IEEE TRANSACTIONS ON MICROWAVE THEORY AND
1104
rECHNIQUES,
38, NO. 8, AUGUST
VOL.
1990
Transverse Resonance, Standing Wave, and
Resonator Formulations
of the Ridge
Waveguide Eigenvalue Problem and Its
Application
to the Design of E-Plane
Finned Waveguide Filters
JENS BORNEMANN,
Abstract—Utilizing
guide
the rigorous
eigemnortes,
this
paper
sign of compact,
low-cost,
band-pass
with
filters
takes into account
higher
order
advantage
eigenvalue
quasi-high-pass
filters
with
the
techniques.
with
in
version
15].
with
&plane
the
lization
for
the related
the
design
gap widths
and
resonator
data
and
are
and
modegiven
E-band
and wide stopband.
of
for
(60-90
The theory
INTRODUCTION
advantages
been
neglect
field
description
for
influence
of
the
thickness
reliable
filter
the
or
used
to
filter
the
ridge
have
designs
unilateral
construct
designs
design
of
MEMBER,
IEEE
fins of negligible
thickness with
layer [12] have been applied for
evanescent-mode
band-pass
filters.
Such
filters [12], [16]–[18] are of particular
importance
due to
several advantages over the conventional
filter types, e.g.
compactness
and wide
waveguide,
stopbands.
capacitive
ments, however,
screw,
are often
and to mass-produce
The
and
difficult
[15]–[18].
usual
round
thick
post
to fabricate
On the other
tric layers cause additional
losses;
tion-loss
design potential
inherent
field
ridge
filter
ele-
at low cost
hand dielec-
hence the low-inserin the finned wave-
[8],
either
based
distribution
of
the
of
finite
both
waveguide
turned
in
on
ridge
metalthe
cascaded
to
be
ridge
waveguide
distributed
[11],
[15].
shunt
More
accuracy
with
respect
to the number
mode-matching
a stable
in
nontouch-
Manuscript
received March 30, 1989; revised February 27, 1990.
J. Bornemann
is with the Department
of Electrical
and Comuuter
Engineering,
University
of Vic(oria,
P.O. Box 1700, Victoria,
‘B. C.,
V8W2Y2
Canada.
F. Arndt
is with the Microwave
Department,
University
of Bremen,
Kufsteiner
Str., NW1, D-2800 Bremen, West Germany.
IEEE Log Number 9036423.
0018-9480/9010800-1
to
be-
separation
constants and normalization
factors
have to be numerically
calculated
before the
procedure
in the propagation
direction.
These
directly influence the complexity,
the numerical
Therefore
capacitances
sensitive
formulations,
cause the
involved
matching
quantities
dimensional
are
disconti-
in the design. The
is extremely
and
all-metal
[12].
description
of the
the modal scatter-
eigenfunction
and the computational
sections
recently,
however,
cross-section
considered,
impor-
capability
block
accu-
sections
out
is the basic building
response,
different
[12]–[14].
handling
a rigorous field theory
is utilized to formulate
of the waveguide-to-ridge-waveguide
which
computed
characterization
effects
with
low-loss
a low power
In this paper,
ridge waveguide
nuity,
discontinuities.
Bilateral
the
ing matrix
finned
field-theory
[7],
are
the
However,
its
of low-cost,
presented
the
wave
include
in
Various
for
found
circuits
and,
discontinuity
for
have
millimeter
these
possibility
designs.
techniques
metallization
and
of
reported
equations
or
circuits
operation
the
waveguide
the
SENIOR
ing unilateral
E-plane
and without
a dielectric
may achieve
waveguide
circuit
thickness.
low-pass
solving
ridge
GHz)
numerical
ARNDT,
guide technology may not be fully utilized. Moreover
thin
unilateral
fins may lead to small ridge gaps, and therefore
ridges,
have
waveguides
often
(8-12
broad-band
ridge
of
finline
for
wave
in microwave
The
thin
approximate
tant
and the
The
design
selectivity
finned
integrated
approaches
rate
fin thickness
of different
standing
X-band
applications
single-mode
the
theory
is demonstrated
filters
classical
and
[1]–[
Most
method
waveguide
fins. The design
discontinuities.
resonance
achieve high skirt
many
devices
of
evanescent-mode
E-plane
by measurements.
lDGED
large
de-
all
problem
I.
R
wave-
Computer-optimized
passbands
GHz), which
is verified
at
and band-pass
is compared
matching
of the ridge
computer-aided
of both the finite
interaction
of the transverse
cross-sectional
distribution
metallic
the influences
AND FRITZ
IEEE,
an accurate
low-insertion-loss
bilateral
mode
field
presents
MEMBER,
reliable
for
terms
the three-
process.
special emphasis
and
of expansion
effort
is placed on the selection
design
method.
Three
of
distinct
mode-matching
techniques,
each having its individual
advantages, are applied to solve the special eigenvalue problem for the purpose of investigating
their suitability
for
the efficient
analysis of finned waveguide discontinuities.
The first is the transverse resonance method, which has
turned out to be well suited to dealing with complicated
waveguide sections [13], [19]–[21]; the second is the conventional
standing
advantage
tions
in the
104$01.00
wave
of independent
01990
subregions;
IEEE
method
[22], which
numbers
and
the
shows
of expansion
third
is the
the
func-
resonator
‘
BORNEMANN
AND
ARND’C
TRANSVERSE
RESONANCE,
STANDING
WAVE,
AND
RESONATOR
1105
FORMULATIONS
,/
bands.
