAR Machine
The past decade has witnessed a tremendous explosion o f research on various
aspects o f parallel processing covering parallel architecture, parallel models and
complexity classes, parallel algorithms, programming languages with parallel construct,
compilers, and operating systems for parallel computers. Also, with the recent advances
in VLSI technology, it has become feasible to build computing machines with hundreds
or even thousands o f processors cooperating in solving a given problem. Computing
machines with various types and degrees of parallelism built into their architecture are
already available in the market, and many more are in various stages o f development.
Examples include CRAY research machines, and Massively Parallel Processors (MPP) .
Parallel computation for electromagnetic problem analysis has been successfully
implemented for many applications [60]-[61].
FDTD technique has an explicit or sem i-im plicit scheme, where parallel
computers provide a good environment to run such schemes. The electric and magnetic
fields components are calculated using their nearest magnetic and electric field
components, respectively. This algorithm avoids the time consuming communication
within the MASPAR machine. The implementation of the parallel program code is
summarized in the following steps. First, a FORTRAN77 code for the problem is
developed, executed on a serial machine, and compared with previously published work.
Second, the serial code is parallelised using the VAST translation software. The parallel
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78
code is executed on the MASPAR, and the efficiency of the code is investigated. Then,
the parallel code is optimized.using MASPAR FORTRAN, which is sim ilar to
FORTRAN90, in order to increase the efficiency o f the code on the MASPAR . Finally,
the computation time on the serial IB M RS6000 cluster machine and that o f the
MASPAR are compared.
3.2.3.1 FDTD Parallel Algorithm
The implementation o f the FDTD on parallel machine requires a different
algorithm for defining the media parameters from the serial code. The media parameter
in FORTRAN77 code are defined for each domain constituting the structure as follows,
c
Compute the media parameters
do 1 m = 1, media
eaf
1
c
- r * sig(m) / epr(m)
ca(m)
= (l.OeO - eaf)/(1.0e0 + eaf)
cb(m)
= ra / epr(m) / (I.eO + eaf)
continue
define the dielectric substrate
do 2 k - 0, ksub-1 ; do 2 i = 0, im a x; do 2 j = 0, jm ax
ixmed ( i, j, k ) = 2
2
continue
But, media parameters can not be defined as domain in MASPAR FORTRAN, they need
to be represented as point parameters. They are written as,
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79
c
Compute the media parameters
e a fl: media )
c
= r * sig( :media ) / epr( :media )
ca( :media )
= (l.OeO - eafl :media )) / (LOeO + eafl .-media ))
c b (: media )
- r a / e p r ( :media ) / ( I.eO + eafl :media ))
define the dielectric substrate
cax( :imax, :jmax, :ksub-l ) = ca(2)
3.23.2 MASPAR FORTRAN versus FORTRAN77
MASPAR FORTRAN is based on the well-known FORTRAN77 standard, with
extensions from DEC FORTRAN and the FORTRAN90 ISO standard.
These
enhancements are designed to take advantage of the massively parallel, SIMD power of
the MASPAR family of computers. The most significant enhancements provided with
MASPAR FORTRAN are in the areas of processing arrays and new' intrinsic functions.
Next, some of these features w ill be discussed.
Their implementation in the
computational program w ill be shown. The differences with the serial code w ill be
addressed. These characteristics are:
(a) Arrays as a First Class Objects
(b) Attribute specification
(c) Array Assignments
(d) Intrinsic functions
In FORTRAN77, operations on arrays are programmed explicitly, using iterative
DO loops. In MASPAR FORTRAN, arrays are first class objects and arrays operations
can be written as simple expressions rather than necessarily as iterative loops. The
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80
M ASPA R FORTRAN compiler generates parallel SIM D code fo r these array
calculations. The following serial code
do 3 k = 1, kmax-1 ; do 3 j = 0, jm a x t; do 3 i = 0, imaxt
m = iymed ( i, j, k )
ey(i,j, k) = ca(m) * ey(i,j, k) + cb(m) *
(h x (i , j, k) - hx( i, j, k -1 )hz(i+1, j, k) + hz(i, j, k) )
3
continue
is changed to the corresponding parallel code as,
ey(: imaxt,:jmaxt, 1: kmax-1) =
cay(: imaxt,:jmaxt, 1:kmax-1) * ey( :im axt
cby(:imaxt,:jmaxt, 1 :km ax-l)*( ( hx( :im axt
jmaxt, 1:kmax-1)+
,:jmaxt, 1:kmax-1 )-
hx(: imaxt,:jmaxt, 0: kmax-2)) - (hz(l: im axt+1,:jmaxt, 1:kmax-1 )+
hz(:im axt,:jm axt,l:km ax-l)))
MASPAR FORTRAN allows to specify attributes of arrays in a compact form as,
real, array ( O'.imax, O.'jmax, 0:km ax) : : ex, ey, ez, hx, hy, hz
instead o f FORTRAN77, which has the following form
real ex ( O'.imax, 0:jm a x, 0:kinax)
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81
M ASPAR includes a FO RALL statements to specify an array assignment
statement in terms of array elements. The FORALL statement is the parallel equivalent
o f a DO loop; it specifies parallel calculation for the included assignment statement.
There are some restrictions on its use. The FORALL statement didn't efficiently work in
our FDTD implementation for replacing the DO loop. Regular assignment expressions
are used instead.
MASPAR FORTRAN offers significant additions to the FORTRAN77 intrinsic
function libraries. Most of these new intrinsic functions are derived from the proposed
FORTRAN 90 standard. The array reduction functions, for example the statement SUM,
are very helpful in writing a compact computer code for many applications. The array
manipulation functions, as CSHIFT, are very useful for finite difference solution
applications, however it didn't work efficiently when used for updating the field
expressions. To perform array operations on certain array elements, WHERE statement
can make array element assignments conditionally.
3.2.3.3 Data Allocation and Array Mapping in MASPAR FORTRAN
The MASPAR
system is composed of a front end and a Data Parallel Unit
(DPU), each of which has its own processors and memory organization. On the DPU,
memory is distributed among each Processing Element (PE) in the PE grid.
Serial
operations are performed on the front end. Parallel operations are performed on the PE
array in the DPU. The key to good performance on a data-parallel machine is to use as
many PEs as possible and to use them efficiently. This can achieved by using
(a) Mapping Directives
(b) Array Mapping
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82
Mapping directives give control over how arrays are allocated on the PE grid.
Arrays for the fields and the media parameters are allocated as follows,
cmpfmap ex(memory, xbits, ybits)
These compiler directives allocate the first dimension to memory of the corresponding
PE, the second dimension in the X dimension, and the third dimension in the Y
dimension. These allocations are changed with the size of the problem in each Cartesian
coordinates or dimensions.
Arrays are mapped onto the PE grid in columns and rows on the MASPAR
machine. The target PE array size is chosen according to the size o f the problem. The
-pesize compilation option specifies the number of parallel processors in the DPU that
can be dedicated for the executed program code.
Table 1 shows the number o f
processors, and the machine size for the available options.
Table 3.1
Number of Processors and Machine Size for available options
Options
Number of processors
Machine size
1
1,024 processors
32 columns x 32 rows
2
2,048 processors
64 columns x 32 rows
4
4,096 processors
64 columns x 64 rows
8
8,192 processors
128 columns x 64 rows
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83
The PE grid is physically two-dimensional. The PE local memory on each processor
provides a virtual third dimension, as it is shown in Fig. 3.15. Layer 0 corresponds to the
PE array, and layer 1 to n represents the local memory. It is crucial to adapt the number
o f processors to the problem size.
layer n
layer 1
ybits
layerO
xbits
Fig. 3.15 Virtual Layers in the processing elements memory.
3.2.3.4 Communication in MASPAR FORTRAN
Communication on the MASPAR machine is performed either through X-Net or a
global route. Programming style and array access have a direct effect on the type of
communication used. Therefore, when PE-PE communication is required the type o f
communication used is determined by how the arrays are aligned. Regular column-to-
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84
column or row-to-row operations are translated into X-Net instructions by M ASPAR
FORTRAN. This is the primary benefit o f the MASPAR FORTRAN array mapping.
The fields arrays and the corresponding media parameters arrays are mapped the same
way in the computational program.
cm pfm ap ex(memory,xbits,ybits)
cm pf map cax(mernory,xbits,ybits)
x
The communication-to-computation time needs to be minimized on the MASPAR to
increase the execution efficiency of the code on the parallel machine.
The FDTD
algorithm exhibits nearest-neighbor communication pattern, which makes the approach
well suited for parallelization.
3.2.3.5 Comparison between Serial and Parallel FDTD Codes
A comparison between the required CPU time to execute the serial code on the
IB M RS6000 and the parallel code on the MASPAR
machines is presented.
The
dimension o f the problem is fixed in the x and y directions to (64x32), while the third
dimension, which corresponds to the z direction, varies from 30 to 120. The number of
time steps in the FDTD calculation is constant for all simulations. The results are shown
in Fig. 3.16. The improvement in the CPU time acquired by running the code on the
parallel machine is clear. The speedup achieved in the FDTD calculations increases as
the problem size increases. It increases from 3.13 for the (64x32x30) problem to 10.39
for the (64x32x120) problem.
The results demonstrate the potential o f parallel
computation for FDTD calculations in electromagnetics.
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□
MASPAR
M
IBM RS6000
30
Fig. 3.16
60
90
3rd dimension size (kmax)
120
CPU time for MASPAR and IB M RS/6000 machines for
problems that have different sizes and equal number of time steps.
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86
3.2.4 FDTD Application to Microstrip Line
A microstrip line with lossless material on isotropic substrate is analyzed using
the FDTD technique. The thickness of the perfectly conducting strip is rigorously taken
into consideration. The strip is presented with adequate number o f mesh points as
previously explained. The calculated effective dielectric constant for the finite lossless
strip is compared with the result obtained using the following analytical expression [92]:
-
p
r*ff
=
er +1 e - 1
T ^ O
—L______ |------1_____
1+
+ 0.04 1 -
('W/h)
-
errff
reff =
£. +
<-)
1
£
-1
—
1+
'■)
1/2
12
12
W
C,
W /h < 1 (3.30)
1 /2
-C ,
W /h ^ 1 (3.31)
(W/A).
and,
C=
e , ~ l t/h
4.6 V WJh
(3.32)
where £r is the relative dielectric constant for the substrate, h is the substrate height, and
W and t are the strip width and thickness, respectively.
Numerical results are obtained for microstrip line with W = 7.5 fim , h = 10 fim ,
and £r = 13. The simulation is performed for zero, 0.5 fdm, and 1 (im strip thicknesses.
These dimensions are selected to allow comparing our results with those obtained in [52],
The calculated effective dielectric constant using the FDTD is plotted in Fig. 3.17. The
results depicted for the zero thickness perfectly conducting strip case shows good
agreement with the previously published ones using the FDTD. The results obtained for
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87
the finite thickness lossless strip are compared w ith the ones calculated using the
analytical expression presented in Eq. (3.31). Excellent agreement can be observed from
the graph. The decrease in the effective dielectric constant with increasing the strip
thickness can be explained by considering the increase in the total power propagating in
the air by increasing the strip thickness. This leads to an increase in the phase velocity of
the wave propagating along the microstrip line, which in turn decreases the effective
dielectric constant. One may note that the slow wave effect of the field penetration inside
the conducting strip is not considered due to the perfectly conducting strip assumption.
9
_
§
88
------------ ,i------------ i,------------,
------------- ------------" r-----------1
-
-
’
thickness = 0 pm
to
__
>
,
-
i
~
—------- •
----------------
8.4
thickness = 0,5
0.5 pm
um
z_______ *________ 0_________
82
----------- .----------- •----------- .----------- •--------- — -------- =
•a
o
w
t
--
I
O
ZJ
'M
O
----------- FDTD
......Analytical
Analytical
......Published
.............
Published ..
.________ .-------------=
thickness = 1 pm
8
0
I1
I1
10
20
I1
I1
30
40
Frequency (GHz)
I1
-----------50
60
Fig. 3.17 Effective dielectric constant (er = 13). Comparison o f the results
calculated using the FDTD with the empirical formula and the results
presented in [52],
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88
3.3 Comparison between FDFD and FDTD Solutions
The finite-difference technique offers many advantages as a modeling, simulation,
and analysis tool, in general. Both the FDFD and FDTD could deal w ith arbitrary
structure geometries. The interaction with an object o f any conductivity either real metal
or perfect conductor can be handled. The main advantage o f the FDTD is its broadband
response predictions capability. However, the time domain method does not have the
capability to distinguish between modes. The technique typically assumes that the only
mode which can propagate is the dominant mode.
Actually, the solution does not
correspond to the exact dominant mode in general, although how great an influence the
higher order modes have on the dominant mode is still to be determined. The FDFD
differentiates between the different modes propagating in the structure. The problem is
formulated in an eigenvalue form, where each mode corresponds to a specific eigen
value. Then, the deterministic problem can be solved for each eigen value or mode.
However, the Fields and currents distributions obtained from the FDTD solution are more
realistic than those of the FDFD. Also, FDTD is most suited to computing transient
responses. Moreover, it may be the method of choice for some problems. Interestingly
enough, interior coupling into metallic enclosures is a situation where FDFD w ill most
likely fail to capture the highly resonance behavior of a metallic enclosure, even when
made at many frequency points. Especially, the nonlinear analysis can only be conducted
using the FDTD and never the FDFD.
Finally, the FDTD is computationally more
efficient than the FDFD. It has an explicit or semi-implicit scheme where no matrix
operation is required. To conclude our discussion, the choice between the FDTD and
FDFD depends on the problem on hand.
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89
Numerical results obtained using the FDTD and FDFD approaches are presented
in Figs. 3.18 and 3.19. The simulation is performed for a microstrip line with 50 £2
impedance. The microstrip line dimensions are: substrate height = 10 /dm, strip width =
7.5 /im , strip thickness = 0.5 /im and 1 /dm. The relative dielectric constant o f the
substrate equals to 13.
The substrate is assumed lossless, since the effect o f the
conducting strip needs to be analyzed. The conducting strip is studied for the lossless,
lossy copper with conductivity o f 5.87 x 107 (S/m), and YBCO superconducting material
with magnetic field penetration depth of 0.223 /im and normal conductivity o f 3.51 x 105
(S/m) at T = 77 K cases. The attenuation constant for the copper and superconductor
strips is presented in Fig. 3.18. As expected, the losses introduced by the copper are
much higher than the losses associated with the superconductor. The losses increase with
the decrease of the strip thickness. This can be explained by the decrease in the number
o f the conducting electrons as the strip dimension decreases. Also, the fringing field at
the edges of the strip increases as the strip thickness decreases. The results depicted for
the attenuation constant are simulated first using the FDTD technique, then duplicated at
10 and 30 GHz using the FDFD approach. The same results are obtained using both
methods.
Fig. 3.19 shows the effective dielectric constant o f the microstrip line for three
different cases: lossless , lossy, and superconductor strips. The lossy and superconductor
strips are copper and YBCO, respectively. The structure is investigated for the 0.5 um
and 1 um strip thicknesses.
As explained before, the effective dielectric constant
decreases as the strip thickness increases. The effective dielectric constant for the copper
strip is greater than the lossless case. This can be explained by the increase in the
inductance o f the transmission line as the field penetration inside the conducting strip is
included. The effective dielectric constant for the superconductor is larger than the
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90
copper conductor. This is due to the addition of the internal kinetic inductance associated
with the motion of the paired superconducting electrons. This kinetic inductance w ill be
added to the total inductance o f the line, and results in an increase in the effective
dielectric constant o f the transmission line. These results are obtained first using the
FDTD technique, then duplicated at 10 and 30 GHz using the FDFD method. Typical
results are obtained using both approaches. Finally, one can conclude that both methods
could be used to calculate the dispersion characteristics of waveguiding structures with
the same accuracy.
0.5
B
CQ
"3
0.4
£3
S
0.3
.o
copper .5 um
copper 1 um
super .5 um
super 1 um
FDFD solution
o
V
c
o
0.2
a
o
0.1
-
* -
10
20
30
40
Frequency (GHz)
50
60
Fig. 3.18 Attenuation constant for copper and superconducting microstrip
lines with different strip thickness using the FDTD and FDFD approaches.
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9.5
super
lossy
lossless
FDFD solution
£
O
8.5
'S
Z
J
>
o
W
thk = luml
7.5
0
Fig. 3.19
10
20
30
40
Frequency (GHz)
50
Effective dielectric constant for lossless, copper and
superconducting microstrip lines with different strip thickness using the
FDTD and FDFD approaches.
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92
3.4 Summary
The FDFD and FDTD approaches are presented to execute HTS transmission
lines with nonuniform dielectric filling. The conducting strip is rigorously modeled to
account for the field penetration effects. This approach is essential to accurately model
and simulate transmission lines which incorporate superconducting material. Slow wave
propagation is observed along the superconducting microstrip line. The increase in the
attenuation with temperature and frequency is clearly demonstrated. The geometric
effects on the propagation characteristics of microstrip line are shown. As the conducting
strip dimensions increase, the field penetration effects decrease.
The nonuniform mesh generator is described. Results show better convergence
and less memory space requirement by using the nonuniform discretization compared to
the uniform one. Implementation of the algorithm on serial and parallel supercomputers
are presented. The improvement in the execution time is examined.
A substantial
speedup is obtained by running the FDTD code on the MASPAR machine. A novel
implementation for the PML ABC is suggested. The approach is suitable to terminate the
simulation domain for structures with metallization. The enhancement in the reflection
coefficient obtained by using the PML ABC compared to the one-way wave equation is
shown.
A comparison between the FDFD and FDTD methods is conducted. The main of
advantage of the FDTD is its broadband response capability. The FDFD is more suitable
for single frequency analysis, where mode differentiation is required. The accuracy
obtained from methods is o f the same order, since both are based on the finite difference
approximations. The technique presented in this chapter are used in the rest of the thesis.
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CHAPTER 4
ANISOTROPIC SUPERCONDUCTORS ON
ANISOTROPIC SUBSTRATES
A common feature o f the family of high temperature superconductors (HTS),
including LaBaCuO, YBaCuO, BiSrCaCuO, TIBaCACuO, is that all have layered
structures. It is generally believed that the two-dimensional Cu 0 2 network is the most
essential building block o f the HTS materials [18].
One expects the quasi-two-
dimensionallity associated with the layered crystal structures to be manifested not only in
the superconducting properties but also in the electronic properties in the normal state.
Anisotropy, associated with HTS, analysis and modeling are also essential for studying
the nonlinearity associated with these materials. Moreover, there has been a growing
interest in the use of low loss anisotropic substrates such as sapphire and boron nitride in
microwave and millimeter-wave integrated circuits [24], Although the sapphire substrate
is anisotropic, the r-cut single crystal sapphire seems to be an appropriate substrate
material for HTS applications.
Crystal lattice matching o f the sapphire with c-axis
oriented YBCO, small dielectric loss and high thermal conductivity can be achieved
simultaneously [27], Rigorous analysis of anisotropic HTS on anisotropic substrate is
essential, regarding both utilization the anisotropy characteristics and elimination of its
undesirable effects.
In general, the analysis o f isotropic microstrip lines on anisotropic substrates has
been reported in a few full-wave studies [93]-[94], Results have been presented for the
tilted optic axis lying on the transverse plane [95]-[96] and the horizontal plane [97]-[99].
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94
For the HTS microwave transmission lines, a full wave analysis which takes into account
either the anisotropy in the HTS material or in the substrate itself based on the spectral
domain/volume integral equation approach (SDVIE) is represented [13]-[14], A ll the
papers employed spectral-domain techniques except [95], who formulated Bergeron's
method in the time domain to deal w ith tilted optic axis.
However, no dispersive
characteristics of microstrip lines were presented. Full-wave analysis of such structures,
which takes account o f both the anisotropy in the metal strip and the dielectric substrate,
is not performed or presented in any of the literature.
In this chapter, we present a technique based on the three dimensional finitedifference time-domain method, to model transmission lines incorporating an anisotropic
superconducting material deposited on sapphire substrate. The anisotropy of both the
HTS material and the sapphire substrate are taken into account simultaneously. The
equations are derived directly from Maxwell's equations. The approach fits the needs for
accurate computation of the dispersion characteristics of an anisotropic superconducting
transmission line. Also, effects o f anisotropy on the field distribution inside the structure
and on the currents distribution inside the HTS are investigated. The model is flexible,
and can be used for any o f the planar transmission line structures containing HTS
material. It incorporates all the physical aspects of the HTS through London's equations.
The electromagnetic characteristics and the required boundary conditions in the structure
are represented using Maxwell's equations. The physical characteristics of the HTS are
blended with the electromagnetic model using the phenomenological two fluid model.
To validate the accuracy of our algorithm, the complex propagation constant is calculated
and the results are compared with the published spectral domain technique data. The
effect o f the anisotropy orientation on the characteristics of the microstrip lines and the
coplanar waveguides are studied.
Interesting comparisons between isotropic and
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anisotropic structures as well as comparison between the characteristics o f the microstrip
lines versus coplanar waveguides w ill be presented.
4.1 Anisotropic High Temperature Superconductor Model
The two fluid model assumes that the electron gas in a superconductor material
consists of two gases, the superconducting electron gas and the normal electron gas [67].
The main parameters o f the superconducting material are the London penetration depth
X l and the normal conductivity a n.
The physical nature o f the superconducting
phenomena is included in the dependence of the charge carrier densities and the effective
masses o f the superconducting and normal states, as well as the normal electrons
relaxation time, on the temperature. The temperature dependence is approximated by the
well known Gorter-Casimir model. In fact, this model is in good agreement with the
measured penetration depth for conventional superconductors, but not for HTS [21]. The
total current density in superconducting material is expressed as follows
(4.1)
where Jn and Js are the normal state and super state current densities, respectively. The
normal fluid current density obeys Ohm's law
(4.2)
The superconducting fluid current density is obtained using London equation [ 6 8 ],
— =
E
dt
\ l 0XL
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(4.3)
96
These expressions are valid assuming that the HTS material is isotropic.
However,
experiments show that the properties o f the HTS materials are anisotropic.
Such
anisotropy has significant effects on the device performance. The designer is faced with
several choices, as the type o f material and film direction, to obtain the optimum
configuration that enhances the characteristics of the HTS material. A common feature of
the family o f HTS, including YBaCuO and TIBaCaCuO, is that they all have layered
crystal structures. It is generally believed that the two-dimensional Q 1O2 network is the
most essential building block o f the HTS materials. Thin-film transmission lines w ill
favor films in which conducting sheets lie in the plane o f the film . The anisotropy for the
HTS can be represented by an anisotropic conductivity for the normal state, and an
anisotropic London penetration depth for the superconducting state.
The diagonal
conductivity tensor cr is given by
0
o'
0
°b
0
0
0
tfc.
(4.4)
Also, the diagonal London penetration depth tensor A is diagonal, and is written as
0
0
0
0
'
0
0
(4.5)
K.
where a , b and c are the principal axes of the anisotropic superconducting material. The
normal conductivity and the London penetration depth in Eqs. (4.1)-(4.3) w ill be replaced
by their corresponding tensors to formulate the anisotropic superconductor model. Note
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97
that the subscripts a, b, and c do not refer to the direction of the magnetic field, but to the
direction, in which the screening currents flow.
Experiments show that, for YBCO, the HTS parameters along the a- and b -axes
are approximately equal.
In our discussion, these parameters are the normal fluid
conductivity onab and the superconducting fluid London penetration depth
The
normal conductivity o nab equals 10-80 times anc, and the penetration depth Xc equals 3-5
times Xab [21]. Experiments also report that the critical current density and the upper
critical field are anisotropic [22], One must note that this superconductor model assumes
that the material has almost constant penetration depth at a temperature w ell below the
critical temperature Tc and the field penetration remains almost unchanged with
frequency. In general, the effective penetration depth is greater than the one depicted in
the two fluid model [ 2 1 ],
4.2 Anisotropic Finite-DifTerence Time-Domain Approach
The finite-difference time-domain (FDTD) solution of Maxwell’s curl equations is
one of the most suitable numerical modeling approach for the electromagnetic analysis of
volumes containing arbitrary shaped dielectric and metal objects.
An arbitrary 3-D
structure can be embedded in a FDTD lattice simply by assigning desired values of the
material parameters (e.g. electrical permittivity, conductivity, and London penetration
depth) to each lattice point.
These parameters are utilized in the calculation of the
respective electromagnetic field components.
These are interpreted by the FDTD
program as local coefficients for the time-stepping algorithm [51]. Specification o f the
media properties in this component-by-component manner provides a convenient
algorithm to represent the anisotropy in a media, and assures continuity o f tangential
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98
fields at the interface o f dissimilar media with no need for special field matching. The
developed three-dimensional finite-difference time-domain scheme is capable of
modeling the finite thickness of the HTS strip. The finite thickness is represented by
adequate number o f mesh points. A graded non uniform mesh generator is used to
discretize the simulation domain [49]. The ground plane is chosen as a perfect electric
w all, for sim plicity.
