close

Вход

Забыли?

вход по аккаунту

?

A hybrid numerical technique for analysis and design of microwave integrated circuits

код для вставкиСкачать
A Hybrid Numerical Technique for Analysis and Design
of Microwave Integrated Circuits
B.S., Tsinghua University, 1985
M.S., Tsinghua University, 1986
A Dissertation Submitted in Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY
in the Department of
Electrical and Computer Engineering Department
We accept this Dissertation as conforming
to the required standard
0 Dr. R. Vahldieck. Sunervisor
Dr. J.'Bornemadn. Departmental Member
Dr. W. Hoefer, Departmental Member
Dr. C^Efa^ley, Outside Member
Dr. V. K. Tripathi, Exjen^l'Exanrlhen.Oregbn State Univ.)
© Ming Yu, 1995
UNIVERSITY OF VICTORIA
A ll rights reserved. This Dissertation may not he reproduced
in whole or in part by mimeograph or other means,
without the permission o f the author.
1 ^ 1
National Library
of C anada
Biblioth&que nationale
du C anada
Acquisitions and
Bibliographic Services Branch
Direction d e s acquisitions et
d e s services bibliographiques
3 f5 Wellington Street
Ottawa, Ontario
K1A0N4
395, rue Wellington
Ottawa (Ontario)
K1A0N4
Your file
O ur file
Votre reference
Notre reference
The author has granted an
irrevocable non-exclusive licence
allowing the National Library of
Canada to reproduce, loan,
distribute or sell copies of
his/her thesis by any m eans and
in any form or format, making
this thesis available to interested
persons.
L’auteur a accorde une licence
irrevocable et non exclusive
permettant a la Bibliotheque
nationale
du
Canada
de
reproduire, prefer, distribuer ou
vendre des copies de sa these
de quelque maniere et sous
quelque forme que ce soit pour
mettre des exemplaires de cette
these a la disposition des
personnes interessees.
The author retains ownership of
the copyright in his/her thesis.
Neither the thesis nor substantial
extracts from it may be printed or
otherwise reproduced without
his/her permission.
L’auteur conserve la propriete du
droit d’auteur qui protege sa
these. Ni ia these ni des extraits
substantiels de celle-ci ne
doivent etre
imprimes
ou
autrement reproduits san s son
autorisation.
ISBN
Canada
0-612-13726-0
IA Z h JC r
Name
Y *
Dissertation Abstracts International is arran g ed by broad, general subject categories. Please select the one subject which most
nearly describes the content of your dissertation. Enter the corresponding four-digit code in the spaces provided.
E-/&C fr~0»i 'cc,
mm u m i
X Ar/e c fn '
SUBJECT CODE
SUBJECT TERM
Subject Categories
THH HUMANITIES AND SOCIAL SCIKKCIS
COKMUKKATIONS AND THE ARTS
Architecture......................................0 7 2 9
Art History........................................ 0 3 7 7
C in e m a
.............................
0900
D a n c e ............................................ 0 3 7 0
Fine A r ts ........................................... 0 3 5 7
Information S cience........................ 0 7 2 3
Journalism.........................................0391
Library S cien ce..................
0399
M ass Communications...................0 7 0 8
M u sic................................................. 0 4 1 3
S peech Communication...............0 4 5 9
T h e a te r............................................. 0 4 6 5
EDUCATION
G e n e r a l............................................ 0 5 1 5
Administration................................. 0 5 1 4
Adult and Continuing.................... 0 5 1 6
Agricultural...................................... 0 5 1 7
A rt.......................................................0 2 7 3
Bilingual and Multicultural
0282
B u sin ess............................................ 0 6 8 8
Community C ollege ................... 0 2 7 5
Curriculum and Instruction
0727
Early Childhood...............................0 5 1 8
Elem entary....................................... 0 5 2 4
F in a n ce............................................. 0 2 7 7
G uidance and C ounseling
...0 5 1 9
H ea lth ................................................0 6 8 0
H ig h e r .....................
0745
History o f
.......................0 5 2 0
Hom e Econom ics............................ 0 2 7 8
Industrial.....................
..0 5 2 1
Language and Literature............... 0 2 7 9
M athem atics.....................................0 2 8 0
M u sic
..................
.0 5 2 2
Philosophy o f .........................
0998
Physical.,...........................................0 5 2 3
Psychology.....................................0525
R eading......................................... 0535
Religious........................................ 0527
Sciences......................................... 0714
Secondaiy..................
0533
Social Sciences............................. 0534
Sociology o f .................................,0340
Special..........................
0529
Teacher Training........................... 0530
Technology
.................... 0710
Tests anaM easurem ents.............. 0288
Vocational.............................
0747
PHILOSOPHY, RELIGION AND
THEOLOGY
LANGUAGE, LITERATURE AND
LINGUISTICS
American Studies
........ 0323
Anthropology
0324
Archaeology.....................
C ultural
........................0326
Physical....................................0327
Business Administration
G eneral................
0310
Accounting
.........................0272
Banking...................................0 7 7 0
M anagem ent.......................... 0454
Marketing
..................... 0338
Canadian S tudies.........................0385
Economics
G eneral......................
0501
Agricultural..............................0503
Commerce-Business............... 0505
Finance
...................0508
History............................
0509
L abor....................................... 0510
, Theory................. ................... 0 5 1 1
Folklore
.............................. 0358
G eography
.........
0366
G erontology.................................. 0351
History
G en eral
........................... 0578
Language
G en eral....................................0679
Ancient...
.......................... 0289
Linguistics................................0290
Modern
..........
C291
Literature
G en eral....................................0401
Classical ..............................0294
Com parative
..................0295
M edieval................................. 0297
M o d ern ....................................0290
A frican.....................................0316
A merican.......................
0591
A sia n ....................................... 0305
Canadian (English)................0352
Canadian (French).................0355
English
.........................0593
Germanic,................................0311
Latin American.......................0312
Middle Eastern........................0315
R om ance.................................0313
Slavic and East European
0314
Phijosophy......................................0422
Religion
............... 0318
G eneral
Biblical Studies ...............,..0321
C lergy......................................0319
History o f ................................. 0320
Philosophy o f ..........................0322
Theology........................................ 0469
SOCIAL SCIENCES
Ancient.....................................0579
M edieval.................
0581
M o d ern ................................. 0582
Black........................................ 0328
African ................................. 0331
Asia, Australia and O ceania 0332
C a n a d ia n
...... 0334
European................................. 0335
Latin American .................... 0336
Middle Eastern
................ 0333
United States
.. .0337
History of Science.........................0585
Lav/./.............................................. 0398
Political Science
General
............................ .0615
International Law and
Relations.........................
0616
Public Administration.............0617
Recreation................
0814
Social W o rk
.........
0452
Sociology
G eneral
..........
0626
Criminology and Penology . 0627
Demography
. .0938
Ethnic and Racial Studios......0631
Individuc1and Family
Studies
.................0628
Industrial and Labor
Relations...............................0629
Public and Social Welfare .,,.0630
Social Structure and
Development.,,.................. 0700
Theory and M othods.............. 0344
Transportation
..................0709
Urban and Regional Planning
0999
Women's Studies
........... ...0453
THE SCIENCES AND ENGINEERING
BIOLOGICAL SCIENCES
Agriculture
G en eral......................
0473
A gronom y
.......
0285
Animal Culture and
Nutrition
.......
0475
Animal Pathology .................0476
Food Science ana
Technology .........................0359
Forestry ana W ildlife
,.0 4 7 8
Plant Culture
.................... 0479
Plant Pathology
.............. 0480
Plant Physiology
..... .,...0 8 1 7
Range M anagem ent
....... 0777
W ood Technology ..............,0 7 4 6
^ n e r a l ....................................0306
A natom y
............... ,..,0 2 8 7
Biostalislics..............................0308
Botany............................
0309
Cell ....................................... 0379
Ecology....................................0329
Entomology.,...................... 0353
G enetics
.......................0369
Limnology ............................... 0793
M icrobiology.......................... 0 410
M olecular................................ 0 307
Neuroscience...............
0317
O ceanography........................0 4 16
Physiology
.........................0433
Radiation
.......................... 0821
Veterinary Science..................0778
Zoology....................................0472
Biophysics .
G en eral....................................0786
M edical.........................
0760
EARTH SCIENCES
Biogeochemistry
................ 0 425
G eochem istry................................ 0996
G eo d esy
................................ 0370
G eology..........................................0372
Geophysics ....................................0373
Hydrology ....................................0388
Mineralogy.....................................0411
Paleobotany ................................. 0345
Paleoecology................................. 0426
Paleontology...................................0418
Paleozoology
......................... 0985
Palynology.................................,..0427
Physical G eography..................... 0368
Physical O cean o g rap h y .............. 0415
HEALTH AND ENVIRONMENTAL
SCIENCES
Environmental Sciences............... 0768
Health Sciences
G en eral....................................0566
Audiology................................0300
Chemotherapy
................ 0992
Dentistry.................................. 0567
Education............................ ...0350
Hospital M anagement............ 0769
Human Development ......... 0758
Immunology.....................
0982
Medicine and S urgery...........0564
Mental Health .......................0347
Nursing ................................. 0569
Nutrition.................................. 0570
Obstetrics and Gynecology ..0380
Occupational Heal‘h a n d
Therapy
................... 0354
O phthalmology..................... 0381
Pathology................................0571
Pharmacology
...... ......... 0419
Pharm acy
.............. . .0572
Physical th e r a p y ....................0382
Public Health........................... 0573
Radiology........................
0574
Recreation............................. 0575
Speech Pathology...................0460
Toxicology
.....................0383
Home Economics .........................0386
PHYSICAL SCIENCES
P u c e S c ie n c e s
Chemistry
G eneral
........................0485
Agricultural............................. 0749
Analytical................................0486
Biochemistry ......................... 0487
Inorganic...................
0488
N uclear ................................. 0738
O rganic....................................0490
Pharmaceutical ..................... 0491
Physical................................... 0494
Polymer
........
0495
Radiation
.................... 0754
Mathematics.................................. 0405
Physics
G eneral................................... 0605
Acoustics................................. 0986
Astronomy and
Astrophysics.........................0606
Atmospheric Science.............. 0608
A tom ic
....................0748
Electronics and Electricity
0607
Elementary Particles end
High Energy.........................0.7-98
Fluid and Plasm a
. .0759
M olecular
................ 0609
N uclear....................................0610
Optics ................................. 0757
Radiation................................. 0756
Solid S ta le
............0611
Statistics....................................... 0463
A p p lie d S c ie n c e s
Applied M ec h a n ic s.....................0346
Computer Science..................... 0984
Engineering
G eneral............
0537
Aorospaco ......
0538
Agricultural ..
0539
Automotive......
0540
Biomedical......
.0 5 4 1
Chem ical.........
0542
Civil
.
- .....................................
0543
-'jf loctronics and Electrical . 0544
Hoal and Thermodynamics.. 0348
Hydraulic.................................0545
Industrial .................................0546
M arino
...................... 0547
Materials S c ie n c a ................ 0794
M echanical......................
0548
M etallurgy..............................0743
Mining
............................ 0551
N uclear................................. 0552
Packaging ............................. 0549
Petroleum ....................
0765
Sanitary and Municipal
0554
System Scionco..................... 0790
Gootochnoloqy
0428
Operations Research . .
0796
Plastics Technology
079,5
Toxlilo Technology ...................
0994
PSYCHOLOGY
G e n e r a l.....................................
B o h av io ra l.............................
Clinical . .
.
Developmental
Experimental
Industrial
Personality
Physiological
.
P sychobiology ..........................
Psychometrics
Social
0621
0384
0622
0620
,062.)
0624
0625
0989
0349
0632
0451
Supervisor: Dr. R. Vahldieck
ABSTRACT
Miniature Hybrid Microwave Integrated Circuits (MHMIC’s) in conjunction with Mono ­
lithic MIC’s (MMIC’s) play an important role in modem telecommunication systems. Ac­
curate, fast and reliable analysis tools are crucial to the design of MMIC’s and MHMIC’s.
The space-spectral domain approach (SSDA) is such a numerically efficient method, which
combines the advantage of the one-dimensional method of lines (MoL) with that of the one­
dimensional spectral-domain method (SDM). In this dissertation, the basic idea of the
SSDA is first introduced systematically. Then, a quasi-static deterministic variation o f the
SSDA is developed to analyze and design low dispersive 3-D MMIC’s and MHMIC’s. Sparameters and equivalent circuit elements for discontinuities are investigated. This in­
cludes air bridges, smooth transitions, open ends, step in width and gaps in coplanar
waveguide (CPW) or microstrip type circuits. Experimental work is done to verify the sim­
ulation.
The full-wave SSDA is a more generalized and he'd theoretically exact numerical tool
to model also dispersive circuits. The new concept of self-consistent hybrid boundary con­
ditions to replace the modal source concept in the feed line is used here. In parallel, a de­
terministic approach is developed. Scattering parameters for some multilayered planar dis­
continuities including dispersion effect are calculated to validate this method.
Examiners
y
--------------------------------------------------- -— —
—\
fp r . R. Vahldiqck, Supervisor
Dr. J. Sfo^femann, departmental Member
Dr. W. Hoefer, Departmental Member
Dr. C. Bra$Ht?y, Outside Member
Dr. V. K. Tripathi,j2$eiHftH3Xarniner (Oregon State Univ.)
Table of Contents
Table of Contents
List of Figures
List of Tables
Acknowledgments
1 Introduction
2
iii
v
vii
viii
I
1.1
Background and G o a ls ....................................................................................... I
1.2
Organization of This Dissertation..................................................................... 7
The Space-Spectral Domain Approach
8
2.1
The Spectral Domain M e th o d .............................................................................8
2.2
The Method of Lines......................................................................................... 10
2.3
The Relationship Between the SDM and MoL.............................................. 14
2.4
The Space-Spectral Domain Approach........................................................... 16
2.4.1 SDM in x-direction.........................
17
2.4.2 MoL in z-direction
.................................................................... 19
2.4.3 The Eigenvalue Solution of a Resonator P rob lem ............................ 25
3 The Quasi-Static SSDA
28
3.1
Why Quasi-static ? .....................
3.2
The Quasi-static SSD A .................................................................................... 28
3.3
On the Nature of the S S D A ............................................................................ 39
4The Full-wave SSDA
28
43
4.1
Eigenvalue A p p ro a ch ....................
43
4.2
Deterministic A p p ro a ch ................................................................................ 48
IV
5 Numerical and Experimental Results
54
5.1
Convergence Analysis o f Quasi-static S S D A ............................................... 54
5.2
Sim ulation Results o f Quasi-static SSDA . .