‘Z=71
mits
Moreover,
the exact design theory
high-precision
techniques
error
manufacturing
without
The
per-
or milling
of postassembly
trial
and
methods.
immediate
teracting
by etching
the necessity
adjustment
presented
modal
structures
S-matrix
includes
combination
the higher
of all in-
order
mode
cou-
pling effects and allows the stopband
characteristics
of
the filters to be taken into account for the filter design.
For computer optimization,
the evolution strategy method
[13], [24], i.e., a suitably modified
direct search procedure,
is applied where no differentiation
step in the optimization
process
minima
mized
(a)
al
r
hence
evanescent-mode
the
problem
Design
E-plane
finned
passbands
at about
11 GHz
and 75 GHz
The
design
11 GHz
uses WR
for
waveguide
for
the input
below
and output
cutoff
of
examples
with
cutoff
7
is necessary;
may be circumvented.
local
for opti-
band-pass
filters
are given.
90 waveguides
sections
for the filter
above
and a WR
section.
42
For the 75
GHz design example, WR 12 waveguides
above cutoff
and a WR 7 waveguide below cutoff, reduced in height,
are applied. The filters achieve high attenuation
over a
wide second stopband.
results
The ‘theory
of an E-plane
90 evanescent-mode
finned
k verified
WR
by measured
62 structure
and a WR
filter.
(b)
H.
A. Field Description
THEORY
with Two Vector Potential
For the computer-aided
E-plane finned waveguide
trix method
[13], [19],
structure is decomposed
I i
I
(c)
continuities:
Fig. 1. Evanescent-mode
bilateral
E-plane finned waveguide band-pass
filter. (a) Input and output waveguide,
below-cutoff
waveguide
with
E-plane fins of thickness t.(b) Cross-sectional
dimensions.
(c) Longitudinal section dimensions
in the filter region.
section,
gular
waveguide
for
junction
which
is reported
to yield
a more
appropriate
numerical
procedure
[23]. In contrast to [22] and [23], in
this paper all three techniques are based on a six-component
representation
the inclusion
The
rigorous
ridge waveguide
an efficient
of the electromagnetic
of TM-
to- TE-mode
modal
scattering
discontinuities
computer-aided
millimeter-wave
coupling
printed
matrix
field
description
is then applied
design
circuit
method
and on
effects.
of
to provide
for a type of
evanescent-mode
wave-
guide band-pass filter (Fig. 1). Based on bilateral all-metal
E-plane fins, the design combines the well-known
properties of the waveguide
E-plane
integrated-circuit
design of the evanescent-mode
filter (Fig. 1), the modal S-ma[20],
into
double
[24] is applied.
two key building
step
above-cutoff
filter
that
method,
the
waveguide
to
junction
the
from
waveguide
and the step discontinuity
the
to the E-plane
corresponding
from waveguide
Components
from
finned
inverse
The filter
block disthe
the rectan-
waveguide.
structure
below cutoff
input
below-cutoff
Note
(e.g.,
to waveguide
the
above
cutoff),
the related
modal
scattering
matrix
is simply
derived by interchanging
the corresponding
submatrix elements of the original
structure.
Combination
with the
known scattering matrices of the corresponding
intermediate homogeneous
waveguide
sections of finite lengths
yields the total
scattering
Since a rigorous
ities involved
matrix
(Fig. 1) requires
as the TM-to-TE
electromagnetic
of the filter.
field description
coupling
field
of the step discontinu-
all field
effects
components
as well
to be considered,
in the subregions
the
i = O, I, II (Fig.
l(c)),
technol-
ogy [7], [11], [12] with the advantages of the evanescentmode concept [15]–[18] and the low-insertion-loss
quality
resulting
from the complete
absence of supporting
dielectrics [24]. The proposed structure, where the finite fin
thickness is included
in the design, achieves relatively
large gap widths (about one quarter
of the waveguide
height in the below-cutoff
section) and leads to filter
characteristics
with
high
skirt
selectivity
and wide
stop-
is derived
from
the
z components
of two vector
poten-
IEEE TRANSACTIONS ON MICROWAVE THEORY
1106
___
of these three
tials,
AVHZ=
~
(JZ&)
“~&(
X, J’)
(JY&)
. T;,(
~> Y)
with
exp(-~k:~Pz)-
standard
field
38,
NO.
theory
AUGUST
8,
1990
methods
is
1) Transverse Resonance Method (TM):
13y considering
propagation
to take place in the x direction,
an inhomogeneous
waveguide
transmission
“[V&
rigorous
VOL.
tively.
“ [ V;q exp ( – jk;H~ z)+’R~qexp(+jk,~.~)]
~
p=l
TECHNIQUES,
particularly
indicated.
The functions
T& and T~~ may
conveniently
be simplified
utilizing
the magnetic wall and
electric wall symmetries at x = a/2 and y = b/2, respec-
q=l
An~z =
AND
RLP exp ( + ~k,~,z)]
(2)
the wave impedances
Z~q = (~pO)/’(k;HQ)
= l/Y~,
Y~P = ((qJ/’(k;~P)
= l/Z~P.
(3)
VA, E and R~, ~ are the TE- and TM-mode
tudes of the forward
and backward waves,
wave amplirespectively,
line
cross section
subdivided
into
can be regarded
homogeneous
as a
subre-
gions [13], [19]–[21]. A transverse transmission-line
matrix
relates the field amplitudes
at the lower and the upper
bounda~
of each subregion.
Finally,
the line resonator
resonance condition
is satisfied by the thus far neglected
bounda~
condition
at the metallic
sidewalls.
For the
ridge or finned waveguide cross section under consideration (Fig. 2(a)), the transverse
boundary
conditions
resonance
to be applied
method
with the
in the x direction
(in
which have to be related to each other at the corresponding discontinuity.
This will
yield
the corresponding
scattering
matrix relations.
The propagation
factors are
our case electric wall at x = O, magnetic wall at x = a/2)
provides the corresponding
homogeneous
system of equa-
denoted
[19]-[21].
by k=, and T~~, T~P are the cross-section
functions
of the corresponding
der consideration.
waveguide
For the rectangular
eigen-
structures
waveguide
un-
sections
(regions i = O,I), the relations of [251 are given for completeness, using the present notation:
1) Input-output
waveguide (region O):
tions,
requiring
the system
The cross-section
determinant
eigenfunctions
of the x- and y-dependent
IIb:
functions
to be zero [13],
are expressed
in terms
of subregions
IIa and
‘iq(xy)=A’c0s[(2m-l)ixl
L
(CC’S*’),(-)
‘:Jx7y)=D~sin[(2m-1)51
”sin%y
‘4)
2) Waveguide
T~q(x,
below
cutoff
(region
I):
1
(C++(A)
1
y) = A;cos
(2rn –l)lX
[
T&(x,
y) = D;sin
(2rn –l)ZX
[
sin
a
2nw
—
b ‘“
‘5)
For the E-plane
finned or ridged waveguide
eigenvalue
problem (region II), the transverse resonance, the standand the resonator
method are used and coming wave,
pared with
each other.
B. E-Plane Finned
Problem
or Ridged Waveguide Eigenvalue
Three different
mode-matching
methods
are investigated to solve rigorously
the cross-sectional
bounda~
value problem of the finned or ridged waveguide (Fig. 2).
As the formulations
directly influence
the solution of the
related discontinuity
problem with regard to complexity,
the computational
effort, and the numerical
accuracy in
relation to the expansion terms considered,
a comparison
(6)
where
superscripts
corresponding
terms
IIa
and IIb
denote,
in the finned
respectively,
waveguide
the
sub–cross
sections (Fig. 2(a)), and the kX’s are calculated by solving
the related cross-section eigenvalue problem [23].
BORNEMANN
AND
ARNDX
TRANSVERSE
RESONANCE,
STANDING
WAVE,
AND
RESONATOR
FoRMur-ATloNs
with M < N,
given by
mag wall
1107
then
the
cross-section
eigenfunctions
are
I
-—- —. ‘
f
decwau
Ea
--L —--+
1
IIb
1
~
‘-c
‘ ‘Y
2
,.[cos[~b-c))pw
●z
ex
(a)
w
ek.
wall
-—.—.
(++(4-=)]
I
—-
——
t
——
IIa
!-
c
~
—-
2
IIc
———— ——.
;/
a
e
i ~Y
49
X2
“sin
(b)
Fig.
2.
Finned
or ridged
cross section.
(a) Subregions
because
with multiple
account
(TRM)
and the standing
wave
for the resonator
method (RM).
an arbitrary
additional
system
transmission-line
insert
subregions.
verse
resonance
number
of subregions
inserts [20]) maybe
in the matrix
simply
matrices
On the other
technique
[13],
(e.g.
easily taken into
by multiplying
the
of the corresponding
hand,
the
due to the transnumber
N
of the
expansion terms must be the same for both subregions.
For very small slot widths of about w/b<
0.1 (cf. Fig.
l(b)), therefore,
the related
limited
number
of expansion terms in this case may lead to a poor convergence
behavior.
However,
nance method
the main
advantage
described
given number
of the transverse
reso-
is the fact that there are no poles
in the determinant
function
matrix equation. Therefore
of the resulting characteristic
the search algorithm
to find a
of subsequent
(Y-c)
for
+
eigenvalues
(i.e. the propaga-
tion constants for a given frequency, or, as in the present
case, the cutoff frequencies)
can be simplified
and the
risk of omitting eigenvalues is very low, in contrast to both
other methods under consideration.
2) Standing Wave Formulation
(SWF): For this formula-
N
~ D;:b(l/
~=1
k;&H
)’
“Sin(k=)sin(+y)o
The transverse resonance procedure reduces the size of
the characteristic
matrix equation
to a quarter
of the
it makes the method
very
original
size. Furthermore,
finlines
[)
mrr
—
b
-//
?–c
waveguide
the transverse
resonance
method
formulation
(SWF). (b) Subregions
flexible
\
\
~=1
ltb
This expression
is essentially
equivalent
to that given in
(6), but with the notable exception that the numbers M
and N of the expansion
terms may be independently
chosen
in each
subregion.
This
advantage,
undermined
by the fact that, particularly
modes,
determinant
matrix
the
equation
yields
function
zeros with
of
however,
for higher
the
is
order
characteristic
poles in the immediate
vicinity. As a consequence,
certain eigenvalues of higher
order modes, which can have a significant
influence
on
the three-dimensional
field distribution
small gap widths), may not be detected,
modal
scattering
matrix
parameters
(especially
for
and the resulting
are no longer
fully
reliable.
3) Resonator
onator method
divided
into
ikfethod (RM): In order to apply the res[23], the cross section (Fig. 2(b)) has to be
three
resonator region:
force the electric
subregions,
one of which
is called
the
IIc. The basic idea of this approach is to
field to be zero at .x= e and Y = (O, c) in
contrast to the two preceding methods, where this condition is met indirectly by adding up a number of expansion
are matched directly at the common interface at x = e (cf.
Fig. 2(a)), which results in tSVOhomogeneous
matrix equa-
terms, Therefore
the propagation
is considered
to take
place in the y direction
for region IIb. By applying the
superposition
principle
and matching the fields of subregions IIb to IIc at Y = c, and those of IIa to IIc at x = e,
tions [221. Assuming
spond to subregions
the resonator functions (IIc) can be expressed
the subregions IIa and IIb, respectively [23].
tion,
the tangential
field
components
EY, HZ or Ez, ~Y
that the numbers M and N correIIa and IIb (Fig. 2(a)), respectively,
in terms of
IEEE TRANSACTIONS
1108
As a result,
U:(X,
the cross-section
eigenfunctions
ON MICROWAVE
common
read
of the
guide:
Y)
THEORY
interface
related
AND
TECHNIQUES,
yields
the modal
discontinuity
38,
VOL.
NO.
8, AUGUST
1990
matrix
(S)
scattering
waveguide
to finned
wave-
( 10)
where
T;;(x,
~
D~~FJ&.
sin
/71=1
insert
filters
and finned
puter-aided
the expressions
FX and
FY are elucidated
in the
Appendix.
The
more
complicated
structure
of
these
functions,
compared
with (6) and (7), is caused by the additional
plane of subregion
interaction
due to the division into
three subsections (Fig. 2(b)). At first glance, the resonator
method seems to be a very attractive technique for solving
this eigenvalue
the independent
in subregions
ary
problem,
as it combines the advantage of
choice of the number of expansion terms
Ha and Hb with the feature
condition
satisfied
does
at
the
plane
depend
considered.
determinant
on
the
However,
function
that the bound-
of
the situation
of the related
vicinity
applying
in [13].
plane
step
function
is minimized
with
III.
expansion
terms
waveguide
a very
pro-
An
error
to a parameter
vector
lengths
as well
RESULTS
ISZII
filter
where the gap width w/b = 0.81 is relatively high. A WR
90 (22.86 mm x 10.16 mm) waveguide
housing is chosen
utilizes
require
respect
method.
Fig. 3 shows the calculated transmission
coefficient
in dB as a function of frequency of a quasi-high-pass
and output
of the zeros to be detected
strategy
which contains the coupling and resonator
as the slot widths (Fig. l(c)).
for the input
associated with the
characteristic
matrix
[13], [24], the com-
out by an optimization
the evolution
guide
is exactly
equation has turned out to be still more severe than that
of the standing wave formulation:
the poles in the immediate
described
of the double
transformers
is carried
which
resonator
number
gram
design
function,
x = e, Y = (O, c)
by the corresponding
not
matrix
single modal scattering
matrices, and the corresponding
relations
are given in [13], [24], and [25]. As with metal
N
where
are already
scattering
discontinuity
of the rectangular
waveguides
of different
cross section, regions O to I (Fig. l(c)), is already derived
in [25]. The series of step discontinuities
for a complete
filter structure is calculated by direct combination
of the
y)
=
the submatrices
The modal
\
n=O
section
housing.
waveguide,
a WR
and the ridge wave-
42 (10.668
mm X 4.318 mm)
In Fig. 3(a), the transverse
resonance
method (TRM)
is used to solve the related ridge waveguide eigenvalue
problem
and parts (b) and (c) of the
figure show the solutions with the standing wave formulation (SWF) and the resonator method (RM), respectively.
Seven
TM
modes
and
12 TE
modes
in
(2)
and
(3),
time consuming
search algorithm.
Despite
this effort,
eigenvalues
may be overlooked
which would otherwise
together with M = N = 9 expansion terms in (4)–(8), have
been used. Good agreement
between the results of the
have been found
three
relatively
easily with the transverse
reso-
different
In order
nance method.
mulations
C. Modal
Scattering Math
In order to calculate the modal
key building block discontinuities
scattering matrix of the
of the filter component
(Fig. 1), the coefficients
A:, D;, Al, and D; in (4) and
(5) and the amplitude
vectors of the expansion functions
A~a,
A;Ib,
D 11,
~ , and D~lb in (6)–(S) are normalized
so that
the
power
F’~
==
carried
J(
F’
.
{
~~
by
X G’
a given
)
~z
mode
is
dF = @~Y;
J(
grad T;)z
propagating
j W,
evanescent
modes
evanescent
TM modes.
–jW,
of the
ridge
techniques
different
waveguide
is given.
cross-section
against
for-
experimental
data, Fig. 4(a) shows the calculated
and measured input
reflection
coefficient
IS1~I in dB as a function of frequency
of a very simple, nonoptimized
filter structure with four
bilateral
E-plane fins in a single WR-62 waveguide (15.799
mm X 7.899 mm) housing (cf. Fig. 4(b)). The thickness of
the fins is t = 0.9 mm, and the gap width w/b = 0.39 is
about half that of the preceding case (Fig. 3), Due to the
smaller
gap width,
the
techniques applied—TRM
dF
F’
1 w,
mode-matching
to check the three
three different
mode-matching
(solid line), SWF (dashed line),
RM (dash–dotted
Iine)—lead
to different
solutions. This
is particularly
true for the RM. The numbers of modes
modes
(9)
In this equation, the subscript M replaces the indices Hq
and Ep for TE and TM modes, respectively.
Matching
the tangential
field components
of regions I and II at the
and expansion terms are equivalent
to those chosen for
calculating
the results in Fig. 3. The mechanical
tolerances of the structure
(which have been measured by a
measuring
microscope)
are taken into account in the
analysis; for the slightly oblique gaps, a mean value for
the corresponding
gap width
has been assumed. Very
BORNEMANN
AND
ARNDX
TRANSVERSE
RESONANCE,
STANDING
WAVE,
AND
RESONATOR
757
60.
?-‘~,
,@ ‘-~/__ _
4$-
T
J/
;
=
I
I
I
\
~
\.
\
1%11
i,,’
dB
1
I
-1_
Tful
1109
FORMULATIONS
cue
\
45.
“\
; = 0.81
0.81
30.
\
i.
\
5.
I
ls
\
i
\
~
0
0
24
20
f/GHz
(a)
#-
o
16
12
12
8
I
I
\
‘%
%
~
/-,\J/--f/GHz
‘
24
20
~
(b)
0
12
20
16
f/GHz
24
~
(c)
Fig. 3. Quasi-high-pass
filter. WR 90(22.86
mmx10.16
mm) input/output
waveguide,
WR42(10.668
mm X4.318 mm)
ridge waveguide
section.
Filter
dimensions
(cf. Fig. 1): f=l
mm, lCI=lC=0.236
mm, 1~1=1~3=1.001
mm,
lc2=lc3=11.429
mm, l~2=l.58mm.
(a) Transverse
resonance method (TRM)solution
of theeigenvalue
problem.(b)
Standing
wave formulation
(SWF) solution
of the eigenvalue
problem.
(c) Resonator
method
(RM) solution
of the
eigenvalue problem.
good
agreement
between
theory
been achieved for theTRM.
For still smaller gap widths,
ence between
the results
and measurements
w/b=
obtained
has
0,116, the differ-
by applying
the TRM
(solid line) and, particularly,
the RM (dash-dotted
line) is
increased further.
Fig. 5 shows the calculated
transmisX-band
sion
coefficient
lSzl I for
an
optimized
chosen were N = 5 for the TRM and M = 4 and N = 9
for the SWF and the RM respectively.
In order to search
the cutoff frequencies
for the related ridge Waveguide,
cross-section
eigenvalue
problem,
the frequency
step’
widths in the numerical
search algorithm
for the SWF
and the RM have been set to 1% of the corresponding
step width
utilized
in the search algorithm
evanescent-mode
E-plane finned waveguide band-pass filter design example with passband at about 12 GHz. As in
In spite of the large increase
a factor of 4 for a complete
Fig. 3, a ,WR 90 waveguide
housing is chosen
input/output
waveguide
and a WR 42 housing
filter
utilized
section.
Seven TE
in (2) and (3}
modes
and
the numbers
12 TM
for
for
the
the
tion),
modes
are
overlooked.
of expansion
terms
important
modes in the eigenvahte
lated by the SWF, and particularly
For
small
SWF and particularly
gap widths,
the RM
for the TRM.
in CPU time (approximately
frequency response calculasolution
by the RM,
therefore,
are considered
formu-
have been
both
the
to be inap-
IEEE
1110
TRANSACTIONS
ON MICROWAVE
THEORY
AND
30
TECHNIQUES,
38,
VOL.
NO.
8,
AUGUST
1990
,
1 ‘s’’’-m;
: :::red
25
1$1
dB
20
60
15
I
Lo
10
20
5
0
L8
0
.+~+
\
‘
/q”M
!’
2il
*
10
I/
12
16
lb
f/GHz ~
//
““L. ./”’
Fig. 5. Computer-optimized
X-band evanescent-mode
E-plane finned
waveguide band-pass filter. WR 90 (22.86 mm X 10.16 mm) waveguide
input and output
housing
WR 42, (10.668 mm X 4.318 mm) belowcutoff waveguide.
Fin thickness
r = 1 mm; gap width w = 0.5 mm.
Design with five E-plane
fins. lCI = 1C6 = 0.2 mm; lR1 = 1R5 = 0.975
16
IL
f/GHz ~
(a)
mm, 1C2 = 1C5= 10.944 mm, l~z = 1~1 = 1.541 mm, 1C3 = 1C4 = 12.120
mm, l~j = 1.491 mm; 12 TE, 7 TM modes. Expansion
terms: N = 5
(TRM),
N =9, M = 4 (SWF, RM). Transmission
coefficient
applying
the transverse
formulation
dash–dotted
resonance
method
(TRM,
solid line),
(SWF, dashed line), and the
line) for the related eigenvahre
the standing
resonator
solution.
method
wave
(RM,
125
100
/
11/’s*,l
dB
75
(b)
Fig. 4. Nonoptimized
simple filter structure,
to verify the theory. Four
bilateral
E-plane fins in a single WR 62 waveguide housing; measured
dimensions:
15.85 mm x 7.925 mm, fin thickness
t= 0.9 mm, gap
width
w = 3.1 mm. Filter dimensions
(cf. Fig. 1): [Cz = Icq = 1.025 mm,
1~1 = l~i = 2.6 mm, 1C3 = 10.5 mm, l~z = 1R3 = 8.95 mm. (a) Calculated
and measured ( + ) input reflection
coefficient
I,S’ll I in decibels versus
frequency.
Transverse
resonance
wave formulation
(SWF, dashed
dash–dotted
line). (b) Photograph
method (TRM,
solid line), standing
line), and resonator
method (RM,
of the realized structure.
50
25.
08
+x-band
prQpriate
problem.
The
is carried
An
for
out
optimized
waveguide
adequately
final
by
solving
the
related
computer-optimized
utilizing
X-band
band-pass
the
design
design,
example
E-plane
with
14
~
therefore,
finned
passband
16
h-band
eigenvalue
TRM.
evanescent-mode
filter
li I
10
f/
GHz
-I
~
Fig. 6. Computer-optimized
X-band evanescent-mode
E-plane finned
waveguide band-pass filter. WR 90 (22.86 mm X 10.16 mm) waveguide
input and output
housing; WR 42 (10.668 mm X 4.318 mm) belowcutoff waveguide.
Fin thickness t= 1 mm; gap width w = 0.975 mm.
BORNEMANN
AND
ARNDK
TRANSVERSE
RESONANCE,
STANDING
WAVE,
AND
RESONATOR
FORMULATIONS
1111
60-
w=146pm
. . ...
\
11/s21t
;,,.t--
\
40
f’-:\
-
K-4?.-1’
- i-
~<
\
dB
/0
\
\
w= 148 pm
/
\
Zd- “
\
T
-.
/
/
o
1
100
80
70
60
Endof
E-bad+W
f./GHz
Fig. 7.
Computer-optimized
E-band
evanescent-mode
E-plane
finned
waveguide
mmxl.549
mm) input and output waveguide;
WR 7 below-cutoff
waveguide which
b =0.668 mm). Fin thickness t = 150 ~m; gap width w = 0.146 ~m.
at about
11 GHz is shown in Fig. 6. The design with three
E-plane
fins of thickness
t= 1 mm, a WR
90 input
and
sions of the filter
(measured
than
for
above
13.5 GHz
A comparison
sponding
E-plane
[24] with
three
significant
and
metal-insert
resonators
improvement
within
the
with the characteristic
coupled
(dashed
line)
in rejection
adjacent
resonator
results
filter
demonstrates
the
input
waveguide
the
and
reduced
below-cutoff
E-plane
output
waveguide
in height
filter
with
a WR
(1.651 mm x 0.668 mm)
section.
fins of thickness
and
This
design,
with
microscope)
This holds also
has significantly
higher
As a consequence,
the validity
off the proposed
realization
these
design
of these filters.
The
of Figs. 7 and 8(a) not only illustrate
the necessity for a reliable
design theory taking into
account all relevant parameters but also show the requirement for high-precision
fabrication
facilities.
IV.
The
for
t = 150 ~m, achieves a minimum
which
tolerances.
in the practical
characteristics
7
three
(3.098
mm,
by a measuring
in the calculations.
10SS curve
to design
demonstrate
technique
filter
given by the E-plane
printed-circuit
evanescent-mode
filter technique.
Fig. 7 shows the results of a computer-optimized
Eband evanescent-mode
E-plane
finned waveguide bandpass filter design example with a WR 12 (3.098 mm X 1.549
mm)
the return
sensitivity
of a corre-
—~
0.2 dB. Good agreement between measurements
and theory is given (cf. Fig. 8(a)) if the actual geometrical
dimenare taken into account
50 dB
1
120
band-pass
filter.
WR
12
is reduced in height (a=l.651
output waveguide,
and a WR 42 below-cutoff
waveguide
achieves an attenuation
in the second stopband of more
Ku-band.
I
110
an
modal
exact
S-matrix
CONCLUSION
method
computer-aided
low-insertion-loss
presented
design
evanescent-mode
of
here
achieves
compact,
low-cost,
bilateral
E-plane
stopband attenuation
of more than 40 dB up to about 110
GHz (the end of the next higher frequency
band, the
W-band). The sensitivity of the performance
of the opti-
finned waveguide band-pass filters with wide stopbands.
The theory, which includes the finite thickness of the fins’
as well as higher order mode interactions
at all disconti-
mized E-band filter to dimensional
tolerances is demonstrated by the dashed curve, which shows the insertion
loss for a fin gap width increased by only 2 ~m.
Fig. 8 presents a comparison
between theoretical
and
nuities, leads to a reliable prediction
of the filter characteristic and allows the stopband characteristic
of the filter
to be taken into account in the optimization
process.
Moreover,
this design leads to relatively wide gap widths,
measured data for the structure currently
tion.
An
easily
fabricated
Ku-band
which may help to meet high power handling
requirements. The numerical
advantage of the transverse reso-
evanescent-mode
E-plane
8(b)) with a fin thickness
finned
under investigathree-resonator
waveguide
filter
(Fig.
of t= 1 mm has been realized
by
a simple milling technique.
A WR 90 waveguide housing
has been chosen for the input/output
waveguide, and the
filter section is a WR 42 waveguide
below cutoff. The
measured insertion
loss over the passband is only about
nance
method
for
solving
the
related
cross-sectional
eigenvalue
problem
of the ridge waveguide
section is
demonstrated
by comparison
with solutions obtained
by
the more classical standing
wave or resonator
modematching
techniques.
The theory shows excellent agreement with the measurements
presented.
IEEE
TRANSACTIONS
ON MICROWAVE
THEORY
AND
TECHNIQUES,
VOL. 38, NO. 8, AUGUST 1990
Mt
b
5 i}
60
(A2)
\
+‘+
measure{
j
Lo
(A3)
1
~,
*.
‘yEPn
20
0
(A4)
1
8
10
12
14
f/GHz
16
REFERENCES
—--+
[1] S.
B. Cohn, “Optimum
formers,”
IRE Trans.
16-21, Apr. 1955.
(a)
[2]
[3]
[4]
[5]
[6]
(b)
Fig. 8. Ku-band
waveguide filter
three-resonator
evanescent-mode
E-plane
finned
with a fin thickness of f = 1 mm realized by a simple
filing technique.
WR 90 input\ output waveguide,
WR 42 waveguide
below-cutoff
filter section. Actual
dimensions:
al X bl = 22.85 mm X
10.15 mm; a x b = 10.66 mm X4.29 mm; f = 1 mm, lCI = 0.25 mm,
1C2= 1C3= 11.45 mm, 1C4= 0.24 mm, 1~1 = 0.99 mm, l~z = 1.6 mm,
l~q = 1.01 mm; WI = W’z= W3 = 0.98 mm. Number of expansion
terms
used N = 9 (cf. eq. (6)); modal scattering matrix calculated with 20 TE
) and measured
(+) filter
and 12 TM modes. (a) Calculated
(—
response. (b) Photograph
of the prototype
under investigation.
[7]
[8]
[9]
[10]
APPENDIX
ABBREVIATIONS
[11]
FOR THE CROSS-SECTION
EIGENFUNCTION
[12]
[13]
[14]
[15]
transmission-line
transTech., vol. MTT-3,
pp.
S. Hopfer,
“The
design of ridged waveguide,”
IRE Trans. iVficrowaue Theoy Tech., vol. MTT-3, pp. 20–29, Oct. 1955.
E. S. Hensperger,
“Broad-band
stepped transformers
from rectangular to double-ridged
waveguide,”
IRE Trans. Microwaue
Theory
Tech., vol. MTT-6, pp. 311-314, July 1955.
J. P. Montgomery,
“On the complete eigenvalue solution of ridged
waveguide,”
IEEE
Trans. Microwave
Theory Tech., vol. MTT-19,
pp. 547-555,
June 1971.
W. J. R. Hoefer and M. N. Burtin,
“Closed-form
expressions for
the parameters
of finned and ridged waveguides,”
IEEE
Trans.
Microwaue
Theory Tech., vol. MTT-30,
pp. 2190–2194,
Dec. 1982,
Y. Utsumi,
“Variational
analysis of ridged waveguide
modes,”
IEEE Trans. Microwaue
Theory Tech., vol. MTT-33,
pp. 111-120,
Feb. 1985.
Y. Konishi
and H. Matsumura,
“Short
end effect of ridge guide
with planar circuit
mounted
in a waveguide, ” IEEE
Trans. Microwaue Theoty Tech., vol. MTT-27,
pp. 168– 170, Feb. 1979.
D. Mirshekar-Syahkal
and J. B. Davies, “Accurate
analysis of
tapered
planar transmission
lines for microwave
integrated
circuits,”
IEEE
Trans. Microwaue
Theory Tech., vol. MTT-29,
pp.
123-128, Feb. 1981.
L.-P. Schmidt and H. Meinel,
“Broadband
millimeter-wave
PINdiode attenuator
with double-ridged
waveguide flanges,”
Electron.
Lett., vol. 18, no. 19, pp. 839–840, Sept. 1982.
P. J. Meier, “Integrated
finline:
The second decade,”
Microwaue
J., vol. 28, no. 11, pp. 31-54, Nov. 1985; also no. 12, pp. 30-48,
Dec. 1985.
A. M. K. Saad, “Novel
Iowpass harmonic
filters
for satellite
application,”
pp. 292-294.
IN (8)
design of stepped
Microwave
Theory
in IEEE
MTT-S
Int.
Microwaue
,$ymp. Dig.,
1984,
Q. Zhang and T. Itoh, “Computer-aided
design of evanescentmode waveguide
filter with non-touching
E-plane
fins,”
IEEE
Trans. Microwace
Theory Tech., vol. 36, pp. 404-412,
Feb. 1988.
J. Bornemann
and F, Arndt,
“Modal-S-matrix
design of optimum
stepped ridged and finned waveguide
transformers,
” IEEE Trans.
Microwave
Theory Tech., vol. MTT-35,
pp. 561-567,
June 1987.
R. R. Mansour,
R. S. K. Tong, and R. H. McPhie,
“Simplified
description
of the field distribution
in finlines
and ridge waveguides and its application
to the analysis of E-plane
discontinuities,” IEEE Trans. Microwaue Theo~ Tech., vol. 36, pp. 1825–1832,
Dec. 1988.
H. F. Chappell,
“Waveguide
low pass filter using evanescent mode
inductors, ” Microwave J., vol. 21, pp. 71-72, Dec. 1978.
BORNEMANN
AND
ARNDT:
[19]
[20]
[21]
[22]
[23]
[24]
[25]
“The
Trans. Microwave
design
of
evanescent
mode
Theory Tech., vol. MTT-25,
pp. 1013-1021,
Dec. 1977.
R. V. Snyder, “Broadband
waveguide
or coaxial filters with wide
stopbands, using a stepped-wall
evanescent mode approach,”
Microwaue.T., vol. 26, pp. 53–58, Dec. 1983.
F. Arndt and G. U. Paul, “The reflection
definition
of the characteristic impedance
of microstrips,”
IEEE Tram. Microwaue Theory
Tech., vol. MTT-27,
pp.724-731,
Aug. 1979.
R. Vahldieck
and J. Bornemann,
“A modified
mode-matching
technique
and its application
to a class of quasi-planar
transmission-lines,”
IEEE
Trans, Microwaue
Theory Tech., vol. MTT-33,
pp.916-926,0ct.
1985.
R. Sorrentino
and T. Itoh,
“Transverse
resonance
analysis of
fin-line discontinuities,’’
ZEEE Trans. Microwave
Theory Tech., vol,
MTT-32,
pp. 1633 -1638, Dec. 1984.
A. Wexler, “Solution
of waveguide discontinuities
by modal analysis, ” IEEE
Trans. Microwaue
Theory Tech., vol. MTT-15,
pp.
508-517,
Sept. 1967.
E. Kiihn, “A mode-matching
method for solving field problems in
waveguide
and resonator
circuits,”
Arch. Elek. Ubertragung.,
vol.
27, pp.511-518,
Dec. 1973.
R. Vahldieck,
J. Bornemann,
F. Arndt, and D. Grauerholz,
“ Optimized waveguide
E-plane metal insert filters for millimeter-wave
applications,”
IEEE Trans. Microwave
Theory Tech., vol. MTT-31,
pp.65-69,
Jan. 1983.
H. Patzelt
and F. Arndt,
“Double-plane
steps in rectangular
waveguides
and their
application
for transformers,
irises, and
filters,”
IEEE
Trans. Microwave
Theo~ Tech., vol. MTT-30,
pp.
771-776, May 1982.
(M’87)was
born in Hamburg,
West Germanv, on Mav
Jens Bornemann
26, 1952. He received
the Dipl.-Ing.
and ~r.-Ing.
degrees:’ both ii
electrical
engineering,
from the University
of Bremen, West Germany,
in 1980 and 1984, respectively.
1113
AND RESONATOR FORMULATIONS
[17]
IEEE
Mok,
WAVE,
G.
[18]
C. K.
STANDING
waveguide bancl-pass filters for a prescrib~d
insertion
loss characteristic,”
IEEE Trans. Microwave
Theory Tech., vol. MTT-19,
pp.
295-308,
1971.
R. V. Snyder, “New application
of evanescent mode waveguide to
design,”
and
RESONANCE,
[16]
filter
Graven
TRANSVERSE
From 1980 to 1983, he was a Research and
Teaching
Assistant
in the Microwave
Department at the University
of Bremen, working
on
quasi-planar
waveguide configurations
and computer-aided
E-plane
filter design. After a twoyear period as a consultin
gengineer, he joined
the University
of L3remen
again,
in 1985, as an
Assistant
Professor.
Since April
1988, he has
been an Associate Professor at the University
of
Victoria,
Victoria,
B. C., Canada.
His current
research activities include microwave system design, active components,
and problems
of electromagnetic
field theory.
Dr. Bornemann
was one of the recipients
of the A. F. Bulgin Premiumof
the Institution
of Electronic
and Radio Engineers
in 1983. He
Hi
Fritz Arndt(SM’83)
received the Dipl. Ing., Dr.
Ing., and Habilitation
degrees from the Technical University
of Darmstadt,
Germany,
in 1963,
1968, and 1972, respectively.
From 1963to
1973, heworked
on directional
couplers and microstrip
techniques
at the Technical University
of Darmstadt.
Since 1972, he
has been a Professor
and Head of the Microwave Department
of the University
of t3remen, Germany. His research activities center on
field problems of waveguide,
finline, and optical
waveguide
structures,
antenna design, and scattering structures.
Dr. Arndt is a member of the VDE and NTG (Germany).
He received
the NTG award in 1970, the A. F. Bulgin Award (together
with three
coauthors)
from the Institution
of Radio and Electronic
Engineers
in
1983, and the best paper award of the ar(tenna conference
JINA 1986
(France).