The computational domain is closed by the PM L absorbing
boundary conditions [82]. The program code for our analysis is written in FORTRAN 90
and is executed in massively parallel machine (MASPAR) environment. The anisotropy
is included in the FDTD code without significantly affecting its efficiency or the required
memory size.
The FDTD applies second-order accurate central-difference approximations for
the space and time derivatives of the electric (E ) and magnetic (H ) field intensities
directly to the differential operators of the curl equations, when the anisotropy of the
material is along its principal axis. The same algorithm can be implemented when the
principal axes o f the anisotropic material are Lilted with respect to the coordinates. In this
case, the solution w ill be carried on the electric ( D ) and magnetic ( B ) flux densities.
The algorithm w ill be extended one step further, to obtain the field intensity using the
appropriate constitutive relations. In our discussion, the YBCO HTS strip is assumed to
be anisotropic along its principal axis (i.e. c, b, and a along x, y, and z respectively). The
r-cut sapphire substrate is considered for two cases at 0 and 90 degrees rotation about its
principal axis (i.e. rotation about x-axis). The configuration simulated in this study
corresponds to the practical structure used in [27]. However, the analysis presented here
is flexible and can be extended to any anisotropic configuration. The follow ing finite
difference equation for Ez is obtained by combining Ampere's law with the two fluid
model,
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99
t
Al-<rzz{ i J , k + l / 2 )
2£zz(i,j,k
+ 1/ 2 )
Al
ezz( i , j , k +
2|
1/2)
^
A t - f f a { i , j , k + 1/2)
2e-z(i,y,Jfc+ l/2)
//;+1/2(/+ 1/ 2 ,7,A'+ 1/ 2) - #;+1/2(/ -
1/2 J , k + 1/ 2 )
A x ( i- l/ 2 , j, k + l/2 )
+1/2,/: +1/2) -
^(Z, j -1 /2 ,£ +1/2)
(4.6)
Ay(/,y -1 /2 , £ +1/2)
J“ v2(;,;.* + V2)]
where the superconducting current density Jsz is obtained from the discretized form of
London equation, which can be written as,
J : ; v2{ i j , k + 1/ 2 ) = J :r l/2( i j , k + 1/ 2 ) + ~
e:
( /j,a - + 1/ 2 )
(4.7)
K
It is obvious that the physical parameters, electric perm ittivity, conductivity, and
penetration depth, of the different materials are defined at each point in the three
dimensional simulation domain. Similarly, the finite difference equations required for Ex
and Ey can be obtained. The finite difference equations for the H field components obey
the modified Yee's algorithm [51].
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100
The developed three-dimensional finite-difference time-domain scheme is capable
of modeling the finite thickness of the HTS strip. The finite thickness is represented by
adequate number of mesh points. A graded non uniform mesh generator is used to
discretize the simulation domain, where the smallest mesh size is chosen inside and
around the HTS strip. The Courant stability condition is based on the smallest mesh size.
The ground plane is chosen as a perfect electric wall, for simplicity. The computational
domain is closed by the PML absorbing boundary conditions. The symmetry o f the
structure is used, and the simulation is carried only on half of the structure.
The
excitation pulse used in the analysis has been chosen to be Gaussian in time.
Unfortunately, it is not possible to excite the structure with a pulse having the field
pattern o f the appropriate mode due to the anisotropy investigated in our analysis.
A
very simple field distribution can be specified at the excitation plane using our knowledge
of the modes in the simulated structure. As long as a suitable distance is allowed for the
Gaussian pulse propagation, the pulse pattern in the transverse direction evolves to its
actual physical form. This can be verified graphically. The Gaussian pulse must be wide
enough in time to cover adequate number of space divisions to obtain a good resolution.
The turn-on amplitude of the excitation ought to be small and smooth.
4.3 Anisotropic Superconductor Microstrip line on Anisotropic Sapphire Substrate
A microstrip line with anisotropic YBCO HTS material on a sapphire substrate, as
shown in Fig. 4.1, is analyzed using the previously described technique. The HTS strip
has penetration depth \ { 0 K ) = 0.14 pm, normal conductivity o n(Tc) = 0.5x106 S/m, and
critical temperature Tc = 92.5 K. The frequency independent penetration depth and the
normal conductivity
equal to 0.2 pm and 1.0xl0 6 S/m, respectively at 77K.
The
dimensions o f the microstrip line are as follows : strip width 2W = 2 pm, substrate height
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101
h = 1 (im, and HTS strip thickness t = 0.5 pm. These dimensions are selected to allow
comparing our results with those presented by Lee et al. [13]. Although, our analysis is
carried out on this stfucture, the FDTD formulation is flexible and can virtually be
applied to any microwave device w ith arbitrary dimensions and/or anisotropic
orientation.
2W
H-----------M
Fig. 4.1 Anisotropic microstrip line on anisotropic sapphire substrate.
The simulation is performed for a microstrip line with isotropic substrate with
dielectric constant £r = 3.9. Two cases of the HTS strip are considered, the isotropic and
anisotropic HTS. The anisotropic characteristics of the HTS are presented by anisotropic
penetration depth, Ac = 5 A ^ , and anisotropic normal conductivity, a nab = 50 o nc [ 2 1 ].
The principal axes of the HTS Film is aligned with the coordinate axes. The c-axis is
assumed to be in the z-direction. The results calculated for the effective dielectric
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102
constant and attenuation constant, for both the isotropic and anisotropic HTS strip on
isotropic substrate are shown in Fig. 4.2.
i
W5
a
8
0.75 O
3
u
§ 2
Si
~a
5
o
>
•3
fcs
w
FDTD
SDVE
0.5
0.25
*
8
I
12
3
C
3
cr.
o
3
o
0
3
c/i
E
?
3
01
CO
A
16
20
Frequency (GHz)
Fig. 4.2 The propagation characteristics of anisotropic HTS on isotropic
substrate using the anisotropic FDTD.
These results are also compared with these obtained by Lee et al. [13] using the SD/VIE
approach. Our results are in good agreement with the previously published ones. A
slight increase in the effective dielectric constant accompanied by a small increase in the
attenuation constant is observed in Fig. 4.2. Obviously, this slight change is due to the
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103
larger field penetration in the FDTD treatment compared to the SDVIE approach and the
way the current distribution is represented in both methods. This clearly demonstrates
the accuracy and the consistency of our model. The effect of the HTS anisotropy, on the
propagation characteristics o f the microstrip line, is negligible due to the orientation o f
the thin film HTS strip. In the microstrip line, the current in the x-direction is small
compared to the other two components.
istropic HTS/istropic SUB
anisotropic HTS/istropic SUB
N
_
0.6
Total current
Super current
SJ
2
0.2
Normal current
0
0.2
0.4
0.6
Half strip width (pttrv)
0.8
1
► x
Fig. 4.3 Normalized normal-fluid, super-fluid, and total current densities
at the bottom surface of the strip, for both the isotropic and anisotropic
HTS cases, on isotropic substrate.
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104
The distribution o f the normalized longitudinal current density component ( / . ) at
the bottom surface o f the HTS strip for the normal and the super gases are shown in Fig.
4.3. The values and the distributions are approximately equal for both cases, isotropic
and anisotropic HTS strip. This can explain the sim ilarity in the effective dielectric
constant and the attenuation constant for both cases, isotropic and anisotropic HTS films.
Also, one can observe that the current is doubled at the strip edge with respect to the
center o f the strip. The superfluid represents almost 80 % o f the total current at 77 K,
according to the two fluid model used in our analysis.
The study o f the anisotropy is carried on an anisotropic HTS strip on sapphire
anisotropic substrate. The anisotropic YBCO HTS strip has the same characteristics as
before. Anisotropic microstrip line with the same dimensions as presented above is used
in this study. The anisotropy of the sapphire substrate is represented by the dielectric
permittivity tensor. For the 0° r-cut sapphire, the relative permittivity tensor is e „ =
10.03, £>y - 10.97, and e „ = 9.4 along x, y, and z directions respectively. The tensor is
£ „ = 10.03, £)y = 9.4 and e „ = 10.97 for the 90° r-cut [27]. The angle is calculated with
respect to the rotation about the x-axis. To assess the effect of the substrate anisotropy, a
comparison w ill be made between the performance of the microstrip line on a sapphire
substrate with a similar structure on an isotropic substrate with a relative permittivity £r
= 10.03. It should be noted that the chosen relative dielectric constant of the isotropic
substrate equals to the element of the anisotropic dielectric tensor £ „. In the microstrip
line configuration, the electric field in the x-direction is the strongest,
thus the
propagating wave velocity strongly depends on the value of the relative permittivity in
that direction.
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105
Fig. 4.4 shows the attenuation constant for the anisotropic HTS on r-cut sapphire
substrate as a function of the rotation angle about the x-axis at two different angles, 0 °
and 90° and for anisotropic HTS on isotropic substrate with er = 10.03. The microstrip
line on anisotropic sapphire substrate with 0° r-cut has the highest attenuation. The 90° rcut produces the lowest attenuation of the three structures. The effective dielectric
constants ereff of the three structures are also depicted in Fig. 4.4. The 0° r-cut sapphire
substrate results in the highest e reff, while the 90° r-cut substrate produces the lowest.
The change in the propagation characteristics can be explained by the value o f the
permittivity tensor element £)y in the y-direction. The y-field component o f the fringing
field increases with the increase of eyy from 9.4 (90° rotation) to 10.03 (isotropic) to
10.97 (0° rotation) which in turn increases the energy stored in the substrate, and results
in the decrease of the propagating wave velocity on the line or increase in the effective
dielectric constant.
The increase in the attenuation constant can be explained by considering the various
current distributions presented in Fig. 4.5. It is shown that the normal current density is
the largest for the 0° r-cut and it is the lowest for the 90° r-cut substrate. This explains
the slightly higher attenuation in the 0 ° r-cut case.
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106
10
9.6
a
o
U
u
•5
o
_CJ
"o
o
>
•a
CJ
S3
W
9.2
0.6
88
0.4
-<
8.4
0.2
8
0
4
8
12
Frequency (GHz)
16
20
Fig. 4.4 Propagation characteristics for anisotropic HTS on different r-cut
sapphire substrates and on isotropic substrate with er = 10.03.
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Attenuation Constant (dB/m)
anisotropic HTS/sapphire SUB (Cf)
anisotropic HTS/isotropic SUB
anisotropic HTS/sapphire SUB (90°)
107
Total current
Super current
T5
anisotropic HTS/sapphire SUB (0 )
anisotropic HTS/isotropic SUB
anisotropic HTS/sapphire SUB (90°)
Normal current
0.1
0
Fig. 4.5
0.2
0.4
0.6
Half strip width (pm)
0.8
1
N orm al-fluid, super-fluid, and total current densities for
anisotropic HTS on different r-cut sapphire substrates and on isotropic
substrate £r = 10.03.
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108
4.4. Anisotropic Superconductor Coplanar Waveguide on Anisotropic Sapphire
Substrate
A coplanar waveguide with YBCO HTS material on sapphire substrates are
analyzed using the previously described technique.
The frequency independent
penetration depth and the normal conductivity of the HTS equal to 0.2 (im and l.OxlO^
S/m, respectively at 77K for both structures. The dimensions of the coplanar waveguide
are as follows : strip width 2W = 2 pm, substrate height h = 1 pm, HTS strip thickness t =
0.5 pm, and slot width o f s = 0.5 pm. These dimensions are selected to allow comparing
our results with those presented for the microstrip line.
The coplanar waveguide
dimensions are demonstrated in Fig. 4.6.
s
2W
s
Fig. 4.6 Anisotropic HTS coplanar waveguide on anisotropic substrate
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109
The anisotropic characteristics of the HTS strip and the sapphire substrates are chosen the
same as the microstrip line case. The effective dielectric constant and attenuation
constant, and the current densities distribution are calculated as the microstrip line case
and shown in Figs. 4.7 and 4.8, respectively. The coplanar waveguide on anisotropic
sapphire substrate with 0 ° cut has the highest attenuation and highest effective dielectric
constant, while the 90° r-cut substrate results in the lowest ones. The change in the
propagation characteristics can be explained by the value of the perm ittivity tensor
element e}y in the y-direction. It is known that the electric field in the y-direction is the
strongest field component in the coplanar waveguide. This results in an increase in the
energy stored in the substrate as the permittivity tensor element £yy increases, which in
turn decreases the phase velocity of the propagating wave along the transmission line.
The increase in the attenuation constant can be justified by considering the
various current distributions presented in Fig. 4.8. It is observed that the normal current
density is the largest for the 0° r-cut and it is the lowest for the 90° r-cut sapphire
substrate. This explains the slightly higher attenuation in the 0° r-cut case.
