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Parallel-plate dielectric waveguide for multilayer microwave/millimeterwave integrated circuits: Analysis and experiments

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PA R A L L E L -PL A T E D IE L E C T R IC W A V EG U ID E
F O R M U LTILA Y ER
M IC R O W A V E / M ILLIM ETERW AVE
IN T E G R A T E D C IR C U IT S:
A N A L Y SIS A N D E X P E R IM E N T S
D ISSE R T A T IO N
S u b m itted in P artial Fulfillm ent
o f th e R equirem ents for th e
D egree o f
D O C T O R OF P H IL O SO P H Y (E lectrical Engineering)
at the
P O L Y T E C H N IC U N IV E R S IT Y
by
G od frey K w ok-C hiu K w an
June 1999
A pproved:
Copy No..
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9936830
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A p p rove d b y th e G uidance C om m ittee:
M a jo r: E le c tric a l Engineering
N iro d K . Das
Associate Professor o f
E le c tric a l Engineering
Thesis A d visor
M em ber:
Professor o f E le c tric a l E ngineering
aM em ber:
Spencer S.P. K u o
Professor o f E le c tric a l Engineering
M in o r: M a th e m a tics
C liffo rd W . M a rshall
Professor o f M a th e m atics
ii
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M ic ro film o r o th e r copies o f tliis dissertation
are o b ta in a b le fro m
U n iv e rs ity M icro film s
300 N. Zeeb Road
A n n A rb o r. M ichig a n 48106
i ii
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V IT A
G odfrey K w o k -C liiu K w a n was b om on June 30, 1963. in Hong Kong.
In 1983. he
studied the French language at La Sorbonne. Paris. In 1984, he studied a t 1 Ecole Superieure d ’[nform a tiq u e-E le ctro n !q u e-A u to m a tiq u e. Paris. He obtained th e B .S c.(F irst
Class Honours) degree in electronic and co m m un ica tion engineering fro m the Polytech­
nic o f N orth London. London, in 1986 and the M S E E degree from the S tate U n ive rsity
o f New Y ork at S to n y B ro o k in 1994. He worked a t P liilip s Hong K o n g as a systems
engineer. From 1987 to 1990. he worked as a software engineer at S IT A H ong K o n g and
as senior HLS program m er a t S IT A New Y o rk fro m 1991 to 1994. In 1994. he jo in e d the
Departm ent o f E le ctrica l Engineering, P o lyte clm ic U niversity, B rooklyn, New Y o rk, and
since then has been w o rk in g tow ard his P h .D . degree. His research interests include com­
p u ta tio n a l electrom agnetics, analytical and experim ental stu d y o f d ie le ctric waveguides
and antennas and m u ltila y e r microwave integrated circuits.
This w ork has been perform ed largely in the E M C A D laboratory. M icrow ave Lab­
o ra to ry at P olyteclm ic U nive rsity's Long Island Campus.
iv
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To my wife, Ming
V
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ACKNOW LEDGM ENT
I express m y deepest g ra titu d e to Prof. N iro d K . Das for liis guidance and s u p p o rt
d u rin g m y research.
T he research experience I have gained under his supervision has
been b o tli enjoyable and enlightening.
I am g ra te fu l to Professors J e rry Slimoys, Spencer K uo. and C liffo rd M a rsh a ll for
serving on m y com m ittee and fo r th e ir encouragement and advice.
Special thanks are due to Professors J. Shmoys and H. Stalzer fo r th e ir keen in ­
terest in m y w ork.
I especially th a n k Prof. C. M a rs h a ll for serving as m y advisor on
M athem atics. I also th a n k Prof. B olle and Prof. T a m ir fo r th e ir interest in m y w ork.
I th a n k a ll m y friends, especially A tanu M ohanty, B ria n M cCabe. W ei-Jen W ang
(J u s tin ). Peter Lee. C heng-C hi Cheng (M a tt), R u th ie Lyle, E dw ard Koretzsky, Joe
Huang, James B u rn e tt. C handra N elira, Jolin Hug and Y oung M in M oon for th e ir help
and su p po rt.
I th a n k m y w ife and b o th m y daughters for th e ir patience and forebearance and
m y fa m ily fo r a ll these years th a t th e y have stood b y me.
vi
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A N A B ST R A C T
P A R A L L E L -P L A T E D IE L E C T R IC W A V EG U ID E
F O R M U L T IL A Y E R M IC RO W AV E/M ILLIM ETERW A V E
IN T E G R A T E D C IR C U IT S:
A N A L Y SIS A N D E X P E R IM E N T S
by
G odfrey K .C . K w an
A dvisor: N iro d K . Das
S u b m itte d in P a rtia l Fulfillm ent o f the R equirem ents for
the Degree o f D o c to r o f Philosophy (E le c tric a l Engineering)
June 1999
T liis d isse rtatio n discusses the analysis, design and experim ents o f the Parallel-Plate
D ie le ctric W aveguide (P P D W ). and its applications in m u ltila y e r m icro w ave/m illim eterw ave
integrated circuits. T h e waveguide’s basic co n stru ction is q u ite sim ple. I t consists o f a
re ctangular d ie le ctric s trip sandwiched between two p a ra lle l m e ta llic plates. The funda­
m ental mode o f th is w aveguiding structure is being used.
M o d a l analysis is used to s tu d y the dispersion ch aracteristics o f th is waveguiding
s tru c tu re .
A m u ltila y e r spectral-dom ain approach is used to analyze the coaxial-to-
P P D W tra n s itio n a n d s lo t-to -P P D W tra n sitio n as means o f e xcitin g , o r coupling from,
the fundam ental m ode o f p ro pa g a tion w ith in th is w aveguiding s tru c tu re .
Based a m u ltila y e r spectral-dom ain approach, in p u t im pedance o f a coaxial-probe
to P P D W tra n s itio n is found using an a pproxim ation o f th e c y lin d ric a l coaxial-probe b y a
t liin s trip o f an equivalent w id th . T liis a p proxim ation re sults in a c irc u it model th a t can
v ii
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be used for efficient design and accurate sim ulation o f perform ance o f p ra ctical P P D W
circuits using c o a x ia l-to -P P D W transitions. S im ilar m u ltila y e r sp e ctra i-d o m a in /im a g in g
approach also leads to an a n a ly tic a l model o f a slot on the P P D W ground plane using
w liich circu it-m o d e l param eters can be extracted. (1) C o a xia l-to -P P D W -to -co a xia l tra n ­
sitions. (2) P P D W -to -m ic ro s trip interlayer couplers using slot-co up lin g , and (3) P P D W
slot-coupled m ic ro s trip patch antennas, have been successfully b u ilt and tested. Theo­
retical and experim ental results are compared. Im p o rta n t fundam ental characteristics
and practical considerations are addressed.
v iii
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E xperim ent escorts us lastHis pungent com pany
W ill not allow an Axiom
A n O pportunity.
Em ily Dickinson
ix
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Contents
ACKNOW LEDGM ENT
vi
ABSTR A C T
vii
1
In tr o d u c tio n
1
1.1
M o tiv a tio n ....................................................................................................................
1
1.2
Past a nd Recent W o rk on M u ltila y e r In te g ra te d C irc u its and M illim e te r
1.3
2
W a v e g u id e s .................................................................................................................
3
O rg a n isa tio n of T liis S tu d y ......................................................................................
4
T h e P P D W g u id e
6
2.1
M o d a l A n a ly s is ..........................................................................................................
8
2.2
W aveguide C haracteristics o f P P D W .................................................................
12
2.2.1
Waveguide Modes and D ispersion C h a r a c te r is tic s ..............................
12
2.2.2
C ha racteristic Im pedance o f P P D W .........................................................
12
2.2.3
M e ta llic Loss o f P P D W
.............................................................................
20
2.2.4
D ielectric Loss o f P P D W .............................................................................
23
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2.2.5
3
4
P P D W C irc u its D e n s i t y ...........................................................................
25
2.3
T h e 5CM2 P a ra lle l-P la te D ielectric Waveguide
...............................................
26
2.4
C o n c lu s io n ...................................................................................................................
32
C o a x ia l-P r o b e T r a n sitio n to P P D W
34
3.1
C o a x-to -P P D W T ra n s itio n A n a ly s is ...................................................................
36
3.1.1
C urrent E x c ita tio n o f the P r o b e ...............................................................
38
3.1.2
Incidence from a P P D W P o r t .................................................................
43
3.1.3
E quivalent C irc u it
.....................................................................................
46
3.2
Results and D is c u s s io n ...........................................................................................
48
3.3
E x p e rim e n ts ...............................................................................................................
51
3.4
C o n c lu s io n ...................................................................................................................
70
S lo t T ra n sitio n t o P P D W
71
4.1
M u ltila y e r Spectral Green's Functions o f a Vertical M agnetic D ipole
72
4.2
A nalysis o f S hort S lo t ..............................................................................................
80
4.2.1
Spectral Green's F u n c tio n s ........................................................................
80
4.2.2
A d m itta n c e o f Short S lo t ............................................................................
85
Parameters o f a P P D W Short Slot T r a n s i t i o n ...............................................
99
4.3.1
S u s c e p ta n c e ...................................................................................................
99
4.3.2
C onductance
................................................................................................
107
4.3.3
E f f ic ie n c y .......................................................................................................
112
4.3
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4.3.4
4.4
4.5
4.6
5
Param eter
................................................................................................
117
P ractical P P D W -S lo t T r a n s itio n s .......................................................................
117
4.4.1
M ic ro s trip -P P D W -M ic ro s trip In te rla ye r C o u p le r ................................
117
4.4.2
S lot-C oupled M icro strip Patch A n te n n a .................................................
132
Analysis o f a Long S l o t ..........................................................................................
139
4.5.1
Spectral Green's F u n c tio n s .........................................................................
139
4.5.2
A d m itta n c e o f Long S l o t ............................................................................
147
4.5.3
Efficiency o f Long-Slot T r a n s it io n ...........................................................
155
C o n c lu s io n ..................................................................................................................
157
C o n clu sio n
5.1
158
Future C h a lle n g e s.......................................................................................................
A A n a ly tic a l M o d e l o f a z-d irected S p e c tr a l M a g n e tic C u rren t S h e e t
xii
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162
164
List of Figures
1.1
Spectrum o f M illim e te r and S u b m illim e te r W a v e s .........................................
2
2.1
Cross-sectional vie w o f the P a ra lle l P la te D ielectric W aveguide...................
7
2.2
D escription o f some o f the sym bols used for defining a u x ilia ry p o te n tia l
functions.........................................................................................................................
10
2.3
Norm alized dispersion curves o f a P a ra lle l-P la te D ielectric W aveguide.
.
13
2.4
Geom etry for e v a lu a tio n o f integrals in the calculation o f guided power
.
15
2.5
V ariation o f c h a ra cte ristic im pedance o f P P D W w ith respect to th e w id th
o f the center d ie le c tric s t r i p ..................................................................................
2.6
Norm alized ch a ra c te ris tic im pedance curve o f the P arallel-P late D ie le ctric
W a v e g u id e ....................................................................................................................
2.7
18
21
Norm alized ch a ra c te ris tic im pedance curves o f the P arallel-P late D ie le ctric
W a v e g u id e .....................................................................................................................
22
2.8
Norm alized a tte n u a tio n coefficients due to m etal loss and d ie le c tric loss .
24
2.9
The decay ra te o f th e field o f P P D W m o d e ......................................................
27
2.10 A tte n u a tio n due to m etal loss fo r 50-12 P arallel-P late D ie le ctric W aveguides 29
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‘2.11 V a ria tio n o f a tte n u a tio n constant due to m e ta l loss and d ie le c tric loss o f
P arallel-P late D ielectric W a v e g u id e .....................................................................
30
2.12 V a ria tio n o f a tte n u a tio n constant in dB p e r guide wavelength due to m e ta l
loss and d ie le ctric loss
............................................................................................
31
2.13 N orm alized a tte n u a tio n constant due to m etal loss o f 50A2 P a ra lle l-P la te
D ielectric W aveguide using substrates o f d ie le ctric constant 1 2 . 0 ..............
33
3.1
G eom etry o f a coaxial to P arallel-P late D ie le c tric Waveguide tra n s itio n . .
35
3.2
G eom etry o f an equivalent problem used fo r analyzing a c o a x ia l-to -P P D W
tra n s itio n ........................................................................................................................
3.3
37
A P P D W mode w ith electric field Ei. in cid e n t on a coaxial probe located
at the center o f the P P D W .......................................................................................
44
3.4
Equivalent c irc u it o f c o a xia l-to -P P D W t r a n s i t i o n ..........................................
47
3.5
Design d a ta for the characteristic im pedance o f a P P D W .............................
50
3.6
C om puted values o f the real p a rts o f th e in p u t im p e d a n c e s ..........................
52
3.7 C om puted values o f norm alized X c o a x ...............................................................
53
3.8 M a xim u m substrate thickness allowed w ith P P D W below T E l l c u to ff . .
54
3.9
55
N orm alized Xcoax
....................................................................................................
3.10 P ro to typ e geom etry o f a tw o -p o rt co a x-P P D W -coax tra n s itio n , used for
e x p e rim e n t....................................................................................................................
3.11 E quivalent c irc u it for the coax-P P D W -coax tra n s itio n o f F ig .3 .10
. . . .
58
59
3.12 Real and im a g in a ry parts o f the in p u t im pedance o f each c o a x ia l-to -P P D W
tra n s itio n , used in the p ro to typ e g e o m e try ........................................................
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60
3.13 Measured and com puted insertion loss o f p ro to ty p e A10 coax-P P D W -coax
tra n s itio n ............................................................................................................
61
3.14 Measured and com puted S m ith-C hart p lo t o f S21 o f p ro to typ e A10 coaxP P D W -coax t r a n s i t i o n ................................................................................
62
3.15 Measured and com puted S l l o f p ro to typ e A10 coax-PPD W -coax tra n s itio n
3.16 Measured and com puted insertion loss o f another coax-P P D W -coax tra n ­
sition (p ro to ty p e A8_50)................................................................................
64
3.17 Measured loss o f p ro to ty p e A10 as com pared to sim u la tio n in case P P D W
is lossless and probe impedance is not inclu d e d .......................................
66
3.18 Measured loss o f p ro to ty p e A10 as com pared to sim u la tio n in case P P D W
loss and probe impedance are b o th excluded...........................................
67
3.19 Measured loss o f p ro to ty p e A10 as com pared to sim u la tio n in case P P D W
is lossless b u t probe impedance is included...............................................
68
3.20 Measured loss o f p ro to ty p e A10 as com pared to sim u la tio n in case P P D W
is lossy and probe impedance is also in clu d e d ..........................................
69
4.1
G eom etry fo r evaluation o f m u ltila y e r Green's fu n c tio n ........................
74
4.2
G eom etry fo r evaluation o f m u ltila ye r Green's fu n ctio n for a ve rtica l mag­
netic dipole inside a dielectric slab
4.3
G eom etry o f a P arallel-P late D ielectric W aveguide rectangular slot
s itio n
4.4
82
tra n ­
86
G eom etry o f an equivalent problem used fo r analysing a P P D W -slot tra n ­
s itio n
87
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63
4.5
C irc u it m odel d e p ic tin g the re la tio n o f p a ram ter np to other param eters
o f an equivalent c ir c u it fo r a P P D W rectangular s lo t......................................
98
4.6
G eom etry o f P P D W guide and slot, n o tations and sym bols.........................
100
4.7
N orm alized Susceptance (param eters v a r y ) ........................................................
103
4.8
Norm alized Susceptance (param eters v a r y ) ........................................................
104
4.9
N orm alized Susceptance (param eters v a r y ) ........................................................
105
4.10
N orm alized Susceptance (param eters v a r y ) ........................................................
106
4.11
N orm alized Susceptance (param eters v a r y ) ........................................................
108
4.12
N orm alized Susceptance (param eters v a r y ) ........................................................
109
4.13
N orm alized Susceptance (param eters v a r y ) ........................................................
110
4.14
N orm alized Susceptance (param eters v a r y ) ........................................................
Ill
4.15
N orm alized C o n d u c ta n c e .......................................................................................
113
4.16
Efficiency (param eters v a r y ) ................................................................................
114
4.17
Efficiency (param eters v a r y ) ................................................................................
116
4.18
Efficiency (param eters v a r y ) ................................................................................
118
4.19
Param eter r i p ............................................................................................................
119
4.20 Physical g e om etry o f aperture-coupled m ic ro s trip -P P D W -m ic ro s trip m u l­
tilaye r coupler...............................................................................................................
4.21
Equivalent c irc u it o f m ic ro s trip -P P D W -m ic ro s trip m u ltila y e r coupler.
120
. . 122
4.22 Measurement a n d com puted values o f S21 o f aperture-coupled M ic ro s trip P P D W -M ic ro s trip in te rla ye r coupler. Single-stub m a tc liin g ...........................
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124
4.23 Measurement and com puted values o f S l l o f aperture-coupled M icro strip P P D W -M ic ro s trip in te rla y e r coupler. S ingle-stub m a tc lu n g ..........................
125
4.24 Measurement and com puted values o f S21 o f aperture-coupled M icro strip P P D W -M ic ro s trip in te rla y e r coupler (w id e b an d ) using double-stub match­
ing ....................................................................................................................................
127
4.25 Measurement and com puted values o f S l l o f aperture-coupled M icro strip P P D W -M ic ro s trip in te rla ye r coupler (w id e b an d ) using double-stub m atch­
ing.....................................................................................................................................
128
4.26 Measurement and com puted values o f S21 o f aperture-coupled M icro strip P P D W -M ic ro s trip inte rla ye r coupler (n a rro w b an d ) using double stub m atch­
ing.....................................................................................................................................
130
4.27 Measurement and com puted values o f S l l o f aperture-coupled M icro strip P P D W -M ic ro s trip in te rla ye r coupler (n a rro w b an d ) using double stub m atch­
ing.....................................................................................................................................
131
4.28 G eom etry o f a P P D W slot-coupled M ic ro s trip p a tch antenna........................... 133
4.29 Tranm ission lin e m odel o f a P P D W slot-coupled M ic ro s trip p a tch antenna
p r o t o t y p e ....................................................................................................................
134
4.30 Measured and com puted S l l o f P P D W slot-coupled M ic ro s trip patch an­
tenna................................................................................................................................
136
4.31 Measured and com puted S m ith C ha rt p lo t o f P P D W slot-coupled M i­
cro strip patch antenna................................................................................................
137
4.32 Measured and com puted S m ith C h a rt p lo ts o f P P D W slot-coupled M i­
cro strip p atch antenna................................................................................................
x v il
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138
4.33 Geom etry for eva lu a tio n o f m u ltila y e r Green's fu n c tio n fo r a v e rtica l mag­
netic dipole above a d ie le c tric slab.........................................................................
140
4.34 Geom etry for eva lu a tio n o f m u ltila y e r Green's fu n c tio n fo r a v e rtica l mag­
netic dipole below a d ie le c tric s l a b . ....................................................................
145
4.35 Efficiency o f P P D W S lo t-T ra n s itio n in w liic h the le n g th o f the slot can be
larger than the w id th , a. o f the P arallel-P late D ie le c tric Waveguide.
5.1
. .
A 3-layer M ic ro s trip -to -P P D W -to -M ic ro s trip coupler c o n fig u ra tio n using
coaxial tra n s itio n s ........................................................................................................
5.2
160
One possible co n fig u ra tio n o f P arallel-P late D ie le ctric W aveguide in a m ul­
tilaye r in te g ra te d -c irc u it environm ent...................................................................
A .l
156
161
Geom etry o f m agnetic dipole layer for sim u la tio n o f a v e rtic a lly directed
magnetic cu rren t s h e e t ............................................................................................
x v iii
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165
Chapter 1
Introduction
1.1
Motivation
The present work has been m o tiv a te d by the growing interests o f M M IC (M o n o litic M i­
crowave Integrated C irc u its) interconnection alternatives fo r m u ltifu n c tio n a l, m u ltila ye r
M M IC system modules. M ore ro b ust and low loss layer-to-layer tra n sitio n s are needed in
order to meet the requirem ents o f the next generation o f liig h density, m u ltila ye r M M IC
systems.
M illim e te r and su b m illim e te r waves have always been o f interest to m any disciplines
o f scientific research due to its strong interaction w ith m a tte r a t the molecular and
a to m ic levels. Frequencies in tliis band (see Fig. 1.1) is loosely defined to be from 30 GHz
(w avelength 1 cm) to 10.000 G H z o r 10 T H z (wavelength 30 /xm). Note th a t far-infrared
is around 1 TH z and n e ar-in fra red is a b ou t 100 TH z. T h is la rg e ly unexploited band o f
frequencies lies between the m icrowave band and the o p tica l band. As o f present, remote
sensing and radio astronom y are s till the p rim a ry users, scattered in the lower end o f
tliis frequency range. In co n tra st, the adjacent microwave spectrum , being used for m any
com m unication and radar a pplications, is already very crowded.
Needless to say, the b a n d w id th available in m illim e te r and subm illim eter-w ave band
1
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2
Visible
Infrared
■. n c u ’ ! i; ir !
\I)
cm
cm
.'m m
~-)GHz
GHz
10 G H z
') • mm
50 um
5 tun
U ltraviolet
-----------------0 3 um
0 03 um
3 0 0 G H z___________________________________________________________
100 G H z
! THz
!0 :T H z
100 T H z
10 ' H z
S nbm illim eter
Microwaves
*5---------3*
w aves
M illim e te r
waves
Fig.l.l
S pectrum o f M illim e te r and S u b m illim e te r Waves
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10 ‘ H z
3
is huge compared to w h a t is available in the m icrowave band. However, a p p lica tio n of
m illim e te r and s u b m illim e te r waves has m e t w ith some m a jo r d iffic u ltie s and has. to
date, rendered th is p a rt o f the electrom agnetic spectrum a very c o s tly m e d iu m available
to o n ly a sm all n u m b e r o f scientific researchers. A m a jo r problem to overcome is Olmiic
losses. M ic ro s trip lines used extensively in th e microwave spectrum become very lossy
at m illim e te r and subm illim eter-w ave frequencies. O the r problem s faced b y these metal
lines are ra d ia tio n and guided wave e xcita tio n .
It is the purpose o f th e present stu d y to explore a waveguiding s tru c tu re , nam ely the
P arallel-P late D ie le ctric Waveguide (P P D W ). T liis waveguide has n o t been used in the
past in n iicro w a ve /n iillim e te rw a ve circuits. We investigate th e o re tic a lly and experimen­
ta lly its p o te n tia l as an interconnection technology for m u ltila ye r m icrow ave/m illim eterw ave
integrated c ircu its. T h o u g h the scope o f a p p lic a tio n s include m illim e te r and subm illim e­
te r frequencies, p ro to ty p e experim ents o n ly in the microwave frequencies are used for
dem onstration due to convenience o f fa b ric a tio n and testing.
1.2
P ast and R ecent W ork on M ultilayer Integrated
C ircuits and M illim eter W aveguides
M u ltila y e r m icrowave integrated c irc u its is a re la tiv e ly new endeaver.
Due to the un­
a va ila b ility o f a waveguide th a t is sp e cifically designed for the m u ltila y e r environment,
a t present, m u ltila y e r m icrowave integrated c irc u its use m a in ly s trip lin e and m icrostrip
teclmologies. However, users are confronted w ith severe lim ita tio n s due to O lm iic losses
and spurious ra d ia tio n . T h e use o f p rin te d c irc u it technologies causes spurious radiation,
re su ltin g in cross-talk, power leakage and package resonance. In a m u ltila y e r environ­
m ent. these problem s are either tolerated b y th e applications or some fo rm o f ad-hoc
solutions are in tro d u ce d . For example, p a ra lle l-p la te ra d ia tio n m ay be contained b y fenc­
in g "problem lo c a tio n s " w ith shorting pins, m e ta l walls or m aterials o f h ig h p e rm ittiv ity .
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4
In the past, rectangular m e ta llic waveguides were usually used in the study o f
m illim e te r and su b niillim eter waves, b u t th e y cannot be easily integrated.
Despite its
bulkiness, rectangular m e ta llic waveguides has been used in radars and even in satellite
com m unications systems m a in ly because there was really no b e tte r a lte rn a tive . Over the
years, a num ber o f dielectric waveguide structures have been investigated and proposed to
be used in the m illim e te r and s u b m illim e te r band. These include th e H -guide [1], the Im ­
age guide [2],[3], the Inverted S trip D ie le c tric guide [4], the Trapped Image guide [5]. the
N on ra d ia tive D ielectric guide (N R D -g uid e ) [6], etc. Some recent research effo rt has been
d rive n again b y radio astronom y as w ell as b y the com m ercial wireless m arket, leading
to the development o f a num ber o f novel transm ission lines such as m o n o lith ic dielectric
lines, m icrosliield lines, lens-supported coplanar lines and silicon m icrom acliined waveg­
uides [7]. These dielectric waveguides can be p o te n tia lly used for m u ltila y e r integration,
b u t s till suffer from m u ltim o d a l problem s, com plexity o f e x c ita tio n and tra n s itio n design,
a n d /o r size problems.
1.3
O rganisation o f T his Study
T he P arallel-P late D ielectric W aveguide(P P D W ). which we propose and investigate here,
is a candidate for m u ltila y e r c irc u its w ith strategic advantages. In chapter 2. a m odal
analysis is perform ed on the P a ra lle l-P la te Dielectric Waveguide.
D ispersion charac­
te ristics are derived and the mode o f operation for the P P D W guide is described.
A
transm ission line model is then developed for the P P D W guide. C h a ra c te ris tic impedance
is defined for the P P D W guide w liic h is p ra c tic a lly m eaningful as w ill be shown in la te r
chapters. Design parameters o f the waveguide and its consequential im pa ct on the waveg­
uide perform ance are examined. M e ta llic and dielectric losses o f th e P P D W guide is also
investigated and comparisons w ith m ic ro s trip lines are presented.
In chapter 3, an alm ost rigorous analysis o f the c o a x ia l-to -P P D W tra n s itio n is then
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carried out using the S p e ctra l-D o m ain approach. The analysis provides us a fo u n d atio n
on which an equivalent c irc u it for th e coaxial-probe to PPDVV tra n s itio n is derived.
P rototype waveguides fed b y the co a xia l probe are fabricated and the measured d a ta are
compared w ith sim ula tio n results.
A rectangular slot aperture on th e ground plane o f P P D W guide is th e n investigated
in chapter 4 using a m u ltila y e r S p e ctra l-D o m ain m ethod. A n equivalent c irc u it is devel­
oped for tliis s lo t-to -P P D W tra n s itio n and subsequently used for designing p ro to typ e s o f
slot-coupled m ic ro s trip -P P D W -m ic ro s trip m u ltila y e r couplers. Values o f c irc u it param ­
eters for P P D W slot tra n sition s are th e n calculated from the sp ectral-dom ain analysis.
These values are com puted for various cases o f geom etrical and m a te ria l properties. T he
P P D W is also used in an antenna feed co n figu ra tio n where a m ic ro s trip patch antenna
is on the other side o f the ground p la n e o f a P P D W guide and fed b y th e P P D W guide
through a slot on the ground plane. F in a lly , the a u th o r carries o u t an analysis o f a longslot tra n sitio n to the P P D W . for w lu c li e x c ita tio n efficiency is presented. T h e results and
conclusion o f o u r s tu d y facilitates m o re advanced development o f P P D W technology in
m u ltila y e r/m u ltifu n c tio n m ic ro w a v e /m illim e te r wave applications.