5.3
Convergence Study o f Full-wave S S D A ........................................................ 64
5.4
Simulation Results o f Full-wave S S D A ........................................................ 64
5.5
Experim ental R e s u l t s ........................................................................................67
6 Conclusion
55
7J
6.1
C o n tr ib u tio n s ..................................................................................................... 71
6.2
Future W o r k ..........................................................................................................72
Bibliography
75
Appendix
79
v
List of Figures
Figure 1.1
Exam ple for an M M IC Circuit
Figure 1.2
M HM IC discontinuities
Figure 1.3
A typical planar discontinuity
Figure 2.1
A shielded microstrip line
Figure 2.2
Cross-section view o f a microstrip line
Figure 2.3
A microstrip discontinuity in a resonator enclosure
Figure 3.1
Planar circuit discontinuities
Figure 3.2
Discretization of a C PW discontinuity
Figure 3.3
The equivalent circuit
Figure 3.4
A CPW Air Bridge
Figure 3.5
Configuration of general transmission line
Figure 4.1
A n eigenvalue approach
Figure 4.2
A deterministic approach
Figure 5.1
Convergence analysis o f the Quasi-static SSDA (w lli= l, cr~9/>)
Figure 5.2
Capacitance o f microstrip open ends.
Figure 5.3
Equivalent capacitance o f a microstrip gap discontinuity. w /h= I,
h=0.508mm, er=8.875
Figure 5.4
S-parameters o f a m icrostrip step. u q -1m m , u’2=0.2.5//////, u*//;=- /,
8 ,- / 0 .
Figure 5.5
Equivalent capacitance o f a CPW open e*id. er=9.6, h -■■0.6.15, ///
d - l , d=w+2.s
Figure 5.6
S-parameter of a CPW airbridge. w ~ l5\ini,s= IO \im , l=.1\Lrn,
h=200[Lm, b=1\un
Figure 5.7
S-parameters o f a CPW step. w^-0,4m m , w j= 0.lm m , Vj -•■0.1mm,
\\'2=0.4mm, er=9.<V, h=0.254mm
Figure 5.8
Equivalent capacitance o f a CPW gap, d {—2,Vj + it'| ,
vi
d2=2s2+u'2,£r=9.<S’, h=0.635mm, W[lh=0.2, u ’j / d 1 =w2/cJ2-0.56,
H'2/ir , =3
Figure 5.9
Equivalent capacitance o f a microstrip and a C PW step/taper. For
CPW, W[=0.8mm, s [=0.1 mm, w2-0.2mm , s2==0.6mm, er=9.6,
li=0.254mm. For microstrip, w \= lm m , w2-0 .2 5 m m , z r=9.6,
h=0.25rnm
Figure 5.10
S-parameters o f a CPW airbridge versus bridge length /.
w=().3mm, s=0.1m m , b=3]im , er=9.6, h=().254mm.
Figure 5.11
Frequency dependent behavior of microstrip step
Figure 5.12
Frequency dependent behavior of CPW step
Figure 5.13
Convergency behavior o f the full-wave SSDA
Figure 5.14
Full-wave S-parameters o f a microstrip step
Figure 5.15
Full-wave S-parameters o f a microstrip step
Figure 5.16
S-parameters for a cascaded step discontinuity separated by a
transmission line o f length 1. w [=0.4 mm, w o = 0 .2 m m , \\'2=0.8m m ,
er=3.<S\ h=0.25mm.
Figure 5.17
Measured and computed S-param eters o f a CPW gap
Figure 5.18
Measured and computed S-param eters o f end-coupled CPW reso­
nators
Figure 5.19
Measured and computed S-param eters o f a CPW step discontinui tv
Figure 5,20
S-parameters o f a CPW end-coupled filter. w = 0 .2 , s = 0 ,1 5 , g a p
width: 25.4\un, resonator length: 2mm.
Figure 6.1
Future application: electro-optic modulator
Figure 6.2
An in-line 3-port discontinuity
Figure 6.3
Arbitrary multi-port discontinuity
vii
List of Tables
Table 4.1.
B oundary conditions46
viii
Acknowledgments
The author wishes to express his acknowledgm ents to his thesis supervisor, Dr. Ruediger Vahldieck, for his guidance, encouragem ent and invaluable suggestions through­
out the course o f this thesis.
Financial support for this research by Dr. R. Vahldieck (through NSERC), Science
Council of British Columbia (through GREAT Award) and M PR Teltech Ltd. is also
gratefully acknowledged. In particular, I would like to thank Dr. J. Fikart and H.
Minkus, MPR Teltech, for the fabrication o f the MHM IC prototypes, w hich were used
to verify the SSDA results.
The author also wishes to extend his thanks to Dr. K. Wu for his invaluable sugges­
tions. The author is also grateful to his colleagues at the Laboratory for Lightw ave
Electronics, M icrowave and Com m unications (LLiMiC), University o f Victoria for
their support and discussion.
Last, but by no means least, the author wishes to express his thanks to his family, espe­
cially to his wife, M ei Li, for their support and encouragement.
I
Chapter 1
Introduction
1.1 Background and Goals
M iniaturization o f microwave circuits is essential ir Mic evolution of modern com ­
m unications system s. In analogy to the miniaturization that has taken place in VLSI (Very
Large Scale Integrated Circuits), M onolithic Microwave integrated Circuits (M M IC ’s)
but also M iniature Hybrid M icrowave Integrated Circuits (M H M IC’s) combine a steadily
grow ing num ber o f microwave components on smaller and sm aller chip real estate.
M M IC ’s as shown in Figure 1.1 [14] are very expensive in the fabrication and are only
justified for large volum e applications. M H M IC ’s are a hybrid technology that is in par­
ticular suitable for small to medium volume applications. While M M IC’s require sem i­
conductor fabrication facilities (circuits are grown on CaAs), which arc capable to
integrate active devices like F E T ’s (Field Effect Transistors) in one process on the same
wafer, M H M IC ’s are grown on alumina substrate as shown in Figure 1.2, and active
devices are wire bonded into the chip in a final fabrication step. The latter technology is
less attractive for large volume applications because of the additional labor involves, but
offers better circuit perform ance and is less expensivu for small to medium applications.
Both M M IC ’s and M H M IC ’s play an important role in modern com m unication systems.
A serious bottleneck in both technologies is the lack of accurate, fast and reliable
design strategies. Although commercial design software is available, the num erical m eth­
ods used are either computationally very inefficient or inaccurate at higher frequencies.
Several fabrication cycles become necessary to trim the circuit so that it satisfies the
design requirem ents. This process is very expensive and tim e-consuming (scvcra1
months). To cut dow n on the processing time and cost, it is necessary to develop accurate
design algorithm s for M M IC ’s and M H M IC’s in order to achieve first-pass success.
2
In generai, numerical methods can be classified into two categories:
1. M ethods which use an eigenmode approach to describe the electrom agneticJicld
<’ g-,
•
Mode M atching M ethod (M M M ) 111
•
Spectral Domain M ethod (SDM ) f 2
o /
2, Methods which discretize a differential operator
•
Finite Difference M eihod (FDM) (6 j
•
M ethod o f Lines (MOL) [7]-[131
The first category of methods use orthogonal modes or basis functions to expand the
field directly.
T he second category of methods is applied directly to either M axw ell’s or the
H elm holtz equations. The first and second differential operator are approxim ated by
finite differences.
Both category of methods can be subdivided further into
Quasi-static techniques which calculate equivalent network parameters.
Microwave circuits are described by lumped elements like capacitors
and inductor' . which are assumed to be constant over frequency.
Full-wave techniques which describe the electromagnetic f eld directly
fro m M axw ell’s equations. Circuits are considered from the fie ld theory
point o f view and may be described by S-parameters, which include
3
the uite faction o f fundam ental and hi»her order modes at discontinuities.
Figure 1.1 Example for an MM1C Circuit
4
COPLANAR
WAVCGUIDE
AIR BRIOCE
COPLANAR
WAVEGUIDE
AIR BRIDGE
SLOTLINE
(a ) C o p la n a r W a v e g u id e T J u n c t i o n
a ir
( d ) S l o t L i n t C o p la n a r J u n c t i o n
br id g e
SLOTUNE
(b ) C o p la n a r W a v e g u id e S l o t l i n e J u n c t i o n
(e ) S lo tlin e T J u n c tio n
COPLANAR
WAVEGUIDE
COPLANAR
WAVEGUIDE
AIR BRIDGE
AIR BRIDGE
COUPLED
SLOTLINE
• SLOTLINE
(c ) C o p la n a r W a v e g u id e / S l o tli n c T r a n s i t i o n
MHUIC
( f ) MliMIC E m b e d d e d I n to
DIODE'
a CPWG S t r u c t u r e
AIR BRIDGE 3
AIR BRIDGE I
BIAS
AIR BRIDGE 2
MIM CAPACITOR
COPtANAR
WAVEGUIDE
-SLOTLINE
' ' — OUTPUT
(g ) C ir c u it C o n f i g u r a t i o n o f U n i p l a n a r MIC B a l a n c e d M u lti p lie r
Figure 1.2 MHMIC discontinuities
5
Quasi-static numerical techniques are traditionally faster than full-wave techniques,
in particular, on the serial machines widely used today. These methods do not benefit
from the availability of parallel processors. For some applications, their accuracy can
rival that o f full wave techniques and, therefore, they arc very useful engineering design
tools.
Theoretically speaking, a quasi-static approach only works at zero frequency. H ow ­
ever, as long as the dimensions of circuits are small compared with the wavelength and
the dispersion of the transmission line system is weak or non-existing, quasi-static m eth­
ods cun work up to the millimeter-wave range. A large number o f comm ercially avail­
able software is built on quasi-static methods. In most cases the equivalent elem ent
values derived from quasi-static methods are assumed to be constant over the frequency.
Furthermore, it is assumed that the discontinu,ties for which they are derived do not radi­
ate or interfere with each other. This assumption becomes invalid the closer microwave
elem ents are placed on the chip. In this case the predicted performance of the M M IC ’s or
M H M IC ’s may deviate significantly from the required performance.
For structures, or frequencies, at which quasi-static methods do not provide accurate
results, either because the circuit density is too high or dispersion effects arc too signifi­
cant, full-w ave modelling of microwave and millimeter-wave circuits becomes neces­
sary. In this dissertation, a generalized Space-Spectral Domain Approach is first
introduced that is suitable for this task. Secondly, a new quasi-static deterministic tech­
nique w ill be presented. Finally, two SSDA full-wave algorithms will be discussed.
Typical generalized full-v/a -e approaches are, e.g., the Finite Difference M ethod
(FDM) or Transmission Line Matrix (TLM) fl6 j method. These methods start directly
from M axw ell’s equations with very little approximation and virtually no analytical pre­
processing. They have almost no structural limitations and provide a high degree o f accu­
racy. B eing very flexible, they often require large amounts o f computer memory and long
com puter run-time, at least on most of today’s available engineering workstations. To
speed these methods up, parallel processor machines are required which, for som e time
to come, will not be commonly used in engineering laboratories because of the special
program languages necessary to fully take advantage of the potential of these m achines.
6
side view
top view
▼x
Figure 1.3 A typical planar discontinuity
W hen dealing with the analysis and design of M M IC ’s and M H M iC ’s, their quasipJanar structure, as shown in Figure 1.3, allows the use o f less generalized but com puta­
tionally more efficient techniques. In the past, the most suitable methods for 3-D planar
circuit analysis have been the Spectral Domain M ethod (SDM), the M ode M atching
M ethod (MM M ), and the M ethod of Lines (MoL) (full-wave and quasi-static).
In the SDM, the Fourier transform is taken along a direction parallel to the sub­
strate, and G alerkin’s teetmique is used to yield a hom ogeneous system of equations. To
determine the eigenvalue problem (propagation constant) in a planar circuit, the 1-D
SDM is well know n for its fast com putational algorithm and minimum m emory require­
ments. But the SD M also requires that the circuit discontinuities fit into an orthogonal
coordinate system and, especially, that the basis functions are chosen carefully. For 3-D
discontinuity analysis, the 2-D SDM requires usually a large number of two-dim ensional
basis functions w hich are not easy to chose and handle and which increase the com puta­
tion time significantly because o f potential convergence problem s [43].
The MoL is a space-frequertcy dom ain method sim ilar to the FDM but uses an
orthogonal transform. To treat 3-D discontinuities, the 2-D M oL is used, which dis­
cretizes the two spatial variables parallel to the substrate plane while an analytical solu­
tion is obtained in the direction perpendicular to the substrate plane. This method
requires only a two-dim ensional discretization for a general 3-D problem. The advantage
o f this method is its easy form ulation, sim ple convergence behavior and the fact that there
are no special basis functions necessary. The disadvantages o f the 2-D M oL is that satis­
7
fying all boundary conditions simultaneously for arbitrarily shaped circuits may be very
difficult or may require a very fine 2-D discretization. In short, when applying the SDM
or M oL to an arbitrary 3-D discontinuity problem, each method by itself encounters a
number o f serious problems which are inherent in the method.
To overcom e the inherent lim itation of each method, a new hybrid numerical
method has been developed by Wu and Vahldieck fl7]. This method is called the SpaceSpectral Dom ain Approach (SSDA) which combines the 1-D SDM and 1-D MoL. This
new m ethod elim inates the shortcomings of the conventional 2-D MoL and 2-D SDM and
takes advantage o f the attractive features associated with the 1-D SDM and the 1-D
MoL. T he SSDA is developed in particular for the analysis of arbitrarily shaped spatial 3D planar discontinuities.
In this dissertation, we first introduce the generalized SSDA concept which is
extended from the work of Wu and Vahldieck [17], Although this analysis can only be
applied to calculate resonant frequencies but not discontinuity S-paramctcrs, it is used to
explain the basic idea of the SSDA. On that basis, a new deterministic quasi-static SSDA
[18], [20] is presented followed by a full-w ave SSDA [19], which is aimed at the calcula­
tion o f S-param eters in structures supporting hybrid modes.
1.2 Organization of This Dissertation
C hapter 2 reviews the SDA and M oL and investigates their relationship, which
forms the basis o f the SSDA. The generalized SSDA is introduced in a planar resonator
problem which form s the basis o': this dissertation.
Chapter 3 introduces the quasi-static SSDA.
C hapter 4 introduces the full-wave SSDA.
C hapter 5 discusses simulation results and their experimental verification.
Chapter 6 concludes the dissertation.
Chapter 2
The Space-Spectral Domain Approach
In this chapter, the Spectral Domain M ethod (SDM), the M ethod o f Lines (M oL) as well
as the relationship betw een both m ethods are first reviewed. T he concept o f the SpaceSpectral Dom ain A pproach (SSDA) is introduced as a com bination o f the SDM and the
MoL. For a three-dim ensional (3-D) electrom agnetic problem, the SD M is applied to the
x-direction, the M oL is applied to the z-direction, and the analytical process is applied to
the y-direction (see Figure 2.1).