Comparison of the anisotropy effects on both the microstrip line and the coplanar
waveguide can be depicted by analyzing Figs. 4.4 and 4.7. One should observe that the
change in the propagation characteristics for the presented coplanar waveguide with the
anisotropy of the materials is more pronounced than the microstrip line.
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11 0
0.0
0.76
■
-0 .7 2
a
o
o
o
CJ
o
>
'Z
o3
&
w
3
7.8 -
“
anisotropic HTS/sapphire SUB ((f)
------------- anisotropic HTS/isotropic SUB
— anisotropic HTS/sapphire SUB (90°)
7 .4 -
0.68
-0 .6 4
7-
-
0.6
- 0.56
0
Fig. 4.7
4
8
12
Frequency (GHz)
16
20
Propagation characteristics for anisotropic HTS coplanar
waveguide on different r-cut sapphire substrates and on isotropic substrate
with £r = 10.03.
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Attenuation constant (dB/m)
8.2 -
Ill
N
y—**
Total current
Super current.
a
P
anisotropic HTS/sapphire SUB (d )
anisotropic HTS/isotropic SUB
anisotropic HTS/sapphire SUB (9(J)
u
■N3
1
1
Normal current
0. 1 -
2
0
Fig. 4.8
0.2
0.4
0.6
Half strip width ( pn)
0.8
►
1
y
N orm al-fluid, super-fluid, and total current densities for
anisotropic HTS coplanar waveguide on different r-cut sapphire substrates
and on isotropic substrate er = 10.03.
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112
4.5 Summary
A full-wave analysis for anisotropic HTS planar microwave structures deposited
on anisotropic substrates, is presented. The FDTD technique, which takes the finite
thickness o f the anisotropic HTS film into consideration, is developed using a graded
non uniform mesh generator. The propagation characteristics of the anisotropic HTS
microstrip line and coplanar wave guide, on r-cut sapphire substrate, are calculated as
functions o f different r-cut angles. It is shown that the 90° r-cut sapphire substrate
structure has lower loss and lower effective dielectric constant than the 0 ° r-cut one for
both structures. These observations are explained by the current distributions on the HTS
strip. The approach presented can be used not only to obtain the characteristics of HTS
microwave structures but also to determine the optimum design that exploits the
anisotropic conductor characteristics on anisotropic substrates.
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CHAPTER 5
FULL-WAVE NONLINEAR ANALYSIS OF MICROWAVE
SUPERCONDUCTING DEVICES
The recent discovery of HTS has fundamentally changed the prospects for
applications o f superconductive electronics and has generated considerable effort to apply
these materials in a number of new areas [29]-[31]. The first successful applications are
in the area o f passive microwave and millimeter-wave transmission structures such as
resonators [34] and [43], filters [32] and [35], and delay lines [28] and [35]. Computer
simulation is needed to analyze and design superconductor components, devices, and
circuits. These simulations must be valid for both low and high power applications.
Most of the theoretical research, developed for superconducting devices, focused
on the linear theory. London equations were simultaneously solved with Maxwell's
equations [4]-[12], Quasi-TEM methods were employed to calculate the propagation
characteristics o f HTS transmission lines [3], [7] and [37], and the resistance, the
inductance, and the current distributions of striplines [38], The nature o f the quasi-TEM
approximation in these calculation restricts its application to the lower frequency range (0
to 10 GHz for 150 um striplines). The impedance boundary condition approach, in which
the strip is assumed to be either much thinner or much thicker than the magnetic
penetration depth, was also used [8]-[14], The application of these methods are restricted
by the dimensions o f the structures.
To accurately model the characteristics o f
superconducting microwave devices over a wider range of frequencies, a rigorous full-
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114
wave approach has to be used. The spectral-domain volume-integral equation method
[14] and the elaborate mode matching technique [11] have been used.
The methods presented before are for the linear response o f superconducting
materials, and are thus valid for low power levels only. However, significant nonlinear
response can be observed in measurements of the resonance curve and the quality factor
in stripline resonators [39]-[40] and [43], microstrip resonators [41], and cavities [44].
Moreover, the high current value existing in some applications may not exceed the HTS
critical current densities of high-quality YBCO films, but they are high enough to drive
the HTS into nonlinear behavior. As an example, HTS transmission line resonators in
narrow band filters have high peak current densities, which result from the high standingwave ratios on the resonator lines [45], The nonlinear characteristics of the HTS result in
generation of harmonics and also spurious products created by the mixing o f multiple
input signals. Therefore, the magnitude and the detailed nature o f the nonlinear effects
must be understood in order to facilitate widespread application of HTS in microwave
and millimeter-wave applications. It is generally believed that a major source o f the
nonlinearity in superconductors is due to the breaking of superelectron pairs in a high
field environment. Better understanding for the dependence o f the penetration depth, as
well as the superconductor electron density, on the electromagnetic field requires a
rigorous full-wave nonlinear model. In addition, accurate modeling of the nonlinearity in
superconducting resonators can serve as a useful tool to characterize the critical
parameters o f superconductor thin films. The nonlinear model must only be developed in
time-domain and never in the frequency domain or phasor form o f fields [83].
The problem of modeling the nonlinearity in HTS microwave and millimeter has
been tackled before using different approaches. An iterative method combining the
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115
spectral domain approach and the impedance boundary condition model is applied in
[100] and [101]. The Ginzburg-Landau (GL) theory was used to predict the nonlinear
behavior in a superconducting stripline resonator as a function of the input current [ 102 ].
These approaches were based on frequency domain calculation. A macroscopic model of
the nonlinear constitutive relations in superconductors is derived from a velocity
distribution assumption [103]. To our knowledge, modeling and full-wave analysis of the
nonlinearity associated with microwave HTS devices were never performed in the time
domain.
In this chapter, a nonlinear full-wave solution, based on the GL theory is
developed using the Finite-Difference Time-Domain (FDTD) technique. The GL theory
is independent of the microscopic mechanism in superconductor and is purely based on
the ideas of the second order phase transition only. The physical characteristics o f the
HTS are blended with the electromagnetic model using the phenomenological two fluid
model. Maxwell's and GL equations are solved simultaneously in three-dimensions.
This time-domain nonlinear model is successfully used to predict the effects o f the
nonlinearity on the performance of HTS transmission lines and filters. This approach
takes into account the field penetration effects. The spatial distribution of the total
electrons and the number of the super electrons compared to the normal electrons vary
with the applied power.
A study o f the nonlinearity effects on the propagation
characteristics, current distributions, electromagnetic field distribution, and frequency
spectrum o f microstrip lines is conducted. The s-parameter o f HTS filte r is predicted.
The nonlinear output input power relation is estimated. The results are compared with
experimental data. The field distribution along the filter resonators is studied. Our model
is flexible and can be used for any of the planar microwave and millimeter-wave devices
that include HTS material. This approach is not only useful to predict the nonlinearity
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116
effects on microwave devices performance but also can be utilized in the characterization
of the HTS materials.
5.1 Time-Domain versus Frequency-Domain Numerical Techniques
Numerical Characterizations and modeling o f guided-wave components has been
an important research topic in the past three decades. When a specific structure is
analyzed, one has to make a choice which method is best suited for the structure.
Obviously, the choice may not be unique. One must make a critical assessment for every
possible method. Analysis and modeling of the nonlinearity imposes some restrictions on
the selected numerical techniques. It is known that the application of a signal to a waveguiding structure including nonlinear material causes frequency mixing to occur. This
results in generation o f harmonics and spurious products.
The frequency domain
approach is based on analysis in the Fourier transform domain. It provides an elegant
tool for the reduction of the partial differential equations of mathematical physics into
ordinary ones, which in many cases are amenable to further analytical processing. The
time-dependent partial differential equation is decoupled into a series o f frequencydependent ones. Hence, the solution is separately carried on each frequency component.
The time-domain solution can be obtained by the superposition of the results calculated at
each frequency components. This approach is widely used in problem containing linear
materials. However, when a nonlinear material is used, the partial differential equation
can not be transformed to the frequency domain.
The equations for the various
harmonics are no longer separable, and the superposition technique is not allowed.
Hence, the equations must be solved in time domain. One should notice that this is a
fundamental issue. It is not a matter of approximation or simplification.
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117
5.2 Nonlinear Full-Wave Superconductor Model
The nonlinear superconducting model, described in Chap. 2, is incorporated into
the three-dimensional full-wave electromagnetic simulator, presented in Chap. 3, using
the two fluid model postulate. The two fluid model assumes that the electron gas in a
superconductor material consists of two gases, the superconducting electron gas and the
normal electron gas [67]. The main macroscopic parameters o f the superconducting
material are the magnetic filed penetration depth Xs and the normal conductivity o n. The
superfluid characteristics depends on the penetration depth, while the normal conductivity
determines the normalfluid characteristics. The physical nature o f the superconducting
phenomena is included in the dependence of the charge carrier densities and the effective
masses o f the superconducting and normal states, as well as the normal electrons
relaxation time, on the temperature. Also, it is generally believed that the origin of the
nonlinear behavior of superconducting materials is the breaking of the superelectron pairs
inside the superconductor w ith the applied field.
This results in inhomogeneous
superfluid and normalfluid currents distributions with the nonuniform magnetic field
associated with most of the superconducting strip used in microwave and millimeterwave applications. Since the current-field relation is assumed to be local for HTS, the
nonlinear phenomena can be modeled by spatial, field, and temperature dependents
magnetic field penetration depth X s{ t ,\h \,T^ and normal conductivity
where r, |//(r)|, and T are the position, magnetic field intensity magnitude and
temperature inside the superconductor, respectively.
The total current density in
superconducting material is expressed as follows
J = 7„ + 7S
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(5.1)
118
where Jn is the normal state current density, which obeys Ohm's law,
(5.2)
and Js is the superconducting fluid current density, which follows the modified form of
GL equation,
dJ, __
dt
1
E
(5.3)
H 0X ] { r \ H ( r ) \ T )
The temperature dependence is approximated by the well known Gorter-Casimir
model. In fact, this model is in good agreement with the measured penetration depth for
conventional superconductors, but not for HTS [18]. The field dependence is obtained
from the solution o f the phenomenological GL equations.
The normalized order
parameter, which corresponds to the fraction of the superfluid electrons density is
calculated. It is field, position and temperature dependent. The normalfluid and the
superfluid electrons densities, nn and ns, are calculated from the conservation o f the total
number of electrons n, in the superconductor. The relations between nn, ns, and n, is
derived based on the superconductor state at T = 0, Tc. The two fluid theory assumes
that the superconducting material is in the superconducting state at T = 0, and in the
normal conductor state at T = Tc, i.e.
n, = n J(0,0) = nn(0,Tc)
(5.4)
(5.5)
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119
x4
n , ( H , T ) _ nt (H ,T )
n,
(5.6)
ns(0,T)
e/
which can be written in form o f the order parameter
4'
¥ (H ,T )
V (H ,T )
V.
■KO,T)
1 -
f r l
{T c j
(5.7)
and using the low field temperature dependence of the London penetration depth [67]
A^o.o)
A f (o .r) = -
r
(5.8)
r r N4\
iT
V \ C/ y
The spatial, field and temperature dependent magnetic field penetration depth Xs for the
superconductor can be calculated from the following expressions :
A^o.o)
y(»(Q.r)
y /(o .r)
Y (
(T
1\
(5.9)
Tc /
where /l/o.o) is the low field penetration depth measured at T = Oand H = 0 and Tc is
the critical temperature for the superconductor calculated at H = 0. The parameters a
and P may be obtained from experimental studies. In our analysis, we chose a = 2 and
P = 4 fo llo w ing G L model fo r the field dependence and Gorter-Casimir for the
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120
temperature approximations. The normalfluid electron density nn is derived from the
following relations
n J H J l= i-iW I)
n,
( 5 . 10)
n,
The corresponding normal conductivity is expressed as follows :
crn(w (F).r)
= crn(nc/Tc) 1 -
y(H(F),r)
\ f / ( o ,r)
1-
(5.11)
T
where o„{nciTe) is the maximum normal conductivity measured either at T = Te or
H = H C and H c is the critical magnetic field for the superconductor calculated at T - 0.
Eqs.
5.9 and 5.11 are one o f the main results of this chapter.
They represent the
dependence of the main macroscopic parameters of the superconducting material on the
normalized order parameter, which in turn depends on the spatial distribution o f the
applied field. The superfluid and normal fluid currents densities can be updated by using
Eqs. 5.2 and 5.4. Their dependence on the applied field fu lly represent our nonlinear
problem.
