close

Вход

Забыли?

вход по аккаунту

?

Analysis and implementation of multi-layered configurations to reduce mode coupling problems in conductor-backed coplanar waveguides for wide frequency-band microwave integrated circuit applications

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI
films the text directly from the original or copy submitted. Thus, some
thesis and dissertation copies are in typewriter face, while others may be
from any type of computer printer.
T he quality of this reproduction is dependent upon the quality o f the
copy submitted.
Broken or indistinct print, colored or poor quality
illustrations and photographs, print bleedthrough, substandard margins,
and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete
manuscript and there are missing pages, these will be noted.
Also, if
unauthorized copyright material had to be removed, a note will indicate
the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and
continuing from left to right in equal sections with small overlaps. Each
original is also photographed in one exposure and is included in reduced
form at the back o f the book.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6” x 9” black and white
photographic prints are available for any photographs or illustrations
appearing in this copy for an additional charge. Contact UMI directly to
order.
UMI
A Bell & Howell Information Company
300 North Zeeb Road, Ann Aibor MI 48106-1346 USA
313/761-4700 800/521-0600
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ANALYSIS AND IMPLEMENTATION OF MULTI-LAYERED
CONFIGURATIONS TO REDUCE MODE COUPLING PROBLEMS IN
CONDUCTOR-BACKED COPLANAR WAVEGUIDES FOR WIDE
FREQUENCY-BAND MICROWAVE INTEGRATED CIRCUIT APPLICATIONS
A Dissertation
by
MARK ALEXANDER MAGERKO
Submitted to the Office o f Graduate Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
May 1996
Major Subject: Electrical Engineering
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: 9634803
UMI Microform 9634803
Copyright 1996, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
300 North Zeeb Road
Ann Arbor, MI 48103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ANALYSIS AND IMPLEMENTATION OF MULTI-LAYERED
CONFIGURATIONS TO REDUCE MODE COUPLING PROBLEMS IN
CONDUCTOR-BACKED COPLANAR WAVEGUIDES FOR WIDE
FREQUENCY-BAND MICROWAVE INTEGRATED CIRCUIT APPLICATIONS
A Dissertation
by
MARK ALEXANDER MAGERKO
Submitted to the Office of Graduate Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Approved as to style and content by:
Kai Chang
(Co-Chair of Committee)
Steve Wright
(Member)
Cam1
mNguyen
(Co-Chair of Committee)
Jianxin Zhou
(Member)
A.D. Patton
(Head o f Department)
Mark Weichold
(Member)
May 1996
Major Subject: Electrical Engineering
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ABSTRACT
Analysis and Implementation of Multi-Layered Configurations to Reduce Mode Coupling
Problems in Conductor-Backed Coplanar Waveguides for Wide Frequency-Band
Microwave Integrated Circuit Applications. (May 1996)
Mark Alexander Magerko, B.S., University of Illinois Champaign-Urbana, Illinois;
M.S., University of Illinois Champaign-Urbana, Illinois
Co-Chairs o f Advisory Committee: Dr. Kai Chang
Dr. Cam Nguyen
Packaged conductor-backed coplanar waveguides (CBCPW) for microwave integrated
circuits (MICs) can support rectangular waveguide and cavity modes within the substrate
regions which can couple with the dominant quasi-TEM mode and produce several
undesirable results. Lateral sidewalls connect the upper and lower ground planes of the
structure together at the substrate edges in a wrap-around configuration using copper tape
or through the package itself. Strong mode coupling can cause the dominant mode field
pattern to spread out across the entire waveguide width instead of being confined to the
slot area. This phenomenon translates into a significant loss of power and the occurrence
of strong resonances in the transmission measurements which limits the bandwidth of the
waveguide even at lower frequencies. Multi-layered configurations with dielectrics loaded
above and below the circuit conductors are employed to reduce the mode coupling without
restrictions on the cross section (S+2W) or the lateral width (2A) of the waveguide which
is particularly useful for MICs.
The above effects associated with CBCPW are
demonstrated and analyzed using waveguide theoiy, experimental data, and the spectral
domain numerical method in one, two, and three dimensions. Leakage calculations for the
dominant mode in the one-dimensional case are related to the mode coupling effects in
two-dimensions. Summary tables and practical guidelines describing design tradeoffs for
wide frequency-band operation are also presented. The inclusion o f lossy damping layers
to reduce the residual resonances associated with the substrate cavity without affecting the
CBCPW resonator circuit Q are described with a three-dimensional procedure.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental data to 40GHz is utilized to verify the bandwidths for several of the
configurations, confirm the explanations o f the mode and cavity coupling effects, and
demonstrate the improved responses with the incorporation of the absorbing material.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
To my parents and my friend Donna
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ACKNOWLEDGMENTS
I would like to express my appreciation to Dr. Kai Chang for his guidance and
encouragement throughout this work. My thanks are also extended to the members o f my
committee especially my co-chair Dr. Cam Nguyen for their interest in this study. I would
also like to thank the U.S. Department o f Education - Areas of National Need Program
through the Department of Engineering at Texas A&M University for providing a
fellowship and the Army Research Office for supporting this work. I am also appreciative
to Lu Fan for his invaluable assistance in this research effort, Rogers Corporation for the
donation of the Duroid™ dielectric boards used in the experimental characterization, and
technical discussions with Professors N.K. Das and A. A. Oliner o f Polytechnic University,
Dr. M.L. Riaziat of Varian Research Center, Professor R. W. Jackson of the University of
Massachusetts.
Finally, I would like to dedicate this dissertation to my friend Donna whose patience
and understanding during our marriage made the goal of obtaining this degree possible and
to my parents John Sr. and Mary Ann for their constant support, guidance, and sacrifice
made throughout my life.
I would also like to thank Nan Vaughn for her emotional
encouragement and assistance during the writing of this manuscript.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
vii
TABLE OF CONTENTS
Page
ABSTRACT........................................................................................................................
iii
DEDICATION....................................................................................................................
v
ACKNOWLEDGMENTS..................................................................................................
vi
TABLE OF CONTENTS..................................................................................................
vii
LIST OF TABLES..............................................................................................
xi
LIST OF FIGURES...........................................................................................................
xv
CHAPTER
I
INTRODUCTION........................................................................................
A.
B.
C.
D.
E.
II
Comparison Between Microstrip and Coplanar Waveguides.............
Overview of Conductor-Backed Coplanar Waveguides......................
Problem Definition and Research Objective........................................
Literature Review and Discussion.......................................................
Dissertation Organization.....................................................................
I
2
4
10
18
20
THEORETICAL ANALYSIS OF CBCPW USING THE SPECTRAL
DOMAIN METHOD......................................................................................
A.
B.
C.
D.
E.
F.
G.
General Spectral Domain Method Formulation..................................
Basis Function Selection......................................................................
Two-Dimensional Packaged Structures..............................................
Spectral Domain Method Field Formulation.......................................
Characteristic Impedance Derivation...................................................
Analysis of Microstrip and Finite Ground Plane CBCPW..................
Spectral Domain Method Numerical Analysis and Results................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
24
33
36
39
43
47
52
v iii
CHAPTER
IE
IV
Page
ONE-DIMENSIONAL ANALYSIS AND DESIGN OF CBCPW
INCLUDING LEAKAGE EFFECTS........................................................
63
A. Experimental Demonstration of the Leakage Effects in CBCPW
B. Explanation o f Leakage Effects in Printed-Circuit
Transmission Lines...............................................................................
C. Spectral Domain Method Including Leakage Analysis......................
D. Numerical Results of Leakage Effects for CBCPW..........................
E. Limitations of the One-Dimensional Leakage Analysis for
CBCPW.................................................................................................
64
90
TWO-DIMENSIONAL ANALYSIS OF CBCPW WITH LATERAL
SIDEWALLS INCLUDING MODE COUPLING EFFECTS.................
96
.......................................................
CBCPW Mode Identification
Mode Coupling Effects in CBCPW with Lateral Sidewalls..............
Additional Examples o f the Mode Coupling Effects in CBCPW
Experimental Results Confirming the Two-Dimensional SDM
Analysis o f CBCPW............................................................................
E. Mode Coupling Analysis o f Finite Ground Plane CBCPW.................
99
105
112
A.
B.
C.
D.
V
67
76
82
115
119
DESIGN SUMMARY INFORMATION FOR THE PROPAGATION
CHARACTERISTICS OF MULTI-LAYERED CBCPW.......................
A. Propagation Characteristics o f Upper Dielectric Loaded
CBCPW.................................................................................................
B. Design Summary Information for Multi-Layered CBCPW MICs
C. Additional Configuration Considerations for Multi-Layered
CBCPW MICs.....................................................................................
D. Minimizing the Mode Coupling Effects in Single-Layer CBCPW
MICs......................................................................................................
E. Design Procedures for Single-Layer CBCPW...................................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
122
124
130
138
146
154
ix
CHAPTER
VI
Page
THREE-DIMENSIONAL ANALYSIS OF MULTI-LAYERED
CBCPW INCLUDING CAVITY EFFECTS............................................
A. Overview of the Cavity Effects in CBCPW........................................
B. SDM Analysis of Three-Dimensional CBCPW Resonator
Circuits..................................................................................................
C. Verification o f the Three-Dimensional SDM Procedure...................
D. Design Examples for Multi-Layered CBCPW Resonator
Circuits with Damping..........................................................................
E. Experimental Demonstration of Cavity Damping for CBCPW
Resonators............................................................................................
VII
158
163
173
179
183
189
CONCLUSION.............................................................................................
193
A. Original Contributions..........................................................................
B. Suggestions for Future Work...............................................................
194
196
REFERENCES...................................................................................................................
197
APPENDIX
A
SPECTRAL DOMAIN IMMITTANCE FORMULATION......................
B
DERIVATION OF THE FIELD COEFFICIENTS FOR THE
204
SPECTRAL DOMAIN METHOD IMMITTANCE METHOD..............
209
C
DERIVATION OF THE POYNTING VECTOR INTEGRAL................
212
D
NUMERICAL METHODS WITHIN THE SPECTRAL DOMAIN
METHOD.....................................................................................................
A.
B.
C.
D.
E.
Bessel Function Approximation...........................................................
Numerical Integration...........................................................................
Determinant of a Matrix......................................................................
Root Searching Algorithm...................................................................
Basis Function Expansion Coefficients Solution.................................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
216
216
216
219
221
223
APPENDIX
E
Page
PROPAGATION CHARACTERISTICS OF IDEAL PARALLEL
PLATE WAVEGUIDE MODES..............................................................
228
F
DERIVATION OF RESIDUE CALCULUS THEOREM.........................
231
G
IEEE PERMISSION TO REPRINT COPYRIGHTED MATERIAL
234
VITA................................................................................................................................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
235
xi
LIST OF TABLES
TABLE
I
Page
Comparison o f the effective dielectric constant for an infinite-width
CBCPW with Ref. [46] for £rl= l, Er4= 13>
h^lOmm,
h4=0.75mm, h5=10mm, h2=h3=h6=0, S=2mm, W=l.5mm.............................
55
Comparison o f the effective dielectric constant with different basis
function for the example of Table I with S=0.6mm, W=0.45mm,
f=20GHz............................................................................................................
55
SDM integration convergence o f the propagation constant for the
example of Table II with tan 54=0.001 (loss tangent).....................................
55
Comparison of the effective dielectric constant for packaged CBCPW
with Ref. [21] for srl= l, er4=9.6, er5=l, h,=3mm, h4=lmm,
hs=3mm, h2=h3=h6=0, S=2mm, W=lmm, A=7.5mm. presented work
refers to an infinite-width structure with the same parameters for
comparison purposes..........................................................................................
58
Comparison o f the effective dielectric constant for the example of
Table IV with Ref. [21] at f =30GHz for the various modes..........................
58
Comparison of the characteristic impedances for packaged CBCPW with
Ref. [43] for erl= l, sr4=I3, er5=l, 1^=5mm, h4= 1mm, h5=5mm,
h2=h3=h6=0, S=lmm, W=0.4mm, A=7.5mm...................................................
60
Convergence results for the effective dielectric constant and impedance in
terms of the number of spectral terms for packaged CBCPW with srl=l,
er4=10.2, erS=2.2, 1^=5mm, h4=0.635mm, h5=0.635mm, h2=h3=h6=0,
S=0.254mm, W=0.89mm, A=12.5mm, f =20GHz..........................................
61
VIII Basis function expansion coefficients for example in Table VII for M=3
and N=2............................................................................................................
61
II
III
IV
V
VT
VII
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xii
TABLE
IX
X
XI
XII
XIII
XIV
XV
XVI
Page
Comparison of the effective dielectric constant for FGP packaged
CBCPW with Ref. [47] for erl=I, sr4=10, er5= l, h,=10mm,
h4=lmm, h2=h3=h5=h6=0, S=lmm, W=lmm, WG= lmm, A=12.5mm..........
62
Identification of PPMs for multi-layered structure of Fig. E.l with
sr4 =10.8, er5 =2.33 and h4=2.54mm, h5=0.71mm at f=20GFIz........................
78
SDM integration convergence o f the leaky propagation constant for
CASE G in Fig. 19 for single-layer CBCPW at 20GHz...................................
81
Comparison of the normalized leakage rate for conductor-backed slotline
with Ref. [31] for sr=2.25, h=8mm (dielectric thickness), f=10GHz as a
function of the slot width d...............................................................................
85
Comparison of critical frequencies up to 36GHz for CBCPW for
measured data, SDM analysis, predicted with maximum dimensional
uncertainties, and predicted with maximum dimensional uncertainties
and air gap at h5..................................................................................................
93
Measured and predicted resonant cavity frequencies for CBCPW
CASEL from Fig. 30 and for T E 0I/b modes.................................................
95
Lower dielectric loaded multi-layered CBCPW MIC design summary.
erl= l, sr4=10.2, er5=2.2 and h,=5mm, h2=h3=h6=0 and 2A=25.4mm
and S+2W<2.032mm, Smax= 10W, Wmin=0.127mm and refer to
Fig. 13 for dimensional parameters and
CASE 20: S=2W=0.635mm CASE 21: S=W=0.635mm
CASE 22: S=10W=1.27mm CASE 23: S=0.254mm, W=0.889mm.........
132
Upper dielectric loaded multi-layered CBCPW MIC design summary.
srl= l, sr3=10.2, sr4=2.2 and h,=5mm, h2=h5=h6=0 and 2A=25.4mm
and S+2W<2.032mm, Smax=10W, Wmin =0.127mm and refer to
Fig. 13 for dimensional parameters and
CASE 24: S=2W=0.635mm CASE 25: S=W=0.635mm
CASE 26: S=10W=1.27mm CASE 27: S=0.254mm, W=0.889mm.........
133
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
x iii
TABLE
Page
XVII Cutoff frequencies for the TM„, rectangular waveguide mode as a
function o f the lateral width or various CBCPW configurations.
Fig. 13 with h6=0 is referenced for these lower loaded multi-layered
examples. *arbitrary height requires h4<2A....................................................
139
XVIII Specification of the dielectric constant for the lower loaded
multi-layered CBCPW structures using the closed form approximation
procedure at. f=20GHz. Fig. 13 is referenced with erl =1, h,=5mm,
h2=h3=h6=0 and 2A=25.4mm..........................................................................
140
XIX
XX
XXI
Dielectric uncertainty effects on the upper usable frequency of
multi-layered CBCPW for S=W=0.635mm and 2A=25.4mm. CASES
40-42 are the lower loaded configuration of example 2C) from Table
XV and CASES 43-45 are the upper loaded waveguide example 2C)
from Table XVI............................................. ...................................................
143
Measured and calculated resonant cavity frequencies for multi-layered
CBCPW of CASEQ from Fig. 42 fortheTM0<1^ modes......................
159
Resonant frequency verification of the TM0 u mode in a packaged
microstrip structure with microwave absorber damping. Reference
Fig. 61 for dimensions with sr4=10.5(l-y'0.0023), sr5 =1,
=21(1-/0.02),
^ = 1 . 1( 1 - 71 .4 ), and h4=1.27mm, h5=10.16mm, h6=1.27mm and
2A=15mm and 2B=24mm.................................................................................
172
XXII Q verification for various modes in a packaged microstrip structure
with doped silicon damping. Reference Fig. 61 for dimensions with
'
15
1
s r4=12.7, sr5= l,
= 12 and «e=4.0 x 10 cm' and h4=0.1mm,
h5=0.4mm, h6=0.25mm and 2A= 12mm and2B=20mm..................................
173
XXIII Resonant Q verification o f the dominant mode in a microstrip structure
with Si damping. Reference Fig. 58 and Fig. 64(b) for dimensions
/
with
2A=12mm,
16
2B=20mm
and
WG=0
with
= 12
and
,
n= 3.0 x 10 cm ', sr2= l, sr4=12.7 and h,=h4=0.1mm, h3=h5=h6=0
and S=0.2mm and 2L=0.775mm......................................................................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
181
xiv
TABLE
XXIV Resonant Q verification of the TM0 22 mode in a CBCPW packaged
structure with loss tangent damping at h5. Reference Fig. 58 and
Fig. 64(a) for dimensions with srl= l, s r4=Er5 = 12.8, 5 4 =0.002 and
h4=h5=0.125mm, h2=h3=h6=0 and S=0.025mm, W=2S, 2L= 1.0mm
and 2A=2mm, 2B=3mm.......................................................................................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Page
183
XV
LIST OF FIGURES
FIGURE
1
2
3
4
5
6
Page
Active device grounding in coplanar waveguides and microstrip, (a) Top
view, (b) Side view..........................................................................................
3
Circuit photograph of GaAs conductor-backed coplanar waveguide
(CBCPW) MMIC distributed amplifier............................................................
5
Basic CBCPW configurations, (a) Infinite-width. (b) Packaged with
shorting sidewalls, (c) Packaged with finite ground planes. S, W, WG,
and 2A are the widths of the center strip conductor, slots, ground plane
conductors, and the lateral sidewall separation respectively. The
conductor-backed plane exists at x=-h and the cover plate conductor at
x=5mm..............................................................................................................
7
Fundamental modes of CBCPW along with the source topologies
including the cross-sectional vector electric field plots for the two-layer
configuration, (a) Dominant or odd mode, (b) Slotline or even mode.
(c) Microstrip-like or parallel plate mode.......................................................
8
Multi-layered CBCPW configurations, (a) Longitudinal view o f
experimental CBCPW with shorting sidewalls, (b) Cross-sectional view
of CBCPW used in the numerical analysis.....................................................
9
Approximate representations o f packaged CBCPW. (a) Ideal rectangular
waveguide, (b) Ideal rectangular cavity.........................................................
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xvi
FIGURE
7
8
9
10
Page
Experimental responses demonstrating waveguide and cavity modes in
CBCPW with shorting sidewalls and 2A=2B=38mm, S=2W=0.635mm,
srl=Er2=er3=er6= l, h,=oo (open structure), h,=h3=h6=0. Refer to Fig. 5(b)
for the dimension parameters. Case A 50Q microstrip line sr4=10.8,
h4=0.635mm and sr5= l, h5=0, CaseB same as Case A for CBCPW,
Case C same as Case B except WG=lmm, Case D same as Case B except
h4=l.27mm and 8^=2.33, h5=0.381mm, Case E same as Case D except
h4=0.635mm and h5=0.711mm Case F same as Case E except broadband
absorber placed in the connector housing blocks. Cases A,B,C,D are
lOdB/div while Cases E,F are 5dB/div and all are referenced to OdB
13
Test fixture illustrations incorporating CBCPWtransmission lines.
(a) Shorting bar application to ground input and output ports of
CBCPW. Connector housing blocks............................................................
14
Alternative coplanar waveguide configurations, (a) Channelized
coplanar waveguide, (b) Via hole structure..................................................
16
Examples of structures easily simulated using the spectral domain
method, (a) Cross-section of a multi-layered planar structure with
multi-metallization layers, (b) Cross-sections o f CBCPWs that are open,
laterally open, and shielded or packaged from left to right,
respectively
23
11
Multi-layered infinite-width CBCPW cross-section.......................................
25
12
Shapes of the electric field basis functions for CBCPW with
S=0.635mm, W=0.3175mm. (a) Eym basis functions, (b) E.n basis
functions...........................................................................................................
35
13
Multi-layered finite-width CBCPW cross-section.........................................
38
14
Additional structures analyzed using the SDM with air above the
conductors at x=0. (a) Microstrip waveguide, (b) Finite ground plane
CBCPW............................................................................................................
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xvii
FIGURE
15
16
17
18
19
Page
J2 idealized current components for the waveguides under analysis.
(a) Dominant mode for microstrip with S=0.635mm. (b) Coplanar
waveguide
mode for
the FGP CBCPW with S=0.635mm,
W=0.3175mm, WG=0.635mm. (c) Coplanar-microstrip mode for the
FGP CBCPW.................................................................................................
50
Flowchart for computing the propagation constant of the infinite-width
CBCPW.........................................................................................................
54
Flowchart for computing the propagation constant of the packaged
CBCPW.........................................................................................................
57
Cross-sectional vector magnetic fields plot for packaged CBCPW. erI= 1,
sr4=10.2, sr5=2.2, h1=3.73mm, h4=0.635mm, h5=0.635mm, h,=h3=h6=0,
S=2W=0.635mm, A=12.5mm, f =5GHz...................................................
59
Experimental data of the leakage effects in CBCPW MIC with open
sidewalls and
2B=38mm,
S=2W=0.508mm, WG=18.492mm,
h,=oo (open structure), h2=h3=h5=h6=0. Refer to
Fig. 11 for the dimension parameters. CASE G 50Q line Er4= l 0.8,
h4=0.635mm, CASE H same as CASE G except h4=0.71mm and
sr5=2.33 for 95Q line, and CASE 1 is the numerical representation
for the dielectric loss of CASE G. All cases are referenced to 0 dB..
66
Conductor-backed slotline leaking power to a parallel plate mode for
(a) the top view and (b) the side view. The angle o f leakage 0 into the
the parallel plate mode of wave number kc is also shown, k is the
transverse wave number in the ^-direction of the excited parallel
plate mode and k2 is the wave number o f the dominant mode guided
along the r-direction......................................................................................
69
Three-dimensional illustration of leakage in a semi-cone region from a
pulse propagating on a coplanar stripline.....................................................
71
Leakage effects in conductor-backed slotline demonstrated by probing
the Ex field distribution transversely cross the j-axis. The leakage
peak angle is 31.8° which translates to a maximum field component at
^ = ±12.6cm. Experimental data approximately confirms the predicted
leakage angle from Ref. [26]..........................................................................
75
e rl=E r2= s r3=Er5= 8 r6= l ,
20
21
22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xviii
FIGURE
23
24
25
26
27
28
Page
Contours of spectral domain integrations Cs in complex k plane. The
respective poles positions P's are indicated. P0 and P'0 are for no loss
case while Pj and P'j correspond to P 0 and P'0 fo ra lossy waveguide.
P2 and P'2 correspond to P3 and P'3 for a highly lossy structure . All
of these poles refer to a nonleaky waveguide and utilize the contour
C, along the real axis. P3 and P'3 and P4 and P'4 refer to leaky
waveguide case and follow the integration path C3 coinciding with the
real axis and deformed round the leaky poles in a residue calculus
sense...............................................................................................................
79
Normalized propagation constants for the CBCPW dominant mode (P)
and the TEM PPM (&.). One-dimensional SDM analysis for
experimental data of Fig. 19 with CASE 2 and CASE 3 corresponding
to CASE G and CASE H, respectively......................................................
83
Normalized propagation constants for multi-layered CBCPW dominant
mode(p) and the TMq PPM (kc ) with erl=l, sr4 =10.8 and h,=5mm,
h2=h3=h6=0 and S=2W=0.635mm.
CASE 4 with srS=6 and
h4=0.635mm, h5=5mm. CASE 5 with srS=2.33and h4=0.635mm,
h5=0.71mm. CASE 6 same as CASE 5 except h4=1.27mm.............
84
Normalized leakage rate comparisons with Ref. [32] and [33] with srl =1
and h,=5mm, h2=h3=h5=h6=0. CASE 7 with er4 =13 and h4=0.2mm
and S=W=0.1mm. CASE 8 with sr4 = 10 and h4=0.4mm and
S=2W=2.0mm................................................................................................
86
Normalized leakage rates (aIk0) for CBCPWs. CASE 2 and CASE 3 are
referenced from Fig. 24. CASE 6 is from Fig. 25. CASE 9 same as
CASE 2 except h4=2.54mm. CASE 2 is scaled by a factor o f three
88
Leakage design curves for various CBCPW examples with erl= l and
h,=5mm. (a) Proper dielectric thickness at 20GHz for upper and
lower loading structures with S=2W=0.635mm. CASE 10 with
sr3 =10.8, er4 =2.33 and h4=0.71mm, h2=h5=h6=0 and references the
upper dielectric thickness scale. CASE 11 is the same as CASE 6
of Fig. 25 with h5 as the variable and references the lower scale
(b) Proper relative dielectric constant at 30GHz for lower loading
waveguide. CASE 12 is the same as CASE 6 except with er5 as the
the variable. CASE 13 same as CASE 7 except h5=0.635mm. Log
scales are used for the leakage rate..........................................................
89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xix
FIGURE
29
30
31
32
33
34
35
Page
Experimental data of the leakage effects in multi-layered CBCPW MICs
with
open
sidewalls
and
2B=38mm,
S=2W=0.635mm,
WG=18.49mm, Brl=sr2=Er3=srt=l, er4=10.8, sr5=2.33 and h,=oo
(open structure), h2=h3=h6=0. CASE I is measured data for
CASE 5 h4=0.635mm, h5=0.71mm. CASE J is the experimental
results for CASE 6 h4=1.27mm, h5=0.71mm. CASE K same as
CA SEJ except h5=0.381mm. Refer to Fig. 11 for the dimension
parameters. CASE K is referenced 5dB down from CASE J S21
and CASE J S,, corresponds to the top scale...............................................
91
Experimental data demonstrating the resonant cavity modes within an
overmoded single-layer CBCPW. CASEL with srl=sr2=er3=er5=sr6=l,
sr4=10.8 and h,= oo (open structure), h2=h3=h5=h6=0, h4=0.635mm
and S=W=0.508mm, A=19mm, B= 12.7mm. Refer to Fig. 13 and
Fig. 5(a) for dimensional parameters.............................................................
94
Idealized dispersion curves for single-layer CBCPW with lateral
sidewalls for CASE 14 and erl= l, sr4=10.8 and h2=h3=h5=h6=0,
h,=5mm, h4=0.635mm and S=2W=0.508mm and A=19mm. Refer to
Fig. 13 for dimensions. The CBCPW mode response is from CASE 2
of Fig. 24 and superimposed onto the graph. Ideal RW are the TM0*
rectangular waveguide modes............................................................................
Example for the variation o f the SDM determinant with fictitious
propagation constants at 7.5GHz for CASE 15 with erl= l, sr4=10.8,
er5=2.33 and hj=5mm, h4= 1.27mm, h5=0.381mm, h2=h3=h6=0 and
S=2W=0.635mm and A=19mm.......................................................................
Cross-sectional vector electric field plots for CASE 15 at 15GHzfor
(a) CBCPW mode and (b) T M ^ mode.....................................................
98
100
102
Cross-sectional vector electric field plots of CASE 15 at 15GHz for
TMqj mode, (a) Ideal rectangular waveguide analysis, (b) SDM
analysis...............................................................................................................
103
Electric field plot of CBCPW and first three waveguides modes of
CASE 15 at 15GHz and x=-0.01mm................................................................
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
XX
FIGURE
36
37
38
39
40
41
Page
Dispersion curves of CASE 15 from Fig. 32 for CBCPW and waveguide
modes demonstrating the mode coupling effects. SDM RW and Ideal
RW refers to rectangular waveguide modes from the two different
analysis. Mode A is the top continuous dispersion curve and Mode B is
the second continuous curve........................................................................
Field plots at various frequencies for CBCPW mode o f CASE 15
demonstrating the field spreading effects due to mode coupling with the
waveguide modes.............................................................................................
107
109
Additional mode coupling effects in two-dimensional CBCPW CASE
15. (a) Characterisitc impedance plots for the modes indicated in Fig 36.
Modes A and B refer to the lower frequency range (8-10.1 GHz),
(b) Calculated coupling coefficient of the waveguide modes to the
CBCPW mode...............................................................................................
111
Additional examples of field spreading mode coupling effects in
CBCPW.
(a) Single-layer structure CASE 14 o f Fig. 31.
(b) Multi-layered structure CASE 16 same as CASE 15 except
h4=2.54mm, hs=0.71mm...............................................................................
113
Multi-layered CBCPW example demonstrating a bound dominant
mode to 40GHz. CASE 5 here is same as CASE 15 except
h4=0.635mm, h5=0.71mm. (a) Dispersion curves, (b) Plot for Ex field
component.....................................................................................................
114
Experimental data o f the mode coupling effects in CBCPW MICs with
lateral sidewalls with erl= l and h,= oo (open structure), h2=h3=h6=0
and 2B=38mm. Refer to Fig. 13 and Fig. 5(a) for the dimension
parameters. For CASE L sr4=10.8 and h4=0.635mm, h5=0 and
S=W=0.508mm and 2A=5mm. CASE M same as CASE 15 with
er4=10.8, sr5=2.33 and h4= 1.27mm, h5=0.381mm and S=2W=0.635mm
and 2A=38mm. CASE N (CASE 16) same as CASE M except
h4=2.54mm, h5=0.71mm. CASE O (CASE 14) same as CASE L
except 2A=38mm. All cases are 50f2 through lines and referenced to
OdB.................................................................................................................
116
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xxi
Page
FIGURE
42
43
44
45
46
Experimental data demonstrating the cavity mode effects in CBCPW
MICs with erl= l, sr4=10.8, er5=2.33 and h,=oo (open structure),
h2=h3=h6=0 and S=2W=0.635mm. Refer to Fig. 13 and Fig. 5(a)
for the dimension parameters. CASE P with h4= l ,27mm, h5=0.381mm
and 2A=38mm, 2B=25.4mm. For CASE Q with h4=0.635mm,
h5=0.711mm and 2A=20mm, 2B=38mm. CASE R same as CASE Q
except 2B=25.4mm. CASE S same as CASE Q except with broadband
absorber in the connector housing blocks. All cases are 50Q through
lines and referenced to OdB.........................................................................
118
Dispersion curves showing mode coupling effects in FGP CBCPW as a
function of the ground plane width (WG) at 30GHz with srl= l, sr4=10.2
and h[=5mm, h4=0.635mm, h2=h3=h6=0 and S=2W=0.635mm and
2A=25.4mm. (a) Single-layer structure with h5=0. (b) Multi-layered
waveguide with srS=2.33 and h5=0.635mm. CPM refers to the
coplanar-microstrip modes...........................................................................
121
Dispersion curve plot of the normalized propagation constants for an
upper dielectric loaded multi-layered CBCPW. CASE 19 with srl= l,
sr3=10.2, sr4=2.2
and
h,=5mm,
h3=0.635mm, h4= 1.27mm,
h2=h5=h6=0 and S=2W=0.635mm and 2A=25.4mm. Refer to Fig. 13
for parameter dimensions. BSW is the boxed surface wave modes and
the IDEAL case corresponds to the grounded dielectric slab layer h3.
LOWER RW’ refers to the maximum propagation constant of
single-layer rectangular waveguide h4.............................
126
Cross-sectional electric field plot for the dominant CBCPW mode of the
upper loaded example CASE 19 at 40GHz. The field components are
normalized to the maximum values.............................................................
127
(a) Experimental data for CBCPW MICs with lower dielectric value
er4=2.33. erl= l and h,= oo (open structure), h4=1.42mm, h2=h5=h6=0
and S=W=0.508mm and 2B=38mm. Refer to Fig. 13 and Fig. 5(a) for
the dimensions. For CASE T h3=0 and 2A=38mm (single-layer). CASE
U same as CASE T except 2A=5mm. CASE V upper loaded structure
same as CASE T except er3=10.8 and h3=0.635mm and
S=2W=0.635mm and with connector problems. CASEW same as
CASE V except with a cutout o f the substrate round the inputs and
output ports and illustrated in (b). All cases are referenced to OdB
128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xxii
Page
FIGURE
47
48
49
50
51
52
53
Cross-sectional vector electric field plot of the TM0 , surface wave
mode for the upper loaded multi-layered CBCPW of CASE 19 at
40GHz..........................................................................................................
129
Bounded mode behavior for lower loaded multi-layered dominant
CBCPW mode, (a) Normalized dispersion curves for CASES 20 and
21 of Table XV and the first waveguide mode calculated by the SDM
and the ideal rectangular waveguide analysis, (b) Cross-sectional
vector electric field plot (Ex and Ey) o f the dominant CBCPW mode
for CASE 20 at 40GHz...............................................................................
135
Characteristic impedance verification with [57] for an air-suspended
multi-layered CBCPW for power-voltage, voltage-current, and
power-current definitions. Refer to Fig. 13 for dimensions with srl=l,
er4 =13, sr5= l and h1=hs=5mm, h4= 1mm, h2=h3=h6=0 and S=lmm,
W=0.4mm and 2A=25.4mm........................................................................
136
Characteristic impedances of the multi-layered CBCPW MICs for the
three definitions, (a) Lower loaded example 2B) o f Table XV.
(b) Upper loaded example 3B) of Table XVI............................................
137
Dispersion effect analysis for the determination o f (S+2W) of the
dominant CBCPW mode for the lower loaded example 2B) o f Table
XV. Refer to Fig. 13 with srl=l, sr4=10.2, sr5 =2.2 and hj=5mm,
h4=0.635mm, h5=0.635mm, h2=h3=h6=0 and 2A=25.4mm. CASE 36 is
S=2W=0.635mm, CASE 37 is S=2W=0.381mm, CASE 38 is
S=2W=0.1905mm, and CASE 39 is S=2W=0.0953mm............................
142
Effects of an air gap between the substrates for the multi-layered
CBCPW MICs with srl=l and hj=5mm and S=2W=0.635mm and
2A=25.4mm at f=20GHz. Fig. 13 is referenced for the dimensions.
CASE 46 is the lower loaded example with er4= l0.2, sr5= l, 8^=2.2
and h4=0.635mm, h5 is the air gap, h6=0.635mm, h2=h3=0. CASE 47 is
the upper loaded example with er2=10.2, sr3= l, er4=2.2 and
h2=0.635mm, h3 is the air gap, h4=1.27mm, h5=h6=0................................
144
Leakage rate versus relative dielectric constant for single-layer
CBCPW. Fig. 3(a) is referenced here with S=2W=0.3175mm and
h= 1.27mm and f=40GHz.............................................................................
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
xxiii
Page
FIGURE
54
55
56
57
58
59
Normalized leakage rate curves for single-layer CBCPW. (a) Plots
versus frequency with er=10.2 and h=0.635mm with CASE 49
S=2W=0.3175mm, CASE 50 S=2W=0.15875mm, and CASE 51
S=2W=0.079mm. (b) Curves versus cross-sectional circuit area with
sr=10.2 and f=40GHz with CASE 52 h=0.3175mm, CASE 53
h=0.635mm, and CASE 54 h=l.27mm. Fig. 3(a) is referenced for the
parameters......................................................................................................
149
Normalized electric field plots for dominant CBCPW mode
demonstrating the mode coupling effects for a single-layer structure at
x=-0.01mm. Refer to Fig. 3(a) for dimensions with er =10.2 and
2A=25.4mm and f approximately 40GHz. (a) Results as substrate
height (h) is decreased with (S+2W)=h/4 for each case. CASE 51
h=0.635mm, CASE 55 h=1.27mm, CASE 56 h=2.54mm, and CASE 57
h=5.08mm. (b) Plots as function o f the cross section with
h= 1.27mm. CASE 58 S=2W=0.635mm, CASE 59 S=2W=0.3175mm,
CASE 55 S=2W=0.1588mm, and CASE 60 S=2W=0.0794mm...............
152
Normalized dispersion curves for single-layer CBCPW. With
er =10.2 and h=1.27mm and 2A=25.4mm and refer to Fig. 3(b) for
parameters, (a) CASE 58 with S=2W=0.635mm. (b) CASE 60 with
S=2W=0.0794mm. The crossed points are the CBCPW mode in each
case.................................................................................................................
153
Normalized electric field plots for the dominant CBCPW mode
demonstrating the mode coupling effects as a function of frequency for
a single-layer structure at x=-0.01mm. CASE 50 with s r=10.2 and
h=0.635mm and S=2W=0.1588mm and 2A=25.4mm. Refer to Fig. 3(b)
for dimensions................................................................................................
155
Cross-section o f the multi-layered CBCPW with damping material for
the upper and lower resonant cavities. The complex permittivities
and complex permeabilities for the lossy layers are shown.........................
161
Application of the damping material with a thickness t on the perimeter
of the symmetric lower dielectric loaded CBCPW to reduce the cavity
Qs . (a) Side view, (b) Top view.............................................................
162
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
xxiv
FIGURE
60
61
62
Page
Equivalent circuit representation for resonance mechanism in a packaged
CBCPW circuit with cavity effects. Three cavity modes exists within
the frequency range o f interest for this example and are modeled using
-RjLjCj lumped elements and mutual coupling with turns ratio «;...................
165
Ideal rectangular cavity representation of the multi-layered packaged
CBCPW. PEC shorting facewalls exist at z=0 and z=2B...............................
168
Attenuation (dielectric) losses for the dominant CBCPW mode
examining the effects of the silicon lossy layer. Refer to Fig. 58 for
dimensions with srI= l, sr4=10.2, sr5=2.2 and h,=5mm, h4=h5=0.635mm,
h,=h,=0 and S=2W=0.635mm and 2A=25.4mm. CASE 61 h,=0 and
/
CASE 6 2 e rt = 12 andh6=0.25mm and«e=8.35 x 10
63
15
,
c m '......................
171
Demonstration o f the damping effects within a rectangular cavity for a
MIC example using doped silicon. Refer to Fig. 61 for parameters
with er4=10.2, s r5=2.2,
= 12 and h4=h5=0.635mm and
2A=2B=25.4mm. Dielectric loss data for h4 and h5 substrates is listed in
16
the text. CASE 63 h6=0, CASE 64
h6=0.1mm and rt =5.2 x 10
cm"3, and CASE 65 h6=0.25mm and «e=8.35 x 10
64
65
66
cm'3........................
174
Top view representations at x=0 of the center-fed half-wavelength
resonators considered in the analysis, (a) CBCPW and (b) microstrip
175
Shapes of the electric field basis functions for a CBCPW resonator as a
function of z with 2L=12.7mm for the right hand slot at _y=(S/2+W/2).
(a)
basis functions, (b) E.n basis functions.........................................
180
Convergence demonstration o f the resonant frequency from the SDM
with additional basis functions in the z-direction for M=N o f a resonator
circuit, (a) Microstrip example CASE 66 with dimensions from Fig. 58
and Fig. 64(b) o f erl= l, sr4=9.4 and h ^ m m , h4=0.6mm,
h2=h3=h5=h6=0 and S=0.575mm,WG=0, 2L=4.5mm and 2A=2B=10mm.
(b) CBCPW example CASE 67 with dimensions from Fig. 58 and Fig.
64(a) of erl= l, er4=10.2, sr5=2.2 and h,=5mm, h4=h5=0.635mm,
h2=h3=h6=0 and S=2W=0.635mm, 2L=6.2mm and 2A=2B=25.4mm.....
182
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
FIGURE
67
Plot of the E space domain field from the SDM for the dominant
CBCPW mode of the resonator circuit o f CASE 67 with 2L=6.2mm at
,y=(S+W)/2 for the right hand slot.............................................................
68
CBCPW resonator and TM,, 3, cavity mode Qs using doped Si layer
(s r = 12). a) Lower loaded multi-layered structure CASE 68 same as
CASE 67 as a function of the Si layer at h6 with doping
16
3
n = 1.0 x 10 cm . b) Upper loaded CBCPW CASE 69 as a function
of the dielectric thickness h4 with fixed Si layer at hs=0.2mm with
16
3
doping n = 2.1 x 10 cm and erl= l, er3=10.2, er4=2.2, h,=5mm,
h3=0.635mm, h2=h6=0 and S =2W=0.635mm, 2L=4mm and
2A=2B=25.4mm. Log scales are implemented..........................................
69
CBCPW resonator and TM q3 , cavity mode Qs for a lower loaded
multi-layered structure using microwave absorber as a function o f the
dielectric thickness h5. Refer to Fig. 58 for the dimensions o f CASE 70
with erl= l, sr4=10.2, erS=2.2, 8^= 21 ( 1-70 .02 ), ^ = 1. 1( 1-71 .4 ) and
hj=5mm, h4=h6=0.635mm, h2=h3=0 and S=2W=0.635mm, 2L=6.2mm
and 2A=2B=25.4mm....................................................................................
70
Dimension parameters for multi-layered CBCPW straight gap-coupled
resonator with damping material, (a) Top view with gap width (G) and
resonator length (L). (b) Cross-sectional view with absorber placed at
layer h6...........................................................................................................
71
Experimental data response with absorber material to dampen cavity Qs
in straight gap-coupled CBCPW resonator. Reference Fig. 70 for
dimensions with G=0.17mm and L= 12.7mm. CASE X with srl= l,
er4=10.8, erS=2.33 and h,= oo (open structure), h2=h3=0, h4=0.635mm,
h5= 1.09mm and S=2W=0.635mm and 2A=20mm and 2B=38mm.
Absorber layer at h6 and electrical data is not known................................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Page
FIGURE
72
Experimental data o f the transmission responses for the microstrip and
the multi-layered CBCPW MIC 50f2 through lines. A comparison with
Fig. 7 indicates CBCPW MIC is now a viable alternative to microstrip.
Refer to Fig. 5 for dimensions with erl=l and h,= oo (open
structure) and 2A=2B=38mm. CASE A is microstrip line with sr4=10.8
and h4=0.635mm and S=0.635mm. CASE B is CBCPW with
er4=10.8, Er5=2.33 and h4=0.254mm, h5=0.71mm, h2=h3=h6=0 and
S=2W=0.635mm with absorber material in the connector blocks. Both
cases are referenced to OdB........................................................................
195
Coordinates and components of the basic TM and TE modes for the
SDM immittance method............................................................................
205
Equivalent transmission line models of the multi-layered CBCPW for
the TM and TE modes................................................................................
206
E. 1
Parallel plate waveguide representation with two dielectric substrates
229
F. 1
Illustrative example of the residue calculus theorem with a number of
poles a,b,c interior to a simple closed curve C..........................................
232
A. 1
A. 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
CHAPTER I
INTRODUCTION
The classical uses of microwaves in terrestrial radio and radar have expanded during
the past decade into many new applications due to technological advances including
satellite and wireless communications, superconducting circuits, vehicular navigational
systems, and phased and active antenna arrays. The use of high-speed pulse circuits for
wideband communications and ultra-fast computers has pushed microwaves into the area
of digital technology. In the early days o f microwaves, circuits consisted of coaxial and
waveguide systems that were produced separately and then connected together by screws
to form the system. These circuits were large, heavy, and not suited for high volume
systems.
The rapid growth of modem microwaves can be attributed to advances in
microwave semiconductors and in hybrid microwave integrated circuits (MICs) and
monolithic MICs (MMICs). An integrated circuit combines all of the active devices and
circuit components in a planar fashion and results in smaller and cheaper systems and are
easily reproducible for large volume applications.
In MICs the transmission line
components are first etched on a circuit board and then the remaining elements are surface
mounted and wire bonded to the substrate.
MMICs process the passive components
(transmission lines, capacitors, resistors, inductors) and the semiconductor devices
(transistors and diodes) together on the same semiconductor substrate. Transmission lines
are one of the most essential elements o f microwave circuits. Analysis, modeling, and
design of transmission lines are important for any component and subsystem development.
The desirable characteristics of any planar transmission line are low dispersion (change in
the effective dielectric constant over frequency), large characteristic impedance range, and
low dielectric and conductor losses. The circuits should provide sufficient circuit surface
area, easily couple to antenna systems, easily integrate active and passive components, and
This dissertation follows the style and format o f the IEEE Transactions on Microwave
Theory and Techniques.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2
avoid vertical discontinuities within the structure.
As more efforts have been spent on
MMICs, interest in coplanar waveguides has been renewed. The primary interest in this
waveguide is the elimination of via holes that are required to ground nodes within the
circuit and this is especially important at higher frequencies of operation.
A. Comparison Between Microstrip and Coplanar Waveguides
For many years MICs and MMICs have predominantly used microstrip as a
transmission line medium. Microstrip is a simple waveguide consisting o f a conducting
strip placed on a dielectric substrate and backed by a conducting ground plane.
This
transmission line is well understood, flexible in terms o f circuit elements, and can operate
to very high frequencies. However, the main disadvantages with microstrip are that it
requires via holes to ground active devices and other nodes as depicted in Fig. 1 and at
higher frequencies the dielectric thickness must be reduced to cutoff the characteristic
grounded surface wave modes. The via holes also require thinner substrates to increase
the yield and behave inductively at high frequencies which can create problems at the
source electrode of a mounted MESFET (metal-semiconductor field-effect transistor).
