MODELING AND DESIGN OF SUPERCONDUCTING MICROWAVE
PASSIVE DEVICES AND INTERCONNECTS
by
LAURENCE H. LEE
B.S., State University of New York at Buffalo, June 1989
S.M., Massachusetts
Institute of Technology, September
1990
Submitted to the Department of Electrical Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1994
©Massachusetts Institute of Technology 1994
All Rights Reserved
Signature of Author
Department of Electrical Engineering and Computer Science
_
Certified by
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_____l
May 6, 1994
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,Professor,
Certified by
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-
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Terry P. Orlando
Electrical Engineering and Computer Science
IY
I
W. Gregory Lyons
-
SrfMember,
Lincoln Laboratory
Accepted by
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JUL 1
2
MODELING AND DESIGN OF SUPERCONDUCTING MICROWAVE
PASSIVE DEVICES AND INTERCONNECTS
by
LAURENCE H. LEE
Submitted to the Department of Electrical Engineering and Computer Science
on May 6, 1994 in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy
ABSTRACT
A spectral-domain volume-integral-equation method is developed for analyzing superconducting planar transmission lines. This method rigorously accounts for the complex
conductivity, the finite metallization thickness, and the anisotropy of superconductors. Calculated effective dielectric constants and quality factors for microstrip lines and coplanar
waveguides show good agreement with measured data. The effects due to the anisotropy of
high-temperature superconductors are found to be negligible for c-axis oriented films. The
characteristics of a microstrip line with a thin buffer-layer and a stripline with a small air
gap are also analyzed. Based upon the current distribution, a quantitative analysis is made
of the power-handling capability of superconducting non-TEM resonators. Among several
commonly used planar transmission lines, microstrip lines are found to have the largest
power-handling capability.
A more efficient full-wave method is developed based upon an equivalent surface impedance
concept. In the formulation, the superconducting strips are transformed to infinitely thin
strips. A full-wave integral equation method is then used to solve the problem with infinitely thin strips. This method reduces the computation time by nearly two orders of
magnitude compared to the volume-integral-equationmethod. The validity of the equivalent surface impedance approach is verified by measurements and by the volume-integralequation method. This method is extended to model single and coupled microstrip lines
on anisotropic substrates with a rotated optic axis. In the development of the equivalent
surface impedance method, a closed-form expression for the current distribution in isolated
superconducting strips is obtained. This closed-form expression is valid for strip thicknesses
less than a few penetration depths.
The equivalent surface impedance method is used to develop a set of computer routines
for modeling and designing superconducting edge-coupled bandpass filters. In the design
procedure and the simulation, the secondary coupling effects are neglected. A sensitivity
analysis is performed on a 4-pole 1% bandwidth microstrip filter centered at 10 GHz and a
4-pole 0.05% bandwidth shielded microstrip filter centered at 2 GHz. The analysis shows that
the filters remain well matched with a shift in center frequency when the superconducting
3
material is replaced by another superconductor or the thickness of the superconductor is
changed. The filter remains acceptably matched with a slight symmetric variation in the
dimensions of the resonators, but random variations must be smaller than 2 ,um for the 1%
filter and 0.25 pumfor the 0.05% filter. The 4-pole 1% microstrip filter design is fabricated
using silver and niobium. The measured results indicate significant secondary coupling
effects.
The characteristics of coupled shielded microstrip lines are studied using the full-wave
method developed in this thesis. The structures are found to be highly dispersive. A parallelplate mode supported by the ground plates greatly deteriorates the shielding property. The
effects of the parallel-plate mode predicted by the analysis are confirmed by experiments
made on several quarter-wavelength couplers. Closed-form expressions for the open-end
equivalent length in single and coupled shielded microstrip lines are derived based upon the
open-end models for the stripline and the microstrip. The equivalent lengths given by these
expressions agree well with those obtained using a commercial field simulator.
Thesis Supervisor:
Title:
Prof. Terry P. Orlando
Professor of Electrical Engineering
Thesis Supervisor:
Title:
Dr. W. Gregory Lyons
Staff Member of Lincoln Laboratory
4
"...I cannot doubt but that these things, which now seem to us so mysterious, will be no
mysteries at all; that the scales will fall from our eyes; that we shall learn to look on things
in a different way - when that which is now a difficulty will be the only common-sense and
intelligible way of looking at the subject."
Lord Kelvin (William Thomson)
discussed the skin effect in 1889.
"The purpose of computing is insight, not numbers."
Hamming commented on numerical analysis.
5
6
To Phoebe
Taylor
Brandon
Justin
7
8
Acknowledgement
I would like to express my gratitude to my supervisors Professor Terry P. Orlando and
Dr. W. Gregory Lyons, and to my former supervisor Dr. Sami M. Ali for years of supports, guidance, and encouragement. Their professional advise and supervision have made
the completion of this thesis possible. I am also very thankful for their caring friendship.
I would also like to thank my former supervisor Professor Jin Au Kong. Professor Kong's
enthusiasm in teaching his graduate electromagnetic wave course motivated me to pursue
research in electromagnetics.
I thank Professor Jacob K. White for taking his valuable time to be my thesis reader,
and for making helpful suggestions.
I express my gratitude to Dr. Timothy R. Dinger and Dr. G. Arjavalingam and the
Superconductivity Group at IBM T. J. Watson Research Center for their hospitality during
my visit.
I would like to acknowledge valuable conversations with Dr. Richard Withers on microwave filters and couplers. I am thankful to Dr. Cheung-Wei Lam for his friendship and
for making numerous inspiring suggestions. His assistance in my job search is deeply appreciated. Dr. Daniel E. Oates and Dr. Jeffrey D. Goettee provided the measurement results
on resonators. I thank Robert P. Konieczka for his assistance in packaging the filters and
couplers measured in thesis, and Douglas M. Dugas for his valuable discussion on filter design.
I would also like to take this opportunity to thank my present and former colleagues
for many enlightening conversations and for their friendship; especially William Au, David
Berman, David Carter, Amy Duwel, Dr. Hsiu Han, Derek Linden, John Oates, Joel Phillips,
Dr. David Sheen, Enrique Trias, Li-Fang Wang, and Dr. Herre van der Zant. Margery
Brothers, Charmaine Cudjoe-Flanders, Kit-Wah Lai, and Angela Odoardi have also helped
in many ways.
Most of all I thank my wife, Phoebe, and my sons, Taylor, Brandon, and Justin, for their
love, supports, and companionship, and for their sacrifices during this difficult effort. They
have brought new meanings to my life.
This work was conducted under the auspices of the Consortium for Superconducting
Electronics with the full support of the Advanced Research Projects Agency under Contract
MDA972-90-C-0021.
Laurence H. Lee
1994
9
10
Contents
List of Figures
13
List of Tables
17
1 Introduction
1.1 Background ........................
1.2 Description of the Thesis .................
19
2 Spectral-Domain Volume-Integral-Equation Method
27
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Introduction.
The Two-Fluid Model ..................
Generalized Formulation .................
Volume-Integral-Equation for Microstrip Lines .....
Method of Moments ...................
Results and Discussions .................
Summary .........................
20
22
...... ..... . .27
...........
..29
.... ....... . .32
........... ..35
...........
..44
........... ..56
. . ..
. . . . . . . . .37
3 Equivalent Surface Impedance Approach
3.1 Introduction . ............................
3.2
Equivalent Surface Impedance ...................
3.3 Current Distribution in Superconducting Thin-Film Strips . . .
3.3.1 Approximate Solution for the Current Distribution . . .
3.3.2 Applications of Isolated Strip Approximation .......
3.4 Superconducting Microstrip on Anisotropic Substrate ......
3.4.1 Integral Equation Formulation and Method of Moments .
3.4.2 Numerical Results and Discussion .............
3.4.3 Summary ..........................
4 Superconducting Bandpass Filter
4.1 Introduction.
............................
4.2
Transmission-Line Resonators ...................
4.2.1 Power-Handling Capability .................
4.2.2 Packaging ..........................
11
59
59
61
62
63
68
70
71
74
88
91
91
92
92
94
CONTENTS
12
Filter Design Procedure ........
Filter Simulation Method .......
Numerical Simulations and Discussion
4.5.1 A Four-Pole 1% Filter .....
4.5.2 A Four-Pole 0.05% Filter ....
4.6 Experimental Results ..........
4.7 Summary ................
4.3
4.4
4.5
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5 Shielded Microstrip Lines
5.1 Introduction.
5.2 Edge-Coupled Shielded Microstrip Lines .......
5.3 Experimental Verification of Parallel-Plate Mode. . .
.. .
5.4 Open-End Discontinuity in Shielded Microstrip Lines
5.5 Summary ........................
6
Conclusions
98
104
107
107
113
123
125
129
....
....
.
. .
129
130
... . 135
... . 139
... . 144
145
A Dyadic Green's Function for Isotropic Media
149
B Dyadic Green's Function for Anisotropic Media
155
Bibliography
157
Publications from Thesis
163
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
The two-fluid model. ...............................
Electric field generated by current sources enclosed in a volume V' ...
Geometric Configuration of a layered medium with embedded current sources.
A superconducting strip is embedded in layer (0) of a layered dielectric medium.
Superconducting microstrip line.
...................
A nonuniform grid is imposed on the cross section of a superconducting strip.
Sommerfeld integration path in k,-plane .....................
Sommerfeld integration path in ks-plane .....................
.
30
32
34
35
37
38
42
42
2.9
Effective dielectric constant
45
of Nb microstrip transmission
lines ........
2.10 Quality factor of superconducting microstrip resonators ............
46
2.11 Longitudinal current density at the bottom of the signal strips of microstrip
transmission lines. ................................
46
2.12 Superconducting microstrip transmission line with a thin dielectric buffer layer
and a perfectly conducting ground plate .
....................
47
2.13 Effective dielectric constant and attenuation constant of a superconducting
microstrip line with and without a thin dielectric buffer layer ........
.
48
2.14 Effective dielectric constant and attenuation constant of an isotropic and an
anisotropic superconducting microstrip line ...................
50
2.15 Effective dielectric constant and attenuation constant of an isotropic and an
anisotropic superconducting microstrip line..........
....
50
2.16 Stripline transmission line with an air gap ....................
51
2.17 Effective dielectric constant and attenuation constant of superconducting striplines
with (lines 1 and 2) and without (lines 3 and 4) an air gap ...........
53
2.18 Configuration of a coplanar waveguide ...
...................
. 54
2.19 Effective dielectric constant of Nb coplanar waveguides.............
54
2.20 Quality factor of Nb coplanar waveguide resonators ...............
55
2.21 Longitudinal current density at the bottom of the center strips of Nb coplanar
waveguides
....................................
56
2.22 Longitudinal current density at the bottom of the ground plates of coplanar
waveguides .....................
57
3.1
3.2
A superconducting microstrip line. .......
Cross section of an isolated superconducting strip ................
13
...
..
61
63
LIST OF FIGURES
14
Current distribution in an isolated superconducting strip ............
AC resistance at 2 GHz of an isolated superconducting strip ..........
Total resistance and center strip resistance of superconducting striplines. .
Superconducting microstrip line with a perfectly conducting ground plate on
an anisotropic substrate. .............................
3.7 Superconducting coupled microstrip lines with a perfectly conducting ground
plate on an anisotropic substrate .........................
3.8 Measured and calculated effective dielectric constant of three Nb microstrip
lines on LaA103. .................................
3.9 Calculated quality factor at 2 GHz for a Nb microstrip line on LaA10 3 as a
function of strip thickness normalized to the penetration depth. .......
3.10 Calculated quality factor and characteristic impedance of a YBCO microstrip
3.3
3.4
3.5
3.6
line on LaA1O3 as a function of frequency.
....................
3.11 Calculated effective dielectric constant at 2 and 20 GHz for YBCO microstrip
lines on M-plane sapphire as a function of the rotation angle 0 of the optic axis.
3.12 Calculated attenuation constant at 2 and 20 GHz for YBCO microstrip lines
on M-plane sapphire as a function of the rotation angle 0 of the optic axis. .
3.13 Calculated real and imaginary parts of the characteristic impedance at 2 and
20 GHz for YBCO microstrip lines on M-plane sapphire as a function of the
rotation angle 0 of the optic axis .........................
3.14 Effective dielectric constant of a perfectly conducting coupled microstrip line
on M-plane sapphire ................................
3.15 Calculated c- and 7r-mode effective dielectric constants of YBCO coupled microstrip lines on M-plane sapphire as functions of the rotation angle 0 of the
optic axis at 2 GHz.
................................
66
67
68
71
73
75
76
77
78
79
80
82
83
3.16 Calculated c- and r-mode attenuation constants of YBCO coupled microstrip
lines on M-plane sapphire as functions of the rotation angle 0 of the optic axis
at 2 GHz
......................................
83
3.17 Calculated c- and r-mode characteristic impedances (real part) of YBCO
coupled microstrip lines on M-plane sapphire as functions of the rotation angle
0 of the optic axis at 2 GHz ............................
3.18 Calculated c- and 7r-mode effective dielectric constants of YBCO coupled microstrip lines on M-plane sapphire as functions of the normalized line separation at 2 GHz. ..................................
3.19 Calculated c- and 7r-mode attenuation constants of YBCO coupled microstrip
lines on M-plane sapphire as functions of the normalized line separation at
.......
2 GHz .................................
3.20 Calculated c- and 7r-mode characteristic impedances of YBCO coupled microstrip lines on M-plane sapphire as functions of the normalized line separation at 2 GHz. ..................................
84
85
86
86
LIST OF FIGURES
15
3.21 Calculated c- and r-mode characteristic impedances of YBCO coupled microstrip lines on M-plane sapphire as functions of reduced temperature T/TC
at 2 GHz
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
......................................
87
Configuration of coupled shielded microstrip lines ................
Configuration of a quarter-wavelength coupled microstrip line. ........
The magnitude of S21 of a quarter-wavelength coupled microstrip line as a
function of the distance from the side wall ....................
Calculated even- and odd-mode impedances of coupled shielded microstrip
lines on 508 gm-thick LaAlO3 for Nb at 4.2 K ..................
Configuration of an edge-coupled microstrip bandpass filter. .........
Equivalent circuit of the edge-coupled bandpass filter ..............
Simulated response of a YBCO (71 K) 4-pole 2% microstrip bandpass filter
on sapphire with a center frequency at 10 GHz. ................
Layout of an edge-coupled bandpass filter. ...................
Network representation of a bandpass filter. ..................
4.10 A schematic of a two-port network.
........................
4.11 Transmission model for a coupled section in a bandpass filter. ........
4.12 Simulated response of a 4-pole 1% Nb microstrip filter on 508 gm-thick LaA103
with a center frequency at 10 GHz .
.......................
4.13 Simulated response of a Nb (4.2 K) 4-pole 1% microstrip bandpass filter on
LaA10 3 with a center frequency at 10 GHz. ..................
4.14 Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter with the substrate thickness reduced by 10%. ........................
4.15 Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter with the substrate thickness reduced by 2m .m ...................
4.16 Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter with the dimensions of the resonators uniformly reduced by 2 um. .............
4.17 Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter with the dimensions of the resonators randomly reduced by 0 to 2 /m
.........
4.18 Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter with the dimensions of the resonators randomly reduced by 0 to 0.25 #m ........
.
4.19 Simulated response of a 4-pole 0.05% YBCO shielded microstrip filter on
127 ptm-thick MgO with a center frequency at 2 GHz
..............
4.20 Simulated response of a YBCO (50 K) 4-pole 0.05% shielded microstrip bandpass filter on MgO with a center frequency at 2 GHz ..............
4.21 Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
with the cover height reduced by 10%. ......................
.
.
4.22 Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
with the substrate thickness reduced by 2 m .................
4.23 Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
with the substrate thickness reduced by 8% ...................
94
95
96
96
98
100
102
103
104
105
105
108
111
112
112
113
114
114
116
117
118
119
119
LIST OF FIGURES
16
4.24 Simulated response of a 2 GHz 4-pole
with the dimensions of the resonators
4.25 Simulated response of a 2 GHz 4-pole
with the dimensions of the resonators
4.26 Simulated response of a 2 GHz 4-pole
with the dimensions of the resonators
0.05% YBCO shielded microstrip filter
uniformly reduced by 2 m .......
0.05% YBCO shielded microstrip filter
randomly reduced by 0 to 2 m. ....
0.05% YBCO shielded microstrip filter
randomly reduced by 0 to 0.25 im. ..
120
121
121
4.27 Simulated response of a 2 GHz 4-pole 0.05% YBCO (50 K) microstrip filter
on 381 im-thick LaA103. .............................
4.28 Measured and simulated response of a 1-pole edge-coupled Nb microstrip filter
with a center frequency at 2 GHz.........................
4.29 Measured response of a 4-pole 1% bandwidth bandpass filter fabricated using
3 um-thick silver
.........................
.........
4.30 Measured response of a 4-pole 1% bandwidth bandpass filter fabricated using
0.4 m-thick niobium. ...............................
4.31 Simulated response of a 4-pole 1% bandwidth bandpass filter by a commercial
simulator IE3D ..................................
5.1
5.2
5.3
5.5
5.6
5.7
5.8
5.9
123
124
125
126
Configuration of an edge-coupled shielded microstrip line ............
Calculated even- and odd-mode effective dielectric constants of Nb coupled
130
shielded microstrip lines on 508 1im-thick LaA10
132
3
at 4.2 K ...........
Calculated even- and odd-mode impedances of Nb coupled shielded microstrip
lines on 508
5.4
122
1 im-thick
LaA10
3
at 4.2 K.
.....................
132
Calculated backward-wave coupling coefficient C of a symmetric shielded microstrip, a microstrip, and a stripline quarter-wavelength coupling sections. .
Calculated backward-wave coupling coefficient C of shielded microstrip quarterwavelength coupling sections. ..........................
Electric field distribution of the even-mode in a coupled symmetric shielded
microstrip line with 2s/hi = 2. ..........................
Electric field distribution of the even-mode in a coupled symmetric shielded
microstrip line with 2s/hi = 10 ..........................
Configuration of a symmetric shielded-microstrip quarter-wavelength coupler.
Calculated transmission coefficient of symmetric shielded microstrip quarterwavelength couplers ................................
5.10 Configuration
of a shielded microstrip.
......................
134
134
136
136
137
138
140
5.11 Calculated open-end equivalent length for shielded microstrip lines with 2w/hl =
1...........................................
142
5.12 Calculated open-end equivalent for shielded microstrip lines with h2 /hl = 6.
143
A.1 A layered medium with a current embedded in layer (n). ...........
150
B.1 Cross section of an anisotropic substrate sandwiched by two perfectly conducting
ground
plates
.
. . . . . . . . . . . . . . . .
.
...........
156
List of Tables
2.1 Parameters for the microstrip lines ........................
2.2 Parameters for the Coplanar Waveguides ..................
44
52
4.1
Quality factor and power-handling capability of superconducting resonating
structures with f = 2 GHz for Nb on LaA10 3 at 4.2 K. ...........
4.2 The magnitude of S21 of a quarter-wavelength coupled microstrip line shown
in Fig. 4.2 as a function of the cover height. ................
.
4.3 Physical parameters for YBa 2 Cu3O7 _. at 50 K and Nb at 4.2 K .......
4.4 Design for a 4-pole 1% Nb microstrip filter on LaA10 3 with a center frequency
at 10 GHz.
4.5
5.1
....................................
93
95
109
109
Design for a 4-pole 0.05% YBCO shielded microstrip filter on MgO with a
center frequency at 2 GHz ............................
Dimensions of symmetric shielded-microstrip quarter-wavelength couplers.
17
115
. 135
18
Chapter
1
Introduction
The low-loss characteristics and high operating temperatures of high-transition-temperature superconductors (HTS's) open up possibilities of building high-performance planar microwave devices whose performance can only be achieved with current technology by using
bulky waveguides or coaxial components. However, many material and engineering issues
associated with HTS have yet to be resolved. One of the problems in designing microwave
circuitry using HTS films is lack of adequate computer-aided-design (CAD) tools [1]. Existing commercial CAD tools employ design rules and assumptions made for normal conductors.
When HTS is used in building high performance devices, the accuracy of the CAD tools becomes an important issue. In addition, mismatch, radiation, and dielectric loss mechanisms
that are trivial compared to conductor losses in conventional devices become important factors affecting the performance of low-loss HTS devices. Also, most existing CAD tools do
not accept complex conductivities which are used to characterize HTS's.
The objective of this work is to develop efficient and accurate methods for the characterization and synthesis of superconducting microwave passive devices. This thesis presents
the first full-wave analysis of superconducting planar transmission lines, which rigorously accounts for the complex conductivity, the finite metallization thickness, and the anisotropy of
superconductors. Based upon the current distribution generated from the analysis, an quantitative analysis is made for the first time on power-handling capability of superconducting
19
CHAPTER 1. INTRODUCTION
20
non-TEM resonators. A more efficient but yet accurate full-wave method is developed based
upon an equivalent surface impedance concept. This efficient method is extended to analyze
single and coupled microstrip lines on anisotropic substrates with a rotated optic axis, and is
used to build a set of computer routines for modeling and designing superconducting passive
microwave devices. In particular, a CAD tool is developed for edge-coupled superconducting
bandpass filters. By applying this CAD tool, sensitivity analyses are performed on highperformance bandpass filters. Using a well shielded planar transmission-line structure can
ease the design of very narrowband filters. The characteristics of coupled shielded microstrip
lines are studied using the full-wave method developed in this thesis. For the first time, the
effects of a parallel-plate mode in the shielded microstrip structures are predicted by the
full-wave analysis and are demonstrated by experiments.
1.1
Background
Conventional superconductors are either pure metals or alloys with superconducting transition temperatures (To) well below 30 K. Examples of the most widely used conventional
superconductors are Nb (T, = 9.2 K), NbN (T, = 16 K), and Nb3 Sn (T, = 18 K). Because of
the low T,, costly and complex cryogenic refrigeration systems with liquid helium as coolant
are required to cool these superconductors to such low temperatures. The high cooling cost
limits the use of superconductors in only a few ultra-high-performance applications such as
microwave radio astronomy and magnetic resonance imaging.
In 1986, the discovery of the new copper oxide compounds with high transition temperature has fundamentally changed the prospects for applications of superconductors [2],
[3]. One of the commonly used high-temperature superconductors is YBa 2 Cu3 O7 _ (YBCO)
with T, = 92 K. Because of the high superconducting transition temperature, simpler and
more reliable mechanical coolers can be used to achieve the desired operating temperatures.
In the cases where the magnetic fields and vibration generated by mechanical coolers be-
1.1. BACKGROUND
21
come a problem, liquid nitrogen which is a much cheaper and more abundant coolant than
liquid helium can be used instead. Consequently, the cooling cost of HTS's is nearly two orders of magnitude less than that of conventional superconductors [4], [5]. High-temperature
superconducting microwave devices are now being considered for use in high-performance
microwave systems where components are needed with higher performance, smaller size, and
lighter weight than devices based on conventional technology. These new high-temperature
superconductors, however, are very difficult to process and to handle because of their complicated crystal structures consisting of a large number of elements.
Although many material, physics, and engineering issues associated with high-temperature
superconductors have yet to be resolved, prototypical HTS devices developed by a number
of research organizations have demonstrated performance advantages over conventional devices [6]-[8]. Passive HTS microwave devices have a reasonable potential for widespread
commercialization [6]. Most passive HTS microwave devices are based upon simple planar
transmission-line structures which are relatively easy to fabricate on HTS thin films using
existing photolithographic methods. The performance of these HTS passive microwave devices depends greatly upon the quality of the films. Difficulties in the epitaxial growth of
HTS materials have posed a major obstacle for fabricating high-performance HTS devices.
Recent advancements in film-deposition methods have enabled fabrication of a number of
passive HTS microwave devices with good performance such as high-Q resonators with low
phase noise [9], [10], multi-pole bandpass filters with narrow bandwidth [11], [12], and long
delay lines with low insertion loss [13]-[15]. A YBCO chirp filter has also been recently incorporated into a real-time spectrum-analysis subsystem [16]. Unfortunately, improvements
in performance of HTS devices have been very difficult partly because of the complexity of
the materials. Another reason is the limitations of existing CAD tools [1].
The BCS theory of superconductivity
[17] was developed in 1957 by J. Bardeen,
L. N.
Cooper, and J. R. Schrieffer. This theory successfully explains the superconducting phe-
CHAPTER 1. INTRODUCTION
22
nomenon observed in conventional superconductors, but it fails to explain high-temperature
superconductivity. At this point, there is no acceptable theory available for high-temperature
superconductivity. Despite its short-comings, the BCS theory provides a reasonable description of high temperature materials up to certain limits. However, because of its complexity,
the BCS theory is rarely applied directly for engineering purposes. The widely used model
for superconductivity is the classical two-fluid model and London equations [18] which provide an intuitive phenomenological description of superconductors. These classical models
are used in this thesis for analyzing superconducting devices.
1.2 Description of the Thesis
This thesis proposes accurate full-wave methods for analyzing superconducting planar
microwave devices. The building blocks for planar devices are transmission-line structures.
A spectral-domain volume-integral-equation method is presented in Chapter 2 for analyzing
superconducting planar transmission lines. This method rigorously accounts for the complex
conductivity, the finite metallization thickness, and the anisotropy of superconductors. The
integral equation for the electric field inside the superconductors is formulated using a dyadic
Green's function for layered media in the spectral domain. Galerkin's method is applied to
solve the integral equation. Because of the rapid spatial variation of the electric field in the
superconductors, roof-top basis functions rather than the simpler pulse basis functions are
used to approximate the transverse components of the electric field, so that large singular
charge distributions are eliminated. The complex propagation constant of the transmission
lines and the current distribution in the superconductors are obtained from the analysis.
This method accurately quantifies the effects due to film thickness, kinetic inductance, and
anisotropic properties of high-temperature materials. Calculated results are presented for
microstrip lines, coplanar waveguide structures, and striplines. Measurement data obtained
from resonators agree well with the numerical results.
1.2. DESCRIPTION OF THE THESIS
23
In Chapter 3, a more efficient full-wave method is developed using an equivalent surface impedance concept. The idea is to transform the superconducting strips with finite
thicknesses to infinitely thin strips using equivalent surface impedances. These equivalent
surface impedances account for the loss and kinetic inductance of the superconductors. The
expression for the equivalent surface impedance is derived based upon a closed-form expression for the current inside an isolated superconducting strip. This closed-form expression for
current is obtained by curving fitting to numerical results, and is valid for strip thicknesses
less than a few penetration depths. The isolated strip approximation is valid only for line
widths being small compared to the substrate thickness. A 2-D dyadic Green's function
for layered media is used to formulate an integral equation for the surface current on the
infinitely thin strip. Galerkin's method with entire-domain basis functions is used to solve
for the complex propagation constants. The characteristic impedances are then determined
from the propagating power and current. The validity of this equivalent surface impedance
method is verified by the more rigorous spectral-domain volume-integral-equation method
and by measurements. The effects of the optic axis orientation and the line width on the
characteristics of single and coupled microstrip lines on M-plane sapphire are presented. The
effects of the line separation and operating temperature on the coupled lines are also shown.