A general iterative scheme for solving the nonlinear problem is suggested. The
fields are firs t initialized using the low field macroscopic parameter o f the
superconductor. The appropriate boundary conditions for solving G L equations are
extracted from the electromagnetic simulator. The nonlinear G L equations are solved,
and the normalized order parameter is calculated.
The nonlinear magnetic field
penetration depth and normal conductivity are updated. Then, the electromagnetic fields
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121
are recalculated. This procedure is repeated until the total current in the superconductor
converges.
5.3 Nonlinear Full-Wave Superconductor Simulator
Experiments show that the HTS macroscopic parameters, the field penetration
depth and the normal conductivity, are anisotropic, as explained in Chap. 4. Moreover,
thin-film transmission lines w ill favor films in which conducting sheets lie in the plane of
the film . So, the superconductor current w ill be considerably high in the direction of the
wave propagation.
Therefore, the nonlinearity effect is expected to arise from this
longitudinal current component much earlier than the other two current components.
Hence, two-dimensional solution for the nonlinear macroscopic parameters in the plane
perpendicular to the direction of propagation is'satisfactory to model the nonlinearity in
superconducting films used in microwave and millimeter-wave planar structures.
The two-dimensional spatial distributions of the macroscopic parameters of the
HTS, Ai (x,y;|H (x,y)|;T) and
(*, y)\; r ) , can be obtained by solving the
Ginzburg-Landau nonlinear differential equations inside the HTS strip.
The time-
independent GL equations w ill be used since the characteristic time scale of the
propagating electromagnetic fields is much larger than the relaxation time o f the order
parameter. In other words, the order parameter responds almost instantaneously to the
time-varying electromagnetic field. It is straight forward to show that the normalized GL
equations for the z-component current density can be sim plified to the following
expressions,
(5.12)
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122
-rV,2H=Hfw2-i+
K
(5.13)
V
where t stands for the transverse x-y direction. The required boundary conditions become
n x { V , x z A z) = n 0H,
(5.14)
» -v ,H = o •
(5.15)
where H t is the normalized magnetic field component tangential to the superconductor
surface. This tangential magnetic field at the boundary of the strip is deduced from the
full-wave electromagnetic simulator.
So, the superfluid and normalfluid currents
densities in the z-direction can be calculated. The currents densities in the x- and ydirections are calculated using the low field London model.
The solution is performed by dividing the signal strip conductor into numerical
grid segments. A nonuniform gridding scheme, explained in Chap. 3, is adopted to
maintain a good resolution near the edges of the strip. Smaller segments are used near
the edges where the changes in the order parameter and the field are rapid. The grid size
is chosen at least one-fifth o f the penetration depth at the edge and increases gradually
towards the center.
Our nonlinear analysis algorithm is summarized in Fig. 5.1. First, the excitation
pulse is launched into the HTS microwave device. Then, the magnetic field components
are updated using the FDTD approach. Next, the tangential magnetic field to the HTS
strip surface are used as boundary conditions for the nonlinear HTS simulator.
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(
START
)
,, t = 0, n = 0
Launch Excitation
t = n.dt
Update Magnetic Fields
Update Boundary Conditions
forGL
Magnetic vector
Potential
Solve Nonlinear HTS model
Update Xs & ^
Order Parameter
Update Electric Fields
No
n-
No
id t> T
n =n +1
Fig. 5.1 Flow chart of the nonlinear analysis algorithm.
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124
The normalized superfluid electron density is calculated as previously described. The
spatial, field, and temperature dependent macroscopic HTS parameters cr„ and Xs are
updated. Finally, the electric field components are updated using the electromagnetic
simulator. Then, a convergence test fo r the current flowing in the superconductor is
conducted.
This procedure is repeated for a time period suitable to analyze wave
propagation characteristics inside the structure. The temporal fields are probed during the
simulation process. The applied power is measured at the first probe.
5.4 Nonlinear Analysis of Superconducting Microstrip Lines
The nonlinear characteristics o f a YBa 2Cu3 0 7 _x superconducting microstrip line,
shown in Fig. 5.2, is analyzed in this section. The microstrip line is an inhomogeneous
transmission line since the field lines between the strip and the ground plane are not
contained entirely in the substrate. Therefore, the microstrip line can not support TEM
mode. A full-wave analysis is required to rigorously analyze these devices.
2\v
Fig. 5.2 HTS microstrip line geometry.
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125
The YBa 2Cu 3 0 7 _x superconducting material has a critical temperature o f 90 K,
critical magnetic flux density n oH c( T ) o f 0.1T, and GL parameter o f 44.8 at 77 K [102],
The superconducting microstrip line has a 50 Q impedance with a strip width o f 7.5 jim ,
and a thickness o f 1 fim .
The substrate thickness is 10 /rm, with er= 13.
The
transmission line characteristics are simply chosen to demonstrate the nonlinear wave
propagation along the line. The maximum rf power, Pcrf, where the HTS microstrip
looses completely its superconductivity, is predicted using the G L solution described in
Chap. 2. Its value equals to 920 W / cm 2. Numerical results are obtained at different
levels of the applied power : 834 W / cm2, 410 W / cm2, and 181.8 W / cm2 denoted by
0.9 PCrf, 0.45 Pcrf, and 0 .2 Pcrf respectively.
The temporal magnetic field propagating along the line are probed at 60 jdm and
150 iim , and shown in Figs. 5.3 and 5.4. The amplitude of the wave is attenuated as the
applied field increases. Also, the slow wave effect can be observed from the figures.
Fig. 5.4 also shows that London's model incorrectly predicts higher pulse amplitudes and
lower attenuation even at high power. This qualitative discussion gives a good insight
into the behavior o f the wave propagating along the transmission line.
Fig. 5.5 compares the effective dielectric constant o f the transmission line at
different applied power
levels.
The effective dielectric constant predicted using
London's equations is smaller than the one obtained from G L solution at low applied
power. This can be explained knowing that the penetration depth predicted by London
theory is smaller than the one obtained from GL model, which agrees well with the
experimental observations [21], The effective dielectric constant increases as the applied
power increases. This slow wave effect is due to the increase in the internal inductance
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126
o f the line introduced by the increase in the field penetration as the applied power
increases.
1
input
London
- - - G L 0 .2 P
- — GL 0.45 I*1 .
- - - GL 0.9 p cr;
erf
0.8
S
o
2
0.6
“o
E
o
0.4
e
o
03
2
•g
s
0.2
o
0
z
-
0.2
0
1
2
3
time (ps)
4
5
6
Fig. 5.3 Normalized tangential magnetic field intensity under the strip
probed at 60 jim and 150 jim.
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127
Normalized Magnetic Field Intensity
0.94
0.93
■London
GL 0.2 P
' erf
— GL 0.45 P .
G L 0.9 P Crf
erf
[-
A '
0.92
//
//
0.91
h
!> /
h .•
j / i
0.9
3.2
3.25
/
\
u
...
\
i
3.3
time (ps)
\
3.35
3.4
Fig. 5.4 Normalized tangential magnetic field intensity under the strip
probed at 150 Jim at different levels of applied power.
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128
8.5
1
...... 1
------
— — ------
“
■
V
8.45
co
U
u
'£
"3
5
o
>
•a
u
is
w
““
------------ London
------ — GL 0.2 P ,
erf
------------GL 0.45 P .
---------- GL 0.9 P crJ
erf
8.4
8.35
------
------
------
------
------
------
—
8.3
8.25
Fig. 5.5
I
1
1
8
12
Frequency (GHz)
16
1
20
Effective dielectric constant for the HTS microstrip line at
different levels o f applied power.
The fractional change in the effective dielectric constant at different power levels
obtained from GL model with respect to the one calculated using London's theory is
drawn in Fig. 5.6. The change in the effective dielectric constant increases with the
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129
increase in the power level up to 0.7 Pcrf. It is approximately constant for higher power
levels, where almost complete field penetration occurs.
Freq = lb GHz
/—s o 9
CO.
a.
< 0.6
0.3
0
0.2
0.4
0.8
0.6
Normalized Applied power (P.pP p
1
Fig. 5.6 Fractional change in the effective dielectric constant for the HTS
microstrip line with applied power w.r.t. the linear model.
The attenuation constant for the HTS at different levels o f applied power is
presented in Fig. 5.7. As the applied power increases, the attenuation constant increases
as well. This can be explained by the increase in the normal electron density in favor of
the super electron density as the field penetrates the HTS. Also, it is observed that the
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130
HTS looses its superconducting characteristics earlier than its static critical power. This
is due to the field singularity associated with the planar microwave and millimeter-wave
devices. It is noticed that the change in the propagation characteristics o f the HTS
microstrip line is not linear with the applied field. The change is more pronounced as the
applied power increases. This is understood from the nature o f the superconducting
material, which deteriorates very quickly as the applied power approaches its critical
value.
100
London
GL 0.2 P ,
GL 0.45 P ,
GL 0.9 P f
erf
10
1
1
0.01
0.001
0.0001
0
4
8
12
16
20
Frequency
Fig. 5.7 Attenuation constant for the HTS microstrip line at different
levels of applied power.
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131
The fractional change in the attenuation constant obtained from GL model with
respect to the value predicted by London model at 10 GHz is depicted in Fig. 5.8. It is
clear that the HTS loses its superconductivity very quickly as the applied power
approaches the electromagnetic critical power. Also, the nonlinearity associated with the
HTS appears very early, even with the material in fairly good superconducting stage.
200
Freq = 10 GHz
160
120
80
40
0
0
Fig. 5.8
0.2
0.4
0.6
0.8
Normalized Applied Power (P y P crP
1
Fractional change in the attenuation constant for the HTS
microstrip line with applied power w.r.t. the linear model.
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132
The change in the losses is faster and more nonlinear than the change in the phase
velocity. This can be explained knowing that the losses are ohmic, which are induced by
the moving particles. On the other hand, the effect on the phase velocity is a variation in
the stored kinetic energy inside the superconductor, which is mostly a wave effect.
Hence, particle related effects can be stronger and faster than the wave related effects.
The effect of the applied field on the current distribution is presented in Figs. 5.95.11. Fig. 5.9 shows the superfluid current distribution at the bottom o f the HTS strip. It
is observed that the London model under estimates the current carrying capacity o f the
HTS material. The HTS strip looses part of its superconductivity at the edges as the
applied power increases from 0.2 Pcrf to 0.45 Pcrf. For 0.9 PCr f case, the partial loss of
the superconductivity is induced across the entire cross section o f the HTS strip. It is
more pronounced at the edge o f the HTS strip, where the superfluid current for the 0.9
Pcrf case is almost equal to the 0.45 Pcrf one.
The superfluid current distribution at the top o f the HTS strip is presented in Fig.
5.10. The current values are less than the one obtained for the bottom o f the strip as
explained before. For the 0.9 Pcrf case, the superfluid current increases at the top surface
because the applied field is less than the critical magnetic field of the HTS material.
Thus, the superconductor redistributes the superfluid as the applied field increases. The
normalfluid current density behavior w ill be opposite to the superfluid one based on to
the conservation in the total number of electrons in the HTS.
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4
a
0
London
— GL 0.2 P
GL 0.45 F*"
- - - GL 0.9 PCTl
erf
-
3.5
Q
w
a
S
3
<3 <nT 2.5
2
§
1o 'O
s
a.
3
2
o
T—
CO X
o
•o
3
co
co
1
0.5
0
0
0.75
1.5
2.25
'y-axis' strip bottom (pm)
3.75
Fig. 5.9 Normalized longitudinal super fluid current density at the bottom
surface o f the HTS strip at different applied power levels.
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134
London
- GL 0.2 Pcrf
GL 0.45 Pcrf
- - - GL 0.9 Pcrf
3.5
u °C
■S 6
'3
SO vo^
Q* O
53 S
^
.X.
-
2.5
2
1.5
•o
0.5
0
0.75
1.5
2.25
'y-axis' strip top (pm)
3
3.75
Fig. 5.10 Normalized longitudinal super fluid current density at the top
surface of the HTS strip at different applied power levels.
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135
Fig. 5.11 shows the superfluid current distribution at the side of the HTS strip.
The superconductivity o f the HTS material decreases ais the applied power increases.
The distribution obtained from GL model at low power levels equals to the one predicted
by the London model. Then, the effect of the nonlinearity on the electromagnetic field
distribution in the microstrip line configuration is studied.
>>
— London
— GL 0.2 P ,
- GL 0.45 Dcrtf
Pcrf
- GL 0.9 P .
3.5
erf
•o
«
S
'S
<
O
VO
c. o
3 i
Vi
2.5
^
X
top
bottom
0.5
0
0.2
0.4
0.6
’x-axis’ strip side (pm)
0.8
1
Fig. 5.11 Normalized longitudinal super fluid current density at the side
surface o f the HTS strip at different applied power levels.