2.1 The Spectral Domain Method
Figure 2.1 is a 3-D version of Figure 1.3; it shows a typical planar circuit discontinuity in
a shielded box. In the Spectral Dom ain M ethod the Fourier transform is taken along the
x-direction for a 2-D problem and alw ays along the x- and z-direction for a 3-D problem .
The analysis in the Fourier transform dom ain was first introduced by Yamashita and M ittra [2] fo r c o m p u ta tio n o f the c h a ra c te ristic im pedance and th e p h a se v e lo c ity o f a
microstrip line based on a quasi-static approach. It is one of the m ost popular and w idely
used numerical techniques for planar circuits. Numerous publications can be found in the
literature, e.g., [ 3 - 5 ] .
For planar transmission line and discontinuity problems, the electric and m agnetic
fields E and /? are often written in term s o f scalar potentials
and ¥ 1 in a Cartesian
coordinate system, shown in Figure 2 .1 (this is called a TEZ/ T M Z form ulation)
VxVxl 4 'c
(2 . 1)
w here z is the unit v ecto r in z-direction.
e,h
- j2
^2
h satisfy the wave equation
c, h
02
(-,/!
+ L- ~ - + - • „
3A
3r
3.v
,2
k
=
+ k 'V ’
= 0
(2 .2 )
2
CO (0.8
A
y
/
Figure 2.1 A shielded microstrip line
The idea o f the SDM is to apply the Fourier transform along the x-diicction in
order to eliminate the space variable x and replace it with a spectral term a x
y
e,
h
=
c
h jax .
f M' c d x
(2.3)
Assuming the problem is a two-dimensional one in x- and v-dircction, with the
propagation constant p in z-direction, equation (2.2) yields
2
a y
e ,h
-,2 e, h
a \u
_2
+ — — P ij/
3y
e,/i
,2
+ k i|/
e ,h
_
=0
(2.4)
The above equation (2.4) can be further simplified as a one-dim ensional normal dif-
10
ferential equation
2 r2
2)
a + (3 - A' I q/
_
=0
(2.5)
By applying the boundary conditions (more details will be given in Section 2.4 ),
one finally obtains an algebraic equation in matrix form
( 2 .6 )
e and e. are the Fourier transforms of the electrical field in the
/
a
-
and r-direction.
and /\ are the Fourier transforms of the current in the a - and r-direction. The unknown
/ ( and / are expanded in terms o f known basis functions with unknown weighting coeffi­
cients. By applying G alerkin’s technique, the propagation constant and field distribution
can be found.
In summary, the SD M has several features:
•
Simple form ulation in the form of algebraic equations
•
Utilization o f a-priori (physical) knowledge of modes
•
Num erically efficient
The SDM is well known for its computational efficiency and m inimum memory
requirement for two-dim ensional problems (1-D SDM) because usually only a few basis
functions are needed. The SDM loses som e of its advantages when applied to spatial
three-dimensional discontinuities (2-D SDM ). In particular when these discontinuities
are arbitrarily shaped, it becomes generally a problem to find suitable basis functions and
to achieve reasonable convergence.
2.2 The Method of Lines
T he Method o f Lines (M oL) was first developed by mathematicians (7 J in order to solve
differential eq u ations. It w as applied to m icrow ave analysis and design problem s by
Pregla and co-authors [8 - 13].
The concept o f the M oL is as follows: for a given system of partial differential
It
equations, all but one of the independent variables are discretized to obtain a system o f
ordinary differential equations so that the whole space is represented by a r umber of
lines. This semi-analytical procedure is very useful in the calculation o f planar transmis­
sion line structures.
To demonstrate the basic steps of the M oL, consider the m icrostrip line cross-sec­
tion in Figure 2.2
x=L
Figure 2.2 Cross-section view o f a microstrip line
Equation (2.2) is to be solved here. T he discretization is done in the .v-direction as
shown above. The figure also shows that two separate line systems are used to represent
£1
lP
/i
and T . This shifting scheme has several advantages: the lateral boundary condi­
tions are easily fulfilled, it allows an optimal edge condition flOj, second order accuracy
[11] and simple matrix formulation.
Let the number o f ¥* and xVn lines in Figure 2.2 be equal to Ari, T he potentials on
all the lines are com bined to form a vector ^
and
respectively. Equation (2.2) can
then be rewritten as
-.2 r A t J l
dx
..2 rA f j l
dy
+L ^ _ +tV *
dz
.
= 0
(2.7)
The first derivative with respect to x is form ed as backward difference quotients for
'V1 and forward difference quotients for XV"
12
d*P'
r —
' • o a r ^ N 11'
(2.8)
>/J
with
1 ... 0
0
-1 ... 0 ...
£> =
(2.9)
1
0 ... 0 -1
In the difference operator [D] the lateral boundary conditions are included (here a
Dirichlet-N eum ann boundary condition is used as an example). The second derivatives
can also be represented by m eans o f the operator [D]
h
232^
Dxx
OA
(2. 10)
202T>/'
hX
ao T T ^ " DX
ox
.
Equation (2.7) can then be written as
32$ ’
(-2
_2
"=
h
;2W
( 2 . 11)
dy
w here
(2. 12)
c ,h
is the eigenvalue matrix and
7, c ,
h the eigenvector belonging to
h which
can be obtained analytically dependent on the different lateral boundary conditions 1111.
For exam ple, for the structure shown in Figure 2.2, the elem ents o f
~,e
.
in n
arc
N
7 ‘." = s,n/v7 T
(2.14)
Equation (2.12) is called the orthogonal transform because rf-n is a symmetrical
matrix and
is an orthogonal matrix. Equation ( 2 .11) is in the transform domain,
f'
which is sim ilar to equation (2.5) in the Fourier domain. However, the way to solve equa­
tion (2.5) and equation (2.11) are different in either techniques. By applying lateral
boundary conditions, a system equation similar to equation (2.6) can be obtained. Apply­
ing the orthogonal transform a new algebraic system equation can be derived in the (orig­
inal) spatial domain ([11] gives m ore details)
(2.15)
T he vector notation is used here to represent discretized quantities. Because equa­
tion (2.15) is in the spatial domain, it can be simplified by rem oving those lines which do
not pass through the m etallization at the interface
0 =
Z.
(2 . 16)
I,
T he propagation constant and field distribution can be calculated by solving the
root o f the determinant o f jzf] , where subscript r signifies that [Zr \ is a residual matrix.
For three-dimensional problem s also the z-variblc is discretized (2-D fvloL).
14
In summary, the 1-D MoL has the following features:
•
No basis function needed
•
Sim ple formulation and efficient calculation
•
No relative convergence phenomenon
However, similar to the SDM, when applied to three-dimensional problem s, the 2D M oL becomes numerically less efficient because o f the two-dimensional discretization.
2.3 The Relationship Eetween the SDM and MoL
From the previous sections, it is quite obvious that both m ethods, the SD M and M oL,
have so m e sim ila ritie s if one co m p ares equations (2 .3 ), (2.5) and e q u a tio n s (2.11),
(2.12). The follow ing analysis shows that the MoL is indeed related to the SD M and that
this relationship helps to com bine the advantages o f both m< ‘hods into one new m ethod,
the Space-Spectral Domain Approach.
T his becomes obvious if one rewrites equation (2.2) for the two-dim ensional trans­
m ission line problem (the superscripts o f T are removed w ithout loss o f generality)
^
+
dx2
ay2
= 0
(2.17)
y
w here (3 is the propagation constant.
In both the M oL and SDM , a transformation is perform ed in the .v-direction
MoL
¥=>$
#=>$
$ = [7] $
(2.18)
SDM
4' => \]f
fg = J
.0 0
w hich leads to a onc-dim ensional normal differential equation that corresponds to
equation (2.5) and equation (2.11).
From equation (2.18) one may deduce that the orthogonal transform in the M oL
represents a discrete Fourier transform in matrix form. A lthough in [11J some analysis is
provided to support this point, there is no clear explanation to prove that the M oL
schem e (discretization and orthogonal transform) and the SDM are truly identical. Fur­
thermore, also the connection between the SDM, the MoL and the SSDA has not been
investigated in detail. The following analysis is intended to fill this gap.
To dem onstrate the relationship between the SDM and the MoL, the structure in
Figure 2.2 is used again. (Dirichlet boundary condition is applied for the electric potcntial T
Q
at a - ( ) and at x=L ) The Fourier expansion in region
L
a
V = jT ^ s in axeLx
n
the
potential
n
L , /'= /
N. + 1
is
in
/-
/
0
If
=
/.] is written as
[0 ,
discretized
into
=:
-D O
N
(2.19)
or
points
in
the
x-direction,
i.e.
N:, and N: spectral terms are used,
N.
ii = i
e . in n
sinN +1
/ = I
N_
( 2 . 20 )
The subscripts are used to represent discrete quantities and discrete spectral terms.
If vectors are used to represent discretization, equation (2.20) can be rewritten as
\\i
where the elem ent of
=
( 2 .2 1 )
J'J
ne
.
in n
( 2 .22)
From here one may compare
M oL
SDA
From equation (2.14)
.if
r "
\U = I rV! t ] ' / ’
^
I.1 J
(2.23)
16
This shows that the MoL is a discrete SDM. The equivalence o f the M oL and the
SDM is established under the same finite discretization schem e. From another point o f
view, because the SDM (theoretically) uses an infinite num ber o f spectral term s, the
SDM gives an infinite number of precise eigenvalues and eigenfunctions which can be
solved using analytical transforms w hile the M oL yields a finite num ber o f approxim ate
eigenvalues and eigenfunctions which can be solved using finite discretization.
In summary the following properties are found com paring the SDM and M oL
•
The SD M and MoL are indeed related to each other.
•
Both o f them are numerically very efficient for 2-D problem s and less effi­
cient for 3-D problems.
Combining the MoL and SDM will take the advantage o f both m ethods to analyze
3-D problems more efficiently. This leads to the invention o f a new method called the
Space-Spectral Domain Approach (SSDA).
2.4 The Space-Spectral Domain Approach
This section describ es the basic p rin c ip le s o f the S p a c e -S p e c tra l D om ain A p p ro a c h
(SSDA). The SSD A was first introduced by Wu and Vahldieck f 17] and further developed
by the author together with Wu and Vahldieck [18 - 20]. First, a generalized introduction
is given for a 3-D planar resonator circuit.
The two techniques combined in the SSDA are the SD M to sim ulate the cross sec­
tion of transmission line structures and the M oL to m odel their longitudinal direction.
The microstrip line step discontinuity show n in Figure 2.3 is taken as an exam ple to dem ­
onstrate the basic steps involved, n this chapter only the hom ogenous boundary condition
is considered (the discontinuity is enclosed in a shielding box).
First o f all a combination o f electric and magnetic lines are introduced to discretize
the structure in the z-direction. This corresponds to slicing the structure in the x-y plane.
Then a set o f conventional basis functions for each slice is introduced which satisfy the
boundary conditions along the x-coordinate. Every slice is o f regular rectangular shape,
so that only well known conventional 1-D basis functions are needed. The Fourier trans­
form is performed to replace the x-coordinate in the Helm holtz equation with the spectral
17
term u . Since the M oL procedure is used in z-direction, the resulting wave equations arc
coupled. The orthogonal transform in the spatial domain is utilized to decouple the sys­
tem equations. T he three spatial variables in the Helmholtz equation are now reduced to
the rem aining y variables and can therefore be solved analytically. The advantage of this
procedure is that fine circuit details such as narrow strips and slots as well as com pli­
cated discontinuity shapes can be easily resolved by discretizing the structure in z-direc­
tion. Furtherm ore, problem s such as complicated basis functions, huge memory space and
long C PU tim e know n from the 2-D SDM or MoL (i.e. 3-D problems) are avoided. The
final steps o f the SSD A are: the boundary conditions between layers at the top and bottom
of the closed structure are transformed into the circuit plane. Satisfying the boundary con­
ditions at that location leads to a set o f equations which are the G reen’s functions by
nature. A fter transform ing these final equations into the spatial domain, Galerkin’s tech­
nique is applied so that a characteristic matrix equation is obtained. By introducing
hybrid boundary conditions, the S-param eters can be obtained. This will be discussed in
Chapter 4.
2.4.1 SDM in x-direction
The c ro ss-se c tio n o f a m icrostrip resonator is shown in Figure 2,3, Although a single
layer structure is draw n for the purpose o f simplicity, the following formulations are also
valid fo r a m ultilayer structure (yk and ym are used here for a generalized formulation).
The ele c tro m ag n e tic field in the p 'h layer can be expressed in terms o f scalar potential
functions accoraing to equation (2.1)
(2.25)
VxVx T i
18
t
y=y«
1
y=yn
2
e r
r
x=a
M/ '
>[/'
‘hi
I
its
Figure 2.3 A microstrip discontinuity in a resonator enclosure
V* and H!1' are the solutions of the partial differential H elm holtz equation pair
L 2 L _ + L 3L _ + L 3 L _ + ^ . y
O.v
dv
d:
‘ = 0
(2.26)
*o = to V e 0
where
ppr is the relative dielectric constant in the plh layer. The Fourier transform is
19
applied in .v-direction
c,
...c ,//
h
(2.27)
CO
V
h
r n , f ’, h j a x
e
= J 1
,
dx
(2,28)
_ 00
where a is the space-spectral variable.
The electric and magnetic field vectors in equation (2.25) will take the following
form in the spectral dom ain
00
r ^
CO
ja x ,
e = J E c^dx
-j-
J=
J E e^dx
(2.2.9)
_oo
The space-spectral dom ain Helmholtz equation can now be written as:
32q/e,/l d2y L' h ( 2
/j,2'') (Uh
~ 7 T - + - T r - 1 a - r oJ
= 0
dv
oz
(2-30)
2 .4.2 MoL in z-direction
A fter the Fourier transform has been applied in .v-direction, the M ethod o f Lines (M oL)
can be applied in z-direction. The structure is sliced in the x-y plane at each r-coordinatc.
T he electric lines and m agnetic lines are introduced to represent the discretized scalar
potentials in the spatial F ourier transform dom ain. A total num ber o f AL lines are used
for different types o f transmission line configuration.
In vector notation the discretized potentials v ^ ' are written as a N .-element vector
as described in Section 2.2 .
\ \ h
\pc' 11
(2.31)
Non-equidistant discretization [15] can be used here to increase the flexibility. The
non-equidistant discretization can also be considered as a linear transform from
(original vector) to cp4’ 1 (non-equidistantdiscretized vector)
20
_s.
<\ h
X|/
—> cp
h
(2.32)
^ Q Jj
The new potential cp ’
is defined as
h
h
<P
(2.33)
v
where
re,h = d iag
(J;M
(2.34)
I A/ >le<hi)
he lli denotes the discretization interval of electric and magnetic lines, respectively.
h() is the limiting case for the discretization interval (equidistant discretization).
The finite difference expression of the first derivative is written in matrix notation
.
3(p‘
D. <P
(2.35)
.
a?"