Chapter 4 discusses issues associated with designs of superconducting bandpass filters.
An quantitative analysis of the power-handling capability is presented for various planar
resonators. The analysis is based upon the peak magnetic field in the superconductors calculated using the spectral-domain volume-integral-equation method and a modified Weeks'
method.
In principle, a nonlinear model such as Ginzburg-Landau theory is required to
determine the power-handling capability of a superconducting transmission-line resonator.
However, because the nonlinearity inherent in superconductors is small, this analysis based
upon linear models is shown to be valid. Packaging issues associated with planar filters are
also discussed qualitatively. A design procedure and a simulation method are presented for
CHAPTER 1. INTRODUCTION
24
superconducting edge-coupled bandpass filters. Both of the design procedure and the simulation method neglect secondary coupling effects such as next-nearest-neighbor coupling
and package coupling. In the design, a lumped-element prototype is first transformed into a
series of transmission lines and impedance inverters. Then this series of transmission lines
and impedance inverters is implemented by a series of quarter-wavelength couplers. The
geometrical dimensions of the couplers are optimized using the equivalent surface impedance
technique. An equation is presented for compensating unequal even- and odd-mode propagation constants and open-end effective lengths in any planar filter design. In the simulation,
the filter structure is broken into elements of quarter-wavelength couplers. The overall filter response is obtained by the response of cascaded two-port networks which represent the
couplers. A sensitivity analysis is performed on a 4-pole 1% microstrip filter centered at
10 GHz and a 4-pole 0.05% shielded microstrip filter centered at 2 GHz. The 4-pole 1%
filter design is also implemented using silver and niobium. Measurement results indicate
significant secondary coupling effects.
In searching for a structure which has a better shielding property than a microstrip, the
characteristics of shielded microstrip lines are studied using the equivalent surface impedance
method. The results are presented in Chapter 5. The good shielding property as predicted
by an quasi-static analysis for the shielded microstrip structures is greatly deteriorated by a
parallel-plate mode supported by the two ground plates. The existence of the parallel-plate
mode is shown in the calculated electric field distributions. The effects due to the parallelplate mode are predicted by the full-wave analysis and confirmed by measurements on several
quarter-wavelength couplers. Preliminary results show that the parallel-plate mode can be
suppressed by making good contacts between the ground plates to ensure an equal potential.
Closed-form expressions for the open-end equivalent length in single and coupled shielded
microstrip lines are derived using a phenomenological approach.
1.2. DESCRIPTION OF THE THESIS
25
Chapter 6 summarizes the results presented in this thesis and recommendations of future
work are also given.
26
Chapter 2
Spectral-Domain
Volume-Integral-Equation Method
2.1
Introduction
Superconducting transmission lines offer many advantages over their normal-metal counterparts. Dispersion is virtually nonexistent for frequencies as high as several tens of gigahertz, circuit packing density can be greatly increased without incurring the penalties of high
loss associated with very dense normal conducting lines, and many analog signal processing
devices can be devised based on very long transmission lines or high-Q planar resonators [7],
[19]-[20]. The prospect for developing superconducting microwave devices that can be used
in actual system applications improved dramatically with the discovery of high-T, superconductors. Because superconducting devices will move from the research laboratory into
complex microwave systems in the very near future, rigorous methods need to be developed
to accurately predict the performance of superconducting microwave device structures over
wide frequency ranges. For instance, in the design of long delay lines, phase velocities of the
propagating signals need to be precisely determined to about 1%. This requires techniques
which can accurately characterize transmission lines with various geometries, dimensions,
and electrical parameters.
Kwon et al. [21] used the two-fluid model and a Monte-Carlo method to calculate the
27
28 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
propagation characteristics of a superconducting interconnect. A quasi-TEM approximation
was employed by Cangellaris [22] to calculate the propagation properties of two coplanar
conducting strips above an infinite ground plate. Pond et al. [23] analyzed a microstrip
structure using a resistive boundary condition for the case where the resistivity is assumed
to be a complex quantity and the superconducting strip is assumed to be thin compared to
the magnetic penetration depth. More recently, Nghiem et al. [24] used the same method
of modeling the conducting strip as a surface current with an equivalent surface impedance
approximated for the two cases when the strip is either much thinner or much thicker than
the penetration depth. In Refs. [25] and [26], a simple and practical method referred to
as the phenomenological loss equivalence method (PEM) is used for characterizing planar
quasi-TEM transmission lines with a thin and narrow normal conductor on the order of the
skin depth or with a superconductor on the order of the penetration depth. In Ref. [27],
Van Deventer et al. used an integral equation approach to calculate the propagation characteristics of high-To thin film superconducting lines at high frequencies. The losses in these
lines are evaluated by replacing the superconducting strips by frequency dependent surface
impedance boundaries and assuming that the superconducting strips have a large width-tothickness ratio. Recently, Sheen et al. [28] applied a quasi-TEM method for the calculation
of the current distribution, resistance, and inductance matrices for a system of coupled superconducting strip transmission lines having a rectangular cross section. More recently, Kessler
et al. [29] investigated a coplanar waveguide structure using an elaborate mode matching
technique. All of the previously proposed methods except [29] are quasi-TEM and/or restricted by the dimensions of the structure, and none of the previous work accounted for the
anisotropy of the superconducting film.
In this chapter, a volume integral equation using a spectral domain dyadic Green's function for layered media [30], [31] is formulated and used to analyze different configurations
of superconducting transmission lines. This method can be easily generalized to analyze
2.2. THE TWO-FL UID MODEL
29
arbitrary number of coupled strips in different layers without imposing any restriction on
the dimensions and electrical parameters of the structure.
Using this formulation, the ef-
fects of the anisotropy of the superconducting films on planar transmission lines are studied.
This spectral-domain volume-integral-equation (SDVIE) method is used to study various
microstrip (MS) lines, coplanar waveguide (CPW) structures, and striplines (SL). The complex propagation constant k
and quality factor Q for these structures are calculated and
compared to measurements. The current distributions are also calculated.
2.2
The Two-Fluid Model
In 1957, J. Bardeen, L. N. Cooper, and J. R. Schrieffer presented a theory which successfully explained superconductivity in conventional superconductors [17]. This theory is
known as the BCS theory. Despite its success in low-temperature superconductors, the BCS
theory fails to explain high-temperature
superconductivity.
Searching for the theory for
high-temperature superconductivity is still a current research area. Nevertheless, the BCS
theory does provide a reasonable description of high temperature materials up to certain
limits. Because of its complexity, the BCS theory is rarely applied directly for engineering
purposes.
The classical two-fluid model and London equations [18]provide a good phenomenological
description of superconductors, and have been widely used in the engineering community.
As shown in Fig. 2.1, the two-fluid model assumes there are two current carriers, namely
electrons and superelectrons, in a superconductor.
At a given temperature, these carriers
penetrate, but do not interact, with each other. The total current is therefore the sum of
both normal current and supercurrent,
J=J +
(2.1)
30 CHAPTER 2. SPECTRAL-DOMAIN VOL UME-INTEGRAL-EQUATION METHOD
Jn
Js
Figure 2.1: The two-fluid model.
The normal current Jn obeys Ohm's law
J, = al(T)T
(2.2)
(T-
(2.3)
where al(T) is given by
(T) =
= an
a (T)
a,
and a, is the normal-state conductivity at critical temperature Tc. The supercurrent J,
obeys the London equations
-B
a
-(A(T)7,)
(2.4)
= V x (A(T)J,)
(2.5)
=
where
A(T) = poA2(T)
(2.6)
and A(T) is the magnetic penetration depth of the superconductor. In the frequency domain,
the two components of the current may be combined and related to the electric field by
2.3. GENERALIZED FORMULATION
31
defining a complex conductivity a,,,
J = SCE= (al + ia2) E
(2.7)
where
1-(
2
T)4
Wo
(2.8)
o2
and o is the magnetic penetration depth of the superconductor at 0 K.
][n general, the superconducting film can exhibit anisotropic properties as in high-Tc
materials [18]. In this case, the complex conductivity is a tensor given by
sc(T) = -1 (T) + i 2(T)
(2.9)
where
1 =-1
F2(T)= o.-A
(T)
(2.10)
and A is a diagonal tensor assuming that the boundaries of the dielectric slab are aligned
with the principal axes of the superconducting film. That is,
Aa(T)
A(T) =
0
0
0
Ab(T)
0
O
O
(2.11)
A,(T)
where A (T) = poA (T) and j = a, b, c.
In the case of YBa2 Cu 3 0 7_x (YBCO), the penetration depths, along the a- and b-axes,
narme:lyAa and Ab, are approximately equal. However,
Ac =
a,b
(2.12)
where a can range from 3 to 5 [32], [33]. The normal-state conductivity along the a-b plane
is almost an order of magnitude higher than the conductivity along the c-axis.
32 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
Z
y
Figure 2.2: Electric field generated by current sources enclosed in a volume V'.
2.3
Generalized Formulation
Suppose that a volume V' enclosed some current sources J exits in an unbounded free
space as shown in Fig. 2.2. The electric field E at r is given by
E=i o
,J()
vJJ
A
(2.13)
where
r,
[I + k2VVJ4r - 'l
is the dyadic Green function for the unbounded free space with permittivity co and permeability o0.Usually, current sources are carried by conductors or dielectric materials. Suppose
1
that the volume V' encloses a dielectric material carrying current J with permittivity c' + i e"
and permeability
o. Equation (2.13) is now no longer correct because the medium is no
longer a homogeneous free space but an unbounded free space with a piece of dielectric. To
find the electric field using Eq. (2.13), the Green's function has to be modified to account
for the dielectric. However, it is very inconvenient because a new Green's function has to be
derived for every differently shaped dielectric. In some cases where the shape of the dielectric
2.3. GENERALIZED FORMULATION
33
is irregular, it would be very difficult to derive the appropriate Green's function.
To overcome this difficulty, we could alternately modify the current sources rather than
the Green's function. In the dielectric, the Ampere's law can be written as
V x H=
- iwe'T = we" - iwe'E
(2.14)
By purposely adding and subtracting the displacement current in free space, Eq. (2.14)
becomes
V x H = [we"- iw(e'- Eo)]TE- iwEoE
(2.15)
Notice that the dielectric material has been mathematically removed and replaced by free
space with an equivalent current
Jeq = [WE" - iw(e' -
0)] E
(2.16)
With this concept in mind, the electric field can be calculated using the Green's function
for free space and with the current J replaced by J,q. If the dielectric material is a good
conductor then Jq
J for as long as the conduction current is much greater than the
displacement current.
In general, both Jeq and E are unknown.
Since Jq
=
,eqE, an
integral equation can be written for T E V', specifically,
JlJJ
di G(r
E = iwyO
) [ffeq()E(f)]
(2.17)
The method of moments can be used to solve the integral equation.
This procedure can be generalized to formulate a system of integral equations for a layered
medium with multiple current sources. In Fig. 2.3, current sources of arbitrary geometry are
shown embedded in a layered medium. The electric field in layer m is given by
Em,(r)=
ip
7E
'| dr(,
)
(
(2.18)
where ,n is the permeability of layer n, Vn is the volume in layer n which encloses the
current sources Jn, and Gmn is the Green's function with observation point in layer m and
34 CHAPTER 2. SPECTRAL-DOMAIN VOL UME-INTEGRAL-EQUATION METHOD
ZL
(n)
vl
' iJ
(1)
I~~~~~~~
I
-I0
9^
i
II,
I A,(
v · !v
1.L,
A'1,
(2)
·
I
F-2
L2
83
93
I
(3)
(N-2)
,
-
E-,
F-
(N-1)
EN- 1
N-2
, RtN-1
-Ilffiealvj
(N)
EN
N
Figure 2.3: Geometric Configuration of a layered medium with embedded current sources.
2.4. VOLUME-INTEGRAL-EQUATION FOR MICROSTRIP LINES
(0)
v////////
--- - --
/7
/1Eo ,
35
gob X
-- --
01
El1 ,
(1)
6i
1
(2)
Figure 2.4: A superconducting strip is embedded in layer (0) of a layered dielectric medium.
source point at layer n. The dyadic Green's function for layered media can been found in
the literature for isotropic media [30], for anisotropic media [34], and for chiral media [35].
Since J,n = anEn for T E V,, N simultaneous integral equations for the electric field can be
obtained. The method of moments can as well be used to solve the system of equations.
In the following sections, this formulation of integral equations is applied to analyze
superconducting planar transmission lines.
2.4
Volume-Integral-Equation for Microstrip Lines
In Fig. 2.4, a superconducting strip is shown embedded in layer (0) of a stratified dielectric medium. Each dielectric layer is assumed to be homogeneous and isotropic. The
superconducting strip has an anisotropic complex conductivity as given in Eq. (2.9). The
structure is uniform along the propagation direction y. The total electric field E(T) in the
superconducting strip can be expressed in terms of a dyadic Green's function of the layered
medium as
E() = iwaoJv
where Jeq =
[sc + i(Eo - )] E
(Goo(,) 7eq(f)
(2.19)
ac E and V' is the volume occupied by the supercon-
36 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
ducting strip.
The dyadic Green's function Co(T,V'), where the observation point is in layer (0) and
the source is also in the layer (0), can be expressed in the spectral domain as [30], [31]
T
zz
k(-
d
-n) dko
)'
ik(00
-
go o(,
(2.20)
z,)
where
k = k + ky;
F = x + y;
= x' + y'
(2.21)
, z, z') is the F ourier transform of
The first term in Eq. (2.20) is the source dyadic and o(k=
the principal value part of Goo(, r ') (see Appendix A).
We assume that the electric field E(T) existing in the strips has x:, y, and z components and propagates in the y-direction with propagation constant kyo. Thus, E( ) can be
expressed as
~E(r) = eikyoE(X, ) = eikyoY[~Ex(x,z) + yEy(x,z) + iE,(x, z)]
(2.22)
Substituting Eqs. (2.20) and (2.22) into Eq. (2.19), we can get
E(x, ) +
iwIpo
k-- sc]z. E(x, z) =
ko2
- o J/
dk ek
dx' dz' e-ik,'zo(k
.(x',
kyo, ziZ)
z')
(2.23)
where S' is the cross-sectional area of the superconducting strips, and [sc]~z is the zz th
element of asc, and (x, z) E S'. This vector integral equation is then solved using a Galerkin's
method to find the complex propagation constant of the superconducting structures.
In the formulation of integral equations for microstrip lines or striplines, the ground plates
of the structure can be either treated as dielectric layers with a dielectric constant %e= ia,c/w
or as equivalent volume sources. Although the later approach rigorously accounts for the
finite width of the ground plates, the former approach is employed to simplify the problem.
For coplanar waveguide structure, both the signal strip and ground plates are treated as
equivalent volume sources since both are embedded in the same layer.
2.5. METHOD OF MOMENTS
37
tg
Figure 2.5: Superconducting microstrip line.
2.5
Method of Moments
In this section, without loss of generality, we shall apply the moment method for the
solution of Eq. (2.23) to the case of a microstrip line with a superconducting strip of finite
thickness above a single dielectric layer and a perfectly conducting ground plate. As shown
in Fig. 2.5, the width and the thickness of the strip are assumed to be 2w and t, respectively.
The dielectric layer is assumed to be isotropic with permittivity
r,,
permeability
o0, and
thickness d.
To apply the method of moments, a 2M x N nonuniform grid is imposed on the cross
section of the superconducting strip as shown in Fig. 2.6. To minimize computation time
and memory requirements, the division of the strip is determined so that the grid is smaller
near the edges where the current density is larger and is changing most rapidly as a function
of distance into the strip. The dimensions of the smallest grid element are first chosen to
be a fraction of the magnetic penetration depth A. This smallest grid element is AM wide
in the x direction and rl =
TN
high in the z direction. The size of each patch is then equal
to a multiplier times the patch size of the smaller neighbor. The multiplier is numerically
determined after the number of patches is decided. In this way, we have adequate resolution
near the edges while the computation time and memory requirements are reduced by using
larger grid elements in the center of the superconducting strip where the current density is
38 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
Fv
r
AM
1
1 A
A
AM
Figure 2.6: A nonuniform grid is imposed on the cross section of a superconducting strip.
nearly uniform.
An appropriate set of local basis functions as shown in Fig. 2.6 is defined based on the
grid to represent the electric field in the superconducting strip. Pulse basis functions are first
used to describe the three components of electric field inside the conductor; compared to
previously published data, however, the results are found to be very different. Similar findings
have been reported in [36]. The inaccuracy caused by using the pulse basis functions for the
transverse field components is due to the singularity in the charge distribution associated
with the pulse functions. Thus, roof-top basis functions are used to represent the x and z
components of the electric field, and the y component is represented by pulse basis functions.
The symmetry of the geometry is used to reduced the number of basis functions.
The components of the electric field are expressed in terms of the local basis functions
P,(x)
and Qn(z) as
M
E(z, z) =
E
m=l n
a,P'(xa)Qn(z);
a = x, y, z
(2.24)
2.5. METHOD OF MOMENTS
39
where amn are the expansion coefficients, and the sum over n is from 1 to N for the x and y
component and from 0 to N for the z component. The explicit forms of the basis functions
are
[(1+ Am); Xm-Am,X
P0(X)
=
.4(1
x -l
< Xm
) ; Xm<IXI <Xm +Am+
(2.25)
;otherwise
0
where the + sign is for x > 0, and the - sign is for x < O0
PY (X)
=
Pm(x)
xm -Am <
otherwise
Qn(z)
-
-
I < Xm
(2.26)
(z)
Q[(z)
{
1
-{
;
1 +
0
Qn(z)
(2.27)
otherwise
- zn < < zn
;
Z
Z
Zn
;
n < Z <
n +
Tn+l
(2.28)
otherwise
0
Letting P(ks,) be the Fourier transforms of Pm(x), where a = x, y, z and
m = 1,2, * , M, we have
Px (kx)=
- |
dx eikxpm(x) =
7r
i
1 [sin kr(xm - Am) - sin kXm]
+ ,,
Pm(k.) =
m
1
[sinkxm+l
-
sinkxm] }
(2.29)
Pm(kr)
_L
dx e-ikzxPm(x)
= -
2
rk-
-
Am
sin x
2
P,(-kx) =
cos k(xm)
(-kx)
A
2
)
(2.30)
40 CHAPTER 2. SPECTRAL-DOMAIN VOL UME-INTEGRAL-EQUATION METHOD
Substituting Eqs. (2.24), (2.29), and (2.30) into Eq. (2.23), we have
M
a
[(1
2
+
Qn(z)
o
dz'g"' (k kz
11dkx e ik=2x
,zc]pp
-o
=x'y'z
2
sC]() P.'W
+.~
zw]2
C'=-,yz m=1 n
o010
) P(k,
0
Qn(z')]
=0
(2.31)
where g(k,
ky, z, z') and b,z is the Kronecker
kyo, z, z') is the acpth element of o(k
delta function.
Applying Galerkin's method, we choose & Pp'(x) Q'(z) as the testing functions for the
Taking the inner products of the testing functions and Eq. (2.31), we
integral equation.
obtain 3MN + M equations. Integrating these equations over the x-z plane, we have
O=X,Y,Z
Z-a-1mn
m= n
[J dk ; P7 (-kx) P!(kx)
af'(k)+
smn] =0
(2.32)
where
1
W
--
Xpqmn
dx
w1o -W
1
3 wyo
1q,nq (p,m [2(nm + Am+l)] + p,m-1Am + sp,mAp}
1
y
d P () P ) Qx() Qx(z
ft
w
ldl(z Pp9() Pmyl
Xpqmn
7q(Z
n z)
2
=
z
Xpqmn
(2.34)
6p,mTqAp
-q,n
t
JW
=
+
1
1
3
wyoo
(2.33)
2,[L
0T 'O
+k2
Iz
· )-w
sc]zz
5
dx
p,mAp
dz Ppz(x) P(x) QQ(z)Q(z)
{6 q,n [2(rn
+ Tn+l)] + cbq,n-lTn + q-l1,nTq}
(2.35)
2.5. METHOD OF MOMENTS
41
and
J
J
t
t
][ |dz'
0
[,c],a
1
oa~o =
+
+I
dz Q (z) gooP(k,kyo,z > z') Q(z')
dz|
q
pLjd
dz' |'
n
dz Q(z) goo(k. kyO,
Z <z) Q(z'
go(k,
dz
q
900\,zt kyo,
n~yO,
·e > z') Q(z')
n\~C~/
""0Q'(z)
dz'
; q> n
) ; <
q
;q = n
The limit of the integral in Eq. (2.32) has been reduced to 0 < k < oo by utilizing the
symmetry properties of the dyadic Green's function, the basis functions, and the testing
functions. Equation (2.32) can be written in matrix form as
B
B
B
-Y
B
B
B
A
.R
-W·
=0
(2.37)
A
The elements of A are the expansion coefficients for the basis functions. Each entity B
in Eq. (2.37) is a submatrix. The explicit form of the elements is
B2 =
j
dkAPp(-kx) P(k)
21(k) + 6 ,Xqmn
(2.38)
Because of the complexity of the integrands in the above equations, the integrals are to be
carried out numerically. There are poles in the Green's function associated with various
modes. The correctness of the solutions depends on the choice of the integration path. In a
three dimensional problem, integrations in the k-plane, where k = k + k2, are necessary.
The Sommerfeld integration path, as shown in Fig. 2.7, gives the proper solutions which
satisfy the radiation condition. For lossless problems, the poles and the branch points are
on the real axis. If the conductors and/or the dielectric are lossy, the poles and the branch
points are above the real axis for positive Re{k,}, and are below the real axis for negative
Re{k,}. For two dimensional problems, integrations are only needed in the k,-plane. In the
ks-plane, the poles and the branch points are on the imaginary axis for lossless problems
42 CHAPTER
2. SPECTRAL-DOMAIN VOL UME-INTEGRAL-EQUATION METHOD
Branch Cut
Im{k s} A
Surface
Wave Pole
Re{ks}
-
k 1
.
a-
l l
*
-
-
-
ko
SIP
Surface
Wave Pole
Branch Cut
Figure 2.7: Sommerfeld integration path in k-plane.
Im{kx}
Surface
Wave Pole
Branch Cut
_!
01
Integration Path
_
I
e Surface
Wave Pole
Branch Cut
Figure 2.8: Sommerfeld integration path in ks-plane.
Re{kx}
a-
2.5. METHOD OF MOMENTS
43
and are near the imaginary axis for lossy problems. A proper integration path can be taken
along the real axis as shown in Fig. 2.8.
The matrix equation for the propagation constant k
can be solved by setting the de-
terminant of the coefficient matrix in Eq. (2.37) equal to zero, i.e.,
det {B(w, k,o)} = 0
(2.39)
The real part of kyo is the phase constant and the imaginary part is the attenuation constant.
Once the propagation constant is found, the the electric fields/currents are obtained by
searching for the eigenvector of the matrix B corresponding to the "zero" eigenvalue.
To calculate the characteristic impedance Zo of a transmission, one needs to determine
the propagating power. Because of the finite volume of the superconducting strips, it is not
a easy task to calculate the power guided by the structure. However, the dissipated power
can by easily calculated from the electric field inside the superconductor. The resistance per
unit length R of the transmission line is given by
iJS
F12 dA
R =ff
2
(2.40)
c7c8JjTdA
where S is the cross section of the superconductors.
By neglecting the dielectric loss, the
characteristic impedance Z, of the transmission line can be expressed in terms of the resis-
tance R and propagation constant kO, specifically
o
2
Im{ kyo}
Re{kyo
(2.41)
Since both R and Relkyo} are very sensitive to the distribution of the electric field, one
has to be careful in applying Eq. (2.41) to calculate Zo. In some cases where the electric
field changes very rapidly inside the conductor and/or where the loss of the transmission is
very small, the accuracy of the impedance given by Eq. (2.41) may be unacceptable.
transmission lines with normal metallization, Eq. (2.41) has found to be quite reliable.
For
44 CHAPTER 2. SPECTRAL-DOMAIN VOL UME-INTEGRAL-EQUATION METHOD
Table 2.1: Parameters for the microstrip lines (all dimensions are in tlm and the impedances
are in Q).
Nb
YBCO
Structure
MS86
MS50
MS15
2.6
Tc
T
9.25 K 4.2 K
77 K
90 K
2w
20
181
1800
t
0.4
0.4
0.4
A(O)
al (T.)
0.07 m 3000 S/#m
0.22 gm
6.56 S/tm
tg
0.4
0.4
0.4
Zo
-86
Z50
~15
Results and Discussions
Microstrip Line
The configuration of the microstrip (MS) structures studied are shown in Fig. 2.5, and
their dimensions are listed in Table 2.1. The parameters used for Nb and YBa 2 Cu307-_
(YBCO) listed in Table 2.1 are consistent with the values given in [9], [28]. The transmissionline geometries considered here have a 508-pm-thick LaA10 3 substrate with Er = 23.5 and
tan 6 = 2 x 10 - 6 at 4.2 K. In the cases studied, the loss in the transmission-line structures
is dominated by conductor loss.
In the calculations for MS lines, the discretizations used for the strips are 18 x 5, 22 x 5,
and 26 x 5 for MS86, MS50, and MS15, respectively, with the smallest patch size in both
x- and z-directions equal to one-sixth of the penetration depth of each superconductor.
Figure 2.9 shows a comparison between the calculated and measured effective dielectric
constants fEf = (Re{kyo/ko})2 for the three Nb MS lines. The measured
Ef
is extracted from
the resonant frequency f of the Nb half-wavelength MS resonator using the mode number
and the physical length of the resonator. Because of the capacitance of the open ends, the
effective length of the resonator is slightly longer than its physical length by 2AL, which is
calculated using the expression in [37].
2.6. RESULTS AND DISCUSSIONS
O
0
22 C
co
U,
4
9
A
45
MS86 - mea.
MS50 - mea.
MS15- mea.
'- 'MS86 - cal.
- MS50 - cal.
I
I
II
MS15- cal.
_
_
-
C
O
O
i
18-
._
'
16-
a)
_-~~~
El' --[3- --El---3
-> 14-
9r._--- -
_
a12
X
12 0
I
5
I
10
Frequency (GHz)
I
15
i
20
Figure 2.9: Effective dielectric constant of Nb microstrip transmission lines.