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136
The normalized tangential magnetic field intensity at the top and the bottom o f
the strip are presented in Fig. 5.12.
The effect of the applied power on the
electromagnetic field distribution is small. It is only observed near the edge o f the strip.
This explains the small change in the phase velocity of the wave propagating along the
line.
*3
5O L5
a
&0
-t1
S
■a
•a
co
0.5
o
Bottom
C3
W
H
*3
O
N
-
n
London
— GL 0.2 P .
GL 0.45 p“
GL 0.9 P " f
e rf
-0.5
g
Top
'z.
0
0.5
1
1.5
2.5
2
3
'y-axis' halt-strip width (pm)
3.5
4
Fig. 5.12 Normalized tangential magnetic field intensity at the top and
bottom surface of the HTS strip at different applied power levels.
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137
Finally, the effect of the nonlinearity on the frequency spectrum o f the wave
propagating along the line is analyzed in Fig. 5.13.
The fractional change in the
amplitude o f the output pulse frequency components increases with the applied power. It
is observed that as the power level increases, the amplitudes o f the different harmonics
change, which is one o f the primary characteristics of nonlinear devices. This confirms
our point that the nonlinearity has to be modeled in the time domain.
The results
calculated form GL model for low applied power are approximately the same as the ones
obtained from the linear London model. This validates our treatment for the HTS as
linear material below the low applied power value.
0.05
London
GL 0.20 P ,
GL 0.45 P "
GL 0.90 P f
0.04
erf
< 0.02
0.01
0
4
8
12
16
20
Frequency (GHz)
Fig. 5.13 Fractional change in the amplitude of the frequency spectrum of
the output pulse w.r.t the dc component at different applied power levels.
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138
5.5 Nonlinear Analysis of Superconducting Filters
High temperature superconductor materials have potential applications in the
design o f small size, light weight, and high quality factor microwave filters [32] and [45].
However, transmission line resonators in narrow band filters have peak current densities,
which result from the high standing-wave ratios on the resonator lines. This high current
value may not exceed the HTS critical current densities o f high-quality YBCO films, but
they are high enough to drive the HTS into nonlinear behavior. The maximum r f power
level that a filte r can handle without changing its characteristics depends on the HTS
properties, the cross-sectional dimensions of the transmission line resonators, and the
bandwidth o f the filte r.
Moreover, most microwave CAD software requires an
approximation in order to analyze filters circuits.
They may fa il to predict the
performance o f resonator array filters, especially for low coupling resonator filters. Thus,
filte r design and analysis need a more rigorous simulator tool.
For high power
applications, nonlinearity associated with HTS materials w ill effect the filte r
characteristics. In this case, the simulator has to be implemented in the time domain. If
such a model is implemented in frequency domain, unreliable results may be obtained.
In this section, the non linear full-wave model, based on the GL theory, is used to
analyze and study the nonlinearity effects on HTS microwave filters. The s-parameter of
microstrip line array resonator filte r is estimated. The nonlinear output input power
relation is predicted. The field distribution along the filter resonators is studied. The
calculated results are compared with the experimental data. This approach is not only
useful to predict the nonlinearity effects on microwave filters performance but also can be
utilized in the characterization of the HTS materials.
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139
5.5.1 Simulation of Microstrip Resonator Array HTS Filter
Simulation for the microstrip resonator array filte r using the full-w ave 3D
nonlinear HTS FDTD electromagnetic simulator, which rigorously includes the finite
thickness o f the HTS film , is performed. The filter configuration is shown in Fig. 5.14.
Fig. 5.14 HTS microstrip resonator array filter structure
The filte r consists of 6 staggered resonators and two feeding lines. The substrate
is LaAlO w ith dielectric constant of 23.5 and loss tangent of 5xlO '3. The dielectric
height is 10 mils. The feeding lines have 50 £2 impedance. The width o f the feeding
lines equal to 3.8 mils. The microstrip resonators have lengths of 163,165 and 166 mils.
Their widths are 9.7, 10.6 and 8.3 mils, respectively. The spacing between the resonators
is 17.7 mils at both ends of the staggered filters. It equals 76 mils between the 4 center
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140
resonators. One must note that the filter is symmetric in the z-direction. The ground
plane of the filter is assumed perfect electric conductor in our simulation. The HTS strip
thickness is considered to be 0.3 pm. The macroscopic parameters o f the YBaCuO films
are as follows: low field penetration depth AJ(0) = 1600A° and normal conductivity
o n(Te) = 1.6 x 10sS / m. The critical temperature Tc for YBCO equals to 89.6 K. The GL
parameter is assumed to equal 44.6. This filter is designed by David Samoff Research
Center. A ll the measurements are performed in their labs.
In general, the frequency dependent scattering parameters, Sy, can be calculated
as follows [56]:
(5.16)
where V; and Vj are the voltages at ports i and j, respectively, and Z 0j and Z0, are the
characteristic impedances of the line connected to these ports. The voltages, V(co), are
obtained by taking the Fourier transform o f the time record of the voltage underneath the
center of the strip. The voltage equals the average line integral o f the vertical electric
field under the whole strip.
The characteristic impedances are calculated using the
following relation, as described in [52]
where I(a>) is defined as the Fourier transform of the time record o f the loop integral of
the magnetic field around the metal strip. Eq. (5.16), for the scattering parameters, takes
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141
into account the variation o f the characteristic impedance of the feeding lines with the
frequency and the applied power, which is the case for superconducting structures.
The S21 parameter for the array filter is calculated from 0 to 24 GHz. The results
are shown in Fig. 5.15.
The FDTD extracts the frequency response over a wide
frequency band in one simulation. The simulation is not only capable to predict the
fundamental resonant response o f the bandpass filter but also to calculate the higher order
resonance. The second resonance peak appears at double the operating frequency of the
filte r due to the periodic design of the array filter, as expected.
The higher order
intermodulation distortion can be easily predicted by the developed electromagnetic
simulator. As a matter o f fact, the 3rd-order two-tone intermodulation performance of a
microwave device is often used as a figure of merit for the linearity o f a microwave
device. The calculated S21 depicted by our simulation is compared with the measured
S21 . The comparison is shown in Fig. 5.16. The agreement between the measured and
calculated scattering parameters is clear. The resonance frequency at 8.67 GHz as well as
the small bandwidth of the bandpass filter are successfully simulated. The discrepancy
between the measured and the calculated S21 at the upper frequency tail o f the filter
response characteristics does not affect the accuracy of the calculation.
The output input power relation for the bandpass filter is calculated and presented
in Fig. 5.17. The predicted power relation is compared with the measured one. The
nonlinearity associated with the HTS filte r appears at approximately 10 dBm.
The
developed nonlinear HTS electromagnetic simulator is able to estimate the nonlinearity in
the HTS microwave filter. It is clear that the filter response deteriorates very quickly, as
the applied power approaches the critical r f power, which is previously defined. The
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142
critical r f power varies for the same HTS material according to the dimension o f the
structure and its configuration.
The electric field distribution in the dielectric along the longitudinal direction of
the filter is depicted in Fig. 5.18. The input signal propagates along the input feeding line
and is reflected at the end o f the feeding line. The wave resonates along the first
microstrip resonator. Part o f the wave transfers to the second resonator, while the
reflected wave returns to the input end. The resonant component transfers to the third
microstrip resonator. The coupling between the resonators continues, and the wave
reaches the output feeding line. Although, the field distribution gives a qualitative picture
for the filte r performance, it is very important for optimizing the filter design. It is clear
that the field levels are very high at the connection between the input feeding line and the
first resonator. The 90° junction at this connection point may not be the optimum design
configuration, especially for HTS materials. It is known that the field distribution is
highly nonuniform for microstrip resonators, usually with very high peaks at the edges of
the line. This behavior is clearly shown in Fig. 5.18. Moreover, one can observe that the
fields at the microstrip resonators close to the input side are higher than the resonators
near the output end. This behavior needs to be taken into account during the preliminary
design o f the filter, especially for estimating the power handling capability o f the filter.
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0.8
0.6
0.4
0.2
0
4
8
12
16
20
24
Frequency (GHz)
Fig. 5.15 The calculated S21 parameter for the HTS microstrip staggered
resonator array filter.
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a 21
ma g
S]_ i
Khh
-2 0 .0
dB
i,
i. 0 . 0
dB/
v
-3.G 9 0 S
dB
3 C 7 3 7 T 5 1 J0
LAM35B2 . 0 +
log
REF
0 .0
dB
log
A
5 .0
dB/
MAG
77
M AR < E R
375
GHz
CENTER
SPAM
Fig. 5.16
8 .7 35 0 0 0 0 0 0
0 .2 50 0 0 0 0 0 0
GHz
GHz
1
Comparison between the calculated and the measured S
parameters lor the HTS microstrip staggered resonator array filter.
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linear resp'
— -*• —calculated
measured
S
CQ
T3
V -/
u
O
&
o
CL,
-10
-20
-30
-30
-20
-10
0
10
20
30
Input Power (dBm)
Fig. 5.17 Output input power relation of the HTS microstrip resonator
array bandpass filter.
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Fig. 5.18 Electric field distribution in the substrate along the longitudinal
direction of the bandpass filter.
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147
5.5.2 High Power Design Consideration for HTS Filters
HTS microstrip line resonator array filter has very high unloaded quality factor as
a result o f the very low surface impedance o f HTS materials. The HTS thin film s
characteristics depend on the operating frequency, temperature, and r f power. A t high
frequencies, fi)> 1012^-1, the macroscopic parameters o f the HTS strip, the normal
conductivity and field penetration depth, become functions of frequency [19]. This limits
the maximum operating frequency to the low millimeter-wave region.
It is well known that the current distribution in microstrip lines is highly
nonuniform. As a matter of fact, the current has a very high peak at the edges of the strip.
For high power applications, the HTS may lose its superconductivity near the edge of the
strip. This may drive the HTS material to exhibit a nonlinear behavior. The nonlinearity
w ill result in the generation o f harmonics and also spurious products created by the
mixing o f multiple input signals.
Also, calculation of multiresonator filters power
handling capability using the multiplicity approach is not adequate, especially for high
power applications. The power handled by each resonator is not equal. The maximum
operating r f power for HTS filters not only depends on the HTS materials characteristics
but also varies with the filter dimensions. The connection between the feeding lines and
the resonators must be carefully chosen. To avoid the nonlinearity effects, the filte r
design must avoid any bend which creates high field region.
Microstrip line resonators have significantly different even- and odd-mode
effective dielectric constants.
This difference w ill result in considerable forward
coupling, which w ill reduce the dimension o f the filte r in the lateral dimension.
However, the small bandwidth of the high quality filters requires a relatively large
spacing between the strips in the longitudinal direction. Hence, microstrip line resonators
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148
are characterized by their small size, even when made o f conventional conductors.
Therefore, the main advantage o f using HTS in designing high quality filters is the better
performance compared to conventional metal. However, the size and the layout o f the
high power HTS filters must be optimized to avoid the nonlinearity effects associated
with the HTS materials.
5.6 Summary
A nonlinear full-wave three-dimension time-domain analysis for the HTS
microwave devices is presented. This approach takes into account the variation of the
macroscopic parameter of the superconducting material with the applied power, position,
and temperature simultaneously. The wave penetration effects are rigorously included
using the features of the three-dimensional finite-difference time-domain approach.
The nonlinearity in the HTS is modeled by the GL equations. The anisotropic
three-dimensional behavior of HTS superconductor is reduced to a quasi two-dimensional
one. The GL solution gives the spatial distribution of the order parameter. The deduced
order parameter is used to update the main macroscopic parameters o f the HTS, the
magnetic field penetration depth and the normal conductivity in the longitudinal
direction. The normal and super fluid current densities are calculated using Ohm's law
and London's equation respectively. The currents in the transverse plane are very small.
They are depicted using the simple linear model. The physical characteristics of the HTS
are blended with the electromagnetic model using the two fluid model. Hence, the
superfluid electron density is calculated at different levels o f applied power.
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149
The maximum r f power for HTS microstrip line is predicted. Numerical results
show that a change in the phase velocityof the wave propagating along the line of about
1.5% occurs as the applied power reaches 0.9 o f the maximum r f power.
The
corresponding increase in the attenuation is dramatic. It increased 170 times compared to
the low power case. The presented results show that the attenuation constant is more
nonlinear than both the phase velocity.
The effect on the electromagnetic field
distribution is studied. It is more pronounced near the edge of the HTS strip. The
superfluid current density distributions change dramatically with the applied field. The
t
change in the frequency spectrum is successfully depicted.