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Chapter 2
The PPD W guide
T h e P arallel-P late D ie le c tric Waveguide consists o f a re cta n gu la r dielectric s trip embed­
ded between tw o p a ra lle l m e ta llic plates. A cross-sectional view o f the waveguide is shown
in Fig.2.1. S im ila r geom etries have also been used in the H -guide [1] and the N R D -g uid e
[6]. However, b o th th e H -guide and the N R D guide chose to use liig h er o rd e r modes o f
th is parallel p la te s tru c tu re . In o u r present study, the focus is on the fu n d a m e n ta l T E \ q
mode in th is waveguide.
The fundam ental m ode o f the P P D W guide propagates down to D C . in contrast
to the H-guide and th e N R D guide where b o th have c u to ff frequencies below w liic h the
guides w ill n o t operate. T h is p ro p e rty o f the P P D W allow s the use o f p h y sica lly th in n e r
(sm aller b) substrates fo r a range o f m icrowave and m illim e ter-w a ve frequencies, w hich
is a desirable feature fo r integrated c irc u it ap plicatio n s.
In term s o f signal transm ission, the P P D W . u n lik e some o th e r waveguide stru ctu re s
w ith parallel m etal planes(striplines. conductor-backed slotline. conductor backed coplanar waveguide), has th e advantage o f confining power w ith in the dielectric s trip and w ill
n o t radiate o r leak in to the p a rallel plate modes. A t th e same tim e, the m etal plates also
serve to provide is o la tio n between one layer and th e n e xt, w liic h is a m a jo r requirem ent
o f m u ltila ye r m icrow ave integrated circuits. The N R D guide and the H -guide also have
6
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y
/
Sc ^ Ss
Ccenter dielectric
s•trip
/
Sc
\ Region I
Regie n 11
/
s5
Region III
\
metal plates
Fig-2.1
Cross-sectional view o f th e P a ra lle l P la te D ielectric W aveguide. a = w id th o f center d i­
electric s trip . 6=thickness o f su b stra te . ec ^ d ie le c tric co n stant o f center d ie le ctric s trip .
= d ie le c tric constant o f o u ts id e d ie le c tric substrate.
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8
the above properties. However, the com plexity involved in designing transitions to the
N R D guide and the H -guide has always been a m a jo r hurdle, where the ra d ia tio n loss
and m u ltim o d a l e x c ita tio n problem s in these tra n s itio n s often become unavoidable.
In the P P D W . a lth o u g h the fundam ental mode is n o n -T E M . its field co n figu ra tio n
in the central region is s im ila r to a q u asi-T E M line. T h is feature allows a P P D W to be
excited w ith o u t com plex tra n s itio n geometries. In fact, a coaxial probe can be placed
sim ply at the center o f the dielectric strip w ith the center conductor o f the probe con­
nected to the opposite m etal plate. We w ill examine in d e ta il the coaxial-probe e xcita tio n
o f the P P D W guide in chapter 3.
2.1
M odal A n alysis
The geom etry o f the waveguide is as shown in Fig.2.1. M odal analysis has been done
on this waveguide s tru c tu re b y selecting a p p ro p ria te vector p o te n tials in each region
and m a tcliin g th e ir b o u n d a ry conditions across the d ie le ctric interfaces.
M odal field
expressions are o b tained for each propagating mode. C lassification o f modes has been
done according to w he the r the mode is transverse e le ctric o r transverse m agnetic w ith
respect to the x -d ire c tio n and also w hether the m ode has even o r odd sym m e try w ith
respect to the x = 0 plane w hich is the bisecting plane o f th e w-aveguide geometry. Sym bols
used in defining the p o te n tia l functions as w ell as some o th e r symbols are ta b u la te d
in Fig.2.2. In itia l analysis sta rte d o u t w ith region I and region I I I containing different
substrates. A p p ro p ria te vector p o te n tia l functions are th e n selected for each o f the three
regions.
A t tliis p o in t, b o un d a ry conditions are m atched across the interfaces and a
set of transcendental equations are derived b y assum ing th a t region I and I I I co n ta in
identical substrates. Solutions o f these equations are evaluated num erically, p ro v id in g a
basis for ca lcu la tion o f pro pa g a tion constants o f a ll th e possible guided modes. The T E 10
solutions are used fo r the c a lcu la tio n o f p ro pagation constant, guided power, conductor
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loss, dielectric loss and ch a ra cte ristic impedance o f th e P P D W guide.
Set o f modes T E to x are found by le ttin g the m agnetic vector p o te n tia l fu n c tio n
.4 to be zero and the e lectric vector p o te n tia l F as follows:
F = uy
•
(2.1)
if’J E\4;JiEW 'n f are electric p o te n tia l functions for regions I . I I . I I I respectively.
O n the other hand, set o f modes T M to x are found b y le ttin g the electric vector
p o te n tia l function F to be zero and the m agnetic vector p o te n tia l .4 as follows:
.4 = uuFM:
H'™-W™ a r e
(2.2)
m agnetic p o te n tia l functions for regions I . I I . I I I respectively. Bound­
a ry co n d itio n for v TE is
= 0 on a perfect e le ctric conductor, ra is the u n it vector
n o rm a l to the boundary. T h e b o u n d a ry co n ditio n for (/’r ‘vf is
= 0 on a perfect e le ctric
conductor.
For T M to x modes, we choose the m agnetic vector p o te n tia l functions to be o f th e
fo llo w in g form.
Region I: x < —|
(2.3)
Region II: —| < x < |
(even sym m e try w .r.t. x):
1 '™ ‘ = C ™ * c o s( k ™ ‘x)sin(k ™ y )c~ * -=
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(2.4)
10
Description
Symbols
rT E t
xl
rTEt
fT E e
’ ^ lIII
Electric field, x-directed component, TE even mode, for regions
I,II,III respectively. Similarly for y and z-directed electric field
components.
TJ TEe IT TEe T_T TEe
n xl » “ i l l » n iI I l
Magnetic field, x-directed component, TE even mode, for regions
I,II,III respectively. Similarly for y and z-directed magnetic field
components.
k I ’ k I I ’ k III
intrinsic propagation constants in regions I, II, III respectively
b TEe 7 TEe K TEe
x l > K x2 ’ x3
propagation constant, x-directed, TE even mode, in region 1(1),
11(2) and 111(3) respectively
k yTlE■>k y2TE’ Kk yJTE
propagation constant, y-directed, TE (even or odd) mode, in
region 1(1), 11(2) and 111(3) respectively.
kz
propagation constant along
mode propagation.
a
width o f central dielectric strip.
b
height o f central dielectric strip, or separation distance o f parallel
ground planes
s-'TEe
s-' TEe
TEe
r,
which is the direction o f waveguide
arbitrary constants, determined from normalization per each
mode.
e abbreviates for even, o abbreviates for odd.
Fig.2.2
D escription o f some o f the sym bols used for defining a u x ilia ry p o te n tia l functions. Shown
here are th e sym bols used fo r transverse electric com ponents. Symbols for the case o f
com ponents transverse m agnetic to x are defined s im ila rly using superscripts or subscripts
T M . T h e y are m o s tly self-explanatory.
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n
(odd s y m m e try w .r.t. x ):
r ™ ° - C j IM‘- s i n ( k ^ ° x ) s i n { k ™ v ) c - ‘ i ---
(2.5)
Region I I I i x > |
= C ™ e -W '\(‘ -V s in (k ™ y )e -^
(2.6)
For T E to x modes, we choose the electric vector p o te n tia l functions to be o f the
fo llo w ing form .
Region I: x < —|
r TE = c T Eelt ™ \(*+ i')cos(kTlEy ) e - ’ t -‘
(2.7)
Region II : —§ < x < %
(even s y m m e try w .r.t. x):
if J Ee = C f Eecos ( k j f e x)cos( k ^ f y ) c ~Jfc-''
( 2. 8 )
(odd sy m m e try w .r.t. x):
i r J Eo =
CfjEos iu ( k ^ 2 ° x )c o s (
( 2 .9 )
y ) c ~ jk - z
Region III:a ; > |
Vm =
(2.10)
F ie ld com ponents are derived fro m the vector p o te n tia l functions. Boundary con­
d itio n s at
the p a ra lle l conducting planes and the dielectric interfaces between
substrates are th e n im posed onto the field com ponents w ith in eachregion
different
in order to
determ ine the relationships among th e p ro pagation constants and the a rb itra ry constants.
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12
2.2
W aveguide C haracteristics of P P D W
2.2.1
W aveguide M odes and D ispersion C haracteristics
The P P D W supports a discrete set o f guided modes. From section 2.1. we obtained a
set o f transcendental equations for the values o f kx2 according to the specific waveguide
mode. Dispersion relation for the fundam ental TEy o mode (the P P D W m ode), w liic li
can be shown to be the same as th a t for an in fin ite dielectric slab o f tliickness a. can be
expressed as [8]:
~ ( W O l = —jy ; e \ = ; -
tan
A*
-
(2.11)
\ j { £ j £ s ) - ( €J es)
As is the wavelength in the o u te r d ie le ctric m edium . ee = (3p/ko)2 is the effective dielectric
constant o f the P P D W mode, where 3P is the propagation constant o f th e P P D W mode,
and ko is the free-space wave num ber. T h e roots o f this equation are solved n u m erica lly by
a ro o t searching algorithm . Based on these values, we plotted the norm alized dispersion
curves o f a P P D W guide having the ra tio o f a to b equals to l(F ig .2 .3 ). In tliis case, we
have used a dielectric constant o f ec= 10 .8 and es =1.0. Notice th a t T E y o mode is the
o n ly m ode th a t propagates down to D C .
2.2.2
C haracteristic Im pedance o f P P D W
In order to define the characteristic im pedance o f the PPDW' guide, we define the voltage
a t a p a rtic u la r point r along the waveguide to be the value o f the e le ctric field a t the
center o f the waveguide across the p a ra lle l plates m u ltip lie d b y the distance between the
plates.
T h e reason for such a choice is based on the simple field co n fig u ra tio n o f the
TEy o mode. T liis fundam ental mode has the p ro p e rty o f an electric fie ld u n ifo rm in the
y d ire c tio n and varies cosinusoidally in the x d irection w itliin the d ie le c tric s trip . The
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13
3.5
PPDW mod
T E m X ^
2.5
TE
TM
TM
TE,
0.5
TE
TM
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Normalized Frequency, a/X,
Fig.2.3
Norm alized dispersion curves o f a P ara lle l-P la te D ie le c tric Waveguide. R a tio o f a to 6
equals to 1. In tliis case, we have used a d ie le ctric co n sta n t o f ec=10.8 and es= 1.0. N otice
th a t T E i o m ode is th e o n ly m ode th a t propagates dow n to DC.
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14
characteristic im pedance is found from th e expression ~p where V is the peak voltage
and P is the average power propagating dow n the waveguide. F irstly, we evaluate the
guided power o f the T E w mode by in te g ra tin g the real p a rt o f E x H* over the e ntire
cross section o f the wraveguide. see Fig.2.4. T h is is given b y the follow ing double in te g ra l
over the x and y directions.
K = j ^ d x f Q dy \
4- R e ( - E yH;)]
( 2 . 12)
For the TEio mode, we have E ^ f = E j / f = E xfn = 0. So o n ly the second te rm in (2.12)
is non-zero, thus g iv in g
/
rb
+oc
1
dx
d y - [Re{—EyH*)}
J0
-oc
(2.13)
~
T h is can be fu rth e r sim plified as follows.
P. = 4 [
Jo
dx f
dy^-[Re(—E yH*)\.
Jo
2.
(2.14)
The last step is obtained from consideration o f sym m e try along the y = 6 /2 plane
as w ell as along the x = 0 plane. Thus, the average powder propagating dow n the waveguide
is given by
(
P: = 4
ra/2
I
r + oc
t i r + J( / 2
\
rb/2
dx) J 0
^
dy ^ ( - E y H : ) }
(2.15)
w hich can be w ritte n in the follow ing form ,
f
ra/2
rb/2
^ j 0
1
r + oo
rb/2
^
)
d y ' - l R * ( - E yH-x )\ + l / 2 ^ J a ‘i y - l R e ( - E yH ; ) ] \ .
(2.16)
D en o tin g the double integrals w ith in the c u rly braces b y / 2 and I3 , we have
K =
4 ( / 2 + / 3).
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(2.17)
15
y
A
center dielectric
strip
integral I 2
metal plates
integral I 3
outside dielectric
substrate
F ig . 2.4
G eom etry for evaluation o f integrals in the ca lcu la tion o f guided power and characteristic
im pedance for the P arallel-P late D ie le ctric Waveguide.
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16
A n d w ith the a m p litu d e o f Ey(x = 0) norm alized to u n ity for easier ca lcu la tio n o f
characteristic im pedance o f P P D W guide and o th e r waveguide parameters, we fin d the
expressions for 4/ 2 and 4 /3 to be.
a b k T El° (
s i 7 i ( k ^ f l0a ) \
4 /2 = — —----- 1 + — r r 0 — ~
4 ^jfi \
kJ2Eloa )
f 2 -1^
and
4 /i
b k ^ El° cos3 ( k ^ f 101 )
= 2 jJfM k l ^ s M k ™ ™ * )
(2.19)
respectively.
Consequently, the guided mode power is g iven b y
p T E ia
=
—
p
=
ab kTEl°
4
(2 .20 )
1+
or alternatively.
-p _ ab kTEl°
4
( 2 .21 )
1+
W ith the value o f the electric field a t the center o f d ie le ctric strip again n o rm alized
to u n ity, voltage V is given by —b times Ey(x = 0). o r s im p ly —6 volts. The ch a racteristic
impedance is then given b y b2/2PZ w hich is
Zc = b
jjfx
kTEl°
1 + |A :Jf101f '
The value o f th e ch a racteristic impedance carries a fa cto r 1
1[k~ and a fa cto r a / { 1 +
) where a =
(2 .22 )
because o f the fa cto r
is the decay rate o f field outside the
dielectric s trip .
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17
I t is seen th a t the ch a racteristic impedance defined in th is way also varies p ro p o r­
tio n a lly w ith the value o f b. the distance between the tw o ground planes. However, fo r
a given thickness o f substrate. 6, the impedance v a ria tio n w ith respect to . a. th e w id th
o f th e dielectric s trip is ra th e r interesting. In fact, there are either two possible values
o f a . o r no so lution a t all. depending on the p a rtic u la r value o f impedance desired. T liis
does n o t yet take in to account the p o s s ib ility o f higher modes o f propagation. In fact,
fo r fixed values o f b and frequency, as the value o f a increases from zero to in fin ity , the
characteristic im pedance rises fro m zero, peaks a t a ce rta in value, and then drops back
slow ly to zero.
T he value o f a a t w liic h the peak o f characteristic im pedance occurs
decreases w ith increasing frequency w liile the peak im pedance value its e lf goes higher
and liigher. So. in order to achieve liig h e r values o f characteristic impedance for a given
substrate tliickness. we have to operate a t a liig h e r frequency range.
In Fig.2.5. we p lo tte d the v a ria tio n o f ch a racteristic impedance o f P P D W fo r a
25 fj,m substrate w ith respect to the w id th o f the center d ie le ctric strip . Five d iffe re n t
frequencies w ith in the range fro m 200 G H z to 1000 G H z are shown for com parison. The
scale o f the x-axis is in lo g l0 (a /6 ) in order to be able to show the variation o f ch a racteristic
im pedance over a very w ide range o f values o f a.
Fig.2.5 also shows th e p o in ts a t w hich higher order modes sta rt to come in to the
p ic tu re and renders the o p e ra tio n o f the P P D W guide dubious at frequencies liig h e r th a n
the c u to ff p o in t. However, depending on the m ethod used in e xcitin g the P P D W guide
mode ( T E X0 m ode), the im m ed ia te higher mode beyond the c u to ff p o in t m ay or m a y not
be excited. T h is is a fa cto r th a t m ay be considered w hen the circumstances demands an
o p e ra tin g frequency w hich is close to o r beyond th e TE^o c u to ff frequency o f the P P D W
guide.
A n o th e r im p o rta n t feature can be observed fro m Fig.2.5 when choosing th e opera­
tio n p o in t o f P P D W guide to a t say 600 GHz and 50 Ohms. There are very in te re s tin g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
140
O
120
O
1000 G H z
points beyond which
higher order modes start to
propagate
100
800 G H z
600 G H z
400 G H z
O
200 G H z
-0.5
logio(a/6), 6=25|jm.
Fig.2.5
V a ria tio n o f ch a racteristic impedance o f P P D W w ith respect to the w id th o f the center
d ie le c tric s trip , a. fo r a 25 /m i-su bstra te o f GaAs (er = 1 2 ). Five d iffe re nt frequencies
w ith in th e range o f 200 G H z to 1000 G H z are p lo tte d fo r com parison.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
differences between choosing the liig h e r value o f a or the lower value. W hile b o th values
can give the same characteristic im pedance value, the range o f values th a t the impedance
can va ry w ith a b u t not operating beyond c u to ff (i.e. in tro d u c in g higher order modes),
is q u ite d ifferent. The higher value o f a operates closer to the c u to ff p o in t and has there­
fore a narrow er "dynam ic range o f im pedance" w ith respect to the variation o f a fo r a
p a rtic u la r frequency. On the o ther hand, the lower value o f a has the advantage o f al­
ways being able to sweep tliro u g h the whole possible range o f impedance values w ith o u t
going beyond c u to ff and w ith o u t passing tliro u g h the peak value o f the characteristic
impedance.
Using the follow ing relations between the P P D W guide propagation constants.
k z = koSf e c\
k ^ f 10 - |fcx l | = \/e e — eska;
kx2 =
^ka
(2.23)
the ch a racteristic impedance o f P P D W can be norm alized as follows.
7
^
J 1- ^
\
= 2t t
(2 .24 )
1+
where r}a is the free space characteristic impedance. A0 is th e free space wavelength. ee
is the effective dielectric constant defined by ( r ^ ) 2- The curve is p lo tte d in Fig.2.6. It
Ro
is seen th a t the normalized im pedance peaks a t around a value o f a / \ Q approxim ately
equal to 0.044 in tliis case.
In Fig.2.7. we p lo tte d the characteristic impedance curve for different cases o f ec
and es. We see th a t even after we have increased the value o f ec 10 tim es as high, the peak
value o f the norm alized impedance increased by o n ly around 36 percent. T liis in effect,
u n fo rtu n a te ly, pose serious lim ita tio n on the range o f impedance values one can o b ta in
w ith any fixed thickness o f substrate. For example, when we use a substrate o f ec= 10.8
and es= 1 .0 and operate the P P D W w ith the values o f a / \ a between 0.05 to 0.25, we have
a ra tio o f the m axim um characteristic impedance to m in im u m characteristic impedance
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
o f 4 to 2 . T h a t is. a ra tio o f 2 to 1 w ill be w hat we can best acliieve no m a tte r w hat
frequency and substrate tliickness we choose to use. One way to overcome th is problem
is to use m u ltila ye rs o f P P D W w ith different tliicknesses. P P D W 's designed in such a
m u ltila y e r environm ent can have altogether a larger range o f impedances available and
can be interconnected tliro u g h probe o r slot tra n sition s across th e different layers.
2.2.3
M etallic Loss o f P P D W
The m etallic loss o f the P P D W guide is found tliro u g h a p e rtu rb a tio n a l analysis. W hen
the c o n d u c tiv ity o f the p a rallel ground planes is fin ite b u t ve ry large, th e powder loss
due to ohm ic resistance can be approxim ated b y 5 / Rs \Ht \2ds . where the in te g ra tio n
is perform ed over the w hole m e ta llic surface confining the fie ld o f th e waveguide mode.
Ra is the surface resistance o f the m etal p la te and Ht is the ta n g e n tia l m agnetic field at
the surface.
Refering again to Fig.2.4 fo r coordinates, the power loss due to the co n d u ctin g w'alls
is given by
l- R a r dx ( l dz \ \H A y = 0 ) | 2 + | H : (y = 0 ) |2 + |H J y = b)|2 -I- | H z{y =
2
J-Oc Jo
L
6 )|2]
(2.25)
The in te g ra tio n is perform ed separately over the inside and outside regions o f the
P P D W guide.
a m p litu d e .
A g a in we assume the electric field at the center o f the guide has u n it
T liis gives us the fo llo w ing expression for m e ta llic loss due to co n du ctin g
walls.
'k T E v
Pxv
= ^ 2
^§ + ,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.26)
21
2 5 r
TE;() mode cutoff
TE™ mode cuto:
z
2
15
0 .2
0
25
Normalized Frequency, a/X,
F ig .2 .6
N orm alized ch a racteristic impedance curve o f the P a ra lle l-P la te D ie le ctric W aveguide.
A0= fre e space w avelength(ni). ?70= fre e space in trin s ic im pedance(Sl). a = w id th o f center
d ie le ctric s trip (m ), 6=thickness o f s u b stra te (m ), Z c= c h a ra c te ris tic impedance(S I).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
-^ec=10.8, £^=1.0
5.0
0 05
1
0*5
0 2
0 25
0 35
Normalized Frequency, a / \ Q
Fig-2-7
N orm alized characteristic im pedance curves o f the P arallel-P late D ie le c tric Waveguide
for cases o f ec= 10 .8. ^ = 1.0: ec= l0 .8 . es= 5 .0 : ec=108, es=1.0 and ec= 1 0 8 , es= 5.0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
or.
(2.27)
The a tte n u a tio n constant
due to co n d u ctin g w all loss is given by
and is
derived to be
Voby/e;
in N epers/m .
1+
(2.28)
*'o
T h e norm alized attenuation constant due to con du ctin g m etal plates,
defined as a^r}ab/ R s. is p lo tte d against a/ X0 as shown in Fig.2.8.
2.2.4
D ielectric Loss o f P P D W
D ielectric loss w ith in a m edium is given by 5 f v ~cje" E
dv where e = e7 — je" is the
com plex p e r m ittiv ity o f the m edium . In case o f th e P P D W guide, the dielectric loss o f
the P P D W guide is given by
(2.29)
where V in t and V ex t stands for volume o f inside d ie le c tric and outside dielectric respec­
tively. ec" is a p pro xim a te ly given by eceQ m u ltip lie d by the loss tangent o f the m a te ria l
at the frequency o f o p eration. S im ila rly for the case o f es".
I f the a m p litu d e o f the electric field is taken to be u n ity a t the center o f the dielectric
s trip and Fig.2.4 is again used for coordinates reference, the power loss per u n it len g th
o f the waveguide due to the dielectrics can be evaluated as follows.
0
-a/2
0
0
a/2
0
(2.30)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
sc=10.8, ss=1.0
c/i
-♦—»
C
03
cn
3 5
.
C
o
metal loss
ct\i r j o b f R y ^
TE;o mode cutoff
TEm 1110(16 cutoff
U
/
3
3
f
<D
H
t
~o 1 5
<U
_N
' /dielectric loss
*3
cxd^o/ tcSc, 5s-0
O
0 5
o.L
0 0 5
0
1
0
15
0 2
0 25
0 3
0 35
0 4
Norm alized Frequency, a/X0
Fig.2.8
N orm alized a tte n u a tio n coefficients due to m etal loss and dielectric loss in P a ra lle l-P la te
D ie le ctric Waveguide. Q ,vf=m etal loss a tte n u a tio n c o n s ta n t(N e p e rs/n i). o;c>=dielectric
loss a tte n u a tio n c o n s ta n t(N e p e rs /m ). Rs=surface resistance(Sl). £=loss ta n g e nt. A0= fre e
space w ave le ng tli(m ). r?0= fre e space in trin s ic impedance(S2). a = w id tli o f center d ie le ctric
s trip (m ). 6 = tliickn ess o f su b stra te (m ). Z c= c h a ra c te ris tic im pedance(£l).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T liis can be fu rth e r s im p lifie d as
^
sin(kx2a)
kx2a
(2 .3 1 )
in VV/m.
A n d the corresponding a tte n u a tio n coefficent
~
qd
O
(i
,
.
ttS-
./*_
=
\
= P d / 2 P z is given by
/
cos2 ' tv
I
-yec - e e ) >
:ry;V'«c—
,
.
-—
^o\/^e
.
.
1
{ 2 .22 )
1 "t" _o_ /
T h e u n it is in N epers/m .
Norm alized dielectric loss a tte n u a tio n constant can be defined as ac,X0lTz8c in the
case th a t 5S is so sm all th a t th e second te rm in eqn.2.32 is negligible. If the d ie le ctric loss
in th e outside dielectric is s ig n ifica n t, the norm alized d ie le ctric loss a tte n u a tio n constant
can be defined as aoXo/Tr. T h e case shown in Fig.2.8 has th e outside d ie le tric as a ir and
negligible loss is assumed.
2.2.5
P P D W C ircuits D en sity
T h e density o f c irc u its th a t can be fa b ricated using P P D W depends la rg e ly on how
close the P P D W guides can be placed n e xt to each o th e r. In order to m in im ize signal
cross-talk between the P P D W guides, i t is necessary to ensure th a t the field stre n g th or
the power density o f the P P D W m ode be kept under a to lerable level when i t is a t a
p a rtic u la r distance away fro m the center d ie le ctric s trip . In tliis respect, Fig.2.9 is useful
in e stim a ting the m in im u m distance th a t can be allowed between tw o p a ra lle l P P D W
guides and can also be used as a ru le -o f-th u m b figure in m ore general s itu a tio n s where
d iffe re n t types o f P P D W ’s a nd bends are involved.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
‘2 6
In Fig.2.9. the decay ra te o f th e field o f P P D W mode is shown as a fu n ctio n o f
the distance d away from the d ie le ctric interface. The ve rtica l axis is in lo g a ritlim o f
the ra tio o f d to a. Three different values o f the field 1%. 5% and 10% are shown for
comparison. As we have seen before, there are in general tw o types o f P P D W s . One has
w id th o f a larger th a n the substrate thickness 6 . and the o th e r has a value o f a sm aller
than b. We shall refer to the form er type o f P P D W ’s as w id e -P P D W ’s and the la tte r
type as tliin -P P D W 's fro m now on. From Fig.2.6. these two types correspond to typ ica l
values o f norm alized frequency f - a b ou t 0.15 and 0.02 respectively. For example, around
^ = 0 .1 5 . w id e -P P D W has values o f Logi0(d/a) o f a p p ro xim a te ly -0.2, 0.08 and 0.25 for
10%. 5% and 1% o f field decay. The corresponding values o f d are 0.63a. 1.2a and 1 .8 a
respectively. For th in -P P D W , around y-=0.02. Logw {d/a) has a value o f about 2 to 2.5
for 1%. 5% and 10% decay o f the field. T liis corresponds to values o f d around 100 a to
300a.
It is interesting to note th a t w id e -P P D W is e le c tric a lly tliin w hile tliin -P P D W is
electrically wide.
For exam ple, a 50-12 w id e -P P D W on 5m m substrate at 2.5 GHz is
electrically about 3a wide o r 45 m m wide. And a 50-12 tliin -P P D W on 25-/m i substrate
a t 500 GHz is e le ctrica lly a b ou t 100 a wide or 2.5 m m wide.
2.3
T he 50-Q P arallel-P late D ielectric W aveguide
In tlu s section, we w ill exam ine the factors governing the design o f the P arallel-P late
D ielectric Waveguide for any given features such as kwv loss, w ide b a n d w id th , w ide
dynam ic impedance range, ease o f fabrication, etc. Very often, these desirable features
pose co n flictin g requirem ents on the design and tradeoffs have to be ca re fu lly balanced in
order to achieve the objectives o f a given design. A n d in the present case, we w ill focus
on the design o f 50-12 P P D W guides. The m etal loss characteristics is also presented
using 2 5 - Ga As substrates as an example.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
27
oo
o
p
o
10 %
0 05
0 25
0 35
Normalized Frequency, a/XQ
Fig.2.9
T h e decay rate o f the field o f P P D W mode as a fu n ctio n o f th e distance d away from the
die le ctric interface. a = w id th o f P P D W guide. Three d ifferent decay values o f the field
1%. 5% and 10% are shown for com parison.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
In Fig.2.8. we saw th a t b o th the norm alized a tte n u a tio n constants fo r m etal and
d ie le ctric loss o f the P P D W increase m o n o to n ica lly w ith frequency. B u t given the con­
s tra in t o f achieving a ch a racteristic im pedance o f 50 olrms. the scenario becomes quite
different. T h is is because the n o rm alized characteristic im pedance is n o t a m onotonic
fu n ctio n o f frequency, or a / A0.