The calculated quality factors Q = Re{kyo}/2Im{kyo} of the three Nb MS and the two
YBCO MS resonators are shown in Fig. 2.10. The measured results on the Nb MS resonators
show reasonable agreement with calculation. The discrepancies can be easily accounted for
by changes in al and the penetration depth A of Nb from film to film. The MS line with
the wider strip achieves higher Q because of a lower current density and therefore lower loss.
The Q of the YBCO resonators, with the parameters given in Table 2.1, is nearly an order
of magnitude lower than that of the Nb resonators because of the larger penetration depth
of YBCO.
The current density in the signal strips of the MS lines is shown in Fig. 2.11, for a total
current I of 1 A. It is clear that the average current density in the widest line MS15 is nearly
two orders of magnitude lower than that in the narrowest line MS86. However, the peak
values are only different by slightly more than one order of magnitude. Since the Q of Nb
MS15 and MS86 also differ by only an order of magnitude, it is clear that the magnitude of
46 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
106
'
'
'
I
O
Nb MS86- mea.
O
Nb MS50- mea.
Nb MS86 - cal.
Nb MS50 - cal.
Nb MS15- cal.
YBCO MS86 - cal.
YBCO MS50 - cal.
-
-
I
...........
[]
'El%..
105
-
C-·
iLU
_-
-
l
_,
a io4
-~~·0
10 3
I
100
I
I
I
I
i
I
!
I
I
I I
100
10
1
I
Frequency (GHz)
Figure 2.10: Quality factor of superconducting microstrip resonators.
1013
.
I
I
I
I
I
I
Nb MS86
- NbMS86
NbMS15
1012
10
m
I=1A
---Nb MS50
E
I
*-
YBCO MS86_
-
NbMS50
-
YBCOMS86
-
NbMS15
11
_
I
ii......................
i
_
_
_
_
_
_
,
I
__---/
_ _
_
1010
I
_
109
-
I
I
0.2
0.3
.
_. .
I II
0.4
_, _
_,l
l
_d
I
0.5
I
I
II
0.6 0.7
X/W
I
0.8
d_,
_
I
0.9
1
Figure 2.11: Longitudinal current density at the bottom of the signal strips of microstrip
transmission lines.
47
2.6. RESULTS AND DISCUSSIONS
I
buffer
(0)
11
layer
\
YBCO
tb$
_
_
2W
_
_
_
_
_
4
_,__..________
mh.x
(2)
d
(3)
it,,,,,,,,,,e,,,,,,,,,/
dielectric
conducting
'\perfectly
X
ground plate
Figure 2.12: Superconducting microstrip transmission line with a thin dielectric buffer layer
and a perfectly conducting ground plate.
the current density near the edge of the strip determines the Q of the structure more than
the average current density. The current density at the edges of the YBCO microstrip lines
is lower than that in the Nb structures because of the larger penetration depth of YBCO.
Microstrip Line with Buffer Layer
A requirement in fabricating high-quality high-T0 superconducting films is a good lattice
match with the substrate and chemical compatibility with the substrate material. A thin
dielectric buffer layer as shown in Fig. 2.12 can be used to provide a good lattice match to
the superconducting film and to physically separate a chemically reactive substrate material
from the superconducting film. The procedure described in the Section 2.4 can be applied
directly to the problem of a superconducting microstrip with a thin dielectric buffer layer.
The effective dielectric constants and the attenuation constants of microstrip lines with
and without buffer layers are presented in Fig. 2.13. The buffer layer is assumed to have
a thickness tb = 100
A and
a relative dielectric constant Eb = 500 at 77 K. Since we are
48 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
c-'
C
6.45
0.08
0.07
c-
o
6.40
0.06 m
co
0.05
.ra,) 6.35
0.04 o
0.03 c
cD
.)
6.30
0.02
A
0.01
6.25
0.0
0
2.0
4.0
6.0
8.0
10.0
Frequency (GHz)
Figure 2.13: Effective dielectric constant and attenuation constant of a superconducting
microstrip line with and without a thin dielectric buffer layer. (2w = 160 m, t = 0.3 m,
d = 500 /m, e = 10, tb = 100 A, Eb = 500, A(OK) = 0.2 m, al(77 K) = 1.0 S/Ym,
T = 93 K, T = 77 K, and AX= 5 Aa,b-.)
2.6. RESULTS AND DISCUSSIONS
49
focusing only on the loss due to the superconducting strip, the buffer layer is assumed to
be lossless. In the calculation, the smallest grid size is chosen to be A/6, where M and
N are respectively 11 and 5. The results show that the effective dielectric constant of the
microstrip line is increased by less than 0.5% when the thin buffer layer is used, despite the
large relative dielectric constant of the buffer layer. The losses of the microstrip lines with
and without the buffer layer are approximately the same.
Anisotropy of High-To Superconducting Films
High-To superconductors, such as YBCO, are generally anisotropic. To study the effect
of the anisotropic properties of the superconductors, two superconducting microstrip lines
with different dimensions are investigated. Both lines are assumed to be anisotropic where
Ac
= 5
a,b.
The c-axis of the superconducting film is assumed to be in the z-direction.
The effective dielectric constants
eff
and the attenuation constants for the two microstrip
configurations are calculated for isotropic and anisotropic superconducting films. The results
are shown in Figs. 2.14 and 2.15. For these microstrip configurations, no noticeable difference
due to the anisotropy properties of the c-axis-oriented superconducting films is found. Since
the z-component of the current is very small compared to the other two components, the
effect of the anisotropy is suppressed. Although only the effects due to the anisotropy of a
c-axis-oriented superconducting film are studied, the method presented can treat cases where
the principal axes of the YBCO crystal are tilted with respect to the coordinate axis.
Stripline with Air Gap
Striplines are commonly used in microwave circuits as transmission lines and resonators.
Epitaxial growth of multilayer dielectric structures on top of YBCO is a challenging technical
problem.
An alternative way to fabricate YBCO stripline is to press a superstrate onto
a YBCO microstrip using a spring-loaded package [11]. However, using this method to
50 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
3.60
0.8
0.7
oC
3.56
0.6
0.5 '
3.52
.g
0 3.48
0.4
O
0.3
=
0.2
3.44
0.1
a)
w
E
3.40
0
0
5
10
15
20
Frequency (GHz)
Figure 2.14: Effective dielectric constant and attenuation constant of an isotropic and an
anisotropic superconducting microstrip line. (2w = 2 gm, t = 0.5 ,m, d = 1 lm, e, = 3.9,
A(0 K) = 0.14
C
0oO
m, crl(Tc) = 0.5 S/jm,
T, = 92.5 K, T = 77 K, and A, = 5
6.50
0.06
6.45
0.05
a,b.)
E
6.40
0.04
m
6.35
0.03
0
6.30
0.02
=:
.r
a)
0
a
w
0
a)
II
a)
6.25
0.01
6.20
0.00
0
2
4
6
Frequency (GHz)
8
<
10
Figure 2.15: Effective dielectric constant and attenuation constant of an isotropic and an
anisotropic superconducting microstrip line. (2w = 160 m, t = 0.3 m, d = 500 m, e = 10,
,(0 K) = 0.2 m, a1(77 K) = 1.0 S//m, T, = 93 K, T = 77 K, and A, = 5 ,b.)
2.6. RESULTS AND DISCUSSIONS
(-2)
51
.
~·////////////~//-//A//
(-1 )
d
Z
YBCO
i~~~
dielectric
"'"""""""-"m.`-'rm
(0)_
(1)
.
d
die
I"'t'
t
e2Wdielectric
__
x
fto-
+~~~~~~~~~~~~
(2)
perfectly conducting
ground plate
Figure 2.16: Stripline transmission line with an air gap. Both ground plates are assumed to
be perfectly conducting.
fabricate stripline introduces an air gap into the structure as shown in Fig. 2.16. The effect
of the presence of such air gaps on the propagation characteristics is investigated.
This
analysis takes into account the thickness of the superconducting film and can handle air
gaps of arbitrary dimensions.
The formulation of the integral equations for the stripline problem shown in Fig. 2.16 is
basically the same as the formulation given in Section 2.4 The electric field inside the strip
can be expressed as
E(f) = iWoLu
0 l,
diG
00 (,)
E E(')
(2.42)
Since the upper half space is bounded by the superstrate and the ground plate, the Fourier
transform of the principal value of oo(,
0
') includes the terms that account for the reflection
from the superstrate (see Appendix A). The procedure of obtaining the matrix equation using
basis functions and the Galerkin's method has been stated in the previous section.
To investigate the effects of an air gap in a stripline structure, the effective dielectric constant and the attenuation constant for two superconducting striplines with strip thickness
52 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
Table 2.2: Parameters for the Coplanar Waveguides (all dimensions are in
impedances are in Q).
Nb
Structure
CPW50-20
CPW50-50
CPW86-20
T:
T
9.25 K
4.2 K
0.07 pm
(0)
2w
20
50
20
t
0.4
0.4
0.4
s
35
87.5
355
m and the
al (T)
3000 S/pm
wg
600
600
600
Zo
.50
~50
,86
of 1.0 and 0.3 m, respectively, are calculated. The thickness of the air gap is assumed to
be the same as the strip thickness and the width of the strips is assumed to be 120 pm. The
results are plotted in Fig. 2.17 together with the results obtained for striplines with no air
gap. While the effective dielectric constant is decreased by a small percentage because of the
existence of an air gap, the effect of the air gap on the attenuation constant appears to be
negligible. Since the kinetic inductance of a superconducting stripline increases as the thickness of the center strip decreases [28], the effective dielectric constant of the stripline with
a thicker superconducting film is relatively small compared to the stripline with a thinner
superconducting film. However, both striplines with no air gap have an effective dielectric
constant that is larger than the relative dielectric constant of the substrate (r = 23.5),
illustrating the effect of kinetic inductance for these geometries [38], [39].
Coplanar Waveguide
The configuration of a coplanar waveguide is shown in Fig. 2.18 and the parameters of
the structures used in the calculation is listed in Table 2.2. In Fig. 2.19, the calculated
es values for three Nb CPW geometries are plotted with the measured values for two of
the structures. The measured er is extracted from the resonant frequency of the Nb halfwavelength CPW resonator without corrections for end effects. The deviation (< 4%) of the
2.6. RESULTS AND DISCUSSIONS
^f
-_
I
c
I
I
I
I
I
53
a
I
I
I
I
I
I
I
I
I
I
I
I
23.7 -
00 23.6 -
-
o 23.5 -
.0------
0---E
----------
0
a)
23.4 -
a)
23.3 -
t
O
23.2 -
line 1
- line 2
-- e--line 3
23.1 -
- --.
._-
o
0
u
o
u
·
Z;.U
. . I . . .
. .
. ..
0
I, ,
1 .
2
4
line 4
. .
.
_
I .
6
. .
8
._
10
Frequency (GHz)
0.25
-e--
0.20
m
o 0.15
--
_
line
1
line
2
--O-line34
-- ~--line
/
/
/
_
C
0
:3
0.10
C
c-
4-
0.05
0.00
0
1
2
3
4
5
6
7
8
Frequency (GHz)
Figure 2.17: Effective dielectric constant and attenuation constant of superconducting
striplines with (lines 1 and 2) and without (lines 3 and 4) an air gap. (2w = 120 jm,
d = 500 im, r, = 23.5, A(77 K) = 0.2 ,um, o-(Tc) = 0.1 S/ilm, Tc = 90 K, T = 77 K, and
A, = 5 A,b. Line 1: t = 1.0 jm; line 2: t = 0.3 pm; line 3: t = 1.0 #m; line 4: t = 0.3 jim.)
2. SPECTRAL-DOMAIN VOL UME-INTEGRAL-EQUATION METHOD
54 CHAPTER
Figure 2.18: Configuration of a coplanar waveguide.
14
C
4-
C
O
0
-
O
X
-
--
CPW50-20- mea.
CPW50-50 - mea.
CPW50-20- cal.
X
CPW50-50 - cal.
13 -
------ CPW86-20
- cal.
0
.,
0
~
a)
{D
0
I
I·
I
-
12 -
-
C)
~
~
x
x
X
X
CIa)
11
-
I
0
i
II
5
10
15
_
20
Frequency (GHz)
Figure 2.19: Effective dielectric constant of Nb coplanar waveguides.
2.6. RESULTS AND DISCUSSIONS
55
I10 3
CPW50-20- mea.
O
_V
\-rA
-
,rvv3U-3U- mea.
;PW50-20 - cal.
'PW50-50 - cal.
,PW86-20 - cal.
104 -
%
0
X
0%
0
103
I
'
l
l
I
-
L
I
II
I
1
10
100
Frequency (GHz)
Figure 2.20: Quality factor of Nb coplanar waveguide resonators.
measured
fes's
from the calculated result might be eliminated if corrections for end effects
were made on the measured eff's. The dependence of es on frequency is very similar for the
measured and calculated results. In the calculations, the number of patches in the signal
strips used is 18 x 5 for CPW50-20 and CPW86-20, and 20 x 5 for CPW50-50. For the
ground plates, 7 x 5 patches are used in all the calculations. The smallest patch size used is
one-sixth of the penetration depth. The Q of the three Nb CPW structures are illustrated
in Fig. 2.20. By increasing the width of the signal strip by 2.5 times, the Q for CPW50-50 is
higher than that for CPW50-20 by approximately a factor of 2. The Q of a CPW resonator
can also be increased by enlarging the strip-to-ground-plate separation, as demonstrated in
the calculated results for CPW50-20 and CPW86-20. The difference in Q results from the
difference in the current distribution in the superconducting strips. Although the current
densities in the signal strips of CPW50-20 and CPW86-20 are nearly identical, as shown in
Fig. 2.21, the wider strip-to-ground-plate separation in CPW86-20 yields lower peak current
56 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
1013
-.
.
I
I
- --
I
I
!
I
I
I
I
!
l
I
m
I
I
I
I
r
II-
I=IA
CPW50-20
CPW50-20
(CPW86-20
CPW50-50
10
- CPW50-50
CPW86-20
------.I
12
'10
F
sr'
1-
10
1010
0.2
I
I
I
I
I
I
I
I
0.3
0.4
0.5
0.6
I
I
0.7
I
I
0.8
I
0.9
I
1
X/W
Figure 2.21: Longitudinal current density at the bottom of the center strips of Nb coplanar
waveguides.
density in the ground plate, as shown in Fig. 2.22, which gives rise to a lower attenuation
constant.
2.7
Summary
A spectral-domain volume-integral-equation method has been developed using a dyadic
Green's function for layered media to analyze superconducting transmission lines. The thickness and the anisotropy of the superconducting film are rigorously considered. This method
allows accurate investigation of the various physical and geometrical effects on the characteristics of superconducting transmission lines. Calculated results have been presented for
standard microstrip lines, microstrip lines with a thin dielectric buffer layer, striplines with
an air gap, and coplanar waveguides.
The calculated ef and Q of the Nb MS lines agree with the measurements.
A thin
2.7. SUMMARY
57
I
1013
I
I
- --
12
----
CPW50-20
CPW50-50
101
I=1A
10 ' 5
10
4
0.001
0.01
0.1
1
X/W
Figure 2.22: Longitudinal current density at the bottom of the ground plates of coplanar
waveguides.
dielectric buffer layer with a high dielectric constant (Er = 500) only slightly modifies the
dielectric constant of a superconducting microstrip transmission line. This result indicates
that the effects of a thin buffer layer made of dielectric with low dielectric constant such as
MgO
(Er =
10) can be neglected. The effects of the anisotropy of the superconductor on the
characteristics of the transmission line are found to be negligible when the superconducting
film is assumed to be c-axis oriented.
The effective dielectric constant of a stripline is
decreased by a small percentage when an air gap is present. The effect of the air gap on the
attenuation constant appears to be negligible. The deviation (<4%) of the measured eff
for Nb CPW resonators from the calculated values would be reduced if the corrections for
end-effect capacitance were made on the measured eff.
The current distribution is more
uniform in the YBCO structures than in the Nb structures owing to the larger penetration
depth of YBCO.
Theoretically, the SDVIE method can be generalized to analyze arbitrary number of
58 CHAPTER 2. SPECTRAL-DOMAIN VOLUME-INTEGRAL-EQUATION METHOD
conductors embedded in layered media as shown in the previous section. Both finite volume and finite conductivity of the conductors are rigorously accounted for. Because this
method is very numerically intensive, it requires long computation times. To be incorporated in design tools, a more efficient and yet accurate method must be developed. The
SDVIE method is used as a basis and reference for the development of more efficient methods.
Chapter 3
Equivalent Surface Impedance
Approach
3.1
Introduction
In Chapter 2, a spectral-domain volume-integral-equation (SDVIE) method has been presented to analyze superconducting microstrip lines and striplines. Several other authors have
also proposed full-wave analyses of superconducting planar transmission lines on isotropic
substrates [23], [24], [29]. Kessler et al. [29] have investigated a coplanar waveguide structure
using a mode matching technique. The SDVIE and the mode matching methods rigorously
account for the finite volume of the superconductors, yielding accurate characterization for
the kinetic inductance and loss. However, both of these methods are computationally inefficient. Pond et al. [23] have used a resistive boundary condition to analyze microstrip
structures with very thin metallization. A similar method has been used by Nghiem et al.
[24] to study microstrip lines and striplines for the cases when the superconductor is either
much thinner or much thicker than the penetration depth. Both Pond and Nghiem have
used the surface impedance derived for a very thin, infinitely extended plate to replace a
superconducting strip. Nghiem has also considered the case of using the surface impedance
for a very thick, infinitely extended plate to replace a superconducting strip. Because of the
different field behaviors in a superconducting strip compared to an infinitely extended plate,
59
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
60
the loss and kinetic effect of the superconducting strip are not correctly determined. Pond
has pointed out that the singular behavior of the currents at the edges of an infinitely thin
strip might increase the kinetic inductance contribution resulting in an enlarged propagation
constant.
In this chapter, an efficient full-wave method is developed to calculate the effective dielectric constant, attenuation, and characteristic impedance of superconducting microstrip
lines and coupled lines on anisotropic substrates. The case of M-plane sapphire is considered
in which the optic axis of the substrate is in the plane of the substrate at an arbitrary angle
with respect to the propagation direction. To increase the efficiency of the method, the
superconducting strips are transformed into infinitely thin strips using an equivalent surface
impedance (ESI) concept. These equivalent surface impedances account for the loss and
kinetic inductance of the superconductors.
A 2-D dyadic Green's function is then used to
formulate an integral equation for the surface current in the strips. Galerkin's method with
entire-domain basis functions is used to solve for the complex propagation constants. The
characteristic impedances are then determined using the power-current definition. To verify
the validity of the ESI approach, numerical results for superconducting microstrip lines on
isotropic LaAlO0
3 are compared to previously published measured data and results calculated
by the SDVIE method. The effective dielectric constants calculated for perfectly conducting coupled lines on sapphire show good agreement with the results by the finite-difference
time-domain method. This efficient ESI technique is then used to study the effects of the
optic axis orientation and the strip width on the characteristics of single and coupled superconducting microstrip lines on M-plane sapphire. The effects of the line separation and
operating temperature on the coupled lines are also investigated.
By utilizing the ESI concept rather than the more rigorous SDVIE method, the computation time is reduced by nearly two orders of magnitude.
3.2. EQUIVALENT SURFACE IMPEDANCE
61
A
A
_
IIx
w
F~~
'
l
4
h
Figure 3.1: A superconducting microstrip line.
3.2
Equivalent Surface Impedance
A superconducting microstrip line with thickness t is shown in Fig. 3.1. To implement
an efficient method, an equivalent surface impedance Z, is used to transform the superconducting strip with a finite thickness into an infinitely thin strip. Since the fields and currents
in a superconducting strip behave differently from those in an infinitely extended superconducting plate, the surface impedance derived for an infinite half-plane superconductor or for
parallel plates [40] is not suitable for the transformation.
Instead, Z, should preserve the
power dissipated and stored in the superconductor. Suppose the thickness of the superconducting strip is on the order of the penetration depth A or less, then the power associated
with the magnetic field can be neglected. The power conservation equation can be simplified
to
J
z. IJs,
2 dx =
1
I
IJ0
)2
dx dz
(3.1)
where J,y is the y-component of the surface current J in the infinitely thin strip, a
is
the complex conductivity of the superconductor, and j(o) is the y-component of the volume
current in the superconducting strip. Both J(O)and J,y are normalized to give a unit current.
The simple two-fluid model [18] has been used to describe the complex conductivity ac with
U
=
+ i/(WioA 2 ).
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
62
Because of the edge conditions, the surface current J,y, is infinite at the edges of the strip
[41], [42]. The surface current for a single infinitely thin strip in homogeneous medium [43]
can be used as the first-order approximation for J,,, specifically
J.,y
cX,
(3.2)
J/1 (x/w)2
However, because the y-component of the electric field (Ey = ZJ,,y) is finite at the edges,
Z, should cancel the singularity of J,,, at the edges. Since Ey << Es and Ez for quasi-TEM
propagation, we can assume a uniform distribution of Ey across the strip without introducing
significant error. Based upon these assumptions, the equivalent surface impedance takes the
following form:
Zs(x) = Z 1 - (x/w)2
(3.3)
j t fWi IJ)dxdz
(3.4)
where
=
The volume current J() remains nearly frequency independent as long as the penetration
depth A is much less than the skin depth S. When the strip is thin and the substrate is thick,
0
) can be well approximated by the current in an isolated thin strip. However, the exact
J?
distribution of current cannot be put in a simple analytic form. An approximate solution is
derived in the next section.
3.3
Current Distribution in Superconducting ThinFilm Strips
London equations and the two-fluid model [18]give a phenomenological description of the
currents and fields in superconductors. Based upon this classical theory of superconductivity,
the problem of the steady-state current distribution in a superconducting thin film has been
discussed by several authors [44]-[47]. Cooper [44] and Marcus [45] suggested solving the
3.3. CURRENT DISTRIBUTION IN SUPERCONDUCTING THIN-FILM STRIPS
63
t
W
-W
Figure 3.2: Cross section of an isolated superconducting strip.
problem by formulating an integral equation for the current, but only numerical solutions can
be obtained. While Rhoderick and Wilson [46] presented an approximate solution for the case
in which the thickness of the film is very thin compared to the London penetration depth,
most applications involve films with a thickness either greater than or comparable to the
penetration depth. A more general closed-form solution has been proposed by Shadowitz
[47]. However, we will show that
Shadowitz's
solution
is incorrect
because of improper
assumptions made in the derivation.
We present here a closed-form expression for the current distribution in an isolated strip
made from a superconducting thin film. This formula is valid for strip thicknesses less than
a few penetration depths. Since the penetration depth is not a function of frequency, the
current distribution remains frequency independent as long as the penetration depth is much
smaller than the skin depth.
3.3.1 Approximate Solution for the Current Distribution
Our goal is to find an expression for the steady-state current flowing in a superconducting
strip of infinite length as shown in Fig. 3.2. The width and thickness of the strip are assumed
to be 2w and 2t respectively. The origin of the coordinate is taken to be at the center of
the cross section with -w
< x < w and -t < y < t. The currents and fields inside the
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
64
superconducting strip satisfy the spatial distribution governed by
V2
-
V 2 J=
VB= -B
-V2-
(3.5)
J
where A is the London penetration depth. A general solution for the axial current is given
by
J,(x, y)
2 + (atn/)2)
Jzxoy)
-anoax/cosh(yV(1/A)
=
an cos(anx/w)
n=o
cosh(tj/(1/)A)2 + (an/W)2)
+ ] bn
+
Z bn cos(any/t)
n=o
cosh(x\(1/A)2 + (a'n/t) 2 )
cosh(wV(1/A) 2 + (a/t)
t2)(3.6)
2)
where an and bn are constants determined by the boundary conditions, and an -= n7/2. To
determine the expansion coefficients an and bn, one needs to find the solutions of the Laplace
equation for the field outside of the superconducting strip, and then to match the boundary
conditions. Because of the rectangular cross section, this is considered to be a formidable
task. Even if one manages to solve the problem, the solution is very likely to be a slowly
converging series.
In Ref. [47], Shadowitz assumed a simple solution to Eq. (3.5) for the axial current
Jz(x, y) = J cosh(x/A) + J2 cosh(y/A)
(3.7)
The ratio of J1 /J2 was then determined by minimizing the kinetic energy in the supercon-
ducting strip,
Uk = I
rw rt
J-wJ-t
-
12
J
*
A dy d =
J-
/
-t 2
loA2'J2 dy d
(3.8)
As noted by Shadowitz, the resulting J 1 /J2 is practically zero. This solution suggests that the
current in a superconducting strip is nearly uniform across its width. In reality, however, the
current has a strong spatial dependence across the width [45], [46]. The failure of Eq. (3.7)
arises from the energy minimization process, in which only the kinetic energy is considered.
The kinetic energy is minimum when the current is uniform and therefore J1 /J 2 is practically
3.3. CURRENT DISTRIBUTION IN SUPERCONDUCTING THIN-FILM STRIPS
65
zero in Shadowitz's expression. To obtain the correct current distribution in a strip of finite
width, the total energy given by
u=L-
2+
[2
-
oA2J172] d
(3.9)
must be considered. Again, because of the rectangular cross section and the sharp corners
of the strip, the magnetic energy outside of the superconducting strip is very difficult to
calculate.
It is extremely difficult to obtain an analytical solution for the current in a superconducting strip. Therefore, we have studied the numerical solutions generated by the modified
Weeks' method [28]. We have found a closed-form expression for the current distribution,
that fits the numerical results well, specifically
cosh(y/A)) [Ccosh(x/l)
Z(y)
1 - cosh(x/12)/ cosh(w/12 )
1-(X/W)2
cosh(w/11)
-lcosh(t/A)
+J2 csh(x/A)
(3.10)
cosh(w/A)
where
J2
1.008
Jx
cosh(t/A)
C
=
(0.506
w/AlX
4/A
- 0.08301A/Al
/I-iA.7
11 = A2A/A,
12 =
0.774A2 /A1 + 0.5152A
and AX = A 2 /2t is the transverse penetration depth. This formula has only been tested for
A/Al > 1. Equation (3.10) breaks down when the strip thickness becomes large compared
to A and its limit is determined by the expression for J 2 /J.
When A/Al is greater than 6.9,
J 2 /J1 becomes a complex number. As suggested in Ref. [46], the axial current J at the center
of the strip has the functional form of [1 - (x/w) 2]- 1 / 2 . In Fig. 3.3, the current distribution
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
66
I
I
101
I
I
I
I
I
I
I
I
I
I
I
I
numerical
o
Eq. (3.10)
x' = 0.085
Os
10
_
_-
x
-,,
..-
v
x'
V_
3 81
"-
I II
.L
.-
31 4
-
10-1
N
4I-'
.