Coplanar waveguide (CPW) has been suggested as an alternative to microstrip [1-2]
but was not widely used initially due to the mistaken assumption that it possesses a higher
conduction loss than microstrip [3], CPW is a surface oriented planar transmission line
suspended in air with two ground planes running parallel to the center strip and all of the
conductors are in the same plane of the dielectric. This waveguide can be viewed as a
coupled-slot configuration with the electromagnetic energy o f the dominant mode guided
by these slots. The dominant mode is guided by the transmission line and should be TEM
or quasi-TEM (transverse electromagnetic with a small field component in the direction of
propagation) since the dominant source mode o f the excitation coaxial measurement
system is TEM. It is also desirable to only have a single propagating mode within the
frequency-band of interest. With the availability o f the ground planes, via holes are not
necessary in coplanar waveguides and mounting o f active devices and other shunt elements
is easily achieved as in Fig. 1. The elimination o f the via holes in GaAs (gallium arsenide)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3
c o p l a n a r waveguide
m ic ro strip
via hole
ground
planes
23
(a)
hole
jvia
.
m
s
s
/
ground
plane
(b)
Fig. 1. Active device grounding in coplanar waveguides and microstrip,
(a) Top view, (b) Side view.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
MMICs can result in dramatic savings in cost as well as improved process yields on the
order of 40% [4], The presence of a ground plane between the strip conductors allows a
compact circuit layout and reduces the crosstalk of adjacent lines. Also, the dielectric
thickness is not required to be thin for high frequency operation and the elimination of the
via holes allows for thicker and structurally stronger substrates.
The characteristic
impedance of coplanar waveguides is determined from the ratio o f the strip to slot widths
so structures with different cross sections can have the same impedance. As the number of
conductors increases, the propagation characteristics become more complex. One major
disadvantage of this waveguide is the existence of an unbalanced or slotline mode that is
excited at discontinuities and asymmetries within the circuit. The effects of this mode can
be reduced through the use of air bridges which extend the cutoff frequency o f this mode
by equalizing the potentials on the top ground conductors. However, these air bridges also
represent a vertical discontinuity and are utilized extensively within a circuit at the vicinity
of the discontinuities to eliminate this mode as demonstrated in Fig. 2 for a GaAs MMIC
amplifier [5],
The darkened connections are the air bridges and maintain all of the
grounded regions. CPWs are not very well characterized in terms of circuit elements and
discontinuities. The effectiveness of coplanar waveguides has been demonstrated in the
work of a 5-100GHz GaAs MMIC amplifier [6] and the Cascade Microtech™ wafer probe
system [7], An additional conductor plane on the other side o f the dielectric is utilized to
support the structure and for heat sinking purposes and forms the conductor-backed
coplanar waveguide (CBCPW) This conducting plane introduces several problems with
regard to mode coupling effects.
B. Overview of Conductor-Backed Coplanar Waveguides
The basic two-dimensional, symmetrical (with respect to the y-direction) CBCPW
configurations with an infinite-length in the z-direction are shown in Fig. 3 and includes an
infinite-width, packaged (shielded or boxed) with shorting sidewalls (connecting the
conductor-backed plane, the ground planes, and the cover plate plane), and packaged with
finite ground planes. If none o f the conductors are electrically connected (total of N=4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 2. Circuit photograph o f GaAs CBCPW MMIC distributed amplifier.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
conductors excluding the cover plate conductor), then three normal modes of propagation
exist (N -l) within such a system [ 8 ] and are depicted in Fig. 4. A mode is a solution to the
system of equations describing the waveguide, satisfies Maxwell's equations and the
boundary conditions, and corresponds to a field configuration supported by the
transmission line. Associated with each mode is a cutoff frequency. If the frequency of
operation is above the cutoff frequency, the mode propagates and carries energy away
from circuit discontinuities. If the frequency is below the cutoff, the mode is referred to as
evanescent and energy is stored at the discontinuity and is attenuated away from the
discontinuity. The dominant or odd mode is a balanced, zero-cutoff, and quasi-TEM one
and corresponds to a magnetic wall placed at y=0. Most o f the electric fields for this
mode are concentrated within the slot areas. If the ground conductor planes at x=0 are not
at the same potential, the even or slotline mode is present. This mode is an unbalanced,
nonzero-cutoff, and non-TEM one and corresponds to an electric wall placed at y= 0. An
example o f how this mode can be excited at a right angle bend discontinuity is also shown
in Fig. 4(b). The path difference between the electromagnetic waveforms exists on the
inner and outer ground planes and translates into a phase difference between the planes.
Air bridges are required to extend the cutoff frequency for this mode.
If the ground
conductor planes are not connected to the conductor-backed plane at x=-h, the
microstrip-like or parallel plate mode can propagate. This mode is a balanced, zero-cutoff
mode with most of the fields concentrated within the dielectric. The cross-sectional vector
electric field plots (Erx and £„)
for these modes are presented in Fig. 4 and are normalized
y
to the largest field component for each case.
The even mode can be eliminated by maintaining symmetry and an example is a
through transmission line. The microstrip-like mode can be excluded by connecting the
ground plane conductors at x=0 and the conductor backing plane at r=-h 4-h 5-h6 along the
line length of the transmission line to achieve a uniform ground connection as in Fig. 5(a).
This configuration also represents the boundary effects o f the CBCPW within a package
and demonstrates the actual physical waveguide. WG extends to the substrate edge unless
noted. This wrap-around connection is achieved using copper tape, conductive epoxy, or
through the package. This structure is modeled with shorting lateral sidewalls at _y=±A as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7
X
air
E
E
m
w s w
y
€r
(a)
air
E
E
in
y
jC
2A
(b)
X
E
E
m
air
w s w wG
f -
f - -,n=a------------
■------2A—
-]
(C)
Fig. 3. Basic CBCPW configurations, (a) Infinite-width. (b) Packaged with shorting
sidewalls, (c) Packaged with finite ground planes. S, W, WG, and 2 A
are the widths of the center strip conductor, slots, ground plane
conductors, and the lateral sidewall separation respectively. The conductorbacked plane exists at x=-h and the cover plate conductor at x=5mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
magnetic wall
(a)
electric wall
(b)
v z
t
K2
y y
1t
i f
I
i
——
— 1
' I l 1 i XI t ' i 1 ' ‘ I I » » 1 I '
ar / i i t i t i j i ii vi vi i» i* »i ii vi \\ vl
l
k 1
k k k V\ '
magnetic wall
(c)
Fig. 4 Fundamental modes of CBCPW along with the source topologies including
the cross-sectional vector electric field plots for the two-layer configuration,
(a) Dominant or odd mode, (b) Slotline or even mode, (c) Microstrip-like or
parallel plate mode.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
9
conductor
X
£n
■
^ »,•-A' r'•,-A' r■-A +•,•A'^T2
■'>V^•/•.#.^•-•A' *r■•>
• • ■A ,*•r■
m
m
m
m
b
Fig. 5. Multi-layered CBCPW configurations, (a) Longitudinal view o f experimental
CBCPW with shorting sidewalls, (b) Cross-sectional view of CBCPW used
in the numerical analysis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
in Fig. 5(b) and resembles a rectangular waveguide with two aperture slots to guide the
electromagnetic energy of the dominant mode. The CBCPW mode or dominant mode will
be referenced interchangeably throughout this dissertation. The structures considered here
have cross sections (S+2W) that are much smaller than the lateral width (2A). Waveguide
modes exist within the dielectric or substrate regions at layers 4,5,6 and/or 1,2,3 in Fig.
5(b) that can couple to the dominant mode and produce undesirable results. Air filled
rectangular waveguides are usually designed with the ratio of the lateral width to the height
(2A/h)=2 for a single mode o f operation [9] as shown in Fig. 6 (a) and requires ten separate
rectangular waveguide bands (each operating in a single mode) to cover the 0.75-40GHz
range.
CBCPW MICs can have a ratio between 15<(2A/h)<60 for sufficient circuit
surface area which translates to a multi-mode operation. The finite-length CBCPW of Fig.
5(a), within a package formed from the connector housing blocks, produces a rectangular
cavity and introduces a resonant structure as demonstrated in Fig. 6 (b). A cavity can be
considered as a volume enclosed by a conducting surface and within electromagnetic fields
can be excited and stored. How well the energy is stored within the cavity corresponds to
the Q (ratio of energy stored to the energy dissipated). With a high Q (quality factor
associated with each resonance) cavity, a significant current can flow into the resonator
and produce significant resonances which can interfere with the CBCPW operation. The
structures considered here have a large electrical size so that many resonant frequencies
exist within the band.
An example for a typical multi-layered cavity structure with
dielectrics s r=10.2 (relative dielectric constant) and er=2.2 with a thickness of 0.635mm
each with dimensions 1.27 (h) x 20 (2A) x 38 (2B) mm possesses 23 cavity modes to
40GHz.
C. Problem Definition and Research Objective
The objective o f this dissertation is to explain, predict, and reduce the effects o f the
waveguide and cavity modes on the operation of the CBCPW MICs for wide frequencyband applications. The CBCPW was intended to be utilized with a millimeter-wave active
antenna array system to eliminate the need for via holes within the circuit.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
11
♦
-C
*
2A
(a)
♦
XL
2A
(b)
Fig. 6 . Approximate representations of packaged CBCPW. (a) Ideal rectangular
waveguide, (b) Ideal rectangular cavity.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
consequence of these modes was demonstrated via experimental measurements on
approximate 50ft through lines to 40GHz with an in-house developed test fixture using
Omni-Spectra Inc. OS-50 connectors as shown in Fig. 7. The measurements were taken
on a HP8510B network analyzer with 401 data points and calibrated at the coaxial cable
ends.
The network analyzer measures the S-parameters or scattering coefficients
(magnitude and phase) for two-port networks. S 21 represents the forward transmission
coefficient or the ratio of the transmitted wave (on output port # 2 ) to the incident wave
(on input port # 1) with the output port terminated in the characteristic impedance o f the
system (50ft) [10]. The magnitude of S21 for a lossless and matched transmission line is 1
or OdB and is e_az with losses where a is the attenuation constant. The conductor ground
planes and the conductor-backed plane along the length o f the CBCPW were connected
with electrical tape from 3M Co. as in Fig. 5(a). The ground contacts at the input and
output ports were obtained using a shorting bar within the fixture and good contacts are
critical for proper mode excitation. The connector housing blocks complete the cavity
with a uniform ground connection as depicted in Fig. 8 . This structure is equivalent to that
presented in Fig. 5(a) with the wrap-around connection (shorting sidewalls) at z=±B. The
dielectrics are Duroid™ 6010 (er=10.8) and 5870 (sr=2.33) from Rogers Corp.
The
experimental results are described in Fig. 7 with CASE A representing a microstrip line for
fixture verification, CASE B shows a typical response for a single-layer CBCPW using
6010 substrate, and CASE C depicts a three coupled-strip (finite ground planes)
transmission line. As demonstrated the single-layer CBCPW MICs do not possess any
appreciable bandwidth. These results were totally unanticipated and were not previously
described in the literature and the presence o f the strong resonances were particularly
disturbing. An entire and important class of MIC transmission lines was unusable even at
lower frequencies.
All of the line parameters were varied to eliminate these undesired
effects but these changes still did not produce favorable results. Such attempts included a
structure with a small cross-section (S+2W=0.75mm) compared to the dielectric thickness
(2.54mm) for the 6010 substrate. The structures considered in this dissertation have a
maximum lateral (transverse in ^-direction) or longitudinal (z-direction) dimension of
38mm and correspond to realistic MICs.
A multi-layered configuration (lower dielectric
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CASE A
MAGNITUDE
S21 ( d B )
CASE S
CASE C
V
\
CASE 0
>
'~0
00
■O
CASE E
tO
CASE F
0
4
8
12
16
FREQ
Fig. 7.
w
O
20
24
28
32
36
40
(G H z )
Experimental responses demonstrating waveguide and cavity modes in CBCPW
with shorting sidewalls and 2A=2B=38mm, S=2W=0.635mm, s ^ s ^ s ^ e ^ l ,
h,=oo (open structure), h;,=h3=h6=0.
Refer to Fig. 5(b) for the dimension
parameters. Case A. 50C2 microstrip line s r4=10.8, h4=0.635mm and sr5= l, h5=0,
Case B same as Case A for CBCPW, Case C same as Case B except WG=lmm,
Case D same as Case B except h4= 1.27mm and srS=2.33, h5=0.381mm, Case E
same as Case D except h4=0.635mm and h5=0.711mm Case F same as Case E
except broadband absorber placed in the connector housing blocks.
Cases
AB,C,D are lOdB/div while Cases E,F are 5dB/div and all are referenced to
OdB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
^
shorting bar
conductor
shorting bar
«o‘
2A
(b)
Fig. 8 . Test fixture illustrations incorporating CBCPW transmission lines,
(a) Shorting bar application to ground input and output ports of
CBCPW. (b) Connector housing blocks.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
15
loaded) improved the experimental results as shown in Fig. 7 CASH D and produced a
26GHz bandwidth, CASE E extended the upper usable frequency of operation above
40GHz but demonstrated the effects o f the cavity resonances and modes especially above
20GHz, and CASE F is the same as CASE E with broadband absorbing material from
Arlon Inc. (LS series dc-40GHz) within the connector housing blocks to dampen the cavity
resonances.
CBCPW must be utilized in one of four configurations for successful operation. The
lateral sidewall dimension can be reduced to cutoff the rectangular waveguide modes and
prevent mode coupling as in Fig. 9(a) for the channelized coplanar waveguide [11], but
the available circuit surface area and frequency range is limited.
The cutoff frequency of
the TE0 j mode (first waveguide mode and the subscript integers pertain to the number of
standing wave maxima in the field solutions describing the field variations in the x and
^-directions, respectively) for a single-layer CBCPW with er=10.2 and lateral width (2A) of
5mm is 9.46GHz.
Also, this configuration requires a grooved fixture to support the
dielectric structurally. An alternative CBCPW utilizes via holes along the transmission line
length to eliminate the microstrip-like mode and almost forms an electric wall to cutoff the
waveguide modes similar to channelized CPW is described in Fig. 9(b) [17], This structure
eliminates the primary advantage of coplanar waveguides over microstrip, increases the
modeling complexity, and enhances coupling effects between the vias. The cross-section
dimension (S+2W) can be made much smaller than the substrate thickness to reduce the
mode coupling effects and can be easily achieved for MMICs. Keeping (S+2W) less than
one-fourth the dielectric thickness for GaAs and one-twentieth the wavelength is a design
estimate o f the leakage rate for the CBCPW mode to achieve a useful
circuit [12],
Leakage is the loss of power (or attenuation) from the CBCPW mode coupled to the
parallel plate modes within the substrate region between the ground plane conductors and
the conductor-backed plane.
width and length.
The power is leaked away if the waveguide is infinite in
For MICs, these line/slot requirements may not be attainable with
respect to circuit etching capabilities or possible for certain applications. Multi-layered
structures can be utilized for these "general purpose" systems and modify the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16
CENTER STRIP
GROUND
PLANES
DUROID
METAL
CHANNEL
(a)
Fig. 9. Alternative coplanar waveguide configurations, (a) Channelized coplanar
waveguide, (b) Via hole structure.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17
propagation characteristics of the waveguide modes and prevent mode coupling effects in
CBCPWs [13-16], Strong mode coupling with the waveguide modes can cause the
dominant mode field pattern to spread out across the entire waveguide width instead of
being confined to the slot area. This phenomenon translates into a loss o f power and the
occurrence of strong resonances in the transmission measurements of the waveguide. Both
lower and upper dielectric loading structures are proposed here as realistic alternative
configurations as described in Fig. 5(b) with the various layers representing dielectrics, air
gaps, bonding film, and damping material. The multi-layered coplanar waveguide is still
referred to as CBCPW since the thickness o f the dielectrics is not so large that the
electromagnetic effects of the conductor-backed plane are still present. The lateral and
longitudinal sidewalls are extended for available surface area and multiple propagating
modes exists within the circuits.
From the experimental data if strong mode coupling
occurs within the CBCPW, absorber or damping material will not remedy this situation.
The disadvantages of the multi-layered waveguides are a more complex structure, possible
air gaps between the substrates, and difficulty in configuring the upper dielectric loaded
CBCPW within a test fixture.
The physical phenomenon behind the above mode coupling problems can be explained
and modeled via the spectral domain method (SDM) in one, two, and three dimensions.
SDM is a very efficient full electromagnetic wave numerical method [18-19] (especially
suited
for
multi-layered
planar
transmission
lines)
that
can
predict complex
propagation characteristics, leakage rates, impedance variations, mode coupling, field
patterns, and resonator Qs for the various modes.
This modeling capability has been
utilized to predict the upper usable frequency (onset frequency of strong mode coupling
between the dominant and first waveguide mode) up to 40GHz, explain in detail the mode
coupling mechanism, select multi-layered structures to extend the upper usable frequency
or bandwidth, and suggest configurations to reduce the substrate cavity Qs without
affecting the CBCPW circuit.
Experimental data is used to verify the upper usable
frequencies, confirm the explanation o f the mode coupling mechanism, and demonstrate
the necessity of
some type of
damping material to reduce the residual resonances
associated with the cavity stmcture even when the dominant mode is bound to the slots.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
To summarize the questions that this dissertation will answer involving the transmission
and propagation characteristics of CBCPW MICs are:
1) explain what mechanisms are causing the effects in the experimental data;
2) explain how and why these mechanisms are interfering with the CBCPW operation;
3) propose realistic alternative CBCPW configurations to minimize the above effects
(multi-layered configurations);
4) develop a numerical model to predict the bandwidth of operation;
5) verify the model with experimental data to 40GHz;
6) present design summary information for multi-layered CBCPW configurations;
7) demonstrate how CBCPW MMICs can work with the mode coupling effects present.
D. Literature Review and Discussion
The coplanar waveguide was invented by Wen [1] in 1969 as an alternative
transmission line for MIC applications, recognized the need for a shorting bar to connect
all of the ground regions, and analyzed the infinite-thickness structure with a quasi-static
(low frequency) method. Pucel [2] suggested the use o f coplanar waveguides for GaAs
MMICs but made no mention of the possible mode coupling problems. The SDM was
utilized by several authors to solve for the propagation characteristics o f air suspended
coplanar waveguides.
[20],
The infinite-width structure was analyzed by Knorr and Kuchler
Fujiki et al. [21] modeled a packaged CPW with the first few higher order
waveguide modes and produced dispersion curves but these modes were not identified and
the coupling effects were not discussed. Davies and Mirshekar-Syahkal [22] developed a
procedure to analyze multi-layered CPW in the SDM but only considered air suspended
structures with no waveguide modes propagating. The air suspended configurations are
not practical because it is difficult to dangle a dielectric or semiconductor in the air. The
analysis of infinite-width CBCPW was demonstrated by Shih and Itoh [23] but the leakage
effects associated with the parallel plate modes were neglected. The coupling effects of
the waveguide modes due to the lateral sidewalls for a GaAs dielectric were investigated
with a mode-matching technique by Leuzzi et al. [24] and demonstrated the field spreading
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
of the dominant CBCPW mode. However, the authors misidentified the modes in the
propagation curves. The realization that losses and dispersion in CPW could be less than
those of microstrip was described by Jackson [3]. Jackson [25] also first discussed the
possibility of the leakage of power from the dominant CBCPW mode and suggested a
multiple dielectric configuration (lower dielectric loading) to prevent this mechanism in the
infinite-width case. The analysis of leakage in conductor-backed slot lines and potential
problems associated with CBCPWs were described by Shigesawa et al. [26],
Mode
conversion in CBCPW discontinuities with finite ground planes from the dominant mode
into the microstrip-like mode was explained by Jackson [27],
Channelized coplanar
waveguide MICs were proposed by Simmons et al. [11]. Propagation characteristics for
several CPW configurations and an approximate closed form relation for the leakage rate
of the dominant mode for a single-layer CBCPW were discussed by Riaziat et al. [12],
The most impressive applications of CPW circuits were demonstrated in a 5-100GHz
MMIC amplifier by Majidi-Ahy et al. [6 ] and the Cascade Microtech™ wafer probe
system [7], MMICs have a distinct advantage over MICs in the capability to make the
cross-section much smaller than the dielectric thickness which reduces the leakage
coupling effects [12], [28], Cascade Microtech™ has incorporated absorbing material in
the probe heads [29] to reduce spurious mode effects and studied surface wave coupling
problems in CPWs [30],
Das and Pozar [31] incorporated the leakage analysis of the
dominant mode into the SDM for several waveguide configurations. Leakage calculations
for CBCPWs were published by McKinzie and Alexopoulous [32] and Chou et al. [33],
Magerko et al. [13] presented experimental data for CBCPW MICs demonstrating the
effects of moding and resonance problems, incorporated a multi-layered (lower dielectric
loading)
to extend the upper usable frequency, and approximately predicted
frequency using a one-dimensional SDM analysis.
The mode coupling
effects
this
of
packaged multi-layered CBCPWs were described in detail by Magerko et al. [14] using a
two-dimensional SDM including field spreading of the dominant mode and impedance
variations of the modes. The mode coupling mechanism was discussed in general terms for
packaged transmission lines by Carin et al. [34], Jackson [35] has established a circuit
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
model to approximate inter-circuit coupling effects due to substrate resonances in
CBCPWs. The resonance effects associated with finite ground planes in a test fixture have
been predicted by Tien et al. [36],
packaged CBCPW MICs
Configuration considerations for multi-layered
including design summary tables, upper dielectric loading
structures, and the analysis of lossy layers to reduce the substrate Qs without affecting the
CBCPW resonator appears by Magerko et al. [15], [28],
Leakage calculations for
multi-layered CBCPWs have been published by Liu and Itoh [16]. Attempts by Yu et al.
[17] have included via holes along the length of the CBCPW to connect the ground plane
conductors and the conductor-backed plane. However, the main advantage of coplanar
waveguides over microstrip has been eliminated.
A commercial CPW test fixture to
20GHz has been offered by Wiltron Inc. [37] but no published or listed information exists.
E. Dissertation Organization
The content o f the dissertation is organized in the following chapters.
In Chapter II a description o f the SDM for multi-layered CBCPWs in one and two
dimensions will be presented. The SDM is the numerical method utilized in this work.
The concept of the leakage o f power from the dominant mode into the parallel plate
modes is introduced in Chapter III and the one-dimensional SDM analysis is applied to
multi-layered CBCPWs.
described.
Dispersion and leakage curves for several structures will be
The one-dimensional analysis (infinite width and length) can approximately
predict the bandwidth of operation from experimental data in many cases but is not
accurate in others.
Also, this procedure provides no explanation o f the measurement
effects above the upper usable frequency.
The effects o f the lateral sidewalls on the CBCPW operation are included in Chapter
IV as a two-dimensional (infinite-length) SDM procedure is presented. Realistically, all
waveguides will be finite in the transverse and lateral directions. Rectangular waveguide
modes are propagating within the substrate regions and can cause mode coupling problems
with the CBCPW mode. These effects include field spreading of the dominant mode and
changes in the characteristic impedances. The two-dimensional analysis can predict the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
bandwidth more accurately and explain the occurrence of the loss of power and the strong
resonances in the transmission measurements.
Design characteristics of various multi-layered configurations for upper and lower
dielectric loading are discussed in Chapter V including the upper usable frequencies,
characteristic impedances,
and
effective
dielectric constants.
Summary tables
demonstrating design trade-offs, air gap sensitivity, dielectric uncertainty analysis,
cross-section dimensions to limit dispersion effects, multi-layered microstrip interfacing,
closed form procedure to determine the lower dielectric constant in a multi-layered
arrangement, and parameter selection for 50Q lines are all presented and discussed. The
mode coupling effects in packaged CBCPWs can be empirically related to the leakage
rate of an infinite-width waveguide and this is also detailed in Chapter V. If leakage is not
present in a transmission line, mode coupling will not occur. Furthermore, a small leakage
rate corresponds to small mode coupling effects. A closed form leakage rate or constant
from [ 12] is verified and dimension recommendations to reduce the mode coupling effects
in terms of this constant are detailed. A design procedure incorporating these concepts is
described. This material will also demonstrate how CBCPW MMICs can operate with the
mode coupling effects present.
The effects of the residual resonances still present in cases when the dominant mode is
bound to the slots must be addressed (see Fig. 7 CASE E). Some of these resonances are
on the order of 2-4dB and are unacceptable for a transmission line.
As mentioned
previously, the CBCPW within a test fixture or package forms a rectangular cavity as a
substructure. The effects of lossy damping layers (doped silicon or microwave absorber)
on the Qs of the CBCPW resonator circuit and the cavity modes are simulated using the
three-dimensional SDM and are developed and discussed in Chapter VI.
To limit the
effects of the damping material on the CBCPW resonator while still reducing the Qs of the
cavity modes, the lower dielectric layer thickness can be increased or the lossy material
applied on the perimeter o f the structure.
Chapter VII concludes the dissertation by summarizing the work, presents the original
contributions produced in the area of multi-layered CBCPW MICs for wide
frequency-band applications, and makes suggestions for future research work in this area.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
CHAPTER H
THEORETICAL ANALYSIS OF CBCPW USING
THE SPECTRAL DOMAIN METHOD
The spectral domain method (SDM) is widely used for the analysis o f planar integrated
circuits (MICs and MMICs) and is a very efficient full-wave electromagnetic numerical
method that can simulate the propagation and transmission characteristics of these
structures. The parameters predicted by the SDM in this work for one, two, and three
dimensions include the complex propagation characteristics, leakage rates, impedance
variations, mode coupling effects, field patterns, and the resonator and cavity Qs. This
method was originally proposed by Itoh and Mittra in 1973 [38], The SDM can be applied
to planar transmission lines including microstrip, slot lines, finlines, coupled strips, and
coplanar waveguides. The analysis can also simulate well-shaped discontinuities such as
transmission line steps, disk, triangular, or ring resonators, and planar periodic structures.
From a mathematical point o f view, the SDM is simply an integral transformation method.
Under special circumstances the differential equations governing the behavior o f a system
can be transformed into a new domain, the spectral or Fourier domain, where the
transformed equations can be easily solved as algebraic equations. The alternative method
to solve for the propagation characteristics o f the waveguide is the integral equation
technique and is performed in the space domain [39], The problem with this procedure is
the Green's functions (functions relating the currents on the strips to the fields in the slots
for the CBCPW) are not available in closed form for the inhomogeneous structure which
translates into very slowly convergent integrals and prohibitively long (computation time)
numerical solutions. One of the most important aspects o f the SDM is the ability to handle
multi-layered structures and waveguides with various metalization levels as pictured in Fig.
10(a). Examples of structures suited for the SDM analysis are depicted in Fig. 10(b). The
main reason the SDM is numerically efficient is that it requires a significant amount o f
analytical preprocessing of the unknown currents on the strips or the fields in the slots.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
Metallization
Substrates and
Su p erstates
(a)
• £r
Er
- £r
(b)
Fig. 10.
Examples of structures easily simulated using the spectral domain method,
(a) Cross-section of a multi-layered planar structure with multi-metallization
layers, (b) Cross-sections o f CBCPWs that are open, laterally open, and
shielded or packaged from left to right, respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
This requirement imposes some restrictions on the applicability of the method.
The
procedure has difficulty with finite conductor thickness especially when the thickness is on
the order of the skin depth (the distance a wave, as part of the field localized in a surface
layer, is attenuated to Me or 36.8% of the initial value) and cannot handle structures with
dielectrics that are perpendicular in the lateral direction. A mode-matching method [40]
would be utilized in the latter case. The conductors in this work are assumed to be perfect
electric conductors (PECs) with infinitesimal thickness and infinite conductivity. Also, the
dielectrics are assumed to be homogeneous, isotropic (electric permittivity e and magnetic
permeability |i are scalar constants and do not vary with position) materials.
A. General Spectral Domain Method Formulation
The general SDM formulation will be developed following [18-19] for the CBCPW of
Fig. 11 showing the cross-section of a symmetric, infinite-width (y-direction) and
infinite-length (z-direction) structure.
The CBCPW consists o f three dielectric regions
below the circuit conductors at x=0 for lower dielectric loaded configurations of heights
(thickness) h6 , h5, and h4 with relative dielectric constants
, s r5 , and s r4 respectively,
and three dielectric regions above the circuit conductors at Jt=0 for upper dielectric loaded
configurations o f heights h3 , h2 , and h, with dielectric constants er3 , s r2 , and srl
respectively. These layers represent dielectrics, air gaps, bonding film, and absorbing or
damping material. CBCPW also includes a center metallic strip of width S, slots of width
W, and ground plane conductors (WG extends to infinity). The relative permeability for all
of the dielectrics is |a.r= l unless otherwise noted.
A conductor-backed plane exists at
x=-h6-h5-h 4 and a cover plate conductor is placed at x=h3+h2+h, to simplify the analysis by
eliminating the radiation effects at discontinuities.
Due to the inhomogenity of the medium, the coplanar modes can not be pure TE
(transverse electric) to z or TM (transverse magnetic) to z. Longitudinal components (£,
and H ,) for the fields exist. These hybrid modes can be expressed as a superposition of TE
to x (TEX) and TM to x (TMX) fields since the discontinuities in the dielectric layer are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
/
/
V
/
✓
/
/ '
/ '
/
Fig. 11. Multi-layered infinite-width CBCPW cross-section.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
26
in the x-direction. The fields in the space domain set are expressed from [9] as
E ——
y \dx
r
2
¥
(2 . 1a)
\d x
d¥
dz
i a 2¥ e
y dxdy
e
l S 'P
y dx dz
£ , „ = -3- -2;—
+ k
+ k
d¥
Hy =
dz
h
-r
H ,= -
dy
e
2
h
, i d ¥
~z d xdy
d¥
dy
I
z dx dz
(2.1b)
(2 . 1c)
for each region i=l,2,...,5,6 where ¥ and ¥ are the electric and magnetic scalar
2
2
potential functions respectively, k = co s r;e 0 prfi0 , y =j(aeris 0 , z=y'cop.rp 0 and j=
V =T.
co is the radian frequency and co=27tf where f is the frequency of operation (FREQ)
and e0=8.854 x 10' 15 F/mm and |i 0=47t x 10‘10 H/mm. The time convention is implied
and the z dependence e*1* is assumed where y is the complex propagation constant in the
z-direction and y=3 -jo..
P is the phase constant and a is the attenuation constant due to
losses in the transmission line. These loss mechanisms can include material or dissipative
(dielectric), modal (leakage), or radiation (open structures at discontinuities). However,
the waveguides analyzed here have upper and lower conductor planes so radiation in the
6
h
x-direction is not present. ¥ and ¥ are functions of x and y and are solutions of the
Helmholtz or wave equation
( 2 .2 )
where £, represents ¥ or ¥ .
In the SDM the potential functions and field components are transformed into the
Fourier or spectral domain. The Fourier transform is taken parallel to the substrates (see
Fig. 11). The transform of a function <)>with a continuous spectrum (infinite-width in the
^-direction and z-direction) can be expressed as
00
%{x,ky) = ^
J <|>(x,y) eikyy dy
(2.3)
-0 0
or in short notation as
$ = F(<f>)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.4)
27
where ~ represents the transformed function and k is the continuous spectral variable in
they-direction. The inverse Fourier transform is written as
oo
<t>(x,y) =
j
$< x,ky)e~jkyy dky .
(2.5)
—oo
Sufficient conditions on <{> and <J) to exist are
00
J l<t>
—00
dy < oo
(2 .6 )
and (j) (x,y) satisfies the Dirchlet conditions over (-oo <y < go) which are:
(i) <t>(*>.V) and ^
{<{>(x,>>)} are piecewise continuous;
(ii) (j) (x,^) has a finite number o f maxima and minima.
Equations (2.1) and (2.2) are transformed into the spectral domain using the following
relations
f ( | 5 ) = ~jkrl
(2.7a)
* ■ < !? « > = -* :? •
(2-7b)
and
The solution to the transformed homogeneous differential equation of (2.2) is expressed as
£, = i?coshK X + rsinhK X
~e
h
where % represents 'F or *F and
k
where
k
2
2
2
= ky + k . - k
(2.8)
2
(2.9)
is the propagation constant in the x-direction and K=jkx and R,
T are constants.
k_ isthe continuous spectralvariable in the z-direction. For aninfinite line with variation
efp and from (2.3), y=k,. The scalar potentials can be written for each region o f Fig. 11
using ( 2 .8) by incorporating the boundary conditions at the upper and lower conductor
planes (tangential electric fields vanish) as;
Region 6 :
~,e
e
Tg = A c o sh K 6 (x + h 6 + h 5 + h 4)
~h
<P6 - A
h
sinhic6(x
+ h 6 + h 5 + h 4)
(2.10)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
Region 5:
Q
Q
Q
5 = B sinhK5(x + h 5 + h4) + C coshK5(x + h 5 + h4)
~
h
^
(2.11)
h
'Rj = B coshic5(x + h5 + h4) + C sinhK5(x + h 5 + h 4)
Region 4:
C
g
Q
i¥ ,4 - D sinhic4(x + h4) + E coshic4(x + h 4)
^
h
¥ 4 =
h
(2.12)
h
D c o sh ic 4 (x + h4) + E sinhic4(x + h 4)
Region 3:
~
e
e
h
h
e
'r 3 = F sinhKjX + G c o sh ic 3x
^
(2.13)
h
¥ 3 = F coshKjX + G sinhK3x
Region 2:
g
g
ft
'f ' 2 = H sinhK 2 (x - h3) + / co sh K 2(x - h 3 )
h
t 2 -H
h
(2.14)
h
coshK2(x - h 3) + / sinhK2(x - h3)
Region 1:
~
e
e
^
h
h
T j = J c o s h K ,( h 3 + h 2 + h , - x )
sinhK 1(h 3 + h 2 + h , - x)
(2.15)
where Ae , Ah, ... , f , J 1 are the unknown coefficients. The electric and magnetic fields in
the spectral domain can be obtained for each region by substituting (2.10)-(2.15) into (2 . 1)
and utilizing the relationships in (2.7).
Twenty unknowns exist and require twenty
boundary conditions to solve for the above coefficients. These conditions result from the
continuity o f the tangential electric and magnetic fields between the dielectric interfaces
and the presence of an electric surface current at the PEC (x=0) as follows:
At x = -h5-h 4
E y6 E yS
for all y
(2.16a)
E=6=E:5
for all.y
(2.16b)
HX>=Hys
for all y
(2.16c)
5
for all y
(2.16d)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
29
At x = -h4
for all y
(2.17a)
Ezr E z4
for all y
(2.17b)
Hy5=Hy4
h z5=h :4
for all y
(2.17c)
for all y
(2.17d)
Ey*=Ey3
for all-V
(2.18a)
E,a =E.3
for all y
(2.18b)
A tx = 0
~ H y4 =JZ
=0
H:3-H :4 = -Jy
=0
H yl
\S/2\ > M aIld Ivl > IS/2 + W|
otherwise
|S/21 > Lv| and \y\ > |S/2 + W|
otherwise
(2.18c)
(2.18d)
At x = h3
for all y
(2.19a)
e z3=e :2
for all y
(2.19b)
£
II
for all y
(2.19c)
e :3=h :2
for all y
(2.19d)
Ey2 Eyl
for all y
( 2.20 a)
Ezr E zl
for all y
(2.20 b)
II
for all y
( 2.20 c)
h z2=h :1
for all y
( 2 .20 d)
a?
Ey3 Ey2
At x = h3+h2
a?
where Jy(y) and J.(y) are the unknown current distributions on the strips at x=0 and the
boundary conditions are specified for the entire range of y. The boundary conditions in
the spectral domain are obtained as the Fourier transforms of the conditions in the space
domain. The unknown currents are related to the unknown electric fields across the slots
(|S/2| < [y| < |S/2 + W|) (or treating these electric fields as equivalent magnetic currents)
through the admittance Green's functions which rigorously incorporates the effects of the
layered medium as
[J\ = [Y\[E\.
(2.21)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
In terms of system theory, the Green's function is basically the transfer function. It is very
desirable that the Green's function be formulated in closed form. In the general solution
approach for finding the Green's functions, the field formulation is derived by solving
4(N-1) unknown coefficients from 4(N-1) coupled equations for an N dielectric layer
structure.
As the number of layers increases, this process becomes substantially more
complicated.
An alternative procedure called the immittance approach [18] enables an
easy solution for multi-layer structures by decoupling the TEXand TMX components. In
the immittance approach, the formulation o f (2 .21 ) is possible without knowledge of the
field coefficients. The basic concept of the immittance method is a special mapping or
decomposition relating the currents and fields in the Fourier domain.
Appendix A
describes this procedure in detail and the dyadic spectral domain admittance Green's
function are written as
( 2 .22 )
(2.23)
(2.24)
where Ye and Y h correspond to the TM and TE modes respectively, and are determined
from (A.5), (A.7), and (A. 14-A.19).
Dyadics are used for representing Green's functions
relating an arbitrarily oriented source to the fields and currents that it creates.
Equation
(2.21) can be written in the spectral domain using (2.22-2.24) in matrix form as
' J y
'
J z
.
'Yyy
V 'V
For instance, the Green's function Yyz relates the ^-directed current on the strips at (x=0, z)
to the source slot electric field in the 2-direction at (jc=0, z). The unknowns of the above
system are the complex propagation constant y and E y,E z,J y, J ..
A large number of
basis functions would be required for the current expansions (J y and J . ) due to the wide
conductor area for the CBCPW (see Fig. 11). Hence, it is more numerically efficient to
apply the procedure across the unknown electric fields over the slots as expressed in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
(2.25). The current expansions can be eliminated by applying the Galerkin method which
is a moment method procedure. In this method the testing functions are the same as the
basis functions. The first step is to expand the unknown electric fields Ey and E, across
the slots at x=0 in terms of known basis or trial functions Em and E;n in the space domain
as
Ey = £ c mE ym(y)
m=\
E . = 2 d nE :n(y)
n= 1
where cm and dn are the unknown expansion coefficients.
(2.26)
The basis functions in the
spectral domain are obtained from (2.26) and utilizing (2.3)
oo
oo
E y= I cmE ym(ky)
m=\
E._= s d nE :n(ky) .
n=\
(2.27)
The basis functions must bemembers of a complete set and the Fourier transforms must
exist,preferable inclosed form.
Any set o f functions can be usedbut those that represent
the physical electric field distribution will reduce the number of functions, enhance the
accuracy of the solution, and reduce the matrix size and computation time. Applying the
boundary condition that the tangential electric fields at a PEC must vanish, each basis
function is chosen so that it is nonzero only in the slot areas ( |S/21 < [y| < |S/2 + W|). Due
to computer limitation in handling infinite size matrices, the number o f basis functions
must be truncated to a finite value and the accuracy of the solution will depend on the
number o f functions. After substituting (2.27) into (2.25) yields
~
~
M
~ N
^ y = ^yy ^ c mEym + Yyz ^ d nE_n
m=l
n= 1
(2.28a)
~ ~
M
J-. = yv X c . E
m=\
(2.28b)
+
~ N
2 </„£,.
n= 1
Let the inner product required for the application o f the Galerkin technique be defined as
< X ,Y >
=
J XY*dky
(2.29)
—oo
where (*) specifies the complex conjugate. The Galerkin technique can now be applied
using (2.29) with the inner product o f (2.28a) with E yp and similarly (2.28b) with E :g and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
yields
oo
_ _
M
E*„Y„
S
< J y 'K > = f
—00
m -\
„
^
C„ £
+
„
£
^
N
2
^
d , E ;n dky
p = 1,2, ...,M
n= 1
(2.30a)
00
= J
-0 0
~
M
£ zg
-0^zv
2“
zy
m -\
~
~ ~ N
~
£?o r _ S , c/„n e :n dfc
n= 1
+
ym
m
q = \,2,...,N.
(2.30b)
The left hand sides of (2.30) are zero using Parseval's theorem because the product of the
currents and the slot electric fields at x=0 is zero since the currents exist only on the strips
and the electricfields exist only in the slots. Parseval's theorem is written as
oo
oo
J J(y)E*(y)dy =
J J (ky)E *(ky)dky = 0 .
-o o
(2.31)
71 - o o
A homogeneous system of equations is now formed and is described in compact form as
c
d
M(y)]
(2.32)
= [0]
w h ere [/1( y)] is a m atrix o f order p+q w h o s e elem ents are listed as
30
Jf
E * Y WE
—oo
yp
y>
ym
dk y
(2.33a)
00
(2.33b)
—00
oo
21
22
f E
J
—00
00
* Y . VE
dk y
zq - y ym
J K *
(2.33c)
(2.33d)
-0 0
Considering (2.32), nontrivial solutions can only occur when the matrix is singular as
det [/4(y>] = 0
(2.34)
where det is the determinant of the matrix and the complex propagation constant or
eigenvalue y can be computed at each frequency co. Associated with each eigenvalue
or mode are the eigenfunctions
Q
and
fa
. An infinite number of modes exist or are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
supported by the waveguide and depending on the structural parameters and operating
frequency, one or more modes will be propagating and attenuating (P positive, real and a
positive, real) in the z-direction. Most of the modes will be evanescent or non-propagating
(P negative, imaginary and a positive, real).
B. Basis Function Selection
The basis functions describe the behavior o f the electric fields across the slots for the
CBCPW. Any type of basis function can be used as long as the function satisfies the
boundary conditions of the PECs (nonzero only on the slots) at x=0.
The numerical
accuracy and efficiency of the SDM depends on the basis function selection.
In an
electromagnetic problem involving strips with sharp edges and modeled by infinitely thin
PECs, some of the field components may possess an unbounded behavior in the vicinity of
these edges [19], In this case, the transverse electric and magnetic field components are
unbounded at the strip edge and approach infinity (singular fields) with |r| 2 variation
where r =(y-S/2-W/2) for the right hand slot of
Fig. 11.
The longitudinal field
I
components are bounded near the edge and vary as |r l 2 . The above conclusions are
general and hold at the edge of an infinitely thin PEC placed between two dielectrics and
the dielectric constants do not enter the given asymptotic expressions. Spurious solutions
(non-physical) may be eliminated by choosing basis functions which are twice continuously
differentiable [19], The set of basis functions should be complete to enable approximation
o f the exact solution to any degree desired by simply increasing the number o f terms o f the
expansion.
In this manner, the numerical solutions o f the SDM can be checked for
convergence.
In recent years, most investigators have employed a combination of sinusoidal [18] or
Chebyshev [32] functions with the edge condition or correction term to model the
unbounded fields at the conductor edges for the basis functions. As described earlier in
Fig. 4(a), the dominant mode for the CBCPW has an nonsymmetrical characteristic in
the transverse direction.
This translates to odd functions for E
(dominant field
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
component) while the longitudinal component E. can be described with even basis
functions. Sinusoidal basis functions are used in this dissertation and are expressed as
cos [(m—1)7t(y-/ft)/W]
cos [ ( aw- 1)7t(y+l>)/W]
J\-[2(y-b)/W ]2
sin [(wi-l)7t(y-l>)/W]
J l-[2(y+b)/W)2
sin [ ( aw- 1)7t(y+6)/W]
J l-[2(y-6)/W ]2
cos [»7i(y-6)/W]
+
m= 1,3,...
(2.35)
m = 2,4,...
Jl-[2(y+b)/W\:
cos [»7r(y+Z>)AV]
aa
J l-[2(y-Z»)AV]2
J l-[2(y+b)/W\2
sin [mc(y-fr)/W] _ sin [AA7t(y+6)/W]
E zn(y) =
^ l-[2(y-£)/W ]2
= 1,3,...
(2.36)
/i = 2,4,...
J \-[2(y+b)W]2
where the first terms are valid for ^>0 (right hand slot) and the second terms represent
><0 (left hand slot) and b=S/2+W/2 and all of the terms are scaled by a factor
which
is utilized to simplify the Fourier transforms and the voltage calculation and presented later
in section E. The selected functions satisfy the criteria earlier described for basis functions.
The shapes of the first three electric field basis functions are depicted in
Fig. 12. The Fourier transform for the functions o f (2.35) and (2.36) can be obtained by
applying (2.3) and are expressed as
js m (k y b)
Eymtty) =
jcos(k b) j 0
m = 1,3,...
^ | W ^ + ( w - l ) 7 t l '~] _
f
f
|W ^ - ( /» - l ) 7 l |
Jo
aw
= 2 ,4 ,...
( 2 .3 7 )
cos(kyb) Jo
E M
=
-sin(AyA) Jo
+ «7tp
+ rm
+ J0
-J n
I W A y - W 7t|
2
|W A ,. - W 7t|
n= 1,3,
n
-
2 ,4 ,...
(2.38)
where J 0 is the zeroth-order Bessel function o f the first kind. The integrals are evaluated
using a substitution method, product-fiinction relations [41], and recognizing that Eym is
an even function across the slots for a w = 1 ,3 ,... and is odd for a w = 2 , 4,... and similarly E.n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
m-1
m=2
m=J
1.0r
0.5
4->
o
c
D
co
co
o
_Q
0.0
-0.5
— 1. 0 *
- 1 .0 - 0 .8 - 0 .6 - 0 .4 - 0 .2 0.0 0.2 0.4 0.6 0.8
1.0
(a)
n -1
»= 2
I.Or
71=3
CO
C
O
0.5
-t->
O
c
D
0.0
CO
*to
o
-Q
-0 .5
N
- 1. 0 *
—1.0 - 0 .8 -0 .6 -0 .4 —0.2 0.0 0.2 0.4 0.6 0.8
y (mm)
1.0
(b)
Fig. 12. Shapes of the electric field basis functions for CBCPW with S=0.635mm,
W=0.3175mm. (a) E>mbasis functions, (b) E,n basis functions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is an even function for w=l,3,... and is odd otherwise. An important relationship aiding
the evaluations of (2.37) and (2.38) with a as a constant is
(2.39)
The only problem with the selected basis functions is the Fourier transform contains the
special Bessel function which is a disadvantage from a computational point of view.
J 0(0)= 1 and JQ(y) looks qualitatively like sine or cosine waves whose amplitude decays as
y
-
1/2
. The number of basis functions for an accurate solution is largely dependent on the
aspect ratios. For small W/S and W/D (where D=h6+h5+h4), only a single basis function is
needed since both the slot coupling and the conductor-backing plane (at r=-h6-h5-h4)
effects are small.
As the aspect ratios increase so does the number o f basis functions
required to accurately simulate the CBCPW and the structure becomes more dispersive.
The Chebyshev basis functions for the aperture fields will be used later to verify the
results of the sinusoidal basis functions and are included here as
(H P )
E„ =
N
( 2(y-b)-) [
E d . U„_,
I W
n= 1
(2.40)
(2(y+b)')
( 2 (y+b)']
r w -J f-\T v T J
(2.41)
where TmA and UnA are the Chebyshev polynomials o f
the first and second kinds,
respectively. The Fourier transform of these basis functions may be written in terms of
integer order Bessel functions [42],
C. Two-Dimensional Packaged Structures
In order to connect the ground plane conductors at x=0 and the conductor-backed
plane at x=-h6-h5-h4 throughout the length of the CBCPW to achieve a uniform ground
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
contact (see Fig. 11), PEC lateral shorting sidewalls are used to model this wrap around
structure. The waveguide is assumed to be placed within a packaged assembly and the
sidewalls are extended in regions 1, 2, and 3 (see Fig. 13) to connect the cover plate
conductor at x=h3+h2+h1. The lateral distance (A) is finite in width and the infinite-length
(r-direction) waveguide is analyzed with a two-dimensional SDM.
For structures with sidewalls, the Fourier transform o f (2.3) is replaced by a discrete
finite Fourier transform where <J>(y) is defined over the interval [-A.A] and satisfies the
Dirichlet conditions o f (2.6) with
A
J l<K*,.y)l dy < 00
-A
(2.42)
with
$(*,kyi) = i
^
A
j § ( x ,y ) J k>'y dy
-A
(2.43)
and the inverse Fourier transform is defined as
Z $(x,/cyt) e Jky y .
=
i
(2.44)
= -o o
The solutions for finite regions (waveguides) are characterized by discrete spectra of
eigenvalues. Accordingly, the integrals with respect to k are all replaced by summations
in terms of discrete values of kyj from /'=-oo to
oo
for (2.30) and (2.33). In the above
expansions, k^ is the discrete spectral parameter and is determined by examining the field
behavior along the ^-direction. Since the field components Ex and Hx are proportional to
and T* respectively as from (2.1a), examination o f these two field components is
sufficient for determining kyi. As mentioned earlier, the dominant or odd mode (E ) for
the CBCPW corresponds to a magnetic wall placed at .y=0 and electric PECwalls exist at
,y = ±A.
Therefore,
is equal to sin
becomes cos
f o r /'= - q o , ...,-5 ,-3 ,-1 ,1 ,3 , 5,
...,q o
and T*
( /7tv')
j . The discrete spectral parameter for the j-direction can be rewritten
as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
Fig. 13. Multi-layered finite-width CBCPW cross-section.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
k >" = (/ + 2> A
for / = - o o , ...,-2 ,-1 ,0 ,1 ,2 ,...,
oo.
(2 45)
Recall from (2.1b) that Ey is proportional to J^T ^and
VJ
Parseval's theorem of (2.31) is also modified in the packaged case as
4
i
J J(y)E*(y) dy = ±
-A
,=0° ~
~
I J(kyt)E*(kyi) = 0 .
i=~™
Allof t.heelements of the matrix [A] of (2.33) are even functions
(2.46)
of k r Yyy,
are
even and Yy,, Y ^ are odd functions of kyj (see A. 8-A. 10) while Eym and E :nare odd
and even functions, respectively. An even function in the space domain (y) becomes an
even function in the spectral domain and similarly for odd functions.
Therefore, the
summations in terms of k are represented as
/ = oo
X
/ —oo
A = 2 Z
/ = —oo
A.
(2.47)
/'= 0
An infinite number of summations terms is not possible with a computer and (2.47) is
truncated and checked for convergence. Likewise, equations (2.33) are similarly modified.
D. Spectral Domain Method Field Formulation
The spectral domain immittance method can be efficiently extended to determine the
electric and magnetic field components of the CBCPW. As stated earlier, this part of the
derivation is not necessary to solve for the propagation constant of the waveguide but is
required for mode identification, demonstration of the mode coupling effects on the
dominant mode, boundary condition verifications, and calculation of the mode
characteristic impedances.
Once the propagation constant y is found from (2.34), the electric field basis function
expansion coefficients (cm and dn ) are derived using (2.32). The spectral domain versions
of Ey and E z at x=0 can be determined via (2.27) and then J y and J . are calculated
from (2.28).
The next step is rewriting the boundary conditions of (2-16)-(2.20) in the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
spectral domain as
At x = -h5-h4
Ey 6 = E y5
£ -6 = £ .j
H y6=H y 5
(2.48a)
(2.48b)
(2.48c)
^ , 6 = #.-s
(2.48d)
£ 3,5 = ^
£ -5 = £ .4
H y, = H y4
H :5= H :4
(2.49a)
(2.49b)
(2.49c)
(2.49d)
E y4 = Ey,j
£ -4 = £ -3
(2.50a)
(2.50b)
(2.50c)
(2.50d)
At x = -h4
A tx = 0
#.-3 - H :4 =
~Jy
At x = h3
E y 3 = E y2
£ .3 = £ .2
^ 3 = ^ .2
H . 3 = H :2
(2.51a)
(2.51b)
(2.51c)
(2.5 Id)
Ey2 = E yl
E : 2= E . 1
H y2= H yX
H ^ = H Zi
(2.52a)
(2.52b)
(2.52c)
(2.52d)
At x = h3+h2
where
and J , are the Fourier transforms o f the unknown current components on the
strips atx=0 and
are given in(2.25).
Using the coordinate transform of the immittance
method of (A. 2),
J y and J z arelinear combinations
of J u and J v andare expressed
from (A.3) as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
_
k ZJ :
y jk y
^
k y j z
J k j
+
k y jy
+
“k
+
(2.53)
k \
z J y
(2.54)
k \
In the immittance approach, the fields are decomposed into the TM, and TE,
components. The electric and magnetic fields are now presented in the spectral domain via
(2.1) and using (2.7) with (2.10)-(2.15) as
TE,
TMX
Ex =
y
In ^
■•n
i
n
•->
I
II
tl
Hx = 0
Ex = 0
(2.55a)
Ey = jT V
(2.55b)
~h
E. = -jR V
(2.55c)
1 —h
Hx = I t
z
(2.55d)
I
II
litT
&y =
H : = jR % e
H. =
■R dT *
z dx
^ h
■T <3T
T dx
(2.55e)
(2.55f)
ve
-yh
for each region i=l,2,...,5,6 and t and T are the transformed electric and magnetic
scalar potential functions given in (2.10)-(2.15) and
ky
+
R
1
T — kz
~ k y2 + kl
(2.56)
Equations (2.53) and (2.54) are written in terms o f the field components utilizing the
boundary conditions of (2.50c) and (2.50d) as
7
“
J
+
NK>
<
\
7
} 4e x4 - - y 3E « ]
i . . 'dH x4
dx
j k 2y + k 2
s h x3'
dx
(2.57)
(2 58)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
Substituting the field components of (2.55) into the boundary conditions of (2.48)-(2.52)
with (2.57) and (2.58), results in two sets o f ten independent equations with ten unknowns,
respectively. This method leads to less complexity in the derivation of the coefficients as
compared to the general brute-force approach regardless o f the number o f dielectric layers.
The final derived coefficients ( A e , A h , ... , f , f ) of the electric and magnetic fields are
described in Appendix B. The field components for Ey and Hy are expressed respectively
as
TM_
-jRKysA sinhic6(x + h 6 + h 5 + h4)
E y*
=
K
= - j R K yS^B coshK5(x +
h5
+ h4) + C sinhK5(x + h s +
(2.59a)
h 4) J
(2.59b)
Ey4 = - j R K y^ D coshic4(x + h 4) + E sinhx4(x + h4)J
(2.59c)
E * = - j R K y3^F coshx3x + G sinhK3x J
(2.59d)
=
-jR K yJ^H coshic2(x - h 3) + / sinhK2(x - h3)J
e
Z fl = jR \cylJ sin h K 1( h 3 + h 2 + h , - x)
(2.59e)
(2.59f)
TE_
's,
n
<—
E y6 = j T A sinhK6(x + h6 + h 5 + h4)
h
h
B coshK5(x + h 5 + h 4) + C sinhKs (x + h 5 + h 4)
(2.60a)
(2.60b)
h
h
Ey* = j T D coshK4(x + h 4) + E sinhx4(x + h 4)
(2.60c)
h
h
Ey3 = j T F coshK3x + G sinhK3x
(2.60d)
n
h
h
H coshK2(x - h 3) + / sinhK2(x - h 3)
~
h
E yl = j T J sinhK,(h3 + h 2 + h , - x)
(2.60e)
(2.60f)
and
™x
H y6 = - j T A coshK6(x + h6 + h 5 + h 4)
(2.61a)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
43
/V
B
C
H v5 = - j T B sinhic5(x + h 5 + h4) + C cosh»c5(x + h s + h 4)
(2.61b)
H y4 =
(2.61c)
sinh>c4(x + h 4) + E coshlc4(x + h4)J
H yi = - j T ^ F sinhic3x + G coshic3x J
(2.6 Id)
H y2 = - j T ^ H sinhic2(x - h 3) + / coshK2(x - h3)J
(2.6 le)
H yX = - j T J coshK1(h3 + h 2 + h, - x)
(2.61 f)
= - j R k :6A co sh K 6 (x + h 6 + h 5 + h 4 )
h
h
H yS = - j R k-3 B sin h K 5(x + h 5 + h 4) + C c o sh K s (.r + h 5 + h4)
(2.62a)
TE.
h
Hy*
=
~ J R K r4
(2.62b)
h
D sinhK 4 (x + h 4) + E c o s h K 4 (x + h 4)
h
h
H y3 = - j R k .3 F sinhK-,x + G coshK,x
h
(2.62c)
(2.62d)
h
H y2 = - j R Kja H sinhK2(x - h3) + I coshK; (x - h3)
(2.62e)
H yi = j R K :lJ coshic^hj + h 2 + h t - x).
(2.62f)
The total field is the sum of the TM and TE components and for E v becomes
„
~ TM
_ TE
E ■= E ■ + E ■
y>
y>
y>
(2.63)
for i = l,2,...,5,6. Finally, the space-domain (x,>/) representation o f the fields can be
obtained for plotting purposes using the inverse Fourier transforms of (2.5) or (2.44)
depending on whether the lateral waveguide dimension is infinite or finite, respectively.
E. Characteristic Impedance Derivation
In microwave circuit design, knowledge o f the characteristic impedance of the
transmission line is critical especially in the development of matching networks or antenna
systems. A wide range of impedances is also desirable for various networks and an easily
achievable 50Q impedance is required to interface with network analysis measurement
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
equipment. The SDM can be extended to calculate the frequency dependent characteristic
impedance of the CBCPW.
For pure TEM structures, the voltage and current in the transmission line model have
the same meaning as the voltage and current in a circuit representation. The characteristic
impedance (Z) is unique. The three possible definitions for the impedance give the same
result and are defined as the power-voltage, power-current, and the voltage-current
respectively [43] as
(2.64a)
(2.64b)
(2.64c)
where the magnitudes o f V and / are the voltage difference between the conductors and
the total longitudinal current on the center signal conductor (S) respectively, and P is the
time-average complex Poynting power flow in the z-direction.
For non-TEM (hybrid-mode) structures such as CBCPW, the characteristic impedance
calculation is more difficult and is not unique. This is due to the fact that, unlike a TEM
transmission line, for a non-TEM transmission line ambiguous voltage and current waves
exist. The voltage cannot be uniquely defined independent of path for a non-TEM line.
Continuity of power across the transition from a TEM to a non-TEM line is always valid.
For a general transmission line the definition o f Z may depend on how the transmission
line is fed [44],
For the dominant CBCPW mode, the exciting source is two voltage
sources oppositely directed across the slots as in Fig. 4(a) and hence definition (2.64a)
would seem to approximate the characteristic impedance most closely.
Also at lower
frequencies, the three definitions of (2.64) should correspond to the quasi-static
(zero-frequency) value.
The current I in the characteristic impedance calculation is defined as [25]
S/2
1 = 2 \ J : (0,y)dy
0
(2.65)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
where •/_ (O^y) is derived from the inverse Fourier transform o f (2.28b) and using either
(2.5) or (2.44) depending on whether the lateral dimension is infinite or finite (packaged
waveguide). Careful consideration in the integral evaluation must be applied near the edge
o f the conductor as the current experiences an unbounded or singular characteristic.
The voltage V in (2.64a) and (2.64c) corresponds to the negative of the path integral
of the electric field across the slots for the CBCPW and is stated for the right hand slot as
V= and E
S/2+W
J
E y(0,y)dy
S/2
(2.66)
is taken from the basis function expansion o f (2.35) for m= 1,3,5,...
is an odd function across the slots for m=2,4,6...........
After
only, since Ey
applying the substitution
method, the integral of (2.66) becomes
M
(m- 1)n
v = - £ c mj 0
m=\
(2.67)
with m=l,3,5,... .
The propagated power P can be found as the integration over the cross-section of the
CBCPW of the Poynting's vector projected onto the longitudinal z-direction and for the
infinite-width lateral case is given by
P = Re
where z
h3+h2+hi
OO
j
( (E xH *).z^*
-(h 6+h5+h4) -°°
(2.68)
is the unit vector in the z-direction, E x H* is a time-averaged quantity
cross-product, the asterisk (*) indicates complex conjugate, and Re ( ) represents the real
part of. By applying Parseval's theorem o f (2.31) to (2.68), the integral is expressed as
J (ExH*) • zdydx =
CS
oo
h3+h2+hi
J
j
K
(_E x H*J • z dx dky
(2.69a)
-(h 6+h5+h4)
00
=
t 'Z
j
-oo
[-^ h 6
+ ^h5
+ ^h4
+
^*h3 +
^h2 +
^h l]
^y
(2.69b)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
where CS is the waveguide cross-section and
_ (hs+h4)
P* =
J
-(h 6+h5+h4)
P
- lu
=
J
P 70a)
.
.
(2.70b)
-(h 5+h4)
0
P h4 = J[Ex4H*y4 -h 4
h3
,
Ey4H*4)d x
(2.70c)
.
P * = J (.2 ,3 * ;, - Ey3H*3)itc
0
h 3 + i> 2
= J
.
(2.70d)
v
(2.70e)
h3
h 3 + h 2 + h
^*hl “
/
' ( X i# ? , - £ , ! # : ■ ) *
h3+h2
(2 70f)
where the field components are stated in (2.55a), (2.55d), (2.59), (2.60), (2.61), and (2.62)
and the final results for (2.70) are presented in Appendix C.
v e
-v h
¥ and ¥
From section C where
are even and odd respectively from (2.10), (2.55d), and (2.55e), it is clear to
see that H x and H y
are odd and even Sanctions, respectively. For packaged CBCPW,
Parseval's theorem o f (2.46) is applied to (2.68) and (2.69a) becomes
.
, oo h3+h2+h,
j( E x H *) • idycbc = — 2 £
J
(jE x
s
i=0 -(h 6+h5+h4)
_
H*J
• z dxdkyi
(2.71)
and utilizing the (2.45) definition for the spectral variable with kyi the calculations of
(2.70).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
F. Analysis of Microstrip and Finite Ground Plane CBCPW
The results o f the analysis applied to microstrip are used to validate the procedure for
calculating the resonant frequencies o f the three-dimensional SDM for CBCPW design
examples in Chapter VI. The finite ground plane (FGP) CBCPW [27] is an alternative
structure in which the ground conductor widths do not extend to the edge o f the substrate
(WG of Fig. 13 and Fig. 3) and can be viewed as two sets of coupled microstrip lines. This
type of waveguide is utilized in MMICs where high line density is required to maximize
circuit functions and minimize costs. These two structures are depicted in Fig. 14. Also,
note that the ground planes at x=0 (WG) for the (FGP) CBCPW are not connected
electrically to the conductor-backed plane at x=-h6-h5-h4.
The analysis o f
the above waveguides can be performed using the SDM with
modifications in the process outlined earlier.
First of all, these transmission lines are
fundamentally different from the conventional CBCPW of Fig. 13.
Now the strip
configuration (total strip area) is significantly less than for the conventional CBCPW and
therefore it would be more efficient to expand the basis functions as currents on the strips
as compared to the electric fields in the slots. This is reflected by modifying (2.25) at x=0
by taking the inverse of the admittance matrix as
A
‘V
E.
A
-1
h
Yzz
'
"V
(2.72)
where the dyadic spectral domain Green's function become an impedance matrix as
(2.73)
i
-y
ys
l
i
i
'
I
A
^
i
t&
s11
Y zz
Zyy
1
A V -1
A
and the elements can be derived from Appendix A. The current basis functions for
microstrip are zero only outside the strip area ([y|>S/2) as shown in Fig. 15(a) and can be
expanded in the spectral domain as
M
„
J z ~ 2 c mJzm(ky)
m= 1
N
Jy~
2
d nJ ,.n(k
) .
(2.74)
n= 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 14. Additional structures analyzed using the SDM with air above the conductors at
x=0. (a) Microstrip waveguide, (b) Finite ground plane CBCPW.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
Substituting (2.74) into (2.72) yields
(2.75a)
m= 1
M
n= 1
„
N
(2.75b)
and the Galerkin procedure is again applied to produce
M
„
„
N
(2.76a)
~
M
„
_
N
(2.76b)
The left hand sides o f (2.76) are zero using Parseval's theorem o f (2.31) and
a homogeneous
system
o f equations
is now
formed
to solve for the complex
propagation constant y. The dielectric losses due to the presence o f dissipation in the
dielectric media can be taken into account by the introduction of a complex relative
constant
(2.77)
where tan 8- is the loss tangent of the dissipative dielectric region and (2.77) would be
substituted into (A. 12), (A. 13), and the k wave number calculation.
For the FGP CBCPW, three strips are involved and the current expressions are
written as
Jy(y) = JyA O) + Jy,2(y) + J y3 0 )
(2.78a)
d : (y) = ^ ,iO ) + J , 2(y) + J =,i(y)
(2.78b)
where J , (y), Jy 2 (y), Jy 3 (y) and J. , (y), J:2 (y), J, 3(y) are the components o f the currents
on the first, second, and third strips respectively (from left to right), and are zero outside
the strip regions.
The above equations are now in appropriate form and each of the
unknown elements in these equations may be expressed in terms o f a set of suitable
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
l.OL
0.5
-0.5
- 1. 0,
y (mm)
(a)
1.0 r
0.5
-0.5
1.0
-
--2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4
0.8
1.2 1.6
2.0
y (mm)
(b)
0.5
M
S
-0.5
-
1.0
y (mm)
<c)
Fig. 15. J „
idealized current components for
the waveguides under analysis,
(a) Dominant mode for microstrip with S=0.635mm. (b) Coplanar waveguide
mode for the FGP CBCPW with S=0.635mm, W=0.3175mm, WG=0.635mm.
(c) Coplanar-microstrip mode for the FGP CBCPW.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
basis functions. In expanded form these equations become
J :(y)
R
S
T
= £ a rg r(y) + £ b s hs(y) +
i C, i,(y)
r= 1
5=1
t= 1
(2.79a)
J y(y)
U
V
= £ d J J y ) + £ e v*v(y)
u= 1
v=l
(2.79b)
W
+£ f j w(y)
W=1
where the basis functions hs(y) and kv(y) are both equal zero for.y outside the center strip
([y| >S/2) and the functions g r(y), it(y), j u{y), IJy) are zero outside the ground strips
(| S/2 + W| < \y\< |S/2 + W + W g |.
Due to the symmetry consideration o f these structures, a magnetic wall can be placed at
y =0 which corresponds to even modes for the dominant current component J. . For the
packaged structure in Fig. 14(b), (2.45) will be invoked. Again, the basis functions for the
currents on the strips include the singular behavior o f the magnetic field components
normal to the stripline edge. For microstrip, the following basis set is utilized [18]
cos [2(/w-l)7ry/S]
) =
iw=l,2,...
(2.80)
« = 1, 2 ,.
(2.81)
,/i-(2y/S)2
sin [2«7ry/S]
Jyn(y) =
/M
W
and note that these expressions only exist over the strip and are zero elsewhere (y > | S/21)
2
and are scaled by a factor of —r-. Also, the boundary condition at the magnetic wall
7lu
(tangential magnetic field H. is zero at >=0) is incorporated into Jy . The idealized current
representation in Fig. 15(a) corresponds to m=\. The Fourier transforms of the above
basis set are
Jzm&y) = Ji
JyM y)
= 7 J.
Sk v
+
Sk v
( m -
+ nit
l) 7 t
Sky
~1
Skv
) '- H
(im - l ) 7 t
- tm
(2.82)
(2.83)
where J0 is the zeroth-order Bessel function of the first kind. For the FGP CBCPW, the
basis functions are derived from [45] with (2.80) and (2.81) utilized for hs(y) and kv(y)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
respectively in (2.79) while
g (y) =
Ml/Wo]
^ l-[2 0 + M )A V (j]J
l,<y) =
(2.84)
(2 85)
y i-[2 (y -W )A V G]2
= s in ^ t^ /W o L
^ l - [ 2 ( y + W )A V G] 2
/ w(y) =
fn t^ Q ^ y W o ],
(2 g7)
l-[2(y - M>)/WG]2
2
where AZ»=S/2+W+Wn/2. All of the relations above are scaled by a factor o f
. Since
°
ttWG
the ground conductor planes at x=0 (WG) are not electrically connected to the
conductor-backed plane at jc=-h6-h5-h4 , an additional even mode exists which also has a
zero-cutoff frequency. This mode is referred to as the coplanar-microstrip (CPM) or
parallel plate one as in Fig. 15(c) and the field characteristic has already been plotted in
Fig. 4(c). The fields o f this mode are concentrated between the conductors at x=0 and
x=-h6-h5-h4. The coplanar waveguide mode for this structure as shown in Fig. 15(b), has
been depicted in Fig. 4(a) and again the fields exist primarily across the slots.
The
idealized current representations in Fig. 15(b) and Fig. 15(c) are for R,S,T=l in (2.79).
G. Spectral Domain Method Numerical Analysis and Results
In order to solve the complex propagation constant y and the expansion coefficients of
the electric field basis functions o f (2.27) to plot the field patterns for the CBCPW,
numerical routines must be invoked. These routines include the polynomial approximation
for the Bessel functions, evaluation o f the integrals in (2.33), calculation of the determinant
of the characteristic matrix [A] in (2.34), the implementation of a root searching algorithm,
and the determination o f the expansion coefficients using a least mean squares method. An
explanation of these numerical routines is presented in Appendix D.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
A detailed flowchart representation o f the SDM applied to the infinite-width CBCPW
of Fig. 11 is presented in Fig. 16. The parameter (IN) from the inputs box refers to the
number of basis functions used to perform the simulation. The program consists of three
main routines: ASYMP, GAMMA, and SOLVE. T h e ® mark continuation point in the
figure will be described in Chapter III. The program was written in Fortran with a
Microsoft™ 5.0 compiler using double precision complex variables and simulated on a PC
Pentium 90MHz system.
Numerical results are presented in Table I and compared with [46] which implemented
a full-wave analysis of multi-layered coplanar lines based on a hybrid/SDM approach. A
common parameter utilized by microwave circuit designers based on the propagation
characteristics of a transmission line is the effective dielectric constant defined as
R2
R2
s <#
T
n = 7T
k\ = -co2|a0s
0•
(288)
The hybrid mode of propagation along the CBCPW leads to a dispersive medium. As the
frequency increases, the fields become more concentrated in the region beneath the strips.
Since the fields are forced into the dielectric substrate to an increasing extent as the
frequency rises, the effective dielectric constant increases with frequency.
As Table I
indicates, good agreement exists between these two numerical methods. Unless noted, all
examples in this dissertation will refer to the propagation characteristics o f the dominant
odd mode of the CBCPW from Fig. 4(a) and utilize a total of 5 basis functions (A/=3 and
N=2) for (2.27). In the table, presented work refers to the results o f this dissertation.
Table II shows the convergence of the effective dielectric constant and compares the
sinusoidal basis functions of (2.37) and (2.38) with the Chebyshev functions of (2.40) and
(2.41). A value of 7.08 was predicted by [46] for this structure. The number of basis
functions is M - N where N=M-1 for the number of E. basis functions. The selection of the
edge-corrected sinusoidal basis functions (used in this dissertation ) is verified.
An example of the integration convergence' for the complex propagation constant y for
the waveguide of Table II with dielectric loss is provided in Table III. INT PTS and ITR
# refer to the number of integration points within each subinterval and the iteration value
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
Q tarQ
inputs
' £ r l » ®t2 > P r3 >
>®i*5
- h , . ^ , h3.A , h5 h*
‘ ° U ~2,
?4’
59
6
dielectnc loss tang
- S, W, IN (# basis mts)
CALL ASYMP
- calculates the asymptotic
SDM integral o fjp .9 )
with upper limit for lower
mtegral region
CALL INITIAL
- 3 initial guesses of y for
root searching routine
Muller's Method (D.25)
CALL GAMMA
-CALLGAULEG
SDM integration procedure
- CALL DARA
SDM dyadic Green's functions o f (2.22-2.24)
- CALL DARB
calculates Fourier Transforms of basis fills
o f (2.37-2.38)
- CALL DARC
computes [A] m atrix of (2.33)
- CALL LUDCMP
finds determinant o f (2.34)
- update y using (D.24)
CALL SOLVE
- write v
- solve for basis fht
coeffs of(2.32) using
has y converged?
increase # integration pts
SVDCMP. S \
Q
I
e n d j^
Fig. 16. Flowchart for computing the propagation constant o f the infinite-width CBCPW.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
within the root searching algorithm (Muller's method) respectively. The units for (3 and a
are rad/mm and nepers/mm, respectively.
TABLE I
Comparison of the effective dielectric constant for an infinite-width CBCPW
with Ref. [46] forerI= l, sr4= 13, erS= l, h^lO m m , h4=0.75mm, h5= 10mm,
h2=h3=h6=0, S=2mm, W= 1.5mm.
f(GH z)
5
Seff
10
15
20
Ref. [52]
4.2
4.6
5.1
5.7
presented work
4.29
4.67
5.16
5.73
TABLE II
Comparison of the effective dielectric constant with different basis function
for the example of Table I with S=0.6mm, W=0.45mm, f =20GHz .
Seff
1
3
5
7
9
sinusoidal
6.921
6.941
6.944
6.945
6.948
Chebyshev
6.921
6.941
6.947
6.948
6.951
# basis functions
TABLE III
SDM integration convergence of the propagation constant
for the example o f Table II with tan 54=0.001 (loss tangent).
INT PTS
ITR#
y
5
4
(1.1030, -5.6078D-4)
5
5
(1.1035, -5.5250D-4)
5
6
(1.1035, -5.5252D-4)
15
4
(1.1038, -5.5279D-4)
15
5
(1.1038, -5.5279D-4)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
Convergence for this structure was very fast and the simulation time was 6 seconds to
execute the solution.
The numerical procedure for the packaged (two-dimensional) CBCPW o f Fig. 13
follows the flowchart o f Fig. 16 with four modifications. The SDM integrals are replaced
with the summations o f (2.46). The integration loopback about the routine GAMMA is
eliminated.
The calculation of the Fourier transforms o f the basis functions (routine
DARB) can be performed outside the GAMMA routine iteration since the spectral variable
values are known and fixed and the Fourier transforms are not a function of y. The
presence of the lateral sidewalls can now support waveguide modes within the CBCPW. If
the lateral dimension is electrically large and the operating frequency is sufficiently high, a
number of waveguide modes can propagate within the structure. To account and identify
all of these modes, a bisection routine (BISECT) is used to bracket these solutions and
replaces the routine INITIAL in Fig. 16.
This routine proceeds by dividing up the
difference between upper (emax) and lower ( s ^ ) dielectric constant values by an arbitrary
integer (MP or mesh points) and generating a number of fictitious dielectric constants and
determinants o f the characteristic matrix o f (2.34). The imaginary part of the determinant
value is inspected for the phase constant o f the complex propagation constant. Dielectric
losses contribute to the real part o f the determinant.
The determinant vanishes at
propagating mode solutions of the CBCPW and the determinant sign is examined to
bracket and identify these roots. The identification and solution o f the fundamental modes
in Fig. 4 by the bisection approach is straightforward. The determination o f the waveguide
modes supported by the packaged CBCPW is more involved and requires substantially
more mesh points to solve. Muller's method is utilized to accelerate and locate the actual
complex roots and the third initial guess value for this method is the bisected value of the
two endpoints. The flowchart for computing the propagation constants for a packaged
CBCPW is included in Fig. 17. The parameter (A) from the inputs box refers to the
dimension o f the lateral width of the package.
The convergence criterion for the
asymptotic summations and the propagation constant is the same as for the infinite-width
case. The routine ASYMP is applied in an identical manner, determining the asymptotic
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
C starQ
inputs
? rl > ? r2
I
1? r 3 >
®r4>
> £ rfi
h , , ^ hj h4 h5 h,
°U§2> ° jj 84, 6,, 06
dielectnc loss tang
S .W .IN .A ^ .e ^ .M P l
CALL ASYMP
- calculates the asymptotic
SDM summations
CALLDARB
- calculates Fourier Transfroms
of basis &ts (2.37-2.38)
CALL BISECT
- bracket/identify possible
mode solns. for y
CALL SOLVE
- writev
- solve for basis frit
J
C END
CALL GAMMA
- CALL GAULEG
SDM summation procedure
-CALLDARA
SDM dyadic Green's functions o f (2.22-2.24)
- CALL DARC
computes \A] matrix o f (2.33)
- CALL LUDCMP
finds determinant of (2.34)
- update y using (D.24)
all y's found?
Fig. 17. Flowchart for computing the propagation constant of the packaged CBCPW.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
contributions of the matrix equations, calculating the upper summation limit, and
predicting the range for the lower summation following (D.9).
The two-dimensional SDM is verified in Table IV and Table V using the results of [21]
for air suspended coplanar waveguides. Table IV analyzes the dominant CBCPW mode
and Table V presents the first two harmonic modes (waveguides modes).
Agreement
between the two different procedures (moment method and SDM) is demonstrated. The
simulation time for the two-dimensional example of Table III (A=10mm) was 4 seconds.
TABLE IV
Comparison of the effective dielectric constant for packaged CBCPW with
Ref. [21] for erl=l, er4=9.6, er5=l, h,=3mm, h4=lmm, hs=3mm, h2=h3=h6=0,
S=2mm, W=lmm, A=7.5mm. presented work* refers to an infinite-width
structure with the same parameters for comparison purposes.
S eff
f (GHz)
1
10
20
30
40
3 .6
4 .3
5
5 .9
6 .7
presented work
3 .7 1
4 .2 9
5 .0 2
5 .8 2
6 .6 2
presented work*
3 .7 5
4 .3 1
5 .0 3
5 .8 3
6 .6 4
Ref.
[2 1 ]
TABLE V
Comparison of the effective dielectric constant for the example of Table IV
with Ref. [21] at f =30GHz for the various modes.
e eff
modes
fundamental
1st harmonic 2nd harmonic
Ref. [21]
5.82
4.37
3.03
presented work
5.819
4.366
3.025
An example o f the cross-sectional vector magnetic field plots (Hx and Hv) is depicted
in Fig. 18 using (2.44), (2.45), (2.55d), (2.55e), (2.61), (2.62) and Appendix B. The plot
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
0.9
0. 6 ,
0.3
0.0
£ -0 .3
0.6
-
- 0 .9
-
1.2
1-5 - 1 . 2 - 0 . 9 - 0 . 6 - 0 . 3 0.0 0.3 0.6
y
0.9
1.2
1.5
(m m )
Fig. 18. Cross-sectional vector magnetic fields plot for packaged CBCPW.
srl= l, er4=10.2, sr5=2.2, h,=3.73mm, h4=0.635mm, h5=0.635mm,
h2=h3=h6=0, S=2W=0.635mm, A=12.5mm, f =5GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
is the real part of the magnetic field components and verifies the boundary condition of the
magnetic wall at y=0 for the CBCPW dominant mode (tangential components of the
magnetic field along a magnetic wall are zero).
Hx (xj^O) vanishes. The field arrow
lengths are scaled to the largest field component. The convergence criterion for the field
plots is the change in the real and imaginary parts less than 5% for increments in the
spectral variable kyi of 50 with a minimum o f 250 spectral terms. Evaluation o f the field
components localized around the source for the CBCPW (electric fields across the slots)
require more spectral terms for convergence than for removed points.
The characteristic impedance calculations using (2.64)-(2.71) and Appendix C are
presented in Table VI for comparison purposes with [43] which analyzed the CBCPW with
a variational conformal mapping technique.
Again, the power-voltage, voltage-current,
and power-current definitions are included. The worst case difference between the two
techniques is 1.3Q.
TABLE VI
Comparison of the characteristic impedances for packaged CBCPW with
Ref. [43]
fo rsrl= l,
sr4=13,
sr5=l, h[=5mm, h4=lmm, h5=5mm,
h2=h3=h6=0, S=lmm, W=0.4mm, A=7.5mm.
Z (fi)
2
10
20
Zpv Ref. [43]
46.5
45.5
42.9
Zpv presented work
45.5
44.2
42
ZPI Ref. [43]
45.9
45.9
43.7
45
45.2
43
Zv, Ref. [43]
45.2
46.3
44.6
Zy, presented work
44.9
46
43.7
ffeq (GHz)
ZPI presented work
The procedure involved to evaluate the current in the characteristic impedance calculations
is complicated. The singularity in the current at the edge of the center conductor at ,y=S/2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
as in Fig. 15(b) is determined by dividing the space-domain integral into two regions. The
second integral being confined directly at the conductor edge.
The variation in the
cos fit y)
summations for the SDM field plot calculations (H ) ranges as G(x,y,kyi, y )
-= —
yjKy,
where G(x,y,kyj ,y) is the corresponding product including the Green's functions while the
summations for the propagation constant o f (2.33) varies as G(x,y,k ,y) j - .
Kyt
This
translates to a significant number of additional spectral terms especially near the x=0 plane.
A convergence check with respect to the number of spectral terms (/') in (2.45) is
demonstrated in Table VII for the effective dielectric constant and the characteristic
impedance (power-voltage definition). Each o f these parameters converge quickly.
TABLE VII
Convergence results for the effective dielectric constant and impedance
in terms of the number o f spectral terms for packaged CBCPW with
erl=l,
er4=10.2, sr5=2.2, h,=5mm,
h4=0.635mm,
h5=0.635mm,
h2=h3=h6=0, S=0.254mm, W=0.89mm, A=12.5nun, f =20GHz.
# spectral terms
ZpV
S e ff
( ^ )
25
50
75
100
125
150
175
200
296.1
99.8
98.2
97.6
96.7
96.2
96.1
96
5.3
5.306
5.31
5.311
5.31
5.309
5.309
5.31
The expansion coefficients for the basis functions from (2.26) for the example of Table VII
are presented in Table VIII for M=3 and N=2 with Cj=1.0.
Results o f the SDM for the finite ground plane (FGP) structure of section F is
confirmed with [47] in Table IX using a total o f 3 current basis functions on each strip for
Jz and Jy . The coplanar-microstrip mode is designated CPM (from [27] and CBCPW is
the coplanar waveguide mode.
Agreement between these two procedures (SDM and
full-wave space domain) is excellent.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
TABLE VIII
Basis function expansion coefficients for example in Table VII
for M=3 and N=2.
coefficients
C!
(1.0+/0)
C2
(0.414+/1.0D-10)
(-0.165+/7.92D-11)
(9.5D-11+/3.76D-2)
(9.5D-ll-y3.22D-3)
d2
TABLE IX
Comparison o f the effective dielectric constant for FGP packaged CBCPW
with
Ref. [47]
for
sri=l,
er4= 10,
er5= l,
h,=10mm,
h4=lmm,
h2=h3=h5=h6=0, S=lmm, W=lmm, WG=lmm, A= 12.5mm.
£eff
2
4
6
8
10
CPM Ref. [47]
7.75
7.94
8.125
8.31
8.5
CPM presented work
7.74
7.97
8.17
8.33
8.49
CBCPW Ref. [47]
5.83
5.88
5.92
6.02
6.11
CBCPW presented work
5.84
5.87
5.93
6.01
6.11
f(GHz)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
CHAPTER HI
ONE-DIMENSIONAL ANALYSIS AND DESIGN OF
CBCPW INCLUDING LEAKAGE EFFECTS
Despite the many attractive features o f CBCPW, design problems exist in the
transmission line. In this chapter, the one-dimensional infinite-width CBCPW is analyzed
with the ground planes not connected to the conductor-backed plane as in Fig. 11. The
problem with a single-layer CBCPW (h2=h3=h5=h6=0) is the leakage of power from the
dominant mode to a zero-cutoff, TEM parallel plate mode in the homogeneous substrate
region. This phenomenon produces nonconventional losses and can generate undesirable
coupling effects and an ineffective circuit.
This leakage (attenuation)
effect
unconditionally occurs for the dominant mode of CBCPW at all frequencies for
single-layer, infinite-width, and isotropic substrates.
Leakage can occur for microstrip
when the substrates are isotropic at higher frequencies and may exist for anisotropic
structures [48],
Similar leakage loss from the dominant mode can occur at higher
frequencies for conventional CPW to conductor-backed dielectric slab modes [49], The
SDM can be extended to calculate the leakage loss for the CBCPW.
Three configurations exist to reduce the leakage rate or to eliminate it altogether up to
a frequency (critical frequency) in CBCPW. The first method requires reducing the circuit
cross-sectional area (S+2W) relative to the dielectric thickness or increasing the substrate
thickness itself [12], However, in MICs the available line dimensions can be restricted and
increasing the dielectric thickness may excite additional parallel plate modes that can
enhance the leakage problems.
An alternative method is using substrates with a low
relative dielectric constant (sr =2.33) but this will increase the overall size o f the circuit as
the effective wavelength is increased. A third configuration is the use of a multi-layered
Part of the data reported in this chapter is © 1992 EEEE. Reprinted, with permission,
from IEEE Microwave and Guided Wave Letters, vol. 2, no. 6, pp. 257-259, 1992 (see
Reference [13] and Appendix G).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
CBCPW [13] to modify and control the propagation characteristics of the dominant mode
or the parallel plate modes to extend the critical frequency and bandwidth of operation for
the intended application.
Both lower and upper dielectric loaded configurations are
analyzed and presented in this dissertation.
The upper loaded structure includes the
placement of the substrate on top o f the ground plane (h3).
The lower loaded
configuration inserts the dielectric between the existing layer and the conductor-backed
plane (h5) forming a inhomogeneous medium between the parallel plates.
For this
waveguide the first order parallel plate mode is non-TEM and has a cutoff frequency. This
method places little restriction on the circuit cross-sectional area but may on the dielectric
thickness, depending on the frequency o f operation. The SDM is employed to predict the
critical frequency (fcrit).
In this chapter, experimental and theoretical data are presented demonstrating the
leakage effects and indicating that the loss mechanism is not due to dielectric loss.
A
physical description of the leakage behavior within CBCPW and other printed-circuit
transmission lines is presented along with the mathematical extensions to the SDM for the
calculation of the leakage rate. Leakage curves for various waveguides are presented and
verified with other published results.
Design examples of multi-layered CBCPW along
with measurement data to correlate the model are introduced. The shortcomings o f the
one-dimensional model for the CBCPW are discussed with regard to the accuracy o f fcrit
and explanation and prediction of the resonance effects in the experimental data.
A. Experimental Demonstration of the Leakage Effects in CBCPW
As a precursor to the development o f CBCPW for the proposed active antenna array
system operating at 3 5GHz, an in-house test fixture was developed to characterize the
transmission line and circuit elements.
The infinite-width CBCPW was approximately
modeled with the physical structure depicted in Fig. 8(a) without the wrap-around
conductor connection (shorting sidewalls) along the length at _y=±A. The perpendicular
interface to the substrates at _y=±A is air.
The ground planes are connected to the
conductor-backed plane only at the input and output ports and the open lateral sidewall
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
dimension is extended sufficiently away from the coplanar slots (A=19mm which is over
five wavelengths at 20GHz for a dielectric susbstrate with sr=10.8) as not to interfere with
the dominant propagating mode which should be confined to the slots area.
Figure 19 presents the experimental results for single-layer CBCPW (refer to Fig. 11)
for both high (CASE G) and low (CASE H) dielectric constant substrates.
CASE G
corresponds to a 50Q through line and CASE H represents a 95Q through line (notice the
standing wave transmission response). The high impedance line of CASE H was necessary
so that the cross-sectional circuit area matched with the coaxial connector dimensions. In
this dissertation alphabetically marked cases refer to experimental data and numerically
marked cases depict modeled simulation results. The low frequency, single-layer CBCPW
impedances were obtained using a quasi-TEM, closed-form relationship from [50].
Problems for these transmission lines exist above 3 and 10GHz for CASE G and CASE H,
respectively.
These results were totally unexpected and were initially assumed to be
connector or grounding problems. However, additional data confirmed this phenomenon
as repeatable and had not been previously reported. All o f the line parameters were varied
to minimize these experimental effects but these changes still did not produce a waveguide
with any appreciable bandwidth. Such attempts included a structure with a small circuit
cross-sectional area (S+2W=0.75mm) compared to the dielectric thickness for the 6010
Duroid™ (h4=2.54mm) as proposed in [12]. The photolithography limitation in the circuit
etching equipment in our lab was 0.25mm for a slot or line width. The transmission line
(CASE G) exhibited high loss and included sharp resonance effects. The resonance effects
were not as pronounced for the low dielectric case. The data was affected by changes
in the transmission line length (z=2B) and width (y=2A) as well as the placement of
absorber material at the substrate sides at ,y=±A. By reducing the length or width, the
basic CBCPW response was similar for CASE G but the resonance effects were spaced
further apart. The effects of the absorber material would smooth out the loss response and
reduce the sharpness of the resonances. This was interesting as the CBCPW response was
a function of the boundary conditions enclosing the structure. CASE 1 in Fig. 19 is the
SDM dielectric loss calculation o f CASE G indicating that some other loss mechanism was
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
C AS E
CD
"O
C A S E
1
G
00
CASE H
o
<
0
4
8
12
16
20
24
28
32
36
FR E Q ( G H z )
Fig. 19. Experimental data of the leakage effects in CBCPW MIC with open sidewalls
and
h,=x
2B=38mm, S=2W=0.508mm,
W0 = 18.492mm, e.|=s.,=sr3=ef5=ero.:=l,
(open structure), h,=h3=h5=h5=0. Refer to Fig. 11 for the dimension
parameters. CASE G 500. line sr4=ri0.8,
h4=0.635mm, CASE H same as
CASE G except h=0.71mm and sr4=2.33 for 95f2 line, and CASE 1
is the numerical representation for the dielectric loss o f CASE G.
All
cases are referenced to 0 dB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
contributing to these measurement effects. The loss tangent (tan 54) for the 6010 substrate
at 1, 5, 10, 15, 20, 25, 30, 35, and 40GHz is 1.4, 1.7, 2.1, 2.6, 3.2, 3.8, 4.5, 5.2, and 5.9
x 10‘3, respectively. The dielectric loss calculation for CBCPW will be verified with an
example later in Chapter VI. The phenomenon existed within the substrate region and
could be initially explained in the one-dimensional case as leakage from the dominant
CBCPW to parallel plate modes [26],
B. Explanation of Leakage Effects in Printed-Circuit Transmission Lines
On a uniform length printed-circuit transmission line, it is assumed that the dominant
mode (source excited mode) is purely bound to the strip area for microstrip and to the
slots for coplanar waveguides. For lossless structures, the propagation constant o f the
dominant mode should be real. Under certain conditions, microstrip, slotlines, coplanar
stripline, CPW, and CBCPW will all leak power above some critical frequency. These
leakage effects typically occur at higher frequencies and are usually assumed important
only for millimeter-wave integrated circuits. However, this is not a proper assumption for
CBCPW MICs. A leaky dominant mode associated with a printed-circuit transmission line
has electromagnetic energy which is not entirely confined to the strip or slot regions and
this mode is no longer bound in this region (flow of power in the substrate away from the
strip/slot region). The performance of a leaky transmission line is significantly different
than that described by ideal analysis as the line would be strongly dependent on the
surroundings even far from the guiding slots (in a two-dimensional configuration). Leaked
energy propagating in an integrated circuit results in undesirable and possibly catastrophic
cross-talk (unintentional coupling between transmission lines within an integrated circuit).
The leakage loss or rate of the dominant mode (power leaked per unit length along the
guiding structure) is usually far greater than the conductor and dielectric loss. Depending
on the particular geometry, power leakage can occur in the form o f a surface wave
(microstrip, coplanar stripline, CPW) on conductor or nonconductor-backed dielectric
slabs with one or two open boundaries, a parallel plate wave (slotline and CBCPW) on a
structure enclosed by PECs, or a space wave (for configurations without a top cover
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
plate). These waves are also referred to as characteristic source free modes which are
supported by the waveguide in the absence o f an excitation source and appear as poles in
the Green's functions. A surface wave is a wave bound or trapped to the surface of the
waveguide and decays exponentially in the direction normal to and away from the guiding
structure. The leakage of power from the strip/slot region occurs in the form of the
lowest surface wave or parallel plate wave from the power of the dominant mode for the
waveguide. At higher frequencies, leakage to higher order surface or parallel plate waves
occurs which can enhance the leakage rate.
The power will radiate away from the
strip/slot region in the lateral direction at an angle to the longitudinal axis and the leakage
rate is dependent on the frequency and dimensional parameters [51],
The propagation
constant becomes complex (non-spectral) for lossless waveguides if the structure is
unbounded laterally or open (infinite extent in the ^-direction). The fields of an infinite
transmission line with leakage propagate and exponentially grow in the transverse (lateral)
direction [31], This exponential field growth phenomenon does not occur for physically
real transmission lines which are finite-length. However, once the leakage mechanism is
properly understood, the undesirable effects may be eliminated or minimized and for
certain applications the leakage can be used to advantage in leaky wave antennas.
For the various printed-circuit transmission lines mentioned, the leakage behavior and
the critical frequency are different but the fundamental physical theory is common to all the
cases. For the waveguides considered in this section, the top cover plate is not present and
does not affect the analysis in a qualitative sense. As an example, from Fig. 20 the top
view of a slot on an air-dielectric configuration with conductor-backing is shown where
this slot can represent the slot of a slotline or one of the pair o f coupled slots for a
CBCPW. Assume the structure is lossless (no material or conductor losss). From the
figure the transverse wave number o f the excited characteristic source-free parallel plate
mode (PPM) in the ^-direction is related to the other wave numbers by
2
2
2
kyN = k CJ - k z
(3.1)
where kc t is the propagation wave number (real quantity) of the relevant PPMs (TEM,
TM, or TE and f = l , 2 , ...,a> depending on the operating frequency) that can be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
y p
(a)
y
air
(b)
Fig. 20.
Conductor-backed slotline leaking power to a parallel plate mode for
(a) the top view and (b) the side view. The angle of leakage 0 into
the parallel plate mode of wave number kc is also shown. kyp is the
transverse wave number in the ^-direction o f the excited parallel
plate mode and k, is the wave number of the dominant mode guided
along the z-direction.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
supported by the structure and k, =y is the propagation constant of the dominant mode in
the z-direction.
The poles of the admittance Green's functions for CBCPW from
(2.22-2.25) occur at kypt . A pole is a singular point o f a function fiz ) at a value of z
where f[z) fails to be analytic. The propagation of the PPM in the y-direction is
.
For lossless structures, k=$ (the phase constant). If p >k c t , kypt is imaginary from (3.1)
and the dominant mode guided by the transmission line in the z-direction is purely bound
and the field decays transversely in the j-direction away from the slot and k_ is real.
However, if P < kct then kypt is real from (3.1) and as Fig. 20 indicates, power leaks from
the dominant mode at an angle 0 in the form of a PPM within the dielectric region bounded
by the conductors. Under leakage conditions, the leaky mode propagates slower than the
dominant mode with respect to the phase velocity..
The leakage occurs within the
dielectric and exists as a continuous function of frequency above fcrit (critical frequency).
The leakage (radiation) peak is at an angle 0 given by
where this approximation is not strictly true for complex k_ but provides a good qualitative
description for small loss [52], The three-dimensional leakage phenomenon is illustrated in
Fig. 21 for a leaky coplanar stripline with a pulse propagating along the z-direction and
producing a semi-cone of leakage within a thick substrate. The power associated with the
pulse is no longer confined to the strip area but spread out with the peak leakage angle
given by (3.2).
Again, the consequence of the leakage mechanism is crosstalk and
interference within the microwave circuit from this propagating energy and the possible
significant loss o f power from the dominant mode pulse. The condition for leakage of
power to occur in transmission lines can then be written as
P < k eJ.
(3.3)
Each mode of a waveguide has a frequency in which the leakage criterion o f (3.3) can
occur and this point is referred to as the critical frequency. At higher frequencies, leakage
to higher order PPMs can also be present. The attenuation constant o f the dominant mode
due to leakage is cll and results in a complex propagation constant o f k=$-j<xL . Under
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
propagation
semi—cone
of leakage
Fig. 21. Three-dimensional illustration of leakage in a semi-cone region
from a pulse propagating on a coplanar stripline.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
this circumstance, (3.1) is modified as
k ypj= J K j - K
= JKj -
p 2 + a l + 2J a L P = ± C ± j D
(3.4)
where C and D are constants and recall the double value o f the square root and kyp t may
lie in the first or third quadrant of the complex ky spectral plane. Under these conditions,
the PPM propagates and exponentially grows in the outward direction and this can be
explained from Fig. 20 for a lossless structure. Points a and b receive the characteristic
parallel plate wave originating from the points a' and b' along the slot following (3.2).
When leakage is present, the electric field of the dominant mode on the slotline has an
exponential decay along the z-direction which results in a larger field value at point b' than
at a'. Hence, the field magnitude at point b tends to be larger than at point a [31], For
such an exponentially growing field component, the Fourier transform in the ^-direction as
in (2.3) does not exist for real values of the spectral argument k (nonspectral solution) and
the implementation within the SDM will be discussed in the next section. With regard to
the definition of impedance on a leaky transmission line, for a finite center strip current for
the CBCPW, the total power flow across the transverse cross section is infinite. This is
due to the growing field in the transverse direction. Using the power-current definition of
(2.64b), the impedance is infinite. It should be noted that a leaky infinite width and length
transmission line does not fundamentally violate the radiation condition which assumes a
finite radiated power in the far field due to a finite input source o f power. This is not the
case for an ideal infinite-length leaky line since the source power itself with an variation
e ~az along the line is infinite at z
= -o o .
The leakage criterion for a general printed-circuit transmission line with loss is
presented here. The dominant mode propagation constant is k_ = (3-y'a and that for the
PPM (or substrate mode) becomes kc t = u -/t and (3.4) is rewritten as
kypj=
J v 2 - X2 -
p 2 + a.2 + 2 j(a |3 - ux) = ± E ± j F .
(3.5)
Again, for leakage to exist and for the PPM to propagate and exponentially grow in the
outward direction, the following conditions must all be met: (1) the pole corresponds to a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
guided PPM with propagation constant kc ( , (2)
Re (K.i) > Re (k. )
(3.6a)
and (3) kypt pole may only lie in the first or third quadrant of the complex ky plane
R e(A :^*Im (*p , ) > 0 .
(3.6b)
The leakage effects will now be explained for various transmission lines including
coplanar stripline, microstrip, CPW, conductor-backed slotline, and CBCPW for
single-layer waveguides [12] and [51],
For the coplanar stripline (CPS) structure of Fig. 21, the region under the strips fills up
with the electromagnetic energy of the dominant mode and under leakage conditions, the
power is contained within this region instead of being leaked away.
Outside the strip
region, the structure consists o f a dielectric slab which supports surface waves. The lowest
order surface wave is the TE0 and has a zero-cutoff frequency. Leakage will occur at fcrit
and this frequency can be extended higher by reducing the dielectric thickness.
The
dominant CPS mode has relatively little dispersion but the surface waves are strongly
dispersive.
At the critical frequency the dispersion curves for the CPS mode and the
relevant surface wave mode cross each other as the phase velocities along the r-direction
between these modes match (phase match). For the conductor-backed CPS, the leakage
mechanism is the same as above except the lowest order zero-cutoff frequency surface
wave is the TMqon the conductor-backed dielectric slab surrounding structure.
When the substrate is isotropic, the dominant mode on a microstrip line may or may not
leak depending on the waveguide parameters and the frequency.
The leakage to the
dominant TMq grounded surface wave usually occurs at higher frequencies and can be
extended by reducing the dielectric thickness. Again, examination of the dispersion curves
for the structure will aid in the determination. If the substrate is anisotropic, leakage to
TE0 surface wave can occur and this is due to the fact that this mode is affected primarily
by e x (the principal crystal axis coincides with the x-axis). For substrates with e j_> e ||,
leakage into the dielectric layer outside the strip region exits since the propagation curve kc
of the TE0 mode will exceed p of the dominant mode on the microstrip line at a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
sufficiently high frequency.
CPW can be constructed from Fig. 3(a) without the conductor-backing at x=-h and
suspended in air. The supporting structure for this waveguide is essentially the same as the
conductor-backed CPS with the conductor-backing reversed and the guiding strips of the
CPS removed. Again, leakage from the dominant CPW mode to the TMq surface wave
mode will occur at and above fcrit.
For the conductor-backed slotline with isotropic substrates (pictured in Fig. 20), the
region away from the slot on both sides (y-direction) resembles dielectric-filled parallel
plate waveguides.
The lowest order mode is a zero-cutoff TEM (E:=H,=0).
A
homogeneously filled transmission line with two conductors at different potentials will
support a TEM mode) which has a normalized phase constant (kc , /k0 ) equal to J e J . The
normalized phase constant of the dominant slotline mode is between 1 < PlkQ < ^ s 7 since
the fields are partly in air and partly in the dielectric medium.
Consequently,
conductor-backed slotline possesses an unconditional leakage effect independent of
frequency, dielectric thickness, dielectric constant, and slot width. Fig. 22 depicts a set o f
measurements published in [26] on a conductor-backed slotline. The slot was excited by a
metal loop that curved over the slot and was fed by a coax line at one end and short-circuit
at the other. Absorber material was placed around the edges to approximately simulate an
infinite-width structure. The field was probed across the far end parallel to the >>-axis to
determine the leaky wave field distribution. The leakage peak angle was calculated to be
31° from (3.2) and the experimental data approximately confirms the relationship.
If the
waveguide was not leaky, then the magnitude of the probed transverse field distribution
should decrease away from the slot.
However, the leakage loss can be reduced by
increasing the slot width or increasing the dielectric thickness [31].
The CBCPW leakage mechanism is analogous to that of the conductor backed slotline.
CBCPW possesses an unconditional leakage effect independent o f frequency, dielectric
thickness, dielectric constant, and strip/slot width. The leakage loss can be reduced by
increasing the dielectric thickness or reducing the cross-sectional circuit area relative to the
dielectric thickness [12],
Leakage from the CBCPW odd mode is much less
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
x
(d»5mm
q(
IQGH z )
conductor
short
xixial cable
9' = 31. 0'
proOe
y
Fig. 22. Leakage effects in conductor-backed slotline demonstrated by probing the Ex
field distribution transversely cross the .y-axis. The leakage peak angle is
31.8° which
translates to a maximum field component at y = ±12.6cm.
Experimental data approximately confirms the predicted leakage angle from
Ref. [26],
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
than from the CBCPW even mode (if excited due to nonsymmetry within the circuit)
because the odd mode electric fields are oriented in opposite directions and produce a
partial cancellation (see Fig. 4).
The propagation characteristics o f the PPMs for
conductor-backed slotline or CBCPW can be accurately approximated from waveguide
theory and are presented in Appendix E.
Unless the CBCPW substrate thickness and
circuit cross-sectional area are much smaller than the wavelength, leakage and the
corresponding cross-talk cannot be analyzed using conventional quasi-TEM analysis but
rather a full-wave analysis such as the Spectral Domain Method
with special
considerations for the non-spectral nature o f leaky modes is required.
C. Spectral Domain Method Including Leakage Analysis
The derivation of the SDM was detailed in Chapter II and the homogeneous set o f
equations to solve for the CBCPW propagation constant were presented in (2.31) and
integrated in the spectral domain as
± f j ( k y) E * ( k y)dky = 0
(3.7)
J ( k y) = E(ky) Y ( k y, k :)
(3.8)
where
is from (2.28) and E (ky) is the Fourier transform of the electric field expansion functions
across the slots, J (ky) is the Fourier transform o f the currents on the strips, Y (ky , k. ) is
the spectral Green's functions and the Fourier transform relating the currents at (x=0, z) to
the source slot electric fields at (x=0, z), ky is the continuous spectral variable, and C is
the contour o f integration in the complex spectral plane of k . For nonleaky transmission
lines (fields bounded in the transverse directions), the contour of integration in (3.7) can be
simply along the real k axis from
-
qo
+jO to
qo
+jO which is the normal procedure.
However, for leaky wave solutions the fields grow exponentially in the transverse direction
from the slot area and the integration contour must be appropriately deformed in the
complex k
plane to account for such nonspectral modes since the Fourier transform o f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
77
(2.3) is not defined [31], The deformation o f the integration path was not recognized in
the first SDM analysis of CBCPW in [53] and no experimental measurement data was
performed.
By not selecting the proper integration path, the phase constant can be
determined somewhat accurately but the leakage loss (coupling effects) is totally
neglected. The deformation of the integration path can be identified by considering the
poles associated with the Green's functions of (3.8) and (2.22)-(2.25) and can be stated in
general terms as
(3.9)
The poles of the Green's function are related to the transcendental equations for the TEr
and TMx modes supported by the source-free layered dielectric PPM structure o f Fig. 11
for CBCPW and can be written respectively as
(3.10)
(3.11)
with m = 0,1,2,... , oo and can be solved from (A.5), (A.7), and (A. 14-A. 19) in Appendix
A. For structures not bounded on the top or bottom by conducting plates, the above
transcendental equations would represent surface waves.
The solutions o f (3.10) and
(3.11) can be verified using (E.6) and (E.8) in Appendix E. Equations (3.10) and (3.11)
will have an infinite number o f roots at kc t corresponding to an infinite number o f poles in
(3.9) at kypt where / is the index for the TM and TE modes. For propagating PPMs in
the z-direction, Ke(kc.i)> 0 and for nonpropagating modes Im(kc l) < 0. For a single layer
CBCPW, (3.10) with rh=0 solves for the TEM and the next order modes are the TM, and
TE, which are degenerate modes (modes with the same cutoff frequency).
For
multi-layered waveguides, the first TM mode is the TMq and the first TE mode is the TE0.
Table X includes an example of the propagating PPMs with effective dielectric constants
greater than 1.0 for a multi-layered structure at 20GHz. The spectral integrand in (3.8) is
the product of the Green's function and the Fourier transform of the electric field
expansion functions. The transforms of the electric field basis functions are analytic over
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
TABLE X
Identification o f PPMs for multi-layered structure of Fig. E. 1 with
sr4 =10.8, sr5 =2.33 and h4=2.54mm, h5=0.71mm at f=20GHz.
e eff
q
9.061
TE0
5.027
TM,
2.261
TM
the entire complex spectral plane as indicated from (2.37) and (2.38) and the integration
contours are determined only by the pole locations of the Green's functions. The poles in
the Green's functions are simple (first order) poles [60], For nonleaky transmission lines
(assume lossless waveguide), P > Re(£c, ) for the PPMs for all t and from (3.4) all o f the
poles {k ,) exist exclusively on the imaginary axis in the complex ky plane. Again, the
Fourier transform of (2.3) exists and the contour of integration can be along the real k
axis. This situation is pictured in Fig. 23 for symmetric poles P0 and P0' (recall the double
value o f the square root for kyp ) with C, as the path for the spectral domain integrations
for various pole cases. Note in Fig. 23 that no branch cuts exist in the complex ky plane
which is always true for structures bounded on the top and bottom by conducting plates
[54] as for the CBCPW o f Fig. 11. Branch cuts are due to multi-valued square root
expressions for outward propagation for infinite mediums and satisfying the radiation
condition. Poles P1 and P,' in Fig. 23 correspond to a lossy transmission line with Re(Arr) >
Re(Arc () and hence a nonleaky condition again exists and the integration contour C, is again
chosen.
These poles are displaced off the imaginary axis to the second and fourth
quadrants proportionally to the attenuation rate. Poles P2 and P2' were originally in the
first and third quadrants o f the complex ky plane but for a sufficiently lossy waveguide
(conductor or material losses), these poles have moved to the second and fourth quadrants
according to (3.5) and do not contribute to leakage. Now assume leakage occurs with the
PPMs corresponding to poles P3 and P3' and P4 and P4' in Fig. 23 and satisfies the leakage
criteria for a lossless structure.
Any integration contour can be chosen containing the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
+ lm
• p;
Fig. 23. Contours of spectral domain integrations C s in complex kv plane. The respective
poles positions P's are indicated. P0 and P'0 are for no loss case while P, and P',
correspond to P0 and P'0 for a lossy waveguide. P2 and P'; correspond to P3
and P'3 for a highly lossy structure. All of these poles refer to a nonleaky
waveguide and utilize the contour C 1along the real axis. P3 and P'3 and P4 and
P'4 refer to leaky waveguide case and follow the integration path C3 coinciding
with the real axis and deformed around the leaky poles in a residue calculus
sense.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
poles at Pj and P3' and P4 and P4' above and below these poles as in C , . By further
deforming C2 , a simple contour C3 that covers the entire real axis and encloses poles at
P3 and P3' and P4 and P4' in a residue calculus sense is selected.
A mathematical
description of the residue calculus theory is presented in Appendix F. By using residue
calculus, (3.7) is expressed in terms of a sum of residues evaluated along the C3 integration
contour as
^
{ J ( k y ) £ * (ky ) dky + j t
C3
E(kypq) Y( k ypqt k:) EM(kypq)
<7=1
0
~j
X
E{ - kypq) Y { - k ypq, k: ) E \ - k ypq) = 0
(3.12)
q= 1
where O is the total number of leaky PPMs, and in the evaluation of the integrals
k v € iR [54], The contour of integration satisfies the existence and validity o f the
transforms of the three components of (3.12). The second and third terms of (3.12) denote
the residue contribution at ky- k ypq for the poles located in the first and third quadrants in
the complex plane, respectively.
Equation (3.12) is represented as an infinite sum of
decaying fields (transversely) for the PPMs plus the exponentially increasing fields for the
PPMs. The PPMs corresponding to exponential growth are the modes to which leakage
occurs.
The SDM algorithm presented in Chapter II and Fig. 16 needs to be amended to
include the calculations of the leakage effects in CBCPW. The continuation mark ® in
this figure corresponds to the determination of the propagation constants for the PPMs for
the structure solved from (3.10) and (3.11). These solutions are identified for the TM and
TE modes as zero crossings for the real part of the functions for effective dielectric
constant values from 0.1 to er max and then the program utilizes Muller's method to find the
complex propagation constants kc t . A check is also made that the identified pole is not a
removable singularity by determining whether a zero in the function occurs at kypl . An
additional verification is included such that procedure does not converge to a previous
solution. The above computation is included in the routine PPMID. The conditions from
(3.4)-(3.6) are implemented to determine whether the PPMs are leaky.
If the poles are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
leaky modes, then the routine GAULEG is modified to include the residue calculus terms
from (3.12). For an assumed lossless structure, the presence o f the leaky modes and (3.12)
cause the spectral domain matrix elements o f (2.32) to become complex which translates
to a complex propagation constant y. As described in Chapter II, the spectral domain
integrand functions in (2.32) and (3.7) are even functions o f ky and the integration interval
exists from 0 < k y <ao.
ensure convergence.
Near the poles, the integration subintervals are increased to
The above enhancement of the SDM integration algorithm is
included in routine LEAKYM. The loss tangent for one o f the dielectrics layers should be
input with a nominal value for the leaky mode analysis to perturb the solution in the
Muller's method to converge to a complex propagation constant.
An example o f the
integration convergence for the complex propagation constant y for the leaky waveguide
of Fig. 19 CASE G is listed in Table XI and required 10 seconds to simulate.
TABLE XI
SDM integration convergence o f the leaky propagation constant
for CASE G in Fig. 19 for single-layer CBCPW at 20GHz.
integration
points
iteration #
Y
5
4
(1.0552, -2.6628D-8)
5
7
(1.0487,-7.6852D-2)
5
10
(1.0459,-1.081 ID-2)
5
13
(1.0951, -4.9788D-2)
5
16
(1.0910, -6.4428D-2)
5
19
(1.0907,-6.4317D-2)
15
4
(1.0829, -6.0784D-2)
15
7
(1.0838, -5.9980D-2)
25
4
(1.0840, -5.9962D-2)
25
5
(1.0840, -5.9955D-2)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
D. Numerical Results of Leakage Effects for CBCPW
The SDM incorporating the leakage effects from the previous section is utilized to
analyze various CBCPWs, present leakage curves for the structures, verify the
mathematical algorithm, and predict some o f the experimental data results. Unless noted,
all o f the numerical examples assume no conductor or dielectric loss and the waveguide
parameters chosen are typical for a CBCPW MIC. The first analysis is the dispersion
curves for the normalized propagation constants o f the CBCPW dominant mode and the
PPMs. This will determine the frequency range o f the leakage effects or fcrjt. Again, recall
that for single-layer CBCPW, leakage occurs unconditionally regardless of frequency. This
is depicted in Fig. 24 for two different dielectric substrates. This figure demonstrates that
k jk 0 > |3/&0 across the entire frequency-band and the leakage from the dominant CBCPW
mode to the zero-cutoff TEM PPM (lowest order mode) is a continuous function o f
frequency. Other PPMs are propagating within the waveguide for this frequency range but
do not contribute to the leakage effects and are not shown. The unconditional leakage
effects partially explain the experimental results o f Fig. 19 especially for CASE G in which
this transmission line is unusable across the entire frequency-band.
A multi-layered CBCPW is proposed here to extend the critical frequency and produce
a waveguide with an appreciable bandwidth. Examples of multi-layered (lower dielectric
loading) CBCPW are presented in Fig. 25. The placement o f a lower dielectric constant
substrate (sr5) between the high dielectric constant substrate (sr4=10.8) and the
conductor-backing effectively modifies the propagation characteristics of the waveguide.
The lowest order PPM is now the TM,, PPM which has a cutoff frequency.
The
propagation constant of the PPMs is reduced below that of the CBCPW dominant mode
up to some frequency. This occurs since the field pattern for these modes is concentrated
between the circuit conductors and the conductor-backing as in Fig. 4(c) and the field
pattern for the bound CBCPW mode exists primarily within the slot region and is not
significantly affected with this additional dielectric layer as depicted in Fig. 4(a).
The
multi-layered CBCPW does not place any restriction on the cross-sectional circuit area but
is a slightly more complex structure to manufacture. CASE 4 o f Fig. 25 demonstrates
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
3.50r
—T—
.Q3.25 ■
3.00 ■
o
<
•4>
2.75
2.50
2 .2 5 2.00 ■
-4?
\
CO.
1.75-
O
V
-9 -9 -
CASE 2 CBCPW
CASE 3 CBCPW
CASE 2 TEM
CASE 3 TEM
20
24
1.50-„V1.251.00-
8
12
16
28
32
36
40
FREQ (G H z )
Fig. 24. Normalized propagation constants for the CBCPW dominant mode (P) and the
TEM PPM (kc ). One-dimensional SDM analysis for experimental data of
Fig. 19 with CASE 2 and CASE 3 corresponding to CASE G and CASE H,
respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
o
CASE 4
CASE 5
CASE 6
~o— CASE 4
CASE 5
-X CASE 6
V
X
CBCPW
CBCPW
CBCPW
TM0
TM0
TM0
o
o
o
■se
0
4
8
12
16
20
24
28
32
36
FREQ (G H z )
Fig. 25. Normalized propagation constants for multi-layered CBCPW dominant mode (0)
and the TMq PPM (kc ) with srl=l, er4=10.8 and h ^ m m , h2=h3=h6=0 and
S=2W=0.635mm. CASE 4 with er5=6 arid h4=0.635mm, h5=5mm. CASE 5 with
sr5=2.33 and h4=0.635mm, h5=0.71mm. CASE 6 same as CASE 5 except
h4=l.27mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
that a lower dielectric constant of sr5=6 is too high even with a thick substrate (h5=5mm)
and leakage occurs throughout the frequency-band.
A nonleaky CBCPW (50Q
transmission line) occurs for CASE 5 for the entire band of interest. A critical frequency
of 21 GHz exist for CASE 6 which is the same as for CASE 5 but with a larger substrate
thickness h4. For the frequency region above fcrit, the leakage o f power continues from
CBCPW dominant mode into TM,, PPM. These numerical results will be compared with
experimental data in section E.
The leakage analysis is verified in Table XII and Fig. 26. In this table for the slotline
example, the SDM analysis from [31] is compared and the results are acceptable (see Fig.
22) for this extreme case.
TABLE XII
Comparison o f the normalized leakage rate for conductor-backed slotline
with Ref. [31] for sr=2.25, h=8mm (dielectric thickness), f=10GHz as a
function of the slot width d.
a/kn
d (mm)
3.75
7.5
15
22.5
Ref. [31]
0.11
0.123
0.1
0.0114
presented work
0.108
0.155
0.074
0.0112
Fig. 26 compares the calculated leakage rates with references [32] (GaAs MMIC example)
and [33]. Good agreement is demonstrated for both cases (except for the peak leakage
point for CASE 8) and validates the leakage results presented here. CASE 8 is an extreme
example and exhibits an interesting behavior as the large cross-sectional circuit area
(S+2W) compared to the dielectric thickness, behaves more like microstrip than CPW.
The fields are concentrated primarily below the center conductor and do not strongly
couple with the PPMs.
The leakage analysis for CBCPW for this dissertation was
developed prior to [32] but not published as it did not fully explain the experimental
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
0.090
0 .0 8 0
0 .0 7 0
• - - CASE
v
CASE
CASE
o
CASE
7 Ref. [323
7 p re se n te d work
8 Ref. 1333
8 p re se n te d work
0 .0 6 0
o 0 .0 5 0
5
0 .0 4 0
0 .0 3 0
0.020
0.010
0.000
20
30
40
FREQ (G H z)
50
60
Fig. 26. Normalized leakage rate comparisons with Ref. [32] and [33] with srl =1 and
h,=5mm, h2=h3=h5=h6=0. CASE 7 with er4 = 13 and h4=0.2mm and
S=W=0.1mm. CASE 8 with er4 = 10 and h4=0.4mm and S=2W=2.0mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
results of Fig. 7.
Leakage curves for CBCPW MIC examples are shown in Fig. 27.
CASE 2
corresponds to CASE G of Fig. 19 and the leakage has a second-order frequency
dependency. CASE G from the experimental data depicts an increase in loss and in the
resonance peaks with frequency but the calculated leakage significantly overestimates this
loss. CASE 6 corresponds to the results in Fig. 25 with a fcrit of 21 GHz and the leakage
increases quickly beyond that point. CASE 3 corresponds to CASE H of Fig. 19 and the
leakage increases linearly with frequency and is much lower than for the higher dielectric
constant of CASE 2. CASE 9 represents a thick dielectric example of CASE 2 and the
leakage is reduced and reaches an inflection point at 30GHz, then increases substantially.
The reason for this phenomenon is the leakage to additional PPMs (TM, and TE,) above
30GHz.
The leakage rate can be directly related to the field coupling or overlap of the CBCPW
mode and the PPMs. Consider only single-layer structures with an air dielectric above the
circuit conductors. The leakage rate increases with frequency because the CBCPW mode
field pattern is concentrated more and more into the higher dielectric substrate. This is
evidenced by the increasing propagation phase constant on a dispersion curve and the
mode interacts strongly with the PPMs.
For a lower dielectric constant substrate, the
leakage rate is smaller than for the equivalent higher dielectric since a greater portion o f
the CBCPW mode field pattern exists in the air above the circuit conductors and does not
couple with the PPMs. For CBCPW with a large cross-sectional circuit area, the leakage
rate increases as the field pattern is spread out over a larger area and more efficiently
couples to the PPMs. Another analogy for the leakage rate for this example is that of an
aperture antenna which for a smaller aperture (S+2W), reduces the radiated energy
available to couple to the PPMs.
Design examples for multi-layered CBCPW using the leakage analysis are demonstrated
in Fig. 28 for various leakage turn-on parameters. Fig. 28(a) is the design of the proper
dielectric thickness to prevent leakage at 20GHz. CASE 10 is for an upper dielectric
loading structure with a dielectric constant o f sr3=10.8 and requires a thickness
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
0 .1 8
0.16
0.14
CASE
CASE
CASE
CASE
2
6
3
9
0.33*ct/k,
0.12
0.10
's* 0.08
0.06
0.04
0.02
0.00
FREQ (G H z )
Fig. 27. Normalized leakage rates (o<Jk0) for CBCPWs. CASE 2 and CASE 3 are
referenced from Fig. 24. CASE 6 is from Fig. 25. CASE 9 same as
CASE 2 except h4=2.54mm. CASE 2 is scaled by a factor o f three.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
l
O
o
•a*
CASE 10 h j
CASE 11 h5
*
I
0Y—
<a
1
O
a OQO OJOOS 0.010 0.015 0.020 0.025 0.030 0.0 3 5
Eo
0.1
0.2
0 .3
0.4
0.5
0.6
0.7
DIELECTRIC THICKNESS (m m )
(a)
CASE 12
CASE 13
(b)
Fig. 28. Leakage design curves for various CBCPW examples with srl=l and h ^ m m .
(a) Proper dielectric thickness at 20GHz for upper and lower loading
structures with S=2W=0.635mm. CASE 10 with er3=10.8, er4=2.33 and
h4=0.71mm, h2=h5=h6=0 and references the upper dielectric thickness scale.
CASE 11 is the same as CASE 6 of Fig. 25 with h5 as the variable and
references the lower scale, (b) Proper relative dielectric constant at 30GHz
for lower loading waveguide. CASE 12 is the same as CASE 6 except with
er5 as the variable. CASE 13 same as CASE 7 except h5=0.635mm. Log
scales are used for the leakage rate.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
of h3 >0.03175mm which is a very thin substrate.
CASE 11 is for a lower loaded
waveguide with a dielectric constant of er5=2.33 and stipulates a minimum thickness o f h5
>0.63mm.
Recall from Fig. 25 that fcrjt was 21GHz with h5=0.71mm.
Fig. 28(b)
demonstrates the design o f the proper relative dielectric constant value to prevent leakage
at 30GHz for a lower loaded CBCPW.
CASE 12 presents er5 <3.6 as the necessary
condition to prevent leakage and CASE 13, for a carrier substrate o f a GaAs dielectric
from CASE 7, requires er5 <6.3. Note for all o f the examples o f Fig. 28, the leakage
response has a sharp rise at the leakage turn-on parameter.
E. Limitations of the One-Dimensional Leakage Analysis for CBCPW
Fig. 29 lists experimental data curves with open sidewalls and A=19mm to simulate an
infinite-width waveguide for multi-layered lower dielectric loaded CBCPW to compare
with the leakage analysis for sr4=10.8, erS=2.33 and S=2W=0.635mm (approximate 50Q
through transmission lines). CASE I corresponds to CASE 5 in Fig. 25 with h4=0.635mm,
h5=0.71mm and no leakage is present (bounded dominant mode response) and is confirmed
from the analysis. CASE J is CASE 6 with h4= 1.27mm, h5=0.71mm and demonstrates a
leakage or unbounded dominant mode characteristic around 25GHz. Note the reflection
coefficient (Sn ) for this example behaves unusually at this corresponding frequency point
of the S21 response.
The data includes the effects o f the connectors as the network
analyzer was calibrated at the cable ends.
The numerical analysis for this structure
predicted fcrit of 21 GHz with a sharp increase in the leakage (attenuation) rate beyond that
point. CASE K is the same as CASE J except with h5=0.381mm and shows a leakage
frequency around 20GHz and the SDM calculates an unconditional leakage effect with
frequency.
These experimental data results and numerical confirmations were initially
published as part of this dissertation from [13],
As shown above, the one-dimensional SDM leakage analysis could only partially
explain the measurement results. The fcrit could be predicted fairly accurately in some
cases and not in others. A qualitative physical description above fcrit was represented only
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
91
CASE I S2i
MAGNITUDE
(d B )
■ CASE J Sn
CASE J S 21
10 . CASE K S 21
4
8
12
16
20
24
28
32
36
FREQ (G H z )
Fig. 29.
Experimental data of the leakage effects in multi-layered CBCPW MICs with
open
sidewalls
and
2B=38mm,
srl=er2=er3=ert=l, sr4=10.8,
h2=h3=h6=0.
sr5 =2.33
S=2W=0.635mm,
and
h,=oo
WG= 18.49 mm,
(open
CASE I is measured data for CASE 5
structure),
h,=0.635mm,
h5=0.71mm. CASE J is the experimental results for CASE 6 h4= 1.27mm,
h,=0.71mm. CASE K same as CASE J except h5=0.381mm.
Refer to
Fig. 11 for the dimension parameters. CASE BC is referenced 5dB down
from CASE J S21 and CASE J Su corresponds to the top scale.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
92
by the leakage rate and could not describe the resonance peaks in the measured data. A
high leakage rate translated to a fairly strong unbounded transmission response and
likewise for a small rate. Also, the measured fcrit selection is somewhat arbitrary. Other
CBCPW examples not shown include h4= 1.27mm, h5= 1.42mm with measured fcrjt of
28GHz and predicted fcrit of 23.5GHz, h4=2.54mm, h5=0.71mm with an experimental fcrjt
of 14GHz and simulated unconditional leakage with frequency, and h4=2.54mm,
h5=0.381mm with measured fcrit o f 12GHz and calculated unconditional leakage with
frequency. One possible reason for the inconsistencies for the critical frequencies could be
attributed to the uncertainty of the dielectric thickness (±0.102mm, ±0.051mm for 2.54mm
and 1.27mm boards, respectively using 6010 er=10.8 Duroid™ substrate and ±0.0254mm
for 5870) and the relative dielectric constants (±0.25 for 6010 and ±0.02 for 5870) and the
possible existence o f a small air gap (0.0254mm) between the substrates (sr4=10.8, er5=l,
sr6=2.33).
Including these maximum dimensional uncertainties with respect to the
maximum critical frequency for CASE K, produce sr4=10.55, sr5=2.35, h4=1.22mm, and
h5=0.41mm. The results of the uncertainty analysis for fcrit are summarized in Table XIII.
The unmarked responses in the table are nonsimulated cases.
The one-dimensional analysis utilized in this section assumes an infmite-width structure
which under certain conditions can constitute a source of power loss from the CBCPW
dominant mode to the PPMs in the absence of lateral confinement. However, this is an
unrealistic physical structure. The substrate edges aty=±A are terminated in some manner.
If the side termination does not allow leakage to occur (loss of energy from the
waveguide), for instance a conducting sidewall, the outwardly traveling PPM will be
reflected and a transverse standing wave will be created. This standing wave can place an
admittance termination to the slot region and influence the value of the dominant mode
CBCPW propagation constant [26], Assume the CBCPW of Fig. 13 with PEC sidewalls
that are not shorting the ground planes and the conductor-backing and a single-layer
structure. The propagating leaky TEM wave becomes totally reflected from the sidewalls
and the waveguide is no longer leaky but instead the CBCPW dominant mode is purely
real. If the sidewalls are shorting then the TEM does not exist and the lateral confinement
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
TABLE XIII
Comparison of critical frequencies up to 36GHz for CBCPW for measured
data, SDM analysis, predicted with maximum dimensional uncertainties, and
predicted with maximum dimensional uncertainties and air gap at h5.
measured
fcrit (GHz)
predicted
CASE G
3
unconditional
unconditional
CASE H
10
unconditional
unconditional
CASE I
>36
>36
CASE J
24
21
—
—
23.6
CASE K
20
unconditional
12
16.5
h4= 1.27mm,
h6= 1.42mm
28
23.5
—
25.9
h4=2.54mm,
h6=0.71mm
14
unconditional
—
7.4
h4=2.54mm,
h6=0.381mm
12
unconditional
—
4
max uncertainty max uncertainty
with air gap
—
—
—
of this waveguide gives rise to mode coupling effects associated with rectangular
waveguide modes. This is the reason why the predicted critical frequencies for the
examples in Table XIII are not always accurate. The leakage analysis of infmite-width
CBCPW provides insight and can correlate the degree of mode coupling effects in a
two-dimensional waveguide and this will be described in Chapter V. For instance, a larger
leakage rate implies a stronger coupling effect with the characteristic waveguide modes.
An example of the sidewall termination effects is presented in Fig. 30 for a single-layer
CBCPW with shorting sidewalls along the waveguide length with the wrap around copper
tape to provide a uniform ground connection within the circuit as in Fig. 5(a).
This
three-dimensional waveguide produces a cavity resonator substructure which explains the
presence of the strong resonance peaks for the measured data as the resonant cavity modes
are above cutoff.
Again, the one-dimensional leakage analysis can not predict these
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
MAGNITUDE
S21 ( d B )
-5
-1 0
CASE
-1 5
-20
-2 5
-3 0
FREQ ( G H z )
Fig. 30.
Experimental data demonstrating the resonant cavity modes within an
overmoded single-layer CBCPW. CASE L with srl=sr2=sr3=er5=ert=l,
er4 = 10.8 and hj=oo (open structure), h2=h3=h5=h6=0, h4=0.635mm and
S=W=0.508mm, A=19mm, B=12.7mm. Refer to Fig. 13 and Fig. 5(a)
for dimensional parameters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
cavity modes. The dominant TE. resonant cavity mode frequencies can be approximated
for a single-layer rectangular cavity o f height a, width b, and length c from [9] as
fe w
l®
+ ^b)
+ (= )
(3 13)
with rh=0,1,2,... , ^=1,3,5,... ,p = 1,2,3,... . The placement o f a magnetic wall excitation
at _y=0 places a restriction on the value of the mode number ft. The identified resonant
mode frequencies to 25GHz from Fig. 30 and the probable equivalent resonant cavity
modes (TE01^,) o f (3.13) are listed in Table XIV.
TABLE XIV
Measured and predicted resonant cavity frequencies for CBCPW
CASE L from Fig. 30 and for TE0>1^ modes.
predicted
T E o .,i
2.2
2.16
TE01.2
3.8
3.79
T E o .1.3
5.5
5.52
T E o .1 .4
7.3
7.29
T E q .1.5
9
9.07
10.9
10.85
TF 0 .1 .7
12.9
12.63
14.6
14.43
T E 0 . i .9
16.4
16.2
T E 0 . , . 10
18.1
18.02
^E 0.i.n
19.8
19.8
T£ 0.1.12
TE0.1.13
21.5
21.61
23.2
23.39
©3
measured
o
JL
00
mode
on
fr(GHz)
The identification of the resonant cavity modes for CBCPW with sidewalls was originally
observed as part o f this work [14],
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
96
CHAPTER IV
TWO-DIMENSIONAL ANALYSIS OF CBCPW W ITH LATERAL
SIDEWALLS INCLUDING MODE COUPLING EFFECTS
The one-dimensional infmite-width analysis of the previous chapter was not adequate to
explain and predict all o f the experimental data effects within CBCPW MICs.
This
discrepancy was due to the finite dimensions of the physical transmission lines.
The
waveguides analyzed in this chapter are again illustrated in Fig. 5 and Fig. 13 with the
ground planes connected to the conductor-backing along the length (r-direction) to
achieve a uniform ground connection within the test fixture.
The waveguide now has
terminating lateral shorting sidewalls in the transverse y-direction and could also model a
CBCPW placed within a package. This structure (with a small cross-sectional circuit area
compared with a large lateral width) resembles a rectangular waveguide with two aperture
slots to guide the electromagnetic energy. Characteristic rectangular waveguide modes
operating above the cutoff frequency are excited at circuit discontinuities (coaxial
connector feed) and will propagate and couple with the dominant CBCPW mode
(multi-mode propagation exists). These mode interactions are different from the leaky
wave situation because the coupling effects occur only at discrete frequencies instead of
existing as a continuous function of frequency. Also, a longitudinal phase match condition
between the CBCPW and waveguide modes does not occur. The presence o f these modes
can cause the CBCPW mode field distribution to spread out across the entire waveguide
width instead of being confined to the slot regions and modify the propagation
characteristics. This phenomenon is similar to the leaky wave case except the propagation
constant of the CBCPW dominant mode is real and the exponentially growing wave in the
transverse direction is not present. The existence of higher-order waveguide modes for the
Part of the data reported in this chapter is © 1993 IEEE. Reprinted, with permission,
from 1993 IEEE Microwave Theory and Techniques International Microwave
Symposium Digest, pp. 947-950 (see Reference [14] and Appendix G).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
97
two-dimensional air-suspended CPW was originally presented in dispersion curves from
[21]. The field spreading within a single-layer packaged CBCPW was published in [24],
however the authors did not precisely identify the modes and did not present any
experimental data demonstrating these effects. According to coupled-mode theory [55],
power can be exchanged between the modes along the propagation path and is dependent
on the transmission line length. Energy outside the cross-sectional circuit area will not be
detected by the output coaxial connector and will resonate in the finite-length cavity
structure. The connector housing blocks complete the cavity at z= 0 and z=2B of Fig. 5(a).
The more this energy component spreads out, the stronger the resonance effects present.
To completely avoid the coupling problems, the CBCPW dimensions can be selected to
cutoff the waveguide modes. This is most easily achieved by reducing the lateral sidewall
separation [11], However, to operate at higher frequencies, the narrow wall separation
(y=±A) drastically reduces the available MIC surface area. For practical circuits the lateral
sidewalls walls must be extended allowing rectangular waveguide modes to propagate.
Conventional air-filled rectangular waveguides are designed for a single mode o f operation
over a designated frequency bandwidth.
An example o f the possible problems due to
propagating rectangular waveguide modes in CBCPW with PEC lateral sidewalls is
depicted as CASE 14 in Fig. 31 for CASE G (single-layer sr=10.8) o f Fig. 19 with
A=19mm. This figure presents the first 17 ideal waveguide modes (no TEM exists since
the top and bottom conductors are electrically connected) following the analysis of
Appendix E and Fig. E. 1 with lateral sidewalls at y=±A. The propagation constants for the
nonzero-cutoff TMXmodes for a single-layer substrate are written as
(4.1)
where M - 0 , 1 , 2 , . . . ,
qo
and ft = 1 , 3 , 5 , . . . ,
the dominant CBCPW mode).
ao
with a magnetic wall excitation at _y=0 (for
The phase constant for the TE modes are described by
(4.1) with th = 1 ,2 ,..., qo and h - 1,3,5, ...,oo and are degenerate modes with the TM
modes.
The TMr and TEr modes are alternative mode set representations for the
waveguide and for the single-layer substrate, the TM ^ h modes are the T E .0 h [9], The
wave number in they-direction becomes
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
3 .6
3 .2
2.8
o
•se
o
o
-se
1.2
0.8
Ideal RW
CBCPW
0 .4
0.0
FREQ
(G H z)
Fig. 31. Idealized dispersion curves for single-layer CBCPW with lateral sidewalls for
CASE 14 and erl= l, er4=10.8 and h;=h3=h5=h6=0, h ^ m m , h4=0.635mm and
S=2W=0.508mm and A=19mm.
Refer to Fig. 13 for dimensions. The
CBCPW mode response is from CASE 2 of Fig. 24 and superimposed onto
the graph. Ideal RW are the TM0h rectangular waveguide modes.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and this term is added to the left hand side of (E.7) and (E.9) in Appendix E to determine
the propagation constants for the ideal multi-layered characteristic rectangular waveguide
modes.
The CBCPW mode response in Fig. 31 is obtained from the Re(£.) for the
infinite-width case of Chapter II.
As demonstrated, many waveguide modes are
propagating which can couple to the dominant mode and produce the unexpected
measurement results of CASE B in Fig. 7.
For instance at 20GHz, 10 waveguide
modes propagate with a slower phase velocity than the CBCPW mode.
This chapter utilizes the two-dimensional SDM to predict the coupling effects for
CBCPW with lateral sidewalls. The SDM is a full electromagnetic wave numerical method
which can directly simulate these effects.
The analysis includes dispersion curves to
properly identify the modes, mode impedances, coupling coefficients, field plots, and the
field spreading due to the mode coupling effects o f the dominant CBCPW mode.
A
qualitative description of mode coupling, improved explanation of the CBCPW
experimental data, and the coupling effects of finite ground plane (FGP) CBCPW are also
presented.
The culmination of the above analysis was an original contribution and
published as part of this dissertation from [14],
A. CBCPW Mode Identification
The SDM for the analysis of the packaged two-dimensional (infinite-length) CBCPW
follows from sections C-E of Chapter II. The spectral variable kyj of (2.45) becomes a
discrete parameter and the integral analysis for the infinite-width structures is replaced by
spectral summations. The bisection method is invoked to properly isolate and identify the
CBCPW and waveguide modes. This routine proceeds by dividing up the propagation
constant values and generating a number o f fictitious propagation constants and
determinants o f the characteristic matrix [A] o f (2.34). The determinant vanishes at mode
solutions o f the waveguide. An example o f this procedure is depicted in Fig. 32 for CASE
15 for a lower loaded multi-layered CBCPW and will be utilized in sections B and C of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
50
CASE 15
40
(det U I )
30
20
CBCPW
spurious
-1 0
£
-20
-3 0
-4 0
.390
2.398
2.406
2.414
2.422
2.430
Re ( k z /k o )
Fig. 32. Example for the variation of the SDM determinant with fictitious propagation
constants at 7.5GHz for CASE 15 with srl= l, er4=10.8, er5=2.33 and h ^ m m ,
h4= 1.27mm, h5=0.381mm, h2=h3=h6=0 and S=2W=0.635mm and A=19mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
this chapter. The second root in the figure is a spurious solution associated with the pole
of the Green's function while the proceeding root is the proper TM q , mode.
This
characteristic response in the determinant is repeated for the other waveguide modes. The
spurious modes can usually be identified by having extremely large basis function
coefficients. The determination o f the rectangular waveguide modes using the bisection
method is not a straightforward procedure. The propagation constants calculated by the
ideal waveguide analysis of Appendix E are a necessary addition to the SDM to locate the
modes. The dominant waveguide mode for the multi-layered structure is the TMX since
2A>h6+h5+h4.
The same SDM basis functions for the dominant CBCPW mode are
employed for the waveguide modes since the same boundary conditions apply and the
fields should fringe across the slots due to the potential difference.
Field plots generated by the SDM are necessary to identify the various waveguide
modes within the CBCPW and the following examples correspond to CASE 15 of Fig. 32.
The vector electric field plots (Ex and Ey) within the cross-sectional circuit area are shown
in Fig. 33 for the CBCPW mode and the TMX0 5 waveguide mode. The field components
for all the plots are normalized to the maximum values. The full waveguide field plot for
the TMq j mode is depicted in Fig. 34. The SDM field plot of Fig. 34(b) is verified by the
inclusion of the fields determined from the ideal two-layered rectangular waveguide
analysis between the ground planes and the conductor-backing using Appendix E, (2.1),
and [9],
The TM waveguide modes possess a predominantly vertical electric field
orientation in the x-direction. The couple of field points at the bottom o f the figure do not
exist outside the waveguide but are just scaled in order to observe the other points. The
field plots for the Ex component o f the CBCPW and the first three waveguide modes are
pictured in Fig. 35.
The second subscript fi in T M m o d e s refers to the number of
half-sinusoids (standing wave maxima) in the transverse direction.
The field response of
the waveguide modes is perturbed in the vicinity o f the slot regions and satisfy the
boundary condition that the tangential electric field does not exist on a PEC at y=±A.
With air above the circuit conductors, the waveguide mode cutoff frequencies are
increased proportionally to the cross-sectional circuit area (S+2W). Also, note in Fig. 35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
1.6
1.2
0.8
0.4
X-N.
E o.o
E
*
Y N r* * "'’ *
— v y t v — *- s / i \ \
0.8
-
-
1.2
-
1.6
—
2.0
n
n
n
u
V V *
V I*
n
- 2 .0 - 1 .6 - 1 .2 - 0 .8 - 0 .4 0.0 0.4 0.8 1.2 1.6 2.0
l/( m m )
(a)
1.6
1.2
0.8
0.4
£
\ \ r /
o.o
E
0.8
-
-
1.2
-
1.6
r\ \ \ \ s v < ' - ' / ' / M M
T TM
\ \ \ ' i >/ / / / rr t
t
T1 T 1 1 \ 1 t i / / ; t r i t
fr f rtr i t t t 1t t i ti t ti rr r? tt ii iT
t
w -0.4
•
I T T
T
I T
' T
T
I I T
T
-2.0.
T
T
- 2 .0 - 1 .6 - 1 .2 - 0 .8 - 0 .4 0.0 0.4 0.8
y
t
t
i
i
t
I
1.2
1.6 2.0
(m m )
(b)
Fig. 33. Cross-sectional vector electric field plots for CASE 15 at 15GHz
for (a) CBCPW mode and (b) TMg 5 mode.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
103
0.0
-
0.2
-0 .4
-0.6
£ - 0.8
-1.0-
-1.6
-1.8
y
(m m )
(a)
1.&
0.8/"~N
0.4
E
E
-0 .4
-0.8
-1.2
-
1.6 1
-2.0
(b)
Fig. 34. Cross-sectional vector electric field plots of CASE 15 at 15GHz for TM „, mode,
(a) Ideal rectangular waveguide analysis, (b) SDM analysis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
104
CBCPW
1.0
TM0.1
TMqj
0 .8
•0— TMo^
0 .6
0.4
Q
2
0.2
CL
0 .0
< - 0 .2
H
^ —0.4
-0 .6
- 0 .8
-
1.0
V
(m m )
Fig. 35. Electric field plot of CBCPW and first three waveguides modes
of CASE 15 at 15GHz and x=-0.01mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
that the CBCPW mode is confined to the slot regions at 15GHz as expected. All o f the
waveguides modes discussed are propagating. The bound CBCPW mode is quasi-TEM
while the waveguide modes are either TM or TE modes.
B. Mode Coupling Effects in CBCPW with Lateral Sidewalls
In
two-dimensional
printed-circuit
transmission lines,
classical
coupled-mode
interaction between the dispersion curves o f the transmission line dominant mode and
those of the supporting substructure is possible. In CBCPW with lateral sidewalls these
modes are rectangular waveguides modified by the slot regions.
Using classical
coupled-mode theory [55], the unperturbed modes associated with the two independent
structures are first identified. The two guiding structures are then combined and the new
perturbed modes o f the composite waveguide are described and represented in terms of the
unperturbed modes of the original independent guides. This technique was presented in
Fig. 31 for CBCPW. Regions o f classical mode coupling in the perturbed modes occur at
and near frequencies at which the dispersion curves o f the unperturbed modes cross. The
composite structure is the finite-width CBCPW placed within a package and the lateral
sidewalls connect the ground planes to the conductor-backing. The unperturbed modes of
the package are those associated with the rectangular waveguide in the absence of the
slots, and the dominant CBCPW mode of the infinite-width structure.
The perturbed
modes are the propagating rectangular waveguide modes in the presence of the slots, and
the dominant CBCPW mode in the presence of the package. The rectangular waveguide
supports an infinite number of modes that are propagating or nonpropagating (depending
on the structural parameters and frequency).
The phase constants of the rectangular
waveguide modes (PRW) are related to the parallel plate waveguide (Ppp) as
(4.3)
with ih = 0,1,2,... ,qo and fi = 1 ,3 ,5 ,..., oo .
The unperturbed transmission line can be
leaky and could a complex propagation constant.
However, the attenuation constant
associated with the leaky dominant mode is usually several orders of magnitude smaller
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
106
than the phase constant (see Table XI in section C of Chapter III). Under these conditions,
classical mode coupling occurs at frequencies where the unperturbed propagation constant
of the CBCPW dominant mode equals that o f the unperturbed package modes.
The
perturbed dominant CBCPW mode has a real propagation constant for a lossless structure
[56],
The dispersion curves for the normalized propagation constants for the multi-layered
CASE 15 are calculated by the SDM for the two-dimensional CBCPW and shown in Fig.
36 and the waveguide modes (TMo,, TMq 3, ...., TM,, 17, TMq 19) are represented from left
to right as the frequency is increased. Above 32.5GHz, the TM l h and TE0 h modes also
propagate but are not included as this would clutter the graph. The SDM RW and the
Ideal RW refer to the waveguide mode propagation constants calculated from the spectral
domain method and from the Appendix E analysis, respectively. As indicated in the figure,
frequencies at which the perturbed modes approach, do not cross as for the leaky wave
one-dimensional case, but rather bend away from each other in a classical coupled-mode
behavior. In this frequency region of coupling, the mode field distributions are dissimilar
to those for the same modes on either side of this transition region and cannot be clearly
identified with either mode as energy is exchanged between the modes. As indicated in
Fig. 36, strong mode coupling between the dominant CBCPW mode and the waveguide
modes happens at frequencies where the dispersion curves approach and have the smallest
separation and occurs approximately at 9.2GHz with TMq, mode, 17GHz with TM03,
21 GHz with TM05,
26GHz with TM0J, and 31 GHz with TM09, respectively.
The
coupling regions are not a continuous function of frequency but are localized.
For
instance, the strong mode coupling region between the dominant CBCPW mode and the
TMqj mode is from 7.5-12GHz. An indication o f the degree o f mode coupling between
the dominant and waveguide modes can be approximated from the dispersion curves. The
greater the separation between the two dispersion curves in the mode coupling region, the
stronger the coupling between the dominant and waveguide mode.
From Fig. 36, the
mode coupling effects between the CBCPW mode and the TMq7 mode is much more
enhanced than the coupling between the dominant mode and the TMq , mode. Mode A in
the figure represents the top continuous dispersion curve response of the CBCPW
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
107
3.0
x
CBCPW mode
SOM RW
o
I d e a l RW
2.6
o 2. 4
2.0
-B
1.0
FREQ (G H z )
Fig. 36. Dispersion curves o f CASE 15 from Fig. 32 for CBCPW and waveguide modes
demonstrating the mode coupling effects. SDM RW and Ideal RW refers to
rectangular waveguide modes from the two different analysis. Mode A is
the top continuous dispersion curve and Mode B is the second continuous
curve.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
108
mode before the mode coupling region and of the TMq^ mode after the coupling. Mode B
represents the second continuous curve from the cutoff frequency of the TM q , mode
before the coupling at 9.2GHz and o f the CBCPW mode after the coupling.
The
propagation constant of the dominant CBCPW mode does not seem to be affected by the
mode coupling mechanism outside the transition region across the frequency-band.
The fields associated within the coupling region become a combination of the dominant
CBCPW mode and the rectangular waveguide mode. The fields are distributed or spread
across the entire waveguide width instead of being confined to the slot regions even
though the propagation constant of the dominant CBCPW mode is real for a lossless
structure (recall for the leaky wave case the propagation constant was complex). The
residual coupling effects on the dominant CBCPW mode for frequencies outside the
transition regions where the modes can still be identified is also important. The cumulative
effect o f the waveguide mode coupling causes the dominant mode fields to spread out
across the waveguide and not bound to the slot regions. The coupling efficiency between
the modes determines how the dominant mode fields spread and resembles a rectangular
waveguide mode distribution (non-TEM). This effect is demonstrated for CASE 15 in Fig.
37 at x=-0.01mm for the CBCPW dominant mode at frequencies before and after the
strong mode coupling regions. The field values are normalized to the maximum values.
Figure 35 showed at 15GHz after coupling with the TMq, mode, the CBCPW mode is
still bound to the slot regions. After coupling with the TM0 3 mode, the dominant
mode energy is starting to spread across the waveguide width as indicated at 20GHz. This
situation is enhanced after coupling to the TMq5 mode at 23 GHz and to the TM q7 mode at
29GHz.
Energy outside the CBCPW slot regions will not detected by the coaxial
connector and will resonate in the finite-length three-dimensional cavity structure. The
analysis would predict the beginning o f the mode coupling effects approximately after
20GHz, higher insertion loss at 23 GHz with some resonant peaks, and probably an
unusable transmission line at 29GHz.
These coupling effects also depend on the
transmission line length. Measurement results verifying the analysis will be presented in
section D later in this chapter. Outside the transition regions, the mode coupling effects
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
109
1.0
0.8
CASE 15
0.6
0.4
3
ti
0 -2
-I
0 .0
CL
-
0.6
-
0.8
— 20GHz
23GHz
- - 29GHz
- 1 .0
V
(m m )
Fig. 37. Field plots at various frequencies for CBCPW mode of CASE 15 demonstrating
the field spreading effects due to mode coupling with the waveguide modes.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
110
on the rectangular waveguides modes due to the CBCPW mode were not observed in the
SDM.
The presentation of the mode coupling effects to the various waveguide modes
with respect to the field spreading was an original contribution of this dissertation o f [14].
Additional demonstrations of the mode coupling effects are illustrated for CASE 15 in
Fig. 38. Characteristic impedances for Modes A and B and the CBCPW mode are shown
from Fig. 36 using the power-voltage definition of (2.64a). This impedance definition is a
proper choice since the rectangular waveguide modes energy is distributed across the
entire waveguide width. The impedance o f the TMXrectangular waveguide modes should
be small since the fields have a large amplitude and are extended across the entire
waveguide width which translates to a much higher power flow.
As indicated in the
transition region, the energy is exchanged between the modes and becomes a combination
o f the modes and the impedance of the two modes become equal at some frequency. In
other words, the fields o f the CBCPW mode and the TMq j mode in Fig. 38(a) become
nearly identical at 9.2GHz. The impedance of the CBCPW mode should decrease with
frequency and the field distribution resembles a waveguide mode from these coupling
effects.
The CBCPW dominant mode impedance varies from 50Q to 20Q over the
frequency-band primarily due to the mode coupling. This impedance demonstration of the
mode coupling effects was first published in [14],
Coupling coefficients between the
waveguide and the CBCPW modes are also pictured in Fig. 38.
The response
demonstrates a peak value near the strong mode coupling regions and a smaller coefficient
outside these frequency points. This behavior follows from the dispersion curve plots of
Fig. 36. The coupling coefficient between modes a and b describing the waveguide
mode field overlaps is calculated using the following power flow relationship from [33] as
Co
(4.4)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ill
x
CBCPW moda
Mode A
Mode B
O 40
I-
0
8.0
5
8.3
10
8.6
15
8.9
20
9.2
25
9.5
FREQ (GHz)
30
9.8
35
10.1
(a)
-g 0.060
o
E
^
0.050
CL
O
g
0.040
O
*4-*
^
0.030
Ll.
LlI
O 0.020
O
2
—I 0.010
CL
u
o 0.000
o
FREQ (GHz)
25
35
(b)
Fig. 38. Additional mode coupling effects in two-dimensional CBCPW CASE 15.
(a) Characteristic impedance plots for the modes indicated in Fig. 36.
Modes A and B refer to the lower frequency range (8-10.1GHz).
(b) Calculated coupling coefficient o f the waveguide modes to the
CBCPW mode.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
112
where t represents the transverse field components, CS is the entire waveguide
cross-section, and
C aa = C bb = 1.
The integrals in the denominator of (4.4) can be
determined from the spectral domain analysis of (2.70) and Appendix C and the integral in
the numerator is evaluated in the space domain.
C. Additional Examples o f the Mode Coupling Effects in CBCPW
To confirm the SDM analysis o f the mode coupling effects in CBCPW with lateral
sidewalls, three additional examples will be presented in this section and compared with
experimental data in section D. Figure 39(a) includes the coupling effects for a single-layer
(er=10.8) CBCPW of CASE 14 in Fig. 31 and jc=-0.01mm. At 5GHz, the dominant mode
has experienced mode coupling with the TM,,, mode. Additional field spreading occurs
at 8GHz after coupling to the TMq 3 mode.
The CBCPW mode has coupled with the
TMq 5 mode and significant field spreading across the waveguide occurs at 12GHz. At
25GHz, the dominant mode has coupled with modes TM q pT M ^, ... ,TM09,TM0 n and is
no longer a quasi-TEM. A multi-layered example of CASE 16 which is similar to
CASE 15 except with thicker substrates (h4=2.54mm, hs=0.71mm) is presented in Fig.
39(b). The mode coupling problems become pronounced after coupling with the TMq 5
mode as indicated at 15.5GHz.
The multi-layered example of CASE 5 (same as CASE 15 except h4=0.635mm,
hs=0.71mm) from Chapter III (Fig. 25) is analyzed with lateral sidewalls in Fig. 40(a). The
normalized dispersion curves demonstrate that no strong mode coupling regions exists
with the waveguide modes to 40GHz. As a result, the dominant mode should be bound to
the slots at 40GHz which is confirmed in part (b) o f the figure. This outcome was stated in
Chapter III for this example as this structure was not leaky to 40GHz. In other words,
mode coupling effects will not occur in a perturbed transmission line if the unperturbed
transmission line dominant mode is bound and not leaky. Also, the phase constant of the
perturbed waveguide modes will always be less than the phase constant o f the unperturbed
parallel plate waveguide modes from which the waveguide is composed as indicated in
(4.3). Therefore, the leaky critical frequency o f an infinite-width structure will always be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
113
1.0t
0.8
CASE 14
— 5GHz
BGHz
- - 12GHz
r— 25GHz
0.6
lu
0.4
p
0.2
Q
a! o.o
0.2
< -
H
C o -0.4
-
0.6
-
0.8
-1.01
(a)
1.0
0.8
CASE 16
12.5GHz
15.5GHz
19GHz
0.6
UJ 0.4
O
0.2
CL 0.0
0.2
< -
H
c*q-o.4
-
0.6
-
0.8
-1.G
-2 0 -16 -12 - 8
-4
0
4
y (m m )
8
12
16
20
(b)
Fig. 39. Additional examples of field spreading mode coupling effects in CBCPW.
(a) Single-layer structure CASE 14 of Fig. 31. (b) Multi-layered structure
CASE 16 same as CASE 15 except h4=2.54mm, h5=0.71mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
114
2.8
2.6
a 2.4
2.2
2.0
c* 1.8
1.2
1.0.
FREQ (GHz)
(a)
1.0(
0.8
40GHz
0.6
UJ 0.4
O
^
0.2
a
o.o
< -
0.2
C ^-0.4
-
0.6
-
0.8
-1.01
(b)
Fig. 40. Multi-layered CBCPW example demonstrating a bound dominant mode to
40GHz.
CASE 5 here is same as CASE 15 except h4=0.635mm,
h5=0.71mm. (a) Dispersion curves, (b) Plot for Ex field component.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
less than the first mode coupling frequency for the two-dimensional structure.
D. Experimental Results Confirming the Two-Dimensional SDM Analysis o f CBCPW
Measurement data of CBCPW through lines is presented to
validate the
two-dimensional SDM analysis of the previous sections and is shown in Fig. 41. The S21
transmission coefficient data was taken with an HP8510 network analyzer using 401
points. CASE L in the figure is the single-layer channelized CBCPW with 2A=5mm and
the cutoff frequency of the first waveguide mode is 10.6GHz which is confirmed in the
figure. CASE M corresponds to CASE 15 and the transmission line characteristics initially
change above 12GHz with the inclusion of a small standing wave pattern which continues
to about 21 GHz. A significant insertion loss and the presence of strong resonant peaks
appear after 27GHz. This response is closely predicted by the SDM in Fig. 37 for this
case. The improvement of the two-dimensional CBCPW model over the one-dimensional
leaky wave analysis for this example is readily apparent from Table XIII o f Chapter III
which stated 0<fcrit<12GHz without an air gap layer between the dielectrics. CASE N in
Fig.^41 is the measurement results for CASE 16 and the data presents problems beginning
at 14GHz and an unbounded behavior occurring above 17GHz. This result is in good
agreement with the numerical plots of Fig. 39(b). The single-layer example o f CASE 14 is
depicted by CASE O and illustrates an initial trouble point at 5GHz with strong resonances
beginning at 9GHz. The data is truncated for responses 20dB down. The experimental
data for CBCPW MICs in which mode coupling effects are present, exhibits a low-pass
filter characteristic with a soft comer frequency.
The coupling o f the CBCPW mode to the TMq5 mode for the multi-layered lower
dielectric loaded examples studied in this dissertation marks the frequency point where the
dominant mode has significant energy spreading, and increases with coupling to the
higher-order rectangular waveguide modes.
This conclusion is verified with the
experimental data. For the single-layer CBCPW, coupling with the TMq3 mode provides
an approximate upper usable frequency limit o f operation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
116
CASE L
S2 i (d B )
CASE M
>
'•a
CASE N
MAGNITUDE
QQ
TO
O
CASE 0
0
4
8
12
16
20
24
28
32
36
40
FREQ ( G H z )
Fig. 41. Experimental data o f the mode coupling effects in CBCPW MICs with lateral
sidewalls with srl= l and h,=oo (open structure) , h2=h3=h6=0 and 2B=38mm.
Refer to Fig. 13 and Fig. 5(a) for the dimension parameters. For CASE L
er4=10.8 and h4=0.635mm, h5=0 and S=W=0.508mm and 2A=5mm. CASE M
same as CASE 15 with er4=10.8, srS=2.33 and h4=1.27mm, h5=0.381mm and
S=2W=0.635mm and 2A=38mm. CASE N (CASE 16) same as CASE M
except h4=2.54mm, h5=0.71mm. CASE O (CASE 14) same as CASE L
except 2A=38mm. All cases are 50Q through lines and referenced to OdB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
117
The effects of the transmission line length in the overmoded CBCPW are pictured in
Fig. 42. CASE P is the same as CASE M of Fig. 41 with a shorter length and the response
is similar except for the lower insertion loss at higher frequencies. CASE 5 o f Fig. 25 is
represented by CASE Q and as indicated the dominant mode is bound to the slots up to
40GHz as described in Fig. 40. Resonances in the data on the order of 2-4dB exist in the
upper ffequency-band. Energy is coupled to the waveguide modes at the coaxial feed or
by the CBCPW mode over the waveguide length. By reducing the cavity (transmission
line) length, the resonances should be shifted upward in frequency and the cavity Qs
modified. This statement is verified for the lower half o f the band for CASE R but the
upper band resonances have effectively disappeared. The resonance at 23GHz (beginning
of the upper band resonances) o f CASE Q could correspond to either TM0112, TM0 3 u ,
or TMgjg modes for an ideal multi-layered cavity with a length (2B) o f 38mm.
The
resonant cavity frequencies are determined using the transcendental expressions from
Appendix E and (4.2) and are written for the TM r
modes as
k
k
c xl - ta n ^ .h s = - P xi
r5
0
r4
tanAri4h 4
0
(4.5)
4
with
^ +(S)
**4 +
where
TM
=(®™J s*eo^o
(f£) + Of)
cocAthj} is the unknown and
(46a)
(4.6b)
fi= 1,3,5,... ,oo and p = 1,2,3,... ,oo.
For
2B=25.4mm the resonant frequencies for these three modes are 32.9GHz, 31.9GHz, and
28.7GHz, respectively which are within the measurement band but are not present in
CASE R. Broadband absorber is placed within the coaxial connector blocks o f CASE S
and reduces the resonances by dampening the cavity for CASE Q and produces a 40GHz
transmission line which is the widest bandwidth CBCPW MIC reported to date [14].
In summary, the two-dimensional SDM can determine the transmission line bandwidth
and explain the mechanism producing the line loss and the strong resonances. However,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
118
CASE P
CO
X>
CN
00
CASE Q
>
-O
•
Q
ID
QD
■O
O
m
CASE R
<
CASE S
0
4
8
12
16 20
24
28
32
36
40
FREQ (GHz)
Fig. 42. Experimental data demonstrating the cavity mode effects in CBCPW MICs
with erl=l, er4=10.8, srS=2.33 and ht= oo (open structure), h2=h3=h6=0
and S=2W=0.635mm. Refer to Fig. 13 and Fig. 5(a) for the dimension
parameters. CASE P with h4=1.27mm, h5=0.381mm and
2A=38mm,
2B=25.4mm. For CASE Q with h4=0.635mm, hs=0.711mm and 2A=20mm,
2B=38mm. CASE R same as CASE Q except 2B=25.4mm. CASES same
as CASE Q except with broadband absorber in the connector housing blocks.
All cases are 50Q through lines and referenced to OdB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
119
the Q of the resonant peaks and the resonant frequencies in the experimental data cannot
be explained by the two-dimensional analysis and a finite-length (three-dimensional)
structure is required to fully predict the measurement results. The SDM will be extended
to three-dimensions for CBCPW and will be presented in Chapter VI.
E. Mode Coupling Analysis of Finite Ground Plane CBCPW
The effects o f mode coupling on the CBCPW mode with finite ground planes (FGP)
will be analyzed in this section. Referring to Fig. 14(b) of section F in Chapter II, two
fundamental zero-cutoff modes (CBCPW and coplanar-microstrip) are supported by the
structure and are shown in Fig. 4(a) and (c).
connected to the conductor-backing.
The ground planes are not electrically
Measurement data for a FGP CBCPW was
illustrated for CASE C in Fig. 7. At lower frequencies, the parallel plate region between
the conductors at x=0 and x=-h6-h5-h4 can be viewed as bounded in the transverse
direction by magnetic sidewalls at y=±(S/2+W+W0). These boundaries form a rectangular
waveguide with magnetic sidewalls and waveguide modes can propagate and couple with
the CBCPW dominant mode and produce similar mode coupling effects as described in
previous sections of this chapter. The cutoff frequencies of these waveguide modes can
be adjusted by the lateral sidewall separation (width of the ground planes WG).
The
wavenumber in they-direction becomes
2
-
where A= S/2+W+WG and ft = 0 ,2 ,4 ,...,
( a )
oo
. This expression would be added to the left
hand sides of (E.7) and (E.9) in Appendix E to determine the propagation constants o f the
multi-layered parallel plate waveguide o f Fig. E. 1 with magnetic sidewalls. This structure
could be analyzed using the SDM for CBCPW with the same basis functions o f Chapter II
section B and the above spectral parameter in (4.7) for k . At higher frequencies, this
magnetic sidewall model is no longer valid and the coupled strip SDM analysis of section F
o f Chapter II must be employed by expanding the unknown current distributions on each
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
strip.
The FGP CBCPW has another interesting phenomenon associated with
microstrip-like resonances associated with the two conductor ground planes behaving like
two-dimensional patch antennas [36].
At these resonant frequencies, most of the
electromagnetic energy is carried by the microstrip-like modes and the dominant CBCPW
mode will not effectively propagate along the slots. These frequencies must be avoided by
adjusting the patch antenna dimensions (three-dimensional problem) so the resonances
occur out of the ffequency-band.
An example of the mode coupling effects in FGP
CBCPW is illustrated in Fig. 43(a) for a single-layer structure CASE 17 at 30GHz. A total
of two and five basis functions on the center strip and the ground planes, respectively were
employed for the Jy and Jz currents. Strong mode coupling exists between the CBCPW
and CPM2 mode (identified as TMq2 mode) at a ground plane width (WG) o f 2.75mm. To
prevent this coupling effect from occurring, the ground plane width should be kept less
than 2mm or a multi-layered lower loading structure could be implemented as in Fig. 43(b)
for CASE 18. As indicated from the figure, no mode coupling exists between the CBCPW
and the CPM2 regardless of the ground plane width. Note that the CBCPW mode now has
a higher propagation constant than the CPM0 mode.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.501
if
i
i
i
i
0.5
1.0
1.5
2.0
2.5
3.0
i_____■
3.5
*.0 4.5
5.0
Wg (m m )
(a)
2.50
2.40
CBCPW
2.30
2.20
s*
2.10
•**
2.00
CPMo
1.90
1.70
1.60
CASE 18
(b)
Fig. 43. Dispersion curves showing mode coupling effects in FGP CBCPW as a function
of the ground plane width (WG) at 30GHz with srl= l, er4=10.2 and h ^ m m ,
h4=0.635mm,
h2=h3=h6=0
and
S=2W=0.635mm
and
2A=25.4mm.
(a) Single-layer structure with h5=0. (b) Multi-layered waveguide with srS=2.33
and h5=0.635mm. CPM refers to the coplanar-microstrip modes.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
122
CHAPTER V
DESIGN SUMMARY INFORMATION FOR THE PROPAGATION
CHARACTERISTICS OF MULTI-LAYERED CBCPW
The previous two chapters demonstrated how multi-layered CBCPW MICs could be
analyzed and employed to produce a transmission line such that the dominant mode would
propagate bound to the slots (quasi-TEM) with a bandwidth depending on the dimensional
parameters.
Additional design criterion is necessary to determine whether the
multi-layered structure (both upper and lower dielectric loaded) is a viable and practical
waveguide. This information for the propagation characteristics of the dominant CBCPW
mode includes the upper usable frequency or bandwidth, characteristic impedance range
and variation, parameter values for transmission lines with 5012 impedances, and effective
dielectric constant range and variation for different configurations.
Designers of
multi-layered CBCPW MICs must also understand the effects o f the possible air gap
between the dielectrics, the dielectric constant uncertainty value on the propagation
constant, dimensions for the cross-sectional circuit area to minimize dispersion, closed
form procedures to estimate the dielectric constant value and thickness for the
multi-layered loaded substrate, parameter values of multi-layered microstrip that may be
necessary to interface within a circuit, and other practical physical suggestions for usage.
An additional matter of interest is whether CBCPW can operate properly with the
mode coupling effects present. The experimental and numerical data already presented for
CBCPW MICs indicated that this was not possible beyond the critical or upper usable
frequency.
The dominant mode resembled a rectangular waveguide mode with a field
pattern spread out across the entire waveguide width.
The multi-layered waveguide was
Part of the data reported in this chapter is © 1994 DEEE and submitted to IEEE for
publication. Reprinted, with permission, from 1994 IEEE Microwave Theory and
Techniques International Microwave Symposium Digest, pp. 1697-1700 (see References
[15], [28] and Appendix G).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
utilized without placing any restrictions on the cross-sectional circuit area (S+2W) or the
lateral width. In other words, (S+2W) could be the same size or larger than the substrate
thickness. The question o f how CBCPW MMICs can function in the presence o f the
coupling effects must be investigated. Recall that a single-layer CBCPW unconditionally
leaks energy to the TEM parallel plate mode which translates to mode coupling effects
existing at lower frequencies for electrically wide two-dimensional waveguides.
For
CBCPW MMICs the line/slot widths can be made much smaller than the substrate
thickness.
This parameter guideline for MMICs reduces the leakage constant for the
one-dimensional structure [12], [32] and reduces the coupling to the cavity modes in
finite-length waveguides [35],
Keeping (S+2W) less than one-fourth the dielectric
thickness and one-twentieth the dielectric wavelength
(A.g /20)
for GaAs is an estimate for
the leakage rate of the CBCPW mode for useful results [12], although this statement was
not quantitatively demonstrated. A circuit designer may not have a two-dimensional SDM
numerical program with field plotting capabilities to analyze the coupling effects (as
presented in Chapter IV). The designer requires a procedure and a closed form expression
in terms of the cross-section, substrate thickness, dielectric constant, and operating
frequency for reduced mode coupling and a usable transmission line for the single-layer
CBCPW. This process will also relate the mode coupling in CBCPW with lateral sidewalls
(realistic two-dimensional structure) to the leakage rate of an infinite-width structure
(nonphysical waveguide) and hence unite the analysis from Chapters III and IV. It has
already been shown that if leakage is not present in a transmission line, mode coupling will
not occur and a small leakage rate corresponds to a small mode coupling effect.
This chapter will demonstrate the propagation characteristics of multi-layered CBCPW
with upper loading, summary tables illustrating operational trade-offs, and additional
design information for CBCPW MICs.
This complete design summary is an original
contribution from [15] and [28] as part o f this dissertation. The basic theory relating
leakage in infinite-width to coupling effects in packaged CBCPW will be described along
with appropriate leakage curves and mode coupling plots. Finally, a design procedure for
CBCPW single-layer structures will be detailed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
A. Propagation Characteristics of Upper Dielectric Loaded CBCPW
The upper loaded multi-layered CBCPW is another configuration alternative to extend
the mode coupling effects to higher frequencies and hence the bandwidth o f the
transmission lines. This CBCPW configuration was an original contribution as part of this
dissertation and published in [15], Referring to Fig. 13, the substrate layer (h3) is placed
on top of the circuit conductors and ground planes. This structure may cause problems in
circuits with elements possessing a vertical thickness component but is well suited to
transmission line elements such as matching and feeding networks, antennas, and filters.
The basic idea behind this structure is the placement o f a higher dielectric constant
substrate above the CBCPW with a lower dielectric constant value and hence raising the
propagation constant of the dominant CBCPW mode above that of the single-layer
waveguide modes. Recall that the conventional single-layer CBCPW has an unconditional
leakage condition in the one-dimensional case and mode coupling effects occurring at
lower frequencies within the two-dimensional transmission line. One additional matter of
concern is the presence of surface waves on the grounded dielectric slab (h3) above the
circuit conductors at x=0 (characteristic modes associated with microstrip) and the mode
coupling effects that can occur to the dominant CBCPW mode due to these modes. These
surface wave modes will limit the frequency range of operation and in the two-dimensional
structure with lateral sidewalls, these modes are referred to as boxed surface wave modes
[56], As the thickness of the upper layer increases (h3), the mode coupling effects with the
surface wave modes exist at lower frequencies. Referring to Fig. 13, the surface wave in
the air region (h,) travels in the r-direction and is attenuated in the x-direction (kxl is
imaginary).
The rectangular waveguide (RW) modes of the lower single-layer
homogeneous structure (h4) will not couple with the dominant CBCPW mode since
P cbcpw >
K.
rw
as l°nS as h3 *s sufficiently thick.
The upper loaded multi-layered
configurations used in this chapter for numerical cases consist of er3=10.2 and sr4=2.2.
A dispersion curve plot with normalized phase constants for the dominant, waveguide,
and surface wave modes for an upper dielectric loaded CBCPW with lateral sidewalls is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
125
pictured in Fig. 44 (CASE 19) and demonstrates no mode coupling effects to 40GHz. The
BSW IDEAL (boxed surface wave) response corresponds to a grounded dielectric slab
(h3) without slots. The LOWER RW refers to the maximum propagation constant of the
modes within the lower single-layer rectangular waveguide (ee(r=2.2).
Note that the
propagation response o f the BSW TM,,, is different than for the rectangular waveguide
modes of Fig. 31 and Fig. 36. A vector cross-sectional electric field plot for CASE 19 at
40GHz is illustrated in Fig. 45 and shows that the dominant mode is bound to the slot
areas as predicted from the dispersion curves.
Measurement data for CBCPW high
impedance through lines with sr4 =2.33 is depicted in Fig. 46(a). CASE T represents a
single-layer structure and problems in the response occur at approximately 10GHz. CASE
U is the same as CASE T except the lateral dimension (2A) is reduced to 5mm and the
cutoff frequency of the first waveguide mode is 22.7GHz and the data shows some
variations at about 33GHz. An upper loaded configuration is applied in CASE V and can
be compared to CASE T. Note at the lower frequencies an approximate 50Q. transmission
line impedance is present and then a large insertion loss occurs which is due to connector
problems associated with placing the upper layer substrate on top of the coaxial feeds.
This was already mentioned as one of the physical shortcomings for this multi-layered
waveguide for circuit elements with height. A modification was made for CASE W by
cutting away a semi-cone in the upper layer substrate (h3) at the port connectors as not to
interfere with the coaxial pin feeds and is diagrammed in Fig. 46(b). Therefore, CASE W
is not exactly modeled in Fig. 44 which predicts no mode coupling to 40GHz.
The
presence o f the air region above the connectors requires a three-dimensional analysis for
this example. The impedance of the transmission line is still high but demonstrates a much
improved response over CASE T. Fig. 47 depicts the cross-sectional vector electric field
plot (Ex and Ey) at a z=constant plane for the TM„ , surface wave of CASE 19 at 40GHz.
As indicated most of the energy associated with the mode is bound to the dielectric and the
air interface with a predominantly vertical field component and the field is attenuated in the
x-direction. This response is compared to the TM0 , rectangular waveguide mode of Fig.
34(b) for a lower loaded CBCPW in which the field exists between the conductor planes
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
2.75
CASE 19
2.50
2.25
V 2.00
o 1.75
—
cbcpw
TMo.1 BSW SDM
x TM0.1 BSW IDEAL
• - - TMq.i LOWER RW (MAX)
1.50
1.251.00
FREQ (G H z )
Fig. 44. Dispersion curve plot of the normalized propagation constants for an upper
dielectric loaded multi-layered CBCPW. CASE 19 w itherl=l, er3=lO,2,
er4=2.2
and
h^Smm,
h3=0.635mm,
S=2W=0.635mm and 2A=25.4mm.
h4=1.27mm, h,=h5=h6=0 and
Refer to Fig. 13 for parameter
dimensions. BSW is the boxed surface wave modes and the [DEAL case
corresponds to the grounded dielectric slab layer h3. LOWER RW refers
to the maximum propagation constant of
single-layer
rectangular
waveguide h4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
1.5
1.2
0.9
0.6
0.3
0.0
H -0 .3
0.6
-
-0 .9
-
1.2
-1 .5
y
(m m )
Fig. 45. Cross-sectional electric field plot for the dominant CBCPW mode of the
upper loaded example CASE 19 at 40GHz. The field components are
normalized to the maximum values.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
CASE T
CASE U
CD
T3
CASE V
Ld
O
3
Z
o
<
2
CASE W
0
4
8
12
16
20
24
FREQ (GHz)
28
32
36
40
(a)
(b)
Fig. 46. (a) Experimental data for CBCPW MICs with lower dielectric value er4=2.33.
erl= l and h^oo
(open structure), h4=1.42mm, h;=h5=h6=0 and
S=W=0.508mm and 2B=38mm. Refer to Fig. 13 and Fig. 5(a) for the
dimensions. For CASE T h3=0 and 2A=38mm (single-layer). CASE U same
as CASE T except 2A=5mm. CASE V upper loaded structure same as
CASE T except er3=10.8 and h3=0.635mm and S=2W=0.635mm and with
connector problems. CASE W same as CASE V except with a cutout of
the substrate around the inputs and output ports and illustrated in (b). All
cases are referenced to OdB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
4 .5
4.0
3.5
3.0
E
E
2.5
2.0
H
1.5
1. 0
-
0.5
0.0
-1 4
-1 0
-6
y
(m m )
Fig. 47. Cross-sectional vector electric field plot of the TMg [ surface wave mode for the
upper loaded multi-layered CBCPW of CASE 19 at 40GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
within the substrates.
B. Design Summary Information for Multi-Layered CBCPW MICs
Two basic multi-layered configurations for CBCPW MICs have been proposed in this
dissertation.
These transmission lines were analyzed using the SDM and verified
numerically and experimentally to extend the frequency of operation and in some cases up
to 40GHz. The lower dielectric loaded case is the preferred structure for circuits with
active devices or elements with a sufficient thickness and the upper dielectric loaded
waveguide is recommended for passive circuits involving feed or matching networks and
antennas. In order for these multi-layered configurations to be an accepted transmission
line class, the dispersive behavior, impedance range and variation, and additional design
information requires investigation. Summary tables of various structures demonstrating
operational trade-offs for wide frequency-band design are necessary.
Examples o f these summary tables for lower and upper loaded CBCPW MICs with
dielectric thickness ranging from 0.254mm to 1.27mm are illustrated in Tables XV and
XVI, respectively. All of the numerical results were derived from the SDM o f Chapter II.
The configurations are limited to two layers with relative dielectric constants 10.2 and 2.2
which approximately correspond to widely available substrates in Duroid™ (6010 and
5870). Four possible cross-sections (S+2W) are considered in this analysis. For Tables
XV and XVI, CASES 20 and 24 correspond to 5 0 0 lines, CASES 21 and 25 have equal
strip and slot widths, CASES 22 and 26 are low characteristic impedance lines, and
CASES 23 and 27 produce high impedances. The minimum strip and slot widths were
chosen at Smin=0.254mm and Wmin=0.127mm, respectively. The maximum cross-sectional
value (S+2W) of 2.032mm was selected based on impedance variations versus frequency.
The examples in these tables correspond to the maximum cross-sectional area that could be
implemented in a multi-layered CBCPW MIC to 40GHz without significant dispersion
effects. The listed frequency (f) is the upper usable frequency to 40GHz in which strong
mode coupling occurs between the dominant CBCPW mode and the first rectangular
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
waveguide or surface wave mode (TMq ,) for all the presented cases.
It was already
shown in Chapter IV that strong mode coupling occurs with the TM0 5 mode for the lower
loaded structure so the table responses are conservative estimates of the bandwidth. The
listed frequency (f) is the point where the CBCPW mode impedance, using the
power-voltage definition Zpv of (2.64a), decreases by 10% and indicates the onset of
strong mode coupling with this mode. This mode coupling effect was shown in Fig. 38(a).
A circuit designer requires a large effective dielectric constant to reduce the circuit size and
small impedance and eefr variations over frequency.
increases, the upper usable frequency decreases.
Referencing Table XV, as h4
This behavior occurs because the
waveguide modes are affected to a larger degree to changes in the vertical properties of
the structure than the dominant CBCPW mode and hence the cutoff frequencies o f the
waveguide modes are reduced. This statement is supported by Fig. 33(a) and Fig. 34. No
substantial change in performance exists by increasing the lower substrate thickness (h5)
since for a bounded CBCPW mode, most o f the electromagnetic energy is located in the
slot regions between the air above the conductors and h4 below. One exception to this
statement is for thin upper substrates such as example 1A). For 40GHz operation with a
lower loaded waveguide, examples 2B) and 3B) are recommended (due the higher
effective dielectric constant values) with 2B) utilizing a smaller overall thickness.
Examples 2C) and 3C) could be used to 20GHz. Results from Table XVI are similar to
those in Table XV. As h3 increases, the upper usable frequency (f) decreases as the cutoff
frequency of the first grounded slab surface wave mode is reduced and couples to the
dominant CBCPW mode at lower frequencies. The propagation characteristics o f this
surface wave mode are primarily influenced by the h3 dielectric while the CBCPW mode is
affected by the air, h3, and h4 layers. In both tables, increasing the thickness of the er =10.2
substrate, increases the effective dielectric constant to a diminishing point. From Table
XVI, an upper usable frequency o f 40GHz could be realized by examples 3B) and 2B)
with 3B) providing a closer 50Q characteristic impedance. Examples 3C) and 2C) could
be selected to achieve at least a 20GHz bandwidth for the CBCPW.
Fig. 48 includes dispersion curves for CASES 20 and 21 of Table XV and a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
TABLE XV
Lower dielectric loaded multi-layered CBCPW MIC design summary.
erl=l, er4=10.2, er5=2.2 and h,=5mm, h2=h3=h6=0 and 2A=25.4mm
and S+2W<2.032mm, Smax=10W, Wmin=0.127mm and refer to
Fig. 13 for dimensional parameters and
CASE 20: S=2W=0.635mm
CASE 21: S=W=0.635mm
CASE 22: S=10W=1.27mm
CASE 23: S=0.254mm, W=0.889mm.
FREQ
(GHz)
f
(mm)
h4
h5
sefr(min)
@ 1GHz
Zp v{min),
Zp v (max)
(Q)
@ 1GHz
Zpy (£2)
CASE 20
@ 1GHz
eefr(min)
@f
Zp v (min),
Zpv (max)
(Q)
@40GHz
40.7,
90.8
1A)
0.254
0.254
40
3.87
30.3,
80.3
48.9
4.34
IB)
0.635
0.254
36.9
5.00
31.5,
82.3
49.3
5.96
1C)
1.27
0.254
4.8
5.46
32.5,
85
50.2
5.56
ID)
2.54
0.254
3.5
5.58
33.1,
86.7
50.9
5.68
2A)
0.254
0.635
40
3.61
35.5,
95.2
55.8
4.21
45.4,
117.8
2B)
0.635
0.635
40
4.78
33.7,
88.5
51.8
5.98
33.8,
88.5
2C)
1.27
0.635
20.8
5.33
33.3,
87.2
51.1
5.97
-
2D)
2.54
0.635
6.4
5.53
33,
87
51
5.68
-
3A)
0.254
1.27
40
3.58
37.7,
102.5
58.5
4.2
45.8,
123.3
3B)
0.635
1.27
40
4.75
34.7,
91.8
53
5.97
33.4,
89.9
3C)
1.27
1.27
24
5.3
33.8,
88.7
51.6
6.08
-
3D)
2.54
1.27
10.3
5.51
33.5,
88
51.4
5.77
-
-
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
133
TABLE XVI
Upper dielectric loaded multi-layered CBCPW MIC design summary.
eri= l, sr3=10.2, er4=2.2 and 1^=5mm, h2=h5=h6=0 and 2A=25.4mm
and S+2W<2.032mm, Smax= 10W, Wmjn =0.127mm and refer to
Fig. 13 for dimensional parameters and
CASE 24: S=2W=0.635mm
CASE 25: S=W=0.635mm
CASE 26: S=10W=1.27mm
CASE 27: S=0.254mm, W=0.889mm.
(mm)
f
I S g 2
Zpv(Q)
CASE 24
@ 1GHz
1A)
0.254
0.254
40
3.21
24.1,
65.7
39.6
3.66
40.9,
92.5
IB)
0.635
0.254
40
3.92
22.4,
60.3
36.2
5.43
32,
76
1C)
1.27
0.254
24.3
4.23
21.7,
58.6
35.3
5.39
-
ID)
2.54
0.254
11.8
4.35
21.5,
58.1
35
4.85
-
2A)
0.254
0.635
40
3.54
32.6,
88.5
52
4.21
45.7,
124
2B)
0.635
0.635
40
4.65
30.3,
80
46.6
6.19
35,
92.5
2C)
1.27
0.635
28.6
5.15
28.7,
75.4
45.1
6.51
-
2D)
2.54
0.635
13.7
5.34
28.3,
74.3
44.7
6.05
-
3A)
0.254
1.27
40
3.71
35.8,
98.5
56
4.33
45.7,
127.7
3B)
0.635
1.27
40
5.0
32.5,
86.2
49.9
6.29
35,
93.8
3C)
1.27
1.27
29
5.6
31.3,
82.3
48.2
6.65
-
3D)
2.54
1.27
14.3
5.83
30.8,
81
47.7
6.35
-
h3
h4
FREQ
(GHz)
eeff(min)
@ 1GHz
eeff (min)
@ f
Zpy (min),
Zpv (m ax)
(Q)
@40GHz
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
134
cross-sectional vector electric field plot at 40GHz for the lower loaded example. A small
variation exists between the phase constants for the CBCPW modes in Fig. 48(a).
At
higher frequencies, the CBCPW mode has a larger effective dielectric constant and more of
the electromagnetic energy will be concentrated in the higher dielectric substrate (h4) as
illustrated in Fig. 48(b) as compared to Fig. 33(a). The term dispersion relates the rate o f
change of the propagation characteristics o f transmission lines with frequency and is
primarily applied to the effective dielectric constant and the characteristic impedance.
Smaller dispersion for a waveguide corresponds to a more straightforward circuit analysis
without significant design compromises.
The characteristic impedances for the
multi-layered configurations must also be analyzed in detail. An impedance verification
example is first presented in Fig. 49 comparing the three SDM impedance definitions from
this work and (2.64a), (2.64b), and (2.64c) with the results o f [57],
Recall that for
non-TEM structures such as CBCPW, the characteristic impedance is not unique and the
power-voltage, voltage-current,
and the power-current calculations are invoked.
Approximately a IQ difference occurs between the impedances in Fig. 49.
The
impedances of example 2B) from Table XV for CASES 20-23 and example 3B) from
Table XVI for CASES 24-27 are pictured in Fig. 50 for the three impedance definitions.
Similar behavior for the two multi-layered configurations is apparent.
A 30-90H
impedance range exists for these cases. The variation in the impedance responses is similar
to those described in [57] and Fig. 49. The larger cross-sectional circuit area in CBCPW
results in a more dispersive impedance behavior and this is the reason for the limitation o f
(S+2W)<2.032mm for these examples.
The impedance curves demonstrate that the
voltage-current has the smallest dispersive effects among the definitions for CBCPW
MICs.
The work presented in this section has shown that the upper and lower dielectric loaded
multi-layered CBCPW MICs are valid transmission lines with wide bandwidths, sufficient
impedance range and effective dielectric constant values, and tolerable dispersive effects.
These structures are produced with no limitation on the lateral width and only a small
restriction on the cross-sectional circuit area and translates to a very flexible transmission
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
135
2.50
2.25
Q
.
2.00
1.75
X
CASE 20
CASE 21
TM0.1 CASE 20 SDM
TMo.1 IDEAL RW
1.50
FREQ (GHz)
(a)
0.9
0.6
0.3
0.0
E -0 .3
0.6
-
-0.9
-
1.2
—1.51
- 1 .5 - 1 .2 - 0 .9 - 0 .6 - 0 .3 0.0 0.3 0.6 0.9
y
(mm)
1.2
1.5
(b)
Fig. 48. Bounded mode behavior for lower loaded multi-layered dominant CBCPW mode,
(a) Normalized dispersion curves for CASES 20 and 21 of Table XV and the first
waveguide mode calculated by the SDM and the ideal rectangular waveguide
analysis, (b) Cross-sectional vector electric field plot (Ex and E ) of the
dominant CBCPW mode for CASE 20 at 40GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IMPEDANCE
( 0)
136
50
49
48
ZPv
Z vi
■- - Z p t
o
Zpy
v
Zvi
x
Zpt
Ref. 157]
Ref. (57]
Ref. [57]
p re se n te d work
p re se n te d work
p re s e n te d work
47
46
CHARACTERISTIC
45
44
43
42
40
20
FREQ ( G H z )
Fig. 49. Characteristic impedance verification with [57] for an air-suspended multi-layered
CBCPW for power-voltage, voltage-current, and power-current definitions.
Refer to Fig. 13 for dimensions with srl= l, er4=13, er5= l and h1=hs=5mm>
h4=lmm, h2=h3=h6=0 and S=lmm, W=0.4mm and 2A=25.4mm
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
137
LU
o
z
<
90
OL'JJ
CASE 23
.............
.................
80
Q 70
LU
0. 60
2
O 50
Ltn 40
o:
lu 30
HO 20
<
oc
< 10
X
o 0
CASE 21
CASE 20
CASE 22
Z pv
Zvi
Z pi
B
12
16
20
24
28
32
36
40
FREQ (GHz)
(a)
Q 80
CASE 27
CASE 25
CASE 24
a:
LU 30
CASE 26
■
■
■
■
12
16
20
■
■
24
■
■
28
FREQ (GHz)
(b)
Fig. 50. Characteristic impedances of the multi-layered CBCPW MICs for the three
definitions, (a) Lower loaded example 2B) of Table XV. (b) Upper loaded
example 3B) of Table XVI.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
138
line platform.
C. Additional Configuration Considerations for Multi-Layered CBCPW MICs
The multi-layered CBCPW has been shown in the previous chapters to be a useful
transmission line for MICs with sufficient bandwidth, impedance range, tolerable
dispersion effects, minimum structural complexity, and minimum restrictions on the lateral
width and circuit cross-sectional area.
This section will include additional design
information including the lateral width requirements to cutoff the waveguide modes, a
closed form procedure to determine the dielectric constant and thickness for a lower
loaded structure, dielectric uncertainty analysis on the propagation constants for the
CBCPW modes, (S+2W) values to minimize dispersion effects for the multi-layered cases,
the effects of air gaps between the substrates, and parameter values necessary to interface
to a multi-layered microstrip feed circuit [15], [28], These tools provide the designer the
necessary information to design CBCPW microwave circuits.
The channelized CBCPW configuration [17] is one method for extending the usable
bandwidth by reducing the lateral width dimension and increasing the cutoff frequencies of
the rectangular waveguide modes. Equations (4.1), (E.6), and (E.7) in Appendix E are
utilized to determine the parameter dimensions for this method. Table XVII includes the
cutoff frequency o f the dominant TMj,, mode for the multi-layered configuration in Fig. 13
and assumes that (h4+h5)<2A and h6 =0. For the lateral width (2A) at 10mm, marginal
bandwidths exist for CASES 29 and 31. By operating at frequencies where the rectangular
waveguide modes are cutoff (nonpropagating), a single mode o f propagation for the
dominant CBCPW mode occurs and the mode coupling effects illustrated in Chapters IV
and V are avoided. However, the low dielectric constant utilized in CASE 29 will increase
the wavelength and the circuit area and CASE 31 with an air suspended configuration is
difficult to realize as a practical waveguide solution.
A closed form procedure for determining the dielectric constant and thickness for a
lower loaded multi-layered CBCPW configuration is now presented using Fig. 13 with
h6=0. The analysis is a low frequency approximation requiring (h4+h5) « X4 where
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
139
TABLE XVII
Cutoff frequencies for the TMg , rectangular waveguide mode as a
function of the lateral width for various CBCPW configurations.
Fig. 13 with h6=0 is referenced for these lower loaded multi-layered
examples. *arbitrary height requires h4<2A.
Cutoff freq (GHz)
5
lateral width 2A (mm)
20
CASE 28
sr4=10.2, h4=arbitrary*
h5=0
9.4
4.7
2.35
CASE 29
sr4 =2.2, h4=arbitrary*
h5=0
20.2
10.1
5.05
CASE 30
sr4=10.2, h4=0.635mm
er5 =2.2, h5=0.635mm
15.46
7.85
3.94
CASE 31
8r4=10.2, h4=0.635mm
sr5=1.0, h5=l.905mm
24.6
13.02
6.58
^
and f is the frequency of operation.
20GHz.
10
= T
^
(mm)
( 51)
With (h4+h5)=X4/5, the approximation is valid to
The dispersive effects associated with the dominant CBCPW mode can be
reduced by stipulating (S+2W)<h4/2 and this requirement will be demonstrated later in this
section.
With this assumption the phase constant for the CBCPW mode can be
approximated as
(p/*o)2 * (er4 + 0 /2
(5.2)
with most of the electromagnetic energy existing within the air (srI=l, h,=5mm in Fig. 13)
and the h4 dielectric region.
The low frequency approximations are invoked in the
determination of the propagation constant for the TMq , waveguide mode from [9]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where x=h5/(h4+h5). The dielectric constant and thickness o f the lower layer is determined
to avoid the mode coupling effects between the dominant CBCPW mode and the first
rectangular waveguide mode. The mode coupling effects are the strongest between two
modes at the phase match condition and this situation is realized by equating (5.2) and
(5.3). Table XVIII summarizes the lower dielectric constant values for various CBCPW
examples including GaAs (er4=13) and a high temperature superconducting dielectric
YBa2Cu30 7^ (YBCO) with sr4=26 [58] utilizing the above procedure.
TABLE XVIII
Specification of the dielectric constant for the lower loaded multi-layered
CBCPW structures using the closed form approximation procedure at
f=20GHz. Fig. 13 is referenced with srl =1, hj=5mm, h2=h3=h6=0 and
2A=25.4mm.
£r5
CASE 32
sr4 =10.2, h4=0.47mm
h5=0.47mm
(S+2W)=0.235mm
<3.86
CASE 33
er4 =2.2, h4=lmm
h5=lmm
< 1.26
CASE 34
er4 =13, h4=0.278mm
h5=0.556mm
<5.69
CASE 35
er4 =26, h4=0.294mm
h5=0.294mm
<9.12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
141
An investigation to determine the values of (S+2W) to minimize dispersion effects in
multi-layered CBCPW MICs is presented in Fig. 51 for the lower loaded example 2B) of
Table XV. From the figure, the following condition on the cross-sectional circuit area
should be made to minimize the influence o f the second or lower layer to maintain the
non-dispersive characteristic o f the dominant CBCPW mode to 40GHz as
h 4/2 < (S+2W) < h4 .
(5.4)
As expected, reducing the cross-section decreases the variation in the phase constant
versus frequency.
The reason for a low dispersive transmission line
is a simpler circuit
design for wide band applications.
The effects of the uncertainties of the dielectric substrate parameter dimensions on the
upper usable frequency are illustrated. The uncertainties in the substrate thickness and
relative dielectric constant were presented in Chapter III for calculating the sensitivity of
the leakage criticalfrequency and are applied to the CBCPW with lateral sidewalls. The
variations again are ±0.051mm for the 1.27mm using the 6010 board (er=10.8 Duroid™
substrate) and ±0.0254mm for 5870 (er=2.33 Duroid™ substrate). The relative dielectric
constant variations are ±0.25 for 6010 and ±0.02 for 5870.
Table XIX depicts the
uncertainties for both upper and lower loaded configurations for examples 2C) from Tables
XV and XVI for CASES 21 and 25, respectively. As indicated in the table, a 4GHz range
for the upper usable frequency is present for the lower loaded structure with the substrate
variations which is significant and this sensitivity analysis should be implemented for
bandwidth calculations.
The effects on the propagation characteristics o f the CBCPW mode due to the possible
presence of an air gap between the substrates of the multi-layered configurations requires
evaluation. This air gap configuration is used in the sensitivity analysis o f the structure.
The results are presented in Fig. 52 for both upper and lower loaded waveguides for
examples 2B) from Table XV and 3B) from Table XVI. The air gap exists between the
substrates (h5) for the lower loaded case and between the top substrate and the circuit
conductor ground planes (h3) for the upper loaded case. As indicated in the figure, the
lower loaded configuration is not affected by the air gap while the upper loaded case is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
142
2 .5
CASE
•• CASE
- - CASE
— CASE
36
37
38
39
2.4
2.3
2.2
FREQ (G H z )
Fig. 51. Dispersion effect analysis for the determination of (S+2W) of the dominant
CBCPW mode for the lower loaded example 2B) of Table XV. Refer to
Fig. 13 with
srl= l, sr4=T0.2, sr5 =2.2
and h[=5mm, h4=0.635mm,
hs=0.635mm, h2=h3=h6=0 and 2A=25.4mm. CASE 36 is S=2W=0.635mm,
CASE 37 is S=2W=0.381mm, CASE 38 is S=2W=0.1905mm, and
CASE 39 is S=2W=0.0953mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
143
TABLE XIX
Dielectric uncertainty effects on the upper usable frequency o f multi-layered
CBCPW for S=W=0.635mm and 2A=25.4mm. CASES 40-42 are the lower
loaded configuration of example 2C) from Table XV and CASES 43-45 are
the upper loaded waveguide example 2C) from Table XVI.
Upper usable ffeq (GHz)
CASE 40
er4 =10.45, h4=l.321mm
sr5 =2.18, hs=0.610mm
19.6
CASE 41
sr4 =10.20, h4=1.27mm
sr5 =2.2,
h5=0.635mm
22
CASE 42
sr4 =9.95,
er5 =2.22,
significantly modified.
23.66
h4= 1.219mm
h5=0.660mm
CASE 43
er3 =10.45, h3=l.321mm
sr4 =2.18, h4=0.610mm
27.9
CASE 44
sr3 =10.20, h3=l ,27mm
er4 =2.2,
h4=0.635mm
29.2
CASE 45
sr3 =9.95,
er4 =2.22,
30.3
h3= 1.219mm
h4=0.660mm
This is explained from transmission line theory from the
discontinuity o f the dielectric constants o f the substrates as presented to the dominant
mode. In CASE 46 the air layer is a close match to the lower substrate layer (8^=2.2)
while for CASE 47 the air dielectric is a large step discontinuity to the upper substrate
(er3=10.2) and the field characteristics are greatly affected. As expected, increasing the air
gap thickness reduces the phase constant of the dominant mode since more of the fields
reside within this dielectric region and the phase constant converges to l<p/£0<1.48. In
the manufacturing of multi-layered CBCPW MICs, a bonding film would be applied
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
144
2.50
fi/k o
2.25
2.00
— CASE 45
- - CASE 47
1.75
0.04
0.08
AIR GAP T H I C K N E S S
0.12
(m m )
0.16
Fig. 52. Effects of an air gap between the substrates for the multi-layered CBCPW MICs
with £rl= l and h[=5mm and S=2W=0.635mm and 2A=25.4mm at f=20GHz.
Fig. 13 is referenced for the dimensions. CASE 46 is the lower loaded
example with sr4=l0.2, er5=l, srt=2.2 and h4=0.635mm, h5 is the air gap,
h6=0.635mm,
h2=h3=0. CASE 47 is the upper loaded example with er,= l0.2,
sr3=l, er4=2.2 and h2=0.635mm, h3 is the air gap, h4=1.27mm, h5=h6=0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
145
between applied between the substrates to strengthen the structure and eliminate the air
gap. A bonding film (model 3001) for microwave circuits is offered by Rogers Inc. for
Duroid™ boards with er=2.28, a high melting point, and low dissipation loss.
This
bonding material would provide a match between the substrates in the lower loaded
configuration presented here with respect to the microwave properties.
Rogers also
provides a ceramic-filled paste material for high dielectric boards and would be ideal for
the upper loaded multi-layered structure to eliminate the air gap and the effects on the
propagation constant of the dominant mode.
In some circumstances, the multi-layered CBCPW MIC may be interfaced to a
microstrip transmission line for a feed network or other circuit applications.
The
microstrip line is referenced from Fig. 14(a) and the strip width is determined by the SDM
to provide an impedance match to the CBCPW transmission line. For the lower loaded
example of 2B) from Table XV, a width of S=2.286mm is required for a 50fi
characteristic impedance line which is wider than the maximum cross-section restriction
from this table of (S+2W)<2.032mm.
An approximate 50Q line in the multi-layered
CBCPW corresponds to a strip to slot width of 2:1. The microstrip width would be
reduced approximately in half to match the center strip width of the CBCPW line and this
modification increases the microstrip impedance. The effective dielectric constant for the
dominant quasi-TEM microstrip mode is 3.62 and 5.66 at 1 and 40GHz, respectively. For
the upper loaded waveguide of example 3B) from Table XVI, a strip width of 3mm is
needed for a 50Q characteristic impedance. This strip dimension is again significantly
above the maximum cross-section width limitation for the CBCPW. Also, the microstrip
mode experiences a large dispersive effect with the effective dielectric constant ranging
from 2.67 to 5.25 at 1 and 40GHz, respectively. In summary, an average match condition
between the multi-layered microstrip and CBCPW for the lower loaded case exists with
respect to VSWR (voltage standing wave ratio) and the effective dielectric constants, but a
poor match is presented in the upper loaded configuration.
A design procedure for multi-layered CBCPW MICs would utilize the summary results
from Tables XV and XVI for both lower and upper dielectric loaded structures with
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146
s=10.2 and er=2.2.
The effective dielectric constant values and the bandwidths are
obtained for various substrates thickness.
The characteristic impedances can be
determined from Fig. 50 for low, middle, high, and 50Q cases.
For other substrates,
(5.1)-(5.3) are implemented with the restrictions (h4+h5)= \4/5 and (S+2W)<h4/2 to
calculate the dielectric parameters up to 20GHz. No closed form relationships exist for the
seff or the characteristic impedance in multi-layered CBCPW.
D, Minimizing the Mode Coupling Effects in Single-Layer CBCPW MICs
Wide bandwidth applications utilizing CBCPW have been successfully developed in
[5-7] without the use of a multi-layered configuration. The single-layer waveguide would
be attractive compared to the multi-layered CBCPW with respect to manufacture
complexity and expense. The question is how the CBCPW circuits (MMICs) can function
properly with the coupling effects present. A single-layer CBCPW unconditionally leaks
energy to the TEM parallel plate mode which translates to mode coupling effects existing
at lower frequencies for electrically wide two-dimensional waveguides.
For CBCPW
MMICs the line/slot widths can easily be made much smaller than the substrate thickness.
This parameter guideline reduces the leakage constant for the one-dimensional structure
[12], [32], decreases the coupling to the rectangular waveguide modes in finite-width
structures [28], and reduces the coupling to the cavity modes in finite-length waveguides
[35], The procedure presented here will also relate the mode coupling in CBCPW with
lateral sidewalls (realistic two-dimensional structure) to the leakage rate or constant of an
infinite-width waveguide (nonphysical transmission line) [28], The method calculates the
leakage constant and plots the effects o f mode coupling on the dominant CBCPW mode to
40GHz.
Parameters recommendations for single-layer CBCPW MICs in terms of the
cross-sectional circuit area, substrate thickness, dielectric constant, and operating
frequency to significantly reduce the mode coupling effects will be described with
supported examples.
This work will also demonstrate that the one-dimensional
(nonphysical structure) analysis can be an important tool in the design of single-layer
CBCPW.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
The procedure follows the one and two dimensional SDM analysis for calculating the
leakage rate and mode coupling effects from Chapters III and IV, respectively. A total of
nine basis functions (five for E
and four for E,n) are employed in the analysis since the
fields are tightly coupled to the slots for small cross sections (S+2W) and the slot fields
strongly interact.
The CBCPW is assumed to be symmetric, lossless, single-layer,
infinite-length and follows the configurations depicted in Fig. 3(a) and 3(b) and an ideal
magnetic wall is placed at y=0 plane to excite the dominant odd mode. For the infinite
width waveguide, a zero cutoff TEM parallel plate mode exists. The dominant CBCPW
mode leaks energy into this PPM at all frequencies for any dielectric constant and substrate
thickness and is no longer bound to the slot area. The CBCPW mode also has a complex
propagation constant. For CBCPW with lateral sidewalls, rectangular waveguide modes
with finite cutoff frequencies are supported. Leakage does not occur but mode coupling
can cause the dominant mode field to spread out across the waveguide instead of being
confined to the slot region and the dominant propagation constant is real.
The SDM
determines the degree of mode coupling by plotting the dominant mode field pattern.
The one-dimensional infinite-width analysis is first considered. The examination o f the
dielectric constant to reduce the leakage rate (a /k0 ) is investigated. Fig. 53 illustrates this
calculation at 40GHz with h=1.27mm for CASE 48 and shows that a lower er is desired to
reduce the leakage constant. The leakage for er=10 is ten times greater than that for er=2.
The rate varies approximately as er
at lower dielectric constant values and becomes more
a linear function with higher dielectric constants.
The desire for a smaller guide
wavelength to decrease the circuit size dictates a larger dielectric constant and an er=10.2
(approximately corresponding to Duroid™ 6010) will be utilized throughout this section.
The leakage rate as a function of the frequency and the cross-sectional circuit area
is shown in Fig. 54. The (S+2W) parameter is decreased in half with h=0.635mm in Fig.
54(a) for the three plot curves (CASES 49-51) while the substrate thickness is successively
increased by a factor of two in Fig. 54(b) at f=40GHz (CASES 52-54). The larger cross
section and higher frequency of operation enhance the leakage. The leakage rate varies
linearly with frequency and approximately as (S+2W)2. A 2:1 strip to slot ratio (S:W)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
148
0 .0 5 0
0.040
CASE 48
0.030
Q
0.020
0.010
0.000
14
Fig. 53. Leakage rate versus relative dielectric constant for single-layer CBCPW.
Fig. 3(a) is referenced here with S=2W=0.3175mm and h=1.27mm and
f=40GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
149
o .i iQr
i
i
x
i
CASE
CASE
CASE
CASE
i
—
49
50
51
50 Eqn. (5.5)
o 0.050
0.050
0.040
0.030
0.020
0.010
0.000
10
15
20
25
30
FREQUENCY (GHz)
(a)
0.150
0.135
0.120 •
0.105 •
Q
------------.........
x
CASE 52
CASE 53
CASE 54
CASE 53 Eqn. (5.5)
/
/
0.090
X
/
0.075
iS
/
S
s
s
0.060
s
/
y
0.045
/
0.030
/
.•••*****
.............
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
S+2W (m m )
(b)
Fig. 54. Normalized leakage rate curves for single-layer CBCPW. (a) Plots versus
frequency with e=10.2 and h=0.635mm with CASE 49 S=2W=0.3175mm,
CASE 50 S=2W=0.15875mm, and CASE 51 S=2W=0.079mm. (b) Curves
versus cross-sectional circuit area with sr= 10.2 and f=40GHz with CASE
52 h=0.3175mm, CASE 53 h=0.635mm, and CASE 54 h=l.27mm.
Fig. 3(a) is referenced for the parameters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
approximately corresponds to a 50H characteristic impedance line.
From Fig. 54(b),
keeping the cross-sectional circuit area small compared to the dielectric thickness and
increasing the substrate height (h) reduces the leakage rate as observed in [12] and [32].
The leakage varies approximately inversely with the substrate thickness as indicated in Fig.
54(b). This inverse relationship between the leakage and the substrate height holds for
CASES 53 and 54 however, CASE 52 does not exactly follow this pattern at such large
attenuation rates. For h=2.54mm, leakage to the TMt and TE, modes (degenerate pair)
exists above 29GHz which will increase the rate.
For h=5.08mm, leakage to the
TM2 and TE2 modes (degenerate pair) also occurs above 30GHz.
For (S+2W) small compared to the wavelength, a closed form relation for the
normalized leakage rate of the dominant CBCPW mode into the TEM PPM has been
derived from [12] using the reciprocity theorem as
a . *Z (S+2 W f
k0 -
4h377>.j.
'
'
where Z is the characteristic impedance, seff is the effective dielectric constant, and Xg is
the guide wavelength.
The accuracy of (5.5) is shown in Fig. 54 and verifies the
relationships between the leakage rate and the cross-sectional circuit area, thickness,
relative dielectric constant, and frequency as determined from the SDM calculations. The
results from (5.5) overestimate the leakage on the order of 20%. As described in Fig. 54
and (5.5), decreasing (S+2W) reduces the leakage constant most dramatically among the
parameters o f the CBCPW. This complete analysis of the leakage rate of CBCPW in terms
of these waveguide dimensions and validation o f (5.5) with the full electromagnetic wave
SDM is an original contribution of this dissertation as part of [28],
The
finite-width,
single-layer
CBCPW
with lateral
sidewalls which
two-dimensional version of the infinite-width structure is analyzed.
is
the
This waveguide is
referenced from Fig. 3(b) with 2A=25.4mm and corresponds to a realistic MIC circuit
dimension. The field plots described here are for the dominant mode at x=-0.01mm. As
mentioned earlier, increasing the substrate thickness reduces the leakage rate but additional
PPMs must be considered. These PPMs now become rectangular waveguide modes. The
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
151
effects of these additional modes coupling to the dominant CBCPW mode are presented in
Fig. 55(a) for various substrate heights. The dominant rectangular waveguide mode is the
TM 0ft with h = 1,3,5,... ,00 . As indicated, the dominant mode spreads out across the
waveguide as the fields become a combination o f the dominant and waveguide modes.
Each curve is normalized to the maximum field component and the plots are for the
dominant mode outside the regions o f strong coupling with the waveguide modes, with
respect to frequency (see Chapter IV). The values o f (S+2W) for CASES 51, 55-57 are
modified so that (S+2W)=h/4 ratio is maintained for each plot to demonstrate the mode
coupling effects with the higher order modes for this example (TM AA and T E ^ for
m = \,2 and h= 1,3,5,...) .
Recall that these higher order TM and TE modes are
degenerate modes. For a constant (S+2W)/h value, the leakage and mode coupling effects
should be identical for each case. With h=2.54mm, the mode coupling effects occur with
the TM W mode and with h=5.08mm the coupling effects are present with the TM, h and
TM 2,h modes. The following relationship is invoked to determine the cutoff frequency of
the next higher order waveguide mode as a function of the substrate height as
2
2
(& )
+ ©
(5'6)
As the lateral width (2A) is narrowed, the degree o f mode coupling is also decreased as
fewer waveguide modes couple with the dominant mode. For 40GHz operation,
h<1.17mm is required to cutoff the T M ,, and T E , , modes. Fig. 55(b) demonstrates the
cross section necessary to minimize the mode coupling for h=1.27mm.
The dominant
mode is plotted around 40GHz between the mode coupling regions before or after the
TMq ,3 mode (the curves are plotted at 36, 37, 38, and 39GHz, respectively). A maximum
cross-sectional circuit area between h/4 and h/8 is necessary to significantly reduce the
mode coupling for this case. It is interesting to note that the mode coupling effects are
lessened as (S+2W) is decreased in CASES 55, 58-60 in Fig. 55(b). The dispersion curves
for CASES 58 and 60 are plotted in Fig. 56(a) and Fig. 56(b), respectively. Concentrating
only in the regions of strong coupling about 40GHz, the closer the dispersion curves
approach each other (dominant and waveguide modes), the smaller the coupling effects.
Strong mode coupling occurs for a large separation between the propagation curves and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
152
1.0i
CASE 51
CASE 55
CASE 56
CASE 57
0.8
0.6
0.4
O
0.2
Q. 0.0
2
0.2
< -H
■**-0.4
-
0.6
-
0.8
- 1.0 "
y (m m )
(a)
1.0i
CASE 58
CASE 59
CASE 55
CASE 60
0.8
0.6
Ui 0.4
yi
r.
0.2
0 . 0.0
2
< - 0.2
-
0.6
-
0.8
-W
y (m m )
(b)
Fig. 55. Normalized electric field plots for dominant CBCPW mode demonstrating the
mode coupling effects for a single-layer structure at x=-0.01mm. Refer to Fig.
3(a) for dimensions with er =10.2 and 2A=25.4mm and f approximately
40GHz. (a) Results as substrate height (h) is decreased with (S+2W)=h/4 for
each case. CASE 51 h=0.635mm, CASE 55 h=1.27mm, CASE 56 h=2.54mm,
and CASE 57 h=5.08mm. (b) Plots as function of the cross section with
h= 1.27mm. CASE 58 S=2W=0.635mm, CASE 59 S=2W=0.3175mm, CASE
55 S=2W=0.1588mm, and CASE 60 S=2W=0.0794mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
153
2.8
2.7
TMo.13
-
2.6
o
2.5
CBCPW ■
r 03CPW
•Si
2.4
o
-V
2.3
-
2.2
TMo,17 ‘
15 36
37
38
39
40
41
42
43
44
4!
FREQ (GHz)
(a)
1 *'1
2.8
1
1
I
1
2.7 . TMo.ii ^
o
*
TMo.13
2.6
TMo.fi^,
2.5
2.4
.
: bcpw
CBCPW
2.3
2.2
2.1
2.
•
36
37
38
TM0,17
yS
‘Mo.fi
39
40
41
42
43
44
45
FREQ (GHz)
(b)
Fig. 56. Normalized dispersion curves for single-layer CBCPW. With sr =10.2 and
h=1.27mm and 2A=25.4mm and refer to Fig. 3(b) for parameters, (a) CASE
58 with S=2W=0.635mm. (b) CASE 60 with S=2W=0.0794mm. The
crossed points are the CBCPW mode in each case.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
154
and this is verified from Fig. 55(b).
Hence, the degree o f mode coupling can be
qualitatively estimated from the dispersion curves.
The frequency for strong coupling
between the CBCPW mode and the TMq 13 mode encompasses a wide band from
3 7-43 GHz in Fig. 56(a) while for CASE 60 this coupling range is concentrated only at
36GHz. The dispersion results from Fig. 56(a) are similar to the responses and mode
coupling effects described in Chapter IV.
To determine the mode coupling versus
frequency, CASE 50 is presented in Fig. 57. This example shows that significant mode
coupling commences between 10 and 15GHz.
E. Design Procedures for Single-Layer CBCPW
On the basis of the examples presented in the previous section and several other
simulations, an empirical estimate relating strong mode coupling in CBCPW with lateral
sidewalls to a high leakage rate in the infinite-width structure can be expressed.
An
estimation of the leakage rate will be suggested with supported cases from the previous
section to minimize the leakage and mode coupling effects. The cross-sectional circuit
area for a given substrate thickness, dielectric constant, and frequency can be
approximated from the closed form relationship o f (5.5). A complete design procedure
will be described to determine the parameters for the single-layer CBCPW for successful
operation with minimal mode coupling effects, without requiring the user to possess a one
and two dimensional SDM algorithm.
A normalized leakage rate for (aJkQ) of approximately less than 0.007 or
a<0.155dB/A,g is recommended for minimum leakage and mode coupling effects. This
estimated value was obtained by observing the mode coupling effects as a function of
(S+2W)/h in Fig. 55(b) which indicated a significant reduction in the field spreading in
CASES 59 and 55. These examples correspond to h/4<(S+2W)<h/8 and the average value
o f (S+2W)=h/6 was selected and then mapped onto the leakage curve plot for CASE 54 of
Fig. 54(b) with an expanded log scale. The mode coupling as a function o f frequency for
CASE 50 was identified from Fig. 57
translated to the leakage graph o f
to occur between 10 and 15GHz which was
Fig. 54(a) (again using a log scale).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
These
155
1.0
0.8
FREQ-10GHz
FREQ-15GHz
FREQ=20GHz
CASE 50
E x
AM PLITUDE
0.6
0 .4
0.2
0.0
0.2
-
- 0 .4
-
0.6
-
0.8
-
1.0
-1 4 -1 2 -1 0 -8 -6 -4 -2
y
0 2
(m m )
4
6
8
10 12 14
Fig. 57. Normalized electric field plots for the dominant CBCPW mode demonstrating the
mode coupling effects as a function of frequency for a single-layer structure
at x=-0.01mm. CASE 50 with er=10.2 and h=0.635mm and S=2W=0.1588mm
and 2A=25.4mm. Refer to Fig. 3(b) for dimensions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
156
examples support the estimate of (S+2W)<^.g/16 to minimize the leakage and mode
coupling effects in a single-layer CBCPW. This design parameter for the leakage rate is
intended for dielectric constants of er =10 which could apply to alumina, Duroid™ 6010,
and GaAs substrates. The parameter specifications of [12] suggested (S+2W)<h/4<X.g /20
to reduce the leakage effects for useful CBCPW operation in GaAs circuits, however no
examples were given to substantiate this guideline as compared to the presented material in
the last two sections o f this work.
The leakage rate of (5.5) overestimates the value derived by the SDM approximately
15-20% so a normalized constant of 0.008 is suggested. For the example o f Fig. 55(b)
with h=1.27mm at f=40GHz, (5.5) calculates a value of h/(S+2W)=5.21. CASE 50 with
h=0.635mm and h/(S+2W)=2 would predict an upper frequency of operation with
minimum coupling effects to 11.8GHz. Both o f these results are confirmed by the field
spreading plots of the dominant CBCPW mode. A characteristic impedance of 50Q and
Eeff
=6 were assumed in the above calculations.
An iterative design procedure to determine the dimensions for single-layer CBCPW
with reduced mode coupling based upon the above work would be as follows:
1) Select a dielectric constant (preferably a high value around 7<sr <13);
2) Generate a set of design curves for the characteristic impedance and the effective
dielectric constant using the closed form (quasi-static) relation from [50] for
various (S/W) and (S+2W)/h values;
3) Determine the upper frequency of operation for the application;
4) Use the cutoff relation for the higher order waveguide modes (5.6) to find the upper
limit for the substrate thickness and then choose a height;
5) Select Z;
6) Select an initial value of (S+2W) and obtain a value for eefr from step 2;
7) Utilize (5.5) with (a /k0 )=0.008 to determine (S+2W);
8) Compare (S+2W) with that estimated from step 6 and verify the characteristic
impedance. If both parameters are sufficiently close, goto step 9. If not,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
157
determine the effective dielectric constant and Z from the design table
of step 2 for the new (S+2W) value and goto step 7;
9) Select a value for W based upon the circuit etching resolution available and then
determine S.
The above procedure will probably require a few iterations to converge but can be
calculated manually. The effective dielectric constant value obtained from the quasi-static
method from [50] can be implemented to higher frequencies. This statement is valid
because the cross sections utilized in these applications are smaller than the dielectric
thickness and the fields will be tightly coupled to the slot region and dispersion effects
should be small. Using the above procedure for h=0.635mm with sr=10.2 at f=40GHz, a
(S+2W)=0.1588mm was predicted. This line/slot restriction demonstrates the necessity for
the use o f multi-layered CBCPW to reduce the mode coupling effects in wide
ffequency-band MIC applications since this resolution is almost five times smaller than the
available dimensions in our etching facility.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
158
CHAPTER VI
THREE-DIMENSIONAL ANALYSIS OF MULTI-LAYERED
CBCPW INCLUDING CAVITY EFFECTS
In an actual circuit implementation with input and output port (finite width and length),
a three-dimensional structure is produced. This situation is pictured in Fig. 8(b) for the
waveguide formed with the coaxial connector blocks. The rectangular waveguide formed
in two-dimensions for CBCPW with lateral sidewalls becomes a rectangular cavity in
three-dimensions with longitudinal PEC shorting walls at z=0 and z=2B as viewed in Fig.
8(b). For a packaged CBCPW, a resonant cavity is produced as a substructure of the
waveguide. A cavity stores electric and magnetic energy in the volume and the energy is
coupled to an external circuit via a probe or aperture. Resonator circuits are very useful
for some applications such as antennas, oscillators, and filter networks but not in the
transmission properties of CBCPW. Coupling to the cavity modes should be avoided for
the CBCPW transmission line application. For electrically large configurations, resonant
frequencies fall within the frequency range of interest (dc-40GHz in this dissertation). The
resonances occur at discrete frequencies (the fields within the cavity exist only at these
frequencies) and will limit the bandwidth of operation. If the resonances are present within
the frequency range (depending on the dimensional parameters of the waveguide), two
basic outcomes are possible. For a through transmission line as shown for CASE Q of Fig.
42, sharp insertion losses are present at several o f the resonant frequencies even though the
dominant CBCPW mode is bound to the slots. A portion o f the power from the CBCPW
circuit couples to the cavity modes instead of propagating the energy to the output port.
The amount o f the insertion loss depends on the coupling between the CBCPW circuit and
the cavity modes (coupling coefficient or factor) and the Q (quality factor) o f the resonant
Part of the data reported in this chapter is © 1994 EEEE and submitted to IEEE for
publication. Reprinted, with permission, from 1994 IEEE Microwave Theory cmd
Techniques International Microwave Symposium Digest, pp. 1697-1700 (see References
[15], [28] and Appendix G).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
159
mode.
The Q describes the efficiency o f the energy storage within the cavity for a
particular mode.
desirable.
For the typical resonator applications discussed above, a high Q is
The second outcome is the coupling/interference effects between isolated
circuits within a system via the substrate resonances for a CBCPW and disturbing the
intended operation. This problem includes coupling between the circuits as a function of
position on the substrate [35], creating locations with enhanced coupling with the cavity.
The residual resonances shown in CASE Q o f Fig. 42 for the multi-layered CBCPW
must be suppressed as the 2-4dB insertion losses are unacceptable for a wide-band
transmission line. The resonances o f this example are identified with possible rectangular
cavity modes which are calculated with an ideal analysis and described in the next section
and are listed in Table XX. This procedure was performed in Table XIV with Fig. 30 for a
single-layer CBCPW. A close correlation between the measured and modeled resonant
frequencies is evident (the small differences between the frequencies are attributed to the
uncertainties in the relative dielectric constants) and not all o f the possible resonant modes
appear in the data.
TABLE XX
Measured and calculated resonant cavity frequencies for multi-layered
CBCPW o f CASE Q from Fig. 42 for the TM0 ,^ modes.
fr (GHz)
measured
calculated
TM o.1.2
5.8
5.6
TM o.1.3
7.5
7.2
TM o.,4
9.3
9
TH ...1,
22.5
21.9
TM o.1.13
TM o.1.15
25.9
25.2
29
28.4
TM o.1.17
31.8
31.4
TM o.1.19
34.3
34.2
TM o.1.21
36.8
36.7
mode
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
160
energy is coupled to the cavity at the coaxial feeds and by the CBCPW mode as it
propagates along the transmission line length. For a bound CBCPW mode, the coupling
from the above possible sources to the cavity should be very small since the fields of the
dominant mode are localized in the slot areas and the cavity mode field representations
extend across the waveguide width and length. Therefore, the low loss dielectrics for this
case produce a sufficiently high Q to draw over half o f the power into the cavity (for
resonances greater than 3dB). CASE R of Fig. 42 showed that the resonance effects were
also a function o f transmission line length and this observation would indicate that the
majority of the energy coupled to the cavity modes was from the dominant CBCPW mode
and not the coaxial feeds. The use o f a damping (lossy) material to reduce the residual
resonances within CBCPW by decreasing the substrate cavity Qs is necessary and this
configuration was demonstrated successfully in CASE S of Fig. 42. The absorber was
placed on the perimeter of the waveguide (within the test fixture blocks outside the coaxial
connectors in CASE S) which may not be feasible in some applications and an alternative
layered (planar) waveguide is suggested. This multi-layered damped waveguide is detailed
in Fig. 58 as the h6 substrate for a lower dielectric loaded CBCPW and as the h, layer for
an upper loaded circuit. A lower cavity is formed below the ground planes encompassing
layers h4, h5, and h6 and an upper cavity is generated above the ground planes between
substrates h3, h2, and h,.
The damping planar layers are represented by complex
permittivities (erI, srt) and complex permeabilites (|a.rI, pr6).
Fig. 59 illustrates an
implementation o f the damping material on the perimeter of a circuit.
With this
configuration, connecting the ground planes to the conductor-backing would probably
require shorting bars on the sides as well as input and output ports of the CBCPW. The
results from the measured data o f the single-layer CBCPW of Fig. 30 depicted resonances
(sharp insertion losses) on the order o f 10-30dB which can be explained due to the mode
coupling effects with the rectangular waveguide modes. The dominant mode field pattern
resembles rectangular waveguide modes and efficiently couples to the rectangular cavity
modes. The coupling coefficients from the CBCPW circuit to the cavity modes are not
addressed in this work.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
161
X
£ rl = 5 r \ ~ l € rl
Mrl = ^ r1 ~ J ^ r l
Sf2
I , 6 uL, t
- V
da mpi ng
material
/ / i .
£ r4
da mpi ng
material
£ r6 = £ ’r 6 ~ J E r€
Mr6 = V r S ~ 3 ^ r t
Fig. 58. Cross-section of the multi-layered CBCPW with damping material for the upper
and lower resonant cavities.
The complex permittivities and complex
permeabilities for the lossy layers are shown.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
162
£n = l
d a m p in g
m a te ria l
y
(a)
y
d a m p in g
m a te r ia l
m
(b)
Fig. 59. Application of the damping material with a thickness t on the perimeter of the
symmetric lower dielectric loaded CBCPW to reduce the cavity Qs . (a) Side
view, (b) Top view.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
The presence of the damping (lossy) material in CBCPW is proposed to reduce the
cavity Qs without significantly affecting the propagation characteristics of the CBCPW
circuits. The layered damping configuration can be easily modeled by the multi-layered
SDM solving for the propagation losses in the two-dimensional case and the resonant
frequencies and the Qs in the three-dimensional structure.
In particular, effects of the
damping material on a half-wavelength CBCPW resonator and the cavity modes are
investigated. The inclusion of a CBCPW resonator in the SDM analysis is an original
contribution from [15] and [28] as part of this dissertation.
This chapter will include a basic overview of the idealized CBCPW cavity resonator
and introduce the damping material utilized in this chapter (doped silicon and microwave
absorber). The electrical properties o f the absorber material used in the experimental data
o f this dissertation (see Fig. 42) were not known so the characteristics of a microwave
absorber example from [59] and doped silicon o f [60] were implemented in the SDM
analysis. Doped silicon is ideally suited within a MMIC application but is included in the
analysis o f the MICs here as a design demonstration.
The extension of the SDM to
three-dimensions for the investigation o f the CBCPW resonator is developed.
The
three-dimensional SDM outputs are verified with a microstrip example and additional
results are presented. Design examples with the damping materials are described for both
upper and lower loaded multi-layered CBCPW configurations. Tradeoffs to maximize the
CBCPW resonator circuit Q and minimize the cavity mode Qs are realized with the lossy
layer thickness, doping, and dielectric heights. This methodology is similar to [59] and
[60] which applied the above damping materials to packaged microstrip circuits and was
suggested in [35] for use in CBCPW circuits. The presentation o f the Q design curves for
CBCPW is also an original contribution from [15] and [28], Finally, experimental data is
included for a straight gap-coupled CBCPW resonator with absorber material and
illustrates the improvement in the response.
A. Overview o f the Cavity Effects in CBCPW
The resonance mechanism in packaged CBCPW is illustrated with the equivalent circuit
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
164
in Fig. 60 from [35], The packaged CBCPW is represented by the circuit and the cavity
modes which are modeled by transformers and series RLC elements. For this example,
three cavity modes are assumed present within the frequency band of interest. Associated
with each cavity mode is an equivalent lumped RLC circuit with the resistance R
accounting for the power loss (conductor, dielectric, radiation, and coupling losses) in the
resonant circuit. The inductance L and capacitance C elements determine the resonant
frequency for the cavity modes. The Q for each cavity mode can be described by the R, L,
and C elements.
The transformers with turns ratio (n) for the mutual coupling,
approximate the coupling coefficient from the CBCPW circuit to each of the cavity modes.
The coupling from the CBCPW circuit to the cavity is a function o f the structural
parameters and the type of discontinuity. Efficient coupling to the cavity occurs when the
dominant CBCPW mode is not bound to the slots and for vertical discontinuities
(dominant rectangular cavity modes have primarily vertical field components).
For the
cases considered here, the turns ratio o f the transformers is very small (light coupling).
For CBCPW, the coupling o f energy to the unintentionally formed rectangular cavity
should be avoided since a loss o f energy in the transmission response of the line exists and
interference effects within the circuit are possible. The RLC equivalent circuit is modeled
with an input admittance. For frequencies away from resonance, the RLC circuit has a low
admittance. At resonance with a high Q cavity, the RLC admittance is very high allowing a
large current to flow into the resonator cavity circuit (YV=I where Y is the admittance
presented at the terminals o f the transformer, V is the voltage across the terminals, and /
is the current within the RLC circuit). The admittance presented at the input terminals of
the RLC resonant circuit model is
n
(6 . 1)
for each cavity mode (/'= 1,2,3) for the example considered here. The resonant frequency
becomes
( 6 .2 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
165
CBCPW
C irc u it
inputt
«
•
UAAJ
1:ni*
R
1 £r,
0
output
«
11
1:no*
C }
1 : n 3*
^3
^ 3 ^3
Fig. 60. Equivalent circuit representation for resonance mechanism in a packaged
CBCPW circuit with cavity effects. Three cavity modes exists within the
frequency range of interest for this example and are modeled using R<LiCl
lumped elements and mutual coupling with turns ratio n y
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
166
and the input admittance of (6.1) is purely real at resonance. The Q or quality factor for
each cavity mode is an important parameter specifying the frequency selectivity and
performance of a resonant circuit. In general terms, the Q is the ratio of the energy stored
to the energy loss in the resonant circuit. The Q discussed here is the unloaded Q of the
resonant circuit and is expressed in terms o f the RLC circuit as
n
Q'
_ 03o.i L i
Ri
1
cd0
O.i CiR i
(6,3)
The use o f damping lossy material to reduce the cavity Qs is modeled by increasing the
equivalent resistance R of the cavity circuit. In the vicinity of resonance, co = co0 + Aco
and the input admittance from (6.1) can be written as
Y = —p
X
----------- =r
/? [ 1 +y'2Q Aco/co0]
where the following approximation has been used
(6.4)
l/(co0 + Aco) » (1 - Aco/co0)/co0 . In
microwave systems, sections of transmission lines or metallic enclosures are implemented
as resonators in place of the lumped parameter circuit. The input admittance is modified to
account for the losses in the resonator by replacing the resonant frequency with an
equivalent complex resonant frequency. The admittance of (6.4) can be described with this
frequency as
Y
(m0/2Q )*
j [co - oo0(l +.//2Q)]
r6 5 ,
and the complex resonant frequency becomes
cor = <00(1 + jl 2Q).
(6.6)
If the cavity is filled with a lossy dielectric material, then the permittivity becomes complex
and is described as
/
er = s r
//
- j & r
/
=
e r(l -ytanS)
(6.7)
and if the damping material has a lossy magnetic susceptibility characteristic then the
permeability is complex by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
167
/
n
Hr = Hr - y 'H r .
(6.8)
The resonant frequencies and the Qs for the cavity modes in the packaged CBCPW can
be approximated with an ideal analysis for the structure in Fig. 61. PEC shorting facewalls
exist at z=0 (cut away to show the internal dielectrics) and z=2B in the figure and no slots
are present at x=0. Instead of solving for the propagation constants for the waveguide
modes for the two-dimensional structure, the resonant frequency is calculated for the
three-dimensional cavity as
cor = co0 + Jco"
(6.9)
and from (6.6), the Q can be determined as
CDn
Q*-T72co
(6.10)
The resonant frequency can be derived for the multi-layered cavity (two dielectric layers to
control the propagation characteristics o f the CBCPW mode and the rectangular
waveguide modes and a damping layer to reduce the cavity mode Qs) of Fig. 61. The
equivalent admittance expressions for the Y4 (TM modes) and Y4 (TE modes) from
(A.14)-(A.16) in Appendix A are utilized as the admittance at x=0 for layers h4, h5, and h6
and with the conductor-backing.
A magnetic wall at y=0 is assumed for the excitation of
the dominant CBCPW mode. The wavenumber in the ^-direction becomes ky - fm/2A
with
fi= 1,3,5,... ,oo
and
that
in the z-direction
becomes
k. = piz!2B
with
p = 0 ,1 ,2 ,..., oo for TE and TM modes. The transverse resonance method is employed
along with Muller's method to locate the complex zeros and hence the resonant frequencies
for the cavity modes supported in Fig. 61.
This approach for the cor calculation is
necessary due to the complexity of the analysis for the three-layer configuration.
The
transverse resonance method is an application o f transmission line theory to the equivalent
circuit of a transverse section of the guiding structure and is applicable to any waveguide
that is uniform in the direction o f propagation. The transverse resonance method leads to
the required eigenvalue equation in a direct manner. For the structure of Fig. 61 with a
^ h
short circuit at x=0, the equivalent impedances are calculated for
Z 4 and Z4.
The
transverse resonance condition requires that the sum of the impedances looking toward the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
168
\N ^ c o n d u c to r
$
jp j
W
,
Fig.61. Ideal rectangular cavity representation of the multi-layered packaged CBCPW.
PEC shorting facewalls exist at r=0 and z=2B.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
169
short circuit at x=0+ and that at the input to the three-layer transmission line at x=0’ for the
T M ^ and TE
modes vanish. The determination of the cavity Qs using (6.10) is
similar to the procedure in [61].
The dominant cavity modes analyzed here for the CBCPW MICs are the T M ^ . The
mode field patterns o f the rectangular cavity are similar to the TM or TE waveguide modes
in a z=constant plane. The Q varies approximately as the ratio o f the volume to the surface
area of the cavity. No conductor losses are included in the cavity analysis in this chapter.
Since the electrical properties of the broadband absorber material used in the
experimental data are not known, a microwave absorber example from [59] was
implemented. The parameters stated for this material for 9-12GHz were er=21( l-y'0.02)
and |j.r= l.l(l-y l.4 ) with h=1.27mm (thickness on the order of MIC dimensions).
Microwave absorber is inexpensive and flexible to various circuit conditions (used in
horizontal or verticai locations and small gaps). The drawbacks with absorber are the
electrical properties are not well characterized and there can be difficulty in configuring
this material as a full width layer (y=2A) within a multi-layered structure especially with
tape shorting sidewalls. The other lossy material utilized in this chapter is doped silicon
/
(Si) with s r = 12.
This lossy volume is well characterized, inexpensive, and provides
damping with a small thickness [60], Damping with doped silicon within a cavity would be
optimally applied to MMICs but is introduced here to demonstrate some o f the design
procedures within CBCPW MIC circuits. The doping level is selected so that the skin
depth (5S) is equal to the substrate thickness at the center of the band of interest (damping
has bandwidth characteristic) and presented as
«. =
where co is the radian frequency, p0 =4te
<«•">
x
10'9H/cm, and a is the conductivity and is
expressed as
a =W „ »e
where
q is the electroncharge
(6-12)
1.6 x 10'19C, |xn is the electron mobility for Si
1350cm2/V-sec, and ne is thedoping level in cm'3. The dielectric loss tangent term is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
with e r = 12 and e0=8.854 x 10'14F/cm. As an example o f the above design procedure
with f=22.5GHz and a Si thickness of 0.025cm, the skin depth is set 5s=0.025cm. A lower
doping level results in less damping while higher doping starts to make the Si layer behave
like a metal withan increase in the enclosure Q.The conductivity is derivedfrom (6.11) as
a =1.8mho/cmand the doping level from(6.12) becomes ne =8.35 x10l5cm'3. Finally
u
/
applying (6.13), s r =12. Silicon has no magnetic properties, therefore p. =1 and
n
Hr
r
=
0.
The effects of the damping layers on the propagation characteristics of the CBCPW
and waveguides modes must first be considered before the cavity analysis. In particular,
the lossy materials should not substantially increase the attenuation loss or enhance the
mode coupling effects (relative dielectric constants for the microwave absorber and doped
silicon are very high). Fig. 62 includes the attenuation loss for the dominant CBCPW
mode without (CASE 61) and with (CASE 62) the doped Si layer. The loss tangents for
the upper dielectric (6010 Duroid™ substrate er=10.2) are listed as a function of frequency
at the end of section A in Chapter III. The loss tangents for the 5870 board (er=2.2) at 1,
10, 20, 30, and 40GHz are 0.6, 1.1, 1.55, 1.9, and 2.25 x 10'3, respectively.
The
attenuation loss calculations are verified by the PCAAMT™ program [62] in Fig. 62. For
CASE 62, 8S=0.025mm and a =1.8mho/cm for f=22.5GHz (maximum damping at center
of band). As indicated from the figure, the attenuation loss for the dominant CBCPW
mode is not significantly increased which is expected since the fields are localized about the
slot areas as depicted in Fig. 48(b).
The propagation characteristics o f the dominant
rectangular waveguide mode TMq j must also be investigated with the incorporation o f the
lossy materials. The effective dielectric constant for this mode in CASE 61 at 40GHz is
5.12.
seff increases to 5.29 with the same Si thickness o f CASE 62.
An interesting
response is observed with the microwave absorber for CASE 61 with h6=0.635mm and
extending the electrical properties of the absorber previously stated between 9 and 12GHz
to 20GHz. An initial assumption would be an increase in the effective dielectric constant
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
7
x
O
o
CASE 61
CASE 61 Ref. C62J
CASE 62
6
5
4
3
Q.
2
0
0
4
8
12
16
20
24
28
32
36
40
F R EQ ( G H z )
Fig. 62. Attenuation (dielectric) losses for the dominant CBCPW mode examining the
effects of the silicon lossy layer. Refer to Fig. 58 for dimensions with
6rl= l, er4=10.2, er5=2.2
and h^Smm,
h4=h5=0.635mm, h2=h3=0 and
S=2W=0.635mm and 2A=25.4mm. CASE 61 h6=0 and CASE 62
and h6=0.25mm and «e=8.35 x 10
cm'3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
= 12
172
from seff=3.79 since e r = 21, however the analysis shows eeff=3.1. This result is due to
/
the presence of the magnetic properties o f the absorber with p. =1.1.
The determination for the resonant frequencies and the cavity Qs o f the ideal
rectangular cavity of Fig, 61 is presented and verified by example cases. The microwave
absorber case is implemented in a packaged microstrip structure from [59],
The
configuration consists of a dielectric layer, air, and damping material within a rectangular
cavity (package). The calculated resonant frequency compares well with the result from
this reference and is presented in Table XXI.
TABLE XXI
Resonant frequency verification of the TMq , , mode in a packaged microstrip
structure with microwave absorber damping.
dimensions
with
er4=10.5(l - y'0.0023),
Reference Fig. 61 for
er5= l,
er6=21(l - y0.02),
prt=l.l(l-yl-4), and h4= 1.27mm, h5=l0.16mm, h6= 1.27mm and 2A=15mm
and 2B=24mm.
complex fr (GHz)
mode
TMo.,.,
Ref. [58]
11.051+/0.249
presented work
11.058+/0.259
An example for the doped silicon case is adopted from [60] for a packaged microstrip
structure and presented in Table XXII. The results compare closely with those from this
reference. The application of the doped Si layer for a CBCPW MIC configuration (with
no slots) is depicted in Fig. 63. CASE 63 of this figure represents only the dielectric losses
for 6010 and 5870 Duroid™
substrates without the doped layer present.
CASE 64
corresponds to a silicon layer thickness o f 0.1mm and CASE 65 for h6=0.254mm. Note
that a thicker Si layer reduces the Qs as an increase in the lossy volume exists within the
cavity. A significant reduction in the cavity Qs is obtained using the Si damping over a
broad frequency-band centered at about 20GHz. Resonances with different mode numbers
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
173
TABLE XXII
Q verification for various modes in a packaged microstrip structure with
doped silicon damping. Reference Fig. 61 for dimensions with sr4=12.7,
sr5= l,
'
15
3
= 12 and n = 4.0 x 10 cm' and h4=0.1mm, h5=0.4mm,
h6=0.25mm and 2A= 12mm and 2B=20mm.
mode
fr (GHz)
Q
Ref. [59]
presented work
Q
H u
10.75
18
17.9
TMo.1.2
14.4
14
13.3
TM o.,.3
19
10
10
TM o.,.4
24
8
7.79
TM q.,5
29.3
6
6.27
TH n*
34.8
5
5.16
but similar resonant frequencies have approximately the same Q.
B. SDM Analysis of Three-Dimensional CBCPW Resonator Circuits
The spectral domain method (SDM) from Chapter II will be extended to simulate
three-dimensional resonator circuits. The finite-length structure is analyzed to demonstrate
the effects of lossy damping layers on the unloaded Qs o f a CBCPW resonator and
rectangular cavity modes, which exist when the CBCPW circuit is placed in a package.
The resonator circuits for CBCPW and microstrip are illustrated in Fig. 64. The microstrip
circuit is introduced to validate the numerical procedure with published results.
The
resonators are ideal half-wavelength center-fed circuits and are shorted on both ends for
the CBCPW structure and are open on both ends for microstrip. The circuits are also
symmetrically positioned with respect to the y and r-directions. The physical resonator
length is 2L from Fig. 64 and the physical length of the ideal package is 2B.
The
multi-layered resonator circuit (see Fig. 58) is then placed within the package as in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
--------CASE 63
..........CASE 64
- CASE 65
o
0.0 5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
FREQ (G H z )
Fig. 63. Demonstration of the damping effects within a rectangular cavity for a MIC
example using doped silicon. Refer to Fig. 61 for parameters with
/
sr4=l0.2, er5=2.2,
= 12 and h4=h5=0.635mm and 2A=2B=25.4mm.
Dielectric loss data for h4 and h5 substrates is listed in the text. CASE 63
16 .
h6=0, CASE 64 h6=0.1mm and n= 5.2 x 10 cm , and CASE 65
h6=0.25mm and rt,=8.35 x 10
cm*3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
175
y
-rS? >/ / / / / / / > s s /
(a)
<
co
2L
2B
(b)
Fig. 64. Top view representations at x=0 of the center-fed half-wavelength resonators
considered in the analysis, (a) CBCPW and (b) microstrip.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
176
Fig. 8(b) connecting the ground planes at (x=0, y,z), the conductor-backing at (x=-h4-h5-h6,
y,z), and the cover plate at (x=h3+h2+hj, y,z) with lateral PEC shorting sidewalls at
z) and longitudinal PEC shorting facewalls at (x j\ z=±B).
For a transmission line is shorted at the end, standing waves develop and both the
electric and magnetic fields have null points at periodic intervals along the line. The input
current for this lossless, ideal transmission line (analogy applied to the center-fed CBCPW
resonator) is Im (z=0) =
where Z is the characteristic impedance, Vs is the
cos
source voltage across the input terminals, and L is the length o f the transmission line from
the source at z=0 to the shorted end. If the goal is to minimize the power transfer to the
resonator circuit (one-port device), then 7in(z=0) should be minimized (P=I 2IZ). This can
be achieved by requiring L=njkgl4 with ^=1,3,5,... and nt is the integer harmonic number
for the resonator. The full length of the resonator in Fig. 64(a) is 2L so that 2L=w/A.g/2
and the half-wavelength resonator is derived («f=l). If the goal is to maximize the power
transfer to the resonator circuit, then 7in(z=0) should be maximized by requiring n ,-2,
For structures with sidewalls and facewalls, the Fourier transform of (2.3) is replaced
by a discrete, finite, double transform where <|> is defined over the interval [-A.A] in the
^-direction and <J> is defined over the interval [-B,B] in the z-direction and satisfies the
Dirichlet conditions of (2.6) with
B A
(6.14)
-B -A
and
2 a 2B -f -f ^ X,y'
-B -A
^ ky'y ^ k 'J “ ^ ^
(615)
(Mx.j/.z) = X
X %{x,kyi, k :l) e -jKk*yy e ^k -iZ .
I = - o o i = -o o
(6.16)
where the inverse Fourier transform is defined as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
177
The solutions for waveguides with finite dimensions are characterized by discrete
spectra of eigenvalues. Accordingly, the integrals in Chapter II are all replaced by double
summations in terms of discrete values o f kyi (spectral variable in the ^-direction) from /=oo to
qo
and in terms of discrete values o f kzl (spectral variable in the z-direction) from /=-
to
oo
for (2.30) and (2.33). The values o f kH are determined by examining the field
oo
behavior o f Ex and Hx along the z-direction. The dominant CBCPW mode for a resonator
shorted on both ends o f Fig. 64(a) has a bounded behavior for the longitudinal components
of the electric field (£_x and EyJ at z=±L for the ends of the slot (see section B of Chapter
II). The E
field is zero outside the slot regions at x-0. The fundamental mode (nt = \)
and odd harmonics (^= 3,5,...) of the CBCPW resonator are excited by a magnetic wall at
z=0. The even harmonics (nl =2,4,6,... o f the CBCPW resonator) are excited by an
electric wall at z=0. The fundamental mode of the resonator is assumed in this analysis.
The basis functions for the unknown electric fields in the slots are even and odd functions
with respect to z for Ey and E„ respectively. PEC walls exist at z=0,±L and
cos
and 'P*
e
is equal to sin (^ jj|)
for I =
- o o , ...,
e
becomes
-5 ,- 3 ,-1 ,1 ,3 , 5,..., oo.
h
¥ and *P are the electric and magnetic scalar potential functions, respectively.
spectral variable in the z-direction can be rewritten
* - - ' = ( / + 2} B
for /=
-q o ,
...,-2 ,-1 ,0 ,1 ,2 ,
and T *.
...,o o .
The
(617)
Recall from (2.1b) that Ey is proportional to
Parseval's theorem of (2.31) is also modified for the three-dimensional
packaged structure with
?
j
i i
{ J(y,z)E*(y,z)cfydz = - L - L
-B -A
^=0°
*=Q0 ~
~
X J(kyi ,k :i)E*(kyi ,k :i ) = 0.
/=-°° /=-<»
(6.18)
£
All of the elements of the matrix [/I] of (2.33) are even functions of
and the
summations are represented by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
178
/—oo
i—co
/=oo
/ — GO
£
£ ,4 = 2 £
/= —oo
/= —oo
£
/= 0
,4.
(6.19)
/= 0
A homogeneous system of equations is now formed following the procedure in section A
of Chapter II and is described in compact form here as
c
d
(6 .20)
= [0]
and nontrivial solutions can only occur when the matrix is singular with
det [i4(co r)] = 0
(6.21)
and the complex resonant frequency cor is solved for each mode and the Q can be
calculated from (6.9) and (6.10).
For the excitation of the fundamental mode and the odd harmonics in the CBCPW
resonator with a magnetic wall at 2=0, the rectangular cavity modes are the TM )hhp and
T E Mhp modes and specified with p =
l,3,5,...,oo.
The poles of the Green's functions in
(2.22)-(2.24) now correspond to the rectangular cavity modes and the resonant
frequencies.
The unknown electric fields E and E, across the slots at x=0 are expanded in terms of
known basis functions E
and E„n in the space domain as
O
O
oo
E . = £ d nE :n(y,z)
n= 1
m=\
where cm and dn are the unknown expansion coefficients. The basis functions E
( 6 . 22 )
and E.n
are separable in the y and z directional variables and are written for the CBCPW resonator
defined only in the slots at r=0 and S/2< [y| < (S/2+W) and 0< \z\ <L by
M
X
m=\
E y ,i( y )
i COS
\
2L )
nz
(6.23)
N
2
n= 1
E z,\
00
L
sin
L
(6.24)
where M and N are the number o f expansion terms for the y and z-directed electric fields,
respectively.
Eyl (y) and E , x (y) are the basis functions used in the one and
two-dimensional SDM analysis from (2.35) and (2.36) and only one ^-dependent term is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
179
utilized with respect to numerical computation considerations [63], This implementation
for the y-directed functions can be justified in CBCPW MICs by the phase constant being
accurately modeled with a single expansion term. The above basis functions are plotted as
a function of z in Fig. 65. The expansion functions in the spectral domain are obtained
from (6.22) and (6.15) with a magnetic wall excitation at z=0 (fundamental and odd
harmonics for the resonator) and are written as
7r fir Ir \ - ¥
£ ,(* * •* -< > - J E
Tr fir \
c mEt .,(* „ )
~
N
E .i k ^ k - i) = Z d nE
^
,Z \
where
\jk
. ^
' sin[(2m-1)tt/2] • cos(*.;L)]
(
*
• sin[(2«-l)7r/2] • cos(£_.L)1
------ V - 1-------------------------------(6.26)
[(2«-1)7t/2] - (>t;/L)2
j (kyi) and E z l (kyj) are the spectral basis functions from (2.36) and (2.37).
The microstrip resonator in Fig. 64(b) and Fig. 14(a) (with shorting sidewalls at _y=±A)
is also investigated to verify the three-dimensional SDM procedure. The basis functions
for the currents on the strip are derived from [63] and defined only on the strip at .v=0 and
0 < [y| <S/2 and 0< \z\ <L as
J;(y,z) =
I
ITl—l
Jy(y,z) =
Z J y , (V) Sin [ ( 2 ^ 1 )
n= 1
00 c o s [(2 ^ -i)jB
Lv
AL-,
(6.27)
y
to ]
(6.28)
where J. j(y) and Jy j(y) are the basis functions used in the one and two-dimensional SDM
analysis. These functions incorporate the singular (unbounded) behavior in the ^-direction
o f the magnetic fields normal to the edges o f the strip and again only one ^-dependent term
is implemented. The SDM for microstrip with the expansion of the currents on the strip
follows from section F of Chapter II.
C. Verification of the Three-Dimensional SDM Procedure
Before design examples of multi-layered CBCPW MIC resonator circuits are presented,
the three-dimensional SDM procedure must be validated.
The analysis will include
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
180
m = 1
m-2
I.Or
m =3
CO
c
o
0.5
o
c
3
co
*o35
jQ
0.0
-0 .5
- 1.0 1
-1 0
2 (m m )
(a)
71= 1
71=2
71=3
I.Or
CO
C
0.5
O
3
co
*o55
-O
0.0
o
c
—1.0
2 (m m )
(b)
Fig. 65. Shapes of the electric field basis functions for a CBCPW resonator as a
function of z with 2L=l2.7mm for the right hand slot at j/=(S/2+W/2).
(a) E
basis functions, (b) E:n basis functions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
181
microstrip cases published in the literature. A microstrip example from [60] is illustrated in
Table XXIII for a center-fed resonator in a laterally and longitudinally open structure with
a doped Si cover plate. The resonant frequency for this example was calculated by the
SDM at 57.23GHz with 75 spectral terms for kyi and
z-direction for the E
and 2 basis functions in the
and Ezn electric fields in the slots (total of four). A frequency of
57GHz was listed in the reference. The calculated Qs correspond to those listed for this
microstrip resonator in the table. The discrepancy for the h2=0.4mm case is due to the fact
that the analysis in [60] includes surface-wave radiation loss (open structure) and the
simulations here are for closed cavities. At 57GHz, radiation losses are significant which
would decrease the Q of the resonator.
TABLE XXIII
Resonant Q verification of the dominant mode in a microstrip structure with Si
damping. Reference Fig. 58 and Fig. 64(b) for dimensions with 2A=12mm,
'
16 i
2B=20mm and WG=0 with e rl = 12 and«=3.0 x 10 cm , er2 = l, s r4=12.7
and hj=h4=0.1mm, h3=h5=h6=0 and S=0.2mm and 2L=0.775mm.
Q
Ref. [59]
presented work
0.2
30
32.4
0.4
90
121
h2 (mm)
The convergence properties of the half-wavelength resonators with respect to the
number of basis functions will be investigated. An example from [63] is presented in Fig.
66(a) for a microstrip resonator (CASE 66) and for a CBCPW structure (CASE 67) in Fig.
66(b). Again, 75 spectral terms were implemented for each spectral variable and M=N=2
for the basis function expansions. The results agree closely with [63] and the convergence
behavior is evident with the additional expansion functions. A similar response is depicted
for the CBCPW resonator in Fig. 66(b). The resonant frequency (fr) can be approximated
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
182
13.0i
Ideal half-wavelength point
N
I
o -•
'—
O
QC
,2-6 -
^
12.4 ■
ui
h2
<
2
O
GO
Ul
x
12.2
presented work
Ref. 1631
•
CASE 66
12. 0 -
^
11.9
TOTAL z - D I R BASIS FUNCTIONS
(a)
10.60
ideal half-wavelength point
N
3
10.45
o
LlJ
QL
Li_
f- 10.30
2
<
2
O
C/0
Ul 10.15cc
CASE 67
10.0Q.
TOTAL z - D I R BASIS FUNCTIONS
(b)
Fig. 66 . Convergence demonstration o f the resonant frequency from the SDM with
additional basis functions in the z-direction forM =N of a resonator circuit,
(a) Microstrip example CASE 66 with dimensions from Fig. 58 and Fig.
64(b) of srl=l, er4=9.4 and ht=5mm, h4=0.6mm, h2=h3=h5=h6=0 and
S=0.575mm, WG=0 , 2L=4.5mm and 2A=2B=10mm. (b) CBCPW example
CASE 67 with dimensions from Fig. 58 and Fig. 64(a) o f s rl= l, er4=10.2,
er5=2.2 and h ^ m m , h4=h5=0.635mm, h2=h3=h6=0 and S=2W=0.635mm,
2L=6.2mm and 2A=2B=25.4mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
183
from the effective dielectric constant for this example.
For the CBCPW of CASE 67,
seff=5.21 at 10GHz. The following relationship is utilized 2L=A.g/2 with 2L=6.2mm, and
fr =10.6GHz. This is the ideal half-wavelength point for the zero total number of basis
functions in Fig. 66. Another confirmation of the numerical method is the tracking of the
resonant frequency as the resonator length is doubled for CASE 67.
The fr should
decrease in half for this configuration and the SDM predicts a frequency o f 5.1832GHz.
The simulation o f CASE 67 with M=N and 75 spectral terms for k and k_t required 30
seconds on a Pentium 90MHz PC. Fig. 67 is a plot o f the Ey electric field distribution for
the right hand slot and is determined by calculating the inverse Fourier transform to
produce the space domain field. The expected response for this field component in the
figure also confirms the SDM analysis for the resonators. One final verification example is
derived from [35] for the TM022 mode for two layers of GaAs with the lower layer
providing the damping (lossy dielectric) to the cavity. The configuration in [35] is not
symmetrical in the y or z-directions so neither a magnetic or electric wall is incorporated at
y =0 or z=0, respectively. The resonant frequency from the SDM of this cavity mode
was calculated at 50.4GHz and f=50GHz in the reference. A comparison of the Qs for
this mode is depicted in Table XXIV and the responses again agree closely. The resonant
frequency for the center-fed dipole of [35] was determined to be 54GHz and the SDM
here predicted a frequency o f 53.3GHz.
D. Design Examples for Multi-Layered CBCPW Resonator Circuits with Damping
The three-dimensional SDM applied to resonators has been developed and verified in
the previous two sections. Now multi-layered CBCPW circuits with cavity damping can
be designed with the goal to maximize the circuit Q for a half-wavelength resonator,
minimize the cavity Qs, minimize the dielectric thickness, and minimize the lossy layer
thickness.
Both upper and lower dielectric loaded CBCPW configurations will be
presented along with the doped Si and microwave absorber planar lossy materials. A
relationship is utilized to approximate the circuit Qs from the dielectric and conductor
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
184
0.2
0.0
Q.
-
0.2
E
o
v
>
- 0 .4
o
<D
-
—
0.6
1.0
-1 5
-1 0
-5
0
5
10
15
Z
Fig. 67. Plot of the Ey space domain field from the SDM for the dominant CBCPW mode
of the resonator circuit of CASE 67 with 2L=6.2mm at _y=(S-i-W)/2 for the right
hand slot.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
185
TABLE XXIV
Resonant Q verification of the TM 0 2 2 mode in a CBCPW packaged
structure with loss tangent damping at h5. Reference Fig. 58 and
Fig. 64(a) for dimensions with srl =1, er4=er5 =12.8, 54 =0.002 and
h4=h5=0.125mm, h2=h3=h6=0 and S=0.025mm, W=2S, 2L=1.Omm
and 2A=2mm, 2B=3mm.
Q
Ref. [35]
presented work
0.01
170
167
0.1
20
19.7
loss tangent 5S
losses for the dominant CBCPW propagating mode from [64] by
71
Q = X g (ctc
+ OLd)
(6.29)
where clc and a d are the attenuation coefficients (calculated by the SDM) of the conductors
and dielectrics, respectively. The above equation is applied to half-wavelength resonators.
Assuming the conductors are PECs, then a c=0. For the MIC examples presented, the
conductor losses should not dominate the dielectric losses in an actual circuit operating
from 9-17GHz.
The effects of a constant doped Si layer on the CBCPW resonator (CBCPW
CIRCUIT) and the cavity modes in a lower loaded configuration for a 9-12GHz
application are studied for CASE 68 in Fig. 68 (a). The fundamental mode of the resonator
(w ^l) is excited with a magnetic wall at z=0 and the TMg 3, mode is the only cavity mode
with a resonant frequency within the band. fr of this cavity mode is 9.53GHz and the
resonant frequency of the CBCPW resonator is approximately 9.6GHz, as shown in Fig.
66(b) with h6=0. CASE 68 is the same as CASE 67 except with the presence o f the silicon
layer.
The doping level was arbitrarily selected for this simulation.
The CBCPW
CIRCUIT IDEAL response in Fig. 68 is derived from (6.29) and the TMo 3 , IDEAL is
obtained from Fig. 61 (no slots configuration). As the Si damping layer is increased, the Q
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
186
CASE 68
CBCPW
X CBCPW
TMq.3.1
* TMq.3.1
0.00
0.05
CIRCUIT (SDM)
CIRCUIT (IDEAL]
(SDM)
(ideal)
0.15
0.10
0.20
0.25
LOSSY LAYER THICKNESS h6 (mm)
(a)
CASE 69
X
A
CBCPW CIRCUIT (SDM)
CBCPW CIRCUIT (IDEAL)
TM(w,i (SDM)
TMoj.t (IDEAL)
1.00
2.00
3.00
0.00
DIELECTRIC LAYER THICKNESS h 4 ( m m )
(b)
f
Fig. 68 . CBCPW resonator and TM,, 3 j cavity mode Qs using doped Si layer (er = 12).
a) Lower loaded multi-layered structure CASE 68 same as CASE 67 as a
e
16
i
function o f the Si layer at h6 with doping «,=1.0 x 10 cm . b) Upper
loaded CBCPW CASE 69 as a function o f the dielectric thickness h4 with
fixed Si layer at hs=0.2mm with doping n = 2.1 x 10 cm'3 and srl=l,
er3=10.2, Er4= 2 .2 , h[=5mm, h3=0.635mm, h2=h6=0 and S=2W=0.635mm,
2L=4mm and 2A=2B=25,4mm. Log scales are implemented.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
187
of the cavity mode is decreased at a faster rate than for the CBCPW resonator.
A Si
thickness of approximately 0 .15mm would be an appropriate tradeoff between the CBCPW
and cavity Qs for this structure. The accuracy of (6.29) is also depicted in Fig.
68 (a). The dominant CBCPW mode is still bound to the slots with the incorporation of the
lossy layers for all the cases presented.
For this frequency band, the loss tangents of
tan84=2.1 x 10'3 and tan6 s= l.l x 10'3 are applied to the er= 10.2 and sr=2.2 substrates,
respectively. An upper loaded structure is implemented for CASE 69 in Fig. 68 (b) and the
lower dielectric thickness is varied for a fixed Si layer in the 12-17GHz range. For this
case 2L=4mm, h4= 1.27mm, h5=0, and the resonant frequencies for the CBCPW, TM03, ,
and TMq j j (from the lower cavity between h4, h5, and h6) modes are 14.46GHz,
12.59GHz, and 16.9GHz, respectively.
The TM031 mode (or the TM0] 3 degenerate
mode) of the upper cavity (formed between h3, h2, and h,) has a fr=17.36GHz and is
outside the band of interest. Otherwise, the Si would also be applied at h, to dampen the
upper cavity for this mode. The Si layer at h5 is designed to maximize the damping effects
at the center o f the band between the two cavity modes. By increasing the thickness of the
lower dielectric layer (h4), the resonator Q increases as shown in Fig. 68 (b). The cavity
mode Qs also increase (but at a slower rate than the CBCPW resonator) as the percentage
of the lossy volume in the lower cavity is reduced with a thicker substrate (h4). Recall that
the electric field distributions of the cavity modes have a predominate vertical component
and are affected more by the damping materials than the CBCPW mode.
A design
thickness of h4=l ,5mm is recommended here. The Q of the TMq 3 3 mode is very close to
the TMqj j .
An example of the microwave absorber is illustrated in Fig. 69 o f CASE 70 (similar to
CASE 68) for a lower loaded CBCPW waveguide. The parameters stated for a 9-12GHz
bandwidth are h6=0.635mm and 2L=6.2mm. The TMq3 j is the only cavity mode within
the range of interest for this structure. As compared to the doped Si layer, the microwave
absorber is a more effective damping material for the same volume. One configuration
proposed to minimize the dielectric loss effects of the absorber on the CBCPW mode,
would be to increase the thickness of the lower substrate (h5). This strategy would reduce
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
188
x
CBCPW CIRCUIT (SDM)
CBCPW CIRCUIT (IDEAL)
O
CM
O
CASE 70
o
o
O
0.00
1.00
2.00
3.00
4.00
5.00
D IE L E C T R IC LA Y E R T H IC K N E S S h 5 ( m m )
Fig. 69. CBCPW resonator and TMq 31 cavity mode Qs for a lower loaded multi-layered
structure using microwave absorber as a function of the dielectric thickness h5.
Refer to Fig. 58 for the dimensions of CASE 70 with srl=l, sr4=10.2, £^=2.2,
ert=21(l-_/0.02), |j.rt= l.l(l-y l.4 ) and h,=5mm, h4=h6=0.635mm, h2=h3=0 and
S=2W=0.635mm, 2L=6.2mm and 2A=2B=25.4mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
189
the field coupling of the CBCPW mode to the lossy layer and further concentrate the fields
into the h4 and h 5 dielectrics. Fig. 69 demonstrates the results for this configuration and
shows the thickness necessary to minimize these effects. The response o f CASE 70 is very
similar to CASE 69 of Fig. 68 (b). A 2.0mm thickness for h5 is suggested as a tradeoff
between the circuit and cavity Qs and the dielectric size.
E. Experimental Demonstration of Cavity Damping for CBCPW Resonators
Some type of damping configuration is necessary within multi-layered wide
frequency-band CBCPW MICs to reduce the resonance effects associated with the
rectangular cavity modes.
A measurement example incorporating absorber (damping)
material within a CBCPW resonator is presented. The circuit is a series gap-coupled
straight resonator and is diagrammed in Fig. 70(a). This circuit is often employed in the
measurement o f the effective dielectric constant for a transmission line and the Q of a
resonator. The gap width (G) of Fig. 70(a) is selected sufficiently wide to lightly couple
energy to and from the resonator and not load the resonator circuit with the measurement
system. The resonator length (L) is designed from the following relationship [65]
L + 2 l eo =
n. Xg
where nt=l, 2 ,3,... is the integer order o f the resonance,
(6.30)
is the guide wavelength, leo is
the length extensions due to the fringing fields at the open ends o f the resonator line. The
implementation o f the damping/absorber material at the h6 substrate is pictured in Fig.
70(b) as a layered configuration.
This is the same absorber utilized in Fig. 42.
The
conductor-backed plane is modified to effectively house the lossy material for this
application as packaging the absorber as a full width layer proved somewhat difficult due
to the shifting or unstableness of this layer.
The effective dielectric constant for the
CBCPW of this section (see Fig. 71 CASE X) from the SDM was 5.43 at 5GHz. The goal
was for the first resonant frequency (nj= 1) to occur near 5GHz. Applying (6.30) with this
information and neglecting the end length fringing term, yielded a resonator length
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
190
■*
L
y
y
(a)
X
'M
y
Fig. 70. Dimension.parameters for multi-layered CBCPW straight gap-coupled resonator
with damping material, (a) Top view with gap width (G) and resonator length
(L). (b) Cross-sectional view with absorber placed at layer h6.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
191
L=12.87mm which was set to L=12.7mm for the measurement data. The experimental
result for this configuration is presented in Fig. 71 and demonstrates the expected response
with a smooth curve to 40GHz. The sharpness o f the resonance reduces for each order
(^=1,2,3,... ) as the losses increase and the Q decreases for subsequent modes. The first
resonance occurs at 4.7GHz which could be predicted when the uncertainty in the
dielectric dimensions is factored into the analysis. The absorber layer traverses the entire
length of the waveguide (z=0 to z=2B). When the damping material was applied only in
the vicinity o f the connector feeds at h6, the presence of the cavity modes on the CBCPW
resonator response was readily apparent.
The absorber volume for this case was not
sufficient to dampen the substrate mode Qs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
192
S2 1
log
REF —1 . 8 0 S dB
7 . 0 dB/
MAG
* u
START
STOP
0 .1 0 0 0 0 0 0 0 0
4 0 .1 0 0 0 0 0 0 0 0
GHz
GHz
Fig. 71. Experimental data response with absorber material to dampen cavity Qs in
straight gap-coupled CBCPW resonator. Reference Fig. 70 for dimensions
with G=0.17mm and L= 12.7mm. CASE X with erl= l, s r4 = 10.8, er5 =2.33
and hj=oo (open structure), h2=h3=0, h4=0.635mm, h5= 1.09mm and
S=2W=0.635mm and 2A=20mm and 2B=38mm. Absorber layer at h6 and
electrical data is not known.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
193
CHAPTER VII
CONCLUSION
This dissertation has demonstrated the configurations necessary to achieve a wide
frequency of operation for CBCPW MICs. Unless the user has access to an etching facility
capable of generating very fine line/slot widths or a circuit that requires very little surface
area that can cutoff the waveguide modes, a multi-layered configuration (either upper or
lower dielectric loading) is needed to reduce the waveguide mode coupling effects and
generate a useful transmission line. The CBCPW with lateral shorting sidewalls and placed
in a package or test fixture resembles a rectangular waveguide in two-dimensions and a
rectangular cavity in three-dimensions.
The leakage and mode coupling effects were
described in detail using both one and two dimensional SDM numerical procedures.
Design recommendations concerning the upper usable frequency, impedance range and
variation, effective dielectric constant range and variation, inclusion of air gaps between
the dielectrics, and uncertainty analysis in the dielectric constant and thickness have been
presented for dielectrics with sr=10.2 and sr=2.2. The propagation characteristics of the
upper loaded structure are quite sensitive to the possible air gap existing between the
circuit conductors and dielectric, and the waveguide is difficult to configure within a test
fixture. The use of some type of damping material is required in many cases to reduce the
residual resonances associated with the cavity structure whose resonant frequencies are
within the band of interest. This damped waveguide was demonstrated experimentally
with broadband absorber placed within the connector housing blocks and as a dielectric
layer.
Simulations using a three-dimensional SDM were presented with the effects of
doped silicon and microwave absorber on the Qs of the cavity modes and the CBCPW
resonator circuit. The experimental data conducted in this research played a vital role in
the identification and understanding of the mode coupling problems in the CBCPW MICs.
Without the measurement data, this research topic would not have been investigated. The
modeling procedure was extended from one, to two, and to three dimensions to more
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
194
accurately predict and describe the experimental effects. Fig. 72 depicts a microstrip
and a multi-layered CBCPW through lines and as compared to Fig. 7, demonstrates that
these two transmission lines are now both available for wide frequency-band MIC
applications. After twenty-five years since the invention by Wen [1], coplanar waveguides
(in particular the conductor-backed form) have developed to the point where they will rival
microstrip as a transmission line choice for MICs. Hopefully, this work has contributed in
some small way to this advancement.
A. Original Contributions
This dissertation has provided an improved understanding of the propagation and
transmission properties of multi-layered CBCPW MICs. Specific contributions from this
work are listed as:
1) introduced measurement data showing the problems with single-layered structures;
2) incorporated multi-layered (lower and upper dielectric loading) to increase the
bandwidth;
3) utilized a one-dimensional SDM to approximately predict this bandwidth;
4) explained the mode coupling effects in detail with a two-dimensional SDM in a
packaged configuration;
5) modeled the finite-length CBCPW as a resonator and identified the cavity modes;
6) presented design summary information for multi-layered CBCPW MICs including the
upper dielectric loaded structure;
7) incorporated lossy layers to dampen the cavity mode Qs without significant affect on
CBCPW mode and simulated this arrangement with a three-dimensional SDM;
8) described the simulation o f a shorted CPW resonator in the SDM;
9) demonstrated how CBCPW MMICs can operate with mode coupling effects and
presented specific and supported guidelines for reduced coupling effects;
10) produced the widest frequency-band CBCPW MIC to date (40GHz).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
195
CASE A
lQ
T)
CN
in
>
LlI
"O
Q
3
CD
TD
CASE Y
O
<
0
4
8
12
16
20
24
28
32
36
40
FREQ (G H z )
Fig. 72. Experimental data of the transmission responses for the microstrip and the
multi-layered CBCPW MIC 5 0 0 through lines. A comparison with Fig. 7
indicates CBCPW MIC is now a viable alternative to microstrip. Refer to
Fig. 5 for dimensions with erl=l and hj=oo (open structure) and
2A=2B=38mm. CASE A is microstrip line with er4=10.8 and h4=0.635mm
and S=0.635mm.
CASE B is CBCPW with er4=10.8, sr5=2.33 and
h4=0.254mm, h5=0.71mm,
h2=h3=h6=0 and
S=2W=0.635mm
with
absorber material in the connector blocks. Both cases are referenced to
OdB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
196
B. Suggestions for Future Work
The required research work on CBCPW is still immense. The majority o f this effort is
and will concentrate on discontinuity modeling [66 ] and [67], Important considerations
involving the discontinuities must be given to the analysis, design, and placement o f the air
bridges associated with the slotline mode and to minimize the coupling to the cavity modes
of the packaged (finite-length) structure. The moment method, SDM, finite-element, and
finite-difference time domain are the numerical procedures that will be utilized in this
endeavor.
However, closed form expressions or simplistic design relations must be
presented from the above fiill-wave procedures for the convenience of the user. Additional
considerations to reduce the number of the air bridges are also extremely important. In the
analysis of this dissertation, it was assumed that a perfect connection existed between the
upper and lower ground conductors with the lateral sidewalls but in reality floating
potential regions within a circuit can occur due to inexact grounding and this issue must be
addressed. Finally, the presence of more than one propagating mode on the calibration and
deembedding procedures associated with the network analyzer reference planes involving
CBCPW MICs must be investigated.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
197
REFERENCES
[1]
C.P. Wen, "Coplanar waveguide: A surface strip transmission line suitable for
nonreciprocal gyromagnetic device applications," IEEE Trans. Microwave Theory
Tech., vol. MTT-17, pp. 1087-1090, December 1969.
[2]
R. Pucel, "Design considerations for monolithic microwave circuits", IEEE Trans.
Microwave Theory Tech., vol. MTT-29, pp. 513-534, April 1981.
[3]
R. Jackson, "Coplanar waveguide vs. microstrip for millimeter wave integrated
[4]
[5]
circuits," in 1986IEEEM TT-SInt. Microwave Symp. Dig., pp. 699-702.
M. Riaziat, private communication, 1991, Varian Research Center, Palo Alto, CA.
M. Riaziat, I. Zubeck, S. Bandy, G.Zdasiuk, "Coplanar waveguides used in 2-18
GHz distributed amplifier," in 1986 IEEE MTT-S Int. Microwave Symp. Dig., pp.
337-338.
[6 ]
R. Majidi-Ahy, C. Nishimoto, M. Riaziat, M. Glenn, S. Silverman, S. Weng, Y.
Pao, G. Zdasiuk, S. Bandy, Z. Tan, "5-100GHz InP coplanar waveguide distributed
amplifier," IEEE Trans.
Microwave Theory Tech., vol. MTT-38, pp. 1986-1993,
December 1990.
[7]
E. Strid, "26GHz wafer probe MMIC development and manufacture," Microwave
J., vol. 29, no. 8, pp. 71-82, 1986.
[8 ]
L. Carin and K. Webb, "Isolation effects in single- and dual-plane VLSI
interconnects," IEEE Trans. Microwave Theory Tech., vol. MTT-38, pp. 396-404,
April 1990.
[9]
R. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York,
[10]
1961, pp. 143-190.
R. Collin, Foundations fo r Microwave Engineering, McGraw-Hill, New York,
1966, pp. 170-179.
[11]
R. Simmons, G. Ponchak, K. Martzaklis, R. Romanofsky, "Channelized coplanar
waveguides: discontinuities, junctions, and propagation characteristics," in 1989
IEEE MTT-S Int. Microwave Symp. Dig., pp. 915-918.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
198
[12]
M. Riaziat, R. Majidi-Ahy, I. Feng, "Propagation modes and dispersion
characteristics of coplanar waveguides," IEEE Trans. Microwave Theory Tech.,
vol. MTT-38, pp. 245-251, March 1990.
[13]
M.A. Magerko, L. Fan, K. Chang, "Multiple dielectric structures to eliminate
problems in conductor-backed coplanar waveguide MICs," IEEE Microwave
Guided Wave Lett., vol. 2, no. 6 , pp. 257-259, 1992.
[14]
M.A. Magerko, L. Fan, K. Chang, "A discussion on the coupling effects in
conductor-backed coplanar waveguide MICs with lateral sidewalls," in 1993 IEEE
MTT-S Int. Microwave Symp. Dig., pp. 947-950.
[15]
M.A. Magerko, L. Fan, K. Chang, "Configuration considerations for multi-layered
packaged conductor-backed coplanar waveguide MICs," in 1994 IEEE MTT-S Int.
Microwave Symp. Dig., pp. 1697-1700.
[16]
Y. Liu
and T. Itoh,
"Leakage phenomena in multi-layered conductor-backed
coplanar waveguides," IEEE Microwave Guided Wave Lett., vol. 3, no. 11, pp.
426-427, 1993.
[17]
M. Yu, R. Vahldieck, J. Huang, "Comparing coax launcher and wafer probe
excitation for lOmil CBCPW with via holes and airbridges," in 1993 IEEE MTT-S
Microwave Symp. Dig., pp. 705-708.
[18]
T. Uwano and T. Itoh, "Spectral Domain Approach" in T. Itoh (ed.) Numerical
Techniques fo r Microwave and Millimeter-Wave Passive Structures. John Wiley &
Sons, New York, 1989, pp. 324-380.
[19]
D. Mirshekar-Syahkal, Spectral Domain Method fo r Microwave Integrated
Circuits. John Wiley & Sons, New York, 1990.
[20]
J. Knorr and K. Kuchler, "Analysis o f coupled slots and coplanar strips on
dielectric substrates," IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp.
541-548, July 1975.
[21]
Y. Fujiki, M. Suzuki, Y. Hayashi, "Higher-order modes in coplanar-type
transmission lines," Electronics and Communications in Japan, Vol. 58-B, no. 2,
pp. 74-81, 1975.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
199
[22]
J. Davies and
D. Mirshekar-Syahkal, "Spectral domain solution of arbitrary
coplanar transmission lines with multi-layer substrate," IEEE Trans. Microwave
Theory Tech., vol. MTT-25, pp. 143-146, July 1977.
[23]
Y. Shih and T. Itoh, "Analysis of conductor-backed coplanar waveguides,"
Electronics Letters, vol. 18, no. 12, pp. 538-540, 1982.
[24]
G. Leuzzi, A. Silbermann, R. Sorrentio, "Mode propagation in laterally bounded
conductor-backed coplanar waveguides," in 1983 IEEE MTT-S Int. Microwave
Symp. Dig., pp. 393-395.
[25]
R. Jackson, "Considerations in the use o f coplanar waveguides for millimeter-wave
integrated circuits," IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp.
1450-1456, December 1986.
[26]
H. Shigesawa, M. Tsuji, A. Oliner, "Conductor-backed slot line and coplanar
waveguide: dangers and full-wave analyzes," in 1988 IEEE MTT-S Int. Microwave
Symp. Dig., pp. 199-202.
[27]
R. Jackson, "Mode conversions due to discontinuities in modified grounded
coplanar waveguide," in 1988 IEEE MTT-S Int. Microwave Symp. Dig., pp.
203-206.
[28]
M. A. Magerko, L. Fan and K. Chang, "Analysis of multi-layered structures to
reduce mode coupling problems in packaged conductor-backed coplanar
waveguide MICs," submitted for publication to IEEE Trans. Microwave Theory
Tech.
[29]
K. Jones, "Suppression of spurious propagation modes in microwave wafer
probes," Microwave J., vol. 32, no. 11, pp. 173-174, 1989.
[30]
E. Godshalk, "Generation and observation of surface waves on dielectric slabs and
coplanar structures," in 1993 MTT-S Int. Microwave Symp. Dig., pp. 923-926.
[31]
N. Das and D. Pozar, "Full-wave spectral domain computation of material,
radiation, and guided wave losses in infinite multilayered printed transmission
lines," IEEE Trans. Microwave Theory Tech., vol. MTT-39, pp. 54-63, January
1991.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
[32]
W. McKinzie and N. Alexopoulos, "Leakage losses for the dominant mode of
CBCPW," IEEE Microwave Guided Wave Lett., vol. 2, no. 2, pp. 65-66, 1992.
[33]
L. Chou, R. Rojas, P. Pathak, "A WH/GSMT based full-wave analysis of the
power leakage from conductor-backed coplanar waveguides," in 1992 MTT-S
Int. Microwave Symp. Dig., pp. 219-222.
[34]
L. Carin, G. Slade, K. Webb, A. Oliner, "Packaged printed transmission lines:
modal phenomena and relation to leakage," in 1993 IEEE MTT-S Int. Microwave
Dig., pp. 1195-1198.
[35]
R. Jackson, "Circuit model for substrate resonance coupling in grounded coplanar
waveguide circuits," IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp.
1461-1465, September 1993.
[36]
C. Tien, C. Tzuang, S.Peng, C. Chang, "Transmission characteristics o f finite-width
conductor-backed coplanar waveguides," IEEE Trans. Microwave Theory Tech.,
vol. MTT-41, pp. 1616-1624, September 1993.
[37]
Wiltron 36804-25C Universal Test Fixture, September 1990, Wiltron Inc. Morgan
Hill, CA.
[38]
T. Itoh and R. Mittra, "Spectral domain approach for calculating the dispersion
characteristics o f microstrip lines," IEEE Trans. Microwave Theory Tech., vol.
MTT-21, pp. 496-499, July 1973.
[39]
R. Mittra and T. Itoh, "A new technique for the analysis of the dispersion
characteristics o f microstrip lines," IEEE Trans. Microwave Theory Tech., vol.
MTT-19, pp. 47-56, January 1971.
[40]
A. Wexler, "Solution of waveguide discontinuities by modal analysis," IEEE
[41]
[42]
Trans. Microwave Theory Tech., vol. MTT-15, pp.508-517, September 1967.
Earl Swokowski, Calculus. Prindle, Weber & Schmidt, Boston, 1992, pp. 463-472.
El-Badawy El-Sharawy and R. Jackson, "Coplanar waveguide and slot line on
magnetic substrates: analysis and experiment," IEEE Trans. Microwave Theory
Tech., vol. MTT-36, pp. 1071-1078, June 1988.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
201
[43]
C. Chang, Y. Wong, C. Chen, "Full-wave analysis o f coplanar waveguides by
variational conformal mapping technique," IEEE Tram. Microwave Theory Tech.,
vol. MTT-38, pp. 1339-1343, September 1990.
[44]
N. Das, "A study of multi-layered printed antenna structures," Ph.D. Dissertation,
University of Massachusetts, Amherst, September 1989.
[45]
B. Janiczak, "Analysis of coplanar waveguide with finite ground planes," AEU,
vol. 38, no. 5, pp. 341-342, 1984.
[46]
C. Chang, W. Chang, C. Chen, "Full-wave analysis of multilayer coplanar lines,"
IEEE Tram. Microwave Theory Tech., vol. MTT-39, pp. 747-750, April. 1991.
[47]
N. Fache and D. De Zutter, "Circuit parameters for single and coupled microstrip
lines by a rigorous full-wave space-domain analysis,"
IEEE Tram. Microwave
Theory Tech., vol. MTT-37, pp. 421-425, February 1989.
[48]
M. Tsuji, H. Shigeswa, A. Oliner, "Printed-circuit waveguides with anisotropic
substrates: a new leakage effect," in 1989 IEEE MTT-S Int. Microwave Dig., pp.
783-786.
[49]
M. Tsuji, H. Shigeswa, A. Oliner, "New interesting leakage behavior on coplanar
waveguides of finite and infinite widths," in 1991 IEEE MTT-S Int. Microwave
Dig., pp. 563-566.
[50]
G. Ghione and C. Naldi, "Coplanar waveguides for MMIC applications: effect of
upper shielding, conductor backing, finite-extent ground planes, and line-to-line
coupling," IEEE Tram. Microwave Theory Tech., vol. MTT-35, pp. 260-267,
March 1987.
[51]
M. Tsuji, H. Shigeswa, A. Oliner, "Dominant mode power leakage from
printed-circuit waveguides," Radio Science., vol. 26, pp. 559-564, March-April
1991.
[52]
D. Kasilingam and D. Rutledge,
"Surface-wave losses of coplanar transmission
lines," in 1983 IEEE M TT-S Int. Microwave Dig., pp. 113-116.
[53]
Y.C. Shih and I. Itoh, "Analysis o f conductor-backed coplanar waveguide,"
Electronic Letters, vol. 18, pp. 538-540, June 1982.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
202
[54]
D. Phatak, N. Das, A. P. Defonzo , "Dispersion characteristics of optically excited
coplanar striplines: comprehensive full-wave analysis, " IEEE Tram. Microwave
Theory Tech., vol. MTT-38, pp. 1719-1730, November 1990.
[55]
J.R. Pierce, Almost All About Waves. MIT Press, Cambridge, MA, 1974, Chapter
6.
[56]
L. Carin, G. Slade, K. Webb, A. Oliner, "Packaged printed transmission lines:
modal phenomena and relation to leakage," in 1993 IEEE MTT-S Int. Microwave
Dig., pp. 1195-1198.
[57]
C. Chang, W. Chang, C. Chen, "Full-wave analysis of coplanar waveguides by
variational conformal mapping technique," IEEE Tram. Microwave Theory Tech.,
vol. MTT-38, pp. 1339-1344, September 1990.
[58]
F. Miranda, K. Bhasin, K. Kong, M. Stan,, "Conductor-backed coplanar waveguide
resonators o f YBa^UjO^g on LaA103" IEEE Microwave Guided Wave Lett., vol.
2, no. 7, pp. 287-288, 1992.
[59]
J. Burke and R. Jackson, "A simple circuit model for resonant mode coupling in
packaged MMICs," in 1991 IEEE MTT-S Int. Microwave Dig., pp. 1221 -1224.
[60]
R. Jackson, "Removing
package effects from microstrip moment method
calculations," in 1992 IEEE MTT-S Int. Microwave Dig., pp. 1225-1228.
[61]
D. Williams, "Damping of the resonant modes of a rectangular metal package,"
IEEE Trans. Microwave Theory Tech., vol. MTT-37, pp. 253-256, January 1989.
[62]
PCAAMT™ , Antenna Design Associates, Inc., Leverett, MA 1990.
[63]
T. Uwano, "Accurate characterization of microstrip resonator open end with new
current expression in spectral domain approach," IEEE Trans. Microwave Theory
Tech., vol. MTT-37, pp. 630-633, March 1989.
[64]
A. Gopinath, "Losses in coplanar waveguides," IEEE Trans. Microwave Theory
Tech., vol. MTT-30, pp. 1101-1104, July 1982.
[65]
T. Edwards, Foundations fo r Microstrip Circuit Design. John Wiley & Sons,
1992, pp. 245-253.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
203
[66 ]
N. Dib, M. Gupta, G. Ponchak, L. Katehi, "Effects of ground equalization on the
electrical performance of asymmetric coplanar waveguide shunt stubs," in 1993
IEEE MTT-S Int. Microwave Dig., pp. 701-704.
[67]
T. Becks and
I. Wolff, "Full-wave analysis o f various
coplanar bends and
T-junctions with respect to different types o f air bridges," in 1993 IEEE MTT-S
Int. Microwave Dig., pp. 697-700.
[68 ]
M. Abramowitz and I. Stegun, Handbook o f Mathematical Functions.
Dover
Publications, New York, 1972, pp. 369.
[69]
W. Press, B. Flanners, S. Teukolsky, and W. Vetterling, Numerical Recipes
{Fortran). Cambridge University Press, Cambridge, 1989, pp. 121-126.
[70]
R.W. Hombeck, Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ, 1975,
pp. 154-159.
[71]
W. Press, B. Flanners, S. Teukolsky, and W. Vetterling, Numerical Recipes
{Fortran). Cambridge University Press, Cambridge, 1989, pp. 31-39.
[72]
,Numerical Recipes {Fortran). Cambridge University Press , Cambridge
1 989, pp. 2 4 0 -2 6 2 .
[73]
,Numerical Recipes {Fortran). Cambridge University Press , Cambridge
1 9 8 9 , pp. 5 2 -6 4 .
[74]
M. Spiegel, Schaum's Outline Series o f Advanced Calculus. McGraw Hill, New
York, 1962, pp. 348-361.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
204
APPENDIX A
SPECTRAL DOMAIN IMMITTANCE FORMULATION
The spectral domain immittance procedure is a powerful technique using the transverse
equivalent transmission line model to relate the components of the currents and fields in
the Fourier domain and is especially suited to multi-layered structures.
The full-wave
spectral domain dyadic Green's functions (immittance functions) are obtained using TMX
and TEXmodes. These functions are based on the transverse equivalent circuit concept as
applied in the spectral domain in conjunction with a simple coordinate transformation. The
basic concept can be illustrated by observing the inverse Fourier transform o f the fields
from (2.5) and multiplying both sides by the exponential factor and recalling that for an
infinite line with variation e'//z, y=k, and
GO
§(x, y)e~j Yz =
J $ (x, k y) e
+
dky .
(A. 1)
—00
From (A. 1), all the field components are a superposition o f inhomogeneous (in x) plane
waves propagating in the direction of 0 relative to the z-axis as described in Fig. A. 1 which
presents the coordinates and components o f the basic TM and TE modes.
The
decomposition of the spectral waves from the (x,_y,z) coordinate system in to the new
system (x,w,v) is written as
cos9 -
-
sin0 =
^y
(A.2)
and
u = zsin0 —_ycos0
v = zcos0 + _ysin0.
(A.3)
For each 0, waves may be decomposed into TMX(EX, E V, H U) and TEx (HX, E U, H V) in
the (x,u,v) system.
The current J v creates only the TM fields because it is associated
with H u and J u creates the TE fields. The equivalent circuits for the TM and TE fields
are found by applying a transverse (x-direction) transmission line analogy as shown in Fig.
A.2 for the six-layer structure. For each basic TM and TE mode, one o f the two models
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
205
k
y
/L.
Hx
~
components
of TE modes
components
of TM modes
X
Fig. A. I. Coordinates and components of the basic TM and TE modes for the SDM
immittance method.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
206
Fig. A.2. Equivalent transmission line models of the multi-layered
CBCPW for the TM and TE modes.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
207
in Fig. A.2 is applicable and the two models are completely decoupled.
For the TM
equivalent circuit, the circuit equations become
H u3 - H u4 = J V
E v-3 = E v4
(A.4)
where
Y' = y \ + y \
^ e
(A. 5)
^ e
and Y3 and Y4 are the driving point wave admittances looking from the location of the
current (x=0) by utilizing conventional transmission line theory.
Similarly, the circuit
equations for the TE fields become
H vi —H v4 = - J u
E u3 = E U4
(A.6 )
where
Y h = YH
3 + Yh4
(A.7)
^ h
and
Y3 and T4 correspond to the input wave admittances for the TE waves.
Once
Ye and Yh are determined, the dyadic spectral domain admittance Green's functions can
be obtained using the transformations of (A.2-A.3).
from the(x,w,v) to the (x,y,z)coordinate
The fields are expressed by mapping
system. E yandE . are linear
combinations o f
E u and E v andlikewise J yand J , are those of J u and J v. The results are described as
+
<A 8 >
<A9)
K
=
+
■
(A .io)
The input admittance is determined by
Y, +Y .tanhK.h,
K- - r « r ^Oi i L a
^ i 1
<A11)
where YL is the load admittance, hj is the thickness of each layer, and T0; is the wave
characteristic admittance defined as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
208
H
> )e ■
i
I'.TE, = T r = n r L = i
(A.i2)
To™ =-%*- = r ^ - = K .
TEi
Eu
/
2
2
2
2
where i refers to the corresponding region and k. = k +fc_ -
(A. 13)
. For the six-layer
waveguide of Fig. A.2, (A. 11) is invoked at each interface. The cover plate conductor
and the conductor-backed plane are taken as short circuits. The following relations are
then derived
&
Y$ = Ctg/K^
Y6 = CtgK.g
(A. 14)
h
~ e
r, =
,
k c t sr 6 + i
^ ^ 6 + C tj
K + c tsr 6
^=<--5-K_.5—H
r
C t5 + ? 6
(a.is)
h
, K / t / 5+ l
Y4 =
-------'
Ky4 y 5 + c t 4
~ e
1 ^y3
= icT y
'
JY
1
1
^
K .+ C t,? 5
r t - K;4 - 2 -------2 - 4
K__4 c t 4 + ? 5
—h
l^j = k:3
K_, +CtjKl
Hr
(A-16)
(A17)
K_-3Ct 3 + T 2
S 3 y 2 + C t3
~ e
,
y,
k 2c t , r +1
= r - H - ---y2
,, v
Ky2Y l +Ct 2
~ e
YX =C tj/K j,,
K0 + c t , r .
r z = * - a — ------- H r
~r.
K ^C tj + f
-A
= C t , K =1
(A. IS)
(A. 19)
/
where Ct; = cothx-hj with i=l,2,...,5,6.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
209
APPENDIX B
DERIVATION OF THE FIELD COEFFICIENTS FOR THE
SPECTRAL DOMAIN METHOD IMMITTANCE METHOD
By imposing the boundary conditions o f (2.48a), (2.48c), (2.49a), (2.49c), (2.50a),
(2.51a), (2.51c), (2.52a), (2.52c), (2.57) and (2.58), the following relations are generated
TM
At x = -h 5-h4
At x = -h.
A tx = 0
i] = v D
Ky*\p
=
c4 + E
A C6 = C
(B .l)
B S5 + C C5 = E
(B.2 )
S 4] = k y3F
L = [ d s t + e .c , - a
Jkj+ k\
(B3)
At x = h3
k y J [ F C 3 + G e5 3] = Ky2H
F S3 + G C3 = /
(B.4)
At x = h3+h2
~K# [ H C 2 + / S 2] =
H S 2 + I C 2 = J C,
(B.5)
J Sx
TE :
h
h
At x = -h5-h4
A S6 = B
k :6A
At x = -h.
B C5 + C S5 = D
h
h
k.j B S 5 + C C5
C 6 = k .5C
(B.6 )
k ,4E
h
(B.7)
A tx = 0
D C4 + E S4 = F
Ju =
At x = h,
f h
h
S 4 + E C< ) "
(B.8)
k 24(Z)
J k j + k\
h
h
F
S,
+
G
C-,
Ka [
*z2J
(B.9)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
210
h
h
h
- k . H hS 2 + I hC2 j = k zXJ
-A° ^C,1
H C2 + I S 2 = J S x
A t.r = h3+h2
(B. 10)
/
where C{ = coshKjh; and
= sinhKjh; for i=l,2,...,5,6. The goal is to solve the
e h
—
coefficients (TM and TE) in terms of A and A respectively, and recall that J v and J u
are known. These equations are derived as
TM :
B
= Krf/Ky5S 6A
(B .11)
C
= C 6A
(B. 12)
D
=
E
—
F
= D S A + £*C 4
(Ky6/Ky5C sS 6 + S $ C 6) j A
+ C s C6~] ^
= D A
~ E A
(B.13)
(B.14)
e
F
= FA
(B .15)
,
C2
^2
K-yl^yX £ + q
F
s2
C,
F
~ ^-yl^y\ £
G
= ~ [Ky3/Ky 2 V 3 C 3 + / M
K y./K ^S ^/^
e
H
=
e
i
q
Ky3/ Ky2 j r
3 /g
F /4
= I 2S 3 - C2~r
= / /4
=
A
e
J
]
_ e £l
F\
C,
S,H
+ C ,/
e
A
• = _F ‘
- ~ I S
(B. 16)
= H A
(B.17)
(B. 18)
(B. 19)
-jJ k fT k f
e
A
A
= ^Ky4/Ky3 D C 4 + E S 4
=
J,
(B.20)
/.+71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-jjfcj+kl
e
A
=
(B.20)
J,
TE.
h
h
(B.21)
B
= Sz A
C
=
D
= [K.g/Kjjj’jC g + C 5S 6] A
(B.22)
k z6/ k z5C 6A
h
E
h
= [ k .j / k ^ ( k ^ / k ^ C s C j + iSj.S'g)]^
h
h
h
=
h
h
= E A
(B.24)
h
Ly = K z4 D S d + E
F
(B.23)
=D A
Cd
h
A
D C4 + E
h
= L-,A
h
(B.25)
S.
C,
L 3 = K jj/k ,! ^ - +
1
Z, 4
C,
5,
= k ^ /k .,^ - +
r H
=
~^-K = ^ K i 2 ^ 2 ^ 4 ^ 3 + ^ 2 ^ i ^ 3 ~ \
k z3J k z2. L .4C ^
h
H
=
h
T 2 ^C- 3 - J<s3 —
r
6
h
I
=
J
J
= -1 C rH
Si
A
h
L$
.h
U A
h
h
(B.26)
(B.27)
= H A
.
u
L 2S 3 - C 3 t ^su
h
A
Ah _
+L^S~
'3 3
h
+ S?I
-j^^kj+ kl „
= -----— ----- 7-----J u
1 T j. 5
^ L l + r6
h
A
h
= I A
h
(B.28)
(B.29)
(B.30)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX C
DERIVATION OF THE POYNTING VECTOR INTEGRAL
The derivation of (2.70) is required for evaluation of the Poynting power flow vector for
the characteristic impedance calculations.
To assist in this process, some mathematical
identities are required [18] and presented as
f
2
*
I |coshKx| cbc = Re(f/sinh jch cosh ich)
(C .l)
f
I |sinhioc| 2cbc = Re(t/coshich sinh * Kh)
(C.2)
J sinh ice cosh*icxc& = i I/|coshich|
+ t/*|sinhich|
- U
(C.3)
| cosh ioc sinh*jcc cbc = }■ U* |coshich|
+ f/|sinhjch|
- U*
(C4)
where
U = —i- r + — -[
K+K
K -tC
and recall if G is a complex function, then G G
*
(C.5)
= \G\ .
Let P h| = E^Hyi - E ^ H ^ = P Ai - P gi for i= 1,2,...,5,6. Following this statement,
the results for the integrals become
PA6 =
- j E K . , 6A ^
P B6 = (rjRKytA
+ jTA )
P A5
R e({/ 6sinhK6h 6cosh*ic6 h 6)
Re (C/6coshK 6h 6sinh K6h 6)
(C.6)
(C.7)
KZ6
B_
5
c
c_
~ «, E) Ji C 51 + U D 52 C 52 + «M D j 2 C 53 + « L)jj C J4
y5
y5
y5
ys
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(C.8)
where
£51
= {~jTBe
- j R K z5B h)
D 52 = {-JTC ~ j R K =5C )
D 53 ={r j R K ySCe + J T C h)
D
= {- jR K y5B e + J T B h)
C 5I = Re({/5coshK5h5sinh*K5h 5)
C$2 = Re(f/ 5sinhK5h 5cosh!|‘K5h J)
C
C 53 -- -<■> f / 5 |coshK5h 5 1
+ t/* |sinhK5h 5 1 - f / 5
2
2
t/* |coshK5h 5 1 + U5 |sinhK5h 5 1 - U*
D_
—
,uu,
D
^41
^41
+
“
^42
^42
>'4
*4
(
R B4
E
h \
u
~
^
*
(
^43
4 )
^41
+
*
D
f
•^44 ^"42 +
«T
V *4
£
“u
+
^ 4 2 ^43
“u
^41
^"44
(CIO)
^4
*
D_
u«T
(
^43 C 43 +
v ^ 4 ^
)
+
M
E_
u■T
D 44 C 44
Vz 4 y
(C .ll)
where
h \ *
d a\
= {~jTD
- j R k z4D )
& 42
=
- i ^ K: 4 ^ )
{ - jT E
^43 = ( - 7 ^ V £
+ J T E h)
D „ ^ { - J R k^ D
+ j T D h)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
214
C 41 = Re(C/4 c o s h ic 4 h 4 sinh*K4 h 4)
C 42
= R e ( £ /4 sinhK 4 h 4 c o s h * K 4 h 4)
2
2
+ U* |sinhK4h 4 1 - U
'C
- 4 3 -- —
o ( / 4 |coshK4h 4 |
2
c
P A3 ~
"K~£*31 (-31
+
~^~£*32 ^ 3 2
*
F B3
~
*7
kZ3 J
£*33
C 31
(
F
^34 -32 + T
\ Z3 J
V
V* 3 ^
£33
U
-
(C. 12)
~ ^ ~ £ * 31 F 34
*
(
F
■3—
+
+ f / 4 |sin h K 4 h 4 |
+ ~ ^ ~ £ * 3 2 F 33 +
♦
< < f\
2
- — U* | c o s h K 4h 4 |
44 — 2
C 33
+
*
G
\ 23
£>34 C 34
J
(C.13)
where
D 31
hV
{~JTF
=
~jR*z3
J
^ )
h\ *
£>32 =
- j R K :3G
D 33 = ( ^ K )l3G e + y7G *)
Z) 34 = (-;/? k ,3F
(-31
+ j T F h)
= R e ( t / 3 c o s h K 3 h 3sinh*K 3 h 3)
C 32 = R e ( f / 3 sinhK 3 h 3 c o s h * K 3 h 3)
2
C 33 -“ —
2 U 3 |coshK 3h3 |
2
+ v \ | sinh k
2
r 34 -- —
2 t / 3 |coshic 3h 3 |
e
P a2
e
% T D 2\ C 2\
=
(
P B2 ~
I
M
3~
\ Z 2J
*
£^23 (-21
+
J
U*
e
D 22 C 23 +
♦
/ /
"3—
V Z2
2
■ '2
"2
I -u ,
+ t / 3 |sinhK3h 3 1 -
e
+ { ^ D 22 C 22 +
2
A
D 21 C 24
♦
D 24 C 22 + H
\ 2>
(C 1 4 )
-^2
#
£*23 ( -
23
\ 2y
£*24 (-24
( C l 5)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
215
where
D 2l = [-JTH
D 2i
=
(
- JR k ^ H
e
[-JT I
h \*
- J R
k
^ I
J
D 23 = { - j R K y2i e + j T I h)
D 24 = { - j R K y2H
C 2l
=
+ j T H h)
R e (f/2coshK 2h 2sinh*K 2h 2)
C 22 = R e ({ /2sinhK2h 2 cosh *K 2h 2)
C
^23 -~ -2 U2 |coshK 2h 2 |
+ t/^|sinhK 2^2 |
i
C
^24 -- -2 U* | c o s h K 2 h 2 |
P M = 4— [~jTJ + j R K :lj ' j
P Bl = (JRKylJ
+
i
U2 | s i n h K 2h 2 1 - U *
R e ( { /iS in h K ,h ,c o s h * K ih 1)
f M*
+ j T J ) "yRe ( £ / , co sh K , h,sinh*K ,h1)
I 2'
e h
with the field coefficients A ,A
-U 2
(C.16)
(C l 7)
eh
taken from Appendix B.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
216
APPENDIX D
NUMERICAL METHODS WITHIN THE
SPECTRAL DOMAIN METHOD
A. Bessel Function Approximation
The polynomial approximation for the zeroth-order Bessel function of the first kind for
the Fourier transformed basis functions of (2.37) and (2.38) is represented as [68 ] for
complex argument y as
0 < [y| <3
J 0(y) = 1 - 2.2499997 (y/3) 2 + 1.2656208 (y/3) 4 - .3163866 (y/3) 6
+ .0444479 (y/3) 8 - .0039444 (y/3)‘° + .0002100 (y/3) ' 2
(D. 1)
3 < [y| < oo
_I
•A)(v)
= y 2/ 0 cos
e0
(D.2 )
where
/ 0 = .79788456 - .00000077 (3/y) - .00552740(3ly)
- .00009512(3ly)
+ .00137237 (3/y) 4 - .00072805 (3ly)
+ .00014476 (3/y) 6
and
Q0 = y - .78539816 - .04166397 (3/y) - . J0003954 (3/y ) 2 + .00262573 (3/y) 3
-.00054125 (3/y) 4 - .00029333 (3/y) 5 + .00013558 (3/y)6 .
B. Numerical Integration
For the infinite-width structure of (2.33a)-(2.33d) and Fig. 11, the elements of the
matrix [A] require the evaluation of integrals to solve for the complex propagation
constant y and these integrals cannot be expressed analytically.
Therefore, numerical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
217
integration (also referred to as quadrature) is necessary to solve for the system of
equations [69], The quadraturemethods are based on the summation of anintegrand at
sequence of points(abscissas) within therange of integration.
a
Thegoal is to obtainthe
integral as accurately as possible with the smallest number of function evaluations of the
integrand. The typical numerical integral of a function is approximated by the sum of the
functional values at a set of equally spaced points and multiplied by certain weighting
coefficients. By appropriately choosing these coefficients, integration formulas of higher
order can be realized which translate to higher accuracy only when the integrand is very
smooth or is well-approximated by a polynomial. The Gaussian quadrature methods also
allow the selection o f the abscissas at which the function is to be evaluated and provides
another degree o f freedom. In other words, given W(x) (some known function), an integer
N, a set of weights B i , and abscissas x( , the integral approximation for the function fix)
becomes
b
N
J W{x)flx)dx * Z B , A x , )
a
(D.3)
/=1
and is exact if/(x) is a polynomial and
b
B, = j W(x) L , dx
a
(D.4)
where Lt (x) is the Lagrange multiplier function which is used for the generation of the
interpolating polynomial for unequally spaced arguments. The arguments x,,...,xv are the
zeros of the M h degree polynomial PN belonging to a family of functions having the
orthogonal property
b
j JV(x)PN P M dx = 0
M*N.
(D.5)
a
Many different orthogonal polynomials can be employed to obtain Gauss-type quadrature
formulas which can very accurately model the integrand. The choice o f the polynomial will
depend on the type o f function to be integrated and the limits o f the integral. In the SDM,
(2.33) is integrated and by analyzing the functional nature o f the zeroth-order Bessel
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
218
functions of the first kind from (2.37) and (2.38), the Legendre polynomials are selected
from [70], The integrals of (2.33) behave like
a* ,
a
which corresponds to Legendre polynomials and the integration formulas obtained are
termed Gauss-Legendre quadrature formulas with W(x)= 1.
The integration procedure
utilized in this dissertation is the Appoint Gauss-Legendre quadrature formula (GAULEG)
from [69], The routine scales the range of integration from (a,b) to (-1,1) and provides
absciassas xj , weights Bt , and uses a special formula that holds for the Gauss-Legendre
case as
- ( i =
? f e
<D 7)
where Ps is the Legendre polynomial of degree N and Newton's Method is employed to
determine the derivative (the prime designation) of the Legendre function in (D.7). A
recurrence relation also exists among Legendre polynomials of three consecutive degrees
(N+ 1)PN+l (x) = (2N + 1)xPN(x) - NPn_, (x) .
(D.8)
The dominant or truncation error term involves the (27/)th derivative o f J[x) and translates
to very high orders.
The best method for evaluating the accuracy of the integral
determined by Gauss-Legendre quadrature is to compare the results with different values
of N. Caution must be employed due to the presence o f one or more singularities in_/(x)
or if/(x) is highly oscillatory. Also, with very large values o f N, roundoff errors can be
significant to deteriorate the accuracy of the integral.
The limits of integration for the computation of the matrix elements o f (2.33) are from
ky = 0 to k y = oo via (2.47) and is separated into two integrals with a lower and
asymptotic contributions. An example for the following element in the [A] matrix is
n
A pm = 2 J {
0
00
+ 2 J { }dky = 4 , + A 2 .
Ni
(D.9)
The Fourier integrals are usually defined and evaluated along the real spectral axis of ky.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
219
The actual limits of integration for the upper integral range are from k = N x to k=N2. iV, is
determined by setting the value initially to 15 and increasing by a factor of 15 until the
change in the complex determinant (real and imaginary parts) o f (2.34) is less than 1%
(convergence criterion) using a 10-point Gauss-Legendre procedure. The routine locates
the integral region where y is no longer a contribution by comparing the determinant of the
matrix for two y initial guesses that are vary considerably. In this acceleration approach,
the asymptotic integral contribution is only calculated once at eachfrequency.
This
approach is justified by analyzing (2.9) with ky large as
k 2 » ky .
(D.10)
The lower integral is broken into subregions o f length 5 (with respect to the spectral
variable) from k= N x to k=N2 and is solved by using an Appoint Gauss-Legendre
integration algorithm. Convergence is obtained by increasing N (the initial iteration sets
N= 5) sequentially by ten until the change in y (real and imaginary parts) is less than 1% as
part of the iterative root searching procedure.
C. Determinant of a Matrix
For a given set of linear algebraic equations written as
m
= i«]
(Dio
where [C] is a square matrix with as many equations (AO as unknowns with x
(j=l,2,...,N). [X] is the column vector of unknowns and [ft] is the right-hand side column
C 12
C 1N
H
C 22
••••
C 2\
*2
_ CNl
C N2
••••
CNN
1
r 2
=
1
1
1
1
X
C l\
C 21
X
1
i
1
vector written as
(D.12)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
220
The formal solution for the unknowns can be found by employing Cramer's Rule. Any
arbitrary unknown x is given by
d e ,[C j]
XJ
=
(D -13)
det[C]
where det is the determinant of the matrix and [C] ] is the matrix [C] with its yth column
replaced by [/?]. If all o f the elements o f R are zero as in (2.32), then the det [C; ]=0 since
one entire column is zero. Then, nontrivial solutions can only be obtained if det [C]=0.
Suppose the matrix [C] can be written as a product o f two matrices
[L][U\ = [C]
(D. 14)
where [L\ is lower triangular (has elements only on the diagonal and below) and [U\ is
upper triangular (has elements only on the diagonal and above). In other words, (D. 14)
can be presented as
a li
a 21
a 22
a M a ,V2
0
0
am
P.l P l 2
0 (322
0
0
IN
li
12
'
'
c IN
2N
-21
22
'
'
C 2.V
A/2
' ••
c .v.v
P.w
(D. 15)
Decomposition is used to solve for the linear equation set. The advantage of breaking one
linear set into two successive ones is that the solution o f a triangular set of equations is
quite trivial using forward and back substitution methods. Given [C], [Z,] and [£7] are
determined using Crout's algorithm which solves for the a's and P's by arranging the
equations in a certain order. In other words the right-hand side of (D. 15) is expressed as
3H 11 3h 12
a 21
22
a Wl
a .V2
- 3k IN
2N
(D. 16)
NN
and Crout's method fills in the above combined matrix by columns from left to right and
within each column from top to bottom. The implementation o f Crout's method includes
partial pivoting. Pivoting is the selection o f the appropriate matrix element for the division
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
221
calculations and is essential for numerical stability purposes.
largest element of each row.
The pivot element is the
Only partial pivoting (interchange o f rows) can be
implemented efficiently. Actually the matrix [C ] is not decomposed into LU form, but
rather it is decomposed into a rowwise permutation o f C. An L U decomposition algorithm
from [71] (LUDCMP) is utilized and keeps track of whether the number of row
interchanges are even or odd. The determinant o f an L U decomposed matrix is just the
product o f the diagonal elements from (D. 16) as
N
det = I Pr
(D.17)
D. Root Searching Algorithm
The root solutions for the propagation constant y o f the CBCPW are determined by an
iterative root search of the determinant equation o f (2.34).
Recall again that the
propagation constant is a complex quantity. Equation (2.34) can be written in general
terms as
A x) = 0
(D.18)
where x is the desired solution or solutions and / is the dimensional function whose
components are the characteristic system of equations. Root finding proceeds by iteration.
Starting from some approximate trial solution, a useful algorithm will improve the solution
until some predetermined convergence criterion is satisfied.
For smoothly varying
functions, robust algorithms will always converge provided the initial guess is accurate.
The algorithm should also bracket a root between bracketing values and then search to
converge to the solution.
Consider a point x0 which is not a root of the function A x), but is reasonably close to a
root. Expanding A x) *n a Taylor series about x0 becomes
A x) = A Xo) + (x ~ x 0) f / (x0) + ^ " 2 °^ • • / / / (* o) +
(D. 19)
where the single and double prime represents the first and second derivatives, respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
222
If7(x) is set to zero, then x must be a root. Unfortunately, the above equation (right-hand
side) is a polynomial of infinite degree. However, an approximate value of the root x can
be obtained by taking the first two terms o f (D. 19) and solving for x. Now x represents
an
improved estimate o f the root and can replace x0 onthe next iteration. The
general expression for this method (Newton's) can be written as
1*1 - ! . = » * ,
= -
(D-20 )
where the subscript n denotes values obtained on the wth iteration and (n+ \) indicates
values found on the (n+ 1,1th iteration. Newton's method will converge to a root for most
functions and if it does converge, it will usually do so extremely rapidly.
The secant method is essentially a modification o f the conventional Newton's method
with the derivative replaced by a difference expression.
This is advantageous if the
function is difficult to differentiate. Also, it is only necessary to provide a single function
subprogram rather than subprograms for both the function and its derivative. Replacing
thederivative in (D.20) by a simple difference representationyields
l» l - i . = 5*,
------------- ^
----------
(D.21)
where / (x„.,) must be saved in this method. This is the value o f / from two iterations
previous to the present one. Since no such value will be available for the first iteration,
two different initial guesses for the root must be supplied. For the secant method to be
effective, the functions must be smooth near a root since the root does not necessarily
remain bracketed. Muller's method [72] generalizes the secant method but uses quadratic
interpolation (solving for the zeros of the quadratic) among three points instead of linear
interpolation through the two most recently evaluated points as in the secant method.
Muller's method is a preferred algorithm for finding complex zeros of analytical functions
in the complex plane. Given three previous guesses for the root xn_2 , xn l , xn and the
values of the polynomial J(x) at those points, the next approximation x^, is produced by
the following
* = &
<D'22a>
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
223
X = q f ( x n) - q( 1 + q ) f ( x n_i) + q 2f ( x ^ 2)
(D.22b)
Y = (2q + 1) / ( x „ ) - ( l + ^ ) 2 / ( x ll_i) + q 2f ( x n_2)
(D.22c)
Z = (1 + * ) /( x B)
(D.22d)
and
y±
2Z
JY2 - 4 x z
(D.23)
where the sign in the denominator is chosen to make its absolute value or modulus as large
as possible. The method allows the iterations to start with any three values of x such as
three equally spaced on the real axis and will locate complex roots.
The three initial
guesses of the solution for the dominant CBCPW mode complex propagation constant in
this dissertation are
Yj = * o V maX (e rl ,Sr2>Sr3,er4,Sr5<er6)
(D.24a)
y 2 - k o ^ (e rl + e r2 + s r3 + e r4 + s r5 + s rt) / 6
(D.24b)
Y3 = (Y, + Y2) / 2
(D.24c)
where max represents the maximum value o f the relative dielectric constants of the six
possible layers of the structure.
Muller's method has shown to be a very robust root
searching procedure. The algorithm is terminated when the magnitude o f the computed
change in the value of the root 5n+, is less than some predetermined quantity p. In this
analysis, p is set at 0.1 and the root convergence requirement for the real part of y is
(D.25)
and is similarly employed for the imaginary part o f y. The above equation is derived from
(D.21) with 6^, being the second term on the right-hand side.
E. Basis Function Expansion Coefficients Solution
After the true value of y is determined from the SDM, the basis function expansion
coefficients o f (2.27) are solved from (2.32). In matrix form, this relation is written from
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
224
(2.30) and (2.33) as
11
11
^ ,1
A 12
11
A 21
A 22
21
AU
21
A N\
11
12
21
21
21
A N2
21
A 12
12
12
A 22
-
a xn
22
22
A NM A VI
A A/2
'M
0
dx
0
22
22
A 22
0
12
12
22
22
0
A 2N
22
A ,2
A 21
“
12
A IN
A
n MN
22
A 2M
12
A M1 A M2
A 1M A n
21
A
12
a mm
21
A 12
21
^21
11
A 2M A 21
11
A M2
12
A IM A n
11
11
A Ml
11
0
A 2N
22
A
nn
0
4N
_
_
(D.26)
where this homogeneous system o f equations and the matrix [A] is a (M-*-N) x (M+N)
matrix. In this case, while no unique solutions are available, certain relationships exist
between the unknowns.
The values of the expansion coefficients (cm and dn), which
determines the relative amplitude o f the eigenvector distribution, depend on the strength of
the excitation of the structure. These coefficients are normalized with respect to the first
expansion function coefficient (c,) which is arbitrarily set equal 1.0. This implementation
modifies (D.26) as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
225
11
A 12
11
A IU
11
A 22
11
Am
21
A 12
11
A 2M
A
21
A N2
A U
im
21
A 2M
21
A NM
12
12
A ,2
A IN
12
12
12
A 21
A 22
A IN
12
A m\
21
A
2!
A 22
11
12
22
^11
12
A M2
A
12
MN
22
22
A 12
A W
22
22
22
A 21
A 22
A IN
22
22
22
A m
A A/2
A NN
11
Am
21
AU
21
I21
21
^.Vl
(D.27)
The [A] matrix is not square (M+N) x (M+N-l). In this case, more equations exist than
unknowns. A method is required to find the least-squares solution to the overdetermined
set of linear equations.
Singular value decomposition (SVD) is the method o f choice for solving most linear
least-square problems.
SVD is a technique for dealing with matrices that are either
singular or else numerically very close to singular. SVD methods [73] are based on the
following theorem o f linear algebra. For a general M x N matrix [A] whose rows M is
greater than or equal to its number of columns N, can be written as the product of an M x
N column orthogonal matrix [U\, a x \ N x N diagonal orthogonal matrix [W] with positive or
zero elements, and the transpose of an N x N orthogonal matrix [V], These relations can
be represented as
[A] = [U] [ W \[ V T\
(D.28)
The matrices [t/] and [V] are both orthogonal in the sense their columns are orthogonal.
The decomposition of (D.28) can always be accomplished. If the matrix [/I] is a square N
x N, then [£/], [V], and [W] are all square matrices o f the same size. The matrix inverses
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
226
are trivial to compute.
[U\
and [V] are orthogonal so the inverses are equal to the
transposes. [W] is diagonal and the inverse is the diagonal matrix whose elements are the
reciprocals of the elements w}(j=l,2,....,N). From (D.28), it follows that the inverse of [A]
is
[A]~l = [V] [diagonal (l/w; )] [ f / 7] .
(D.29)
The problem with this matrix construction is if one of the wjs is zero or numerically so
small, the value o f (D.29) is dominated by roundoff error.
Now consider the general set of simultaneous equations
[A] [x] = [B]
(D.30)
where [A] is a square matrix and [5] and [x] are column vectors. Equation (D.30) defines
A as a linear mapping from the vector space x to the vector space B. A matrix is singular if
the condition number (ratio of the largest of the w's to the smallest of the w's) is infinite
and is ill-conditioned if the condition number is too large (approaches the computer
floating point precision). If [A] is singular, then there is some subspace of x, called the
nullspace, that is mapped to zero as [/4 ][x]=[0 ], There is also some subspace of B which
can be related by A , in the sense that there exists some x which is mapped there. This
subspace of B is called the range of A. SVD explicitly constructs orthogonal bases for the
nullspace and the range of a matrix.
Specifically, the columns o f [U\ whose
same-numbered elements Wj are nonzero and are an orthogonal set of basis vectors that
span the range. Also, the columns o f [V] whose same-numbered elements w. are zero and
are orthogonal basis for the nullspace.
The important point from (D.30) is whether the
vector B on the right-hand side lies in the range of A. If this is true, then the singular set
of equations does have a solution [x]. In fact, more than one solution exists since any
vector in the nullspace can be added to x in any linear combination. To select one single
2
solution of this set, an appropriate choice would be the one with the smallest length |x| .
Using SVD, the Vwjs are replaced by zero if w= 0. The solution is computed working
from right to left as
[x] = [V\ [diagonal ( 1/w,)] [ l / 7’][£].
(D.31)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
227
Equation (D.31) will be the solution vector of smallest length. The columns o f [V] are in
the nullspace and complete the specification o f the solution set. If B is not in the range of
the singular matrix [A], then the set o f equations o f (D.30) has no solution. However,
(D.31) can still be used to construct a "solution" vector [x], This vector will not exactly
solve (D.30) but among all possible vectors x, the SVD method will find the least-squares
compromise solution. This solution is the one that minimizes the residual of (D.30). SVD
must be implemented carefully by deciding the threshold to zero the small w's ( 1.0 x 10-6
chosen in the routine). First, the SVD of a matrix [A] is calculated from a call to routine
SVDCMP. The solution vector [x] is obtained for a right-hand side vector [B] via (D.31)
with a backsubstitution routine SVBKSB.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
228
APPENDIX E
PROPAGATION CHARACTERISTICS OF IDEAL
PARALLEL PLATE WAVEGUIDE MODES
For an ideal homogeneously filled parallel plate waveguide formed by conductors
covering the *=0 and x=-h4 in Fig. E. 1 with h5=0, the wave functions for the TEM, TMr ,
and TEXare
TE
(E d
TM
- cosl hf) e
(E2)
TEM
~ jk
TEM
vp0
= *
Z
(E.3)
c0
and the propagation phase constant for the TMm and TEm modes with m = 1,2, . . . , o o
becomes
and that for the TEM
TEM
k c,0
=k =
C O jS re o ^ P o
(E.5)
.
A multi-layered CBCPW configuration proposed in this dissertation can be employed to
prevent the leakage effects up to a critical frequency. From Fig. 11 two inhomogeneously
filled parallel plate waveguides exist above and below the circuit conductors/ground planes
at x=0 in the region outside the cross-sectional circuit area ( [y| >S/2 + W). Consider only
the lower ideal parallel plate waveguide here in Fig. E.l.
Most of the modes are now
hybrid with both E, and H: components. TEM mode does not exist and the characteristic
equations for the mode propagation constant of the TM^ and TEr modes are respectively
[9]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. E. 1. Parallel plate waveguide representation with two dielectric substrates.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
with m = 0, 1, 2 , . . . ,oo. The above transcendental equations for the TM and TE modes are
solved for the mode propagation constants (kc).
Unlike the homogeneously filled
waveguide, the TM and TE modes will have different cutoff frequencies.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
231
APPENDIX F
DERIVATION OF THE RESIDUE CALCULUS THEOREM
The derivation o f the residue calculus terms of (3.12) for a leaky waveguide is
presented [74]. Assume fiz) is analytic within and on a simple closed curve C except at a
number of poles a,b,c,
interior to C and C is traversed in a positive (clockwise) sense as
shown in Fig. F. 1 then
| J{z) dz = 271/ {sum of residues of f(z) at poles a, b, c, ...}.
C
(F. 1)
By constructing cross cuts from any point on C to any point on C,, C2, C3,.... the following
relation can be derived
j f[ z) dz =
C
| ftz)dz +
C,
j J(z)dz + | J(z)dz + ....
C2
(F.2)
C3
where J{z) is analytic in the shaded area. If f(z) has a pole of order n at z=a but is analytic
at every other point inside and on a circle C with center at a,then (z-a)nfiz) is analytic at
all points inside and on C and has a Taylor series about z=a such that
<yV
Cl—ri+\
y
(7 -1
+ (^ 5 F
+
■+
v
/
v
+ a ° * a ' (z ~ a) + a 2 ( z ~ a)
^
+
(F .3)
and is
called the Laurent series forf(z). By integrating (F.3)yields
fiz)dz = |
I
a~\ -dz + | -- - - - - d z + ... + <f ■- -~ dz +
I W
c (.z-d)
I
| j a 0 + a , ( r - a ) + a 2( z - a ) 2 + ...Jcfe
c
= 27ij a_x.
(F.4)
This is valid from Cauchy's Theorem stating that fora simple closed curve C, if f(z) is
analytic within the region bounded byC as well as on C then
| j{z) dz = 0.
C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(F.5)
Fig. F. 1. Illustrative example of the residue calculus theorem with a number
o f poles a,b,c interior to a simple closed curve C.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
233
The following relationship is also invoked in (F.4) as
f _ d z _ = j 27zj if n= 1
1
I (?-a)n
j 0
if «=2,3,4,... J
where C is a simple closed curve bounding a region having z=a as an interior point. The
point a , in (F.4) is the residue o f f{z) at the pole z=a. The above theory can be applied
similarly for poles b and c and hence (F. 1) is derived.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
234
A P PE N D IX G
IEEE PERMISSION TO REPRINT COPYRIGHTED MATERIAL
IE E E
OPERATIONS CENTER
T H E I N S T I T U T E O F E L E C T R I C A L A N D E L E C T R O N I C S E N G I N E E R S . INC.
■US HOES l a n E. P O. 9 0 X 1331. PISCATAWAY. NJ 08855-1331. 'J.S.A. TEL. !908) 981-0060 TELEX: 333233 PAX: 1908) 981-0027
D IR E C T NU M BER (908) 562- 3 9 5 5
April 18, 1996
Mr. Mark Magerko
11500 Jollyville Road
#1121
Austin, TX 78759
Dear Mr. Magerko:
This is in response to your letter of April 10 in which you have requested permission to
reprint, in your upcoming dissertation, four of your IEEE copyrighted papers. We are
happy to grant this permission.
Our only requirement is that the following copyright/credit line appears prominently on the
first page of each reprinted paper, with the appropriate details filled in:
® 199x IEEE. Reprinted, with permission, from (full journal name;
volume, issue, and page numbers; month/year of issue).
Sincerely yours,
William J. Hagen, Manager
Copyrights and Trademarks
WJH:lp
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
235
VITA
Mark Alexander Magerko was bom in Colorado Springs, Colorado on August 2, 1962,
the son of John Sr. and Mary Ann Magerko. After finishing his work at Kaneland Senior
High School in 1980, he entered Northern Illinois University in DeKalb, Illinois to
complete pre-engineering courses. Afterward, he transferred to the University o f Illinois at
Champaign-Urbana and received his B.S. and M.S. in Electrical Engineering in 1984 and
1987, respectively. From 1984-1995, he was employed with several companies including
Magnavox Electric Systems in Fort Wayne, Indiana, Cascade Microtech, in Beaverton,
Oregon, Raytheon Research Labs in Lexington, Massachusetts, and Dell Computer in
Austin, Texas. He joined the graduate school o f Electrical Engineering at Texas A&M
University in 1989 where he worked as a research assistant and assistant lecturer in the
Electromagnetics Group under Dr. Kai Chang. He was awarded a Fellowship from the
Department of Education - Areas of National Need Program at Texas A&M University.
Permanent Address:
138 Neil Road
P.O. Box 529
Sugar Grove, Illinois 60554.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
9 365 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа