UPPER
CUT-OFF FREQUENCY
NEW LEAKY
WAVE
OF
THE
BOUND
WAVE
ON
THE
SLOTLINE
AND
Ján Zehentner, Jan Macháè, Maurizio Migliozzi
Czech Technical University, Technická 2, 16627 Prague 6, Czech Republic
slotline, with its influence on the upper limit of
ABSTRACT
the dominant
This
profound
behaviour.
providing
bound
paper
contributes
understanding
A
simple
the
wave
of
upper
to
closed-form
cut-off
propagating
over
the
band, and
more
will present closed-form formulae for this limit
suitable for CAD. The paper is a contribution to
formula
frequency
wave frequency
circuit
a
uniplanar
bound
of
the
slotline
is
the
general
open
planar
discussion
on
transmission
characteristics
lines,
particularly
of
the
slotline.
presented. A new leaky wave which brings down
this limit due to the overlapping of the boundand
leaky-mode
reported.
This
regions
wave
is
can
identified
deteriorate
and
circuit
performance, especially in the mm-wave band.
Fig. 1 Cross-section of the slotline
Conclusions made for the slotline hold generally
We have analysed the open slotline, Fig.
for open printed-circuit lines.
1, by the method of moments modified as in the
Galerkin testing procedure in the spectral domain
with successive
INTRODUCTION
integration
Planar
microwave,
transmission
millimetre
lines
wave
used
and
in
optical
deformed
residue
complex root searching. The
path
from
in
the
the
spectral
real
contributions
axis
domain
to
associated
was
include
the
with
the
investigated
propagation of the corresponding surface waves.
intensively over the last fifteen years, e.g. [1-5].
The accuracy of our calculation was checked by
Besides the attempt to minimize natural losses in
comparison
them, attention has been paid to leakage that can
b/k
result
neighbouring
with their plotting published in [4].
circuit,
program provides
integrated
in
circuits
have
cross-talk
components
or
been
between
portions
of
the
in
a
lowering of their quality factors, or in outright
0
of
the
normalized
phase
constant
and the normalized leakage constant a/k
us
with
all
0
Thus, the
solutions
known
and published until now.
loss of power. Leakage has a significant effect on
circuit performance. Shigesawa, Tsuji and Oliner
[4] have summarized
effects which
all
earlier
findings
THE
2ND
may occur in open printed-circuit
However,
dispersion
lines.
In this paper we will deal with a new,
previously
unidentified
leaky
wave
LEAKY
WAVE
about
on
the
besides
equation
of
the
these
solutions
slotline
also
the
has
a
further solution that is new and interesting. Let
us denote its already known complex solution as
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the
1st
leaky
mode,
and
the
new
complex
solution, which we are reporting here, as the 2nd
leaky
mode.
The
behaviour
of
the
latter
influence
UPPER
BOUND
[4].
frequency of the TE
2nd
leaky
mode
sets
constant
of
2
the
solution
2nd
becomes equal to that of the TE
1
leaky
wave
surface wave,
the leaky wave is nonphysical in the sense of the
generalized condition of leakage [5]. By chance
f
3
(0.2695)
is
here
only
slightly
lower
and
3
f
6
CUT-OFF FREQUENCY
are
too
OF
THE
WAVE
To
is
3
the
f
when
and f , at which point
2
phase
in,
cut-off
and f
1
improper real. Between f
the
the
surface wave f
1
the
Between
since
close together, Fig. 5.
taken
from
negligible,
is
illustrated in Fig. 2, and belongs to the slotline
over
is
than
f
4
leakage
avoid
it
is
trouble
necessary
caused
to
by
know
the
surface
upper
frequency cut-off of the dominant bound wave.
In the case of a spectral gap this is given by f
4
denoted
hereafter
as
f .
When
c
simultaneous
propagation of the bound and the 1st leaky wave
occurs, the upper frequency cut-off f
c1
is equal to
(0.2780), the uper cut-off frequency of the pure
f5 as in Fig. 5. If the region of the bound and the
dominant
2nd leaky wave overlap, the cut-off frequency f
bound
wave.
frequency gap from f
to f
4
to
the
presence
of
The
2nd
frequencies higher than f
and TM
1
TE
1
0
leaky
wave.
surface waves. Leakage to the
surface wave occurs at a lower angle than to
surface wave.
the TM
0
constant a
2
Corresponding
leakage
is greater than a , Figs. 2,3,4.
1
When w/h increases, e.g. to 0.5, f
3
distinctly lower than f
is
and the bound and the
4
2nd leaky wave are present simultaneously in this
overlapping region, Fig. 3. From f
6
is nonphysical since b/k
to f
7
the wave
does not comply with
0
the generalized condition of leakage [5].
These newly-revealed characteristics are
more
expressive
on
the
higher
permittivity
substrate, e.g. when e =10.8, w/h=0.236 (w=0.15
r
mm
and
h=0.635
overlapping
f =59.9
interval
GHz
4
mm),
is
from
purposes.
The
improper
real
2,3,
5,
Figs.
always
frequency f
1
sets
in
4.
Now
f =46.0
GHz
3
sufficiently
practical
Fig.
Fig.
2nd
significant
solution
or
improper
at
the
TE
1
the
has
greater
influence
(h/l )
2,3,4,
while
for
a
wider
of
f
depending
c
on
the
dimension w/h and permittivity e
is
r
relative
shown
in
Fig. 6. Designers will appreciate the closed-form
formula
(
h/
λc ) =
[
A1 A 2
1
e.
2
Owing
r
3
to
]
A5
+ A 3 (w / h)
A4
where A , A , A , A , A
4
limited
5
(1)
are simple functions of
space
in
this
paper,
the
complete records will be given in the talk. Eqn. 1
holds
when
0.01£w/h£1
and
2£e £16.
r
These
intervals of validity are the same also for Eqns. 3
and 5.
The equation
(
h/
λ
cb1
)=[
0.76388
.
+ 0.20999 .
(
)
w / h - 0.493
0.08212
]
(2)
24.589
either
complex
cut-off
on
the
satisfies
slotwidth
its
the
boundary
wave
condition
line
f =f
c
separating
region
from
and
c1
the
the
controls
left
pure
right
the
bound
region
of
simultaneous propagation of the bound and the
1st leaky waves.
Similarly,
behaviour of the slotline of a narrower slotwidth
Figs.
instead
c
for
and has the same value as the 1st
much
3
The normalized upper frequency cut-off
to
leaky wave at this split-off point. The 2nd leaky
wave
c2
is given by f , e.g. Fig. 4.
At
energy can leak into
3
the TE
now disappears due
5
the
conventional
f
c1
normalized
(h/l
)
c1
instead
is also plotted in Fig. 6. The formula
0-7803-4603-6/97/$5.00 (c) IEEE
of
(hh) =[B,+B2(w/h)-B3]2
describes
fC1. B ~, Bz,
functions
will
Bq,
be specified
The
the
(3)
G, dependent
in the talk.
condition
REFERENCES
produces
fC=fC2
the
equation
(h/
[0.00816+ 0.14095.
kct32) =
(4)
. (w/h
defining
the
pure bound
-0.037)0G8218]0”4941’
boundary
wave
simultaneous
line
region
propagation
the 2nd leaky wave,
The
the
the right
region
of the bound
left
of
wave and
Fig.7.
normalized
(h/XC2) instead
separating
from
upper
frequency
of fC2 is also drawn
participation
in this research. The work was done on the
SP-2 computer at the Joint Supercomputer Center of the
Czech Technical
University,
University
of Chemical
Technology and IBM Prague.
cut-off
in Fig. 7 and
[1] A. A. Oliner: Leakage from higher modes on microstrip
line with application to antennas. Radio Science, Vol. 22,
1987, Number 6, pp. 907-912.
[2] H. Shigesawa, M. Tsuji, A. A. Oliner: Dominant mode
power leakage from printed-circuit
waveguides, Radio
Science, Vol. 26, 1991, Number 2, pp. 559-564.
[3] A. A. Oliner, D. R. Jackson: On spectral gaps at the
transition between bound and leaky modes. Proceedings
of the 1995 International
Symposium on Electromagnetic
Theory, URSI, St. Petersburg, Russia, May 23-26, 1995,
pp. 764-766.
[4] H. Shigesawa, M. Tsuji, A. A. Oliner: Simultaneous
Propagation of Bound and Leaky Dominant Modes on
Printed-Circuit
Lines: A New General Effect. IEEE Trans.
Microwave
Theory
Tech., Vol.
43, December
1995, pp.
3007-3019.
(h/&2)
= [C, +C2(w/h)-c’]c4
where
Cl,
functions.
limit
C3,
Their
It stands
leaky
C2,
reading
to reason
wave
C4,
of the bound
+C5
C5
will
are
down
s,
be given
that either
brings
(5)
dependent
in the talk.
[5] D. Nghiem, J, T. Williams,
D. R. Jackson, K. A.
Michalski: The Choice of Integration Path in the SpectralDomain
Solution
of
Leaky
Modes.
Personal
communication
about the viewgraphs presented at the
URSI
National
Radio
Science
Meeting,
Boulder,
Colorado, January 1995 by D. R. Jackson,
the 1st or the 2nd
the
upper
frequency
1.6
wave.
1.5
X“
~1.4
CONCLUSIONS
1.3
We believe
on the
slotline
may
printed-circuit
encoded
findings
in the
also
lines
in
dispersion
that the new effects
the
and
equations.
mm-wave
the desired
for
band.
operation
wave
other
paper
It also
particularly
frequency
1.1
open
generality
solution
the behaviour
CAD,
only the bound
This
on
possess
multi-valued
regarding
formulae
occur
1.2
revealed
of
their
presents
new
1.0
improper
0,0
0.1
0.2
0.3
2-
L
,
I
r
0.0
0.1
0.2
0.3
0.4
0.5
hlLO
,
A
of the slotline
provides
wmplex
0.9
useful
in determining
band
in which
can be propagated.
ACKNOWLEDGMENTS
0.4
0.5
hik,
We gratefully
acknowledge
the scholarship
granted by the Faculty of Electrical Engineering of the
Czech Technical University in Prague to M. Migliozzi,
a
student of the University of Ancona, Italy, supporting his
Fig. 2 The normalized
slotline
with
normalized
w/h=O.4
phase and leakage
and
frequency (h/XO).
0-7803-4603-6/97/$5.00 (c) IEEE
E
=2.25
r
constants
for the
as a function
of
1,75
I
improper
x“
&
\.
F:
1.50
,:proper
complex
(leaky
I
11
2.0
I
1
b’
/’f’~\\
real
x“
h
1-
mode)
improper
1.5
real
/
1.25
..
. . . .. .. . . . . . . . . . . . . . . . . . . .
1.0
.... . . . . . . . . . . . . . . . . . . . . . . . . .
1.00
0.0
0.1
0,2
0,3
0,4
0.0
0.5
0,1
0.2
I
t
0.3
0.4
0.5
hlk,
t-s.
r.
I
1
,
3
0.3
0.4
0,5
1 .0,.
0.2
0.1
0.0
0.1
0.2
0.0
h/LO
Fig. 3 A plot similar
slotwidth w/h=0.5.
to that in Fig, 2 but for
,
‘:m
Y“
-s
00
,
wider
0,1
Fig. 5 A plot similar
slotwidth wlh= 1.
0.2
I
,
0.3
0.4
i
0.5
wk.
to that in Fig. 3 but for still wider
5
l“’’’’’
x“
a.
4
I!mproper
complex
”’’’’’’’’’”1
1-
.......... ......
/
4
3
1!
2
...................
0.1
1
r
0.00
0.05
0.10
0.15
hi).,
1.0
X“
-2
,
w=O.15
0.5 -
,
(),0 ~
0.20
0.00
,
Fig. 6 The normalized
dominant bound wave
0.00
15
f
2
\
(
0.05
0.15
to that in Fig. 2 but for narrower
and higher substrate permittivity
C=10.8.
of the 1st leaky
~ 0.3 .--?
2
d“
k 0.2
0.1
Fig, 7 A plot similar
propagation
1“’’’’’’’’’’’’”’”1
0.20
hl)u,
Fig. 4 A plot similar
slotwidth
w/h=0.236
1.00
wave (h/ACl).
,
0.10
0.75
upper frequency cut-off of the
determined by the spectral gap
(h/XO) and by simultaneous
h= O.635 mm
0.0
050
W/h
mm
sr=10.8
0.25
. .. .. . . .. . . . .. . .. . . .. .
.. . .. . . . . .. . . . . .. . . . . ..
. ... . .. ... . . . .. . . . . .------
... . . . .. . . .. . .. . . . .
to that in Fig. 6 but the normalized
hilcz
upper
frequency
cut-off
(h/kCz) is now determined
the 2nd leaky wave .
of the dominant
by simultaneous
bound
wave
propagation
of
+
0.00
0-7803-4603-6/97/$5.00 (c) IEEE
0.25
0.50
0.75
1.00
W/h