--
~
--
x'= 250
x'= (w - x)/X
10 -2
I
I
I
I
I I
I
I
~~~~~~~~~~~~~~~~~~~~~~~~~I
I
I
I
II
I
I
0.05 0.1 0.15
I
-0.2 -0.15 -0.1 -0.05
I
0 0.05
(m)
0.1 0.15
. _
0.2
Figure 3.3: Current distribution in an isolated superconducting strip. Comparison is made
between the closed-form expression Eq.( 3.10) and the results generated by the modified
Weeks' numerical method [28] (w = 50 pm, t = 0.2 ,im, A = 0.2 ym, ol = 2.0 S/pm).
3.3. CURRENT DISTRIBUTION IN SUPERCONDUCTING THIN-FILM STRIPS
10 4
-
x
numerical
from Eq. (3.10)
Shadowitz
10
10 2
I~
10
I
I
I~
5
_
---
tim
w
10
67
-
0
500
M
.I
I.
.I
I.
.I
I.
I.
0.5
1
1.5
2
2.5
3
3.5
t/k
4
Figure 3.4: AC resistance at 2 GHz of an isolated superconducting strip for various strip
widths and thickness-to-penetration depth ratios. Comparison is made between calculations
from the closed-form expression Eq. (3.10), from the expression proposed by Shadowitz [47],
and by the modified Weeks' numerical method (t = 0.5 #m, al = 2.0 S/pm).
calculated from this closed-form expression shows good agreement with the numerical result
generated by the modified Weeks' method.
The current distribution in a superconducting strip remains frequency independent as
long as the penetration depth is much smaller than the skin depth. Take YBa2 Cu3 07
as
an example. The typical value for the superconducting transition temperature Tc is 90 K,
the penetration depth at 0 K is 0.22 tm, and the real part of the complex conductivity at TC
is 6.56 S/pm. Assume the operating temperature to be 77 K. The ratio of the penetration
depth to the skin depth is approximately 1:6 at 20 GHz. Therefore, Eq. (3.10) is still valid for
microwave frequencies. Figure 3.4 shows the ac resistance at 2 GHz versus strip thickness
of an isolated superconducting strip for various strip widths. The results calculated by
Eq. (3.10) agree well with the numerical results by the modified Weeks' method. The error
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
68
I
A
't.V
A
f
__
X
I' -
'I'1
1
1
111
I
I I'"1
I
1 11
3.5E
LO
3.0-
0
2.52.0-
30
1.0-
0
I
-7
W
-0.6
E
-0.5
O
E
II
R. ... X . X
0
0
0
0.0-- .
I
111.11
I
I
.....
....
.
.
......
.
1
................
i
x
R
O
R
-0.3
CS
101
d/w
-0.2
- by Eq. (3.10)
.
I
.
t,
,ll
.
I
I
10 0
3:
0
x"o·TO
TO
I
-0.4
0
0.5-
101
m
w = 5 i0 ~rn
1.5n-
w
= 5
w=5im
X
V.
I
"
102
I
..
.
i
I
.
;
.
YL
^
J
-U.1
10 3
Figure 3.5: Total resistance R and center strip resistance R of superconducting striplines
calculated by the modified Weeks' method for various strip widths and ground plate separations at 2 GHz. The resistance estimated by the closed-form expression Eq. (3.10) falls
within the 5% margin bar of R (t = 0.2 gm, A = 0.2 #m, al = 3.5 S/ym).
of the closed-form expression runs from less than 1% to approximately 5% for different strip
widths and thickness-to-penetration
depth ratios. As shown in Fig. 3.4, the ac resistances
calculated using Shadowitz's equation, which postulates a more uniform current distribution
than the actual profile, are only 1/3 to 1/2 of the results given by Eq. (3.10) or those by the
modified Weeks' method.
3.3.2
Applications of Isolated Strip Approximation
Although an actual isolated strip does not exist in reality, it may be used to estimate
characteristic parameters of actual structures. One application of the closed-form expression
for the current density presented here is to approximate the resistance and inductance per
unit length at 2 GHz of superconducting striplines and microstrip lines. Figure 3.5 shows
the resistance per unit length of an air-filled (, = 1) superconducting stripline calculated
3.3. CURRENT DISTRIBUTION IN SUPERCONDUCTING THIN-FILM STRIPS
69
by the modified Weeks' method for various strip widths 2w and ground plate separations 2d.
The thickness of the strip and the ground plates is 0.4 #m, while the width of the ground
plates is taken to be 6350 pm. The typical parameters for YBa 2 Cu 3 0 7_ at 77 K are used
in the calculations. As the separation of the ground plates decreases, the resistance of the
center strip Res decreases because the current becomes more uniform. Equation (3.10) fails
to give a good approximation for R,, when d/w is less than 4 for w = 5 m and less than 2
for w = 50 m, respectively. However, the total resistance R of the stripline remains fairly
constant because of an increase of the resistance of the ground plates which compensates the
reduction of the center strip resistance. As shown in Fig. 3.5, the resistance calculated by
the closed-form expression for an isolated strip falls within the 5% margin bar of the stripline
resistance.
Since the current in a superconducting stripline or a microstrip line can be well approximated by the closed-form expression presented here, one could use this formula to derive
an equivalent surface impedance to transform the center strip with finite thickness into an
infinitely thin strip while preserving the loss and kinetic inductance. The resulting problem
involving an infinitely thin strip can be efficiently solved by the integral equation method.
This idea is carried out to develop an efficient and accurate full-wave technique for analyzing
superconducting planar transmission lines in the following sections.
In applying Eq. (3.10), one has to be careful about the limits of approximations.
For
example, Eq. (3.10) is not suitable in the limit in which, Ai used in the calculation of the
edge-barrier effect on critical current density, as discussed below, is much greater than A. The
critical current density J for a superconducting thin film, which is given by the measured
critical current divided by the cross-sectional area of the film, has been found to increase
systematically as the width of the film decreases [48]. An edge barrier is believed to be the
cause of the increase in J. By assuming the current density decays exponentially from the
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
70
edge with a characteristic length Al for the case w > Al > A, Tahara et al. [48] found that
(3.11)
J = Jc(aA±/w)+ J.
where J and J are the critical current densities corresponding to the edge barrier and to
bulk pinning, respectively, and a is an adjustable dimensionless parameter. As we discussed
earlier, the current density in a thin film has a functional form of [1 - (/w)2]
-1
/2 instead of
the exponential function assumed by Tahara. In the given limit in which w > A > A, the
expression given in Ref. [46] should be used instead of Eq. (3.10). By employing the proper
description of the current distribution given in Ref. [46], the dependence of Jc upon w is
found to be
+ e /27
J,= J, [2(1- e1/2)aA
2
W
Notice that the dependence of Jc upon w is dominated by the term
+ J!
(3.12)
aAX/w instead of aAl/w
suggested by Eq. (3.11). Despite the different dominant terms in Eqs. (3.11) and (3.12), both
equations suggest that the critical current density Jc approaches the bulk-pinning value when
the width of the strip is greater than a few Al's.
3.4
Superconducting Microstrip on Anisotropic Sub-
strate
The performance advantages of superconducting microwave devices have been demonstrated in the form of very high Q planar resonators, long delay lines, and multipole narrowband filters with low insertion loss [7], [8]. The proper choice of substrate materials is a
key to the performance of these devices. The uniform and low-loss dielectric properties of
sapphire make it attractive for high-temperature superconducting passive microwave device
applications. The performance of microwave devices on sapphire has been demonstrated in
the form of a low-loss 9-ns delay line [14] and a 4-pole microstrip filter operating at 10 GHz
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
71
2w
a - oo
Figure 3.6: Superconducting microstrip line with a perfectly conducting ground plate on an
anisotropic substrate.
[15]. However, the anisotropic dielectric property of sapphire requires special attention in
the design of these devices.
In Fig. 3.6, a superconducting strip is shown on an anisotropic substrate for which the
optic axis is in the x-y plane at an angle 0 with respect to the y-axis. The thickness of
the substrate is assumed to be h. The thickness and the width of the strip are t and
2w respectively.
The structure is uniform along the propagation direction.
conducting ground plate is assumed.
The permittivity tensor
E for
A perfectly
the substrate in the
chosen coordinate system can be written as
e
e.lco
(ell-
2
0
+
) sin
0
where eL and
e ll sin2 0
cos 0
(ell-e.L) sin cos0
sin 2 0 + Ellcos2 0
0
0O
0
(3.13)
fJ
l are the permittivities perpendicular and parallel to the optic axis, respec-
tively.
3.4.1
Integral Equation Formulation and Method of Moments
Figure 3.6 illustrates the configuration of a superconducting microstrip line. The superconducting strip with finite thickness is transformed into an infinitely thin strip by using
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
72
the ESI concept, as described in Section 3.2. A 2-D dyadic Green's function !9(k,, ky) for
layered, anisotropic media (see Appendix B) is used to formulate an integral equation for
the surface current in the strip,
Zs(x)Js(x) = 20
X
27r
oo
dk (k,ky) . Ys(k)eikx
(3.14)
where k, 0 is the complex propagation constant and Ys(kx) is the Fourier transform of J,(x).
To apply the method of moments, the surface current is expressed in terms of a set of
orthogonal basis functions,
7
,(x) = o 0=[ixanXn(x) + b,Y,n(x)] where
Xn(~) =
sin[nmr(~+ w)/2w]
(3.15)
Yn(~) = cos[nr(~ + w)/2w]
(3.16)
V1-
for 161 < w.
(=/w)2
=o[an(kx) +
= E n(k=a)
The Fourier transform of J,(x) is given by
ybnYn(kr)] where
X~(kx)
n)
2w
Yn(kx) = (i)
- e(3.17)
e-ikw(-1)n
k2
(3.17)
[Jo(kxw + 2 ) + (-l)nJo(kw
-
2 )]
(3.18)
and JO is the Bessel function of the zeroth order.
By applying Galerkin's method, the integral equation for the surface current is transformed into a set of linear equations for the coefficients of the basis functions,
n=0
mn
mn1
for m = 0, 1, -. , N. The elements of the matrix equation are given by
K
q
= f|
dkx
Pm(-k)gpq(kx,
If Pqn
00
5 pq i2 ZR,
kyo)Qn(kx) + bpq
W/10
l~ ZRP
(3.20)
(3.20)~~~~O
where p = x,y; q = x,y; Pm = Xm, Ym; Qn = X, y,,; 5pq is the pqth element of the g; 6pq
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
.4
1ZZ ZZZ.-
d/r
2w
z
Z
,
2s
0F4
Z.
-
Z.-.
73
2w
.2
go
d,.
Z
d
Z dZ
dd,
4
00
Figure 3.7: Superconducting coupled microstrip lines with a perfectly conducting ground
plate on an anisotropic substrate.
--J
(mr
)
is the Kronecker delta function; Z is defined in Eq. (3.4); and
r
i 2m
im-n"
RXRy=
2
m+n)r
mn
27I2
,m = n
,m ± n are even
(3.21)
m ± n are odd
Rmn =(
m + n are even
2
)
i-0(
, m ± n are odd
(3.22)
Here Jo and J1 are the Bessel functions of the zeroth and first order, respectively.
The complex propagation constant kyois determined so that the matrix equation has a
nontrivial solution. The surface current is then obtained by searching for the eigenvector of
the coefficient matrix corresponding to the "zero" eigenvalue. The electric field E(k,, z) and
magnetic field
(k., z) in the spectral domain are obtained in terms of the surface current.
The characteristic impedance Zc of the line is found by calculating the propagating power,
1 fo~
Z = 1 o
.0. (
x
"-)dz dk,
(3.23)
The method of solution for the coupled microstrip lines, shown in Fig. 3.7, is similar to
the one for the single microstrip line. The surface current of the coupled lines is expressed
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
74
in terms of a set of basis functions,
N
J , ,(x)
=
a()X,,(x - s - w) + a(2)X,(x + s + w)
(3.24)
b()Y,( - s - w) + b(v)Yn(x+ s + w)
(3.25)
n=O
N
J, .,(x) =
n=O
where a(') and b(l) are the expansion coefficients for the microstrip line 1 centered at x = s+w,
a(2) and b(2)are the expansion coefficients for the microstrip line 2 centered at x = -(s + w),
Xn and Yn are expressed in Eqs. (3.15) and (3.16), and 2s is the separation of the strips.
The complex propagation constants for the two fundamental modes, c and 7rmodes, are
determined using Galerkin's method. The characteristic impedances, Z 1 , Zc2, Zcrl,and Z2,
of the two lines for the two modes are calculated using the power-current definition,
Zm,,n
c
zmZ2
1
21
L° f00
21 ff
II
dz dk.2
* (Em x !)
12 - RI2m2
x *)dzdk.
lI2 l2- ill2/R
*(Em
(3.26)
(3.27)
where Emand WI, are the electric and magnetic fields for the m = c, r mode in the spectral
domain, I
IlI,/I2I2.
and I
are the longitudinal current in the lines for the m mode, and R =
For symmetric lines (R = -1), Eqs. (3.26) and (3.27) reduce to an equation for
the m = even (c mode) and m = odd (r mode) impedances.
3.4.2
Numerical Results and Discussion
In the calculations, a low-order (N) series of basis functions is found to be sufficient for
the frequency range studied in this work. For a single line on an isotropic substrate, N = 0
is sufficient. Converged solutions are obtained by using N = 2 for single and coupled lines
on anisotropic substrates.
Figure 3.8 shows good agreement between the calculated effective dielectric constants
e6f = (Re{kyo/ko})
2
using the ESI approach discussed here, and the measured data for
three Nb microstrip lines on the isotropic substrate LaAlO3 ( = 23.5 and tan 6 = 2 x 10 -6
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
a 21
,, 20
I
-
I
'
I
I '
I
_
]j
-
C
0
75
19
000 18
C1
4)
17
0
16
a
0
C.)
0)
_
m
Line 2
-
15
,
.
I
l
Line 3
_
14
0-
Uj
13
,
0
I
5
,
10
I
15
,
I
20
,
25
Frequency (GHz)
Figure 3.8: Measured and calculated effective dielectric constant of three Nb microstrip
lines on LaA10 3. Calculations were performed using the equivalent surface impedance (ESI)
approach ( = 0.0715 #m; al = 128 S/ym; h = 508 #m; t = 0.4 gm; line 1, 2w = 1800 jm;
line 2, 2w = 181 m; and line 3, 2w = 20 gim).
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
76
I~~~~~~~~~~~~~~~
50000-
I
II
II
I
I
I
'
II
40000-
a
3000020000
-
I-
-
10000
A
v
-
I
I
0
I lI
I
I
1
2
l I
I
I
l
3
t/x
I I
I
4
l
-
I
I
5
6
Figure 3.9: Calculated quality factor at 2 GHz for a Nb microstrip line on LaA1O3 as
a function of strip thickness normalized to the penetration depth. Comparison is made
between the ESI approach, the parallel-plate surface impedance (PSI) approach, and the
spectral-domain volume-integral-equation (SDVIE) method (A = 0.0715 /um,01 = 128 S/pm,
h = 508 #m, 2w = 20 um).
at 4.2 K). The measured e~ is extracted from the resonant frequency and mode number of
the Nb half-wavelength microstrip resonator. Because of the capacitance of the open ends,
the resonator is effectively slightly longer than its physical length. The effective length of
the resonator is calculated using the expression published in [37].
To demonstrate the necessity of the equivalent surface impedance, Fig. 3.9 shows the
calculated quality factor Q = Re {ky0 }/(2Im (ky0 }) of a Nb microstrip line on LaA103
as a function of film thickness at 2 GHz. The results by the ESI approach agree well
with the results calculated using the SDVIE method.
As we have expected, the results
calculated using the surface impedance derived for a superconducting parallel-plate to replace
the superconducting strip do not agree with the results by ESI or those by SDVIE, except
when the strip thickness is thin compared to the penetration depth. The expression for the
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
77
94
10000
93
7500
92
0 5000
91
N
90
2500
89
88
0
0
5
10
15
20
25
Frequency (GHz)
Figure 3.10: Calculated quality factor and characteristic impedance of a YBCO microstrip
line on LaA1O3 as a function of frequency. Comparison is made between the ESI and the
SDVIE approaches ( = 0.323 #m, al = 3.5 S/#m, h = 508 #m, t = 0.4 pm, and 2w = 20 #m).
superconducting parallel-plate surface impedance used in the calculation can be found in
[40],
;=
=w
coth (t ( /o ))
(3.28)
which properly accounts for the superconductor in a parallel-plate configuration. For a very
thin plate t < A, Eq. (3.28) approaches Z, = 1/(ta,¢). When t >
the limit for an infinite half-plane, Z =
-iwpoac.
, Eq. (3.28) approaches
To implement the parallel-plate
surface impedance (PSI) approach, Galerkin's method cannot be used because the integral
equation for the surface current expressed in Eq. (3.14) becomes nonintegratable. Thus, the
more general method of moments is applied. However, more basis functions (N = 8) are
required for the PSI method to obtain convergence than for the ESI method.
In Fig. 3.10, the quality factor Q and the characteristic impedance Z, of a YBCO microstrip line, calculated using the ESI approach, are shown together with the numerical
78
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
7.4
o 7.2
0a0
o
0
7.0
0
6.8
D
6.6
6.4
6. 6.2
2
0
20
40
60
80
100
0 (deg.)
Figure 3.11: Calculated effective dielectric constant at 2 and 20 GHz for YBCO microstrip
lines on M-plane sapphire as a function of the rotation angle 0 of the optic axis. The
normalized strip width is used as a parameter ( = 0.323 pm, al = 3.5 S/pm, h = 430 pm,
and t = 0.4 pm).
results obtained using the SDVIE method. The comparison again shows good agreement,
thereby validating the ESI technique. The ESI approach achieves a great saving in the computation time, requiring nearly two orders of magnitude less time than the SDVIE method.
This efficient full-wave method using the ESI approach is now applied to study YBCO
microstrip lines (t = 0.4 pm) on 430-pm-thick M-plane sapphire substrates (
= 9.34 and
ell = 11.6). Typical values of AO= 0.22 m, al(Tc) = 6.56 S/pm, and the superconducting
transition temperature Tc = 90 K for YBCO are used, where Aois the penetration depth at
T = 0 K, and al is the real part of oa,. The operating temperature is assumed to be 77 K.
The calculated effective dielectric constant ea, attenuation constant ac, and characteristic
impedance Z, at 2 and 20 GHz for various 2w/h ratios are illustrated as functions of 0 in
Figs. 3.11-3.13.
Because of the larger dielectric constant along the optic axis and the fact
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
0.0100
1.00
E
0.95 -
I
N
N
C
E
m
co
D 0.0095
I
79
0.0090
0.90. 0
cuj
0.0085
0.85
0
C
o
0.80 :
= 0.0080
0)
.*I
a.
0.0075
l
0
20
40
60
80
U. /D
100
6 (deg.)
Figure 3.12: Calculated attenuation constant at 2 and 20 GHz for YBCO microstrip lines on
M-plane sapphire as a function of the rotation angle 0 of the optic axis. The normalized strip
width is used as a parameter (A = 0.323 m, al = 3.5 S/pm, h = 430 pm, and t = 0.4 ym).
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
80
58
56
54
a:
N
cc
52
50
48
46
44
0
20
40
60
80
100
0 (deg.)
0.60
I
N
C(
' '
'
'
II '
'
'
I'
I
-U.u
_
' '
-5.5
0.552w/h = 0.8
CN
0.50-
-
0.45-
f=2 GHz
f = 20 GHz
-5.0
N
oc
<o
-4.5
2w/h = 1.0
N
E
' I '
0.40 -
N
2w/h = 1.2
-4.0 r
0.35
0
20
r
_n
V *
8
I
40
60
80
100
.
0 (deg.)
Figure 3.13: Calculated (a) real and (b) imaginary parts of the characteristic impedance at
2 and 20 GHz for YBCO microstrip lines on M-plane sapphire as a function of the rotation
angle 0 of the optic axis. The normalized strip width is used as a parameter (A = 0.323 ,m,
a = 3.5 S/,um, h = 430 ,m, and t = 0.4 #tm).
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
81
that the transverse electric field is much larger than the longitudinal field Ey, the inductance
L and capacitance C of the lines increase as 0 increases from 0° to 90° . The effective dielectric
constant, which is a function of the product of L and C, shows a greater dependence on
0 compared to the attenuation constant and the characteristic impedance, which depend
on the ratio of L and C. Since the capacitance increases more rapidly with 0 than the
inductance, the a increases with 0 while the Zc decreases. The rotation of the optic axis
about the z-axis changes the effective dielectric constant "seen" by the x-component of the
fringing electric field. As the signal strip becomes narrower, the fringing field becomes more
significant. Consequently, the effects of the rotation of the optic axis on the characteristics
of the lines are greater for smaller 2w/h. The es, a, and Zc become less dependent on 0 as
the frequency increases from 2 to 20 GHz. This occurs because the fields for microstrip lines
are more confined under the signal strips at higher frequencies.
For symmetric coupled lines, the fundamental modes are called even mode where the
longitudinal currents in the lines are in the same direction, and odd mode where the currents
are in the opposite direction. In the case of coupled symmetric lines on anisotropic substrates,
the field symmetry of the coupled lines no longer exists when the optic axis is not aligned
with one of the coordinate axes (i.e., the permittivity tensor has off-diagonal elements). The
fundamental modes for coupled lines on anisotropic substrate are generally labeled c and r
corresponding to the typical lack of perfect field symmetry. In principle, the impedances of
the two coupled microstrip lines are different because of the distortion in the field symmetry.
However, in the case of M-plane sapphire the optic axis is in the plane of the substrate. Since
the longitudinal component of the electric field is much smaller than the other components,
the field symmetry is only slightly distorted when the optic axis is rotated about the z-axis.
As a result, the difference in the impedances for the two lines (line 1 and line 2) on M-plane
sapphire is negligible. The distortion of field symmetry is much greater in the case where the
optic axis of the substrate is in the plane normal to the propagation direction. Although this
82
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
.
8.0-
I
I
I
I
I
I
Cr
g
7.5-
, 1%
·r-t7. 7.0 -
cmode
.
a)
6.5a)
,;
6.0-
I mode
a)
5.5-
0
I
I
I
I
I
I
5
10
15
20
25
30
35
Frequency (GHz)
Figure 3.14: Effective dielectric constant of a perfectly conducting coupled microstrip line on
M-plane sapphire. Comparison is made between the ESI approach and the finite-difference
time-domain (FDTD) method [49]( = 450, 2s/h = 1, 2w/h = 1, and h = 430 Mm).
case is not studied here, the ESI approach can analyze such a structure using the appropriate
Green's function.
Figure 3.14 shows the effective dielectric constant for perfectly conducting coupled microstrip lines on M-plane sapphire. The results calculated using the ESI method compare
well with those generated by the finite-difference time-domain method [49]. In applying the
ESI method, the equivalent surface impedance Z, is set to zero for the perfectly conducting
strips. The effect of the rotation of the optic axis on the characteristics of YBCO coupled
microstrip lines on M-plane sapphire are then studied using the ESI method. The parameters
for YBCO and sapphire used in the calculations are those used for the single microstrip lines.
The calculated
, ac, and Re{Z 0} at 2 GHz for the two fundamental modes are illustrated
in Figs. 3.15-3.17. The results show that the effects of the orientation of the optic axis on
the characteristics of coupled lines are similar to the effects on the single lines.
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
-7
I .0f-
C
0oCu
I
I
I
I
c mode --
7.05- .---
C
I
I
_
83
0
2w/h = 0.85
2w/h = 0.95
2w/h
----= 1.05
6.5 -
0 6.0G)
._
icmode
-
a,
,.,=
5.5
.
*
*.
.1
. .
. .I .
I
.,
.
40
20
0
.
,.
.
I.
. .
.-_
100
80
60
O (deg.)
Figure 3.15: Calculated c- and r-mode effective dielectric constants of YBCO coupled microstrip lines on M-plane sapphire as functions of the rotation angle 0 of the optic axis at
2 GHz (A = 0.323 pm, a1 = 3.5 S/pm, h = 430 m, 2s/h = 1, and t = 0.4 pm).
fn
u.
I
I
I
n44
i
B
I
i
r-
~
- I
1j
I
I
I
I
I
I
I
I
I
I
I
I
1
I
I
0.0110
0.01050.0100-
_---_
C
0 0.0095c-
c
4(D
0.0090 - _
0.0085 - C
____--------__
7: mode
fn
I
V%1
'
--2w/h = 0.95
------ 2w/h = 1.05
1irI
-.I
2
0
n...
?s
- I - ,-
0
ff%.7
I III
IL,
A 0 C .7
W/vll = nQ
.u
,r
I
c mode
0.0080V.
. ,r/
20
i
40
0
60
I
.
80
I
100
(deg.)
Figure 3.16: Calculated c- and 7r-mode attenuation constants of YBCO coupled microstrip
lines on M-plane sapphire as functions of the rotation angle 0 of the optic axis at 2 GHz
(A = 0.323 pm, rl = 3.5 S/pm, h = 430 plm, 2s/h = 1, and t = 0.4 pm).
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
84
65
I
I
II
I
I
I
I
I
I
I
I
I
I
60
55N
a:
cr:
c mode
0
50-
2w/h =
7rmode
-
.85
-2w/h = 0.95
-----2w/h = 1.05
45
,, . ,,
40
,I
I,,
2
0
20
40
60
80
100
0 (deg.)
Figure 3.17: Calculated c- and r-mode characteristic impedances (real part) of YBCO coupled microstrip lines on M-plane sapphire as functions of the rotation angle 0 of the optic
axis at 2 GHz (A = 0.323 m, a = 3.5 S//m, h = 430 m, 2s/h = 1, and t = 0.4 m).
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
85
7.0C
e--
0o
6.5-
e
md
0
.-
a)
6.0-
2w/h = 0.85
-----
4· 5.5-
'
0
I
II
II
I - 2w/h = 0.95
I
_
2w/h = 1.05
aU)
5.00
1
2
3
4
5
6
2s/h
Figure 3.18: Calculated c- and 'r-mode effective dielectric constants of YBCO coupled microstrip lines on M-plane sapphire as functions of the normalized line separation at 2 GHz
(A = 0.323 ,pm, a = 3.5 S/pm, h = 430 m, t = 0.4 pm, and 0 = 0).
Figures 3.18-3.20
show the variation of c- and 7r-mode e,
ac, and Re{Zc} with strip
width 2w/h and separation 2s/h for superconducting YBCO coupled microstrip lines on Mplane sapphire at 77 K. The optic axis of the sapphire is again assumed to be aligned with
the propagation direction. The characteristics of the 7r mode behave similarly to coplanar
strips. The ef and Re{Zc} for the lr mode increase with the separation, while the a decreases
because of the more uniform current distribution as the separation increases. For the c mode,
the calculated results approach the values for two single lines as the separation increases.