F irs t o f a ll. the norm alized c h a ra cte ristic impedance
peaks a t a value o f a/ X0 a p p ro x im a te ly equal to 0.044 fo r the case o f ec=10.8. es=1.0.
A n d as the w id th a changes, the a tte n u a tio n due to m e ta l loss also reaches m inim u m at
a p a rtic u la r frequency for 50-12 P P D W guide o f a given thickness. A t the same tim e,
there is a lower end o f the frequency at w liic h the P P D W guide w ill no longer be able
to acliieve a value o f 50 12. T h is p a rtic u la r feature is also visib le fro m Fig.2.5 where we
can see the impedance curve for th e case o f 400 GHz was b a re ly able to make it over 50
O lim s and the curve for 200 G H z tops a t a b o u t 25 12.
For the p a rtic u la r case o f designing a 50-12 P P D W guide u sin g 25 fin l GaAs sub­
strate. Fig.2.10 shows the a tte n u a tio n constant due to m e ta l loss in the frequency range
up to 1600 GHz. In F ig . 2 . 10 . th e m e ta l loss a tta in s a m in im u m value o f approxim ately
0.29 d B /m m for a value o f a a t a ro u n d 6.5 fim. A t the same tim e , there is another value
o f a. 73.6 fin l, a t w liich the P P D W guide is also 50 12. It is also n o te d th a t th e a tte n u a tion
a tta in s m in im u m over a re la tiv e ly b ro ad range o f frequencies. F or illu s tra tio n purpose,
we picked a value o f a=12.5 fin l. betw een sample points num ber 13 and 14. w liich is very
close to the m in im u m a tta in a b le value o f m etal loss b u t is alm ost double o f th a t value o f
a = 6 .5 fim. U sing this value o f a. Figs.2.11 and 2.12 are p ro vid e d to show the frequency
characteristics o f the a tte n u a tio n due to b o th dielectric and m e ta l loss o f such a P P D W
guide. T he 50-12 operating frequency is determ ined to be a t a b o u t 476 G Hz. Same d a ta
are being p lo tte d on Fig.2.11 and Fig.2.12 where the form er shows a tte n u a tio n in term s
o f dB per n m i and the la tte r in te rm s o f dB per guide w avelength.
To sum m arize, the present 50-12 P P D W guide works a t 476 G H z. Design dim en-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
29
W idth a(jim ) at each sam ple point
50-Q
M icrostri:
0.89
5.22
22.47
1.13
1.39
1.77
6.43
7.84
25.97
29.76
9.45
33.66
2.21
11.50
37.68
2.74
3.38
13.76
16.32
41.82
46.02
4.22
19.26
50.05
54.01
26
27
57.76
61.39
64.78
29
30
31
32
67.89
70.83
73.61
76.16
50-Q PPD W
200
400
600
800
1000
1200
1400
1600
Frequency (GHz)
F ig .‘2.10
A tte n u a tio n due to m etal loss for 50-12 P arallel-P late D ielectric Waveguides using 25fini G aAs(er = 1 2 ) substrates. T he sam ple p o ints are for different w id th s o f the center
d ie le c tric s trip o f P P D W at w hich the corresponding frequency w ill be fo r a characteristic
im pedance o f 50 Ohms. Shown also fo r com parsion are m etal loss a tte n u a tio n o f 50-11
M ic ro s trip lines using same GaAs substrates. M e ta l is assumed to be G o ld o f c o n d u c tiv ity
4.1e7 S /m . For example. P P D W losses are 0.317dB /m m a t 400 G H z fo r w id th a= 18.75
finu o r 0 .5 6 d B /m m at same frequency 400 G H z fo r a=50.25 /m i.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
PPDW dimensions:
a=12.5/im, 6=25pm, ec=12.0
£
M icrostrip total loss.
PPDW total loss.
r —■
o
PPDW metal loss
c
y
<
PDW dielectric loss
200
300
4 0 0
500
600
700
30 0
9 0 0
100G
Frequency (GHz)
Fig.2.11
V a ria tio n o f a tte n u a tio n constant due to m e ta l loss and dielectric loss o f P arallel-P late
D ielectric Waveguide. P P D W param eters are a=12.5 ^ n i, 6=25 /in i. ec= 12.0 (GaAs).
50A2 frequency = 476 GHz. M e ta l is assumed to be Gold. co n d u c tiv ity= 4 .1 e 7 S /m .
D ielectric loss tangent 8 = 0.004. A lso shown for comparison is 50-f I M ic ro s trip on 25-/xm
substrate o f s im ila r m a te ria l properties.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
PPDW dimensions:
a=12.5p.m. 6=25pm, e„-=!2.0
0.4
M icrostrip total loss
PPD W total loss
PPD W m etal loss
<
PPD W dielectric loss
0 05
200
300
SCO
600
300
Frequency (GHz)
Fig.2.12
V a ria tio n o f a ttenuation constant in d B per guide w avelength due to m e ta l loss and
d ie le c tric loss o f P arallel-P late D ie le c tric W aveguide. P P D W param eters are a = 12.5 p.m.
6=25 /jm . ec=12.0 (G aAs). 50-it frequency = 476 G H z. M e ta l is assumed to be G old.
co n d u c tiv ity = 4 .1 e 7 S /m . D ie le ctric loss tangent 5 = 0.004. Also shown fo r com parison is
50-12 M ic ro s trip on 25-pm substrate o f s im ila r m a te ria l properties.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
sions are a = 1 2 .5 ^mi. 6=25 /xm. ec= 12.0 (G a A s). M e ta l is assumed to be gold w ith a
c o n d u c tiv ity o f 4.1e7S/m . Loss tangent o f th e d ie le ctric substrate o f the center s trip is
taken to be 0.004. T h e to ta l loss o f th e P P D W is a b ou t 0.43 d B /m m fo r th is design.
(In cid e n ta lly, tliis value is very close to the value o f 0.4 d B /m m o f an open-structure
d ie le ctric waveguide re p o rte d to operate a t 490G Hz [7]. Also shown fo r com parison is
the 50-J2 m ic ro s trip lin e using same 25 fim su b stra te o f s im ila r m a te ria l p ro pe rtie s. The
to ta l loss o f the m ic ro s trip is about 1.2 d B /m m a t 476 GHz.
W hen m e ta l loss is being considered fo r 5 0 -il designs a t thicknesses o th e r th a n 25/zm. wre m ay make use o f the chart (Fig.2.13) w liic h has been p lo tte d using a norm alized
frequency w ith respect to substrate thickness.
2.4
C onclusion
We hereby have o b ta in e d some im p o rta n t param eters concerning the perform ance of
the P a ra lle l-P la te D ie le c tric Waveguide. We have also shown th a t, w ith p ro p e r design
param eters, the P P D W guide can be p ra c tic a l and is also expected to w o rk in the m illim eterw ave frequency range. A special p ro p e rty o f the P P D W guide shows th a t it can
designed to be e le c tric a lly t liin o r wide depending on requirem ents o f frequency o f oper­
a tio n. tolerance on a tte n u a tio n and im pedance considerations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
0.35
g
m
-o
03
a
W idth a in X0 at each sam ple point
2
a,
025
7?
02
a
"cs
§ 0
1
2
3
4
0.0047
0.0053
0.0059
0 0067
5
6
7
0.0075
0.0084
8
15
0.0093
0 0105
9
10
U
12
13
14
15
16
0.0119
0.0133
0 0149
0.0167
17
18
19
20
0.0188
0.0211
0.0236
0.0265
21
22
23
24
0.0297
0.0333
0 0375
0.0420
0.0471
0.0528
0.0593
0.0665
25
26
27
28
29
30
31
32
0.0747
0.0837
0.0940
0 1055
0.1183
0.1327
0.1489
0 1671
o
o
T3
0.1L
s r 8"5 0 -n P P D W
o
CQ
=
m
0 05
0
0.005
001
0 015
0 02
0 025
0.03
0 035
0.04
Normalized Frequency b f (metre-GHz)
Fig.2.13
N orm alized a tte n u a tio n constant due to m e ta l loss o f 50-12 P a ra lle l-P la te D ie le ctric
Waveguide using substrates o f dielectric constant 12.0. T h is is p lo tte d against a frequency
norm alized w ith respect to substrate thickness b. For a given thickness, the chosen oper­
a tin g point gives the 50-12 frequency in GHz calculated o ff the value on the x-axis, w hile
the value on the y-axis can be denorm alized to provide the actual a tte n u a tio n constant.
N ote th a t the b o tto m o f th e curve does n o t necessarily mean a m in im u m o f m e ta l loss
due to the frequency dependence o f surface resistance R s.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 3
Coaxial-Probe Transition to PPD W
C oaxial tra n sition s have been successfully used to excite m ic ro s trip lines and re cta n gu la r
waveguides. In tliis chapter, we w ill analyze the coaxial-probe as a means to excite the
P P D W guide from across one o f its ground planes. T h e fields o f the fundam ental mode
in the P P D W guide has an electric field th a t is u n ifo rm in the direction p e rpe n d icu la r to
the parallel plates, w ith m a xim u m electric field a t th e center o f the waveguide. Because
o f tliis property, the probe is to be positioned in th e center where the electric fie ld is
m axim um .
The geom etry o f the P P D W guide together w ith a coaxial probe tra n s itio n is shown
in Fig.3.1. a is the w id th o f the central d ie le ctric s trip and b is substrate thickness or
height o f the P P D W guide. In o rd er to be co m p atib le w ith the coordinates used in the
analysis late r, in Fig.3.1 the y d ire ctio n is the d ire c tio n o f propagation o f th e waveguide
mode and the x-z plane is the cross-section plane o f the P P D W guide. T h is co o rdin a te
system is thus different from the coordinates m ore n a tu ra lly assigned and used d u rin g
the m odal analysis o f chapter 2. Using the new coordinates described here, th e TE]_q
mode now has field com ponents
Hy and Hz.
34
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35
SIDE VIEW
TOP VIEW
coaxial
probe
central ✓
dielectric
guide
metal
plates
END VIEW
Fig.3.1
G eom etry o f a coaxial to P arallel-P Iate D ie le c tric Waveguide tra n sitio n . a = w id th o f
center d ie le c tric s trip . 6=thickness o f substrate. ec = d ie le c tric constant o f center d ie le ctric
s trip . es = d ie le c tric constant o f outside d ie le ctric substrate.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
3.1
C o a x -to -P P D W T ransition A nalysis
In o rd e r to incorporate the co a xia l probe in to a c irc u it m odel, we have analyzed the
probe using th e spectral-dom ain m e th o d o f [9] and [10]. A p p ly in g image th e o ry to the
c o a x ia l-to -P P D W tra n s itio n g e om e try o f Fig.3.1. the problem is transform ed in to another
p ro b le m o f an in fin ite ly long w ire embedded halfw ay inside a dielectric slab o f in fin ite
e xte n t, as shown in Fig.3.2. T h e c u rre n t o f th e coaxial probe is assumed to be u n ifo rm .
T liis a p p ro x im a tio n is expected to be good when the P P D W substrate tliickness 6 is sm all
com pared to A = A0/ y/e,I, w avelength o f a T E M line o f d ie le ctric constant same as the
P P D W s u b s tra te s . A reasonable lim it on b w ill be about 0.1 o f a wavelength. However,
experim ents show th a t the a p p ro x im a tio n seems to be s till q u ite good up to tliicknesses
o f a b o u t 0.16 o f a wavelength. D u rin g o u r course o f spectral-dom ain analysis, we focus
m ost o f the tim e only on the fields a t plane r = 0 because o u r electric cu rren t exists o n ly
on th is plane.
In order to understand the efficiency o f e x c ita tio n /c o u p lin g o f a co a x-to -P P D W
tra n s itio n , and the inp u t-im p ed a n ce characteristics, we p e rfo rm a detailed th e o re tica l
analysis using a spectral-dom ain m e thod. We assume th a t the PPDW ' has been p ro p e rly
designed, based on the earlier discussions, to ensure a single-m odal o p era tio n .
The
g e o m e try o f Fig.3.1 is a tlire e -p o rt c irc u it, w ith probe in p u t as one o f the p o rts, and the
o th e r tw o P P D W ports are s y m m e tric to each other. F or a com plete ch a racteriza tio n
o f th e th re e -p o rt c ircu it, we firs t analyze w ith a current I q injected on th e probe in p u t,
w ith th e o th e r two P P D W p o rts m a tc h te rm in a te d (equivalently, having an in fin ite -lo n g
P P D W exte n d in g in fin ite ly in b o th ± y dire ctio n s.)
In th e second step, we excite one
o f th e P P D W ports w ith an in c id e n t P P D W mode h a vin g an u n it voltage, w liile the
o th e r waveguide p o rt is m a tch te rm in a te d , and the probe p o rt is te rm in a te d w ith a
s h o rt-c irc u it. Under tliis c o n d itio n , fo r s im p lic ity the waveguide m ay also be p h ysica lly
extended to an in fin ite le n g th in b o th sides. T h e results o f the above tw o solutions, w ith
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
37
thin current strip
,
V
-r A
thin strip
SIDE VIEW ;
/ '
ground
planes
_
N
//
image slab
dielectric slab
image strips
thin current strip
original problem
; END VIEW
equivalent problem
Fig.3.2
G eom etry o f an equivalent p ro b le m used for analyzing a co a x ia l-to -P P D W tra n sition .
T lie equivalent geom etry is o b ta in e d v ia im aging over the p a rallel ground planes o f
the P P D W . The coaxial probe is approxim ated as a t liin s trip o f an equivalent w id th .
T liro u g li im aging, the th in s trip is extended in to an in fin ite ly long s trip o f current in
the x d ire ctio n , placed a t the center o f a dielectric slab o f in fin ite la te ra l extent, w ith
thickness a.
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38
proper sy m m e try considerations to tre a t e x cita tio n from, th e second P P D W p o rt, w ill
com pletely describe the c o a x ia l-to -P P D W tra n sition . The tw o steps o f the solution w ill
be presented separately in the fo llo w in g subsections.
3.1.1
Current E xcitation o f th e P robe
In order to s im p lify the analysis, we replace the coaxial probe o f a c y lin d ric a l cross-section
w ith a cu rre n t s trip (henceforth, interchangeably referred to as a cu rre n t s trip or probe),
o f an equivalent w id th W = ttt, where r is the radius o f the c y lin d ric a l probe. T liis
should be a reasonable treatm ent, expected to m a in ta in the essential characteristic trends
o f the o rig in a l cylin d rica l probe. T h e n , the m ethod o f im ag in g is a pplied to the current
p ro b e /s trip , w hich leads to in fin ite num ber o f reflections a b o u t the tw o parallel m etal
planes.
As a result, the c o a x -to -P P D W tra n s itio n geom etry o f Fig.3.1 is transform ed
in to an equivalent problem having an in fin ite ly -lo n g s trip , embedded halfway inside an
in fin ite d ie le ctric slab o f tliickness 6. The equivalent geom etry is shown in Fig.3.2. The
actual surface current. Ja, on the s trip is assumed to flow o n ly along the lo n g itu d in a l
d ire ctio n x . w ith u n iform m ag nitu d e over the w id th as w ell as the le n g th o f the s trip .
T liis is know n to be a good a p p ro x im a tio n fo r narrow strips (or e q uiva le n tly th in probes)
and th in P P D W (small b).
7
r
\
f %w :
0:
|y| <
0 < x < 6.
b | > f ? 0 < 1 <6.
.
(3A >
where the u nknow n param eter 10 is th e to ta l in p u t current in to the coaxial probe. A fte r
the im ag in g o f the actual current, th e equivalent problem as shown in Fig.3.2 can be seen
as a rib b o n cu rren t o f in fin ite e x te n t in the x d irection, o f w id th W in the y d ire ctio n
and zero tliickness in the ~ d ire c tio n . T h is in fin ite cu rren t s trip , Je, can be represented
as:
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39
(3 ' 2)
In the spectral dom ain, th is equivalent c u rre n t can be represented as its Fourier transform .
Je (
x •
) *
r-\-rxj
r
Je(kx. k y ) = /
dx j
J —OO
J—•—
r
A. ur
dy x ~ c ~ j{klX+kyV) =2'K8{kj;)I 0 si7Lc{^L- —)x.
KK
(3.3)
Z
Let Ea and Ee denote th e a ctu a l and equivalent electric fie ld generated by the
currents Ja and Je. respectively (F ig .3 .2 ). L e t G g j denote the sp ectral dyadic Green's
fu n c tio n for an electric field E. produced due to an electric current J placed on the x —y
plane o f the probe z = 0. W e can evaluate the electric field due to the cu rren t Je as
follows.
_
r+ o o
1
=
=
—b
f
4 /1
J —oc
^
r "
r + oc
(Ik* [
J —o c
2;
-
,
dkyGgj - x2K$(kx)[0s iu c ( ^ ~ -) c j{k‘x+k*}J)
*
d k y G g jik ; = 0 -ky)-
Z
X
10 S h l C ( ^ )^ ^ .
Z7T J —o c
(3.4)
Z
T h e in p u t impedance, ZirL, seen a t the in p u t o f the probe by a delta-gap voltage source,
can be expressed in term s o f th e to ta l com plex power, Ps, delivered b y the cu rren t strip .
~ K ol2 “
M 2i
d y g ° ‘ J° -
| / „ p L y d y E ‘ ’ J‘ '
(3’5)
In te g ra tio n w ith respect to x has been effected im m ediately, because th e integrand is
independent o f x w itlu n region 0 < x < b. Also, w itliin the region o f th e p a ra lle l plates,
th e a ctu a l electric field Ea produced b y the th in current s trip is equal to E e o f the
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40
equivalent problem . S u b s titu tin g the expression o f Ee from (3.4) in (3.5), and using Je
fro m (3.2). we have.
d k y G g j( k x = 0 , k y ) - x E s i n c i ^ - y ^
- ( i ^
)
f +°° dky
,, \%
r,
„ , * - - , k VW A ~
/
G g j { k x = 0, ky) ■xstnc{ —-— ) - xsinc{ —— )
J —oc
'
2
2
[
dkyGExJi(kx = 0. ky)sinc2{ ^ ~ - ) .
\Gj?j (kx. ky) - .t] • x
G ( k x. ky).
(I
(3.7)
G Exjx (kx, ky) is the spectral Green's fu n c tio n com ponent for the x-d ire cte d electric field
on th e r = 0 plane, produced due to a rr-directed electric cu rren t placed on the same
plane. T h e m edium under consideration is the dielectric slab o f the equivalent structure
o f Fig.3.2. w ith tliickness a. B o th the d ie le ctric slab and the cu rren t s trip are p a rallel to
the x — y plane. T h e Green's fu n ctio n G ex.jx (kx, ky) o f the equivalent problem can be
found fro m the m u ltila y e r Green’s fu n c tio n approach o f [9. 11].
GE*.ix{k-x = 0, ky) = - (
,d1cos(,d1a /2 ) -F j,d2sin(,51a /2 )
zoi
ji3ism(fiia/2) -1- ,d2cos(,dia/2)
3i = \Jka2 ec — a 2. a 2 < ka2 ec :
a2 = k
2
3i = —j \ J a 2 — k 0 2 ec, a 2 > k 2 ec .
+ ky 2 = 0 + ky2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.8)
(3.9)
(3.11)
41
S u b s titu tin g the Green's fu n ctio n o f (3.8) in to (3.6). we o b ta in the fo llo w in g spectral
inte gra l fo r the in p u t impedance o f the coaxial probe.
b_ p
tn
^
d1cos(,/31a /2 ) 4- j d 2s in ( A a /2 )
47t7-oc ‘ 3y )[j f c m { $ la / 2 ) +
8 2
cos{8 la / 2 ) ]
2
(
i2)
{
}
(3
2
]
y'
T liis in te g ra l is evaluated num erically, b u t care m ust be given in order to deform the
contour around the singularities o f the Green's fu n ction, and include the re s u ltin g residue
co n trib u tio n s . It can be shown, th a t under the co n ditio n o f a single-m odal o p eration o f
the P P D W . the Green's function G e ^.i* (& r = 0. ky ) has a p a ir o f singular p o in ts ky = ± 3 P
on the real axis. 3P = fc0v/e^ is the propagation constant o f the PPDW' waveguide mode,
w hich is same as the fundam ental T E \ 0 mode o f the dielectric slab s tru c tu re in the
equivalent problem o f Fig.3.2. The residue c o n trib u tio n . Zres, can be found an alytica lly,
and then added to the rest o f the p rincipal-value integration. Ze. obtained num erically.
Zin — Z Tes 4- Z e .
(3.13)
In order to avoid any num erical d iffic u ltie s in co m p utin g the principal-value integration,
it m ay be useful to extract o u t the sing u lar p a rt [12, 10] from the integrand, leaving a
well-behaved, sm ooth fu n ctio n for convenient num erical integration. A p p ly in g residue
calculus to (3.12) [13]. the residue c o n trib u tio n Z res can be derived and s im p lifie d as:
Z r e ,
=
- b j
=
•
si.7iC2( - ~ - )
2f3p^V^f nr}ob
,s?nc ( —
)(—
Z.
where
R c s [G e ^
Ao
)(--
(kT = 0.ky)]hj=^ p
y/l - (es/ee) ,
(3 .1 4 )
(3 .1 o )
1 + ^ ' l V ee _ e s
ee = (3p/ k a ) 2 is called the effective d ie le ctric constant o f TEio m ode,
T)a is free
space wave impedance, and A0 is the free-space wavelength. The w id th W o f the s trip
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42
(or. e q uiva le n tly tlie d ia m e te r o f the probe) is n o rm a lly m uch smaller than (A /
th a t
such
<C 1. U nder tliis co n ditio n , for a ll p ra c tic a l purposes, the sine 2 fu n c tio n in
(3.15) can be a p pro xim ate d as 1.
^ A
Ao
f ,
1+
(el ( f L ) — €s
(3.16)
The residue p a rt Zrea can be inte rpre te d as due to th e e x c ita tio n o f the T E iq surface-wave
mode o f the d ie le c tric slab in the equivalent s tru c tu re o f Fig.3.2. w liich corresponds to
the e x c ita tio n o f the P P D W guided mode in the o rig in a l stru ctu re o f Fig.3.1. It m ay be
seen fro m (3.16), th a t Z rea is a real num ber for loss-less dielectrics, w ith es < ee < ec.
T he p rin c ip a l-v a lu e c o n trib u tio n . Z e, on the o th e r hand, is a complex num ber w ith b o tli
real and im a g in a ry parts.
Z e
—
R ra d
T ./A -c o a x
•
(3 -1 7 )
The real p a rt, Rrad: is c o n trib u te d b y the power ra d ia te d into the to p and b o t­
to m open m edia in Fig.3.2. or equivalently to the pa rallel-p la te mode in the m edium
su rro u n d in g the P P D W (w ith dielectric constant es). Hence, Rrad am ounts to undesired
loss, in co n tra st to Zres w liic h is a ttrib u te d to useful guided-m ode power excite d in the
P P D W . T h e im a g in a ry p a rt o f the im pedance. ATCOttC, determ ines the a d d itio n a l reactive
loading produced b y the coaxial probe. We m ay define the efficiency, p, o f th e tra n s itio n
as the ra tio o f the useful guided power to th e the to ta l power excited b y th e probe.
p
=
-
Z Tes 4* Rrad.
(3.18)
T he above d e fin itio n is applicable for a tw o-sided e xcita tio n , as shown in Fig.3.1. and
a cco rdin g ly m ay be changed for single-sided e xcita tio n .
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43
3.1.2
In cid en ce from a P P D W P o rt
Fig.3.3 shows th e s itu a tio n when one o f the P P D W p o rts is excited by an in c o m in g
TE:io wave. EiC^ y. w h ile the second waveguide p o r t is m a tch term inated and th e p ro be
in p u t is te rm in a te d w ith a short c irc u it. As shown in Fig.3.3, a current
[0
is induced on
the probe, assumed in the x direction (see Fig.3.1 and Fig.3.2 for coordinates), w liic h
produces the reflected field R E i e ~ ^ y re tu rn in g back to the in p u t p o rt. Due to sym m e try,
an outgoing scattered field R E iej ^ ,y, o f the same m a g n itu d e as the reflection, is also
produced at the second P P D W p ort. T liis adds to th e in cid e n t field Eiei^,y to c o n s titu te
the to ta l tra n s m itte d field T E i(P ^y.
T=l+R.
(3.19)
For convenience, let us normalize the incid e nt fie ld Ei, such th a t the incid e nt v o lt­
age, Vi. is negative one v o lt (positive E ^.)
Vi = - b E ^ i z = 0) = - 1 .
or. in o th e r words, th e
x component o f th e electric field, E ^ , o f
(3.20)
the incident wave, a t
the center o f th e P P D W ( r = 0 plane) is equal to 1 /6 V /m . Let us find the guided-w ave
electric fields, p ro p a g a tin g in the
+y
d ire ctio n , pro du ce d due to the induced cu rre n t
x Iq
on the probe. T h e x com ponent o f this field can be equated to th a t o f the reflected wave,
R E xi = R /b V /m . T liis problem can be tre a te d u sin g the same form ulation used e a rlie r
for the firs t p a rt o f th e analysis in the Section 3.1.1. T h e equation o f the electric fie ld E e
in (3.4) can be used here again. I f o n ly the guided-wave p a rt o f the field, p ro p a g a tin g to
the in p u t p o rt ( w ith a v a ria tio n <?~J/3py) is needed, residue th e o ry can be applied to (3.4).
T aking o n ly th e x com ponent o f (3.4), and p ro p e rly d e riv in g the residue c o n trib u tio n ,
we obtain:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
ground planes
/
Parallel-PIate D ielectric W aveguide
x
A
<W\A♦x/W '- TE,
e,
REi ~ Escal
I
—
A/W*
- i-
V
conducting post inside dielectric guide
Fig.3.3
A P P D W mode w ith e le ctric field Ei, incident on a coaxial probe located a t th e center
o f the P P D W . The o th e r p o rt o f the P P D W is m atch te rm in a te d . The in p u t p o rt o f
the probe is sh o rt-circu ite d , a llo w in g an induced cu rren t x Iq to flow on the c o n d u ctin g
probe.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
3 W
R
RExi = j t o s i n c ( - E£ - ) R e s [ G Exj x (kx = 0. ^ ) ] fcy=_ ^ = — .
(3.21)
Now. com paring the expression o f Z res in (3.15) w ith (3.21). a nd a p p ro x im a tin g the above
sine fu nction to u n ity for sm a ll IV. the follow ing sim ple re la tio n s h ip betw'een To. R. and
Zres can be obtained.
R = - I QZre s.
(3.22)
In a d ditio n , the b oundary c o n d itio n on the conducting post m ust be ensured.
- .,
W
Ee + E i ( 3 ^ y = 0; 0 < x < b : \ y \ < — ,
where Ee is the scattered fie ld (o f the image equivalent pro ble m ) due
(3.23)
to the induced
probe current x[0. T liis c o n d itio n can be enforced b y a te s tin g procedure as follows.
6
6
—
J dx J ~^r dy Ee ■J* +
Using
—
dx J ~
w dy Ei • J* = 0.
(3.24)
(3.5.3.20.3.2) in (3.24), we cam o b ta in the fo llow ing re la tio n s h ip between the in p u t
impedance. Zin. seen by the probe in the first p a rt o f the s o lu tio n (in Section 3.1.1). and
the cu rren t
[ 0
now induced o n the probe.
- Zin\ F f + / 0* = 0; I 0 =
The unknow n cu rren t
[ 0
"Wn
-
(3-25)
m ay be e lim inated fro m (3.22) and (3.25) to o b ta in .
R
=
_ Z r e 1
Zin
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3 .2 6 )
46
3.1.3
Equivalent C ircuit
The tw o -p a rt analysis presented above provides a com plete representation o f th e coax-toP P D W tra n s itio n . Based on a ll the governing equations (3.13.3.17.3.19,3.26). we can de­
rive an equivalent c irc u it for the c o a x -to -P P D W tra n s itio n o f Fig.3.1. T h e equivalent cir­
c u it is shown in Fig.3.4. For the c irc u it o f Fig.3.4 to be consistent w ith (3.13.3.17,3.19,3.26),
the value o f the characteristic im pedance, Z c. o f the P P D W is required to be approx­
im a te ly twice th a t o f Zres. A sim ple transm ission-line c irc u it th e o ry m ay be used to
ve rify
Zc =
2
Z res .
(3.27)
T liis re la tio n sliip . together w ith th e expression o f Z res in (3.16). provides the follow ing
design equation fo r the ch a racteristic impedance Z c o f the PPD W '. norm alized to Voj^-
Zc
2 Z Tea
(es / ee)
—r =
r ~ 27t---------------■~— ■t = = .
7°
7oy^
I -{-f-Ti^(ee/ e a) — I
f
^
(3.28)
Once the effective dielectric constant ee is know n fro m (2.11), the norm alized expression o f
the characteristic impedance in (3.28) can be com puted independent o f any specific value
o f the guide thickness b. Observe th a t th e characteristic im pedance varies p ro p o rtio n a lly
w ith tliickness 6. T h is allows design o f higher levels o f im pedance ju s t by increasing the
substrate tliickness b. However, th e upper lim it for 6 is constrained by co nditions for
higher-order m ode excitation.
It is useful to show th a t the above expression (3.28) o f th e ch a racteristic impedance
also follows the power-voltage d e fin itio n .
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.29)
47
Parallel-Plate Dielectric
W aveguide
coax
ra d
equivalent circuit
o f coaxial probe
Fig.3.4
E quivalent c irc u it o f c o a x ia l-to -P P D W tra n s itio n o f Fig.3.1.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
where P is the to ta l cross-sectional power, and V is the in te g ra tio n o f the electric field
across the p a rallel plates, com puted along the center o f the guide (~ = 0) (equivalent
voltage). For the P P D W mode, the fields have no va riation in the x direction, and so the
voltage V can be s im p ly calculated as V = —bEx. B y d e rivin g the field expressions for
the P P D W mode ( T E w mode), and using the propagation constant o f the guide from
(2.11), P can derived as the inte gra tio n o f EXH * over the cross section. The expression
o f P then can be used in (3.29). and sim p lifie d to show th a t the same expression o f Z c as
in (3.28) is obtained. It may be m entioned here, th a t the power-voltage d e fin itio n (3.29)
o f the characteristic impedance is com m only used for T E M or q u a s i-T E M transm ission
lines. Though the P P D W mode is n o t a T E M o r q u asi-T E M mode, in th is respect the
P P D W line m ay be treated like a T E M transm ission line for c irc u it m odeling purposes.
3.2
Results and Discussion
F irs t, we present results for the equivalent characteristic im pedance o f the P P D W from
(3.28). as “ seen” b y a coaxial tra n sitio n . In Fig.2.6. we have a p lo t o f the norm alized value
o f Z c — 2 Z res. as a fu n ctio n o f
w ith ec = 10 and es = 1.0. T h e c u t-o ff boundaries
o f the T E -20 and T E 3 0 modes are also shown in Fig.2.6. in order to id e n tify regions o f
single and m u lti-m o d a l operation. The T E 2 0 and T E 30 modes begin to propagate when
^
> 0.16 and —^ > 0.32. respectively, independent o f the substrate thickness b. The
c u to ff frequency o f the T £ u mode, however, is a fu n ctio n o f th e ra tio a/6, and w ill
determ ine p roper design o f the thickness 6 in order to avoid th e T E u mode. It m ay be
seen from Fig.2.6. th a t for any desired value o f the characteristic impedance, there are
in general tw o possible values o f a th a t can meet such a requirem ent. The characteristic
impedance peaks a t around -^=0.044. In th is region o f
th e characteristic impedance
is re la tiv e ly less sensitive to a a n d /o r A0, w liic h m ay be useful for an o p tim u m -b a n d w id th
design.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
In Fig.3.5, we p lo t the ch a racteristic im pedance for different values o f ^
(in incre­
ments o f 0.025). also showing the region o f e x c ita tio n o f the T £ u m ode in a d d itio n to
the T £ 2o and T £30 modes. As m entioned, the characteristic im pedance increases pro­
p o rtio n a lly w ith j k
However, for a given value o f
there is a m a x im u m value o f
and hence a m a xim u m lim it for the ch a racteristic impedance, th a t can be used w ith o u t
e xcitin g the T £ u mode. T liis m a x im u m lim it o f — is shown in F ig.3.8. using which
the T E u lim its fo r the characteristic im pedance in Fig.3.5 have been calculated.
For
ec = 10.8 and es = 1. 0 . the absolute m a x im u m value o f characteristic im pedance one can
o b ta in is about 55012. w liich corresponds to ^ = 0 .3 7 5 and ^ = 0 .0 2 7 .
In Fig.3.5. single-m odal o p era tio n is n o rm a lly guaranteed in the region lim ite d by
the T £20 and T £ n modes. However, in a p plica tio n s where there is even s y m m e try o f the
electric field a b o u t the r = 0 plane, the m ode T £20 (w liich requires an o d d sym m e try o f
Ex about ~ = 0) w ill not be excited. U nd e r such circumstances, the re gion lim ite d b y the
T E 3 0 and T E 11 modes can be considered the p ra c tic a l range o f single-m odal operation.
Consequently, under the physical sym m e try, m ore fle x ib ility is exercised in the design
o f the characteristic impedance. For a p ro b e -to -P P D W tra n s itio n , th e probe is ideally
placed a t the center o f the guide ( r = 0 ), w liich means the electric fie ld has an even
sym m e try about the r = 0 plane. Therefore, Fig.3.5 m ay be used to design the P P D W
covering a broader range o f param eters, lim ite d b y the T £30 mode, n o t b y the T £20
mode.
Follow ing the d e riv a tio n presented in the Section 3.1, we com puted Zres, Rrad, and effi­
ciency p (as defined in (3.18)), as a fu n c tio n o f
for ec=10.8 and es= 1 .0 . T h e results
are shown in Fig.3.6, w ith Z TC!t and Rrad norm alized w ith respect to (■r)0 b/Xo ). I t can be
seen th a t the efficiency attains a m a x im u m o f a b o u t 95%, fo r 0.05 <
< 0.2. T liis sug­
gests, in order to design an efficient co a xia l tra n s itio n , one should design th e P P D W to
operate in the above range o f f-. N otice th a t in the absence o f the P P D W (e quivalently
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
sc=10.8 es=l.O
600
increasing
b /X o
T E 20
TE30
cutoff
cutoff
500
TE„
cutoff
region of
single mode
operation
guaranteed
300
200
0
0.05
0.1
0.15
0.2
0.25
0.3
0.4
Normalized Frequency. a / X 0
F ig . 3 .5
Design d a ta fo r th e ch a racteristic impedance o f a P P D W , p lo tte d for different values o f
6 /A 0 in increm ents o f A ( 6 /A 0) = 0.025. T h e u p p e r lim it for the ch a racteristic impedance
is governed by the T E n cu t-o ff. The T E 2 0 and T E 30 c u t-o ff lim its are also shown, th a t
lim it the m a x im u m value o f
th a t can be used. T he region o f single-m odal o p era tio n
is shown.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
a/Ao = 0) there is significant ra d ia tio n , w inch is successfully suppressed b y the P P D W
due to to ta l in te rn a l reflection from th e d ie le c tric interfaces.
Fig.3.7 shows the normalized reactance. X coax, o f the tra n s itio n o f Fig.3.6. b u t w ith
d ifferent values o f
The reactance is seen to be re la tiv e ly insensitive to th e w id th a o f
the P P D W . except when a is too sm all. T liis is expected, because X coax is co n trib u te d
by the reactive near fields o f the coaxial probe, w liic h is affected by the m edium o n ly
in the im m ediate v ic in ity o f the probe. However, when a is too sm all, the probe tends
to “see" the a ir. w liic h results in an increase o f the reactance, as seen in Fig.3.7. The
va ria tio n o f the reactance as a fu n ctio n o f the norm alized w id th
o f th e probe is shown
in Fig.3.9. C onsistent w ith Fig.3.7. the reactance in Fig.3.9 shows re la tive in se n sitivity
to
In a d d itio n , the reactance is seen to ra p id ly increase for too narrow probes, w liich
suggests the use o f such narrow probes should be avoided.
The region bounded by the T E 2o and T E u c u to ff curves is where single mode oper­
a tio n o f the waveguide can be guaranteed. T he m o tiv a tio n b e liin d w hy we included also
the c u to ff o f the T E 3 0 mode in our charts is because in some cases o f ap plicatio n s where
the e x c ita tio n has an even sym m e try a b o u t the ~=0 plane o f the waveguide, the mode
TEo 0 . having an odd sym m etry, w ill therefore n o t be excited. Under such circumstances,
the T E 3 0 m ode c u to ff together w ith T E u mode c u to ff w ill then become the lim itin g fac­
tors. N otice th a t the excitation by a coaxial probe does have even sym m e try about r = 0
plane. It im plies th a t it is possible to design a P P D W guide up to the c u to ff frequency
o f TE^o m ode w ith o u t encountering th e p ro b le m o f m ultim ode propagation.
3.3
E xperim ents
Fig.3.10 shows the geom etry o f a p ro to ty p e coaxial-P P D W -coaxial tra n s itio n th a t we
have designed and fabricated.
The P P D W is designed w ith a w id th a =
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14.4 m m ,
52
W /a * =0.017
sc=10.8
es=L.O
2.5
efficiency■/
"O
N"
Z
°-5
-
0.05
0.15
0.25
0.2
Normalized Frequency,
0.2
0.3
a !a.q
Fig.3.6
C om puted values o f the real parts o f the in p u t impedances. Z res and Rrad- norm alized to
( 7706/A 0 ), as seen b y the in p u t probe, p lo tte d as a fu n ctio n o f a / A0. A reasonable value
o f W / A0 = 0.017 is used in the calculations. Z Tes is c o n trib u te d due to e x c ita tio n o f the
guided wave along the P P D W . whereas Rrad is the due to ra d ia tio n loss into the parallel
pla te mode in the su rro u n d in g m edium . T h e efficiency o f e x c ita tio n , as defined in (3.18)
is also p lo tte d , showing a m axim um o f a b ou t 95% for a double-sided e xcitation.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
V
o
o
cj
N
0.05
0.2
0.25
N orm alized Frequency. a l \
Fig.3.7
C om puted values o f norm alized X coaj.. fo r the probe tra n s itio n o f Fig.3.6. p lo tte d against
norm alized frequency. D ata fo r three d iffe re nt values o f W j Aq are p lo tte d fo r com parison.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
0.6
0.5
x
r-
0.4
C3
o
0.3
0.2
0.05
0.15
( a / X o)
0.2
0.25
0.3
0.35
0.4
TEn
cutoff
F ig .3.8
M a xim u m su b stra te thickness allowed w ith P P D W below T E l l cutoff. ( b/\Q)niax for
T E n cut-off. versus ( a / A0) for the P a ra lle l-P la te D ie le c tric Waveguide o f F ig.3.5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
00
0.04
0.02
0.01
N o rm :
0.03
dlizcd Frequency. W /a ,
N o m l z e d X „ « fo r th e same parameters o f
Fig.3.7. b u t p lo tte d as a fu n c tio n o f £
d ire e values o f aj X q ( = 0.1, 0.15 and 0.2).
Further reproduction prohibited without permission.
•
Reproduced with pemrissron of
0.0
the copvrigW owner. Further rep
56
thickness b = 5.08 m n i. and d ie le ctric constant ec =
outside dielectric is a ir (i.e.
es= 1 .0 ).
10.8 for the central guide. T h e
From Fig.3.5. th is set o f parameters is selected
to provide a ch a racteristic im pedance o f about 50X2, a t a design frequency o f 2.5 GHz.
for convenient m a tc h in g to 50X2 in p u t ports. M e ta l plates are used to form s h o rt-c irc u it
planes for the P P D W . using w hich s h o rt-c irc u it stubs are made fo r tuning. T h e stu b
lengtlis, L l = L2. are designed to be quarter guide wavelengths a t 2.5 GHz. in order to
result in open-circuits a t the planes o f the coaxial probes. In th is case, they are 38 nun.
T liis p ro totype o f above specified dimensions w ill be referred to as p ro to typ e A 10.
The equivalent c irc u it o f Fig.3.4 for a single tra n s itio n is extended in Fig.3.11 to
provide an equivalent c irc u it for the tw o -p o rt p ro to ty p e o f Fig.3.10. T h e probe reactance.
X coax. and the ra d ia tio n resistance. Rrad- rigo ro u sly account for the effect o f the probe.
X coax, Rrad and Zres (w h ic h is a b ou t h a lf o f the ch a racteristic impedance Z c) for the
prototype design are co m p uted over the frequency range o f 2 to 3 GHz. as shown in
Fig.3.12. The actual co a xia l probe is cylin d rica l in shape w ith a diam eter d — 1.27 m m .
As discussed in the analysis section, the w id th W o f the equivalent strip used in o u r
model is (7r/2)d. It can be seen in Fig.3.12 th a t Rrad is less than 1.5X2. compared to the
guided-wave im pedance Z Tt,s o f ab ou t 25X2, w hich translates to an expected value o f about
97% (0.13d£) for the efficiency" ( = 2 Z rcsj{2ZTes + Rrad)• fo r a single-sided e xcita tio n ) o f
excitation for a single tra n s itio n .
The insertion loss in term s o f the S 2i o f the co a x-P P D W -co a x tra n s itio n is measured
on a netw ork analyzer, and the results are com pared w ith the theoretical co m p u ta tio n in
Fig.3.13(insertion loss) and F ig .3 .1 4 (S m ith C h a rt p lo t o f S2i). There is good agreement
between the theory" and experim ents in both m a g nitu d e and phase, follow ing a ll detailed
characteristic features. For a proper evaluation o f th e perform ance o f the p ro to typ e , we
have also included in th e calculations the m a te ria l loss due to the m etal and d ie le ctric
media, using available values o f the conductivity" o r loss tangent o f the respective m edia
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
(th ro u g h a p e rtu rb a tio n m odel o f loss). T liis is because, the le n g th o f th e PPDVV between
the two p o rts o f the p ro to ty p e is significant in o rd e r for the loss to be ignored. C alculated
S21 w ith m a te ria l loss is found to be —1.89 dB a t 2.5 GHz. w liile calculated So1 w ith o u t
loss is —1.52 dB . T h is am ounts to about 0.37 d B o f loss in th e co n d u c to r and dielectric
s trip o f the waveguide itself.
Measurement a t 2.5 GHz shows a b o u t 0.48 dB o f to ta l
insertion loss, w hich is a c tu a lly b e tte r th a n the calculated value o f 1.89 dB. It m ay be
noted, th a t higher value fro m the th e o ry is due to standing waves, n o t m aterial loss, as
evident fro m the ripples in th e com puted result o f Fig.3.13. I f one excludes the above
calculated m a te ria l loss o f 0.37 dB in the waveguide, the e xp e rim e n ta l value o f 0.48 dB
translates to 0.11 dB o f to ta l in s e rtio n loss due to th e tw o coaxial probes a t the two ports,
or e q uiva le n tly to o n ly 0.055 d B (negligible) o f ra d ia tio n loss p e r one coaxial probe. T h is
m ay be com pared to the estim ated value o f a b o u t 0.13 dB fro m co m p utatio n , discussed
earlier.
In Fig.3.15. a com parison o f the com puted and measured S u is shown. A lth o u g h
there is a general agreement between sim u la tio n and the m easurem ent, the discrepancy
at m id band seems to be to o large to be a ttrib u te d to e xp erim e nta l errors. Therefore,
there is yet an undiscovered source o f error and p robably a lim ita tio n o f the present
model.
Fig.3.16 shows the results o f another coax-P P D W -coax tra n s itio n designed for 4.0
G H z.
T he ripples in the response is a result o f the PPDVV s h o rt-c irc u it stubs w hich
are o n ly quarter-guide-w avelengtlis when used near the design frequency o f 4.0 GHz The
measurement, however, has been done over a large b a n d w id th o f 4 G H z.
From Figs.3.17 to 3.20, we p lo t the power loss measured fo r p ro to ty p e A10 in term s
o f the fo llo w in g expression,
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58
top view
PPDW
metal plate
short circuit
plate
Port 2
coaxial
probe
connectors
Port 1
o
cross-section
B -B ’
cross-section
A-A’
Fig.3.10
P ro to ty p e geom etry o f a tw o -p o rt coax-PPDVV-coax tra n sitio n , used for experim ent.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
PPDW
short circuit stub
PPDW waveguide
PPDW
short circuit stub
■rad
|equivalent circuit
tof coaxial probe
50 9 7
Port 2
F ig .3 .1 1
Equivalent c irc u it for the coax-PPDVV-coax tra n s itio n o f Fig.3.10.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
coax
res
a
2 coax
^res
R-rad
J ^ c o a jc
= coaxial probe input im pedance
2.4
2.6
Frequency (GHz)
Fig.3.12
Real and im a g in a ry parts o f the in p u t im pedance o f each c o a x ia l-to -P P D W tra n s itio n ,
used in the p ro to ty p e geometry. P P D W dim ensions: a = 14.4 m m . b = 5.08 m m .
ec = 10.8, es = 1.0. D iam eter o f coaxial probe = 1.27 mm. W id th o f th e equivalent s trip
a p p ro x im a tio n W = 1.2 7 | mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
M a n ly
2 5 .0q
lag
MAG
5 . 0 dB/
- 0 . 4 - 8 3 7 dB
*-
20. 0 -
MARKER
1
theory
measured
15.0
10. 0 -
m
TD
:
5' 0 "
0 . 0 -----
C\J
-5 .0 -
co
:
-
10.0
-1 5 .0
- 20. 0 -
2.20
2.60
F r ©q
2.00
3.9 0
CGHz )
Fig.3.13
M easured and com puted insertion loss o f p ro to ty p e AlO coax-PPDVV-coax tra n s itio n .
PPDVV dim ensions: a = 14.4 m m . b = 5.08 nm i. ec = 10.8.
= 1.0. S tu b lengths.
L l= L 2 = 3 8 m m . L e ng th o f the con ne ctin g PPDVV Ld=140 m m . Loss ta n g e nt o f center
d ie le c tric strip = 0 .0 0 2 8 . M etal c o n d u c tiv ity (brass) = 1.58c7 S /m .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
Theor\f
M easured
Markers are in
increments o f 0.1 G H z
Fig.3.14
Measured and com p uted S m ith -C h a rt p lo t o f Sot ° f p ro to ty p e A10 coax-P P DVV -coax
tra n s itio n , dim ensions as specified in Fig.3.13.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
sn
lo g
REP 0 . 0
A
V
-----------
mag
dB
10.0
___________
dB /
-2 S .3 S 6
Theory
. ,
,
M easured
dB
MAFKER
T .5
START
STOP
2 .0 0 0 0 0 0 0 0 0
3 .0 0 0 0 0 0 0 0 0
GHz
GHz
Fig.3.15
Measured and com puted S n o f prototype A10 coax-P P D W -coax tra n s itio n , dim ensions
as specified in Fig.3.13.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
EEE
C.C
dB
np.
20. 0 -
th e o r y
15.0-
m e a s u re d
-5.0
-
20. 0
-
-25. 0
2.00
3.-00
4. 00
F r 0 q
5. 0 0
6.00
C G H z )
Fig.3.16
Measured and co m puted in se rtio n loss o f another co a x-P P D W -co a x tra n s itio n (p ro to ty p e
A8J50). PPDVV dim ensions are a = 5.0 m m . b = ‘2 .54 m m . ec = 10.8.
= 1.0. S tu b
lengths, L l= L 2 = 1 0 .0 m m . L e n g th o f the connecting P P D W ' L d = 2 2 5 nm i. Loss ta n g e nt
o f center dielectric s trip = 0 .0 0 2 8 . M etal c o n d u c tiv ity (brass)=1.58e7 S /m .
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
and compare them w ith losses sim ulated under d iffe re n t assum ptions. We hereby consider
th e com bination o f cases where m a te ria l loss in P P D W and ra d ia tio n plus reactance due
to probe are either excluded o r included in the sim u la tio n .
Fig.3.17 shows th e case where the PPD W ' substrate is lossless and the probe
im pedance is excluded. T h a t is. the waveguide is lossless, no ra d ia tio n loss is taken in to
account and the probe reactance is zero. O f course, s im u la tio n in tliis case gives zero loss
a t a ll frequencies.
In Fig.3.18. we have a com parison between th e measurement and the sim u la tio n
where the PPDVV' su bstrate loss is accounted fo r b u t the probe impedance is n o t yet
included. We notice th a t th e levels o f loss between measured value and s im u la tio n is
q u ite close at m idband. S im u la tio n data and m easurem ent generally deviates a t o th e r
frequencies except near 3 G H z.
In Fig.3.19. we have included the probe im pedance b u t d id not include any loss
th a t is due to the PPDVV substrate. We see th a t in tills case the trend o f b o th curves
fo llo w each other b u t the level o f loss deviates fro m th e experim ent, especially when the
frequency is away from m id b a n d where less pow er is being tra n sm itte d .
A n d finally,
w hen measurement is com pared w ith sim ulation in Fig.3.20 where m aterial loss and probe
im pedance are b o th accounted for. b o th the levels and trends follow each o th e r q u ite
w ell. T liis shows the good m e rit o f the probe m odel as in explaining the losses in the
coaxial-PPDVV-coaxial tra n s itio n p ro to typ e A10.
However, we do remember th a t the
com parison between S l l in measurem ent and th e o ry was n o t too good near the m id b a n d
region (Fig.3.15).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
0.0
S
- 2 .5
TJ
w
-5 .0
- - Simulation without probe
impedance and PPDW is lossless
-1 2 .5
measured
-1 5 .0
-1 7 .5
-
20.0
-2 2 .5
2.20
2.60
2.00
3.00
F r ©q CGHz)
Fig.3.17
Measured loss o f p ro to ty p e A10 as com pared to sim ulation in case P P D W is lossless and
probe impedance is n o t included. P ro to ty p e dimensions are as specified in Fig.3.13.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
0.0
- 2 .5
-
6.0
-7 .5 -
Simulation without probe
impedance but PPDW is lossy
-Q —1 0 .0
measured
0 -1 5 .0
-1 7 .6
-
20.0
-
22.6
2.20
2.6 0
2.00
3.00
F r ©q CG H z)
Fig.3.18
Measured loss o f p ro to typ e A10 as compared to s im u la tio n in case PPDW loss and probe
im pedance are b o th excluded. P ro to typ e dimensions a n d loss param eters are as specified
in Fig.3.13.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
- 7 .5 -
CD- 10. 0 “O
CO-1 2 .5 -
CO
O -
_j
Simulation with probe impedance
but PPDW is lossless
16. 0 -
:
measured
-1 7 .5 -;
-2 0 .0 -j
-2 2 .5 -25.0 i i i i i i i i i | i -ri i i i i i i | i i i 'i ~rr n i~p i i i i i i i i | i i i i i i i i i 1
2.00
2 .2 0
2.4 0
2.60
2.60
3.00
F r g q CGHz)
Fig.3.19
M easured loss o f p ro to ty p e A10 as com pared to sim u la tio n in case PPDVV is lossless
b u t probe impedance is in c lu d e d . P ro to ty p e dim ensions a nd probe param eters are as
specified in Figs.3.13 and 3.12.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
0. 0
- 2 .5
- 5 .0
-7 .5 -
-Q —1 0 .0
CO-1 2 .5
0 -1 5 .0 -
Simulation with probe
impedance and lossy PPDW
-1 7 .5
-
20.0
measured
-2 2 .5
2.00
Freq
3.00
CGHz)
Fig.3.20
M easured loss o f p ro to typ e A 10 as com pared to sim u la tio n in case P P D W is lossy and
probe im pedance is also inclu d ed . P ro to ty p e dimensions and probe parameters are as
specified in Figs.3.13 and 3.12.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
3.4
Conclusion
We showed th a t the c o a x ia l-to -P P D W tra n s itio n is a sim ple and efficient m ethod o f
e xcitin g the Parallel-Plate D ie le ctric Waveguide (P P D W ). We conducted a theoretical
investigation o f the coaxial tra n s itio n by approxim ating the probe as a tliin conducting
s trip . Design data for the ra d ia tio n loss, efficiency, and in p u t im pedance o f the coaxial
tra n s itio n were com puted. Based on the theoretical data, an e xp erim ental p ro to typ e o f a
tw o -p o rt c ircu it, consisting o f tw o coaxial transitions at the tw o ends o f a P P D W guide,
was designed and tested.
As th e experim ental and theoretical results indicate, a low
insertion loss in the order o f 0.1 d B can be acliieved from a c o a x ia l-to -P P D W tra n sition .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 4
Slot Transition to PPDW
The design o f PPDVV laye r-to -la ver tra n s itio n s using coaxial-probes has been discussed
in chapter 3. These tra n sitio n s serve as "jo in ts " o r "ju n c tio n s " between PPDVV' guides in
different layers o f a m u ltila y e r in te g ra te d c irc u it. Using sim ila r designs o f coaxial probes,
m icro strip lin e s and o th e r p o p u la r transm ission lines can also be jo in e d to the PPDVV'
guides in this m u ltila ye r en viro n m en t. T h e im ple m e ntatio n o f probe tra n s itio n s, though
successfully dem onstrated in ch a pte r 2. is n o t always convenient as it u su a lly involves
d rillin g and soldering, o r the m a k in g o f via-holes using a m icro e le ctro n ic process.
In
tliis chapter, we move our a tte n tio n to a nother class o f PPDW ’ tra n s itio n s , nam ely the
"ground-plane slot tra n s itio n s ". T liis class o f tra n sition s can lead to sim ple methods o f
im ple m e ntin g inter-layer inte rcon n e ction s in a m u ltila y e r in te g ra te d -c irc u it environm ent.
In p a rtic u la r, the aperture u n d e r in ve stig a tio n is a rectangular slot h a v in g its longer side
p e rpe n d icu la r to the d ire ctio n o f p ro p a g a tio n o f the PPDVV m ode. T h e slot is centered
a t the "m id d le " o f the PPDVV guide. VVe shall refer to th is slot as th e P P D W transverse
rectangular slot, "transverse" m e a n in g th a t the longer side o f th e s lo t lies in a plane
transverse to the d ire ctio n o f p ro p a g a tio n o f the PPDVV guide. VVe c a ll it a "sho rt slo t"
i f its le n g th (o f the longer side) is w ith in the extent o f the central d ie le c tric strip . It is
referred to as a "long slo t" w hen its le n g th extends outside o f the c e n tra l d ie lectric s trip
o f the PPDVV guide.
71
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72
4.1
M ultilayer Spectral G reen ’s Functions o f a
V ertical M agnetic D ip o le
G e o m e tr y a n d N o ta tio n o f P r o b lem
Before we p e rfo rm a fu ll spectral analysis o f th e P P D W slot, we need to find th e spectral
Green's fu n c tio n o f a ve rtica l m agnetic d ip o le in a m u ltilayered m edium . For d e fin itio n
and b o u n d a ry co n ditio n s o f a z-directed m a g ne tic c u rre n t sheet, the reader is re fe rre d to
appendix A fo r m ore details. W hat follow s and up to the end o f tliis section is dedicated
to the d e te rm in a tio n o f the coefficients o f th e m u ltila y e r spectral p o te n tia l fu n ctio n s by
evaluating th e ir p ro pe r values p e rta in in g to each o f the functions. W ith o u t loss o f general
co n tin u ity , one m ay skip the follow ing and fo llo w on to the next section fro m here.
Follow ing the n o ta tio n o f [11]. we s ta rt w ith th e fo llo w in g expressions fo r th e electric
and m agnetic field components in the spectral dom ain.
(4.1)
(4.2)
(4.3)
(4.4)
( 4 .5 )
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Fig.4.1 shows the geom etry o f a m u ltila ye red m edium . A r-d ire cte d m agnetic current
sheet is located a t the r = 0 plane. Spectral dom ain p o te n tia l functions in th e layers 11
and 21 are given by:
(4.7)
(4.8)
-4=21 =
F, n =
4- T F n e - j0llz) f }i i i
F -.21 = (?-■’* '= +
r ra<J*>=)/21,
(4.9)
(4.10)
where 3 n and d2i are the propagation constants o f layers 11 and 21 respectively in the r
d ire c tio n for the spectral p o te n tia l functions and T n i and T f-2 i are reflection coefficents
at the boundaries.
To fin d the coefficients a n . a 2i , / n and / 21, we need to make use o f th e b o un d a ry
conditions on E and H. and therefore, on each o f th e spectral dom ain p o te n tia l functions
.4=n , .4; 21i F I 11 and Fz21 at the interface. In the fo llo w ing section, we w ill make use o f
eqns (4.1-4.6) to o b ta in four equations in term s o f the bo un d a ry conditions o f the electric
and m agnetic fields, and thus w ith the b o u n d a ry conditions o f the vector p o te n tia ls, tlus
allows us to solve fo r the 4 coefficients a U i <1211 /11 and /21 when the b o u n d a ry conditions
are applied in the rig h t form .
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74
z
layer 22
layer 21
z = 0
layer 11
z-directed magnetic
spectral current sheet
layer 12
Fig.4.1
G eom etry fo r eva lu a tio n o f m u ltila ye r G reen's fu n c tio n .
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75
S e t o f E q u a tio n s for V e c to r P o te n tia l C o e ffic ie n ts
F irs t, we d iffe re ntia te eqns (4.7) to (4.10) w ith respect to r , w liic h gives
- j 3 l u F ALle - ^ ' - ) a n -
oz
<9c
d F zn
ar
=
(- J 0 2 I
f ' 3* ”
+ / 3 2 I , r . i 2 1 eJ* > = ) a 2 1 .
(4-11)
(4 .1 2 )
= ( ; 3 t i ^ u= - 7'5u . r F u e -J'Al=) / u .
(4.13)
=
(4-14)
( ~ j 3 2 i e - j * - lZ + j 3 2 i . T F 2 i e i ^ l Z ) f 2 i -
A t the interface w iiere the m agnetic current sheet is located, i.e. at r = 0 . the fo llo w in g
relations are d irect results o f eqns (4.7) to (4.10) and eqns (4.11) to (4.14)
-4=n|—0 = (1 + r>ui)<Zii,
(4-15)
-4=211—0 = (f + r .4 2 l)a212
(4.16)
£ : u |==0 = (1 4" r f l l ) / l l >
(4-17)
^=2l|—0 = (1 4- rf'2l)./21 •
(4-18)
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76
3-4-n
3r
3.4 =21
dz
(4.19)
= —jS-zi ( 1 — r.4 2 l)a2l-
(4-20)
—j,3 n ( l - T f i O / u ,
(4.21)
= —7.321( 1 - r F2I) / 21.
(4.22)
==0
dF-=ii
dz
==o
!
C fZ
— j 3 u ( l — r A11)a u .
==0
I 2=0
Above 4 pairs o f equations effectively provide us w ith the b o undary conditions o f
the vector p o te n tia l fu n ctio n s in term s o f the un kno w n coefficients.
Now. when tliis
set o f vector p o te n tia l b o u n d a ry conditions is related to th e electric and magnetic field
b o un d a ry conditions, we w ill be able to o b ta in equations in term s o f o n ly the unknow n
coefficients and the e le ctric and m agnetic field b o u n d a ry conditions.
S u b s titu tin g re la tio n s (4.15) to (4.22) in to eqns.
(4.1,4.2) for E x , Ey and eqns.
(4.4.4. 5 ) for Hx, Hy a t above and below the r = 0 plane, we arrive a t the follow ing
expressions for the ta n g e n tia l electric and m agnetic fields above and below the r = 0
plane where the source sheet is located.
Ex 2l
= - j k y Fz21
==o
= - ] k yFz11
Exu
:= 0
= jk-xFz21
E y2,21
==0
^
2=0
kx 3A-2X
^621 d z
(4-23)
==0
3 A -u
2=0
_|,
==o
^£11
dz
(4.24)
==o
3-4=2i
i -----------
ky
^^21
3r
(4 .2 5 )
==o
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77
- y ll
+
— jkxFzll
==0
H,x 21
==0
= jkyAz 21
=o
==0
= j k yA zll
H xu
kx dFz2i
jJH dz
==o
(4.27)
kx dF zll
jjfi dz
(4.28)
+
z= 0
, ky dF z 2 1
= —j k xA z21
H
==0
I ==0
z=Q
(4.29)
^
r=0
= —j k xA z 11
7/yu
(4.26)
—o
==0
y2i
hy <9.4=n
~oea d z
==0
==0
C'~
,
9 F -U
"I---------- ^
dz
==o
(4.30)
We then m u ltip ly eqn. (4.25) by kr and eqn. (4.23) by ky, substract the la tte r result
from the form er re su lt to o b ta in the first o f the fo llo w in g relations, nam ely eqn(4.31).
A ll the o th e r relations from (4.32) to (4.38) are obtained by a sim ilar procedure. The
results are as follows.
k x^y2
rE, l
kXE;y ti
- L,y £~'x21
E
==0
—
k xFJx\i
+ kyHy 21
==0
==0
+ kyHyH
=
;=0
==0
(4.32)
z= o
z= 0
z=0
(4.31)
— j(k* + ky)F. 2i
= j(k* + k 2y)Fzll
kyE x i\
= =0
k xH,x 2 l
0
k*
+ ky
dz
k”x2 4-■ k y2 d F =- 11
ix)fi
dz
(4.33)
z=0
(4 .3 4 )
==o
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78
k rE,x 2 1
==0
-(- kyEyll
==0
==0
kyHj; 11
k J2r +' k-yl
‘-'JCll
_
~=0
— kXHy2l
==o
<9A=ll
_____________
==0
(4.36)
dz
==0
(4.37)
==0
==o
(4.35)
dz
:ue2i
==0
x ll
kykfx2l
_ kl + kl d.L=21
+ kyEy 21
==0
= j ( ^ + ^ )-4 = n
(4.38)
==o
Then, b y s u b tra c tin g (4.32) fro m (4.31) and s im ila rly fo r the o th e r 3 pairs o f equations,
we fin a lly o b ta in the fo llo w in g relations, in spectral dom ain, between the boundary con­
d itio n s o f th e ta n g e ntia l fields and the b o undary co n d itio n s o f the vector potentials.
k-r
h/21
== 0
— £ 3/11 ==0 —k„ E.x 2 l
= j(kl + kl)
Hx 21 I
0 — 77*11
Fai Z=0n -
==0
4-k.
kx + kl dFz21
JL>H L dz
==0
£ x ii
:= 0
(4.39)
;=0
y21 i ;=0
dF zll
dz
~
==0
(4.40)
==0
E x 21 i ==0 — £ 7.11 I ==0 +fcy ^y2i I _=0 — Eyn
kl + kl
LiJ
dz
==0
1 d A zy\
1 <9.4-21
.621
==0
==o
eu
3~
==0
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(4 .4 1 )
79
ky #*21
\ z=0
“ #*11
= M
I ==0
—k, Hy2i I s=0 - # / U | c=0
+ kl) A- :21 -=o - -4=ll I ==0
(4.42)
A p p ly in g t h e B o u n d a r y C o n d itio n s
Using the b o un d a ry co n d itio n s found fo r a v e rtic a lly d ire cte d m agnetic current sheet
(eqns A.25 to A .29), we can tra n s fo rm the set o f equations (4.39 - 4.42) involving the
unknow n vector p o te n tia l coefficients as follows.
The c o n tin u ity co n ditio n s o f E between layer 11 and layer 2 1 . according to eqn(4.39).
gives the re la tio n F ~21
z=0
— Fzn
z= 0
= 0. And from eqns. (4.17,4.18). we deduce th a t
(1 + T p it ) / i i — (1 + T f 2i ) / 2 i-
(4-43)
Boundary co n d itio n o f H and eqns. (4.21.4.22) applied on eqn. (4.40) gives
— j — $ 21(1 — r F2i ) / 2i + d u ( i — r F U ) / n .
( 4 .4 4 )
C o n tin u ity o f E and eqns. (4.19,4.20) applied on eqn. (4.41) gives
^ (1
^21
- T 421) 0.21 = - — (1 - r . 4 u ) a 11.
en
(4.45)
A n d bo un d a ry c o n d itio n o f H and eqns (4.15,4.16) applied on eqn (4.42) gives
(1 - b r .4 ii) a n = (1 + r.4 2 i)o 2 i-
Equations (4.43) and (4.44) can now be solved to give
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(4.46)
80
fn —
hi —
(1 + T f2i)
(4 .4 7 )
T„
(1- + I Y n )
T„
(4.48)
where Te is given by
Te — j ,d2i(i — r f 2 l ) ( i + r f n ) ■+• 5 u ( i — r f n ) ( i + r ^ i )
(4.49)
Equations (4.45) and (4.46) can be solved to give
a2l 1321 ( i - r . « i ) + — ( i - r 41i ) 1 + r -A21 = 0.
.e2i
eu
i + r 4 II.
(4.50)
Since the fa cto r w ith in the square brackets cannot be id e n tic a lly zero, we conclude th a t
d2i — a n = 0.
(4-51)
which im plies th a t .4= = 0 o r E z = 0 .
We have th u s obtained the coefficients a u .a 2i . / u
and / 2i necessary to com plete
the d e fin itio n s o f the spectral dom ain p o te n tia l functions in eqns. (4.7) to (4.10).
4.2
4.2.1
A n alysis o f Short Slot
S p ectral G reen ’s Functions
We w ill now go o n to specialize our p o te n tia l functions to the case o f a m agnetic d ip o le
embedded inside a d ie le ctric slab o f tliickness a. T h is p ro ble m geom etry w ill be used in
our spectral analysis o f the PPDW r short slot. R eferring to Fig.4.2, we use two co o rdin a te
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81
systems in w hich one is centered a t the m iddle o f the dielectric slab (labeled r axis fo r
present configuration) and the o th e r is centered on the the m agnetic dipole located a t
z = z' (labeled z° axis for canonical configuation).
Wave impedances and adm ittances are defined as follows.
^A21 = —
?^.422 = —
Z An = —
. Z M2 = —
jJ€2i
-o!eu
22
-
(4-52)
.
(4.53)
^ 1 2
yp2i = — .Yf22 = — -
(4-54)
^■u = —
(4-55)
-J/J.o
. n - 12 = — u;/x0
Since we are interested in F F2i and T f l l o n ly (a21 and a n being zero for a r-d ire cte d
m agnetic dipole), we w ill not be needing the impedances d e fin itio n (4.52,4.53). In term s
o f the r° coordinates. r f -21 and F m are given by
r
, =
" 2l
. *V2i . -y>-22 =
Y,-2l + YF22
- 2 iA t ( !- = ') . - % - 322
32l + 322
r , . . , = <1-2iA i(S +=/) . ^
~
= c-2 iA i(|+ = ') . 3 n ~
*U
Y m + YF12
3 n + 3l2
where z is in z coordinates.
I f e21 = eu = ec . e22 = el2 = es, we have
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.56)
V
(4 57)
82
Z
layer 22 |
£ 22“
a/2
layer 21 -{
£ 21—£
■
^ r F2i
I
t r Fl
4
*
layer 11
layerl21
-a/2
S l2 = S :
Fig.4.2
G eom etry for evaluation o f m u ltila y e r Green's fu n c tio n fo r a v e rtic a l m agnetic dipole
inside a d ie le ctric slab.
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83
3-22 ~ $12
= 3-2 ■ $21 = $11 = $1-
(4-58)
U nder these special conditions, vve can fu rth e r w rite
r f -2i = T ^ e - j ^ a - e2j0'~
r fii =
(4.59)
• c ~2jA=/
(4.60)
r V n = I V 21 - e~4j^l='
(4.61)
r4 - ir r r $1 4- $2
(4-62>
where
Now. in term s o f z° coordinates. Fzn and F z21 can be expressed as follows.
A n = ( ^ u=° + I Y u < r jV3“ =°)/iL,
A 21 =
+ r V 21e ^ = ° ) / 21.
- ( f + $ ) < -° < 0:
(4.63)
0 < c° < | - r '.
(4.64)
Expressing field points in te rm s o f the r coordinates as w ell where z = z° + z'. we have
A n = (<$>*»<=-=') + r f . iie - J'A l -=-=')) / u .
A 2i =
U sing th e relation T f -u = T ^21 ■
+ r F2l< 3 ^ ~ - ^ ) f 2u
- -
< r C r ':
~
(4.65)
(4-66)
(4.61), we can s im p lify the expression fo r Te,
g iv in g
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84
Te — 2ji3i(l — T F n r F21).
(4-67)
In tliis way. we can w rite
£11 = ( ^ ' " - y r n
i e ^ - - 1)
^ (1 ^ F ™ )
ZOi\ l — L f i l l
£ 2i =
+ r m
(4.68)
P 21)
^ ' " - 1) - a - f (1 + VpF
ZUi( L — L FiiL F2i)
4
.
< r <
4
(4.69)
Because the slot length L is less th a n a and so the equivalent cu rre n t source has coordi­
nates r such th a t | ~| < | . we o n ly need to evaluate field points for w hich \z\ < f for the
purpose o f calculating the a d m itta n ce o f a short slot. The spectral p o te n tia l functions
Fzu and F z 2 1 found above are therefore sufficient for our present purpose.
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85
4.2.2
A d m ittan ce o f Short Slot
Fig.4.3 shows the geom etry o f a rectangular slot on a P P D W ground plane. T h e dielectric
constant o f the central dielectric guide o f the P P D W is ec and we assume th a t th e m edium
on the o th e r side o f the slot has a dielectric co nstant o f ems. W hen the length o f the
co u pling slot, L. is sm all compared to a wavelength, the electric field over the slot opening
can be a p proxim ated by a sinusoidal d is trib u tio n . We shall define Va as the voltage taken
across the center o f slot. The w id th o f the slot. W . is assumed to be less th a n 0.1 tim es L.
Using the equivalence principle, the electric fie ld over the slot is replaced b y a m agnetic
surface current and the a d m ittance o f the slot lo o k in g fro m the voltage feed Va in to the
P P D W layer is found by in te g ra tin g the P o ynting vector over the slot as follows.
L
y in = ~
J
W
j
H{x = 0 , y , z ) - \ r d y d z
(4.70)
== _ i y= _ ^
where the equivalent m agnetic surface cu rren t M is given by
jV 7 =
K .s m M iz T ll,
W
and ke is equal to ka
(4 .T I)
sin ke^
- T liis is the "a c tu a l" m agnetic cu rren t w liic h has an a m p li­
tude M a = ^ as opposed to the "im aged" currents whose am p litu d e s are doubled.
Based on th e equivalence p rin c ip le and b y w ay o f im aging over the P P D W ground
planes, the p ro ble m geom etry is transform ed in to an equivalent problem (Fig.4.4) h a vin g
an in fin ite a rray o f m agnetic current strips v e rtic a lly oriented and embedded w ith in a
d ie le ctric slab o f in fin ite extent. We firs t focus o u r a tte n tio n on a p a rtic u la r h o rizo n ta l
plane where z is fixed, say z ' . From Fig.4.4, p a rt o f the to ta l m agnetic field over the
slot surface can be evaluated by sum m ing over th e c o n trib u tio n s fro m a ll the ^-directed
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36
TOP VIEW
SIDE VIEW
central ^
d ie le ctric
guide
m e tal
p la te s
END VIEW
Fig.4.3
G e o m e try o f a P arallel-P late D ie le c tric W aveguide rectangular slot tra n s itio n . a = w id th o f
center d ie le ctric strip . 6 = thickness o f substrate, ec = d ie le c tric constant o f center dielectric
s trip . es = d ie le c tric constant o f ou tside d ie le ctric substrate. L = le n g th o f re cta n g u la r slot.
W = w id th o f rectangular slot.
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87
T his strip has a thickness
p erpendicular to this page
o f value a
Images
2M,
central
dielectric
guide
2M,
2M ,
m etal
r;----plate
cu rren t elem ents on
the sam e
plane
D ielectric slab o f thickness a
m agnetic current
ribbons
substrate
boundary at
z = -a /2
F ig .4 .4
G eom etry o f an equivalent p roblem used fo r analyzing a P P D W -s lo t tra n s itio n . The
equivalent geom etry is o b ta in e d via im aging over the pa rallel ground planes o f the PPDW'.
T liro u g h im aging, the th in m agnetic cu rren t s trip projects in to an in fin ite array in x o f
v e rtic a lly oriented (in r ) s trip dipoles embedded w ith in a d ie le ctric slab o f in fin ite lateral
extent o f thickness a.
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88
current elements on th a t p a rticu la r z=z' plane.
T h e field due to these elements w ill
be found tliro u g h the spectral Green's fu n c tio n o f r-d ire cte d magnetic c u rre n t dipoles.
F in a lly, the to ta l magnetic field is given by in te g ra tin g the c o n trib u tio n s fro m these
elements on the z= z‘ plane where z' now varies fro m —L f 2 to +L/2.
F irs t o f all. equivalent current elements on z = z ’ plane can be expressed as
ciri k (L — | ~/ h
+°°
e; 2 L - - d z ' - Uy{W) - Y . 6 ( x - 2nb)
sin ke j
n=-oo
dMe(z') = zM'a
where ITv ( tF ) is the rectangular pulse fu n c tio n o f w id th W centered over zero.
(4.72)
The
spectral representation o f dMe(z') on the same z = z' plane is required in ord e r to evaluate
the field due to th is r-directed m agnetic c u rre n t sheet. Consequently, the a m plitudes o f
the spectral current sheets on : = z' plane is given b y
d \ I e(z') =
J J d M e(z')c-]k^ c - jhyydxdy
—OO —OO
-
-W
sinfcef
where the factor o f 2 in M'a = ^
s in c ^ )
2
•I
6
g
S (k . -
’f )
b
(4.T3,
is due to image over ground plane a t slot. N ote th a t
a lthough the cu rren t in the current sheets used for spectral evaluation has a d ire ctio n
norm al to the spectral sheets, the a ctu a l equivalent magnetic surface c u rre n t in the
o rig in a l pro ble m are tangential to the slot surface.
We can now fin d the magnetic field due to the m agnetic current elements on : =
plane
as follows,
OC
cIH(z') = dH(x, y, z, r ') = - L
J
OO
J
d g ^ d ^ ^ ^ ^ d ^ d k y
—OO —OO
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.74)
where G ^vfc-') designates the tensorial spectral G reen's fu n ctio n fo r m agnetic field H
due to spectral m agnetic c u rre n t sheets M ( z f) located a t z = z/ plane. We w ill o n ly need
the value o f H on x — 0 plane and tliis is given b y in te g ra tin g a ll th e c o n trib u tio n s o f
the spectral c u rre n t sheets fro m r 7 = —
ff(x = 0,y,z) =
j
to r 7 =
OC
dH(z') =
OO
^ JJ j
d S t f ( s )d-tie(z')eik' ydkxdkydzf.
—oo —o c -j —_ L
(4.75)
S u b s titu tin g d M e(z/) fro m (4.73), we get
H ( x = 0 .y. z) =
7b I
J
■sinc[ky 1^ - )
I
- o o - o c , / = _C .
]T
S(kx - 7-^ ) c jk»ydkxdkydz'.
n ~ —oa
e 2
(4.76)
Now. (4.76) and (4.71) can be used fo r evaluating Yin as in (4.70). A n d a fte r effecting
the in te g ra tio n w ith respect to y, we have
L
L
Z g hzm , sin ke(— - \z \ ) s in £ e( - - \z | )
e 2
W
7 > ir
-sinc2{ky — ) Y , b(kx - — )dzdz'dkxdky.
^
(4.77)
n = —oo
A lth o u g h the in te g ra tio n w ith respect to c and z' m a y be carried o u t num erically, th is
can be c o m p u ta tio n a lly intensive ta k in g into account o f integrations on th e spectral kx-
ky plane as w ell as th e in fin ite series in t l. We w ill hereby determ ine the closed form
expressions for these integrals in order to be able to co m pute the spectral in te g ra l o f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
the slot a d m itta nce more efficiently. Before we proceed, we w ill explore some fundam en­
ta l properties o f the spectral Green's functions o f a v e rtic a lly oriented m agnetic d ip o le
embedded w it liin a dielectric slab, the geometry o f w liic h is shown in Fig.4.2.
The r r com ponents o f the spectral Green's fu n c tio n G g^j(_t) for layers 21 and 11
are Gh.m . 2 i{~^~) and
~) respectively . B y the p rin cip le o f re cip ro city, we
have the fo llo w in g re la tio n sh ip between them.
G hz\
~
)
= Gh . \ r-u (~ - ~).
(4-78)
A n d from sym m etry, we also know that
G h . m . i i ( z'. -) =
- • - ) = G hzm . h (—z ,
(4-79)
C om b in in g these two relations, we get
GV/.;vf. 21 ( - /- - ) = G h.M: u ( —
—:r)-
(4.80)
Based on above properties o f the spectral Green's fu n c tio n , we can derive a re la tio n
concerning the integrals o f the spectral Green's fu n ctio n s in layer 11 and 21. T liis re la tio n
can then be used to s im p lify our task o f evaluating the Yin integral. F irs t o f a ll, using
(4.80), we can re w rite th e inte gra l w ith respect to r and ~ in layer 21 as follows.
L2
7=
j
L2
j
G Hzmz2 i{~, z ) s i n k e( ^ - |:| )s in fc e( ^ - \z'\)dzdz
Z' =~L. ===/
L
2
=
L
^
J J
j-^
G h . m =i i {—~ . —-) sin A:e( ^ — | r | ) sin fce( — — |~'| )dzdzr
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.81)
A n d then by a change o f variable from
to
we a rrive at
L_
/= j
J
z ) s in k e( ^ — |r| )sinA:e(-^ — [ r ' | ) - —dzdz ' .
(4 .8 2 )
L -z = z '
A bsorbing the - sign in to the r lim its , we get
= j
/
J
—
—1
~)sinfce( ^ - I-| ) s i n k e( ^ - \z'\ )dzdz .
(4.83)
_L_
O
B y another change o f variable from —z' to z'. we have
-z' = k
r=
z=z>
J
—
/
G hzmzi i (~- r)s in fc e( ^ - |r| )sinifce( ^ - |r'| )dz - - d z '
(4.84)
G HzMzn( z ' . z ) s i n k i:( ^ - |c| )sinA:e( j - \z'\)dzdz'.
(4.85)
n—
■_L_
n
[= j
j
VVe have now obtained th e fo llo w ing relationship
L
k
j
= j
— o“
j GH;M;2i(z\z)smke{^-\z\)smke(I^-\z'\)dzdz'
j
GH;M:ii{z'.=)smke( ^ - \ z \ ) s i n k e( ^ - \ z ' \ ) d z d z .
(4.86)
_L
2
In spectral dom ain, the ~ com ponent o f the m agnetic field can be w ritte n ([14] o r see
appendix A ) as follows
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92
Hz =
JUJfi
+ k* + k± F z = G h . m ,
(4.87)
JUJfl
T liis is the m agnetic fie ld spectral Green's fu n ctio n fo r a r-d ire c te d m agnetic dipole at
the origin. In the present case where the spectral sheet is located at the r = z' plane,
see Fig.4.2, we have
C
-
*(--
:
JUJll
1
k ‘ :--------+ kl f
J'jJfX
where from the spectral Green's fu n c tio n o f a ve rtica l m agnetic dipole embedded in a
dielectric slab, it has been shown th a t ( eqns (4.68.4.69)). fo r ~ < ~ . we have
F; =
F=n
=
m
^ 11^
1)
1
2 0 i(l — 1 F lli F2l)
(4.88)
and s im ila rly Fz = Fz2l fo r z > z' such th a t
F 21 =
(4.89)
llD \[1
—
L f i l l
f 2 l j
where
3l = s j e t f - k l - k l
- fcj -
(4.90)
and T f 21. T f a and T j are as defined in eqns(4.59.4.60,4.61). As the slot is com pletely
inside the central d ie le c tric guide, or com pletely w itliin the d ie le c tric slab in the equivalent
problem, Green's fu n c tio n s F~u and Fz2i are already sufficient fo r our present purpose.
Now, by e x tra c tin g the p a rt o f the integral o f (4.77) th a t is respect to z and z' and
designating its value b y Ic, we have
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
= j
lc
j
Gh;m A~- ~)sinfce( ^ - |r| ) s i n k e{ ^ - |r'| )dzdz'
£
=
J
/
£
G h :M ;u(z'. z) sin k e( - - \ z \ ) d z + J
sin ke( ^ - \ z ' \ ) d z +
j
j
G HzMz2 i{~- ~) s in ke( - - \z\ )dz
------ — sin-k e( ------ j ~ |) s in ke( ------- | r ' j )dzdzr (4.91)
jujfi
2
2
=
1 - G .z jiz ' +
fc .s
where
r, ;,zjiz'
=
j
J
L•>
+
/
Gff.M.n{z', c) sin ke{— — - I ) sinfcef — — |~'| )dzdz'
—
-*
/
G H..vf.2i ( r ' . r ) s i n W j - | : | ) s i n * « ( | - | = ' | ) r f r < t ' .
(4.92)
and f c .6 stands for th e tliir d double integral in (4.91). We know from (4.86) th a t the
second integral te rm in (4.92) evaluates to the same value as the first integral te rm , we
therefore o n ly have to evaluate the first te rm kn o w in g th a t the integral Ic.z^z’ can be
expressed as
G hzm J~ ■~) sinA:c( —
=
2
j
J
I.
Gh;M;\i{~. z )s in k e{ ^
|r| )s in /c e( - - |z'| )dzdzr
l - l )s in fc c( — - |-'| )dzdz'.
Lt
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(4.93)
94
Now. we proceed to evaluate tliis in te g ra l w ith respect to z and z' for layer 1 1. The
inte gra l becomes
Ic.z-t-j =
J
= ) s i n k j - - \z \ ) sin k e( - - |r'| )dzdz'
J
r '= - 4 z =
= 2
[
j
,/
__i
•*
[
J
■**
+
£
i
2 u l {i
j u j ii
—
• s in k e( ^ - |-| )sinfce( ^ - |r'| )dzdz'.
r F11rF21j
(4.94)
Let us introduce the a u x ilia ry re la tio n
(4.95)
Using tliis relation and b y m a k in g use o f eqns. (4.59.4.60). we w rite
-2 j73i :
(4.96)
r>2l = $ 3 e 2* * -
(4.97)
and
1 — T ^ u r f2 i = 1 —
j.
A p a rt o f the integrand o f (4.94) can then be re w ritte n as follows.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.98)
95
= (<J'As 4 - 4>j • e- j^ s)(e-j01- +
= (1 + 4>j )2 cos d t r c o s ie r 7 + (1 —
§ 3
■c-j3' z')
sin 3 iZ sin 3\~
+ .7(1 — $ 3 ) sin 3iz cos 3\z' — j ( 1 — 4>|) c o s ^ r s i n ^ r ' .
(4.99)
Consequently, we can w rite
G
(4.100)
2ujfi3i(l - $ 2 j)
To s im p lify the to ta l integral Lq%
z±z* we introduce the follow ing fo u r integrals.
fn-,i
= J
tc2 =
_
sin 3i ~ sin
J
r
f
cos 3\~ sin &e( — — | r 7| )
L
( -^ — | r 7| )
^G3
= J
£
cos A r s i n / c e( — - | r ' | )
‘ G4
=j
s i n f t r 's i n A e (^ - \z ' \ )
cos 3yZ sin. ke(-
2
L
f
sin d t r s in ke(
_ c
L
L
(4.101)
|r| )dz c/r7.
(4.102)
|~| )d~ d r'.
(4.103)
|~ |)d : dz'.
(4.104)
2
L
/ sin to c s in A:e(
L-= _ t
2
f
|r| )dz d r'.
c o s 3 i z s m k e(
L
2
A n d I —/ can be w ritte n as
hz> = (1 + $/3)2/ g i - f (1 — *&3)2^G2 + j ’( l _ § 3 )(I g3 — I ga),
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.105)
96
where Igi was fo u n d to be id e n tic a lly zero. The integral
is found to be as
follows
— ”
-i
~
L-i
• sin ke( — — | r [ ) sin
— |; '|
)dzdz'
(4.106)
k ' +kl ^ L ,
l-4>5)
A n d integral fc takes on th e fo llo w ing form .
I g = ^G.z^z' + fc.6
(4.107)
where integrals [G\ and Ig 2 — Ic 4 are e x p lic it functions.
S u b s titu tin g tliis result in to (4.77). we arrive at the follow ing s p e ctra l inte gra l for Yin.
suitable for num erical c o m p u ta tio n .
-t-oc
-hoc
[
J
[
J
^
7i7T
*in = - r ^ T
2-nb
. A — • [G • sinc 2 (ky — ) £
sin k£7
2
„ =_oc
S(k
)dkxdky
fc£ = - O C fcy = —OC
L_
f
‘h r b t jf i sin~ k^k
J
[
J
^ fcr = — o c k y = —oo
di
kk
- sinc2(ky-—)
7ci + j ( 7g3 — 7c4)
^2 5(kx
j (r
7t7T
—)dkxdklJ.
Integrand factors Ici, and / C3 — [ G4 are each evaluated to be as follow s:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sin
(4.108)
97
fc i = 2
r
r
1
I c 3 - lc 4 =
2 fc<L ■
(u L
^
- cos fli
(4.109)
| - ) 2.
^
_ } Ake sin 3 \^ cos ke^ — ke sin 3iL — 3i sin keL
i3t ^sin A:eL
32 _ k2 Y ^ — g
( COS fee ^
Are)
/
+
{31
e
-
k lf~
'
(4.110)
Assum ing P P D W mode is the o n ly p ro p a g a tin g mode in the waveguide, the singularities
in th e integrand o f (4.108) occur at ky = zfcjtya(i.e. ±,dp). and are associated w ith the
te rm where the denom inator equals to 1 —
The c o n trib u tio n o f these poles to the
spectral integral o f Yin can be ca lculated separately using residue calculus and is found
to be
Yn , c = ( ~ ) 2 Zc-. r°! L v6 ^ /z '
s u ke%
(4.111)
T h is p a rt o f the slot a d m itta n ce is real and is associated w ith the guided mode power th a t
propagates
along the P P D W guide. Its value can also be characterized in a transmission
line model, seeFig.4.5. where th e slot is treated as a series elem ent connected
across a
transm ission line o f ch a racteristic im pedance equal to Z c as defined for the P P D W guide
in (3.28). T h is model can be shown to be consistent by fo llo w in g an analysis sim ilar to
the one used for d e rivin g the co axial-probe tra n s itio n equivalent c irc u it in chapter 3. The
equivalent c irc u it has a tra n sfo rm e r element. The turns ra tio tlp o f tliis transform er can
be deduced from (4.111) such th a t
ILp — Z £
kys
2 Z ckys
\ / 2 / l ___________
ke^
bojfi sin ke^
ke(cos ke — cos 3 1 4 )
31 ~ kl
(4.112)
A cco rd in g to the c irc u it m odel, we have the follow ing re la tio n s liip between Yresc and n p,
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98
£.
Top View
ec
- A'
“ ' f in
Equivalent circuit o f
slot. P P D W side
Adjacent side
iM ic ro s m p antenna side'
slot
/
Cs
\
\diacenr side
: Microstnp antenna side)
PPDW !
vde
:
re su
PPDW
side
V p-G r - jB P N'"-
“ ^rcsO t ^ xsG_-^c'
°m s
-U A A ; ■
*
<
.-N■-> /-»
,■
1 ! , r !nD
slot
£c
•
Side V iew
.icross A A ’
Physical Geometry
Circuit Model (PPDW side)
Fig.4.5
C irc u it m odel d e p ictin g the re la tio n o f param ter rtp to other param eters o f an equivalent
c irc u it for a PPD W ' rectangular slot. ec = d ie le c tric constant o f center d ie le ctric s trip . es
= d ie le c tric constant o f outside d ie le ctric substrate. enis = d ie le c tric constant o f substrate
on the o th e r side o f P P D W slot.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
99
yresc = ^ - -
(4-113)
Yin — Yp + Yresc = Gp + j B p -f- Yresc-
(4.114)
A n d we w rite
where Yp is the p rin c ip a l value o f the spectral inte gra l (4.108).
4.3
Param eters o f a P P D W Short Slot T ransition
4.3.1
Susceptance
The to ta l a d m ittance o f the slot can be seen to consist o f two parts. One p a rt th a t looks
from the slot opening into the P P D W guide its e lf and another p a rt th a t looks fro m the
slot opening in to a m icrostrip antenna, another P P D W guide, or other electrom agnetic
entities on the adjacent side o f the slot. A diagram illu s tra tin g various dimensions and
symbols is shown in Fig.4.6. In a ll experim ental setups, the tliickness o f the ground plane
is less th a n 0.0006Ao and is assumed to be zero in a ll calculations. T liis is also an
assum ption in o u r analysis where the thickness o f the ground plane is assumed to be
zero. For ca lcu la tion purposes, the to ta l a d m itta n ce o f the P P D W slot w ill be found as
a sum o f the follow ing two adm ittance values. YlTl fo r adm ittance o f the P P D W side and
Ymii for adm ittance o f m icrostrip side o r o th e r e n tities (P P D W . air, etc). T h a t is,
Ysi0 t,totai = Yin + Yms = Yp -h- Yms 4- Yresc.
(4.115)
The two values, Yp and Yms. are calculated separately for a given set o f param eters, and
then added together w ith YresG to give th e final value o f the to ta l slot ad m itta nce . Yp
and Yms are n o t to ta lly independent o f each other. B u t in m any cases, as long as the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
medium above slot
slottotal
Y ms-^ Y {n
ms
rectangular slot
ms
metal plates
center dielectric
strip
Fig.4.6
G eom etry o f P P D W guide and slot, n o ta tio n s and symbols.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
current d is trib u tio n o f the equivalent m agnetic surface c u rre n t is not d ra s tic a lly affected
by slight changes o f param eter values on e ith e r side o f the slot, th e y can be treated
as independent o f each other. T h is means th a t one value o f Yp calculated fo r one set
o f parameters m ay be added to a value o f Yms ca lculated for a s lig h tly d iffe re nt set o f
parameters to a p p ro xim a te the value o f the to ta l slot a d m itta n ce , under the assum ption
th a t the set o f param eters used for fin d in g Ynis has negligible effect on Yp and vice versa.
T liis p ro pe rty w ill be taken advantage o f in o b ta in in g some o f the sim u la tio n d a ta . For
the purpose o f a p p lica tio n s in a 50-J2 system, we have presented the susceptance and
conductance data n o rm alized to Ya = 1/50 = 0.02 M ho.
We present here tw o sets o f results fo r characterising the coupling between a P P D W
guide and entities on the o th e r side o f the ground plane th ro u g h a rectangular slot on the
common ground plane. These entities m ay be m ic ro s trip antennas, o th e r PPD W ' guides,
o r ju s t a homogeneous m edium fillin g the h a lf space. T h e classification o f the results in to
these two sets is m a in ly fo r a p plicatio n purpose. T h e difference between th e tw o d ifferent
sets o f results comes o n ly from the difference in th e value o f the dielectric constant used
for the m a te ria l on the o th e r side o f the P P D W slot in question. ems has been used to
designate th is d ie le ctric constant w ith the su bscript intended to stand for m ic ro s trip . It
w ill however be used to designate the d ie le ctric co n stant o f any m a te ria l on th e adjacent
side o f the slot opening. W ith th is in m in d , we see th a t the result can be in te rp re te d for
other applications as long as the dielectric constants in question bear the s im ila r values
th a t are used in the results being presented.
The firs t set o f results, case P P D W ^-slot-M icrostrip. is calculated for th e typ ica l
case o f ems=2.2 and represents the p a rt o f K, in e q n .(4 .115) for evaluating the to ta l slot
adm ittance in case o f co u pling to objects such as m ic ro s trip antennas. The second set o f
result, case P P D W -s lo t-P P D W . is calculated for th e case o f emi.=10.8, the same d ie le ctric
constant as th a t o f the PPD W r guide. A n d in tliis case, Ynis w ill assume the same value
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
as Yp. T liis p a rtic u la r value is useful for fin d in g the to ta l slot adm ittance o f a slot on a
common ground plane between two P P D W guides, one on each side o f the slot, w hich is
given by ju s t the value o f 2 (Yp 4- YresC).
The adm ittance value Yp consists o f 2 parts. One p a rt. Gp. relates to the power
th a t is lost to ra d ia tio n as p a rallel-plate modes in the P P D W layer, or ra d ia tio n to space
above and below the slab in th e equivalent problem . A n d the p a rt j B p is due to reactive
fields surrounding the slot. T liis is summarized in the fo llo w in g equation.
Yp = Gp + j B p.
(4.116)
ca se 1: P P D W -s lo t-M ic r o s tr ip
In Fig.4.7. we p lo tte d the v a ria tio n o f slot susceptance versus norm alized slot length for
the case ec=10.8. es= 1 .0 . ems—2.2 and ^ = 0 .0 0 8 3 . T w o cases o f ^ are shown. However,
the susceptance curves fo r various cases o f ^ are alm ost on top o f each other, suggesting
a m in im a l effect o f a on Bp as long as the extent o f the slot lies w ith in the dielectric
o f the P P D W guide. T h e curve for each value o f ^
is p lo tte d for values o f L up to a
only, owing to the c o n s tra in t th a t L < a. In o ther words, th e curve for -^= 0 .0 5 ends at
^ = 0 .0 5 , etc.
In Fig.4.8, we see th a t the variation o f the d ie le ctric constant on the side o f the
m icro strip substrate has a sm all effect on B p. It can also be seen th a t in this case, the
susceptance curves fo r various cases o f
are again alm ost on to p o f each other.
In Fig.4.9, 3 cases o f substrate tliickness are shown.
In Fig.4.10.
4 cases o f ^Ao are shown.. We can see th a t the w ider the slot, the less
°
sensitive the susceptance Bp is to slot-length or frequency.
Bp also becomes zero a t a
liig h er value o f s lo t-le n g th o r frequency as the slot gets w ider. However, it is to be noted
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
103
sc= 10.8 ss= l.O sms=2.2
W/Ao=0.0083
a/Xo=0.05,....0.3
^
0 .5
cn
■o
o
N
-0 5
o
2
0 05
0 25
Normalized Slot Length L/XQ, where L<a.
Fig.4.7
Norm alized Susceptance.
for the case of: ec= l0 .8 . es= 1.0. ^ ,,.= 2 .2 . ^ = 0 .0 2 .0 .0 4 2 3 .
t^=0.0083.
Value o f fA-o varies from 0.05 to 0.30 in steps o f 0.05.
Aq
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
104
sc=10.8 £s=1.0
W/A.o=0.0083
b ! \ o= 0 M 2 3
tf/A.o=0.05......0.3
c
w
Q.
D
CJ
VI
O
Z
0 05
0.2
0.25
0.3
Normal i zed Sl ot Length L/XQ, where L<a.
Fig.4.8
N orm alized Susceptance. p 2. for the case of: ec= 10.8. es= 1.0. ^= 0 .0 2 .0 .0 4 2 3 . 1-^=0.0083.
Value o f
varies fro m 0.05 to 0.30 in steps o f 0.05. enis= 2 . 2 and 4.0.
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105
Sr”*10.8 8.= l.0 sm=2.2
b /X 0
= 0 .0 8 ,
W/Ao=0.0083
a/Xo=0.3
/Ao=0.04
05
«
o
c
b / X o= 0 . 0 2
3
Q.
u
CJ
'/I
3
C/D
T3
1)
N
-5
-05
o
-1 5 l
0.05
0.1
0 15
02
0 25
Normalized Slot Length L!XQ, where L<a.
Fig.4.9
N orm alized Susceptance.
fo r the case of:
ec=10.8.
es= 1.0. ems=2.2.
V-=0.0083.
fAo =0.02. 0.04 and 0.08.
Ao
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-^ = 0 .3 .
106
3i--------------------------------------------------------^
s c=1 0 . 8
s s—1.0
sms= 2 .2
//
a/XQ= 0.3
/ /
y /
/ / y
b/X0= 0.0423
-
23
^
^—
—
s
B.
rj
CJ
zn
Z3
i
1
i!
i
zn
TD
rj
_N
/
/
/
w /l^ o .o o 8
r—
i“
o
/
i
/
W/Ao=0.004
i
/
W/Ao=0.002
-4
i
l
l
0 05
0 1
0.15
i
02
0.25
. j
0 3
Normalized Slot Length L/XQ, where L<a.
Fig.4.10
N orm alized Susceptance.
iA -o =0.0423.
Ao
for the case of: ec=10.8. es= 1.0 . em i=2.2.
=0.002. 0.004. 0.008 and 0.012.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-^= 0 .3 .
107
th a t the susceptance c o n trib u tio n fro m the o th e r side o f the slot needs to be taken into
account in order to determ ine w hether the to ta l susceptance is zero o r n o t.
F ig .4 .1 1 shows the effect o f change o f ec on susceptance Bp. T he h ig h e r the dielectric
constant ec. the higher the value o f B p. m a kin g the slot adm ittance m ore capacitive.
ca se 2: P P D W - s lo t -P P D W
Fig.4.12 shows the va ria tio n o f susceptance B p w ith respect to tw o d iffe re nt cases of
d ie le ctric constants for ec. We see th a t the increase o f ec has in general the effect of
increasing the susceptance o f the s lo t, m a k in g it m ore capacitive. However, in the case
where ec=10.8. we see th a t the slot becomes h ig ld y in d u c tiv e as L approaches an effective
le n g th o f lialf-w avelength w ith in the d ie le ctric guide o f the P P D W . T h is is n o t observed
in th e results for the case o f P P D W -s lo t-M ic ro s trip because ema is re la tiv e ly low in that
case w ith the assumption th a t the substrate o f the m ic ro s trip c irc u it on the o th e r side of
the slot has dielectric constants o f 2.0 to 4.0 only.
In Fig.4.13. we again see th a t the susceptance becomes less sensitive fo r w id e r slots.
A n d the slot behaves alm ost like a short c irc u it as the slot length approaches the value
o f a h a lf effective-wavelength.
Fig.4.14 can be compared w ith Fig.4.9 where ems=2.2 instead o f 10.8 as in th is fig­
ure. T h e tre n d o f the susceptance is again s im ila r except th a t there is cle a rly a p a rtic u la r
value o f slot length where the slot im pedance becomes extrem ely ca p acitive or inductive,
behaving alm ost like an open c irc u it.
4.3.2
C onductance
T he conductance term Gp in eqn(4.116) corresponds to the real p a rt o f Yin in the spectral
in te g ra l o f (4.108). Under the assum ption th a t o n ly the fundam ental m ode is propagating,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
108
es=l.O
Sms=2.2
W/a.o=0.0083
6/?io=0.0423
afXQ—0.05......0.3
if
D
3
Q.
D
5
0
0 5
0 .1 5
0 25
Normalized Slot Length L/X0, where L<a.
F ig .4 .1 1
N orm alized Susceptance.
for the case of: 6., = 1.0. e„lA= ‘2 .2. ^ = 0 .0 0 8 3 . -^=0.0423.
value o f
varies from 0.05 to 0.30 in steps o f 0.05. ec=10.8 o r 5.4.
Aq
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
109
D
tJ
53
c.
V
rJ1
-o
Si
-1C
case PPDW-slot-PPDW
-14'r
0 25
0 15
Normalized Slot Length L/X0. where L<a.
F ig.4.1‘2
N orm alized Susceptance. ^4. fo r the case of:
ec= e nUi= l0 .8 o r -5.4.
es= 1.0. ^ = 0 .0 0 8 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-^ = 0 .0 4 .
^ = 0 .3 .
110
W / X o= 0 .0 0 2
W/a.o=0.004
W/a.o=0.008
-4
-J
C
J
r“
53
Q.
3J
3
yJ
i
-6
W//,o=0.012
-8
__ *
zr.
“O
D
-1C
ss= 1.0
e ms= 1 0 . 8
C“
o
Z
-12
-14
b/Xn= O M
a/XQ= 0.3
case PPDW-slot-PPDW
-18
0
0 05
0
1
0
15
0 2
0 25
0
LA
-1 6 -
Normalized Slot Length L/X 0, where L<a.
Fig.4.13
Norm alized Susceptance.
for the case of: ec= e ms = 10.8. es= 1.0. -^ = 0 .0 4 . ^ = 0 . 3 .
^ = 0 .0 0 2 . 0.004. 0.008 and 0.012.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
O
o
o
0
1' n<
II
CO
s c= s ms= i 0 .8
4
II
O
I ll
W/Xo=0.008
i)
/
/
/
afXo= 0 3
,
CJ
3
!
t
1-
I
b/Xo=0.04
V
________
i
i
!
cTi
~o
1
N
s
\
O
Z
j
1
i
i
!
6/^0=0.021
_
case PPDW -slot-PPDW
-
1
1
1
1
1
s
1
1
1
1
1
>
'
i
'
i
0 05
01
3 15
C2
0 25
Normalized Slot Length L/XQ, where L<a.
Fig.4.14
N orm alized Susceptance. f? . fo r the case of: ec= e ms = 10.8. es= 1.0. ^ = 0 .0 0 8 . ^ = 0 . 3 .
f =0.02, 0.04 and 0.08.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
112
the integrand in (4.108) takes on p u re ly im a g in a ry values for all n except when n = 0
where i t w ill assume a com plex value. As a re s u lt, the o n ly te rm th a t co n trib u te s to the
real p a rt o f Yin comes fro m the n = 0 te rm o f (4.108). A n d w ith n = 0 . the in te g ra n d o f
(4.108) becomes independent o f b. However, th e pa rallel-p la te separation b s till factors
in the denom inator outside o f the integral, m a k in g the real p a rt a fu n ctio n inversely
p ro p o rtio n a l to b. [t is therefore possible to no rm a lize the conductance by thickness b
under th is special circum stance. However, we keep the norm alization o f Gp to be w ith
respect to ^ = 0 .0 2 in order to be the same as w ith n o rm a lizatio n o f susceptance.
Fig.4.15 shows the v a ria tio n o f the conductance Gp o f a typical case o f co n fig u ra tio n
PPD YV -S lot-P PD W where ec= e ms=10.8. es= 1 .0 . g = 0 .0 0 2 to 0.012. ^ = 0 .0 5 to 0.3 and
^•= 0 .0 4 . N otice th a t the effect o f ^ va ria tio n over the p ra ctica l range again has ne gligible
effect on the conductance a nd the curves due to th is va ria tio n cannot be distin g u ish e d
from one another.
4.3.3
Efficiency
We m ay define efficiency o f the slot as the ra tio o f the power th a t goes in to th e guided
mode o f P P D W layer to the pow er th a t goes in to th e guided mode power plus the ra d ia te d
power in to parallel-plate modes, i.e.
fi -
* re s
i
(4-U 7 )
T he data presented here is for th e e x c ita tio n efficiency o f slot to the P P D W layer. If
the slot is used in a c o n fig u ra tio n o f P P D W -s lo t-P P D W tra n sition , the overall efficiency
in th a t case w ill have the same value as presented in these data. I f the slot is used in
a co n fig u ra tio n where one side is a P P D W layer and the o th e r side is. for exam ple, a
m ic ro s trip layer: the overall efficiency o f the slot w ill be given by the efficiency o f the slot
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
113
0 06r
sc=ems=10.8 85= 1.0
case PPDW-sIot-PPDW
W/Xo=0.002 to 0.012
A/A^=0.04
0.05-
u
C-)
G3
o
3
•o
c
o
U
"O
U
.X 1 0 4
c
aJ'K,=0 2 5 /
004
a/K,=0
.
0 C3P
I
m/ a.,,-0
1?
, a !A,,=0 05
"si
o
Z
o
a h = Q
2
a /A < ,= 0 .3
/
0 02-
a.
001
-
tf/A^O.O:), 0.1, 0.15
a /A ^ = 0 .2
a /^ o = 0 .2 D
N orm alized Slot Length LIXQ. where L<a.
Fig.4.15
Norm alized Conductance. ^ E. for the case of: ec= e ms = 10.8. es=1.0. -^= 0 .0 0 2 to 0.012.
f-=
Ao 0 .0 5 to 0.3. fAo =0.04.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
114
sc= e ms= 1 0 .8
es= 1 .0
0
W/Xo=0.002 to 0.012
9 9 5
0
0
9 9
9 8 5
0
0
9 8
9 7 5
04
0
0
9 6 5
0
0
9 7
9 6
9 5 5
case PPDW-slot-PPDW
0 05
0 .1 5
0.2
0 2 5
Normalized Slot Length L/XQ. where L<a.
Fig.4.16
E fficiency fo r the case of: ec= e „IS= 10.8. es= 1.0. ^ = 0 .0 0 2 to 0.012. -^= 0 .0 5 , 0.1. 0.15.
0.2. 0.25. 0.3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
fo r the P P D W side (d a ta presented here fo r th e case o f P P D W -s lo t-M ic ro s trip ) m u ltip lie d
by the efficiency o f the slot for m ic ro s trip side.
Fig.4.16 shows th a t the efficiency is liig h e r than 95 percent fo r the case o f a PPDVVs lo t-P P D W tra n s itio n where ec= e ms=10.8. es=1.0,
A (j
varies fro m 0.002 to 0.012. and ~
A
q
varies between 0.05 and 0.3. T h e curves fo r different cases o f ^Ao cannot be distinguished
in tills ch a rt as they larg e ly overlap on one another, except fo r th e case where ^ = 0 . 3 . a
slig h t spread is ju s t about noticeable at the lower end o f n o rm a lized slot length.
Note th a t the efficiency is independent o f b when b is less th a n the m axim um
allow able value o f 6 before the liig h e r order T E u mode sta rts to propagate. It is n e gligib ly
affected b y W over the p ra c tic a l range o f ^ = 0 .0 0 2 to 0.012 as the value o f the sine
fu n c tio n rem ains close to one in the range k y= 0 to ka.
Fig.4.17 shows th a t the efficiency va ria tio n when the d ie le ctric constant o f the
m ic ro s trip substrate varies from the value 1.0 up to the value 10.8 where it becomes
equivalent to the case P P D W -s lo t-P P D W (see Fig.4.16). In th a t case, the param eter ke
takes on the same value and the c u rre n t d is trib u tio n in those tw o cases becomes identical.
In th e case o f -^ = 0 .3 and when ^
is close to tliis value, the efficiency increases w ith ems.
show ing th a t less power is ra d ia te d in to parallel-plate modes. T lu s dependence is not
obvious by ju s t loo kin g a t the spectral inte gra l. However, one possible e xplanation is th a t
as e„ls increases, the equivalent c u rre n t d is trib u tio n is m ore confined w itliin the dielectric
waveguide, and we thus see an increase o f coupling between th e equivalent source and
the P P D W mode. T h a t is to say, we see an increase o f c o u p lin g between the slot fields
and the P P D W -m ode fields. T liis effect is more pronounced when the w id th a o f the
guide is w ide and the length L o f the slot is approacliing the guide w id th .
In Fig.4.18, we show the efficiency fo r the case es= 4 .0 . A s com pared to the case
ems= 2 .2 shown in one o f the cases in Fig.4.17, we can see a s ig n ifica n t drop in efficiency
in a ll cases o f f-. T h is is a d ire ct re su lt o f the lesser am ount o f re fle ction th a t is occuring
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
116
£ c= I 0 . 8
s s = 1.0
W /A o= 0 .0 0 2 to 0 .0 1 2
1
a/Ao=0.3
0 9
a/ko=0 . 2
a/Ao=0.l5
0.8
0
0
9 9 5
7 0 .9 9
0 6 0
0 5
9 8 5
0
0
9 8
9 7 5
0 4
0
9 7
0 3
0
02
01
9 6 5
0
0
9 6
9 5 5
0
0 5
0
2
0
2 5
0 :
case PPDW-slot-MS
0 05
01
02
0 25
0 3
Normalized Slot Length L/X0. where L<a.
Fig.4.17
E fficiency fo r the case of: ec=eA.=1.0. ems varies from 1.0 to 10.8.
f-= 0 .0 5 , 0.1. 0.15. 0.2, 0.25. 0.3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
=0.002 to 0.012 and
117
over the dielectric interface between substrates o f es= 4.0 and ec=10.8 th a n the interface
between substrates o f es= 1 .0 and ec=10.8.
4.3.4
Param eter n v
As a result o f the fact th a t the characteristic impedance Z c (3.28) o f P P D W is p ro po r­
tio n a l to b. we can see th a t param eter 7ip. as shown in eqn(4.112). is independent o f b.
We also see th a t Up is a m onotonic fu n c tio n o f the length o f a slot. However, Fig.4.19
shows th a t rip takes on sm aller values fo r the case when -^= 0 .0 5 . This can be a ttrib u te d
to the fact th a t the field o f P P D W m ode o p era tin g near o r below -^= 0 .0 5 becomes more
and more spread o u t. effectively decreasing the am ount o f power th a t can be coupled
from the slot fields to the guided mode fields. np is n e gligib ly affected by W in the range
o f v"=0.002
to 0.012 for sim ila r reasons discussed before.
Ao
4.4
P ractical P P D W -S lo t Transitions
4.4.1
M icrostrip-P P D W -M icrostrip Inter layer C oupler
The P P D W has tw o p a rallel ground planes and are n a tu ra lly taken advantage o f for
aperture coupling to transm ission lines on the o th e r side o f the ground plane using slots.
Because o f the presence o f two ground planes at the same time, tliis feature enables
the P P D W guide to w ork as a coupler in the m u ltila y e r integrated c irc u its environm ent
where signal can be transferred from one layer to another tliro u g h an inte rm e d iate layer
of PPDW .
The geom etry o f such a device is shown in Fig.4.20. O peration o f th e coupler is
as follows. Signal is fed a t the in p u t p o rt o f the m icro strip line on the b o tto m layer. A
slot on the ground plane com m on to th e b o tto m m ic ro s trip layer and the m id d le P P D W
layer allows signal to be coupled in to the P P D W layer. A n o th e r slot w hich is com m on
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
118
ss=4.0
0.98
0 96
0 94
0 92
0 88
0 86
0 84
0.05
02
0 25
0:
case PPDW-slot-MS
0 05
0 25
Normalized Slot Length L/X0. where L<a.
Fig.4.18
Efficiency for th e case of: es=4.0. ec= 10 .8. erriJj= 2.2. ^ = 0 .0 0 2 to 0.012. ^ = 0 .0 5 . 0.1.
0 .1 5 .0 .2 .0 .2 5 .0 .3 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
119
case PPDW-siot-PPDW
IJ 4
£c ^ms 10.8
0 35 ' ss=I.O
W/Xo=0.002 to 0.012
Parameter n
a !7^=0 2 5
0 05
0 02
0 06
0 OS
a/7^=0.05
0
0 5
0 2 5
Normalized Slot Length L/X0. where L<a.
Fig.4.19
Param eter n p for the case of: ec= e rns = 10.8. es= 1 .0 . ^ = 0 .0 0 2 to 0.012. ^ = 0 .0 5 . 0.1. 0.15.
0.2. 0.25. 0.3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
extra stubs in d e v ic e i f double-stub
matchim: used
Microstnp lute on
upper layer
PPD W coupler
section
Cross section A -A '
A
-c --i
P ort
Port 2
; I-# -.,;...
slot between upper layer
and PPDW
upper microstnp
substrate
Port 2
PPDW
-coupler
- laver
lower microstnp
substrate
Port
Cross-section BB '
Fig.4.20
Physical geom etry o f aperture-coupled m ic ro s trip -P P D W -m ic ro s trip m u ltila y e r coupler.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
121
to the top n iic ro s trip layer and the P P D W layer provides coupling o f signal to the top
layer.
The impedance values calculated is incorporated in to our c irc u it m odel, the equiv­
alent circuit o f w liic li is as shown in Fig.4.21.
D efinition o f sym bols in Fig.4.21 are as follows: Z siot is the im pedance o f the slot,
length of slot L = 12 m m . w id th o f slot W — 1 mm, iim is coupling param eter o f m ic ro s trip
to slot. A value o f 0.62 is used in the present calculations. np is co u pling param eter
o f P P D W to slot.
are lengths o f P P D W waveguide sections.
I3 is fixed to be
24 mm. /m l. lm2.1m3 and /m4 are lengths o f m ic ro s trip lin e sections. lm 1 = lm3=92 mm.
W id th s of m ic ro s trip lines are 4.4 nm i. /s is the le n g th o f open c irc u it stub in case o f
double-stub m a tc liin g and in th a t case the stub is located at a distance di before the
slot.
4 w ill be th e le n g th o f the open c irc u it stu b im m ediately a fte r the slot. km is
propagation constant o f m ic ro s trip lin e . Zni is ch a racteristic impedance o f m icro strip lin e ,
Z a and Zm are b o th 50 Ohms. 3P is propagation constant o f P P D W waveguide, and Z c
is characteristic im pedance o f P P D W waveguide. W id th o f P P D W guide a = 14.4 m m
and height o f P P D W guide b is 1.27 m m (50 m ils).
Dielectric constant o f P P D W substrate is 10.8. Tliickness o f m ic ro s trip substrate
is 1.524 mm(60 m ils) w ith dielectric constant ems equal to 2.33.
Wre perform ed 3 experim ents on tlus coupler configuration. The lengths h - h - l m 2
and lm 4 w ill be p a rt o f th e design parameters. A n d in case double-stub m a tc liin g is used,
the lengths ls and di w ill also have to be adjusted.
In order to e ffic ie n tly couple signal from the b o tto m layer to to p layer, we perform ed
sim ulations on the perform ance o f b o th single-stub m a tcld ng and doub le -tu b m a tc liin g
designs. O p tim iz a tio n o f the designs involves the tra d e o ff o f b a n d w id th w ith co u pling
efficiency. The choice o f thickness for the P P D W layer has also been chosen ca re fu lly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Upper Microstnp Laver
Port 2
Aw3
In
hnX
3 -o
* lu U u /T „
M ic r o s ^
open circuit
I
stub
P PD W Lav
U JU U U
norm p
PPDW
short circuit
stub
'slot
An"5
UJUUU
o—[
O—{
Port 1
Lower Microstrip Layer
F ig .4 .2 1
E quivalent c irc u it o f m ic ro s trip -P P D W -m ic ro s trip m u ltila y e r coupler.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
in our present co upler co n fig u ra tio n under the co n stra in t o f having to m atch to 50A I
coaxial lines a t b o th th e in p u t and o u tp u t ports. T h e results th a t follows are for the cases
o f single-stub m a tch ing , double-stub m a tc liin g (w ideband) and double-stub m a tch ing
(narrow band).
S in g le-S tu b M a tc h in g
Design o f single-stub m a tc h in g for tliis coupler c o n fig u ra tio n is re la tive ly simple. There
are m ainly o n ly tw o param eters th a t we can a d ju st.
N am ely the lengths o f the short
c irc u it P P D W stubs in th e P P D W layer, li and l2, and the lengths o f the o p en-circuit
m icrostrip stubs at th e co u p lin g slots. lm 2 and /m4. O u r analysis gave a design o f / i = / 2= H
mm. and /m2= G i 4 = H n u n . Because o f the re s tric tio n o f single-stub m a tcliin g and cou­
p lin g efficiency, we have designed the coupler to operate near the cu to ff frequency o f
the T E 2o mode. T h is is a c tu a lly not a serious p ro b le m as the present e xcita tio n o f the
P P D W guide has an even sym m e try and w ill not couple to the T E 2 0 mode w liich has an
odd sym m etry even i f we do have to operate above th e T E 2q c u to ff frequency.
Measurements and sim ulations are shown in Fig.4.22 and Fig.4.23.
The coupler has a 3 dB b a n d w id th o f ab ou t 0.8 G H z a t a center frequency o f 3.5
GHz. The m in im u m in se rtio n loss is a tta in e d at frequency 3.68 G H z has a value o f 1.69
dB . T liis includes tw o coaxial to m icro strip tra n s itio n s and two m icrostrip to P P D W
layer to layer tra n s itio n s . T h e reflected power measured a t th is frequency is negligible a t
about -20 dB . As a re su lt, the insertion loss o f a single P P D W -s lo t-M ic ro s trip tra n s itio n
is measured to be less th a n 0.85 dB for tliis device.
S im ula tio n a t 3.68 G H z gives a value o f 1.18 dB inse rtio n loss. T liis includes 0.22
dB o f m ic ro s trip lin e loss, 0.23 dB o f P P D W line loss, 0.09 dB o f loss to ra d ia tio n a t
slot into P P D W laye r and m ic ro s trip layer o f w liic h 0.005 dB is due to ra d ia tio n loss to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
124
?2 1
-JS
R E F C . 0 dB
i
1 0 .0
dB/
V - 1 . S 3 5 4 - dB
Theory
Measured
MAL>
50 .0
_r
M A R K E R {1
3 . 6 8 GHz
30.0
20.0
m
“O
1 0 ,0
C\J -10.0
CO
-
20.0
-5 0 .0
2.00
3.00
4.00
5.00
Frequency(GHz)
Fig.4.22
Measurem ent and com puted values o f S21 o f aperture-coupled M ic ro s trip -P P D W M ic ro s trip interlayer coupler. Design dimensions are / 1= / 2==H- nun. and Zm2 = /m4= H
m m . Single-stub m a tch in g is used on top and b o tto m m ic ro s trip layers.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
125
Theory
Measured
^
! a g MAG
REF 0 . 0
dB
^
1 0 .0 dB/
- 5 ( 1 . 1 Cl A
V
dR
i
MARKER ! l
3 . S8iGHz
38.0
2 0.0
s-
q
10.0
CO
-50.0
1.00
2.08
3.00
Frequency(GHz)
Fig.4.23
Measurement and co m p u te d values o f S i L o f aperture-coupled M icrostrip-P P D V V M ic ro s trip interlayer coupler. Design dimensions are / 1=Z2= 11 m m . and / m2 =Zm4= l l
m m . Single-stub m a tc liin g is used on to p and b o tto m m ic ro s trip layers.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
p a ra lle l-p la te modes in P P D W ' layer. S l l at tliis frequency is -9.1 dB .
I f a ll losses are excluded fro m sim ula tio n. S12 at 3.68 G H z w ould have been around
0.64 d B . S ubstractng tliis value fro m 1.18 dB gives the value o f 0.54 dB which is the sum
o f m ic ro s trip lin e loss. P P D W lin e loss, and losses due to ra d ia tio n a t slot. The amount
o f loss in measurement th a t s im u la tio n is not able to account for is about 1.69 - 1.18
=0.51 dB .
In a ll loss calculations, th e loss tangent o f m ic ro s trip substrate is taken to be 0.0012
and the loss tangent o f P P D W substrate is taken to be 0.004. O th e r losses not taken
in to account in sim ulation are c o a x ia l-to -m ic ro s trip tra n s itio n losses w liich may explain
some o f the unaccounted for losses.
D o u b le -S tu b M a tch in g , W id e b a n d
Design o f double-stub m a tc h in g fo r tliis coupler involves the v a ria tio n o f the lengths o f the
s h o rt-c irc u it P P D W stubs in th e P P D W layer, Li and l2: the lengths o f the open-circuit
m ic ro s trip stubs at the co u p lin g slots. l m 2 and /m4: the lengths o f the e xtra open-circuit
m ic ro s trip stubs before the c o u p lin g slots Ls. and th e ir distance d\ from the slots. T liis
design have considerable degrees o f freedom and an non-exhaustive search o f solutions has
yielded the follow ing design in te rm s o f o p tim ize d b a n d w id th . li=l2= \ § m m . Im2=dm4= l l
n u ii, Ls= 9.4 mm, ciL= 8 m m . M easurem ent and sim ulations are shown in Fig.4.24. and
Fig.4.25. Measured values m a tc h reasonably well w ith the sim ula tio n.
In se rtio n loss a t 3.52 G H z is measured to be ab ou t 1.98 dB . T liis is about 1.29 dB
m ore th a n the predicted value o f 0.69 dB . E xclu d in g a ll m a te ria l losses and ra d ia tio n
losses, sim ulation gives a value o f 0.18 dB inse rtio n loss. P redicted loss a t this frequency
is thus a b ou t 0.69 - 0.18 = 0.51 d B . T h e e x tra loss o f 1.98 - 0.51 = 1.47 dB is therefore not
accounted for by present sim u la tio n . T h is unaccounted-for loss is sig n ifica n tly more th a n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
S2 1
log
Theory
Measured
MAC)
REF
0 . 0 dB
1 0 .0 dB/
- 1 . 9 8 9 1 dB
50.0
T
I----
M A R K E R 11
3 . 5 2 ' GHz.
20.0
a
ID
„
10*0
-J ____
-30.0
-40.0
4.00
1 .00
Frequency(GHz)
Fig.4.24
Measurement and com puted values o f S21 o f aperture-coupled M ic ro s trip -P P D W M ic ro s trip interlayer coupler (w ideband) using double-stub m atching. Design dimensions
are ^= ^ 2 = 1 6 nm i, and /m2=^m 4= H rum . LS—9A m m . c4=8 nun.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
----------- Theory
------------ M easured
S j_
lag
REF 0 . 0 dB
1 0 .0 d B /
- I S . 354 d8
MARKER
3 . 52
1-------
MAG
G H z:
_r
,
- 4
_
— I-------
—
j
T ‘
.
28.0
...j
V- - r t ------- t f —Ji-
-<40.0
3.00
4.00
00
Frequency(GHz)
F ig .4 .2 5
Measurement and com puted values o f S l l o f aperture-coupled M ic ro s trip -P P D W M ic ro s trip inte rla ye r coupler (w ideband) using double-stub m atching. Design dimensions
are / i= Z 2=16 n m i. and /m2 =Zm4= l l n im . Zs= 9.4 nm i. c /i= 8 mm .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
129
the unaccounted-for loss o f 0.51 dB in th e previous case o f single-stub m a tch in g . T h is
m ay be due to th e presence o f the e xtra open c irc u it stubs introduced before th e co u pling
slots. A n d again we have not accounted fo r th e loss due to the tw o c o a x ia l-to -m ic ro s trip
transitions.
N ote th a t th e 3-dB b a n d w id th in th is case is alm ost 1 G H z a t a center frequency
o f 3.2 GHz.
D o u b le -S tu b M a tc h in g , N arrow B a n d
In the two previous designs, they b o th operate near the liigher end o f th e frequency
band.
I t is the a im o f tliis design to t r y to operate the coupler a t a low er frequency.
Design o f double stu b m atching fo r tliis co u ple r again involves a d ju stin g li. lo. lm2- lm4-
ls. and d\_. A non-exhaustive search o f so lu tio n s yielded the follow ing design in w hich the
coupler is p re d icte d to w ork at a lower frequency o f 2.5 GHz. f 1=Z2= 2 4 m m . im2 =^m4 = ^
n m i. Zs= 16 n m i. d i = 13 m n i. Measurements and sim ulations are shown in Fig.4.26 and
Fig.4.27.
It is seen th a t o u r design objective o f try in g to operate at a lower frequency has cost
us quite a b it in te rm s o f b o th b a n d w id th and in se rtio n loss. In fact, measured inse rtio n
loss at 2.31 G H z has a value o f 3.3 dB w h ile the b a n d w id th has decreased to a b o u t 0.2
GHz. T h e higher in se rtio n loss again m a y be due to the double-stub m a tc liin g design
w hich m ay be m ore lossy in lower frequency range. Since the predicted perform ance at
2.31 G H z has s ig n ific a n t reflection, values a t the frequency 2.5 G H z are used here fo r the
purpose o f c a lc u la tin g the losses at 2.31 G H z. A t 2.5 GHz, S21 w ith loss is ca lcu la te d to
be -0.69 d B . w liile c a lcu la tio n w ith o u t loss gives a value o f -0.09 dB . We th u s estim ate
th a t the loss a t 2.31 G H z is around 0.6 d B . T liis gives us a loss o f 3.3 - 0.6 = 2.7 dB
w liich is n o t accounted for b y sim ulation. T h is is b y far the largest discrepancy in term s
o f loss. I t is also th e lowest frequency th a t we are o p era tin g the in te rla ye r coupler w ith .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
S2 1
58.0
‘ —g
Theory
Measured
MA G
R E F 0 . C dB
1
10 . G dB /
V - 3 . 3 2 7 dB
M A R K ER i 1
I
3 . 3 1 GHz'
38.0
20. 0 -
-q
1 0 .0 “
CM-10.0
CO
- 20. »
Tt —
-50.0
2.50
3.00
f r i r T 'T
3.50
4.00
Frequency(GHz)
Fig.4.26
M easurement and com p uted values o f S21 o f aperture-coupled M ic ro s trip -P P D W M ic ro s trip in te rla y e r coupler (narrow band) using double stub m atching. Design d im en­
sions are l {=l 2= 24 n u n . and /m2=^m4= l l nun. ls= 16 m m . d \ = 13 m m .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
60.0
5
log
R EF 0 . 0 dB
1
1 0 .0 dB/
v - 1 7 . 7 2 5 dB
mar
Rer 7 l '
MAG
Theory
Measured
.
2 . 3 1 ! GHz
10.0
CO
30.0
50.0
3.60
2.60
4.00
Frequency(GHz)
Fig.4.27
M easurem ent and com puted values o f S l l o f aperture-coupled M ic ro s trip -P P D W M ic ro s trip inte rla ye r coupler (narrow band) using double stu b m atching. Design d im e n ­
sions are Li=l2—24 m m . and /m2=Cn4= l l nun. ls= 16 m m . d\ = l3 mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
T liis p a rtic u la r factor to g e th e r w ith errors in fa b rica tio n o f p ro to ty p e may e xp la in the
larger loss at th is frequency.
VVe have designed the aperture-coupled m ic ro s trip -P P D W -m ic ro s trip coupler in
three different cases. In a ll cases, the experim ents agree w ith the th e o ry in general. In
the single stub m a tc iiin g case, we acliieved an insertion loss o f b e tte r th a n 0.85 d B per
m icrostrip-PPD VV tra n s itio n .
4.4.2
Slot-C oupled M icrostrip P atch A n ten n a
Fig.4.28 shows the geom etry o f a M ic ro s trip patch antenna th a t is fed from a slot th ro u g h
the ground plane o f a P P D W guide. T h e P P D W guide is being excited by a coaxial
probe inserted th ro u gh the second ground plane o f the PPDVV layer. The PPDVV guide
is short-circuited on b o th ends w ith some p a rtic u la r chosen lengths away fro m the probe
and slot locations. Fig.4.29 shows a transm ission line m odel fo r the p ro to ty p e o f a PPDVV
slot-fed M ic ro s trip patch antenna.
In this p rototype, the corresponding equivalent c irc u it o f the coaxial feed and c irc u it
parameters e xtra ctio n has been discussed in detail in previous chapter. The le n g th o f the
PPDVV s h o rt-c irc u it stub L\ is chosen to be about a q u a rte r guide-wavelength so th a t an
open c irc u it is seen a t the coaxial-probe. The slot-fed patch antenna is seen as a series
impedance
- ^ YA + c r + j B r
( 4
' l l 8
)
across the PPDVV guide where YA is the adm ittance o f the slot-coupled patch antenna as
seen from the slot [15]. Gp-\-jBp = Yp is the adm ittance on th e PPDVV side o f the slot and
is com puted according to eqn(4.108). T h e length o f the PPDVV s h o rt-c irc u it stu b L 2 is
designed so as to m atch the antenna impedance to the the PPDVV guide im pedance. For
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133
TOP VIEW
Side View B-B’
ET
microstrip patch
coupling
slot .
PPDW
guide /
Short-circuit
plate
Coaxial
probe
a
Cross-section A-A’
Fig.4.28
G eom etry o f a PPDVV slot-coupled M ic ro s trip p a tch antenna. a = w id th o f center d i­
electric s trip . 6=thickness o f substrate. ec= d ie le c tric constant o f center d ie le ctric s trip .
e5= d ie le c tric co n sta n t o f outside dielectric substrate. f mji= thickness o f M ic ro s trip sub­
strate. ems= d ie le c tric constant o f m ic ro s trip su b stra te . L and W = le n g th and w id th
o f re cta n gu la r slot. Lm and W m= le n g th and w id th o f M ic ro s trip p a tch antenna.
Ld, L i. L 2 = ie n g th s o f PPD W ' guide and PPDVV s h o rt-c irc u it stubs.
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134
Admittance above slot
PPDW
short circuit stub
P p
Admittance below slot
PPDW guide
<f" j X c o a x
P p
I
SPA
PPDW
short circuit stub
equivalent circuit!
iof coaxial probe ;
50 Q / / port i
Fig.4.29
Tra n m issio n line m odel o f a P P D W slot-coupled M ic ro s trip patch antenna p ro to typ e as
shown in Fig.4.28.
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135
present prototype, the P P D W guide characteristic impedance at 2.40 G Hz is designed to
be 50 Ohms by using a = 1 4 .4 m m , 6=5.08 m m . ec= 10.8 and es= 1.0. For a patch antenna
o f size 37.5 m m x 20.0 n m i w dtli ems=2.2. f ms=1.58 m m and a slot o f dimensions 12 m m
x 1 m m . we are able to m a tch a t around 2.48 GHz by a d ju stin g L? to a value o f 27 m m
according to sim ulation.
Measured resonant frequency, see Fig.4.30. o f the p rototype
antenna is at around 2.46 G Hz.
The measured and com puted S m ith chart p lo t o f this antenna feed configuration is
shown in Fig.4.31. A lth o u g h we are able to m atch well in term s o f the resonant frequency,
the theory is not able to p re d ic t the antenna performance accurately in the region close
to resonance. T he a u th o r, however, is aware o f a possible source o f error from the fin ite
thickness o f the ground plane o f the slot which has n o t been included in the analysis.
L im ita tio n o f the probe m odel is another possible source o f error.
In order to test the th e o retica l results in a broader perspective, we have perform ed
two more experim ents by v a ry in g the length o f the P P D W s h o rt-c irc u it stub. The results
together w ith the previous experim ent are shown together for comparison purpose in
Fig.4.32. In a ll the results, they a ll show a general agreement w ith regards to resonant
frequency b u t the same e rro r between theory and measurement seem to be present in all
three cases.
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136
M a n ly
REF 0 . 0
V
log
Theory
Measured
MAG
dB
1 0 . 0 dB/
_ 1 5 . 9 7 6 dB
1—
MARKER
2 . 46
i
ST ART
STOP
2.200000000
2.S 0 0 0 00 0 0 0
GHz
GHz
Fig.4.30
Measured and com puted S l l o f PPDVV slot-coupled M ic ro s trip p atch antenna. PPDVV
dim ensions: a = 1 4 .4 mm. 6=5.08 m m . L i = 13.0 m m , £,^=69.0 m m , L 2=27.0 mm. ec=10.8.
es= 1 .0 . PPDVV slot dimensions: L x W = 12.0 m m x 1.0 m m . M ic ro s trip patch dim ensions:
LTnx W m—37.5 n u n x 20.0 m m . ema= 2 .2 , £ „„ = !.5 8 n m i .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
L37
Theory
—O-—
Measured —&
Fig.4.31
Measured and com puted S m ith C h a rt p lo t o f PPDVV slot-coupled M ic ro s trip patch an­
tenna. PPDVV’ dim ensions are: a = 1 4 .4 m m . 6=5.08 m m . L t = 13.0 m m . L<f=69.0 mm.
L 2=27.0 mm. ec=10.8. es= 1 .0 . PPDVV’ slot dim ensions are: LxVV= 12.0 n m i x 1.0 mm.
M ic ro s trip patch dim ensions are: L nix W ni=3T.5 m m x 20.0 m m . e„,s= 2 .2. £ ^ = 1.58 mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
138
s ABT
S TG P
2 .2 0 0 0 0 0 0 0 0
2 . 600000000
GHz
GHz
a) L2=24 mm
START
2 .2 0 0 0 0 0 0 0 0
GHz
S TG P
a.S 00000000
GHz
b) l_2=27 mm
Markers V are at
0.05 G H z intervals
\
«
START
2 .3 0 0 0 0 0 0 0 0
GHz
V C P
2 .S O O O 0 C O O O
GHz
Markers o
are at 2.47 GHz
c) l_2=30 mm
Fig.4.32
Measured and com puted S m ith C h a rt p lo ts o f P P D W slot-coupled M ic ro s trip pa tch
antenna. L 2 for diagrams a), b) and c) are 24.0 nm i, 27.0 m m and 30.0 m m respectively.
P P D W dim ensions are: a=14.4 n u n , 6=5.08 m m . L 1= 13.0 mm . Lrf=69.0 m m . ec=10.8.
es= 1 .0 . P P D W slot dimensions are: L x W = l 2 .0 m m x 1.0 m m . M ic ro s trip patch
dim ensions are: LmxW ni=37.o n m i x 20.0 m m . ems= 2 .2 . tms = 1.58 m m .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
139
4.5
A n alysis o f a Long S lo t
In tu itive ly, we w ould im agine th a t the efficiency o f the short-slot w ill s ta rt to dro p as the
extent o f the slot gets close to the d ie le ctric b o u n d a ry o f the PPDVV guide. T h is trend
is not seen in last section. Furtherm ore, th e results o f analysis o f the PPDVV slot w ith a
length shorter th a n th e w id th o f the PPDVV guide ind ica te th a t the e x c ita tio n efficiency
o f PPDVV slot is e xtre m e ly liig h even when th e slot is as long as the w id th o f the PPDVV
guide. T h is result, being against in tu itio n , has m o tiva te d us to c a rry o u t an analysis o f
a slot w ith a length extending beyond th e PPDVV central dielectric s trip . T h a t is. the
length o f the slot. L. is to be greater th a n a. the w id th o f the P P D W guide.
4.5.1
S p ectral G reen’s F unctions
The derivations in th is case are more involved as we have to account fo r sources and fields
bo th inside and outside o f the central d ie le c tric s trip . In previous section . we have found
the Green’s fu n c tio n fo r the fields inside a d ie le c tric slab produced by a spectral sheet o f
vertically-oriented m agnetic current located inside the slab. For the fields outside o f the
dielectric slab (see Fig.4.33 ). they are g ive n b y
(4.119)
for fields above th e slab.
And
a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.120)
140
layer 21 ^
£ 2 l —£
layer 11 1 ^ 11 £
FI 1
laver 12
P> 12- Pi
laver 13
0
j
_o
S l 3 — £:
Fig.4.33
Geom etry for e va lu a tio n o f m u ltila ye r Green's fu n c tio n fo r a ve rtica l m agnetic dip o le
above a d ie le ctric slab.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
141
A n = ( < ^ = - = '> + r F n e - ^ ~ - ' > ) f l l .
~\<~<
(4-121)
for fields inside slab (F ig.4.2) as found before in sh o rt slot analysis, where / u . / 2i are
same as given in eqns(4.65,4.66). A n d
A 12 =
.
-0 0 < r <
(4.122)
is for fields below the slab (F ig.4.34). Coefficients fo r / u and / 21 are known and have been
found in eqns (4.68.4.69). / i 2 and / 22 are to be fo u n d as follows. The c o n tin u ity o f Fz
across the interface between layers 11 and 12 requires th a t F~u(z = —| ) = F -12( r = —| )
w hich gives
= fi2-
(4.123)
Therefore, the spectral p o te n tia l function for fields below the slab is
—oc < r < —
( 4. 124)
From synm ietry. we know th a t
A 22( r /. r ) = R l2 ( - r ' . - r ) .
(4.125)
Using above relation, we see th a t
F=22=
+ r Fn\=,__sf
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.126)
142
which is equivalent to
(4.127)
VVe have thus com p le te ly determ ined fields everywhere fo r a ll source points w itliin the
dielectric slab.
VVe now go on to find th e fields for source p o in ts located above the slab. Note th a t
our layer numbers (see Fig.4.33) are now different fro m the case when the source sheet
is inside the slab. A n d because o f tliis . we need to be careful when applying re cip ro city
and m aking use o f previous results derived for source p o in ts inside the slab. F irs t, we
w ill determ ine fields inside the slab. VVe use here a su p erscrip t o f greater than sign " > "
on symbols th a t are associated w ith source p o in ts above th e slab.
The re cip ro city p rin c ip le im plies th a t
(4.128)
T liis gives us the fields inside the slab.
The fields below the slab is given by
a
—oc < ~ < —
where F p 13= 0 . 3
is
0 2
a
>3
(4.129)
- and F3l 3 becomes
—oc < r < ——.
2
f
2
(4.130)
can be found b y using firs t the boundary c o n d itio n between layers 13 and 12 and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
143
then the re cip ro city re la tio n in eqn(4.128), w hich gives
= - f) = £ * ( -
/5 =
z').
(4.131)
As a result, the fields b e lo w the slab is given by
F>3(z'. --) = ^ * !=+ S>F=22(.-'
(4.132)
The fields above th e slab are determ ined separately for the region above th e spectral
current sheet, region 21. and below the spectral c u rre n t sheet, region 11. For the fields
above the spectral sheet o f cu rren t, we have
f? 2I(.-'. r) =
(4.133)
Now. n o ting th a t the b o u n d a ry condition between layers 21 and 11 is th a t o f a m agnetic
current sheet, we have th e fo llo w ing relation s im ila r to eqn(4.43).
(i +rJuJ/S = (i + r ? , , ) / £ .
(4.134)
However, we have in th is case, r ^ 21=0, and
r>u =
a
(4.135)
%+
where T ^-12 is given by
T > 1 2 = e - 2 j^ aTp.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.136)
144
r F ll can be s im p lifie d e ith e r as
■ I2
-
?2
d f 4- 3 i - 2 7
f 'u
d 2 co t 3 xa '
(4 .1 3 7 )
or in term s o f T j as
F>u = e
- «
^
- [T 2 eL2j0ial { -
(4-138)
and FT2l can thus be found by using eqns (4.68,4.69) and s u b s titu tin g q u a n titie s
3\ 1.^21 b v d2. r f i x b y F f U . and
by r ^ - T liis allow s us to w rite the field p o te n tia ls
in term s o f F f U as follows.
(4.139)
F i - ( < r * * - = '> +
4 }± Ih i\
VVe have thus co m p le te ly determ ined the fields fo r sources above the slab.
(4.140)
T h e field
potentials we have found so far w ill allow us to d e te rm in e the fields everywhere produced
by a spectral c u rre n t sheet inside o r above the slab. For completeness sake, we w ill use
sym m e try and re c ip ro c ity relations to express th e field p o te n tia ls o f fields produced b y
a current sheet be lo w the slab in term s o f those fo r above th e slab. A n d the q u a n titie s
associated w ith tliis case w ill bear the less th a n sign " < " . T h e layering indices are as
shown in Fig.4.34. A n d th e field potentials are as follow s,
F < ,( r '. -) = £
>
, -z),
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.141)
145
I
1
o 4^
laver 23 i
F22
laver 22
laver 2
F2
layer 1
Fig.4.34
G eom etry for evaluation o f m u ltila y e r Green's fu n c tio n fo r a vertical m agnetic dipole
below a d ie le ctric slab.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
146
F<21(r'. =) = F>n ( - r ' . - z ) .
(4.142)
F < 2(='. z) = F > 2( - z '. —- )
(4.143)
F < 3( z ' . z ) = F > 3( - F . - z ).
(4.144)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
147
4.5 .2
A d m ittan ce o f L ong Slot
N ow th a t we have the spectral Green's functions o f a v e rtic a l m agnetic dipole below,
inside o r above a dielectric slab, we are ready to ca rry o u t a spectral analysis o f a long
slot. R eferring again to Fig.4.3 fo r slot geom etry and to Fig.4.4 fo r equivalent problem
geom etry. B u t th is tim e the le n g th o f the slot is extending b e yond the central dielectric
s trip . The a d m ittance o f the lo n g slot can be derived fro m eqn(4.70). repeated here for
convenience.
K:n = - ^
j
j
H(x = 0 ,y ,z)-M * d y d z .
==-■§ y= -4 However. because the long slot extends in to the dielectric s u b stra te outside o f the cen­
tra l d ie le ctric s trip o f the PPDVV guide, the m agnetic equivalent cu rre n t is now b e tte r
represented b y the follow ing d is trib u tio n .
^ s i n A : e l( % - |r|)<f(x)5 :
|?/| <
|~| < §
k, i
' T # - ^ 5 -sinfc e2( 4 - l - I M M i :
\y\ < f . § < |-| < §
sin /c«i «
^
M
sin
(4' 14o)
where
Ka =
(4.146)
and th e phase m a tc liin g c o n d itio n a t the interface between th e inside and outside dielec­
tric requires th a t
Lp = ^ - ( L - a) + a.
Kei
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(4.147)
148
ema is the dielectric constant of the substrate on the other side of the slot, the adjacent
layer. It may be a microstrip layer or another PPDVV layer, depending on the particular
application. If we denote the quantity
by Ma' . the image current elements on : =
plane are given by
d M .O') = ^
M'
^ E Ismkel( ^ - \ z '\ ) d = 'U y(W) £ S (x-2nb)S : |r'| < §
'
»=-“
t S (x -2 n b )z : § < |--'| < f
z
"
n = —oc
(4.148)
These current elements are then expressed in the spectral domain as follows.
d M Jz') = <
6 sin fcei
sin kei( ^ - | z'Ddz1■sinc{ky^ )
Wo*. sinfce2( f - |z'\)dz' ■sinc{ky\ )
6 sin k ci -£
E
£ (k * -!f)z :
n = —oc
•oo
£
^ ) f:
| < l-'l < 2
(4.149)
We will also need to determine the magnetic field at the slot opening. Tliis can be
expressed in terms of the spectral Green’s functions of source points in the corresponding
regions as found in the previous section. These Green’s functions, designated here by
Gf{
for the general case, can be applied to the spectral magnetic current sheets of
(4.149). giving us the fields everywhere. And in particular, for x=0. we have
OO
ff( x = 0 ,y ,z )=
j
dH (z') =
j
L
oo
J
J
d g ^d ^U z ^^d k ^d z '
—oc —oc
(4.150)
wliich can be separated into tliree contributing parts as
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149
OC OC f
a
£
2
2
ff(x = 0.y.z) = -±z J ] \ J dg!li(s,)dAIe(z')dz,+ J
-O C -O C
-/ = _ A
(4.151)
?j k v y d k x d k y
according to the regions where the spectral source sheets are located. First term is for
source sheets below the slab, second term for source sheets within the slab and the tliird
term is for source sheets above the slab.
Using (4.151) for magnetic field in (4.70). admittance of the long slot can actually
be written in nine separate parts. After reqrouping equivalent terms due to reciprocity,
admittance of long slot can now be written as
=
+ ‘2 /
/
2 = - -2
a.
+2
‘27r&
f
-J—
———
n
G
T-OC
+CO
l
l
g
{
2
J
■Mtfdzdz' 4- 2 j
a.
a
f G hw i-') ’ M-xdzdz' 4-
/
* ~
• siTic2(ky —
- -^33dzd='
f
——L
*Z='2'“ —
->
~
£ -/ — <£
L
) ^
8
G
■Mi^dzdz'
*
~
a
/
G h m <~') ’ ^fadzdz'
-— ^
— £
(kx
—)dkxdky
*
(4.152)
n = —oc
VVe designate here quantities associated with regions below', w'ithin or above the dielectric
slab by 1, 2 and 3 respectively. Subscripts Rij are then used where i,j= l, 2 or 3. For
example. R33 is used for a quantity associated with source points in region 3 and field
points in region 3, or both source and field points are above the slab.
M r33 =
• 2 ) r smke2( ^ - \z'\)smke2( ^ - |cr|).
snr kel-f*
*
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(4.153)
150
(4.154)
- V /f ll3 =
(4.155)
-V^R33-
(4.156)
(4.157)
s in - ite lf
2
2
Using symmetry relations, the expression for admittance can be further simplified into
the following form
Y
[2 ( / r 33+ ^R13)+4/r23+^R22]
snic~(ky — ) ^ 5(kx
rc=—oc
—)dkxdky
(4.158)
fpt22 is the reaction integral between source and fields within dielectric slab. / r 2 3 is the
reaction integral between source inside of slab and fields above slab,
is the reac­
tion integral between sources and fields above slab whereas 7/ } 13 is the reaction integral
between sources below slab and fields above slab.
Coupling integral [ r3 3 is given by
^R33 = tRZZ.zf-z' + Ut33,<5
where
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(4.159)
151
A R 3 3 .:jiz ' =
Q
-J
j'jJpL
282
+ j ( f A3 ~
A
4 3
+
(A
4 1
+
A ll +
^A4 — A w ) —
1
A ? 33.6
—
A .42 +
A4
2
) +
J2& .3A \J ( J . \ l
/ L —a
^ ,3 .4 ( A a i +
+
A \l)(A .4 2 +
A 4 I — ( A .4 2 +
(4.160)
/ ' A 2)
sinke2 (L — a)
ju jfi v 4
4&,e 2
Pa2) )
(4.161)
)•
The constituent integrals of A?33 are subscripted with A and are evaluated to be as
follows.
A 41 +
^2
"b ^ 2 + (^e2 —d2) c o s 2 ^e2
f 'A
4- [,\2 +
i
A 42
—2ke2(ke2 cos p2 ^ p cos Are2 A ^ 4- d2 sin,d2
-
^
sin ^e2
2 ) 2
(4.162)
A u
+
Aa
i
— (A 4 2 +
— FTT? ( ^e2 cos ^2<2 cos2 /ce2 —~z
2
(d2
2 — kio) \
-K 2 P2
sin d2a s i n &e2(A—a ) —2A:22 c o s 8 2 —
-A12)
do cos d2a sin2 Are2 —----—4- kz2 cos d2 A
2
cos
— —
ke2—— - + 2 ke 2 ,3 2 sin
8 2
——— sin k,
^ e 2 ~
2
(4.163)
A 43 — A w
sin ke2^ f 82
— k8e.2-
2 k,c2
L —a
k e2
4- 2 k ,e l "
+
A w
sin 82
—
A 43
cos k e2 A7 5 — 8 2 cos 82
sin ^ 2
~ yl
(Pi ~
(4 .1 6 4 )
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152
\/V M +
—1
(0 2
+
ru
fce2)
- ? a
i L ~ a ,
( ke2 sin
0 2 —cos ke2 —
h
^
(ke2 cos 8 2 —cos ke 2
2
[,A 2)
- ,
L ~
do 2 c o s dq2-a sin
ke2 —
2
— 8 >sin 8 2 —sin ke2 - ^
2
a
2
,
■ ?
ke2
sm
32 — )
— ke2 COS
2
(4.165)
and
=
ej i h °
31 - 31
3* 4- 3* — 2j 8 1 8 2 cot
(4.166)
Coupling integral / fll3 is given by
I r13 =
,- jW# _ (&e2 cos &e2
- j ( l 4- r ^ ) V 7-'32_^ i)a Are2e J -2
CT
ju jfi
2 8 1 (1
—,/J2sin &e2
j3- 2
— $*))
(4.167)
Coupling integral //£>3 is given by
2
Ift23 =
'( / e l l 4- 7^012) “^ ^ ^ ^ 2^ | ( l + c^.j)^C21+2^>.j^C22+.7(l —$ 3 )/c 2 3 | (4.168)
■ & )
where the consitituent integrals, subscripted with C. are
£ e2 ( c o s
C U
6
*2 ^
- c o s
82^ )
=
ke2 sin 32
- d2 sin ke2 ^
rCn = ------------- 3*~ 1 & ----------
„
1 C fU
( 4 ' 1 6 9 )
(4’1/0)
2/cel cos kei^f c o s d if 4- di sin ke l L? —s in 8\a — fcel(l 4- c o sd ia ) cos kel^ a
f a , = -------------------------------W - J^ ----------------------- ------------------(4 .1 7 1 )
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153
(4 .1 7 2 )
^C22 — ^C21
—2ke\ cos A:e l- ^ s in J i f + ke]_cos ke i
/c 2 3 =
Lp2 a sin 3 i a —3 \ ( l —cos 3 \ a ) s in ke
'
-------------------------------------------------------------
(4.173)
And finally, coupling integral [ r 22 is given by
£ fl2 2
~
^ R 2 2 .z jiz ' +
(4.174)
Ir2 2 .6
where
a~
I r 22.z ^ z' =
1+
1-
I El +
(4.175)
j(^E3 ~ ^Ea)
and
sin kei ( L p — a) —sinkei L p '
[r.22.6 = ~z~ (a +
2uJfi
2ui
ll V
\
kei
(4.176)
,
with constituent integrals subscripted with E given as follows
IE1
3 i sin 8i% sin kei Lp» g —
= 20
cos 3 i § cos
+ kei c o s k ei :
3? - k 2el
and
^E3 — [ ea —
0i
32
i ~ &
s in k e iL p — s i n k ei ( L p — a)
a
2 k ei
2
Lp — a
/ - 2 1, 2 ( - keiPi sin keiLp +- ^kei 0i cos (3A cos ke i
sin ke ^
(01 - ke l)2 K
2
2
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(4.177)
154
(4.178)
Integrand I r22 reduces to fc of (4.107) when L=a.
Assuming that all modes are below cutoff except the fundamental mode or PPDW'
mode, the integrand of the spectral integral for Ylrx is singular only for ky =
and
the contribution to the integral due to these poles can be found using residue calculus.
Writing the total value of the residue in terms of four different parts, we write
(4.179)
where YresA- YresB■ Yresc and YresE are parts of the residue that come from the coupling
integrals [rszJriz- [ r 23 and I r 22 respectively. These partial residues have values given
below'.
cjA,<1^ (6 c — es) sin2 3u a
res
(4.180)
J2
1
(4.181)
where conmion factor Cres between YTesA and KesB is
(4.182)
C,r e s
Also.
YresC ~ M rea • [-fc2l]/3i=^1(, ' [-^Cll + 7 ^ 1 2 ]# ,= # ,,
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(4 .1 8 3 )
155
Kr„ E = A /r„ - l [ / E1]A = J i_
(4.184)
and finally the common factor among all partial residues is
M rea
= -------- '2± ---- -j—s iii( ? k vs — - , kys^ s
bujfi s in
(l+jft,§)
v 2
(4. 185)
It is noted that the guided mode power due to currents within dielectric strip can be
accounted for by the residue of
denote by
Y r e s .in t
Y in
with integration limits ~=-^ to
wliich we will
and can be found as follows.
V r e s .tn t
=
Y re a E
+ ^
(4-186)
Y re s C .
Whereas the guided mode power due to currents outside of dielectric strip is accounted
for by residue of Yin with - = ^
to
K es.ext —
4.5.3
r = | to
Y j-e s A +
Y re a B
+ -
Y re a C
-
(4.187)
Efficiency o f L ong-Slot T ransition
Efficiency of the PPDW long-slot transition is same as defined by eqn(4.117) as in shortslot analysis. Fig.4.35 shows the efficiency of a case where the inside dielectric constant
of the PPDW guide is 10.8 and the outside dielectric is assumed to be air. It is seen
that the efficiency is very high for a wide range of slot lengths and PPDW widths. Even
when the slot is physically larger than the PPDW width, the efficiency can still be liigh.
implying that we really do not have to place the slot completely inside the PPDW' guide
in order to suppress the parallel-plate mode excitation.
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156
0.9
0.8
L / ao= 0.05 to 0.50
0.7
c
0.6
.22
,*22
0.5
04
0.05
0.25
0.3
N orm alized Frequency, a/X,
Fig.4.35
Efficiency of PPDW Slot-Transition in which the length of the slot can be larger than
the width, a. of the Parallel-Plate Dielectric Waveguide.
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157
4.6
Conclusion
We have performed a rigorous analysis of a rectangular slot transition on the ground
plane of a Parallel-Plate Dielectric Waveguide. General agreement between theory and
experiment shows the validity of analysis and reliability of the simulation data. Analysis
indicates that the efficiency of such a slot is extremely liigh, more than 99 percent is
acliievable. In particular, for the present case of PPDW parameters and dimensions, as
the slot length varies from zero up to the extent of the width of the central dielectric
strip of the PPDW of 14.4mm, theory shows that the efficiency varies from 99.77 to 99.80
percent. This corresponds to about 0.013 dB of loss from the transition. Although tliis
kind of efficiency has not yet been acliieved by our present prototypes, the results are
very encouraging. Tliis kind of slots on the PPDW ground planes provides another class
of transitions with which the PPDW guides can either interconnect with one another or
with other types of transmission lines like the microstrip lines over different layers of a
multilayer integrated circuit. The extremely liigh efficiency of the slot transition shows
that low loss and large number of layers is simultaneously possible to be acliieved in
multilayer microwave and millimeter wave integrated circuits. The calculated efficiency
of PPDW long-slot suggested that pratical PPDW-slot transitions may not be limited
only to slot dimensions witliin the dielectric strip.
If necessary, slot lengths can be
extended beyond the width of the central dielectric strip of PPDW guide and acceptable
performance of the transition can still be expected.
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Chapter 5
Conclusion
We have completed an investigation on the Parallel-Plate Dielectric Waveguide.
Its
dispersion characteristics and equivalent characteristic impedance for a PPDW circuit
model have also been examined. The metal loss and dielectric loss characteristics of
the waveguide have been investigated, covering microwave and millimeterwave frequency
range.
Loss characteristics in tliis frequency range is predicted to be better than a
microstrip line. The comparison, however, has been done only up to about 600GHz and
is rather conservative in the way that metal loss in microstrip due to edge effect can be
much more prohibitive at these frequencies than the values actually used for comparison
with PPDW.
Transmission line models for the PPDW guide and devices have been developed
and used for simulation of PPDW devices performance. Agreement between simulation
data and measurements of PPDW prototypes are found to be good. Tliis shows the
validity of the analytical approach as well as the versatility of circuit models developed
for the waveguide, the coaxial-to-PPDW transition, and the slot-to-PPDW transition.
The use of the PPDW in a multilayer environment is a natural choice. Firstly, each
PPDW layer is electrically isolated from the other layers by the parallel metal plates of
the waveguiding structure. Secondly, transitions from layer to layer can be done using
158
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159
either probes or slots. Both of these transitions have been investigated theoretically and
demonstrated experimentally. Specific prototypes include (1) two-port PPDW guides
using coaxial transitions. (2) microstrip-to-PPDW-to-microstrip 3-layer coupler using
PPDW-to-sIot transitions, and (3) microstrip patch antennas fed by a slot on the PPDW
ground plane. The results agree well with the circuit model predictions. On the other
hand, a microstrip-to-PPDW-to-microstrip 3-layer coupler can be implemented also by
using coaxial transitions instead of slot transitions as shown in Fig.5.1.
W ith these
PPDW transitions readily available by design, use of the PPDW guide in a multilayer
environment is depicted in Fig.5.2 where the PPDW layer functions as an interconnection
medium between two electrically isolated layers of active/passive devices. These two
layers in tliis case are respectively the top and bottom layer of the complete multilayer
structure. The top and bottom layers are particularly suitable for positioning active
devices where heat dissipation may be a primary concern. The middle layer(s) are to be
PPDW-based and will take on the role of interconnection as well as power distribution. Of
course, the middle layer (PPDW layer) can be expanded to many layers for the flexibility
of design and interconnection.
PPDW is fundamentally a dielectric waveguide. This property itself positions the
waveguide as a good candidate for applications in millimeter/submillimeter wave frequen­
cies. The loss characteristics of PPDW up to a few hundred gigahertz has started to show
some advantage compared with the microstrip line. One can expect tliis trend to con­
tinue in favor of the PPDW guide in even higher frequencies where metallic waveguides
will become too lossy due to skin effect and Ohmic losses.
5.1
Future C hallenges
The Parallel-Plate Dielectric Waveguide is a building block just as any other waveguide.
By itself, the best value it can give may not be much more than a pedagogical one.
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160
microstnp
lower microstnp
substrate
coaxial probe
microstnp
side view
upper microstrip
substrate
PPDW laver
microstnp
coaxial
probe
cross section A - A ‘
Fig.5.1
A 3-layer Microstrip-to-PP D W-to-Microstrip coupler configuration using coaxial transi­
tions.
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161
top layer
active devices and/or
heat dissipation
p a s s iv e d e v ic e s
PPD W
layer
lowloss
bottom
layer
passive devices,
power distribution,
top layer to bottom layer interconnects.
active devices and/or
passive devices
heat dissipation
Fig. 5.2
One possible configuration of Parallel-Plate Dielectric Waveguide in a multilayer
integrated-circuit environment.
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162
Therefore, more devices and subsystems based on PPDW's need to be designed and
improved. We have already demonstrated that the PPDW guide does have properties
that allows it to be used in building circuits and devices in a multilayer environment. And
as mentioned before, we also anticipate that PPDW can perform well in the millimeter
and submillimeter wave spectrum.
The fundamental stage of studies on the PPDW guide has been completed suc­
cessfully. It has shown clearly that the PPDW is a very good building block for the
implementation of multilayer integrated circuits. As for millimeter/submillimeter wave
circuits, the determination of more suitable dielectric substrates for PPDW to operate
at those frequencies is very important and the need is imminent.
Further study on the PPDW long-slot transition is also needed in order to provide
guidelines for the design of devices based on such transitions. The use of slots on the
PPDW ground plane as radiating elements has also been noted and should be a part of
future research effort. In tliis respect, we tliink an improvement of the slot modeling is
also possible when the finite tliickness of the ground plane can be taken into consideration
during the analysis stage. The assumption of zero tliickness of ground plane in some cases
may have introduced errors into the circuit model that could cause discrepancy between
experimental values and theoretical predictions. Improvement of the slot model can
better support future multilayer work around PPDW guide. The present analytical model
of the coaxial-probe to PPDW transition has not been able to predict some experimental
details and seems to be related to a particular limitation of the model. Furthermore,
no experiment on the PPDW guide has been done in the millimeterwave band, and this
effort should also be a priority.
As for immediate challenge, some exploratory work on multilayer PPDWr circuits
using probe transitions are being carried out here at the Microwave Laboratory of Polyteclinic University. Multilayer PPDW circuits using slot transitions are also being in-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
163
vestigated because of its potential for the implementation of practical many-layer microwave/millimeterwave integrated circuits.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A
Analytical Model of a 2 -directed
Spectral Magnetic Current Sheet
In order to carry'’ out a spectral analysis of the slot on a PPDW ground plane, we first
need to find the spectral Green's function of a vertical magnetic dipole in a stratified
medium [II] [14].
Referring to Fig.A.l . a positive and a negative magnetic charge layer are symmet­
rically centered above and below the r=0 plane. These two charge layers together are
referred to as the magnetic dipole layer. Height of the dipole layer is Ah. Each sheet
of the dipole layer carries a magnetic surface charge density of pm = pm(x.y). Magnetic
field inside the dipole layer is uniform and is given by
ff- =
h
.
(A.l)
Consider a rectangular contour abed lying on the y — z plane. At y = —
magnetic surface charge density is pm while at y =
density is given by pm +
the
the magnetic surface charge
Applying Ampere's circuital law around contour abed,
we have
164
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165
positive m agnetic
charge layer
= -A w /2
medium 2
Aw
\ /
direction o f
equivalent
m agnetic current
medium I
H z magnetic
negative m agnetic
charge layer
field due to dipole
x - clxis
is c o m in g out
o f the p a p e r
F ig.A .l
Geometry of magnetic dipole layer for simulation of a vertically directed magnetic current
sheet in the limiting case when the layer is infinitely thin.
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166
H d l = £ = electric f lu x enclosed.
(A.2)
J abed.
Contour abed encloses an amount of electric flux proportional to the area of the con­
tour. The electric flux denisty is assumed to be uniform witliin the small area. That is.
eqn.(A.2) can be written as
C — -
f
rc — —
rd
[ H dl + 1° H dl +
Ja
J c
Jb
Hdl + [ a Hdl = £ = O (A w A h).
(A.3)
J d
where
f b H dl = Hy lA w
Ja
(A.4)
£ H d l= Hz{b)Ah
(A.5)
£
(A.6)
Hdl = - H y2A w
J ' Hdl = - H z{a)Ah
p H z(b) = - p m -
pH z{a) = —pm
(A.7)
(A.8)
(A.9)
Substituting eqns(A.4) to (A.7) into (A.3), the integral becomes
Hdl = {H y l - H y2)A w + { - H z{a) + H jb )) A h = 0 (A w A h ).
J abed
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(A.10)
167
In the limit, to first order approximation. (A .10) becomes
(Hyl - Hy2)Aw + (~ H : (a) + Hz{b))Ah = 0.
(A .ll)
And by relations (A.8) and (A.9). we further obtain
- Hy2)Aw 4- (pm - pm — ~^JLA w )A h - 0.
(A.12)
That is.
- Hy2) -
= 0.
(A .13)
Now. consider a cylindrical surface S enclosing part of the positive magnetic charge
laver. we have
£
J
mvds
Jm
uds
d
J J J"Qmvdv
(A. 14)
c y lin d r ic a l s u r fa c e
where qmv is the volume magnetic charge density. We see that
Jnwd'S
Jnvv(is
(A.Id)
c y lin d r ic a l s u r f a c e
and because the charges inside volume enclosed is in fact contributed by a charge sheet
of density p m over an area of S ' , we have
-§ ;I J J q""'dv=-§ iL pmds'Together, we get
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(Al6)
168
f
Js'
Jmvds' =
f
pmds'.
at Js'
Since S' is arbitrary, we must have Jmv = —
fA.17)
and because Jmv is opposite in direction
to + r. which is tiie direction of Jni. we have
Q
Jm. —
Pm —j'dpTrf
(A..18)
Using relation (A. 18) into (A. 13), we get
(A .19)
or
- Hy2) =
As A h
(JmA h ) .
(A.20)
0. the volume current is squeezed into an infinitesimally tliin layer. VVe
define Jmjj = JmA /i in such a way that as A h —>■0 and Jm —>■oc, Jms remains constant,
thus
hasthe meaning of a surface current density. Jma has a direction normal to
r=0 plane and is a normal surface current density, note that as A /i —> 0. Jm —> oc; we
require pm —»• oc meaning also ff~ —»• oo. But f f z is still uniform inside the dipole layer
within the distance of A h.
To summarize, we have,
- Hy2) =
(A.21)
Similarly, if we consider a contour lying in the x —z plane instead of the y — z plane, we
will be able to obtain the relation
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169
~
H
=
x 2 )
ox
(A.22)
For Jms = 8(x)£(y)~. J ms = 1- 5. and the boundary conditions in the spectral
domain becomes.
2 — H y l) = —jky
(A.23)
and
ju ;ti{H x2
= ~ jk x.
(A.24)
In sunmxary. the spectral boundary conditions for a vertically directed magnetic current
sheet are
2-
= ---Jjjl
(A.26)
Hyi ~ Hyi =
E 'x 2
L' E y 2
H z 2
=
H
(A.25)
E y l ,
z l-
(A.27)
(A.28)
and
E z 2
=
E zl
= 0.
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(A.29)
170
Starting with the Helmhotz equation for the magnetic vector potential function F with
a given magnetic current density A/.i.e.
V 2F + k 2F = —A/.
(A.30)
we have, for a r-directed magnetic dipole, the following relation between the z-component
of the magnetic current and the z-component of the magnetic vector potential.
(V 2 4 - k2)Fz = - M z = -6{x)S{y)S{z).
(A.31)
Expanding in rectangular coordinates, we get
/ d2
S i2
+ ^
cF
9\
+ a ? + * ) F= = S W M W -
(A-32)
That is.
^ F - _ + k>F: = -S (X)S(y)S{z) - ( J 1 + | 1 ) F „
(A.33)
And in spectral domain over kx and ky at ~=0 plane, we get
— ^ + ^F-_ = -S (z) + (k l + kl)F,.
(A.34)
From spectral analysis, we also know that the spectral magnetic field can be related the
the vector potential function as follows (see eqn(4.6),
+ k2Fz = j ‘j JiiH z.
(A.35)
As a result, the magnetic field can be expressed in terms of the delta function and the
potential function as follows,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
171
h _= J M
jujfl
+
jujfl
A
(A.36)
wliich is the r component of the magnetic spectral Green's function of a r-directed
magnetic dipole.
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Bibliography
[1] F.J. Tischer. H guide with laminated dielectric slab. IEEE Transactions on Mi­
crowave Theory and Techniques. M T T -18:9-15, January 1970.
[2] R.M. Knox. Integrated Circuits for the Millimeter through Optical Frequency Flange.
Proc. Symp. Submillimeter Waves. pages 497-516. 1970.
[3] S. Sliindo and T. Itanami. Low-Loss Rectangular Dielectric Image Line for
Millimeter-Wave Integrated Circuits. IEEE Transactions on Microwave Theoiy and
Techniques. MTT-26:745-751. October 1978.
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[5] T. Itoh. Trapped Image Guide for Millimeter-Wave Circuits. IEEE Transactions on
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[8] R.F. Harrington. Time-Hannonic Electromagnetic Fields. "McGraw-Hill Book Com­
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173
[12] N. K. Das and D. M. Pozax. Multiport Scattering Analysis of Multilayered Printed
Antennas Fed by Multiple Feed Ports, Part I: Theory; Part II: Applications. IEEE
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IMAGE EVALUATION
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