The linear London model underestimates the field penetration inside the HTS
material. It also overestimates the current crowding effects. Thus, the nonlinearity
associated with.the HTS material is successfully modeled. A complete study concerning
the effects o f the nonlinearity on the performance of HTS transmission lines is presented.
The full-wave 3D nonlinear HTS electromagnetic simulator is successfully used
to simulate microstrip resonator array filter. The scattering parameter, S21 , is predicted
and compared with the measured data. The results show a good agreement for the
resonant frequency and bandwidth. The output input power relation is also depicted. The
nonlinearity associated with the HTS filter appears at approximately 10 dBm. The field
distribution in the dielectric is presented. The high field values at the input o f the filter,
as well as, the edges of the resonator are shown. The dimension and the layout o f high
power HTS filters needs to be optimized in the design cycle to avoid the nonlinearity
effects.
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CHAPTER 6
N O VE L N O N LIN E A R P H E N O M E N O LO G IC A L
TW O F L U ID M O D E L
The greatest use of phenomenological models, such as the two fluid model, is to
reveal experimentally verifiable interrelations among physical properties, such as the
temperature dependence of the penetration depth. Then, it is the task of the microscopic
theory to justify the phenomenological model. The phenomenological model can also
help in a qualitative visualization of experimental phenomena. The two-fluid model in
superconductivity has had only limited success in these respects. The point here is that
the Gorter-Casimir and related two-fluid model were set up as to have the observed
thermodynamics properties [62], But the model gave no information on, and indeed were
not intended to have anything to do with, the hydrodynamic or electrodynamics aspects
of the two fluids, the superconducting and normal fluids. F. London and H. London
developed
an electrodynamics model for superconducting
materials
on a
phenomenological basis [ 6 8 ]. The theory gives a consistent description of essentially all
the electromagnetics properties of superconductors. However, London model lent itself
primarily to the electrodynamics, and not to the thermal properties of the superconducting
materials. Thus, the Gorter-Casimir two-fluid and London models did not consider the
bidirectional coupling between the thermodynamics and electrodynamics in a
superconducting system. However, the expressions obtained from those models are
elegant and simple in general, and are in use today as a qualitative model. The next level
o f phenomenological description were presented by the GL theory o f superconductivity
[63], The G L phenomenological model ties the thermodynamics and electrodynamics of
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151
the superconductor intimately together. It represents a plausible extension o f the London
theory and two-fluid model to situations where the super electrons density is field- and
position-dependent. But, the theory is encapsulated in a set of nonlinear equations. The
solution o f G L equations is very involved, which makes them unsuitable for a computeraided-design environment.
One important aspect of the physics of the HTS is their layer structure and the
associated large anisotropy. It turns out that the structure and dynamics of flux-lines in
such layered systems are markedly different from those o f isotropic conventional
superconductors. Another important aspect is the unusual range o f parameters, such as
short coherence lengths, large penetration depths, and high operating temperatures
allowed by the HTS. In view of this, it is not surprising that a rich variety o f unusual
behavior is found. A microscopic theory describing the physics o f HTS is unavailable at
the present time. Numerous issues are still in part controversial, especially those dealing
with phase transitions of the vortex lattice, explained later.
In this chapter, we present a novel nonlinear phenomenological two-fluid model
for superconducting materials.
The model is based on experimental observation of
superconductors. The temperature and field dependence are taken into consideration
simultaneously. The main nonlinear macroscopic parameters for superconductors are
derived. An empirical formula for the surface impedance of HTS that agrees very closely
w ith experimental measurements for YBCO superconductors is developed.
These
compact models are validated and verified by comparing the calculated results with data
obtained from experimental measurements. This model combines the physics associated
w ith the G L phenomenological model and the required sim plicity obtained from the
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152
linear London's model. This new model fits the need for nonlinear consideration in
computer-aided-design of microwave and millimeter-wave HTS devices and circuits.
6.1 Nonlinear Phenomenological Two fluid model
HTS are classified as type II or mixed state superconductors. The first successful
theoretical explanations of type II superconductors was derived by Abrikosov in 1957
[103]. The flux is observed to enter a type II superconductor in a discrete array of entities
known as vortices. Each vortex has a quantized amount of flu x d>0 = h/2e associated
with it, where h is the Plank's constant and e is the electron charge. As the vortices enter
the superconductor, it has already been confirmed experimentally that they form a
triangular lattice with a lattice constant a0 =(<E>0 / B)yi [104]-[105]. The radius of the
vortex, called the coherence length £, is another im portant length scale in
superconductivity. As the applied magnetic field increases, the density o f the vortex
lattice increases. A t the upper critical field the normal cores of the vortices overlap, and
the material becomes normal. The formation of the vortex lattice fundamentally changes
the way that externally applied currents flow through the superconductor. Instead o f
being confined to a penetration depth A near the surface, the currents flow uniformly
throughout the superconductor, thereby greatly increasing the current carrying capacity of
the material.
HTS are characterized by small coherence lengths and large penetration depths.
The typical values of the coherence length £ and of the penetration depth A in HTS are of
order C = 10 A 0 and A ~ 1000 A°, respectively. This yields a G L parameter
k
= A/C of
about 100, i.e., the HTS materials are extreme type II superconductors. The structure o f a
flux line is depicted in Fig. 6.1.
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153
-X
X
Fig. 6.1 Variation of the number of superfluid electrons and the magnetic
field H near a flux line.
The magnetic field decays over a length scale set by the penetration depth A. The
superfluid electrons density is zero at the center o f the flux line and approaches rapidly its
equilibrium value over distances of order the coherence length £
When the coherence length £c in the c-direction is shorter than the layer spacing s,
the structure discreteness becomes important and one expects a crossover from
anisotropic 3D behavior to quasi 2D behavior. The structure of the vortex cores quasi 2D
behavior occur for almost all temperatures T < Tc. This condition holds for temperatures
o f about 4% below TC(B=0) for YBa 2Cu 3 0 7 .x [ 6 6 ]. When the magnetic field is applied
parallel to the CuO-layers, the vortex cores fit between the superconducting layers. This
situation is rather similar to that of a single wide Josephson junction in a parallel
magnetic field [71]. The HTS could be represented by a stack of Josephson coupled
layers. The Josephson vortices differ from the usual Abrikosov vortices mainly in that
they have no normal vortex core, since the superfluid electron density is not modified by
the presence of the Josephson vortex. In conclusion, the central point o f the London
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154
theory, that the supercurrent is always determined by the local magnetic field, is an
appropriate postulate for HTS materials.
The main macroscopic parameters o f superconductors are the magnetic field
penetration depth and the normal conductivity. GL theory results in a spatial, field and
temperature dependent macroscopic parameters. The magnetic field penetration depth
can be calculated form the following expression:
(6.1)
where /l/o.o) is the low field penetration depth measured at T = 0 and H - 0. The
normalized order parameter is obtained from
4“
¥ (H ,T )
W{H, T)
V~
¥ ( 0 ,T)
(
t
)
1 -
{ T c)
where Tc is the critical temperature for the superconductor calculated at H = 0. Hence,
the magnetic field penetration depth can be written in the following form
/i,(o.o)
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(6.3)
155
The parameters a and /3 may be obtained from experimental studies. In our full-wave
analysis, we chose a = 2 and ( 3 - 4 following G L model for the field dependence and
Gorter-Casimir for the temperature approximations.
Gorter and Casimir assumed that the fraction of the conduction electrons in the
superfluid state ns varies from unity at T = 0 to zero at the temperature o f the transition to
the completely normal state Tc. They found that the best agreement w ith the thermal
properties o f conventional superconductors was obtained when this fraction was chosen
to have the form
f j. \ 4
^
-
=
1-
(6.4)
vT
-w
where nsx is the number of superfluid electrons at zero temperature and zero applied
field, which is equal to the total number o f electrons in the system. The magnetic field
dependence was ignored in their formulation.
A low field condition is assumed.
However, GL theory coupled both the thermodynamics and electrodynamics o f
superconductors together. This results in simultaneous spatial, field, and temperature
magnetic field penetration depth dependence, as shown in Eq. (6.3).
It is known that fraction of the conduction electrons in the superfluid state ns
could be assumed to vary from unity at H = 0 to zero at the field of the transition to the
completely normal state H c. This postulate is valid at temperature T = 0. The critical
magnetic field H c is computed at T = 0.
This argument is expected to lead to a
dependence similar to the Corter-Casimir thermal dependence. Then, the fraction o f the
superfluid electrons can be obtained from the following expression
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The parameter a may be obtained from experimental studies. A temperature and field
dependence formulation can be obtained by combining Eqs. 6.4 and 6.5, and using the
result obtained from GL theory.
The fraction o f the conduction electrons in the
superfluid state ns can be written as
H
1-
(6.6)
H C(T)
where
2“
H c{T )~ H a
1
-
f T]
for T < T
(6.7)
{TcJ
where H co is the thermodynamic critical field at zero temperature. Actually, H C( T ) w ill
be assumed as a given experimental quantity.
Having provided a plausibility argument based on experiment describing the
presence o f the superfluid electrons as a function of temperature and magnetic field, we
next consider the electrons as particles. Flux quantization inside a superconductor allows
us to deduce that the carrier, called a superfluid electron, is in reality a correlation o f two
actual electrons. This observation is consistent with the notion of a Cooper pair. Using
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157
the conservation o f electrons law, the fraction o f the normalfluid electrons can be
deduced from the following expression
(6 .8)
The factor 2 doesn't appear in this expression because we are dealing w ith ratio of
superfluid electrons, not absolute values.
Eq. ( 6 .8 ) can be rewritten in terms of
temperature and field dependence as
n
1-
f ~
T V5'
\
H
1-
H C(T)
\
Tc J
(6.9)
6.2 Macroscopic Model of Nonlinear Constitutive relations in HTS
London's equations are used to derive the constitutive relations for
electromagnetic fields.
The London's equations are derived from the fundamental
Newtonian dynamics and the Meissner effect. They can also be derived from quantum
mechanics by introducing a canonical momentum [62], I f the Lorentz force due to
magnetic field is not considered, the linear London's equations are valid from the Drude
model and the Newton's second law.
E .
dt
E_
A
Vx / =-
B
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(6.10)
(6. 11)
where E and B are the total electric and magnetic fields, respectively, Js is the current
density due to the superfluid electrons, and
A=
(6.12)
where ms and qs are the mass and charge o f the superfluid electrons respectively, ns is
the number of the superfluid electrons. The nonlinearity w ill be included in
A (H ) = n X ( H )
(6.13)
The spatial, field and temperature dependent magnetic field penetration depth Xs
for the superconductor can be calculated from the following expressions :
(6.14)
where A,(o,o) is the low field penetration depth measured at T = 0 and H = 0,
h = H ( r ) / H c(T ), t = T /T c, and Tc is the critical temperature for the superconductor
calculated at H = 0. The parameters a and /? may be obtained from experimental
studies.
The norm alfluid current density is deduced from Ohm's law, and the
corresponding normal conductivity is expressed as fo llo w s:
(6.15)
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159
where
o „ { hcit c)
is the maximum normal conductivity measured either at T = TC or
H - H c and H c is the critical magnetic field for the superconductor calculated at T = 0.
Eqs. 6.14 and 6.15 represent the dependence o f the main macroscopic parameters o f the
superconducting material on the spatial distribution o f the applied fie ld and the
temperature simultaneously. Their dependence on the applied field fu lly represent our
nonlinear tw o-fluid model. The constitutive relations for time harmonic fields can be
represented using either the complex conductivity or complex dielectric concepts.
6.3 HTS Nonlinear Surface Impedance
The proposed nonlinear two-fluid model is verified using the experimental results
obtained for the surface impedance. In our calculation, the surface impedance formula of
a good conductor is adopted
(6.16)
Using the complex conductivity concept for superconductors
( T = ( T „ + (T,
(6.17)
ZJ =0.5co2n 2
0X3an + ja> ngA.
(6.18)
Eq. (6.16) becomes
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The real part gives the surface losses per unit area per unit surface current-density
amplitude, and the imaginary term represents the surface inductive reactance o f the
superconductor. Then, the surface resistance Rs can be approximated as
(6.19)
Rs =0.5Q )2n ; l 3Gn
The field, temperature, and frequency dependent surface resistance is expressed as
follows
R,(h,t,a>) = 0 . 5 ^ ] ( 0 , 0 ) a n{H J T c)co
where h = H ( r ) / H C(T ), t - T/T c, Tc is the critical temperature for the superconductor
calculated at H = 0, H c. is the critical thermodynamic magnetic field depicted at
temperature T, and co is the operating frequency.