When the strips are close to each other, the characteristics for the c mode show interesting
coupling effects. The interaction of the fields causes the current at the inner edges to peak
less, which slightly reduces the attenuation as shown in Fig. 3.19. As the separation of the
lines becomes small, the pattern of the electric and magnetic fields approaches the pattern
for a single line, and the fringing fields between the strips are reduced. Consequently, the
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
86
0.015
0.014
E 0.013
2 0.012
0 0.011
= 0.010
0.009
0.009
W
0.008
0.007
0
1
2
3
4
5
6
2s/h
Figure 3.19: Calculated c- and r-mode attenuation constants of YBCO coupled microstrip lines on M-plane sapphire as functions of the normalized line separation at 2 GHz
(A = 0.323 pm, a1 = 3.5 S/pm, h = 430 pm, t = 0.4 pm, and 0 = 0°).
70
65
,c mode
60
55
-
N
50 _I-
a)
cc
45
__
40
__
,__
- mod
G
-
x mode
------ 2w/h = 1.05
2w/h = 0.85
2w/h = 0.95
_,'
35
30
0
1
2
3
4
5
6
2s/h
Figure 3.20: Calculated c- and r-mode characteristic impedances of YBCO coupled microstrip lines on M-plane sapphire as functions of the normalized line separation at 2 GHz
(A = 0.323 pm, a1 = 3.5 S/pm, h = 430 pm, t = 0.4 pm,and 0 = 0°).
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
57.05-
,,,
-43.70
,1,,I
I,1
87
')
o
.-43.65
2 57.00o
-
r
-43.60
_
a 56.95-
-43.55
N
' 56.90-
N
)
-43.50z
= U.4
.~ I~ ~~
, ,
56.850
~~
I . . . I , ,
0.2
0.4
T/T
0.6
Im
l
. . -43.45
~~~~~~~
. . .1 I
0.8
1
Figure 3.21: Calculated c- and r-mode characteristic impedances of YBCO coupled microstrip lines on M-plane sapphire as functions of reduced temperature TITC at 2 GHz ( 0O
= 0.22 m, al(Tc) = 6.56 S/#im, Tc = 90 K, 2s/h = 1, 2w/h = 1, h = 430 um, and 0 = 0).
,es and Re{Zc}, shown in Figs. 3.18 and 3.20, are larger for smaller 2s/h.
As shown in
Fig. 3.18, when the enhancement of the fringing fields at the outer edges becomes greater
than the reduction of the fringing fields at the inner edges, the 'e of the coupled lines starts
to decrease as the separation is further reduced.
The reliability of superconducting microwave devices depends on their sensitivity to temperature fluctuation.
Figure 3.21 shows the effects of temperature on the c- and ir-mode
impedances of YBCO coupled microstrip lines on M-plane sapphire.
In the calculations,
the simple two-fluid model is used to describe the temperature dependencies of the pene-
tration depth, A(T) = A0 +/1- (T/TC)4, and the real part of a,c, o1(T) = o1(Tc) (T/T,) 4.
By assuming the critical temperature Tc is 90 K, operating at 77 K gives T/TC = 0.86. The
impedances of the lines are very sensitive to temperature fluctuation as shown in Fig. 3.21.
As we have quantified in our calculations, decreasing the operating temperature from 77 to
CHAPTER 3. EQUIVALENT SURFACE IMPEDANCE APPROACH
88
60 K would dramatically reduce the sensitivity of the YBCO devices to temperature fluctuation. Furthermore, increasing the thickness of the strips would also reduce the temperature
dependence because the kinetic effect of superconductors becomes less for thicker strips.
3.4.3
Summary
A closed-form expression is derived for the steady-state current distribution in an isolated
strip made from a superconducting thin film. We have shown that this formula is valid for
strip thicknesses less than a few penetration depths and for frequencies up to tens of gigahertz. The ac resistance of an isolated superconducting strip calculated using this formula
agrees well with the numerical results generated by the modified Weeks' method. We have
demonstrated that this closed-form expression can be used to approximate the resistance
per unit length of superconducting striplines. Based upon this expression, equivalent surface
impedances are derived for superconducting strips. These equivalent surface impedances
transform the superconducting strips into infinitely thin strips while preserving the loss and
kinetic inductance of the superconductors.
A computationally efficient full-wave method utilizing the ESI concept is developed for
characterization of single and coupled superconducting microstrip lines on anisotropic substrates. The case of M-plane sapphire is considered in which the optic axis of the substrate is
in the plane of the substrate at an arbitrary angle with respect to the propagation direction.
The calculated results for microstrip lines on isotropic LaA103 show good agreement with
previously published measured data and with the more rigorous volume-integral-equation
method. The calculated c- and 7r-modeeffective dielectric constants for perfectly conducting
coupled lines on M-plane sapphire compare well with the results generated by the finitedifference time-domain method. The effects of the rotated optic axis of M-plane sapphire
on the characteristics of superconducting microstrip lines and coupled lines are presented
for various strip widths. The effects of the line separation and operating temperature on
3.4. SUPERCONDUCTING MICROSTRIP ON ANISOTROPIC SUBSTRATE
89
the coupled lines are also reported. Since coupled transmission lines are the fundamental
building blocks for filters and couplers, the ESI method formulated in this chapter are used
to develop a computer-aided-design tool for superconducting bandpass filters. Modeling and
design of these devices will be presented in the following chapters.
90
Chapter 4
Design and Simulation of
Superconducting Bandpass Filter
4.1
Introduction
The low-loss characteristics and high transition temperature of thin-film high-Ta superconductors (HTS's) enable compact planar microwave devices to be built with performance
that could previously only be achieved by using bulky waveguides or coaxial components.
One promising near-term application for HTS's is compact narrowband filter networks. Planar filters based upon various superconducting transmission-line resonators have been reported in the literature [6], [13], [51]-[54]. The reported performance, however, has not yet
attained the level that can be achieved using state-of-the-art cavity or dielectric filters. Issues
such as power-handling capability, packaging effects, and design accuracy must be resolved
in order to improve the performance of superconducting planar filters. In this chapter, we
attempt to address these issues.
A set of computer-aided design and simulation tools, numerical superconducting filter
design (NSFD) routines, is developed for superconducting microstrip bandpass filters. The
backbone of these routines is the equivalent surface impedance (ESI) method that has been
discussed in the previous chapter. Accurate filter design details, such as the proper model
for resonator open-end effects, are included and used to illustrate the fraction of a percent
91
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
92
accuracy required in the design of high-performance narrowband superconducting filters.
Based upon our analysis we also point out the advantages of edge-coupled microstrip and
edge-coupled shielded microstrip filter structures compared to coplanar and stripline struc-
tures.
4.2
4.2.1
Transmission-Line Resonators
Power-Handling Capability
Applications such as output-filter networks demand resonators with high power-handling
capability. The power-handling capability of a superconducting resonator is determined by
the magnitude of the peak magnetic field inside of the superconductor. Electromagnetic numerical analyses can be used to determine the best planar resonator structures for high power
applications. The absolute maximum power-handling capability of a passive microwave superconducting device is limited by the critical magnetic field of the superconductor. When
the peak magnetic field is greater than the critical field, flux penetrates the superconductor
causing a dramatic increase in loss. Even for magnetic fields less than the critical field,
nonlinear behavior is observed in superconductors.
In principle, a nonlinear model such as Ginzburg-Landau theory [18] is required to determine the power-handling capability of a superconducting transmission-line resonator. However, because the nonlinearity inherent in superconductors is small [55], linear models, such
as the spectral-domain volume-integral-equation (SDVIE) and the modified Weeks' methods
[28], can be used to estimate the power-handling capability of superconducting devices. Compared to the results in Ref. [55], the power-handling capability of a superconducting stripline
estimated with the linear models used in the analysis done here is only approximately 12%
smaller than that given by Ginzburg-Landau theory when the peak magnetic field is equal
to 70% of the thermodynamic critical field. This means that the current distribution in the
4.2. TRANSMISSION-LINE RESONATORS
93
Table 4.1: Quality factor and power-handling capability of superconducting resonating structures with resonant frequency f = 2 GHz for Nb on LaA103 at 4.2 K. Q and Qd are the
conductor and dielectric Q, respectively, RE is the ratio of the total stored energy to the
peak energy density, and R' is the normalized RE with respect to the value for SMS50.
MS15 is a 15-Q microstrip, SMS50 is a 50-Q symmetrical shielded microstrip, CPW86-20 is
a 86-Q coplanar waveguide with a 20-pm-wide center strip, and SL40 is a 40-4 stripline.
Structure
MS15
MS50
MS86
SMS50
CPW50-50
Zo
15
50
86
50
50
Qc
1.06
2.02
4.80
1.55
4.66
x
x
x
x
x
4.40 x
1/(1/Qc + 1/Qd)
106
105
104
105
10
4
4
3.40
1.44
4.38
1.18
4.26
x
x
x
x
x
4.04 x
105
105
104
105
104
RE (cm3 )
RE
10 - 5
10 - 6
10- 7
10 - 6
10 - 7
4.6
1.2
0.23
1.0
0.30
0.23
0.13
0.54
0.29
1.3
3.3
6.5
2.8
8.5
x
x
x
x
x
CPW86-20
CPW50-20
86
50
2.06 x 104
1.98 x 104
6.5 x 10 - 7
3.5 x 10- 7
SL40
SL50
40
50
1.20 x 105
7.14 x 104
9.68 x 104
6.25 x 104
1.5 x 10 - 6
8.0 x 10 - 7
10
10 4
superconducting films changes very little even at large magnetic fields (70% of the critical
field) and validates the analysis done here.
The SDVIE and the modified Weeks' methods are applied to analyze several commonly
used transmission-line resonators with 0.4-pm-thick Nb on 508-pm-thick LaA103 substrate
with er = 23.5 and tan 6 = 2 x 10- 6 at 4.2 K. For striplines (SL's), the separation of the ground
plates is 1016 m. The results are shown in Table 4.1. A ratio (RE) of total energy stored
to peak energy density is used to compare the power-handling capability of these resonating
structures. Since the power-handling capability of a superconducting structure is inversely
proportional to the square of the peak magnetic field and is proportional to the total energy
stored in the structure, a larger RE for a structure implies its power-handling capability is
higher. For the 50-Q structure, a microstrip (MS) resonator has more than four times the
power-handling capability of a SL structure.
Coplanar waveguide (CPW) structures can
have similar power-handling capability to MS structures, but the large strip-to-ground-plate
separation for such CPW structures may make them impractical for certain applications.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
94
A4'
0
£
2w
o 2s >
2w
h2
formed on a quarter-wavelength
coupled
MS
in Fig.
::::::::::::
....:-:::::x:: :-.......::
:::
: -: line,
-:::::* as
: : shown
e ::*
:::: 4.2, commonly seen in
::::::::::::::::::::::::::::::
......................
: ....* ¢*:
..............
::::::::::
:::::::::::::::::::::::::
:::::::::::::::::::::::::::-:-:....
::::-:::-:::::::::::::::..
~ ~ ~~ ~~~i:~~~~i:.. .*.:::. . . . ...........................................................................
......
:::::
:::.............*.*::.*.
....... ..... :-:::::::: :: ::: :-:*:.*:.::::.:::::::
:::::::::::::::::
lines are described by the same parameters without an upper ground plate.
Thus MS resonators are superior to the other transmission-line resonators considered here.
For a symmetrical shielded microstrip (SMS) structure as shown in Fig. 4.1 with h1 = h2,
the power-handling capability is very close to that of MS resonators.
4.2.2
Packaging
Packaging is a very important issue especially for high-Q circuits. Ideally, filters are built
using well-shielded transmission line elements so that secondary effects, such as package
interactions, can be neglected. Secondary effects can quickly ruin a high-performance filter
design; therefore, accuracy of significantly better than 1% is required.
MS resonators are the most widely used structure for building superconducting filters.
This is because MS structures are the most thoroughly analyzed structures in the literature,
and they are relatively easy to fabricate. On the other hand, MS structures have a relatively high radiation loss. By enclosing a resonator inside a box with modes at frequencies
different from the resonance frequency of the resonator, the radiation loss can be prevented.
However, coupling from the resonator to the enclosure can pose a problem for designing
high-performance filters. An analysis using a commercial package EM [56] has been performed on a quarter-wavelength coupled MS line, as shown in Fig. 4.2, commonly seen in
4.2. TRANSMISSION-LINE RESONATORS
95
/////,/://1/11/1://I//
jr
-
j
-
3520
A
_t
T
t
mS
h2
.
*-i
.. .i....
-:-
- ......
.v....
..............
.. >>..
..
> >> >> .> ..
.
.....
........
x ..........
-:.
; . .. ..
.
00I..- - ......
....
........................
>
......
>...
> > > > >> >
......
> ..........
|....
......
.
:..-::::::..
.
>
.. ::.....
...............
...............................
... . . . . . . .
........
..... >>
..........................
.
.........................................
..... ................ ............
!::::::S:'::-'
; :::'::w
:S:: -S>>:x >:x:
.; .. .......
Jf
/of
Of
Jf/7os 7/Zi.
........
Hrsxosw~~~~~~oozzzzzzzzz~~~~ai
I
/zuim////////////////////''
_A
_
,
:::
Top View
Cross Section
Figure 4.2: Configuration of a quarter-wavelength coupled microstrip line.
Table 4.2: The magnitude of S21 of a quarter-wavelength coupled microstrip line, shown in
Fig. 4.2, as a function of the cover height h2. The results are calculated using a commercial
simulator EM [56].
h 2 /hl
tS21 1
filter structures.
10
0.0180
20
0.0192
50
0.0205
100
0.0187
1000
0.0163
The results are listed in Table 4.2. By changing the cover-height of the
enclosure, the S21 of the coupled line varies by more than 20% even though the minimum
cover-height used was more than 10 times the substrate thickness. Coupled microstrip lines
are therefore very difficult to model individually without taking into account the package
enclosure.
As shown in Fig. 4.3, the S21 of the coupled line also varies as the side wall
of the enclosure becomes close to the lines. Because of the mirror image projected on the
side walls, the characteristics of the coupled line change rapidly as the distance from the
side wall becomes comparable the spacing of the coupled line. To avoid coupling to the
side walls, circuits should be separated from the side walls by a distance of several times of
the greatest characteristic length in the structures.
In the case of narrowband filters, the
greatest characteristic length is the separation of the resonators.
With an upper ground plate, SL resonators are shielded from the effects due to the
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
96
0.01925
-
-
I
. .
I .-
I-
-
I
I
- .
·-
I
·
.·
I
I
·
·
. ·
S-
0.01920
0.01915
0.01910
cm
CD
0.01905
I
I i I I
I Ii
i. i
1 1 1 1
I
I
I
I
1.6
1.8
2.0
I1
.
I
a
I
I
I
0.01900
0.01895
0.01890
0.01885
0.8
I
I
I
1.0
1.2
1.4
I
2.2
2.4
d / 2s
Figure 4.3: The magnitude of S21 of a quarter-wavelength coupled microstrip line with
h2 = 20hl as a function of the distance d from the side wall. The configuration of the
coupled line is shown in Fig. 4.2.
58
0
C:
56
C
au
n 54
E
U
4-'
0
a>
cr
0
L..
52
50
CZ
0
48
0
5
10
15
20
25
Frequency (GHz)
Figure 4.4: Calculated even- and odd-mode impedances of coupled shielded microstrip lines
on 508 #m-thick LaA103 for Nb at 4.2 K. (2w = 150 m, t = 0.4 tjm, A = 0.281 ,im, al =
2.53 S/lm,
h1 = h 2 .)
4.2. TRANSMISSION-LINE RESONATORS
97
enclosure as long as the side walls of the box are kept far enough from the filter resonators
as compared to the separation of the resonators. The assembly of a SL resonator requires a
substrate with an upper ground plate be placed on top of the center strip, usually creating a
small air gap. The effects of the air gap were found to be large enough to deteriorate the filter
response of 1%-bandwidth superconducting filters, and a difficult packaging technique was
needed to reduce the effects [51]-[52]. The existence of the air gap would limit the ultimate
performance of filters based upon SL resonators.
CPW structures require coating only one side of the substrate and thereby simplify the
fabrication and assembly. Unfortunately, for many applications air bridges must be used to
connect the ground plates to avoid the slotline mode. The additional air bridges complicate
the implementation of CPW structures, and the capacitance induced by the bridges can
create design difficulties. Radiation from an open CPW structure can create further design
difficulties.
The configuration of a SMS structure is similar to a SL structure, but its superstrate
is an air layer. Thus, a SMS does not have the air gap problem associated with a SL.
Consequently, the characteristics of a superconducting SMS line can be accurately analyzed
by full-wave methods such as the equivalent surface impedance (ESI) technique. Because of
its upper ground plate, a SMS has better shielding characteristics than a MS. Based on our
criteria for a well-shielded structure with good power-handling capability, it would appear
that SMS structures are the clear structure of choice for high-performance superconducting
filters. However, some of the shielding advantage of the SMS disappears at high frequencies
because of dispersion effects. In Fig. 4.4, the dispersion effects calculated using the ESI
method on the even- and odd-mode impedances of coupled SMS lines are shown. In order to
take the advantage of the shielding provided by the upper ground plate, the SMS structures
are more suitable for applications at lower frequencies. Because of the parallel-plate mode,
interesting characteristics have been observed in the shielded microstrip structures. A more
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
98
Figure 4.5: Configuration of an edge-coupled microstrip bandpass filter.
detailed discussion will be presented in the next chapter.
4.3
Filter Design Procedure
Our objective is to build superconducting filters with very high-performance by using
accurate modeling techniques. To eliminate complications, a simple edge-coupled configuration, shown in Fig. 4.5, is adapted.
The standard filter design procedures have been
discussed in great detail in several textbooks [57]. The design procedure for a superconducting filter does not significantly differ from the one used for a conventional filter. First,
a lumped-element filter prototype is designed for the desired response. Then the proper
transformation (e.g., Richard's transform and Kuroda's transform depending upon the type
of lumped-element filter) is applied to obtain a prototype which can be implemented with
transmission-line structures.
To design a N-pole bandpass filter, first the g factors, which are the normalized reactive elements of a lumped-element filter, are first obtained by the following equations. For
Butterworth filters, the g factors are given by
go
=
1
gk
=
2sin
(2k-I)r]
2N
k = 1,2,...,N
(4.1)
gN+l = 1
For Tchebyscheff filters with R dB pass-band ripple, the g factors may be computed as
follows:
4.3. FILTER DESIGNPROCEDURE
99
first compute
=
aY
ln[cot (17-37)]
= sinh2N)
= sin
ak
[(2k1)r]
=12
N
k = 1,2,...,N
bk = a2 +sin2 (),
then compute
go=
1
gl =
2a3
ct
4 ak-lak
gk
a
,a
k = 2,3,..., N
bk-lgk-l'
f 1,
for N is odd
coth2
9N+1
(4.2)
(42)
(,)
for N is even
Lumped-element bandpass filters consist of series and shunt resonators. Since it is very
difficult to implement both series and shunt resonators with distributed structures in a same
circuit, direct extension from the lumped-element prototype to the distributed one is not
possible. However, impedance inverters can be used to eliminate either the series or the
shunt resonators. Figure 4.6(a) shows an equivalent circuit of the edge-coupled bandpass
filter using impedance inverters Ki. The values for the impedance inverters are calculated
in terms of the g factors [57],
Z.
7V 2gog'
K1
T0w
rW
Zo
Ki
Z,
Z,,
KN+1
(4.3)
1
1
,
2 V/gjTg23
-
2 gNgN+l
7r
j = 2,3,..., N
(4.4)
(4.5)
(4.5)
where w is the fractional bandwidth of the filter. An ideal impedance inverter has a constant
image impedance K and an image phase shift y of -90° at all frequencies. The image
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
100
(a)
.- .
I-H
'-IiI
!(;I~~~~~~~
7
'-oe
I
I
M
u
I
,-I
M
- I
7
L'-oo
(b)
(c)
Figure 4.6: Equivalent circuit of the edge-coupled bandpass filter.
impedance K of a two-port network is defined as the input impedance of the infinite chain
of the two-port network [57].
The next step is to implement the filter prototype shown in Fig. 4.6(a) using transmission
lines. It can be shown that the filter element shown in Fig. 4.6(c) is approximately equivalent
to that of Fig. 4.6(b). The image impedance Z and the image phase shift y of the coupled
line shown in Fig. 4.6(b) are given by
ZI
[Z e
cos-y =
Zoe cot
Zoe csc
Zoo-2ZoeZo
he
¢e
+ Zoo cot 0,
+ Zoo csc 0
(csC )eCSC0, + cot 0 e cot o)] 2(46)
o
(4.7)
where Zo0 and Zoo are the even- and odd-mode impedances, and qS and q, are the phase
shift for the even and odd modes, respectively. For the ideal impedance inverter shown in
4.3. FILTER DESIGNPROCEDURE
101
Fig. 4.6(c), the image impedance and the image phase shift are given by
4Z
-. +
K
Z
cos
= 2
21
-I
/
(
1
Z
2 sin X
= (
)
1(1(
Zo
+
sin2
K+Z
0
+'
-'---
Z
(4.9)
K)cos§
By equalizing the image impedance of the quarter-wave parallel coupling transmission-line
section to that of the impedance inverter with quarter-wavelength lines on either side, we
find
Zoe = Zo [1
Z + (
zOO = Zo[I
Zo(
Zo]
(k+
, ko) +
2
(4.10)
+
i/2
Zoe
(4.11)
(4.12)
Zoo) (kko)
Zo + Zoo
2
where I is the coupling length of the transmission line, and ke and ko are the propagation constants for the even and odd modes, respectively. Hence, the even- and odd-mode impedances
and the coupling length of the edge-coupled transmission lines are obtained.
However, Eq. (4.12) does not take the open-end effects into account.
As the center
frequency of the filter moves toward higher frequencies, the effective length due to openend effects becomes significant. To include the microstrip open-end effects into the model,
Eq. (4.12) needs to be modified, specifically,
[(
l + Alo
2
(Zoeke
Zooko (
Zo,,k + Zooko
-
(4.13)
2
where Ale and Alo are the even- and odd-mode effective lengths, respectively.
tion (4.13) can be applied to any planar narrowband filter structure.
Equa-
To illustrate the
importance of the open-end effects, Fig. 4.7 shows the simulated response of a 4-pole 2%
microstrip bandpass filter on sapphire with a center frequency at 10 GHz designed using
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
102
0
-10
-20
2
-30
-40
-50
-60
9.4
9.6
9.8
10
10.2
Frequency (GHz)
10.4
10.6
Figure 4.7: Simulated response of a YBCO (71 K) 4-pole 2% microstrip bandpass filter on
sapphire with a center frequency at 10 GHz. The curves labeled (a) and (b) are the ISl I
and IS21 l of a design using the proper open-end model shown in Eq. (4.13). The curve (c) is
the ISu11of a design using the single-microstrip-line open-end model.
two different open-end models. The simulation for the design using Eq. (4.13) to properly
account for the open-end effects shows an ideal filter response. The other design adapted the
open-end model for a single microstrip line. Although the difference between the two designs
in the length for the first coupling section is only 0.4%, the center frequency is shifted by
1% and the return loss is increased by 10 dB. The filter simulation method will be discussed
in the next section.
By incorporating this design procedure, NSFD routines are developed based on the ESI
method and the multidimensional Newton's method. By using these NSFD routines, the
geometrical parameters of the coupled lines are optimized to give proper impedances and
coupling lengths. The definition of the geometrical parameters is shown in Fig. 4.8. This
particular design procedure based upon individual coupled-line elements is valid only if
4.3. FILTER DESIGNPROCEDURE
103
I
;sj
I
I1 e
-4
I
AL 1
L
52
I
AL
AL 2
2
AL2
AL2
-
AL2
2
AL,
I 1'
C_41
.a
_ i
AL 1
L1
d
4-
- -
I
s
1
w
I
Figure 4.8: Layout of an edge-coupled bandpass filter.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
104
Figure 4.9: Network representation of a bandpass filter.
the next-nearest-neighbor coupling and coupling to the enclosure are negligible. Therefore,
choosing a transmission-line configuration with good shielding properties such as SMS is key
to the success of building high-performance filters.
4.4
Filter Simulation Method
To simulate the response of a bandpass filter, one could solve the electromagnetic wave
problem for the whole filter structure, so that the effects due to the next-nearest-neighbor
coupling and coupling to the enclosure can be accounted for. However, it is very difficult to
obtain an accuracy result for such a problem and is a very time consuming process. For a
well shielded structure where the second-order coupling can be neglected, an edge-coupled
bandpass filter can be modeled by cascaded two-port networks as shown in Fig. 4.9. Each
of these two-port network can be characterized by a scattering matrix S. The elements of
the S matrix are the ratios of the input and output voltages at the ports, specifically
[
b2
S21
S2 2
a2]
(4.14)
where ai and bi, for i = 1,2, are the input and output voltages normalized to the square root
of the reference impedance Z, as shown in Fig. 4.10. For a reciprocal network, S12 = S21;
while S 11 = S22 for a symmetrical network. If the network is lossless, then the power is
4.4. FILTER SIMULATION METHOD
105
a2
ao
Zo
-_
N
1
.b
bi
2
Zo
b2
Figure 4.10: A schematic of a two-port network.
I.
Z0
I
v
z
Vi
VV
V2
I
2b
Z=0
%.
V
Zo
I
Z
Zoe, Zoo, ke, ko
Figure 4.11: Transmission model for a coupled section in a bandpass filter.
conserved, i.e.
2
IS.ll2 + IS2,
1 = IS22 12+ IS[212= 1
(4.15)
This scattering-matrix representation of a two-port network can be easily generalized to a
N-port network.
To obtain the S parameters for the two-port networks representing the coupled sections
in the bandpass filter, we use the transmission-line model to describe the coupling sections
as shown in Fig. 4.11. The input and output signals of the coupling section are written in
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
106
terms of the reflection coefficient R and transmission coefficient T,
Vi(z) = V(eik' + Re- 'k z)
I,(z) = V(e 'kz - Re-'ikz)/Zo
{
(4.16)
(4
= VTeik( - )
Io(z) = VTeik(-)/Zo(4.17)
V(z)
The coupling section is characterized by the even- and odd-mode impedances Zoe and Zoo,
and by the even- and odd-mode propagation constants ke and ko, respectively. The electrical
length of the coupling section is assumed to be . The signal in the coupled line is given by
+ v-e-ikz
+ V-e-ikez+ V+eikoz
(Z) = V+eikez
V-eiko
z 1ke[+
[v+ee _-z
I () =
k
°
V2 (z) = V+eikez+ Veikez - V+e z - Ve -(ikoz
- Vloeikoz]
1 [VO+eikz
Veikez]
[V+eikezI 2(Z) =Z
]
(418)
In Eqs. (4.16)-(4.18), there are totally six unknowns but we are only interested in R and T.
To solve for these unknowns, six boundary conditions are needed. At z = 0, we have
I = V(1- R)/Zo,
V 1= V(1 + R),
The boundary conditions at z =
I, = 0,
I2 = 0
(4.19)
are
V2 = VT,
12 = VT/Zo
(4.20)
By elimination, the reflection and transmission coefficients are obtained
R =
T =
a2 -
2
+4
_
)2 p2
2) 2 -3
(C- -4i/
2
(4.21)
(4.22)
where
a
= [Zoecot(ke)+ Zoocot(ko~)]/Zo
(4.23)
f
= [Zoecsc(ke) )- Zoocsc(ko)] /Zo
(4.24)
4.5. NUMERICAL SIMULATIONS AND DISCUSSION
The electrical length
107
of the coupling section is given by the physical coupling length plus
the equivalent length due to the open-end effects. Since the even- and odd-mode open-end
equivalent lengths are different for microstrip and shielded microstrip structures,
takes two
different values for the two modes.
The S parameters for the coupling section are related to the reflection and transmission
coefficients, specifically
S 11 = S2 2 = R
(4.25)
T
(4.26)
S12 = S 21 =
To characterize a cascaded network, it is more convenient to use the transfer scattering
matrix T rather the S matrix. It is because in the transfer scattering matrix presentation
the variables in port 1 are treated as the independent variables while the variables in port 2
are treated as the dependent ones.
a2
]
[
T2 1
T2 2
bl
(4.27)
The T parameters can be expressed in terms of the S parameters as the following:
Tll = (S 21 S 12 -S11S22) I/S
12
(4.28)
T12 =
S22/S12
(4.29)
T21 =
-S 1 /S
(4.30)
T22 =
1/S12
12
(4.31)
The T matrix for a cascaded network is simply obtained by multiplying the T matrices of
the individual networks.
4.5
4.5.1
Numerical Simulations and Discussion
A Four-Pole 1% Filter
The first high-performance high-temperature superconducting (HTS) bandpass filters was
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
108
0
-10
-20
D
-30
-40
-50
-60
9.7
9.8
9.9
10
10.1
Frequency (GHz)
10.2
10.3
Figure 4.12: Simulated response of a 4-pole 1% Nb microstrip filter on 508 m-thick LaA103
with a center frequency at 10 GHz (Design: Nb/LAO-4pl%bw).
reported in 1991 [58]. That particular filter was designed as a 4-pole Tchebyscheff bandpass
filter at 5 GHz with 1% bandwidth. The reports on the latest developments in HTS bandpass
filters [15], [51]-[54]do not show much improvement in performance after a few years of active
research by a number of research organizations. To have a better understanding of how to
improve the performance of filters, an analysis on the sensitivity is needed.
To design a multipole narrowband filter, accuracy in the design model is very important.
The NSFD routines are used to design a 4-pole 1% microstrip filter on 508 m-thick LaA103
(Er
=
23.5). The metallization is assumed to be 0.4[m-thick Nb films. The physical param-
eters for Nb used in the simulation are listed in Table 4.3, and the dimensions of the filter
are shown in Table 4.4. Figure 4.12 shows the simulated response of this filter. To demonstrate the kinetic inductance effects, Figure 4.13 shows the simulated S2 I of this 4-pole 1%
microstrip filter together with three other simulations by the NSFD routines. One of these
three simulations assumed the thickness of Nb metallization is reduced to 0.14 1am, one as-
109
4.5. NUMERICAL SIMULATIONS AND DISCUSSION
Table 4.3: Physical parameters for YBa 2 Cu3 0 7_- at 50 K and Nb at 4.2 K
A
al
YBCO
Nb
0.2 gm
9.62 S/pm
0.0716 Pm
26.2 S/pm
Table 4.4: Design for a 4-pole 1% Nb microstrip filter on LaA10 3 with a center frequency at
10 GHz. (see Fig. 4.8 for layout description.)
Design: Nb/LAO-4pl%bw
Metallization = Nb at 4.2 K
Film thickness = 0.4 pm
Substrate = LaA103 (,e = 23.5)
Substrate thickness = 508 pm
i
Li (#m)
wi (#m)
Si (m)
ALi (pm)
1
2
3
1706.1
168.5
539.5
116.8
1719.3
181.1
2616.5
109.1
1719.2
181.2
3049.5
109.1
110
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
sumed YBa2Cu 3 07_. (YBCO) metallization, and the other assumed perfect conductor. The
ESI method used in the NSFD routines gives more accurate values for the kinetic inductance
and loss of the superconductors compared to the methods based upon parallel-plate surface
impedance [24]. The physical parameters for YBCO used in the simulations are listed in
Table 4.3. By reducing the film thickness, the kinetic effects are enhanced. Thus, the simulated response for the filter with a thinner Nb metallization show a slight downward shift
in the center frequency. An even greater downward shift in the center frequency is observed
in the YBCO results because of the greater penetration depth for YBCO. An upward shift
is observed for the results assuming perfectly conducting metallization because of the absence of the kinetic effects. Although the difference is small between all the simulations with
different physical parameters for this particular filter design, the difference is greater for a
more aggressive filter design, and that is to be discussed in the following section.
To understand the sensitivity of this 4-pole 1% filter design, we have examined the effects
on the filter response by varying some physical dimensions. For 508 um-thick substrates,
variation in thickness by 50 #m is common. With the state-of-the-art machining technique,
the variation can be reduced to 2 #m. Figures 4.14 and 4.15 show the response of the filter
with substrate thickness reduced by 50
m and by 2 m, respectively. The reduction in
substrate thickness by 50 m increases the return loss by nearly 10 dB and shifts the center
frequency. The effects due to the 2 m reduction are negligible for a 1% filter.
Even with a very accurate filter design, the limited resolution of the standard photolithography technique can cause error in linewidths and linelengths of a filter design. Suppose the
dimensions of the resonators in the 4-pole Nb filter are uniformly reduced by 2 #m from
all four edges. As shown in Fig. 4.16, except for a small shift in the center frequency, the
coupling elements of the filter remains properly matched. However, if the dimensions of the
resonators are randomly reduced by 0 to 2 gm from all four edges, the performance of the
filter is greatly deteriorated as shown in Fig. 4.17.
111
4.5;. NUMERICAL SIMULATIONS AND DISCUSSION
0.00 -
I
-0.10 -
. ...
C- -0.20 -
cm
-0.30
I
-
-
;I
-0.50 -
Figure 4.13: Simulated
-----
I- I '
I-
-----
YBCO(t=0.4rm)
Perfect Conductor
Nb (t=0.1432g.m)
,1...... i| -|
j~~~~
.
9.9
-
-
-:~iI lNb (t=0.4gm)
. : l_
-0.40 -
I
9.95
.
.
10
10.05
Frequency (GHz)
10.1
response of a Nb (4.2 K) 4-pole 1% microstrip
bandpass
filter on
LaA103 with a center frequency at 10 GHz (Design: Nb/LAO-4pl%bw). Comparison is made
between results generated using the proper Nb parameters, using the YBCO parameters at
50 K, and using the perfect conductor assumption to demonstrate the kinetic effects on the
filter response. Te effects of the film-thickness are also shown.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
112
0
-10
-20
m
55
-30
-40
-50
-60
9.7
9.8
9.9
10
10.1
10.2
10.3
Frequency (GHz)
Figure 4.14: Simulated response of a 10 GHz 4-pole 1% Nb microstrip
Nb/LAO-4pl%bw) with the substrate thickness reduced by 10%.
filter (Design:
0
-10
-20
V
-30
O3
-40
-50
-60
9.7
9.8
9.9
10
10.1
Frequency (GHz)
10.2
10.3
Figure 4.15: Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter (Design:
Nb/LAO-4pl%bw) with the substrate thickness reduced by 2 tim.
4.5. NUMERICAL SIMULATIONS AND DISCUSSION
113
0
-10
0
D-
-30
-40
-50
-60
9.7
9.8
9.9
10
10.1
10.2
10.3
Frequency (GHz)
Figure 4.16: Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter (Design:
Nb/LAO-4p1%bw) with the dimensions of the resonators uniformly reduced by 2 am.
With an improved photolithography technique, the random variation in the dimensions
can be reduced to 0.25
m. Figure 4.18 shows the simulated response of the 4-pole Nb filter
with random variation in the dimensions of the resonators up to 0 to 0.25 jtm. Such a small
change in the dimensions virtually has no effect on the filter.
4.5.2
A Four-Pole 0.05% Filter
High-temperature superconductors (HTS's) provides the potential for building low cost
filters with very narrow bandwidth and low insertion loss. A possible high-volume commercial
application for HTS's is in cellular communication. Multiplexing units with very narrowband
filters can increase the number of users supported by the cellular system. Four or five-pole
filters with 0.05% bandwidth centered at 2 GHz are very desirable for those multiplexing
units. For such a narrowband filter, the coupling strength of resonators is very weak, which is
in the range of -50 to -60 dB. Consequently, the spacing between the resonators is very large.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
114
0
-10
-20
m
-30
cO
-40
-50
-60
9.7
9.8
9.9
10
10.1
Frequency (GHz)
10.2
10.3
Figure 4.17: Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter (Design:
Nb/LAO-4pl %bw) with the dimensions of the resonators randomly reduced by 0 to 2 m.
0
-10
-20
m
cnv
-
-30
I.-
-40
-50
-60
9.7
9.8
9.9
10
10.1
Frequency (GHz)
10.2
10.3
Figure 4.18: Simulated response of a 10 GHz 4-pole 1% Nb microstrip filter (Design:
Nb/LAO-4pl%bw) with the dimensions of the resonators randomly reduced by 0 to 0.25
Am.
4.5. NUMERICAL SIMULATIONS AND DISCUSSION
115
Table 4.5: Design for a 4-pole 0.05% YBCO shielded microstrip filter on MgO with a center
frequency at 2 GHz. (see Fig. 4.8 for layout description.)
Design: YBCO/MgO-4p0.05%bw
Metallization = YBCO at 50 K
Film thickness = 0.4 m
Substrate = MgO ( = 9.68)
Substrate thickness = 127 m
Cover height = 1016 gm
i
Li (m)
wi (m)
si (m)
AL (m)
1
2
3
14624.9
124.5
341.1
35.9
14624.6
124.8
1607.7
35.8
14624.6
124.8
1820.3
35.8
Because of the weak coupling strength between the resonators, any coupling from the filter
to the package would greatly deteriorate the performance. In order to reduce the secondary
coupling effects, a better shielding structure rather than the standard microstrip structure is
needed. Stripline structures have much better shielding properties. However, because of the
air gap, stripline structures are difficult to be characterized accurately. Shielded microstrips
with thin substrate are believed to be a better structure for building filters with 0.05%
bandwidth.
The NSFD routines are used to design a 4-pole YBCO shielded microstrip filter centered
at 2 GHz with 0.05% bandwidth.
The substrate is 127
m-thick MgO with a dielectric
constant of 9.68. This filter is designed with 0.4 m-thick YBCO films and with an upper
ground plate located at 1016 m above the top of the substrate. The physical parameters
of YBCO used is listed in Table 4.3. The impedance of the filter is 50 Q. The dimensions
of the filter is listed in Table 4.5. Figure 4.19 shows the simulated response of this bandpass
filter.
For an aggressive filter design, properly including the kinetic inductance effects of thin-
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
116
0
-10
I-
-20
cl
-30
-40
-50
-60
1.995
1.9975
2
2.0025
2.005
Frequency (GHz)
Figure 4.19: Simulated response of a 4-pole 0.05% YBCO shielded microstrip filter on
127 #m-thick MgO with a center frequency at 2 GHz (Design: YBCO/MgO-4p0.05%bw).
film superconductors is crucial for a successful design. The responses shown in Fig. 4.20 are
for the 4-pole 0.05% bandpass filter designed with the parameter for 0.4 #m-thick YBCO
at 50 K. The simulations shown are performed using 1.1 gm-thick YBCO films at 50 K,
using the parameters for Nb at 4.2 K as listed in Table 4.3, and using perfect conductor
assumption.
The kinetic inductance effects cause the filter responses to shift totally out
of the band. The simulations generated with Nb parameters and perfect conductor show
very good filter behaviors beside the frequency shift. That illustrates that the responses of
the elements in the filter are changed systematically by the kinetic effects and the elements
remain to be properly matched.
A sensitivity analysis similar to the one discussed in the last section is performed on
the 4-pole 0.05% YBCO shielded microstrip filter. By decreasing the cover height by 10%,
the center frequency is shifted by 0.1% and the return loss is increased to above -20 dB.
The overall response of the filter, as shown in Fig. 4.21, is acceptable.
The effects due
4.5. NUMERICAL SIMULATIONS AND DISCUSSION
117
0.00
-10.00
-20.00
m
- 30.00
-40.00
-50.00
-60.00
1.998
2.0006
2.0032
2.0058
2.0084
2.011
Frequency (GHz)
Figure 4.20: Simulated response of a YBCO (50 K) 4-pole 0.05% shielded microstrip bandpass filter on MgO with a center frequency at 2 GHz. Comparison is made between results
generated using the proper YBCO parameters at 50 K, using the Nb parameter at 4.2 K, and
using the perfect conductor assumption to demonstrate the kinetic effects on filter response.
The design has assumed 0.4 gm-thick YBCO film. The Nb film is 0.4 um thick and one of
the simulations for YBCO is assumed 1.1 pm thick film. The substrate thickness is 127 #m
and the cover height from the top of the substrate is 8 times the substrate thickness.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
118
0
-10
-20
co
-30
-40
-50
-60
1.995
1.9975
2
2.0025
2.005
Frequency (GHz)
Figure 4.21: Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
(Design: YBCO/MgO-4p0.05%bw) with the cover height reduced by 10%.
to the variations in substrate thickness are shown in Figs. 4.22 and 4.23. A reduction in
the substrate thickness by 2
m only shifts the center frequency by less than 0.1%. The
response of the filter is still nearly ideal as shown in Fig. 4.22. When the substrate thickness
is decreased by 8%, the return loss is up to -15 dB and a large frequency shift is observed.
The resulting response is marginally acceptable. For 0.05% bandwidth filters, the variation
in substrate thickness must be less than 8%.
As discussed in the last section, the dimensions of the resonators in a filter can be deviated
from the optimal values because of an inaccurate design and of the limited resolution in
the standard photolithography technique. Figure 4.24 shows the simulated response of the
0.05% filter with the dimensions of the resonators uniformly reduced by 2 #Im from all four
edges. Beside a shift in the center frequency, the response of the filter is nearly ideal. Once
again, it shows that a slight systematic change in the filter layout does not cause the filter
to be mismatched even for a very aggressive filter. A random variation up to 2 lm in the
4.5. NUMERICAL SIMULATIONS AND DISCUSSION
0
I
-10
I
mv-1
O3
I
II
I
-
I
I
I
I
Il
-20 -1
119
I
Ia
I~8
I
a
a
If
-a
I
-r
30
-40
I
I
t I
1%
-DU-
f_
%
I
-50 _
f
a
a
I
I
I
.
-
-
--I -.
I
i.
I
I
1.9975
1.995
I ?
............
.
2
._
. I
--
2.005
2.0025
Frequency (GHz)
Figure 4.22: Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
(Design: YBCO/MgO-4p0.05%bw) with the substrate thickness reduced by 2 #m.
I
0
10 m
Ia
I I
r~~~~~
I-I
m
-
I
aI
20
'o
O3
I
II
30
-
I
55
a
-
-40
i
-I
III
\
I!~~~~
%
1I
- If
-
%
I
-50 m -a
-I
4
A
m IIIp"~L I
-vv
B
_
I
1.99
I
I
I
,
I
1.992
,
I
,
I
I
I
II
1.994
1.996
I
I
I
I
I -
_
I.998
I
1.998
2
Frequency (GHz)
Figure 4.23: Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
(Design: YBCO/MgO-4p0.05%bw) with the substrate thickness reduced by 8%.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
120
0
-10
-20
mo
30
-40
-50
-60
1.995
1.9975
2
2.0025
2.005
Frequency (GHz)
Figure 4.24: Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
(Design: YBCO/MgO-4p0.05%bw) with the dimensions of the resonators uniformly reduced
by 2 gm.
4.5. NUMERICAL SIMULATIONS AND DISCUSSION
121
0
-10
-20
-30
-40
-50
-60
1.995
1.9975
2
2.0025
2.005
Frequency (GHz)
Figure 4.25: Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
(Design: YBCO/MgO-4pO.05%bw) with the dimensions of the resonators randomly reduced
by 0 to 2
m.
0
-10
-20
o
ry
-30
-40
-50
-60
1.995
1.9975
2
2.0025
2.005
Frequency (GHz)
Figure 4.26: Simulated response of a 2 GHz 4-pole 0.05% YBCO shielded microstrip filter
(Design: YBCO/MgO-4p0.05%bw) with the dimensions of the resonators randomly reduced
by 0 to 0.25 m.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
122
0
-10
-20
m
-30
-40
-50
-60
1.995
1.9975
2
2.0025
2.005
Frequency (GHz)
Figure 4.27: Simulated response of a 2 GHz 4-pole 0.05% YBCO (50 K) microstrip filter on
381 #m-thick LaA10 3. By increasing the YBCO film thickness to 1 /m and by decreasing
the impedance to 10 Q, the insertion loss of the filter is reduced to 0.2 dB.
dimensions, however, can destroy the filter performance as shown in Fig. 4.25. If the random
variation can be reduce to 0.25 gm by a better photolithography technique, the performance
of the filter is still deteriorated but is acceptable. Figure 4.26 illustrates the effects due to
the random variation up to 0.25 m in the dimension of the resonators.
For a very narrowband filter such as those with 0.05% bandwidth, the quality factor of
the resonators must be high enough to have an acceptable insertion loss. The 0.05% filter
discussed here is design with 0.4 #m-thick YBCO film. The operating temperature is assumed
to be 50 K. The impedance of the filter is 50 Q. The insertion loss shown in the simulations
is approximately 2 dB which is slightly too large. To reduce the insertion loss, one needs
to increase the quality factor of the resonating elements. For planar resonating structures,
the quality factor can be increased by increasing the film thickness and by decreasing the
filter impedance. Figure 4.27 shows a simulated response of a 2 GHz 4-pole 0.05% YBCO
4.6. EXPERIMENTAL RESULTS
123
0
-5
-10
-15
co
cm
Cn
-20
-25
-30
0
1
2
3
4
5
6
7
8
Frequency (GHz)
Figure 4.28: Measurement and simulation of a 1-pole edge-coupled Nb microstrip filter with
a center frequency at 2 GHz. The second and the third harmonic responses of the filter are
also shown.
microstrip filter on 381 jim-thick LaAlO3 . The operating temperature is still 50 K. The
YBCO film thickness is increased to 1 im and the impedance is decreased to 10
. As a
result, the insertion loss of the filter is only 0.2 dB.
4.6
Experimental Results
To validate the simulation method presented in Section 4.4, a 1-pole Nb microstrip filter
with a center frequency at 2 GHz is fabricated and measured using a Hewlett-Packard network analyzer. The measured response is illustrated in Fig. 4.28 with the calculated result.
The parameter of Nb used in the simulation is listed in Table. 4.3. The calculated result
matches very well to the measured response. Notice the frequency of the second and the third
harmonics is shifted. The shift in the frequency is because of the dispersive characteristics
of microstrip lines.
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
124
0
-10
-20
-30
-40
-- -50
-60
-70
-80
9
9.5
10
10.5
11
Frequency (GHz)
Figure 4.29: Measured response of a 4-pole 1% bandwidth bandpass filter (Design:
Nb/LAO-4p1%bw) fabricated using 3 tm-thick silver. The design of the filter is listed
in Table 4.4.
In Section 4.5.1, a sensitivity analysis is presented for a 4-pole 1% bandwidth Nb microstrip filter with a center frequency at 10 GHz. The design of this filter, listed in Table. 4.4,
is fabricated using 3 g#m-thick silver and 0.4 #jm-thick niobium. The measured responses of
the two versions are shown in Figs. 4.29 and 4.30. The silver filter is measured at room temperature, while the niobium filter is measured at 4.2 K. The large insertion loss of the silver
filter is because of the high conductor loss, and the downward frequency shift is partly due
to the greater internal inductance. The large ripple appeared in the passband of the niobium
filter indicates the effects of secondary coupling. A zero transmission is found at the skirt of
the passband, and that is a clear signature of the next-nearest-neighbor coupling. A commercial field simulator IE3D [59] is used to simulate the filter response. The metallization
is assumed to be perfectly conducting in the simulation. The result is shown in Fig. 4.31.
In the simulation, the enclosure of the filter is neglected. From the simulated response, the
4.7. SUMMARY
125
0
-10
-20
co
-30
_- -40
'n
-50
-60
-70
-80
9
9.5
10
10.5
Frequency (GHz)
11
Figure 4.30: Measured response of a 4-pole 1% bandwidth bandpass filter (Design:
Nb/LAO-4pl%bw) fabricated using 0.4 m-thick niobium. Comparison is made with the
simulated response by the cascaded two-port networks approach. The design of the filter is
listed in Table 4.4.
effects due to next-nearest-neighbor coupling is clear shown. In designing high-performance
filters with a fractional bandwidth less than 1%, the secondary coupling must be accounted
for. Because of the tolerance of the fabrication and packaging techniques, tuning is likely
required for very aggressive filters.
4.7
Summary
A set of computer-aided design and simulation tools, numerical superconducting filter
design (NSFD) routines, has been developed for narrowband superconducting microstrip
filters. These routines are developed based upon the equivalent surface impedance (ESI)
method that has been discussed in the previous chapter. The design procedure and fine filter
design details, such as the proper model for resonator open-end effects, are discussed. An
CHAPTER 4. SUPERCONDUCTING BANDPASS FILTER
126
0
-10
-20
m
-30
-
-40
'0
cm
So
-50
-60
-70
-80
9
9.5
10
10.5
11
Frequency (GHz)
Figure 4.31: Simulated response of a 4-pole 1% bandwidth bandpass filter (Design:
Nb/LAO-4pl%bw) by a commercial simulator IE3D [59]. The metallization is assumed
to be perfectly conducting and the relative permittivity of the substrate is 23.5. Comparison
is made with the measured data.
equation is presented for compensating unequal even- and odd-mode propagation constants
and open-end effective lengths in any planar filter design.
By studying the power-handling capability and packaging of several commonly used
transmission-line resonators, a shielded microstrip structure is found to be the most promising element for building high-performance superconducting bandpass filters especially at
lower frequencies when compared to coplanar and stripline structures.
The NSFD routines are applied to design a 4-pole Nb microstrip filter centered at 10 GHz
with 1% bandwidth, and a 4-pole YBCO shielded microstrip filter centered at 2 GHz with
0.05% bandwidth.
The kinetic inductance effects have been studied by simulating the re-
sponses of the filters with different material parameters.
Beside a shift in frequency, the
performance of the filters is only slightly affected by the change of metallization even for
4.7. SUMMARY
127
the aggressive 0.05% filter. Sensitivity analysis has also been performed for these two filters.
For the 1% filter, the design is robust to substrate variations up to 10%. A slight uniform
variation in the dimensions of the resonators has a little effect on the performance. However,
a slight random variation in the dimensions can greatly limit the performance. For the 0.05%
shielded filter, variations in the cover height are acceptable up to 10%, while substrate thickness variations must be less than 8%. Similar to the 1% filter, a slight uniform variation in
the dimensions of the resonators only shifts the center frequency. A slight random variation
in the dimensions can destroy the filter performance.
The 4-pole 1% bandwidth microstrip filter is fabricated using 3
m-thick silver and
0.4 #m-thick niobium. The silver filter is measured at room temperature, while the niobium
filter is measured at 4.2 K. A large ripple is found in the passband of the niobium filter, and
a zero transmission appears at the skirt of the passband. The measured responses clearly
shown the significance of the secondary coupling effects. In designing high-performance filters
with a fractional bandwidth less than 1%, the secondary coupling effects must be accounted
for. Because of the tolerance of the fabrication and packaging techniques, tuning is very
likely required.
128
Chapter 5
Shielded Microstrip Lines
5.1
Introduction
In the previous chapter, we have presented a design procedure and a simulation method
for edge-coupled bandpass filters. The filter structures are broken into elements consisted of
quarter-wavelength edge-couplers in both the design and the simulation. Such an approach
works well for filters with moderate performance. For high-performance filters with a fractional bandwidth less than 1%, the filter performance is very sensitive to secondary coupling
effects such as next-nearest-neighbor coupling and package coupling. As we have discussed
in the last chapter, it is very difficult to design a filter which accounts for the secondary coupling effects. By using transmission-line structures which have a better shielding property
than microstrip lines, design procedures with the filter broken into elements would still be
acceptable for high-performance filters.
With an upper ground plate, stripline structures have a much better shielding property
than standard microstrip lines. However, the assembly of a stripline filter requires a substrate
with an upper ground plate be placed on top of the center strips, that usually creates a small
air gap. Because of the air gap, the stripline structures cannot be accurately characterized.
As mentioned in the previous chapter, the effects of the air gap were found to be large
enough to deteriorate the filter response of 1%-bandwidth superconducting filters [51]-[52].
129
130
CHAPTER 5. SHIELDED MICROSTRIP LINES
go,
..
...S.......S
<2w
S SS S
2s
S . ......'
2w
h2
..
Figure 5.1: Configuration of an edge-coupled shielded microstrip line.
The configuration of a shield microstrip line, as shown in Fig. 5.1, is very similar to a stripline.
Shielded microstrip lines also have an upper ground plate, but the dielectric between the
center strips and the upper ground plate is replaced by a layer of air. Therefore, shielded
microstrip lines do not have the problems associated with the small air gap in the stripline
structures. When the distance of the upper ground plate to the top of the substrate is
the same as the substrate thickness, i.e., hi = h 2, the shielded microstrip line is called
symmetric. In the quasi-static limit, the electrical characteristic of a symmetric shielded
microstrip line is very similar to a stripline. The characteristic impedances of a coupled
symmetric shielded microstrip line have the same functional form as those of a coupled
stripline [37]. Therefore, one would expect that the symmetric shielded microstrip lines have
similar shielding properties as the striplines. However, full-wave analysis shows that the
shielded microstrip lines are very dispersive, and their shielding capability is greatly reduced
by the parallel-plate mode supported by the ground plates. In the following sections, we will
apply the equivalent surface impedance method to study the characteristics of the shielded
microstrip lines. Experimental results support the numerical analysis.
5.2
Characteristics of Edge-Coupled Shielded
Microstrip Lines
Figure 5.1 shows the configuration of a coupled shielded microstrip lines. The width and
5.2. EDGE-COUPLED SHIELDED MICROSTRIP LINES
131
the separation of the lines are 2w and 2s, respectively. The thickness of the substrate is
hi and the relative permittivity is e,. The distance between the upper ground plate to the
top of the substrate is h2 . The ground plates are assumed to be infinitely extended. The
equivalent surface impedance (ESI) method presented in Chapter 3 is used to analyze the
characteristics of the shielded microstrip lines.
The even- and odd-mode effective dielectric constants of Nb coupled symmetric shielded
microstrip lines are calculated and plotted in Fig. 5.2. Typical parameters for Nb at 4.2 K
are used. The substrate is taken to be 508 m-thick LaA103. The results show very dispersive characteristics. The change in effective dielectric constants is greater than 20% over the
20 GHz frequency span. For coupled microstrip lines, the variation in the effective dielectric
constant over 20 G1Hzrange is only about 14%. The highly dispersive characteristics of
shielded microstrip lines can also be observed from the characteristic impedances. The calculated even- and odd-mode characteristic impedances of the Nb coupled symmetric shielded
microstrip lines are shown in Fig. 5.3. Two major factors are believed to be the causes of the
highly dispersive characteristics in the shielded microstrip lines. Firstly, in the quasi-static
limit, the effective dielectric constant
of a symmetric shielded microstrip is given by [37]
e=
2+
(5.1)
The fields are about the same in the air region as in the substrate. As the frequency increases,
fields tend to concentrate in the region with higher dielectric constant. Hence, the effective
dielectric constant increases with frequency. For microstrip lines, although there is a large
amount of fringing fields in the air, most of the fields concentrate in the substrate between the
strip and the ground plate. Since the symmetrical shielded microstrip has more fields in the
air than the microstrip at low frequencies,the ratio of the field strength in the substrate to
the field strength in the air is likely to change more rapidly for the shielded microstrip as the
frequency increases. Secondly, the two ground plates in the shielded microstrip structures can
CHAPTER 5. SHIELDED MICROSTRIP LINES
132
17
cC
.
-
.
.
.
I
I
.
.
.
.
.
.
..
.
.
.
. . .
.
I
16 -
._
C)
0C00
.r
15 -
0a
0
0
14 2s = 2000 gm
13 -
- - -2s
a)
I.LL
I .
B
I.,
0
I
.
5
I
.
I
.
.
.
.
I
I
I
10
15
= 3000 m
.
.
I
.
.
2
20
25
Frequency (GHz)
Figure 5.2: Calculated even- and odd-mode effective dielectric constants of Nb coupled
shielded microstrip lines on 508 m-thick LaAlO3 at 4.2 K. (2w=150 m, t=0.4 Im,
A=0.281 m, oa1=2.53S/pm, hi = h2 .)
58
a)
Ci
C)
E
C.)
L..
56
54
52
50
c-
0
48
0
5
10
15
20
25
Frequency (GHz)
Figure 5.3: Calculated even- and odd-mode impedances of Nb coupled shielded microstrip
lines on 508 ym-thick LaA1O3 at 4.2 K. (2w=150 pm, t=0.4 im, A=0.281 #m, a1=2.53 S/ylm,
hi = h2.)
5.2. EDGE-COUPLED SHIELDED MICROSTRIP LINES
133
support an addition parallel-plate mode other than the fundamental even and odd modes.
From our analysis, this parallel-plate mode deteriorates the coupling characteristics for the
coupled shielded microstrip lines.
For a stripline quarter-wavelength coupling section, the forward-wave coupling coefficient
is zero because of the equal even- and odd-mode propagation constants, and the backwardwave coupling coefficient C is given by [57]
C=
Zo - zoo
Z
(5.2)
Zoe + Zoo
where Zoe and ZO, are the even- and odd-mode characteristic impedances, respectively. Because of the unequal even- and odd-mode propagation constants, the backward-wave coupling
coefficient of a microstrip or a shielded microstrip quarter-wavelength coupling section is different from the one for the stripline, but Eq. (5.2) gives a good approximation. Figure 5.4
shows the calculated backward-wave coupling coefficient of a symmetric shielded microstrip,
a microstrip, and a stripline quarter-wavelength coupling sections. The coupling coefficient of
the stripline roll off much quicker than the microstrip as the separation of the lines increases.
That demonstrates the better shieldingproperty of the stripline. Based upon the quasi-static
analysis [37], the backward-wave coupling coefficient of the symmetric shielded microstrip
coupler should be the same as that of the stripline. The full-wave analysis indicates that the
coupling coefficients for the symmetric shielded microstrip and the stripline are the same for
small line separations. As the line separation increases, the coupling coefficient of the symmetric shielded microstrip line deviates from that of the stripline, and a "shoulder" appears
at around -54 dB in the coupling curve. By decreasing the frequency, the "shoulder" in the
coupling curve appears at a weaker coupling strength as shown in Fig. 5.5. As shown in the
same figure, the "shoulder" can also be pushed to a weaker coupling strength by rising the
upper ground plate.
The "shoulder" appeared in the coupling curve for the coupled shielded microstrip is
CHAPTER 5. SHIELDED MICROSTRIP LINES
134
-10
-20
-30
c
O
-40
O
V-1
-50
-60
-70
1
10 2s/h
100
1000
1
Figure 5.4: Calculated backward-wave coupling coefficient C of a symmetric shielded microstrip, a microstrip, and a stripline quarter-wavelength coupling sections. The metallization is assumed to be perfectly conducting and the line width is 79 m. The substrate is
127 m-thick and its Er=12. The separation of the ground plates in the stripline structure
is 254
m.
-10-
.
-20-
,%1.\
V
,C)
_
l·___·
I
_
I
__..
l
I
.
......
X.
C
I
~~ h2=hl,
IIf=2 GHz
- _\_
-30m
. . , ,,
_
-h2=h1
-
4050-
f=
- -- h=6h1,f
0 .5GHz
2GHz
I
V'11-_
60-
70 - [-
,
, , , , , ,1
-80
I
1
10
2s / h1 100
,\, ,,
1000
Figure 5.5: Calculated backward-wave coupling coefficient C of shielded microstrip quarter-wavelength coupling sections. The metallization is assumed to be perfectly conducting
and the line width is 79 m. The substrate is 127 gm-thick and its er=12.
5.3. EXPERIMENTAL VERIFICATION OF PARALLEL-PLATE MODE
135
Table 5.1: Dimensions of symmetric shielded-microstrip quarter-wavelength couplers.
Structure
Coupler
Coupler
Coupler
Coupler
f (GHz)
1
2
3
4
2
2
5
10
2w (m)
142
142
146
158
2s (m)
1524
2540
1524
1524
(m)
10304
10305
3893
1764
caused by a slight increase in the even-mode impedance and a slight decrease in the oddmode impedance. The changes in the impedances are because of the additional parallel-plate
mode supported by the two ground plates. The existence of the parallel-plate mode can be
easily identified from the electric field distribution of the even-mode. Figure 5.6 shows the
electric field distribution of the proper even-mode in a shielded microstrip line when the line
separation is not too far. As the line separation increases far enough so that the "shoulder"
appears, the electric field distribution, as illustrated in Fig. 5.7, clearly shows the field
associated with the parallel-plate mode.
5.3
Experimental Verification of Parallel-Plate Mode
To verify the effects due to the parallel-plate mode on the characteristics of coupled
shielded microstrip lines, four quarter-wavelength couplers are designed and fabricated. The
operating frequencies of the couplers are 2 GHz, 5 GHz, and 10 GHz. The configuration of
the couplers is shown in Fig. 5.8 and the dimensions are listed in Table 5.1. Because the
parallel-plate mode is a property of the shielded microstrip line and it is not related to the
material of the metallization, we have used 3 m-thick silver to fabricate the couplers. The
substrate is 508 gm-thick LaAl03 and the area of the substrate is 2.54 cm by 1.27 cm. The
couplers are placed ill a microwave package with two SMA connectors. The transitions from
the shielded microstrips to the connectors are made by wire-bonding. The ground plates
of the shielded microstrip structures are tightly connected at the input and output ports to
CHAPTER 5. SHIELDED MICROSTRIP LINES
136
,
'
t
t
t
t
t
t
,
·i
t
t
p
*
t
t
'
t
t
p
·r *
'
?t
/
I
4
4
*@0
/~~~
4
4
4
t
p
t
/
\t~
,
,
&
4
4
&
Figure 5.6: Electric field distribution of the even-mode in a coupled symmetric shielded
microstrip line with 2s/h 1 = 2. The metallization is assumed to be perfectly conducting and
the line width is 79 im. The substrate is 127 #m-thick and its er=12.
* ttt
*
99 .
.
.
·
·
I
III
4
V
·
I~~~~~~~~~~~~~
·
.
·
ttt
·
·
II
Ott
. tt
t)
a
I
.
I
·
I .
..
.
ap
·a
.
..
·
Ii
·.
4
9
·
·
a
v
a
a
44
4
4
9
*
,
A
,,
I
a
I
5
a
4
.
I
*
44 4
IS
Figure 5.7: Electric field distribution of the even-mode in a coupled symmetric shielded
microstrip line with 2s/hl = 10. The metallization is assumed to be perfectly conducting
and the line width is 79 /m. The substrate is 127 gm-thick and its er=12.
5.3. EXPERIMENTAL VERIFICATION OF PARALLEL-PLATE MODE
137
Top View
sapphire
silve
... ...............................
silver
....................................
groundplates
plates
508 gm-thick
3 gm-thick
silver strip
LaAIO
508 kgm-thick
LaAIO
3
3
spacer
substrate
Cross-Sectional View
Figure 5.8: Configuration of a symmetric shielded-microstrip quarter-wavelength coupler.
CHAPTER 5. SHIELDED MICROSTRIP LINES
138
0
-10
-20
-30
-40
-50
-60
-70
-80
1
10
100
2s/h
Figure 5.9: Calculated transmission coefficient of symmetric shielded microstrip quarter-wavelength couplers. Comparison is made with measured data. ( - measured data
at 2 GHz, O - measured data at 5 GHz, A - measured data at 10 GHz.)
ensure proper transitions.
The simulated and measured S211of the couplers are shown in Fig. 5.9. The simulations
are based upon Eq. (4.22). The measurements are made with a Hewlett-Packard network
analyzer.
Because of the package resonances, the responses of the couplers are not very
clean over a 20 GHz frequency span. Fortunately, no resonance is found near 5 and 10 GHz.
However, a strong package resonance is found near 2 GHz. In order to measure the two
2 GHz-couplers, two strips of absorber are place along the sides of the package in the air
region of the shielded microstrip structures to remove the package resonance at 2 GHz. The
reasonably good agreement between the measured and simulated results has confirmed the
existence of the parallel-plate mode in a coupled shielded microstrip structure. The parallelplate mode in shielded microstrip structures is not desirable for narrowband filters. In order
to take advantage of the good shielding property that shielded microstrip lines potentially
5.4. OPEN-END DISCONTINUITY IN SHIELDED MICROSTRIP LINES
139
provide, one must somehow suppress the parallel-plate mode. Preliminary results based on
measurements and simulations by a field simulator, EM [56], the parallel-plate mode can be
suppress by making good contacts between the ground plates at all four sides to ensure an
equal potential.
5.4
Open-End Discontinuity in Shielded Microstrip
Lines
Open-end discontinuity of planar transmission lines can be commonly found in many
microwave circuits such as resonators, matching stubs, and edge-coupled filters.
In the
pervious chapter, we have demonstrated the importance of the open-end effects in filter
designs. To improve the performance of these microwave components, the open-end effects
have to be quantified and accounted for in the circuit designs. The excess capacitance at
the open end is usually modeled by an equivalent length of transmission line [37]. Closedform expressions for the open-end equivalent length for striplines and microstrip lines have
been previously presented by several authors [61]-[67]. Altschuler and Oliner [61] derived
an expression for stripline structures based upon their measured data. Garg and Bahl [62]
modified a formula for microstrip lines derived by Hammerstad [63] based upon numerical
results computed by Silvester and Benedek [64]. By matching to the numerical results
published in Refs. [65] and [66], Kirschning et al. [67] obtained a more accurate equation for
the open-end effects in microstrip lines. In Ref. [68], Kirschning et al. presented closed-form
expressions for the even- and odd-mode open-end equivalent lengths in coupled microstrip
lines based upon the numerical solutions computed by the mode-matching method [66].
Bedair and Sobhy [69] derived a closed-form equation to describe the equivalent length in
shielded microstrip lines using the open-end model for striplines [61]. However, we have
found that the equivalent length given by Bedair's equation does not approach the proper
limit when the cover height in the shielded structures becomes large.
CHAPTER 5. SHIELDED MICROSTRIP LINES
140
h2
....
...
2w_
::
,...:::...:::.
. . . .....
> , c::
.. . .. . . . ... . . . . .. .:.........>....
... .. .. ...........
.. ......
................
.......
.
..........................
... . . . ......
.. .>......
... .
>
~~~~~~~~~~~~~~..
...>>>
.>>
>..
>.....
>...
>>............
~~~~~~~~~~~~~....
.... ............
........
~~~~~~~~~~~~~~~~~~~~~~..........
~~~~~~~~~~~~~....... ... ..........
c
: ---- :. : -:
. ...............
.::.
-- -. e -- .. .: .. :. e--.:.:.
: -:-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........
:
- .
:.:.
.>..
-: -
v~~~~~~~~~~~~~~~~.
.-........
. ......
.....
>.
. .,
..
............
...
. .............. . . . ..
...........................-.
........
,:
v.-.
::
...>...
>. >......
..........
.............>>
. ......>.
.. ..>
........>>.>.>.
...........
.
. . . .>
v......
Figure 5.10: Configuration of a shielded microstrip.
Here a phenomenological approach is taken to find closed-form expressions for the openend equivalent length in single and coupled shielded microstrip lines. For a given value of
strip width 2w and substrate thickness hi the characteristics of a shielded microstrip, shown
in Fig. 5.10, change gradually from a symmetrical shielded microstrip to an unshielded
microstrip as the ratio of cover height to substrate thickness, g = h 2 /hl, increases from 1 to
oo. This property allows us to write the open-end equivalent length /lsms(g) of a shielded
microstrip in terms of the one for a symmetrical shielded microstrip and an unshielded
microstrip, specifically
Alsms(g) = Alsms(X) + q4[/Alms- /lsms(1)]
(5.3)
where q is an unknown function, and Alm, is the equivalent length for an unshielded microstrip. In the quasi-static limit, the equivalent length of a symmetrical shielded microstrip
Alsms(l)
is equal to the equivalent length of a stripline Ala. The equations for AlI and Alms
are found in Refs. [61] and [67], respectively. In general, the unknown function qj is obtained
by matching to rigorous experimental or numerical results. Alternatively, we attempt to
derive an approximate expression for ql without performing extensive experiments or numerical calculations, and then show that our result compares well with those of a commercial
simulator.
5.4. OPEN-END DISCONTINUITY IN SHIELDED MICROSTRIP LINES
141
The open-end equivalent length Al,,sm(g) for a shielded microstrip line is given by
Alsms(g) = C (g)
(54)
C(g)
where Cf (g) is the excess capacitance because of the fringing fields at the open end, and C(g)
is the capacitance per unit length of the shielded microstrip. The capacitance C(g) can be
expressed in term of the effective dielectric constant
e(g) and the characteristic impedance
Z(g)= Z(g)/ ee(g),
C(g) =
(g)
e(l) + qe()[Ee(OO)- Ee(1)]
cZ (g)
(5.5)
= qC() +qC(oo)
where c is the speed of light, q(g) = tanh(1.043 + 0.121g - 1.164/g) [70], and
(Z ()
(5.6)
(5.7)
qco = qe(g) (Zo(g)
The open-end capacitance Cf(g) is an unknown function. However, the first-order approximation for Cf(g) can be taken as
(5.8)
Cf(g) = qclCf(1) + qcooCf(o)
With Eqs. (5.4) and (5.6)-(5.8), the open-end equivalent length of a shielded microstrip can
be written as
ge()e,--
lsms(g) = (1 - Qe(g)) e (1) Alsms(1) + qe(g)
sms()
(5.9)
(59)
By comparing the terms in Eqs. (5.3) and (5.9), we obtain
q, =
Eqg
e(oo) s(+qe(g)Ee(oo)
Fe(l)+ q(g)[Ee(oo)- e(1)]
(g)
e'(g)
-(1±qe(g)Ee(OO)
(5.10)
CHAPTER 5. SHIELDED MICROSTRIP LINES
142
0.42
0.40
0.38
0.36
en
E
_)
i,
<:
0.34
0.32
0.30
0.28
0.26
1
2
3
4
5
h /h
2
6
7
8
1
Figure 5.11: Calculated open-end equivalent length for shielded microstrip lines with
2w/hl = 1. Comparison is made between the results given by Eq. (5.9), those given by
Bedair's expression [69], and those generated by EM [56].
This argument is also applied to derive the even- and odd-mode equivalent lengths for
the coupled shielded microstrip lines. The resulting expression is identical to Eq. (5.3) with
the variables replaced by the ones corresponding to the proper mode, and
q"j(g) -
(5.11)
- ei(l)]
fe i(1) + qe,i(g)[Ee,i(oO)
where i = e, o stands for even- and odd-mode, respectively. In Ref. [70], the expressions
qe,i(g) are given as
= tanh [1.626 + 0.1079- 1.733
{ tanh
I
[9575-
2.965+ 1.68 - 0.311g2] ; for g
; for g
<
7
7
Figures 5.11 and 5.12 show the calculated open-end equivalent lengths given by Eq. (5.9).
The results agree reasonably well with the numerical results generated by a commercial
5.4. OPEN-END DISCONTINUITY IN SHIELDED MICROSTRIP LINES
I
0.60-
1
I
I
I
I
I
I
I
I
I
x
0.50 -
EM
I
-
I
' '
-
-
II
I
I
.
I. .
..
.
I.
.
.
= 2
Eq. (5.9)
Bedair
0.55 -
I
III I
143
r
,-
=4
0.45 0
E
(,
I
0.40 -
i
£ = 10
0.35-
r
0.300.25-
il.
0
l.........
I
I
1
2
...........I
iii-
-
I
I
I
I-
3
4
5
6
7
2w/h 1
Figure 5.12: Calculated open-end equivalent for shielded microstrip lines with h2 /h l = 6.
Comparison is made between the results given by Eq. (5.9), those given by Bedair's expression
[69], and those generated by EM [56].
simulator, EM [56]. The results calculated using Bedair and Sobhy's equation [69] are also
shown in the figures. Bedair and Sobhy assumed that the open-end capacitance in a shielded
microstrip could be approximated by those in striplines with ground-plate separations of 2h2
and 2h1 . However, their assumption is valid only when h 2 /hl is small and/or 2w/hl is large.
Consequently, the results generated by their equation deviate from the correct values for
large h2 /hl and for small 2w/hl. Compared to results by EM, Bedair's equation gives better
approximation for small h 2 /hl where its assumptions are valid; while Eq. (5.9) provides
a better overall functional form for the equivalent length, especially when h 2 /hl is greater
than 3. Because of their simplicity, these closed-form expressions for the open-end equivalent
lengths are well suited for computer-aided design tools.
CHAPTER 5. SHIELDED MICROSTRIP LINES
144
5.5
Summary
The characteristics of coupled shielded microstrip lines are studied using the equivalent
surface impedance method.
Coupled shielded microstrip lines are found to be highly dis-
persive and the coupling strength of the structures is not the same as the coupled striplines
as predicted by the quasi-static analysis. The full-wave analysis shows that the coupling
strength of the shielded microstrip lines drops to a certain level and then levels off as the
separation of the lines increases. The leveling of the coupling strength is caused by the
parallel-plate mode supported by the ground plates.
The existence of the parallel-plate
mode has been proven by simulated electric field distributions and by experiments. Preliminary results by simulations and measurements suggest that the parallel-plate mode can be
suppressed by tightly connecting the ground plates together to ensure an equal potential.
Closed-form expressions for the open-end equivalent length in single and coupled shielded
microstrip lines are derived using a phenomenological approach. These equations are obtained based upon the open-end models for the stripline and the microstrip. The calculated
results agree well with those obtained using a commercial simulation package. Because of
their simplicity, these formulas can be easily incorporated into computer-aided design packages.
Chapter 6
Conclusions
This thesis presented accurate full-wave numerical methods for analyzing superconducting microstrip planar circuits. Chapter 2 described the development of a spectral-domain
volume-integral-equation method which rigorously accounts for the complex conductivity, the
finite metallization thickness, and the anisotropy of superconductors. In Chapter 3, a more
efficient full-wave method was developed based upon an equivalent surface impedance. Chapter 4 described a design procedure and a simulation method for superconducting edge-coupled
bandpass filters based upon the equivalent surface impedance method. The power-handling
capability of various superconducting resonators was also discussed. The characteristics and
shielding properties of shielded microstrip lines were presented in Chapter 5. The following
list summarizes the results presented in this thesis.
(1) A spectral-domain volume-integral-equation method was developed for characterization of superconducting planar transmission lines. In the model, the superconductors
were described by complex conductivity tensors, so that the anisotropic properties of
the high-temperature superconductors could be studied. The current in the superconductors and the complex propagation constant of superconducting transmission lines
were rigorously solved. (Chapter 2)
(2) The effective dielectric constant and the quality factor were calculated for several mi145
CHAPTER 6. CONCLUSIONS
146
crostrip lines and coplanar waveguides made of niobium. Good agreement was obtained
when the calculated results were compared with data extracted from measurements
performed on resonators (Section 2.6).
(3) The characteristics of a microstrip line with a thin buffer layer and a stripline with
a small air gap were presented. The effects of the anisotropy of YBa2 Cu 3 O7 , were
found to be negligible on the characteristics of the transmission lines fabricated on
c-axis oriented films. (Section 2.6)
(4) An efficient full-wave integral-equation method was formulated using an equivalent
surface impedance method concept. In the formulation, the superconducting strips
were replaced by infinitely thin strips. The characteristics of the superconductors were
described in terms of the equivalent surface impedances. This method reduced the
computation time by nearly two orders of magnitude compared to the more rigorous
spectral-domain volume-integral-equation method. (Chapter 3)
(5) The validity of the equivalent surface impedance approach was verified by comparing
numerical results for superconducting microstrip lines with measured data and calculated results using the spectral-domain volume-integral-equation method. (Section 3.4)
(6) The equivalent surface impedance method was extended to model single and coupled
microstrip lines on anisotropic substrates with a rotated optic axis. Calculated results
for perfectly conducting coupled lines showed good agreement with results generated
by the finite-difference time-domain method. The effective dielectric constants and the
characteristic impedances for single and coupled microstrip lines on M-plane sapphire
were presented for various line widths, line separations, optic-axis orientations, and
operating temperatures. (Section 3.4)
(7) In the development of the equivalent surface impedance method, a closed-form expres-
147
sion for the current distribution in superconducting strips was obtained by curve fitting
to numerical results. (Section 3.3)
(8) A quantitative analysis was presented on the power-handling capability of various
superconducting transmission-line resonators. The structures under consideration were
microstrip lines, shield microstrip lines, coplanar waveguides, and striplines. Among
all these structures, microstrip lines were found to have the largest power-handling
capability. (Section 4.2)
(9) Based upon the equivalent surface impedance method, a computer routine was developed for designing superconducting edge-coupled bandpass filters. An equation was
derived for compensating unequal even- and odd-mode open-end effective lengths in
structures such as microstrip lines where the even and odd modes propagate at different
velocities. The response of the filters was obtained by modeling the filter as cascaded
two-port networks consisting of quarter-wavelength couplers. In the design procedure
and the simulation, the secondary coupling effects such as next-nearest-neighbor coupling and package coupling were neglected. (Sections 4.3 and 4.4)
(10) A sensitivity analysis was performed on a 4-pole 1% bandwidth microstrip filter centered at 10 GHz and a 4-pole 0.05% bandwidth shielded microstrip filter centered
at 2 GHz. The analysis showed that given a perfect design the filter remained well
matched if the superconducting material was replaced by another superconductor even
for the very aggressive 0.05% bandwith filter, except there was a shift in center frequency because of the kinetic inductance effect. The filter performance remained acceptable with a slight symmetric variation in the dimensions of the resonators, but
random variations must be smaller than 2
0.05% filter. (Section 4.5)
lmfor the 1% filter and 0.25 #im for the
CHAPTER 6. CONCLUSIONS
148
(11) A 4-pole 1% bandwidth niobium microstrip filter centered at 10 GHz was designed
using the routines developed in this thesis. The design was implemented using silver
and niobium. The measured results indicated significant secondary coupling effects.
(Section 4.6)
(12) The characteristics of coupled shielded microstrip lines were studied using the equivalent surface impedance method. The structures were found to be very dispersive. A
parallel-plate mode supported by the ground plates was found in the shielded microstrip
lines. This parallel-plate mode greatly deteriorated the good shielding capability potentially provided by the shielded microstrip. Preliminary experimental and numerical
results showed that tightly connecting the ground plates together could suppress the
parallel-plate mode. (Secions 5.2 and 5.3)
(13) Closed-form expressions for the open-end equivalent length in single and coupled
shielded microstrip lines were derived based upon the open-end models for the stripline
and the microstrip. The calculated results agreed well with those obtained using a commercial field simulator. (Section 5.4)
The secondary coupling effects were found to be important in narrowband filters with a
fractional bandwidth less than 1%. The tools developed in this thesis work cannot account
for such effects. To develop an optimization subroutine which accounts for all these secondary
effects is very difficult. An alternative
way is to develop an efficient simulation method which
rigorously models the secondary coupling effects. So that fine tuning on the filter design is
possible. With a very well shielded structure, the secondary effects can be reduced. The
shielded microstrip line is a very good structure if the parallel-plate mode can be suppressed.
Preliminary results showed the possibility of suppressing the parallel-plate mode. Further
experimental and theoretical studies are required.
Appendix A
Dyadic Green's Function for
Isotropic Layered Media
The electric dyadic Green's function Gm(T,F)
for isotropic layered media as shown
Fig. A.1, where the observation point is in layer (m) and the source is in the layer (n),
satisfies the following relation
(V x V x I- k)
)
Gmn(
--
76(f -t), m -n
(A.1)
With a current source in the layer (n), the Green's functions with the observation point at
layer (m), for m = 1, 2, 3, ..., satisfy the following boundary conditions
x Gmn,,=
lm
i x V x Gmn
=
x G(m+),,,
1
x V x G(m+),
/Pm+l
(A.2)
(A.3)
at the interface between layer (m) and layer (m+1).