D. <P
where
D.
(2.36)
[D] is defined in equation (2.9). The second order derivatives can be written as
D_ <P
(2.37)
D
<P
where
D.
Because [/T
D.
D.
D
D.
D.
(2.38)
is a symmetrical matrix, the following transforms can be applied
21
to transform this matrix into a diagonal form:
i r v
Tc
h,o
d
:
=
7
6ce
2
T
=
7
52
(2.39)
ll0
Similar to Section 2.2 , a new potential in the (orthogonal) transform domain is
defined as
s.<\ //
T
Ah
.A /l
<P
(2.40)
The final 1-D Helm holtz equations in the transform ation dom ain are derived from
equation (2.30)
a v
dv2
(2.41)
d 2^ f .2
2
2) /,
~ ~ 2 \ Ith + CX - £r k o ) V = 0
dv
The analytical solutions o f the above N , decoupled equations can be expressed as
transmission line equations from point y m to y k. T he /^'com ponent is
V*I
V,
coshy .d. - sin h y d
el i y ei1,1 i
dV:
yel.sinhy cid.
V,.
(2.42)
dV,
coshy eid.
.
dy.
.
V
coshM
=
a //
dy
^ /usinhM
Y/(,-sinhy/((.d.
co sh y /(/d.
dV
(2.43)
d y.
where
2
£ 2
2 p .2
y ei, = 6 „ + a - B r k Q
2
2
2
„ 2
y>ih = 6/,/,+ a - e/ o
dl =
(2.44)
22
T he equation (2.42) and (2.43) can also be represented in matrix form as:
pe
r
bp"
By
=
BP
By
BPe
By
^ h
1/ _
(2.45)
Y p.
BP"
By
V
h
w here [Qp] is a 4N : by 4N 2 matrix
Ce
L°J
^
c,
\P p.
M
3h2
[o] H
[o] M
= d ia g (c o sh y .d.)
S el
= d ia g ( sinh (y(,.t/(.) / y j
3cl
= d ia g (y ^ sin h (y e|.d .))
(2.46)
[o]
[o]
C,
= diag (coshyh.d.)
’/ii
= diag ( sinh ( y/( .</.) / y /|(.)
(2.47)
h2 = diag (y,„.sinh (y/l/r//) )
If one uses ex, ez and hx, hz to represent the components of e and Ti defined in equa­
tion (2.29), then from equation (2.1), the transverse electromagnetic field in the p ll‘ layer
can be expressed in the spectral domain as
_
a
d\yc d\\rh
(oeQeP
r" d :
e, =
^
dy
' Sa 2\v <
i>.2 e
— T + e, k oV
j(£>£0£P
r
_. _
lx ~
a By
By + m p 0 dz
( -a h
jth
cop V s V dz
N
/
(2.48)
23
A pplying the non-equidistant discretization, equation (2.33), and the orthogonal
transform, equation (2.40), the fields in the transform dom ain can be written in matrix
notation as
b =
ab
toe0e r
P 1,2
$ =
_dP'
dv
y-
££pe
P
j a>£0£r
.
dP_
- I =
' 3v
a 5, y copn
J ',. 2 *
r 0 ~ °/j /i f / 1
J l =
00(X
(2.49)
w here
t
=
jjl
-1
_rh_ K
1 =
I
b~z
=
- 11
k
f.
=
7J[
r,
r
-I
(2.50)
'h
N ote that the vector form ulation is introduced to represent the M oL discretization.
U sing block matrix notation
f
R,
ih z
-L
dPh
dy
dP"
dy
a /'
w here [R/;] is a 4NZ by 4NZ matrix
(2.51)
24
a b l,e
-0 [o]
[o]
[o] [o]
[o]
0
[o] [o]
cop.
[o]
[°] -[/]
p
to e 0 £ r
P .2
j,
r 0 "
ee
j<oeQeP
r
R.
-
(2.52)
6
a 5 ,/l
cop0
By com bining equation (2.45) and equation (2.51), the field relationship betw een tw o lay­
ers is
J t.
LR P_
fip j
R.
(2.53)
jljz
- L 'h
-hx
Tli'* expression for m ultiple layers can be obtained by cascading the respective
matrices.
With equation (2.53) one can always transform the electrom agnetic field from one
layer
another. By transform ing fields into the layers o f metallization arid applying the
boundary conditions at those interfaces, a matrix equation sim ilar to equation (2 .6) and
equation (2.15) can be obtained
e
~,v
\
n lx
= LZ
X! \
b
~z
Lk
[zj ‘s a ^
’z
(2.54)
by 4M, m atrix. Transforming the electric fields and currents back into
the original domain by using the same orthogonal transform introduced in equation
(2,40), the spectral domain algebraic matrix equation becomes
25
\
X
= ITzl
/d
Jx
\
Jz
(2.55)
w here [Z] is a 2NZ by 2/V, matrix
-l
> 1 [o ]
[z] =
[o]
1r
T
1-1
(2.56)
tTC
In summ ary, the following transforms have been utilized in the above analysis
—> Fourier transform -> non-equidistant discretization —> orthogonal transform
i.e.
(2.57)
( Ex . y - > ex,y)
( J X,
V
j'x,
y)
(Zx,y->2jc.y)
( a - , .V - *
I
The system equation (2.55) is obtained by following the reverse procedure.
In summ ary, equation (2.1) to (2.57) represent the generalized procedure of the
Space (from the M oL)-Spectral Domain, (from the SDM) Approach (SSDA). Although in
this chapter, the formulation is lim ited to a resonator problem, the foundation o f the
SSDA for scattering param eter calculation, which will be discussed in Chapter 3 and
Chapter 4, is laid.
2.4.3 The Eigenvalue Solution of a Resonator Problem
T he resonant frequency of a planar resonator can be calculated by finding the roots o f the
determ inant o f the system equation [17 ].
G alerk in ’s technique is applied to obtain the chaiactcristic matrix equations. The
26
first step is to expand the elements o f unknown j x and /_ in term s o f known basis func­
tions with unknow n coefficients a . and a[.
N,
N - 1
/
h i
=
£
/
/
a x i X\ x i
=
h i
/= 1
/
Z
(2.58)
/=0
where / represents the /th basis function, N x is the total num ber o f basis functions, / repre­
sents the /th line and
(2.59)
-ah order B essel’s function. Or, in vector notation
where >t>;- is the strip width. J0 is the 0tn
1
/
(2.60)
n
"ipr
■n,I
/
'H.wv,
/
V
Calculating the inner product betw een basis functions and each elem ent of the sy s­
tem equation (2.55) (further details can be found in Chapter 3) yields
f
K ' <
da = f
da
(2.61)
the right side o f equation (2.61) is alw ays zero. The left side can be written as
!/■ '(/)]* = 0
(2.62)
*s the result o f the inner product and cl represents the coefficient vector, f
denotes the resonance frequency which can be obtained by solving the zeros o f the deter­
minant
d e ( (/•'(/> ]] = 0
(2.63)
27
To extend the SSDA to calculate the S-parametcrs o f planar discontinuities the
deterministic quasi-static SSDA and the full-wave SSDA with hybrid boundary condi­
tions are introduced in Chapter 4 and 5 , respectively.
28
Chapter 3
The Quasi-Static SSDA
3.1 Why Quasi-static ?
Q uasi-static num erical techniques are traditionally faster than fu ll-w a v e techniques in
particular on the serial m achines widely used today. These m ethods do n o t benefit from
the availability o f parallel processors. For som e applications their accuracy can rival that
o f full-w ave techniques and, therefore, they are very useful engineering analysis tools. In
the recent literature, the quasi-static analysis has again received m ore attention [21 - 24],
because M M IC ’s and M K M IC ’s are usually sm all in dim ension com pared w ith the o p er­
ating wavelength and, therefore, dispersion is norm ally weak. In [21] a quasi-static spec­
tral do m ain approach (SD A ) was used to c a lc u la te m ic ro strip d is c o n tin u itie s . T h is
m ethod is num erically efficient but requires co m p licated 2-D basis fu n c tio n s, w h ich
som etim es may be difficult to find. A q u a si-sta tic finite d ifferen ce m eth o d (F D M ) is
described in [22] to analyze CPW discontinuities. This method can treat arbitrary discon­
tinuities, but at the expense of large com puter memory. In [23] the quasi-static m ethod o f
lines (M oL) is employed to analyze cross-coupled planar m ulticonductor system s. T his
m ethod does not use basis functions and is faster than the FD M but still requires signifi­
cant am ounts of mem ory and is difficult to apply to arbitrary d iscontinuities. T he d e te r­
m inistic SSDA eliminates these problems and will be introduced in the next section.
3.2 The Quasi-static SSDA
T his approach utilizes the basic idea behind the SSD A but avoids solv in g an eigenvalue
problem by using a new deterministic technique instead. To m inim ize errors in the calcu­
lation o f the capacitance param eters, the excess charge density [24] has been used and
calculated in the space-spectral domain in one step via G alerkin’s m ethod. T his approach
29
leads to an algebraic equation for the equivalent circuit parameters o f the discontinuities
and is com putationally very stable, requires little memory space and is very fast on serial
com puters. This m ethod is capable o f treating arbitrarily shaped planar circuit discontinu­
ities. Figure 3.1 illustrates the type of discontinuities this method has been applied to.
M icrostrip Open End
CPW Step
Microstrip Gap
CPW Open End
CPW Tai c
■.vfSSKtSTIS*?.;?
IT T
SZ
V
3
Microstrip Step
CPW Gap
CPW Air Bridge
Microsirip Taper
Figure 3.1 Planar circuit discontinuities
A CPW discontinuity illustrated in Figure 3.2 is used as an example to demonstrate
the theory. This discontinuity contains three regions (1, 2, 3) with thicknesses It/, It,, hi
and is shielded by a metal housing. The three regions arc defined as:
1.
hj+li2<y<h]+h2+ht
2. hj<,y<ch i+hi
3. 0<y<hj
A s mentioned before, discretization of the structure in z-direction corresponds to
slicing the structure in the x-y plane at each z-coordinatc. Therefore, the potential for
each slice must satisfy the 3-D Laplace’s equation
)2
I2
)2
JL y + JLv + JLv =o
r).v
0y2
(3I)
r):
In this case k=() (o>=0, compared with equation (2.2)). The task here is to simplify
L aplace’s equation which depends on the three spatial variables. The electric lines (solid
lines) are introduced to represent a discretized electric potential ip,which is independent
30
of the magnetic potential. T he dashed lines are used to represent the m agnetic potentials
as used in the conventional M oL (the magnetic potential is not o f interest here because it
is independent o f the quasi-static electric field). The shift in both lines is necessary to
reduce the discretization error and can be derived from [8 ]. Similar to C hapter 2, the first
step is to transform the electric potential function 'F into
via a Fourier transform
along the x-direction. Here the superscript is om itted because only the electric potential
is of interest. The spatial variable x becomes a spectral variable a . The next step is to
discretize \(/ by using Nz lines in the z-direction which leads to the vector \p. The
tapered region is enlarged in the left part of Figure 3.2, which dem onstrates how a smooth
transition is theoretically discretized and approximated by a sequence of abrupt steps.
Top View
Cross Section
Enlarged discontinuity
m-line
e-line
Figure 3.2
Discretization o f a CPW discontinuity
By utilizing the basic steps o f the SSDA in C hapter 2 with non-equidistant discreti­
zation in the z-direction, Laplace’s equation can be decoupled
^ - j - y 2P = 0
dy
where
(3.2)
31
y
2
_2
= o +a
2
(3.3)
I']*
D ue to the discretization, Laplace’s equation (3.1.) is now reduced to only one spa­
tial variable, y. 5 is the eigenvalue o f D ,, . ['/’] is the eigenvector matrix. 6 and |y j
are defined in Chapter 2 (note: the superscript e is omitted here because only electric
potentials are used in the form ulation, y has only two terms instead of three as in Chap­
ter 2 because k=0). Solutions to the above 1-D simplified Laplace’s equation can be
expressed in terms of the sum o f hyperbolic functions, and the relationship of the electric
potentials between any two adjacent layers can be expressed in the same way as
described in equation (2.42) o f Chapter 2, that -s
X
—
dV,
dy
V, is the
co sh y .d - sinhy.J
1 i yi
' i
y .sin h y ^ , coshy(c/(
V*
ilhelem ent o f P and
I-',
()\\
(3.4)
c)v
corresponds to the ilh line o f discretization. Because
equation (3.2) is decoupled, each line is represented by the same form of normal differen­
tial equation. W ithout loss o f generality the subscript / can be removed in the following
analysis. Instead, the subscript is now used to represent the potential in the different
dielectric region.
For Laplace’s equation, there is always y = y, = y 2 = y.? because k„ 0 in equa­
tion (2.43).
T he boundary condition at the interface is as follows:
at
y = l>2 + ^h
at
31/,
£ ,-^
v = h .j
(3.5)
Sv2
E 0 -=—
-'2 dv
- — L
at
y = h 2 -f //3
(3.6)
dV2
"2 3v
w here
tion
q is
5 dy
at
the charge density in the transform dom ain. At the top and bottom m etalliza­
32
at
V{ = 0
v = h { + h2 + /i3
(3.7)
V3 = 0
v = 0
at
Substitute equation (3.7) into equation (3.4) yields
dV~
Y
tanh y h2
dy
dV3 _
dy
v = h2 + /i3
(3.8)
y
tanh y/i3 ' 3
at
v = h-.
Com bine equation (3.4) - (3.8) provides
(3.9)
8(y) Vi
where
£2y
c 2y
S(Y) =
tanhy/t 2
+
e iY
tanhy/i,
tanhy/i 2
e 2y
e3y
tanhy/i 2
tanhy /?3
(3.10)
To characterize a discontinuity, one needs to find the solution for the electric charge
belonging to the discontinuity part. This is usually achieved by subtracting the total
charge o f the discontinuity area and the charge belonging to the connected transmission
line. Since both quantities are often quite small, the errors arising from the subtraction of
two electric charges, which are close in magnitude, can be significant. To avoid these
errors, the excess charge technique [24] is used. This technique can briefly be sum m a­
rized as follow s; the 2-D transmission line problem is solved first in the spectral domain
on either side of the discontinuity, i.e. solving equation (3.2) (homogeneous transmission
line in z-direction) and analytically subtracting the charge distribution o f the fictitious
hom ogeneous transmission line from the charge distribution of the corresponding discret­
ization line o f the transmission line containing the discontinuity. Based on the above for­
mulation, the 2-D problem can be solved by using the solution given in equation (3.9)
with 5 = 0 , i.e. y = a
33
K (o c )l', = ~
£.
(3.U)
wnere <■/„ is the charge density o f a fictitious hom ogeneous transmission line. The
excess charge density p is defined as
(3.12)
Now the excess charge technique is applied, which m eans the two quantities g(y)
and g(a) are subtracted
= U (Y ) - A'( « ) ) l ~
(3.13)
and the results are transformed back into the original dom ain
M' =
(3.14)
CP
a is the excess charge vector in the original domain and
H
= H H
Mag [g (y) - q (a )] ‘ ’‘ [?J'
(3.15)
where d iag[g(y)-g(a)l includes the difference [g(y)-g(a)J from all the lines. T he size o f
the matrix is Nz by N z.
G alerkin’s technique is now applied to obtain the characteristic matrix •equation.
The first step is to expand the elem ents o f the unknown a in term s of known basis func­
tions with unknown coefficients c/
I I
a =
(3.16)
w here N x is the number of basis functions, N, is the num ber o f discrete lines and
1
r J
4, = | V
ja x
j
dx
(3.17)
34
T|/ and <5/ are Fourier transform pairs of basis functions. For CPW circuits, they
take on the follow ing form for the center conductor with width
ivj
(also for microstrip cir­
cuits with w idth i»’j)
cos
( 2 1 - 1) tcI.
aw.
nu'j
•n, = - r
\
I = 1, 2 ,
(3.18)
( aw.
/=
1 ,2 ,...
For (CPW ) g round (sym m etrical) conductors (iv/( is the ground conductor width,
(3.19)
is
the x-coordinate o f the ground conductor center (one side))
cos
( x + b w!)
In-
(v x - b 117.)'
cos In
wu
H’i/
II
I = 0,2,...
n x+ bj
2 ( x + bwt)
1-
-
w i1
J
wu
x <0
Sill
.v> 0
(x + b J
sin In-
1/
" ’u
2 (X + ! U )
11\
wu
x<0
(3.20)
(x ~ bJ
/ tc IV
J
J
11
/ = 1,3,
-
IV ,
n
/
.v>0
J11/ = —n w
x -"c o s a b, wi
I
=
0, 2,
...
(3.21)
a iv 1(. - I n
11/ =
s i n a 6 w/
/ = 1,3, ...
F or different microstrip or CPW discontinuities, one only needs to adjust ivp b wi
and w u of each line instead of changing the form of the basis function. Thus the disconti­
nuity shape can be a rb itra ry . This is an advantage of the SSD A and makes it possible to
develop contour driven software.
Similar to Chapter 2, the inner product between basis functions and each elem ent
of the system equation (3.14) is calculated
/ = 1 ,2
N.
(3.22)
In quasi-static analysis, the excitation potential is always a constant across the m et­
allization. This property can be utilized to achieve a sim ple deterministic solution
through the use o f Parseval’s theorem
J tj/ • 'q ^ a = 2 k J lF • t; clx
(3.23)
where
1!
\
1' =
(3.24)
I
V1
k
'F is the discretized electric potential (inverse Fourier transform of ij/) and is c o n ­
stant across the metallization. If this constant is defined as V' , the left side of equation
(3.22), which is further processed in equation (3.23), can be written as
J m> • n d a = 2 u V (, | % d x = 2 n V tj \
I« = 0
(3.25)
Unlike the full-wave resonator analysis described in C hapter 2, the left side o f sy s­
tem Equation (3.22) is known as an • gebraic equation. The determ inistic solution can be
obtained by matrix inversion. Rewiiting equation (3.22) in m atrix form, which contains
Nx independent equations, will yield
36
[c]® J
2 nV,
’1
-J
A,
da
(3.26)
a =0
By using equation (3.16) and replacing the continuous integral by a discrete sum ­
mation, equation (3.26) can be written as
(3.27)
where
(3.28)
Z| = B lo c k
Aoc
a =0
1 1
2 *
Nx
G km% n
i 2
2 2
^ 2
G km XSm
km
/v.
1
T1/n
(3.29)
Nx N,
A„
^kni^nt^m ^ km ^m ^m
^knt^m
^
A a is the step w idth of the discrete Fourier integration. [Z] is a ,V_/Vr by N:N y. block
matrix, which contains Nz by Nz submatrices. Each submatrix fZ ]^ , is a N x by N x
matrix. G \m is the (k, m) elem ent of [G],
The charge density coefficient vector d is defined as
d =
1 2
<i a L
..
Nx 1 2
*2 Cl7
A.
1
2
ClNz V
A.
-
"A’
(3.30)
Front equation (3.27), it is evident that only a one step m atrix inversion is now
required
37
(3.31)
In contrast to finding zeros o f a determinant through an iterative procedure (equa­
tion (2.63)), the quasi-static SSDA is a determ inistic approach and provides the results
by a one-time m atrix inversion.
The total charge Q can be obtained by integrating the charge density over the dis­
continuity area
(3.32)
The equivalent circuit for different discontinuities is shown in Figure 3.3. The gap
discontinuity is characterized by a n network. Open end, step in width, tapered disconti­
nuities and air bridge are approximated by a shunt capacitor. The capacitances Cpl, Cp2,
C s or C p are then calculated from C=Q!Vc assum ing different excitation voltages Ve (even
mode Fe l= l, Vt2~ 1, odd m ode Fel= l, Vc2= - 1 for a n network, VQ=\ for a shunt capacitor)
at both ports of the strip. Ve=0 is chosen for the ground conductor, The s-paramctcrs can
be derived by using netw ork theory.
P
Step, Open End
Airbridge
Gap
Figure 3.3 The equivalent circuit
To calculate the shunt capacitance o f an air bridge, the above formulation must be
slightly modified. A C PW air bridge is shown in Figure 3.4.
The three region form ulation from equation (3.2) to equation (3.31) can still be uti­
lized if the air bridge is approximated as a patch sitting above the CPW as shown in Fig­
ure 3.4. This approxim ation is only valid when h 2 is very small (h<w/5), which is true in
most cases. Also the boundary condition of equation (3.6) is changed to
38
dV ,
dv2
at
27 7 =
dV2
;2 a 7
dV3
8377
v = h2 + h3
(3.33)
, ch
at
V = !u
Cross Section
ill
ill
h3
approximate
Figure 3.4 A CPW A ir Bridge
Where q { is the charge density (in the transform ed dom ain) o f the bridge and q2 is
the charge density o f the CPW area. Following a procedure sim ilar to that described by
equations (3.5) - (3.15), a new system equation can be derived
(3.34)
Es (r)]
Y\
where
39
e 2y
tanhy/i 2
- ? 2Y
tanhy/i„
tanhy/i,
(3.35)
- e 2y
tanhy/z.,
-s 2y
s 3y
tanhy/i 2
tanliy/^
Finally the system equation in the transform domain has the same form as equation
(3.14), but in a block m atrix form ulation. [G\ from equation (3.15) becomes
-if./
[c] =
[g (y)] " [g (ex)
0
(3,36)
0
E ach submatrix is a 2x2 m atrix. The total rank o f fG| is 2NZ by 2N/:
3.3 On the Nature of the SSDA
It has been show n in C hapter 2 that the MoL and SDM arc indeed equivalent if the same
d iscretization schem e is used, i.e., the number of lines in the MoL equals the num ber of
spectral term s in the SDM . T his equivalency forms the basis of the SSDA. A fter having
introduced the quasi-static SSD A , it is worthwhile to look back and study the nature of
the S S D A . T his se ctio n w ill show that by using a T E y/T M y form ulation, the SSDA
includes the 2-D SD M and 2-D MoL.
In the analysis o f planar transmission line problems the most common approach is
to express the electrom agnetic field in terms o f Ez and Hz (i.c, TEZ and TM Z wave),
w hich gives a coupled TEZ and TM Z wave formulation as described in Chapter 2. But
w hen the field is expressed in term s o f Ey and Hy, a transmission line type of formulation
can be obtained by coordinate rotation (in the x-z plane) to a u-v coordinate system,
which, for the SDM, led to the immittance approach [4| or a simplified formulation in
the M oL [13], [25]. If a T E y and T M y wave formulation is used in the SSDA, som e inter­
esting results can be obtained.
Starting from M axw ell’s equations
40
Vx£ = -yoopj)
(3.37)
V yj) = jm z tl
and rearranging the above equation yields a more su itable form for the purp o se o f this
section
d2
.2
d2 . d
dxdy J^d=
d2 . a
dydz -/ 0)fia.v
X
57+ *
E
.
(3.38)
,2
d
X
?7 +* u2_
. a
a2
_/ Ea= a ja ?
.
a
a2
a ^
//.
A variable transformation in x- and z- direction (this can be either the F ourier trans­
form o f the SDM or the orthogonal transform of the M oL) is now introduced as follow s
a
dx
T z ^ - ja *
d2
dx
2
d*
(3.39)
2
— =- —> -e x .
d z2
U sing the low er case to rep re sen t the field com ponents after the transform , eq u ation
(3.38) is written as a decoupled TE/TM to y formulation
where
0
-a> |i
coe
~j T
ay
°
0
0
-j
.d
(3.40)
a x and a , can be spectral term s o r eigenvalues of the transform matrix, d ep ending on
which m ethod is used in the x- and z-direction, respectively (note: when an eigenvalue is
used, it should be m ultiplied by j) . T he U matrix corresponds to the coordinate rotation
from (x, z, y) to (u, v, y) as shown in Figure 3.5.
Based on the above form ulation, the SSDA is really a 2-D SDM when a
are spectral terms. Similarly, the SSDA becomes a 2-D M oL when a v and a
and a ,
are eigen­
values o f the transform matrices (discretization). In the SSD A the SDM and Mol can be
applied separately to the x- and z-direction, respectively, or one can use any one o f the
two m ethods. It is worthwhile to point out that the reason behind the form ulation is that
the TEy and TM y modes are independent (not coupled anymore!). On the other hand, this
equivalency is only valid from the formulation point of view. The SSDA has its unique
style in solving discontinuity problem because it uses neither 2-D discretization as in the
MoL nor 2-D basis function as in the SDM.
Figure 3.5 Configuration o f general transmission line
In sum m ary
42
•
The SSD A combines the M oL and SDM which can be derived from each
other
•
T he SSD A can be form ulated from the TEy/TM y wave expansion using a
coordinate rotation.
•
the 2-D SDM or 2-D M oL are the special case o f a general hybrid method:
the SSDA.
45
Chapter 4
The Full-wave SSDA
This chapter focuses on the full-wave SSDA. Two alternative approaches arc presented:
an eigenvalue and a deterministic approach.
The foundation of the full-wave SSDA is laid in chapter 2, where only a hom oge­
neous boundary condition is used. To calculate the S-param eters o f planar circuits, inhomogeneous boundary conditions must be included. This chapter describes two different
approaches to implement the inhom ogeneous boundary conditions and to extract Sparameters.
4.1 Eigenvalue Approach
The eig e n v a lu e approach em ploys the concept o f se lf-c o n siste n t inhom ogcncou.s (o r
hybrid) boundary conditions at the end o f feed lines which are connected to cither side o f
the discontinuity. This approach m akes it possible to sim ulate the whole structure via an
eigenvalue equation in which the solution is the reflection coefficient of the discontinuity.
The hybrid boundary conditions have been used before in [ 15] and 116], but in the first
case to model the forward and reflected waves individually and in the second case to find
the total field at the launching point by using a modal source approach. In the m ethod p re ­
sented here, the reflection coefficient (or S (j) is obtained directly,
If a 2-port discontinuity (Figure 4.1) is under investigation, it is assumed that at
some distance from port I of the discontinuity, there will be a standing wave o f the funda­
mental mode only consisting of incident and reflected waves:
44
<«
<■( -/Pr'
V = %,{<*
yP|*)
- r(1 J
h
h( - f t i 2
jp A
V = v 0{ c
+rc J
where |3| is the propagation constant at the boundary o f port 1 calculated separately
by using the SD M , r is the voltage reflection coefficient and
are the incident T E /
TM potentials at z = 0, which are solutions o f the homogeneous connecting transm ission
line.
Connecting
Transm ission Line
Discontinuity
Region
Connecting
Transmission Line
1
z=0
Discretization
Figure 4.1
N,
An eigenvalue approach
The inhom ogeneous boundary (z-direction) conditions can be derived indepen­
dently without considering the spectral dom ain factors (x-direction). With reference to the
matched, open- and short-circuited conditions, at port 2 three different cases for the
boundaries exist, these are the Dirichlet, Neumann and hybrid boundary conditions. F or
the matched condition at port 2 , there are two choices for the discretization schem e
depending on w hether to assign an e or h line as the first line. In the following, the d is ­
cretization schem e begins with an m (m agnctic)-line (open-circuit). When using an mline as first line, only the boundary condition for \^'0 is specially treated, while the
boundary condition for
is implicitly included. For the same reason, only the bound­
45
ary condition for vjc(J is specially treated w hen using an c-line as the first line.
In case of the matched and opcn-circuit conditions, the hybrid boundary condition
at port 1 can be expressed as:
.„ (
d '/
;lv
/( V ' l
h
(4 .2 )
jx V(>
-/*<•
and at z = 0.5 h
combining equation (4.2) and equation (4.3)
/0.5/ip,
d \ |/
Tz
•o e
J V\
- 0 .5 h
e
./os/ip.
-re
" +
70.5/i|t,
h
70 5/at.Vi
. A
(4.4)
equation (4.4) can be simplified as
By
dz
h
s
(4.5)
z = 0.5//
where
" =
l-y-ctan(0.5/iP1)
1+/
~r~—j'ta.i (0.5/,f5,)
T= T 7
The voltage reflection coefficient r is thus explicitly involved in the hybrid bound­
ary conditions. At port 2 the matched condition corresponds to:
J '
= -,/T V lV
(4-7)
where (32 is the propagation constant at port 2 if a two-port circuit is considered.
The propagation constants
and p 2 can be derived from the l-D SDM or MOL. Note
that the matched condition corresponds to the discretization schem e o f the opcn-circuit
condition. In a sim ilar way, the hybrid boundary conditions obtained for the short-circuit
situation is as follows:
46
3\}/
dz
e
= -v y ,
(4.8)
’ = (Nz + 0,5) h
w here
x -./ta n (0 .5 /j P ,)
v = / P i 71-yxtan (0 .5gi,B~~
/;p [)
1 + /.
T
=
7—
T
1 - r:
(4 -9)
Obviously, the potential functions and their first derivatives constitute the character­
istic solutions o f the w hole circuit. It is interesting to see that the com plex functions of
the inhomogeneous boundary conditions at the input described in the above equations are
not only expressed in term s of the propagation constant Pi but also in terms of the dis­
cretization interval h and the unknown voltage reflection coefficient r (or ,vn ). In other
words, the inhom ogeneous boundary conditions are no longer “static” and strongly
depend on the unknown scattering parameter, which in turn depends on the geom etry o f
the structure o f interest as well as the operating frequency. This is why the inhom oge­
neous boundary conditions are said to be self-consistent.