The surface resistance may be
normalized to the material characteristics factor Q .5jjlA.](0,0)an(H c/T c), that yields to a
general normalized surface resistance phase parameter expression,
(6.21)
This expression is valid for all superconductor materials, and is independent o f their
physical characteristics. The surface resistance phase parameter is only dependent on the
operating conditions. The dependence o f HTS on the temperature and field can be easily
deduced from this expression.
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161
An empirical formula proposed by Pippard that agrees very closely with
experiment measurements for conventional superconductor [67] gives
However, this expression fails to predict the temperature dependence for the surface
resistance o f the YBCO HTS. The formula that fits mostly with the experimental results
depicted for the temperature and field dependence o f the YBCO HTS, as w ill be shown
later, can be expressed as follows
(6.23)
where a and ft are obtained empirically. Their values depend on the operating region of
interest. The function ( l - h a) is plotted in Fig. 6.2 as a function o f h for different
values o f the variable a . The rate o f change in the function value increases with the
decrease in a close to h = 0. It increases with the increase in a near h = 1. The value
of the function itself decreases with the decrease in the value of a .
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162
1
a =4
a = 2
0.8
a = 3/2
0.6
a =3/4
0.4
0.2
a = 1/2
0
0
0.2
0.4
0.6
0.8
1
h
Fig. 6.2 The function (l - / t “ ) as function o f h for different values o f the
variable a .
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163
6.4 Nonlinear Model Validation and Verification
The surface resistance o f YBa 2 Cu3 0 7 -x HTS film with critical temperature o f
86.4 K , and magnetic field penetration depth Ai (0) equals 0.167 fim is estimated using
the proposed nonlinear two fluid model The calculated results are compared with the
measured data in [106]. Fig. 6.3 shows the critical magnetic field, calculated using Eq.
(6.7), as a function o f the normalized temperature. Although, this formula is assumed for
conventional superconductor, it fits well the measured data for the YBCO in this
operating region.
The temperature dependence for the surface resistance at zero applied magnetic
field is calculated using Eq. (6.23), with (5 = 3/2. The operating frequency equals to 1.5
GHz. The results are compared with the measured data presented in [106], and shown in
Fig. 6.4. It is seen that fair agreement is obtained between t = 0.6 and t = 0.95. This
operating region includes the liquid nitrogen boiling temperature, 77 K, where all the new
HTS material operates.
The surface resistance as a function of both the temperature and magnetic field at 1.5
GHz is depicted in Fig. 6.5. The surface resistance is calculated using Eq. (6.23), with
a = 3/4 and [5 = 3/2. A comparison between the calculated results and the experimental
data is conducted and presented in Fig. 6.5. A good agreement between the measured
and the calculated data in the practical operating region of the YBCO HTS can be
observed.
As the operating temperature decreases a slight difference between the
measured and the calculated results appears. This is due to the imperfection associated
with the HTS material, which is not taken into consideration in the analytical formula.
The presented results show that the effect of the magnetic field on the HTS material is
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164
more pronounced than the temperature effect. The material looses its superconducting
characteristics much faster with the increase in the magnetic field intensity.
1400
— Calculated
1200
o
O
— Measured
1000
800
600
400
■c
u
200
0.5
0.6
0.7
0.8
0.9
1
Normalized Temperature ( t = T /T ,)
Fig. 6.3 Comparison between the calculated and the measured [106]
critical magnetic field for YBCO HTS as a function of temperature.
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165
7
Calculated
Measured
PeakH f = 0
6
cs
'o
r—<
5
X
o
u
4
o
o
u
3u
a
3
CA
on
2
1
0
0.5
0.6
0.7
0.8
0.9
Normalized Temperature (t = T/T )
Fig. 6.4 Comparison between the calculated and the measured [106]
surface resistance for YBCO HTS as a function of temperature at zero
magnetic field.
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14
12
-8 4 K
81 K
— Calculated
- - Measured
78 K
10
8
63 K
6
53 K
4
2
0
0
100
200
300
400
500
600
700
800
PeakHr f (Oe)
Fig. 6.5 Comparison between the calculated and the measured [106]
surface resistance for YBCO HTS as a function o f temperature and
magnetic field.
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167
6.5 Summary
A novel nonlinear phenomenological tw o -flu id model is proposed fo r
superconducting materials. A macroscopic model for the nonlinear constitutive relations
is also suggested. The nonlinear main macroscopic parameters for superconductors, the
superconductor magnetic field penetration depth and the normal conductivity, are
derived. An empirical formula for the surface impedance of HTS that agrees very closely
with experimental measurements fo r YBCO superconductors is developed.
These
compact models are validated and verified by comparing the calculated results with data
obtained from experimental measurements. A fairly good agreement is seen in the
practical operating region of the new HTS materials. This model combines the physics
associated with the G L phenomenological model and the required sim plicity obtained
from the linear London's model. It is observed that the HTS looses its superconducting
characteristics faster with the applied field than the temperature. This new model is very
useful for computer-aided-design of microwave and millimeter-wave HTS devices and
circuits.
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CHAPTER 7
CONCLUSIONS
This dissertation presents a comprehensive analysis of linear and nonlinear planar
microwave devices using a full-wave electromagnetics approach, the finite-difference
technique. The technique was applied to High critical Temperature Superconducting
material (HTS). The goal of the research is to conduct a nonlinear analysis of microwave
HTS devices using a full-wave electromagnetics simulator, which incorporates the
anisotropic o f the superconducting strip and the substrate simultaneously. This study is
crucial for high power applications of HTS microwave and millimeter-wave devices. A
nonlinear phenomenological model for HTS materials is also developed. This model w ill
be very useful for CAD.
The preliminary material of chapter 1 introduced the goals, the importance o f the
research, and set guidelines for achieving our objectives. Chapter 2 described the present
linear and nonlinear phenomenological models for HTS. The solution for the nonlinear
Ginzburg-Landau (GL) was presented for bulk and thin film s HTS to provide better
understanding of the nonlinearity and the RF power handling capability associated with
HTS materials. Chapter 3 provided the finite difference approach as it is the numerical
technique used in the dissertation. Solution of nonlinear problems using the Newton's
SSOR method was presented. The finite difference frequency (FDFD) and time (FDTD)
domains
compared.
algorithms as applied to electromagnetics problems were discussed and
Chapter 4 introduced the analysis and sim ulation o f anisotropic
superconductors on anisotropic substrates.
The anisotropic FDTD was applied to
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169
microstrip line and coplanar waveguide structures. Chapter 5 showed the full-wave
nonlinear analysis o f HTS microwave HTS devices. The two fluid model blended the G L
nonlinear solution with the FDTD electromagnetics simulator. A complete study on the
nonlinear effects o f HTS microwave devices was conducted.
In chapter 6 , a novel
nonlinear phenomenological two fluid model was suggested. An empirical formula for
the nonlinear surface impedance o f HTS thin films was also proposed.
7.1 Summary of Findings and Conclusions
This thesis has reported the first rigorous effort in modeling the nonlinear
characteristics o f superconducting microwave and millimeter-wave devices in the time
domain using a full-wave technique. The anisotropic three-dimensional behavior o f the
HTS was reduced to a quasi-two-dimensional analysis for applications operating in the
low gigahertz range. G L theory has been used to model the nonlinear mechanism o f
superfluid electron pair breaking. The longitudinal superfluid current along the direction
of wave propagation was calculated using the nonlinear model. The currents in the
transverse plane, which are small for planar microwave structures, were considered with a
linear model.
The anisotropic characteristics of HTS and the effect o f anisotropic
substrates on the performance o f HTS devices were also studied. This work presented the
first attempt to handle the anisotropy both in the conducting strip and in the dielectric
substrate simultaneously. A novel phenomenological nonlinear two fluid model and an
empirical formula for the nonlinear surface resistance have been reported. This model is
the first simple model that combines the electrodynamics and thermodynamics o f the
superconducting material.
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170
The major results in the dissertation are summarized below:
1.
It has been shown that the nonlinearity behavior must be modeled in the time
domain, and not in the frequency domain. A full-wave simulator was suggested
for the nonlinear analysis of anisotropic HTS.
2.
The nonlinear Ginzburg-Landau model could be applied to the new HTS
materials, despite the anisotropy associated with them. The anisotropic threedimensional behavior o f HTS could be reduced to a quasi-two-dimensional
behavior for HTS used in planar microwave and millimeter-wave applications.
3.
Numerical results for HTS strip used in microwave applications showed that the
number of superfluid electrons decreased near the edges w ith the increase in the
applied power, indicating the breaking o f superfluid electrons pairs.
The
shielding ability of the HTS was weakened near the edges, resulting in enhanced
penetration o f the magnetic field. The linear model underestimates the magnetic
field penetration and overestimates the edge enhancement o f the current density.
4.
The power handling capability o f HTS used in microwave and m illim eter
applications depend not only on the material characteristics but also on the
structure under investigation. In this case, the maximum power is determined by
the rf critical power.
5.
The complete nonlinear study of HTS materials showed that the phase velocity
and the attenuation o f the wave propagating along HTS transmission lines
changes nonlinearly with the applied power. The change in the losses is much
larger than the change in the phase velocity. The change in the electromagnetic
fields is more pronounced near the edge of the HTS strip. The superfluid current
density distributions change dramatically with the applied power. The variation
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171
in the frequency spectrum of the applied signal resulting from the nonlinearity
was obvious.
6.
HTS microstrip resonator array filte r was successfully simulated using the
developed nonlinear approach. Results showed that the dimension and the layout
o f high power HTS filters need to be optimized in the design cycle to avoid the
nonlinearity effects.
7.
A novel nonlinear phenomenological tw o-fluid model fo r superconducting
materials was proposed.
This model blended the thermodynamics and the
electrodynamics properties of HTS.
8.
An empirical formula for the nonlinear surface resistance o f HTS was suggested.
The results calculated using this formula was successfully compared with
experimental data. A good agreement is obtained in the region o f interest.
In order to achieve the goal of this dissertation, several numerical aspects for the
finite difference algorithm are considered. The main features, that are tackled in the
thesis, are:
1.
A general wave equation for nonuniform dielectric structures was derived. There
is no need to impose unnecessary boundary conditions anywhere inside the
structure.
2.
Finite thickness conductors in planar microwave and millimeter-wave devices was
rigorously modeled using nonuniform mesh generator for the finite difference
scheme. Results showed that field penetration effects were successfully predicted
in our analysis.
3.
The Perfectly Matched Layers (PML) absorbing boundary condition was modified
and successfully applied to microwave devices that include finite conductivity
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172
conductors. Results presented the improvement in the finite-difference timedomain lattice truncation by using the PM L compared to the one-way wave
equation ABC.
4.
A parallel finite-difference time-domain algorithm was demonstrated. Results
showed that a tremendous run time improvement is obtained by using parallel
machine environment compared to powerful serial machine as the mesh size
increases.
5.
An anisotropic finite-difference time-domain scheme was suggested.
This
technique was able to take the anisotropy in the conductors and the substrates
simultaneously. Results showed the importance of considering the anisotropy for
the optimum design o f microwave and millimeter-wave devices.
7.2 Recommendations for Future Research
Extensions or possible improvements in this work are numerous.
This
dissertation had contributions in several areas such as numerical techniques, microwave
and millimeter-wave devices, nonlinear phenomena analysis, computer-aided-design o f
microwave devices, analysis and modeling of HTS materials. In the computational
aspect, the perfectly matched layers (PML) absorbing boundary conditions can be
extended and applied to truncate lossy materials, where no appropriate absorbing
boundary conditions exist. A study for the use of the PML at the side and top walls of the
finite-difference lattice that include microwave devices can be conducted. A rigorous
dispersion analysis for the nonuniform mesh generator used with the finite difference
approach can be performed. The nonuniform mesh finite-difference algorithm can be
applied to model and study other phenomena besides the wave penetration effects, such
as the diffusion or injection o f electrons in thin oxide in active semiconductor devices.
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173
The anisotropic finite-difference time-domain technique can be applied to study other
materials where the anisotropy can not be neglected or approximated such as ferrite
materials.
Application o f the full-wave simulator to other HTS microwave and
millimeter-wave devices could be performed. Effects of the nonlinearity and anisotropy
associated with HTS materials on the performance o f those devices can be studied. In the
HTS materials characterization, the nonlinear two fluid model needs to be verified with
measurements from different samples. Empirical formulas fo r the nonlinear surface
impedance o f different HTS materials can be obtained using the same algorithm
r
presented in the dissertation.
Also, a macroscopic model o f frequency dependent
constitutive relations in HTS needs to be developed.
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