The dyadic Green's function
dyadic Green's function
nn,can
be expressed as a superposition of the unbounded
due to the primary excitation and a scattered dyadic Green's
function Vnn [30]. Hence for any layer (m)
Gmn = -- n. + 9mn
(A.4)
where mnis the Kronecker delta function. The unbounded dyadic Green's function 9 can
149
APPENDIX A. DYADIC GREEN'S FUNCTION FOR ISOTROPIC MEDIA
150
(n-l)
dn-1
£n-1 ,
IF
I II
n-1
(n)
current
IF
source
ax
En, Rn
II
(n+1)
dn+l
-
£n+ 1, ,n+ 1
IF
0
0
Figure A.1: A layered medium with a current embedded in layer (n).
be expressed in terms of a scalar Green's function g, specifically
(= [I+ kV]
where k2 =
2 Jem
.
g(, )
(A.5)
This scalar Green's function g satisfies the wave equation
2g(,
V
)+kg(,')
=-')
(A.6)
By applying the Fourier transform to Eq. (A.6), we obtain
1
)
IL00
g(f'I') = (2;)7
where kmz =
12ik.(-')
kz -kLZ
(A.7)
k - k2 - k with the imaginary part greater than zero, and k = ikx + yky +
kz.For solving problems with layered media, it is more convenient to carry out the
For solving problems with layered media, it is more convenient to carry out the kzintegration in Eq. (A.7) and express the Green's function in terms of a double integral. The
151
singularities exist in the ks-integration are the poles occurring at
(A.8)
kz = ±kmz
(a-)ek
) 1-eikl
To properly account for the contributions due to the poles, the integration path is deformed
upward for z > z', and is deformed downward for z < z'. Thus, after the integration, we
have
-)
~T~)
2 (2r)2
where
IL
k = k + ky;
dk
,
00
(A.9)
_
L,
. = ix + y;
s
= ,'xx' + y'
(A.10)
By substituting Eq. (A.9) into Eq. (A.5), the unbounded dyadic Green's function 5 is
obtained
(r~,')
- -
km
i -A')
+ =2
T2
f [h(kmz)h(kmz) +
Jf
1
dk8 eik a(rr; )kmz
z > z'
O(kmz)(kmz)eikmz(zz'),
(A.11)
[h(-kmz)h(-kmz)
+
(-kmz))(-kmz)]e
' - (z z'),
z < z'
where the unit vectors h and v are defined as
h(±kmz) =
k
0(±kmz) = T: kmk
+
k
The unit vector h is in the direction of the electric field for a horizontally polarized TE wave
and the unit vector
is in the direction of the electric field for a vertically polarized TM
wave. In Eq. (A.11), the TE dyad h(kmz)h(kmz)exp[ikmz(z-
z')] for z > z' is formed by two
unit vectors. The anterior vector is related to the observation point where the wave vector
has a positive z-component, and the posterior vector is related to the source point where
the wave vector also has a positive z-component. The term exp[ik,,z(z - z')] accounts for
APPENDIX A. DYADIC GREEN'S FUNCTION FOR ISOTROPIC MEDIA
152
the phase shift as the wave propagates from the source point at z' to the observation point
at z. For z < z', the wave vector at both the observation and source points has a negative
z-component. Therefore, the TE dyad is h(-kmz)h(-kmz)
exp[-ikm(z
- z')]. The TM dyad
has the similar properties.
V,,(?,
r') from the physical picture
It is possible to construct the dyadic Green's function
of the wave radiated from the source and reaching the field point. In the spectral-domain,
we have
-6(-')
n(,')=
I8eik's(r,-)(ks Z)
dk
+8
The first term in (A.12) is the source dyadic and
the principal value part of Gnn(T,T'). The term ,~(k,
(A.12)
nn(k, z, z') is the Fourier transform of
z, z') is a function of the observation
point, at z, and the source point, at z'. For z > z', the explicit form of n(k, z, z') is given
by
TE
[A(knz)eiknz+ RTEh-knz)eiknz(2dn-z)]
=
[h(knz)e-iknz' + R Eh(-knz)eiknzz']
TM
k ~[o(ki2)eiz
+[ Rk_
(-
)eiknz(2dn-z)]
+ R TijM(-kz)eiknz']
i(knz)e-'k
[
(A.13)
and for z < z', we have
TE
nn(
Z< Z
nz
[h(-knz)eiknzz + Rnn(knz)eknzz]
*[(-kn)eiknzz' + R TEjh(knz)eiknz(2dn-z')]
+ __ [(-knz)eiknzz
+Rn
f(knz)eikZ]
* [(-kn,)eiknzz'
+ RuTM(knz)eiknz(2dn-z')]
(A.14)
where d is the thickness of layer (n), and
2 iknzdn
rTE = (1RR
RTE=TE
)
e
nRnn
(1 -
rnM
rTM
=
TM TM e2iknzdn -1
(A.15)
(A.16)
153
The factors
respectively.
rTE
and
rTM
account for the TE and TM multiple reflections in layer (n),
In (A.13) and (A.14), RA, and R"n are, respectively, the global reflection
coefficients at the lower boundary and at the upper boundary of the layer (n), which can be
obtained recursively as
MY
+ RRl
2ik(t+l)d+l
+ n(t+l)
R'l = R(l+l)
I 1 + R+l) R(l+l)e2 ik(+1)d +l
R'Y1..1 ) + R (1_1)ezik (1d 1
-
1I +R
' R1
+ L(t-1)-
R-u' 1
1)
2ik(lj ldt-j
(A.17)
(A.1)
(A.18)
where y = TE, TM, I = 0, 1, 2,..., and R7. is the Fresnel reflection coefficient at the interface
of the i th and the j th layers. The explicit forms are
IIE
R':= -jkiz
ikjz
p z+LRt7jz
+ iijZ
(A.19)
For the microstrip problem, there is no boundary above the strip, therefore, n = 0 and
the global reflection coefficients Ro are equal to zero in (A.13) and (A.14).
154
Appendix B
Dyadic Green's Function for
Anisotropic Media
Figure B.1 shows an anisotropic substrate sandwiched by two perfectly conducting ground
plates. The optic axis of the substrate is in the x-y plane. The permittivity tensor
E for
the
substrate in the chosen coordinate system can be written as
ep oI
o
I Cos20 +
( e ll -e)
sin
0
11sin2 0(ll
cos
E)
e sin2
0
+
sinell cos8
cos 2
0
0
ex
O
(B.1)
where qe and ell are the permittivities perpendicular and parallel to the optic axis, respectively. The 2-D dyadic Green's function
(k, ky) with surface currents at z = h can be
derived using the method proposed in [50]. Let k0o be the k-vector in the free space. Define
k 2 = k 2 + k2, I¢§ = k
k2a and K2 = k2-e.
ko2. The explicit form of (k,
( ' 8)( [° f2O
) Q + °, p
where
fi
=
f2
= Ko coth[Ko(d - h)]-kkp2r +
-k2
coth[Kfo(d- h)]K+
155
1
ky) is given by
(B.2)
156
APPENDIX B. DYADIC GREEN'S FUNCTION FOR ANISOTROPIC MEDIA
~ -- oo
~z
%0, O
d~~~~
.... d............
. .
.
.....
;- l....
/j.......................................
:.
.....
:../I.......
:..j ? /
:;.···...;.....;;-::e.·.
_ -.
:
.·:·::.
:x>::;
>
:::::r::::i::i:·:
:--* -:: ::::::::>:>:
:
x.-;. e e~~~~~:5:i:::::i:
:K::i:::x::: :::::x:::iE|^
:: -:: ::¢:
:.:
-:.:. ~
: ~~~~~~~~~~::::::.
: ..........
:::
:::--::.
::::>::x::
:::::::::::
::::-::::
* ,;..... ~::
.................
.................
^:j::: d8~~~~~~~~~~8~~:
~w://///95··_·······1_····-·_·_·····C
::~~::::*:.::
x:::~:>:::: : :::: :: :::::::
: : : : :~~~~~~~~.
:--:...:tf..-...
....
... ~~~...........
_:.....
.---///////////9
-:.x;.:
:>:·:
:::.::·:
x
::.j:
: :::::
::::j::::j::j:
-:::>x
e.:.::·:···:::::e>-
:: : : :::::::
i~:~:~:.~~8~:~~3~:~:~~~:~:.......... ......................
."
..........:....................
_
Figure B.1: Cross section of an anisotropic substrate sandwiched by two perfectly conducting
ground plates.
r
=
A2
1
1I
,2
A2 [A2 coth(A
2 h)
-
A1 coth(Alh)]
2
A2, [A
1 coth(A 2h) - A
2 coth(Al h)]
and A2
are the 'CI~U
eigenvalues3Vof
1 and A2
2 "C~I
B=
"i
EJ
0I
-ko
[ko(
0
-1
EP
For an open structure where d -- oo, fi and f2 in Eq. (B.2) are
fi=
-k2
+_ (
f2 = K'o-k r+
(B.3)
Bibliography
[1] R. B. Hammond, G. L. Hey-Shipton, and G. L. Matthaei, "Designing with superconductors," IEEE Spectrum, vol. 30, pp. 34-39, Apr. 1993.
[2] J. C. Ritter, M. Nisenoff, G. Price, and S. A. Wolf, "High temperature
superconductivity
space experiment," IEEE Trans. Magn., vol. 27, pp. 2533-2536, 1991.
[3] A. I. Braginski, "Progress and trends in high-Ta superconducting electronics," FED J.,
vol. 1, 1990.
[4] J. D. Doss, Engineer's Guide to High-Temperature Superconductivity. New York: Wiley,
1989.
[5] S. A. Long and J. T. Williams, "High temperature superconductors," IEEE Potentials,
vol. 10, no. 4, pp. 37-40, 1991.
[6] N. Newman and W. G. Lyons, "High-temperature superconducting microwave devices:
fundamental issues in materials, physics, and engineering," J. Superconduct., vol. 6,
no. 3, pp. 119-160, 1993.
[7] R. S. Withers and R. W. Ralston, "Superconductive analog signal processing devices,"
Proc. IEEE, vol. 77, pp. 1247-1263, Aug. 1989.
[8] W. G. Lyons and R. S. Withers, "Passive microwave device applications of high-Tc
superconducting thin film," Microwave J., vol. 33, pp. 85-102, 1990.
[9] D. E. Oates, A. C. Anderson, D. M. Sheen, and S. M. Ali, "Stripline resonator measure-
ments of Z, versus Hrf in YBa 2 Cu3 07_ thin films," IEEE Trans. Microwave Theory
Tech., vol. 39, pp. 1522-1529, 1991.
[10] M. S. Schmidt, R. J. Forse, R. B. Hammond, M. M. Eddy, and W. L. Olson, "Measured
performance at 77 K of superconducting microstrip resonators and filters," IEEE Trans.
Microwave Theory Tech., vol. 39, pp 1475-1479, 1991.
[11] W. G. Lyons, R. R. Bonetti, A. E.Williams, P. M. Mankiewich, M. L. O'Malley, J. M.
Hamm, A. C. Anderson, R. S. Withers, A. Meulenberg, and R. E. Howard, "High-T,
superconductive microwave filters," IEEE Trans. Magn., vol. 27, pp. 2537-2539, 1991.
[12] S. H. Talisa, M. A. Janocko, C. Moskowitz, J. Talvacchio, J. F. Billing, R. Brown,
D. C. Buck, C. K. Jones, B. R. McAvoy, G. R. Wagner, and D. H. Watt, "Low- and
high-temperature superconducting microwave filters," IEEE Trans. Microwave Theory
Tech., vol. 39, pp. 1448-1454, 1991.
157
BIBLIOGRAPHY
158
[13] W. G. Lyons, R. S. Withers,
J. M. Hamm, A. C. Anderson,
P. M. Mankiewich,
M.
L. O'Malley, R. E. Howard, R. R. Bonetti, A. E. Williams, and N. Newman, "Hightemperature superconductive passive microwave devices," IEEE MTT-S Int. Microwave
Symp. Dig., pp. 1227-1230, 1991.
[14] G.-C. Liang, R. S. Withers, B. F. Cole, and N. Newman, "High-temperature superconductive devices on sapphire," presented at IEEE MTT-S International Microwave
Symposium, June 1-5, 1992, Albuquerque.
[15] G.-C. Liang, R. S. Withers, B. F. Cole, S. M. Garrison, M. E. Johansson, W. Ruby, and
W. G. Lyons, "High-temperature superconducting delay lines and filters on sapphire and
thinned LaA1O3 substrates," IEEE Trans. Appl. Superconduct. vol. 3, pp. 3037-3042,
Sept. 1993.
[16] W. G. Lyons, D. R. Arsenault, M. M. Seaver, R. R. Boisvert, T. C. L. G. Sollner, and R.
S. Withers, "Implementation of a YBa2 Cu 3 07_. wideband real-time spectrum-analysis
receiver," IEEE Trans. Appl. Superconduct., vol. 3, no. 1, pp. 2891-2893, 1993.
[17] J. Bardeen, L. N. Copper, and J. R. Schrieffer, "Theory of superconductivity," Phys.
Rev., vol. 108, pp. 1175-1204, Dec. 1957.
[18] T. P. Orlando and K. A. Delin, Foundations of Applied Superconductivity. Reading, MA:
Addison-Wesley,
1991.
[19] J. C. Swihart, "Field solution for a thin film superconducting strip transmission line,"
J. Appl. Phys., vol. 32, pp. 461-469, Mar. 1961.
[20] R. L. Kautz, "Picosecond pulses on superconducting striplines," J. Appl. Phys., vol. 49,
pp. 308-314, Jan. 1978.
[21] O. K. Kwon, B. W. Langley, R. F. Pease, and M. R. Beasley, "Superconductors as
very high-speed system-level interconnects," IEEE Electron Device Lett., vol. EDL-8,
pp. 582-585, Dec. 1987.
[22] A. C. Cangellaris, "Propagation characteristics of superconductive interconnects," Microwave Opt. Tech. Lett., vol. 1, pp. 153-156, July 1988.
[23] J. M. Pond, C. M. Krowne, and W. L. Carter, "On the application of complex resistive
boundary conditions to model transmission lines consisting of very thin superconductors," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 181-190, Jan. 1989.
[24] D. Nghiem, J. T. Williams, and D. R. Jackson, "A general analysis of propagation
along multiple-layer superconducting stripline and microstrip transmission lines," IEEE
Trans. Microwave Theory Tech., vol. 39, pp. 1553-1565, Sept. 1991.
[25] H. Y. Lee and T. Itoh, "Phenomenological loss equivalence method for planar quasitem transmission lines with thin normal conductor or superconductor," IEEE Trans.
Microwave Theory Tech., vol. 37, pp. 1904-1909, Dec. 1989.
[26] 0. R. Baiocchi, K. S. Kong, H. Ling, and T. Itoh, "Effects of superconducting losses
in pulse propagation on microstrip lines," IEEE Microwave Guided Wave Lett., vol. 1,
pp. 2-4, Jan. 1991.
BIBLIOGRAPHY
159
[27] T. E. Van Deventer, P. B. Katchi, J. Y. Josefowicz, and D. B. Rensch, "High frequency
characterization of high-temperature superconducting thin film lines," In IEEE/MTT-S
Int. Microwave Symp. Digest 1990, pp. 285-288, Dallas, TX, May 1990.
[28] D. M. Sheen, S. M. Ali, D. E. Oates, R. S. Withers, and J. A. Kong, "Current distribution, resistance, and inductance for superconducting strip transmission lines," IEEE
Trans. Applied Superconduct., vol. 1, pp. 108-115, Jun. 1991.
[29] J. Kessler, R. Dill, and P. Russer, "Field theory investigation of high-T, superconducting
coplanar waveguide transmission lines and resonators," IEEE Trans. Microwave Theory
Tech., vol. 39, pp. 1566-1574, Sept. 1991.
[30] S. M. Ali, "Topics in electrodynamics," Lecture notes, Massachusetts Institute of Technology, Cambridge, MA, 1988.
[31] J. F. Kiang, S. M. Ali, and J. A. Kong, "Integral equation solution to the guidance
and leakage properties of coupled dielectric strip waveguides," IEEE Trans. Microwave
Theory Tech., vol. 38, pp. 193-203, Feb. 1990.
[32] L. Krusin-Elbaum, R. L. Greene, F. Holtzberg, and A. P. Malozemoff, "Direct measurement of the temperature-dependent magnetic penetration depth in YBaCuO crystals,"
Phys. Rev. Lett., vol. 62, pp. 217-220, Jan. 1989.
[33] G. J. Dolan, F. Holtzberg, and T. R. Dinger, "Anisotropic vortex structure in YBCO,"
Phys. Rev. Lett., vol. 62, pp. 2184-2187, May 1989.
[34] J. K. Lee and J. A. Kong, "Dyadic Green's functions for layered anisotropic medium,"
Electromagnetics,
vol. 3, pp. 111-130, 1983.
[35] S. M. Ali, T. M. Habashy, and J. A. Kong, "Spectral-domain dyadic Green's function
in layered chiral media," J. Opt. Soc. Am. A, vol. 9, pp. 413-423, Mar. 1992.
[36] R. A. York, R. C. Compton, and B. J. Rubin, "Experimental verification of the 2Drooftop approach for modeling microstrip patch antennas," IEEE Trans. Antennas
Propagat., vol. 39, pp. 690-694, May 1991.
[37] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines and Slotlines. Norwood, MA:
Artech House, 1979, p. 131.
[38] J. M. Pond, J. H. Claassen, and W. L. Carter, "Kinetic inductance microstrip delay
lines," IEEE Trans. Magn., vol. MAG-23, pp. 903-906, Mar. 1987.
[39] J. M. Pond, J. H. Claassen, and W. L. Carter, "Measurements and modeling of kinetic
inductance microstrip delay lines," IEEE Trans. Microwave Theory Tech., vol. MTT-35,
pp. 1256-1262, Dec. 1987.
[40] R. E. Matick, Transmission Lines for Digital and Communication Networks. New York:
McGraw-Hill,
1969.
[41] J. Meixner, "The behavior of electromagnetic fields at edges," IEEE Trans. Antennas
Propagat., vol. AP-20, pp. 442-446, July 1972.
[42] J. Geisel, K.-H. Muth, and W. Heinrich, "The behavior of the electromagnetic field at
edges of media with finite conductivity," IEEE Trans. Microwave Theory Tech., vol. 40,
pp. 158-161, Jan. 1992.
BIBLIOGRAPHY
160
[43] R. E. Collin, Field Theory of Guided Waves, 2nd edition. Piscataway, NJ: IEEE Press,
1991.
[44] L. N. Cooper, "Current flow in thin superconducting films," Proceedings of the Seventh
International Conference on Low Temperature Physics. Toronto: University Press, 1961,
pp. 416-418.
[45] P. M. Marcus, "Currents and fields in a superconducting film carrying a steady current," Proceedings of the Seventh International Conference on Low Temperature Physics.
Toronto: University Press, 1961, pp. 418-421.
[46] E. H. Rhoderick and E. M. Wilson, "Current distribution in thin superconducting films,"
Nature, vol. 194, pp. 1167-1168, June 1962.
[47] A. Shadowitz, "Rectangular type-II superconducting wires carrying axial current,"
Phys. Rev. B, vol. 24, pp. 2841-2843, Sept. 1981.
[48] S. Tahara, S. M. Anlage, J. Halbritter, C.-B. Eom, D. K. Fork, T. H. Geballe, and
M. R. Beasley, "Critical currents, pinning, and edge barriers in narrow YBa2 Cu3 07-_
thin films," Phys. Rev. B, vol. 41, pp. 11203-11208, June 1990.
[49] C.-W. Lam, S. M. Ali, and T. P. Orlando,
"Full-wave analysis of microstrip
lines on
anisotropic substrates using the finite-difference time-domain method," to be submitted
to IEEE Trans. Microwave Theory Tech.
[50] R. Marques and M. Horno, "Dyadic Green's function for microstrip-like transmission
lines on a large class of anisotropic substrates,"
1986.
IEE Proc. H, vol. 133, pp. 450-454, Dec.
[51] G. L. Matthaei and G. L. Hey-Shipton, "Novel staggered resonator array superconducting 2.3-GHz bandpass filter," IEEE MTT-S Int. Microwave Symp. Dig., pp. 1269-1272,
1993.
[52] G. L. Matthaei and G. L. Hey-Shipton,
"High temperature
superconducting
8.45-GHz
bandpass filter for the deep space network," IEEE MTT-S Int. Microwave Symp. Dig.,
pp. 1273-1276, 1993.
[53] A. Fathy, D. Kalokitis, V. Pendrick, E. Belohoubek, A. Pique, and M. Mathur, "Superconducting narrow bandpass filters for advanced multiplexers," IEEE MTT-S Int.
Microwave Symp. Dig., pp. 1277-1280, 1993.
[54] R. Weigel, M. Nalezinski, A. A. Valenzuela, and P. Russer, "Narrow-band YBCO superconducting parallel-coupled coplanar waveguide band-pass filters at 10 GHz," IEEE
MTT-S Int. Microwave Symp. Dig., pp. 1285-1288, 1993.
[55] C.-W. Lam, "Modeling of superconducting transmission lines and three-dimensional
high speed interconnects," Ph. D. Dissertation, Dept. EECS, Massachusetts Institute
of Technology, Cambridge, MA, 1993.
[56] EM is a product of Sonnet Software, Inc., Liverpool, NY 13090.
[57] G. L Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching
Networks, and Coupling Structures. New York: McGraw-Hill, 1964.
BIBLIOGRAPHY
161
[58] W. G. Lyons, R. S. Withers, J. M. Hamm, R. H. Mathews, B. J. Clifton, P. M.
Mankiewich, M. L. O'Malley, and N. Newman, "High-frequency analog signal processing
with high-temperature superconductors," OSA Proc. PicosecondElectronics Optoelectronics, vol. 9, pp. 167-173, 1991.
[59] IE3D is a product of Zeland Software, Inc., San Francisco, CA 94112.
[60] M. Kirschning and R. H. Jansen, "Accurate wide-range design equations for the
frequency-dependent characteristic of parallel coupled microstrip lines," IEEE Trans.
Microwave Theory Tech., vol. MTT-32, pp. 83-90, Jan. 1984.
[61] H. M. Altschuler and A. A. Oliner, "Discontinuities in the center conductor of symmetric
strip transmission line," IRE Trans. Microwave Theory Tech., vol. MTT-8, pp. 328-339,
May 1960.
[62] R. Garg and I. J. Bahl, "Microstrip discontinuities," Int. J. Electron., vol. 45, pp. 81-87,
Jul. 1978.
[63] E. O. Hammerstad, "Equations for microstrip circuit design," 5th European Microwave
Conference, pp. 268-272, 1975.
[64] P. Silvester and P. Benedek, "Equivalent capacitances of microstrip open circuits," IEEE
Trans. Microwave Theory Tech., vol. MTT-20, pp. 511-516, Aug. 1972.
[65] R. H. Jansen and N. H. L. Koster, "Accurate results on the end effect of single and
coupled microstrip lines for use in microwave circuit design," Arch. Elek. Ubertragung
(AEU), vol. 34, pp. 453-459, Nov. 1980.
[66] R. H. Jansen, "Hybrid mode analysis of the end effects of planar microwave and
millimeter-wave transmission lines," IEE Proc. H, vol. 128, pp. 77-86, Apr. 1981.
[67] M. Kirschning, R. H. Jansen, and N. H. L. Koster, "Accurate model for open end effect
of microstrip lines," Electron. Lett., vol. 17, pp. 123-125, Feb. 1981.
[68] M. Kirschning, R. H. Jansen, and N. H. L. Koster, "Coupled microstrip parallel-gap
model for improved filter and coupler design," Electron. Lett., vol. 19, pp. 377-379, May
1983.
[69] S. S. Bedair and M. I. Sobhy, "Open-end discontinuity in shielded microstrip circuits,"
IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 1107-1109, Oct. 1981.
[70] S. L. March, "Empirical formulas for the impedance and effective dielectric constant
of covered microstrip for use in the computer-aided design of microwave integrated
circuits," 11th European Microwave Conference, pp. 671-676, 1981.
162
Publications from Thesis
1. L. H. Lee, W. G. Lyons, and T. P. Orlando, "Open-end model for edge-coupled shielded
microstrip Lines," submitted to IEEE Trans. Microwave Theory Tech.
2. L. H. Lee, T. P. Orlando, and W. G. Lyons, "Current distribution in superconducting
thin-film strips," IEEE Trans. Applied Superconduct., vol. 4, pp. 41-44, Mar. 1994.
3. L. H. Lee, W. G. Lyons, T. P. Orlando, S. M. Ali, and R. S. Withers, "Full-wave analysis
of superconducting microstrip lines on anisotropic substrates using equivalent surface
impedance approach," IEEE Trans. Microwave Theory Tech., vol. 41, pp. 2359-2367,
Dec. 1993.
4. L. H. Lee, W. G. Lyons, T. P. Orlando,
"Microwave power-handling
capability
of su-
perconducting planar transmission lines and resonators," Lincoln Laboratory Quarterly
Technical Report on Solid State Research, no. 4, 1993.
5. L. H. Lee, S. M. Ali, W. G. Lyons, R. S. Withers,
and T. P. Orlando,
"Full-wave
analysis of superconducting microstrip lines on sapphire substrates," IEEE MTT-S
Int. Microwave Sym. Dig., pp. 385-388, 1993.
6. L. H. Lee, S. M. Ali, W. G. Lyons, D. E. Oates, and J. D. Goettee,
"Analysis of su-
perconducting transmission-line structures for passive microwave device applications,"
IEEE Trans. Applied Superconduct., vol. 3, pp. 2782-2787, Mar. 1993; presented at
Applied Superconductivity Conference, Chicago, IL, Aug. 1992.
7. L. H. Lee, S. M. Ali, and W. G. Lyons, "Full wave characterization
of high-T, super-
conducting transmission lines," IEEE Trans. Applied Superconduct., vol. 2, pp. 49-57,
Jun. 1992.
8. L. H. Lee, G. Arjavalingam,
S. M. Ali, and T. R. Dinger, "Hybrid-mode
analysis of
coplanar strips and coplanar waveguides," presented at Topical Meeting on Electrical
Performance of Electronic Packaging, Tucson, Az, Apr. 1992.
9. L. H. Lee, S. M. Ali, and J. A. Kong, "Hybrid-mode
analysis of superconducting
transmission lines," presented at Progress in Electromagnetics Research Symposium,
Cambridge, MA, July 1991.
163