In sumrr ry, w hen the load o f port 2 is m atched, open or short, the corresponding
boundary conditions are listed in the following table (homo=homogeneous boundary con­
dition, inhomo=inhom ogeneous boundary condition)
port 2
M'*1
v hi
H'8 2
A
m atched
hom o
inhomo
inhomo
homo
open
hom o
inhomo
homo
homo
sh ort
in h om c
homo
homo
hom o
Table 4.1. Boundary conditions
It is noted that only the case of inhom ogeneous boundary conditic-P need to be spe­
cially treated here, because the case of hom ogeneous boundary condition is discussed in
Chapter 2. Sim ilar to C hapter 2, the determ inant equation will be derived from the SSDA
procedure. The solution o f this determinant equation is the unknown reflection coeffi­
cient r. The matched condition is taken as an exam ple in the following analysis. The inhomogcncous boundary conditions are
47
V
dz
= "V i
2 = 0 . 5 ,'i
(4.10)
chjr
dz
: = (A /i+ 0.5)
h
'Va -2
in which u and v are the coefficients defined in equation (4.6) and (4.9). in order to
m aintain the essential transformation properties (known from the M OL procedure), sym­
m etric second-order finite-difference operators are required to deal with the Helmholtz
equation and, in particular, the field equations tangential to the interfaces. Using the con­
cept and algorithm described in Chapter 2, the electric and magnetic potential vectors in
the original discrete dom ain are normalized by diagonal matrices:
<P
=
(4.11)
with
1
J ii h
1
f //] =
l/’J
r
!_
(4.12)
1
Jvh
Therefore, the first and second derivatives o f the potential functions arc approxi­
m ated by formulae sim ilar to the ones in Chapter 2. Note that the unknown voltage reflec­
tion coefficient r is directly involved in the first elem ent of [/X',/,J and its related matrices.
Applying the continuity condition at each dielectric interface leads to a matrix rela­
tionship between the tangential field components o f two adjacent subregions in the inter­
face plane. Next, by successively utilizing the continuity condition and multiplying the
resulting matrices by the transmission line m atrices associated with the m ultilayer subre­
gions, the boundary conditions from the top and bottom walls can be transformed into
the interface plane o f the discontinuity. This leads to a kind of space-spcctral Green’s
function in the transform dom ain which must be transformed back into the original
dom ain. This step can be performed by the conventional MoL and SDA procedures inde­
pendently. From the m athematical viewpoint there is no difference which procedure is
applied first. However, applying the MoL first leads to a better physical understanding
and easier mathematical treatment. As a result, the matrix elements of the resulting
48
G reen ’s function in the space-spectral domain are once again coupled to each other
through the reverse transformation back into the original domain
\
it
.
ex
b
\
A
\
A
-
..
G alerkin’s technique is again used together with an appropriate choice o f basis
functions which will be defined on the conductor surface for each slicing line in the zdirection. This leads to a characteristic matrix equation system which m ust be solved for
the zeros o f its determinant, whereby the determinant is a function o f the reflection coeffi­
cient r.
(4.14)
F or irregularly shaped discontinuities, the geom etric param eters becom e a function
of the z-coordinate and, therefore, are different for each line. In general, this does not
com plicate the analysis of planar structures at all, as long as the circuit contour can be
described mathematically or by a set of coordinates. In addition, singularities o f the cir­
cuit in the x-direction are automatically considered in the form ulation o f the basis func­
tions. O nce the voltage reflection coefficient r is known, an arbitrary constant for the first
elem ent o f the x-oriented current coefficients can be assumed. Applying a singular value
decom position technique yields all the current coefficients for the chosen basis functions
assigned to each discrete line. Therefore, the S-param eters can be extracted from incident
and reflected currents on the strip.
4.2 Deterministic Approach
A lthough the eigenvalue approach for the full-wave SSD A has the advantages that no 2D field distribution calculation is required, the root o f the determ inant, w hich is derived
from a large matrix, must be found. For practical applications, a “o n e -s te p ” solution is
most desirable. In this section, a deterministic full-wave SSDA is presented, which avoids
solving an eigenvalue equation by iterative computation.
A deterministic approach was used earlier in the 2-D M oL [12] [26]. In [12] a
“three-step” approach was presented rather than a “one-step” approach because open and
49
short conditions were utilized. In 126] inhomogeneous boundary conditions were intro­
duced. B ut based on the a u th o r’s experience, the resulting algorithm does not provide a
stable solution because a good m atching condition for S-parametcr calculation can not be
realized.
T he key steps in the follow ing approach is to express the field distribution on the
connecting transmission line (far from the discontinuity) as a superposition of incident
and reflected waves, then derive the inhomogeneous boundary conditions for incident
and reflected waves (sim ilar to the previous section, the eigenvalue approach), and
finally com bine the incident and reflected waves to satisfy the tangential field condition
at the m etallization plane. By solving the 2-D transm ission line problem first, the inci­
dent w ave distribution is know n. The reflected wave distribution is derived by using the
know ledge o f the incident wave.
F igure 4.2 illustrates the deterministic approach. Region B t and C together arc
called region B. Port 2 is alw ays matched.
Region B
Connecting
Transmission L ine
Region A
Discontinuity
Region B j
Connecting
Transmission Line
Region C
2
z =0
Nz
Figure 4.2
A deterministic approach
It is assum ed that only one propagation mode exists on the transmission line connected
to port 1, w hich is so far from the discontinuity region B j, that the higher order modes
excited by the discontinuity have vanished at port I . The same assumption also applies to
50
region C. Using i and r to represent the incident and reflected waves, the inhom ogeneous
boundary conditions are expressed as
<V'
incident
IPi°-5/' h
-yPjf
dz
portl
-yp, 0 .5/1 h
= .iP f
Vj
dy
reflected
vL
z = 0.5/1
(4.15)
z = 0 .5 h
-yp2o.5/.
in c id e n t
= -JP2C
z
reflected
port 2
-yp20 5/«
3 i|/
dz
V,\.\-
= (Nz + 0.5) //
= -./(V
( 4 .1 6 )
v,Vr
: (yvz + o .5 )/i
Similar to equation (4.10), equation (4.15) and equation (4.16) can be simplified as
f
i, r
chy
dz
= “
h
Vi
z = 0 .5 h
(4.17)
r
Tz
z = (A 'z + 0 .5 )
h
where
,.
u
= y P j-c
-yp,o.5/i
,•
u
=
yp 0 .5/1
-jfif
- / p . 0 .5/1
v =
- / (32 c
( 4 .1 8 )
As shown in Chapter 2, two system equations can be obtained for incident and
reflected waves respectively
Z
(4.19)
Rearranging equation (4.19) and using subscripts A and B to represent fields and
currents in different regions, yields
51
h
11
\i
Ja
i
_ A H_ vl
hi
•/
A11
Ja
Jn
Once again G alerkin’s technique is applied (described in chapter 2 and 3) to
expand the unknow n incident and reflected currents in terms o f known basis functions q
and unknown coefficients C ',r
\ ‘' r
./
=
1]
(4.21)
Calculating the inner product of basis functions and using equation (4,20) yields
r
-|
w'
_ 11_
w‘
L 21
-i
.
w‘12_
ii
<*■'#'w
w‘
_ 22]
\yr
L
C'a =
.
M/r
12
K
w r21
wr
Yy27
_ < 4 V
k
C \4 is the know n (from 2-D SDM ) coefficient for the current distribution at the connect­
ing transm ission lin e at port 1 (region A), C rA is the unknown coefficient of the reflected
current distrib u tio n at port 1. A lthough i and r are used here, both C'n and C n are the
unknown coefficients o f the outgoing current distribution at port 2 .
In region A o r B, on the m etallization, the total tangential electric field must be zero
e+ er = 0
(4.24)
Com bining equation (4.22), (4.23) and (4.24) yields
.
w
*I
<1 +
. . w‘21. 4
+
wr
.
w1
. 12
-
*L
\\h21.
W*22
-
,
w r12
) c »
+
w 22
r y„+
wr12
wr12
52
where
c‘r = c' +cr
(4.26)
C " ^ is more useful fo r determ ining the S-param eters. Since C 'A is know n, equation
(4.25) can be solved by eliminating C B,
W\
IV
w",12
W<21
IV,.
i v ] !w '
1V22
^
-
d t n 2i
IV11
C
(4.27)
where
IV.
iv 12
IV12
iv:22
iv:
22
(4.28)
In equation (4.27), C A and C'rB are unknown. Because C B is valid in region B t
and C, equation (4.27) can be further simplified by replacing C ‘rB by the known coeffi­
cients of region C (solution o f a homogeneous transmission line) multiplied by an
unknown factor. Thus the number of unknowns is reduced to I in region C no m atter
how many lines are used there. Applying the same principle to region A will also reduce
the number of unknow ns in region A from NA to I.
Finally, C A and C 'rB can be obtained by simple matrix algebra which is described
in the appendix.
Since the coefficients of the reflected current at port 1 and outgoing current at port
2 are known, the currents can be calculated accordingly. They are defined as I' j and I '2.
Also the incident current obtained from a 2-D SDM analysis is defined as 1+
The S-
parameter can then be calculated by
S.. =
(4.29)
^12 =
(4.30)
53
w here Z; is the characteristic impedance o f the connecting hom ogeneous transmis­
sion line at port / ( /= / , 2).
Chapter 5
Numerical and Experimental Results
5.1 Convergence Analysis of Quasi-static SSDA
A c o n v e rg e n c e analysis is perform ed for a m ic ro strip open-ended d isc o n tin u ity , as
show n in Figure 5.1 (er is the dielectric constant and h is the thickness o f the substrate).
Nx is the num ber o f basis functions in the x-direction. Nz is the num ber o f lines in the zdirection. The convergence behavior depends on the num ber o f spectral term s, Nz, and
Nx. W hen the num ber of basis functions is increased, the number o f spectral term s must
be increased accordingly. A good convergence behavior is obtained w hen N z >40, N x>2,
and the spectral term is greater than 80.
dashdot:
tiz= 40
N x= l
Nz=40
Nx=2
Nz=40
Nx=3
solid:
Nz=60
Nx=l
Nz=60
Nx=2
Nz=60
Nx=3
LLf
N z=J0
Nz=10
Nz=10
N x=2
Nx=3
dashed
Nx=l
M icrostrip Open End !
100
30
# ol spectral terms
Figure 5.1 Convergence analysis of the Quasi-static SSDA (ir//;= /, e r=9.<5)
55
5.2 Simulation Results of Quasi-static SSDA
First o f all the microstrip (M S) and CPW open end, gap and step in width as w ell as the
CPW air bridge are analyzed by the quasi-static SSDA. Results are shown on Figure 5.2
to Figure 5.8. Those results are used to validate the SSDA.
Figure 5.2 shows the equivalent capacitance o f a microstrip open end. The solid
line is calculated by the SSDA. A ll markers are results from literature.
10
x x x Full-W ave[33|
o o o Quasi-static[ 22 ]
+ + + SD M | 19 J
er= l 6
10
ar=9,6
+
o
- 4 *'X
io'
* * * SDM [39]
SSDA
10“
10
w
10
w/h
10’
Figure 5.2 Capacitance of microstrip open ends.
Figure 5.3 shows the equivalent capacitances o f a microstrip gap, which is repre­
sented by a n network as shown in Figure 3.3. The circles represent the SD M sim ulation
and stars represent measurements [21]. The solid line is the results of the quasi-static
SSDA sim ulation.
56
45
SSDA
* * * M easurem ent [19]
o o o SDM [19]
40
35
30
GT25
0.20
15
10
I
10
20
30
g a p width (x0.01 mm)
40
50
60
Figure 5.3 Equivalent capacitance o f a microstrip gap discontinuity. w/h=l h =0 508m m
er=8.875
09
0 8
S12
V>
&0 7
SSDA
W,
0 6
W2
o o o Full-w ave[28|
0 ,5
S11
0 4
0 3
10
.
.
1-
15
20
25
30
35
F requency (GHz)
, 4. , ,
40
....
t
..
45
50
Figure 5.4 S-param eters o f a m icrostrip step. w ,= l/n m , \\'2=0,25mm, w /h= l,
57
Figure 5.4 illustrates the S-parameters of a microstrip step discontinuity, which was
derived from network theory, and its shunt capacitance value. The circles represent the
full-w ave SD M simulation [30]. The solid line is calculated by the quasi-static SSDA.
Figure
5.5 presents the equivalent capacitance o f an open-ended eoplanar
waveguide com pared with published results. The solid line is calculated by the quasi­
static SSDA. The circles and stars represent finite difference simulation and m easure­
ment [22], respectively.
Figure 5.6 shows the S-parameters for a eoplanar air-bridge. The circles represent
the results from a full-wave analysis using the Frequency Domain Finite Difference
(FDFD) m ethod [31].
Figure 5.7 illustrates the S-parameters of a CPW step discontinuity analyzed in
[32], The solid line is calculated by the SSDA. The circles arc calculated by the fullwave SD M [32].
Figure 5.8 presents the S-parameters of a CPW gap analyzed in |2 2 |. The solid line
is calculated by the SSDA. The circles represent the results from the quasi-static SDM
[22],
In summ ary, Figure 5.2 to Figure 5.8 present a variety o f planar discontinuities.
The SSD A sim ulation agrees well with either published results or measurements.
58
30
25
SSDA
o o o Finite Difference [22]
* * * M easurem ent [22]
tr
20
n3
o(tJ
Q.
03
%
c 15
w
10
02
03
04
0.5
0,6
0.7
0,8
0.9
w/d
Figure 5.5
0 06
Equivalent capacitance o f a CPW open end. er=y.6, h=0.635, /;/</=/,
cJ=\y+2s
I
0 ,0 5
t
SSDA
o o o FDFD [29]
0 04
£2-o 03
w
0 02
0 01
°0
10
20
30
40
so '
60
F re q u e n c y (GHz)
70
8 0 9 0
100
Figure 5,6 S-param ctcr of a CPW airbridge. w = l3\im , x= l()\lm , I=3 [On, h~200[W i,
b~3\xni
59
S21
SSDA
o o o SDM [30|
S11
28
30
32
38
36
34
40
F re q u e n c y (GHz)
Figure 5.7
S-parameters of a C PW step. wj-OAimn, u
w2=0.4 mm, e r~0.S, h=0.254mm
Sj-O .lm m ,
40
J
SSDA
o o o [221
35
tSo*
W,
30
w2
sl
25
c<d 20
O
§
Cp2
15
O
10
5
Cs
0
0 05
Figure 5.8
i.
01
i
Cp1,
,
0 15
02
0 25
GapWidth/SubstratoThicKnoss
0*3
Equivalent capacitance o f a CPW gap, (/|»2s j t-H-j,
h~0,635mm, w^lh-O 2, \ v \ l d l -w ^ d l'-O A d ,
u'2/ " 7
2+tt’2.‘V'* y
"4
60
The main reason why the quasi-static SSDA works extrem ely well for M H M IC ’s is
because the circuit dimensions are small compared with the wavelength, which m eans
the dispersive effect is normally very small (the dispersive effect m ay become obvious at
very high frequencies). When this condition is not valid (Figure 5.12), visible discrepan­
cies to full-wave techniques may occur at lower frequencies.
The com putation time of the quasi-static SSDA is typically a few m inutes on a
SUN SPARC-2 station (this will give the results over the whole frequency range). It is
noteworthy to p oint out that results shown from Figure 5.2 to Figure 5.8 do not only d e m ­
onstrate the good agreement with published results, but also show that this method is flex­
ible. The accuracy o f this deterministic quasi-static SSDA com pares well with full-w ave
methods and measurements.
To show the ability of modeling arbitrary discontinuities, microstrip and C PW
M HM IC tapered discontinuities are calculated. The transitions are represented by their
equivalent capacitances as illustrated in Figure 5.9. When the length o f the slope, W s,
increases, the equivalent capacitance decreases. This makes perfect physical sense.
When the length o f the slope equals zero, it represents an abrupt step in width (this was
proved by calculating the capacitance o f a step). The whole capacitance curve is quite
smooth which show s the good numerical stability of this method.
The S-param eter o f a CPW air bridge is also calculated in terms of air bridge
length, which is show n in Figure 5.10. One can observe that when the air bridge length L
decreases, S j ( decreases too, because the capacitance to the ground decreases.
Figure 5.11 and Figure 5.12 show the limitation of the quasi-static SSDA. F igure
5.11 shows the S-param eters of a microstrip step discontinuity up to 100GHz. A lthough
at frequencies below 50GHz the comparison with full-wave techniques is excellent,
higher order m odes start to propagate on this structure at 60G H z which is not accounted
for in the quasi-static method. This leads to an increase of S n and a decrease in S2i at
about 50GHz.
20
wi
Ws
substrate.
Q.
Microstrip
—
w
CPW
Slope Length Ws (m m )
Figure 5.9 Equivalent capacitance o f a m icrostrip and a CPW step/taper. For CPW
tv1=0.8mm, s ,=0.1 mm, w2=0.2mm, s2==0.6mm, er=9.6, h=0.254m m . For microstrip,
w 1=1 mm, w2=0.25mm, s r=9.6, h=0.25mm
62
<N
20
40
60
80
100
Frequency (GHz)
Figure 5.10 S-param eters of a CPW airbridge versus bridge length /. \\'=03m m ,
s-O .lm m , b=3\lm, s r- 9 ,6 , h=0.254mm.
1.0
—■
-------r
" -t
’
i ... -r ..t ... 1 ......
o .y
- S
'
0.8 -
0.6
„
.............
Full Wave TLM [33]
0.5 •
0.4 0.3
-
1 Quasi-Staiic SSDA
0.7 -
***
,,
s ,,
-
-
-
|
0.2 -
Wi
‘■
”1
...........
Wi
0.1
nn
•
t
,
,1
— I---------- 1--------
w 1=0.25 mm
w2=lmm
h=0.25
Er=9.6
i
Frequency (GHz)
Figure 5.11
Frequency dependent bcba\ lor of microstrip step
63
1.0-1 f ‘
1 ■ 1 I <1 1 ■ I
»
‘
I
>"l
» I < '!- [ i
i I I I I
I i
I j
| - | - | T - |- |
I i
I I
I I
I
i 1 i
i
i i
r~pr~rT-r-w-
0.8-
- SSDA
S -P a ra m e te rs
- MMM 1121
0 .6
•FDTLMI3.1]
-
S11
r. —
0 .4 £r=9.6, h=0.254
0.2
-
0.L
0.0
-
0.8
10.2
I I I I I 1II l l l l l l l l l l l l l l . l l l l . i l
10
20
30
■ l-i-i i
40
I i
i i,, i,_
50
Frequency (GHz)
Figure 5.12 Frequency dependent behavior of CPW step
64
5 .3
C o n v e r g e n c e S tu d y
o f F u ll-w a v e S S D A
A c o n v e r g e n c e an alysis is perform ed for a microstrip step discontinuity as sh o w n in F ig ­
ure 5 .1 3 .
1
........................ t
-------
•*-T--.................
t
■- -
~"T.......... .
*---- -t- - -*
09
0,8
-
■
......
0.7
t2
06
w
—
1-
- - *■—
■*
----- r
..........
-1 - -
.......r -----------T -.
;— . ■ * : ..................... r i - ~ - ■
....
.
■ -t
n z= 3 0
nz=40
n z= 5 0
o o o
+ + +
[29)
[2 9 ]
s 21
n z= 6 0
1 05
rS
09 0 .4
------— - ---------------
—
----------- -----
_
03
S 11
..,* 1 1 1
0.2
W i
' n
W 2
0.1
- .................— -t-—
°c
2
- - J-------
4
...
F igu re 5 .1 3
x.......................X ------- --------- i~ ~ ---------
6
8
- X ............
10
12
Freq (GHz)
i
.............
14
J
16
, - _
18
-----------x. ...
20
,
C on vergen cy behavior o f the fu ll-w ave S S D A
w 1—1mm, w2=025m m , h=0,25mni, er=l()
W h en the num ber o f lin es (in the propagation direction z) is greater than 5 0 , co n ­
verg en ce is a ch ieved .
A s pointed o u t in C hapter 4, using the approach presented in [26] (u sin g hybrid
boundary co n d itio n s) w ill lead to an unstable solution, because the m atching co n d itio n is
applied to o n ly o n e line. U sin g the fu ll-w ave S S D /i, a stable solution can a lw a y s be
obtained.
5.4 Simulation Results of Full-wave SSDA
A co m p a riso n o f S-param eters obtained by the fu ll-w a v e S S D A and by others (i.e
|3 0 |) fo r a m icrostrip step d iscontinuity are show n in Figure 5 .I 4 and F igu re 5 . 15. A
g ood agreem en t can be ob served o v er the frequency range up to 20G H z.
65
1
09
SSD A
[29]
08
ooo
07
+ + + [29]
S 12
J
j
j
1
|
cD
o0 6
C
CD
§ 05
C_D
Q
^04
W,
.
,
L....
w2
j
}
ii
i
03
0 2-
+L,_„__~ .......
01
Su
4
i
2
4
6
8
10
12
Fieq (GH/)
14
1r>
18
,*i)
Figure 5.14 Full-w ave S-parameters of a microstrip step
W j ^ h n m , u’2=0 .5mm, 11=0 .25nun, e,.= I0
i .™~—
----- r... . . T, .... . . T’
r
•*
" *-*----- ‘r
s I2
os
08
t
A
-*»
wi
*r•
m k:
■
z n
. ;. AA*-. 1
- i*1* ..... .
...
...
r
r
+
♦
SSDA
[29]
fOOl
ooo
111
0.7
t2
a» 0.6
OJ
§ 0.5
ro
a.
«• 0.4
t
r
w2
..
S i,
0.3
0.2
0.1
0D.. --- A2 • .... A4 ....
X
6
i
8
i
*
10
12
Freq (GHz)
.
14
j
16
i
18
Figure 5.15 Full-w ave S-paramcters o f a microstrip step
w2=0.25mm, h=0.25mm, er =l()
i
20
66
Transmission characteristics o f two closely spaced m icrostrip step discontinuities
arc shown Figure 5.16 (from 2 to 40 GHz). It is evident from the Figure 5.16 (a) that
there is a strong interaction between both steps since their separation is less than half a
guided wavelength. This type o f structure is widely used in filter design. In general, it
behaves like a low-pass filter as in the case of l=1.20mm. The interconnecting stub also
contributes to the dip around 5GHz for the case o f 1=1.20m m. T he shorter the length 1, the
further the dip moves towards higher frequencies (as shown in the case o f 1=0.15mm).
Hence the stop band effect occurs only at higher frequencies. A lso the Q factor decreases
because the coupling between two hom ogeneous transm ission lines increases. The phase
characteristics o f the S-parameters are show n in Figure 5 .16(b).
0 .6 -,
0.5-
- - I = 1.20 rnm
... I = 0.60 mm
— I = 0.30 mm
I = 0.15 mm
0.3 -
0.2
f (G Hz)
(a)
100-,
50-
X,
o’
100-1---10
20
fto ii/,
•b)
40
Figure 5.16 S-parameters for a cascaded step discontinuity separated by a transm ission
line of length 1. W]=0.4mm, \\’o=0.2mm, \v2=0.Smm, er=3.8, h -0 ,2 5 tn m .
(a) M agnitude of S ,,, (b) Phase of S , j
67
5.5 Experimental Results
By using a netw ork analyzer at microwave and millimeter wave frequencies, it is
often im possible to directly measure the scattering parameters of a device under test
(DUT). D e-em bedding techniques must be used to obtain correct S-parameters (36 - 42].
Som e M H M IC discontinuities have been fabricated for experimental investigation.
Figure 5.17 to Figure 5.20 show the calculated (solid line) and measuremcnt(dashed line)
results perform ed at the University of Victoria. Five M HM IC’s with via holes are tested.
The de-em bedded S-param eters are compared with quasi-static SSDA simulations. All
M H M IC ’s are built on 0.254 mm conductor-backed substrate with dielectric constant
9.6. Ground plane via holes are used to suppress the microstrip mode. Figure 5.17 shows
the S-param eters o f a single CPW gap. Figure 5.18 investigates CPW double gaps with
two different resonator lengths. The SSDA calculation shows how the resonant peaks
move w hen the resonator length changes. Figure 5.18 illustrates the S-param ctcrs o f a
CPW step discontinuity.
68
[ i i-n-| T
i"i r
—
j"i—
i-i i | in
i| in i| i
i | i i i i | i i i-i-q
i i
(dB)
Experim ents
SSDA
S-parameters
0.0254
C i-9.6, h= 0.254m m
0
10
20
Frequency (GHz)
30
40
Figure 5.1.7 M easured and com puted S-parameters o f a CPW gap
SSDA L=3mm
SSDA L=1,5mm
Experiment U 3mm
Experiment L= 1.5mm
S12
(dB)
0.0508, L |
er=9.6, h=0.254nim
Figure 5.18 M easured and com puted S-param eters of end-coupled CPW resonators
69
S11
(dB)
SSDA
Experiment
er=9.6, h=0.254mm
1 0 0 -<L *
i
i i l <i i i I i i u
I i i i i i ■ ■ ■ ■i i
..............
Frequency (GHz)
Figure 5.19
M easured and computed S-parameters o f a C PW step discontinuity
70
/S
-10
-20
*o
--3 0
V
\
-40
SSDA
-60
3 □ El □ O l
- - M easurem ent
-50
26
27
28
29
\
l
31
32
Freq (GHz)
Figure 5.20 S-param eters of a CPW end-coupled filter. it'=(J.2, s= 0.I5, gap width:
25.4\im, resonate, length: 2mm.
Figure 5.20 show s the response of a CPW end-coupled filter. Via holes are also
used to short-circuit the ground-plane and the back-metallization. The SSDA simulation
exactly predicts the position and magnitude of all four resonant peaks which are gener­
ated by this 4-pole filter.
In summary, all experim ental results agree with our quasi-static SSDA simulation
very well.
71
Chapter 6
Conclusion
6.1 Contributions
A fter introducing the generalized form o f the S p ace-Spectral D om ain Approach
(S S D A ), a new d eterm inistic quasi-static S S D A has been d ev elo p ed to an alyze planar cir­
cuit d iscon tin uities.T his n ew approach extends the S S D A to calcu late q u asi-static capaci­
tances and S-param eters o f arbitrarily shaped planar d iscon tin u ities, T h e d iscontinuity
param eters are derived from an algebraic matrix equation instead o f an e ig e n v a lu e matrix.
T w o new approaches based on the fu ll-w a v e Space-S p ectral D o m ain Approach
(S S D A ) have been proposed to calculate scattering param eters for three-dim ensional d is­
continuity problem s in M M IC ’s and M H M IC ’s. T h e theory presented in this thesis d em ­
onstrates how to im p lem ent self-con sisten t hybrid boundary c o n d itio n s and how to derive
the determ inistic approach.
A com parison o f the results with other m ethods and m easu rem en ts sh o w s e x c e l­
lent agreem ent up to 40G H z. R esults o f so m e com p licated structures su ch as a M HM IO
m icrostrip and C P W taper, C P W air bridge and C P W en d -co u p led filter arc g iv e n . Exper­
im ental validation o f the theoretical results are presented. A lth ou gh the lim ited num ber
o f exa m p les g iv en is not representative for all type o f d isco n tin u ities, this technique by
nature can treat arbitrary planar two-port circuit contours, T h is is an advantage o f this
new m ethod w hich m akes it an attractive C A D tool for en g in eerin g ap p lication s.
72
In sum m ary, the major contributions o f this dissertation arc,
•
A g en eralized S S D A w as developed, and a com plete field form ulation w as
g iv en ,
•
A d eterm in istic quasi-static S S D A w as introduced.
•
T h e nature o f the S S D A w as investigated.
•
T he s e lf c o n siste n t hybrid boundary con d ition s have been d ev elop ed and
im plem ented into the full-w ave S S D A .
•
A determ inistic fu ll-w ave SS D A w as d evelop ed .
•
E xperim ental w ork w as done to verify the S S D A and to investigate other
planar structures.
•
A user frien dly com puter-aided design (C A D ) software package, M H M IC
2 .0 , w as d e v e lo p e d based on the work o f this thesis.
6.2 Future Work
A lth ou gh e x te n siv e sim ulation and validation w ere performed in this th esis and a
variety o f d iscon tin u ity m o d u les were included in our C A D software, a uscr-oricnted
so ftw a r e can be d e v e lo p e d for m ore com plicated discontinuities. Another approach is to
use the S S D A algorithm to generate libraries for certain popular CA D p ack ages such as
T O U C H S T O N E ™ or S U P E R C O M PA C T ™ .
T h e quasi-static S S D A is a very efficient tool to analyze most planar circu its. It can
be ex ten d ed to o p to -e le c tr o n ic applications, e.g., for the analysis o f field distribution o f an
electro -o p tica l m odulator as sh ow n in Figure 6.1.
Electrode
Waveguide
F igu re 6 . 1 Future application: electro-op tic modulator
73
S in c e the S S D A in its present form d ocs not include the effect o f finite m etallization
thick ness and conductor lo sse s, tw o different approaches arc proposed to incorporate
finite m etallization:
•
u tilize the concept o f surface im pedance as it is d on e in the SD M : add sur­
fa c e im pedance into the G reen ’s function to incorporate finite m etallization
and conductor loss; the form ulation w ill stay m o stly the sam e as in the
SSD A .
•
u sin g the m ode m atch ing m ethod (M M M ) in conjunction w ith the M oL:
u se M M M in the transverse sectio n to incorporate finite m etallization and
conductor loss, in the propagation direction (u su ally called /.-direction) the
M oL is used as sam e as in the S S D A .
T h e S S D A presumed in this th esis w as applied to a 3-port application in the c a se o f
the qu asi-static S S D A only (C hapter 5 ). Its form ulation w as changed significantly. W ith­
out m o d ify in g the form ulation, both quasi-static and full w a v e S S D A can be used to ana­
ly ze in -lin e m ulti-port d iscon tin u ities as show n in Figure 6 .2 , because the discretization in
the z-direction need not to be changed.
z
Figure 6.2
An in-line 3-port discontinuity
To analyze arbitrary m ulti-port discontinuities as shown in Figure 6,3, the z-direetion-discrctization scheme is not suitable anymore. A possible alternative m ay be to
choose a small region A and devclope a discretization schem e that takes into account the
special boundary conditions surrounding region A. Then m atch the field b etw een the d is­
con tin u ity region and all conn ected transm ission line separately.
Figure 6 .3
Arbitrary multi-port d iscontinuity
75
Bibliography
ii]
A. W exler, “Solution of W aveguide discontinuities by modal analysis,” IEEE
Trans, on M icrowave Theory Tech., vol. MTT-15, pp. 508-517, Sept. 1967.
[2]
E. Y am ashita and R. M ittra, “Variational method for the analysis of microstrip
line,” IEEE Trans, on M icrow ave Theory Tech., vol. A777-16, pp. 251-256, Apr.
1968.
[3]
R. M ittra and T. Itoh, “A new technique for the analysis of the dispersion
characteristics of microstrip lin es”, lEEETruns. on Microwave I'heory Tech,, vol
M T T -1 9 ,N o .l, pp. 47-56, Jan. 1971.
[4]
T. Itoh, “Spectral dom ain im m ittance approach for dispersion characteristics o f
generalized printed transm ission lines,” IEEE Trans, on Microwave Theory
Tech., vol. MTT-2S, pp. 733-736, July 1980.
[5]
T. U w ano and T. Itoh, “ Spectral domain approach,” in Numerical Techniques fo r
M icrow ave and M illimeter W ave Passive Structures, C h.5, T.ltoh (Ed.), John
Wiley, N ew York, pp. 334-380, 1989.
[6]
K.S. Yee, “Numerical solution of initial boundary value problems involving
M axw ell’s equation”, IEEE Trans, an Antenna and propagation, Vol. AP-14,
pp302-307, May 1966.
[7]
O, A. Liskoverts, “The M ethod of Lines,” Review, Differ. Uravneniya, vol. I ,
pp. 1662-1678, 1965.
[B]
U, Schulz and R. Pregla, “A new technique for the analysis of the dispersion
characteristics of planar w aveguide,” Arc/i. Elek. i/hertragung, Band 34, pp. 169173,1980.
[9]
S.B. W orm and R. Pregla, “H ybrid mode analysis o f arbitrarily shaped planar
m icrow ave structure by the m ethod o f lines”, IEEE Trans, on Microwave Theory
Tech., vol. MTT-28, pp. 191-196, Feb, 1984.
[10] U. Schultz, "C the edge condition with the method of lines in planar
w aveguides,” A. A Elek. Ueherlragung., vol. 34, pp. 176-178, 1980.
[11] R. Pregla and W. Pascher, “T he Method of Lines,” in Numerical Techniques fo r
M icrow ave and M illimeter W ave Passive Structures, C h.5, T.ltoh (Fid.), John
Wiley, N ew York, pp. 381 -446, 1989.
76
,l 2J S' o '
“Fuli:Wave ana]ysis o f discontinuity in planar waveguides by the
method o f lines using a source approach,” IEEE Trans, on M icrowave Theory
lech., vol. MTT-38, pp. 1510-1514, Oct. 1990.
°
fJ3j R. Pregla and W Pascher, “Diagonalization o f difference operators and system
ma nces ,n the Method of L ines,” IEEE M icrowave and G uided Wave I I t e r s
Vol. 2, No. 2, pp. 52-54, Feb. 1992.
’
f141 p T e!,i“ * A ‘ Couturier- C - R um elhard, C. V ernaeyen, P. C ham pion, and D.
-ttyo!
com pact, m onolithic m icrow ave dem odulator-m odulator for 6 4 0 A M
“
Si « f ' T n m - " " M ~
r W l' Tech"
™
115| H. Diestel anJ S. B. Worm, “Analysis of hybrid field problem s by the m ethod o f
lines with non-equidistant discretization,” IEEE Trans, on M icrowave Theory
lech., vol. MTT-32, pp. 633-638, June 1984.
'
fl6J ^
/• R‘ Hoefer- t r a n s m i s s i o n line matrix (TLM ) m ethod”, in N um erical
Undpp.
MUlimeU’r
(Ed.), John Wiley, New York,
496-591,WaW
1989.Passive ^ u e tu r e .s , Ch.5, T Itoh
1171 2
R Vahldieck, “A new method of m odeling three-dim ensional M IC /
.
° ircuits- The sPace-spectral domain approach,” IEEE Trans on
M icrowave Iheory and Tech., Vol. MTT-32, No.9, pp. 13-09-1318, Sept. 1990.
[18] M. Yu, K. Wu and R. Vahldieck, “A determ inistic quasi-static approach to
microstrip discontinuity problem in the space-spectral dom ain” IEEE M icrowave
and G uided Wave L e tt.y ol. 2, No. 3, pp. 114-116, M ar. 1992.
[19J K. Wu, M Yu and R. Vahldieck, “Rigorous analysis o f 3-D planar circuit
discontinuities using the space-spectral domain approach (SSDA) ” ™
Im iis. on Microwave Theory and Tech., Vol. MTT-40, No. 7, pp. 1475-1483, July
120] M. Yu R. Vahldieck and K. Wu, “Theoretical and experim ental characterization
I E
E
E
T
m
m
-
"
"
“ " <>
[21] J. M artel, R. R. Boix and M. H om o, “Static analysis o f m icrostrip discontinuity
using excess charge density in the spectral dom ain," IE E E Trans on M icrowave
Iheory and Tech.. Vol. MTT-39, No.9, pp. 1623-1631, Sept. 1991.
I22J M. Naghcd and I. Wolff, “Equivalent capacitance o f coplanar w avesuide
difference m ethod”
131
269, M arch 1990.
T
0* " *
‘
funte
“ * R,',Pre?la' “C° Uplin8 ° f Crossed W ™ m ulticonductor
' ""
n m * a m lT ” * - M TT-38, no. 3, pp. 265-
[24] P. Silvester and P. Benedek, “Equivalent capacitance of microstrip open end,"
IEEE Trans, on Microwave Theory and Tech., MTT-20, pp. 511-516, Aug. 1972.
[25] L. U rshev and A. Stoeva, “ Application of equivalent transmission line concept to
the m ethod o f lines," Microwave and Optical Technology Letters, Vol. 3, No. 10,
Oct. 1990.
[26] Z. Chen and B. Gao, “Deterministic approach to full-wave analysis o f
discontinuities in M lC ’s using the method of lines”, IEEE Trans, on Microwave
Theory Tech., vol 37, pp.606-611, March 1989.
[27] M . Y u, K. Wu and R. Vahldieck, “Analysis of planar circuit discontinuities using
the quasi-static space-spectral dom ain approach,” Proceedings o f International
Sym posium o f IEEE Microwave Theory and Tech., Albuquerque, USA, June, pp.
545-848, 1992.
[28] H. Yang, N. G. Alexopoulos and D. R. Jackson, “Microstrip open-end and gap
discontinuities in a substrate-superstrate structure,” IEEE Trans on Microwave
Theory le c h ., MTT-37, pp. 1542-1546, Oct. 1989.
[29] G. G ronau and I. Wolff, “A simple broad-band device de-embedding method
using an autom atic network analyzer with time domain option,” IEEE Trans, on
M icrow ave Theory and Tech., MTT-37, pp. 479-483, Mar. 1989.
[30] N.H .L. K oster, and R. H. Jansen, “The microstrip step discontinuity: a revised
description”, IEEE Trans, on Microwave Theory and Tech., MTT-34, pp. 213223, Feb. 1986.
[31] K. B eilenhoff, W. Heinrich, and H.L. Hartnagel, “The scattering behavior of air
bridges in coplanar M M IC’S ”, Proceedings o f the European Microwave
Conference pp. 1131-1135, 1991.
[32] C. Kuo, T. K itazaw a and T. Itoh, “Analysis of shielded coplanar waveguide step
discontinuity considering the finite metallization thickness effect”, Proceedings
o f 1991 International Symposium o f IEEE Microwave Theory and Tech., vol.2,
pp. 473-475, 1991.
[33] H. Jin and R. Vahldieck, “Calculation of frequency-dependent s-paramctcrs of
C PW air-bridge considering finite metallization thickness and conductivity” ,
Proceedings o f 1992 International Symposium o f IEEE Microwave Theory and
Tech., Albuquerque, NM, USA, 1992.
[34] Y. R. Sam ii, T. Itoh and R. M ittra, “A spectral domain analysis for solving
m icrostrip discontinuities problem s,” IEEE Trans, on Microwave Theory l ech ,
M TT-22, pp. 372-378, Apr. 1974.
[35] M , N aghed, M. Rittwegcr and I. Wolff, “A new method for calculation of the
equivalent inductances of coplanar waveguide discontinuities” , Proceedings o(
1991 International Symposium o f IEEE Microwave Theory and Tech., pp. 747750, 1991.
78
|3 6 | E. F, Da Silva, M. K. McPhun, “Calibration of an autom atic network analyzer
system using transmission lines o f unknown characteristic impedance, loss and
dispersion”, Radio Electron. Eng., Vol. 48, No. 5, pp. 227-234, May 1978.
(371 P.R. Shepherd and R. D. Pollard, “ Direct calibration and measurement o f
microstrip structures on gallium arsenide,” IEEE Trans, on M icrowave 'Theory
Tech., Vol. MTT-34, pp. 1421-1426, No. 1986.
|3 8 | G. R. Engen and C. A. Hoer, “Thru-Reflect-Line: An im proved technique for
calibrating the dual six-port autom atic network analyzer,” IEEE Trans, on
Microwave Theory Tech., Vol. M TT-27, pp.987-993, Dec. 1979.
[39] R. R. Pantoja, M. J. Howes, J. R. Richardson and R. D. Pollard, “Improved
calibration and measurement o f the scattering param eters o f microwave
integrated circui ts,” IEEE Trans, on M icrowave Theory Tech., Vol. MTT-37, pp.
1675-1680, Jan. 1989.
(40] R. Lane, “De-embedding device scattering param eter,” M icrow ave J., Vol.27, pp.
149-156, Aug. 1984.
]41] H. J. Eul and Shiek, “Thru-M atch-Reflect: An improved technique for calibrating
the dual six-port automatic netw ork analyzer,” Technical Report, RuhrUniversity Bochum, Institut fu r H och and flockstfrecpienztechnik, Bochum,
Germany.
[42] M. Yu, “ M icrowave Device D e-em bedding Technique,” Project Report o f
ELEC454, M icrowave and O ptical Communication System s, Department of
Electrical and Com puter Engineering, University o f Victoria BC, Canada, 1992.
[43] A. Hill and V. K. Tripathi, “An efficient algorithm for the three-dimensional
analysis o f passive microstrip com ponents and dicontinuities for microwave and
millimeter-wave integrated circuits,” IEEE Trans, on M icrow ave Theory Tech.,
Vol. M TT-39, pp. 83-89, Jan. 1991.
71
Appendix
The total num ber o f lines in the z-dircction:
N = N ,+ N /() Nfl=Nfl,+Nf
(F.4)
In region A, one unknown is enough to represent C / because o f the traveling wave
assumption. In region C, the same principle applies. The total number o f unknowns:
C /: I
C/jir: Nfl/-f 1
C / and C flir can be expressed as
/[}//
e '«
r: -
a2'
(K5)
r
aN, +\'(lN„i +2 1.
('
'|W'-
(Kt>)
Com bining all left terms of equation (4.27) into one matrix | W ,) yields
(K 7)
M* - K K
where
M'',.'i
MV'
A = \a,t a2, a i,...,aN +2 j is the unknown coefficient vector.
When N/t>N ^/+2, equation (4.27) can be solved.
(m
VITA
Surname: Yu
Place of Birth:B eijing, C h in a
Given Names;Ming
______
_____________
Date o f Birth:June
1962
_____
1iducational Institutions Attended;
University of Victoria, Victoria, BC, Canada
Tsinghua University, B eijing, P.R. China
Degrees Awarded:
• M S. in Electrical Engineering, July 1986, Tsinghua University, Beijing, P.R. China
Thesis: 11GHz D ual-m ode Linear Phase Filter
•
B.S. with Honor in Electrical Engineering, July 1985, Tsinghua University,
Beijing, P.R. China
1 l o nou rs and Awards:
-
G.R.E.A.T. Aw ard, B.C . Science Council ( i 992-1993)
•
Graduate Studies Fellow ship, University of Victoria (1990)
•
Graduate Teaching Award, University of Victoria (1991)
•
Graduate Teaching Fellowship, University of Victoria (1991,1992)
Publications:
ReFereed Jo u rn a l P a p e rs Published
1, M . Yu, R. Vahldieck and K, Wu, “Theoretical and Experimental Characterization of
Coplanar Waveguide Discontinuities”, IEEE Trims. on Microwave Theory and Tech.,
MTT-41, No.9, Sept. 1993.
2, K. Wu, M . Yu and R. Vahldieck, “Rigorous analysis of 3-D planar circuits disconti­
nuities using the space-spectral domain approach (SSDA)”, IEEE Trans, on Micro
wave Theory and Tech., MTT-40, No,7, pp. 1475-1483, July 1992.
3, M . Yu, K. Wu and R. Vahldieck, “A Deterministic Quasi-static Approach to Micros*
trip Discontinuity Problem in the Space-Spectral Domain”, IEEE Microwave and
(lid d ed Wave Led., Vol.2, No.3, pp. 114*116, Mar. 1992.
Partial Copyright License
I hereby grant the right to lend m y thesis (or dissertation) to users o f the U n iversity o f
Victoria Library, and to m ake sin gle co p ie s o n ly for such users or in resp on se to a
request from the Library o f any other university, or sim ilar institution, on its b eh alf or
for o n e o f its users. I further agree that p erm ission for ex ten siv e c o p y in g o f this thesis
for scholarly p u ip o ses m ay be granted by m e or a m em ber o f the U n iv ersity designated
by m e. It is understood that co p y in g or publication o f this thesis for financial gain shall
not be allow ed w ithout m y written perm ission.
T itle o f T hesis/D issertation:
^ Hybrid Nuniei ical rechni(|iio lor the Analy­
sis and Design of Microwave Integrated Circuits
Author
(Signature)
M IN G Y U
(N am e in B1o_ k Letters)
(D ate)
Документ
Категория
Без категории
Просмотров
0
Размер файла
3 242 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа