Rigorous Mode-Matching Method of Circular to Off-Center Rectangular Side-Coupled Waveguide Junctions for Filter Applications

Abstract

An accurate and efficient method for analyzing circular to multiple off-center rectangular side-coupled waveguide T-junctions is developed based on a rigorous mode-matching technique. The method is very general and not limited by symmetry or other dimensional constraints to the junction. A new way to match fields on the curved interface between subregions is described in detail. The computer code developed based on this theory is highly efficient. The numerical results match with the results obtained by other methods including experimental ones. The method is used to design a side-coupled circular waveguide dual-mode filter.

Full text

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 20072365
Rigorous Mode-Matching Method of Circular to
Off-Center Rectangular Side-Coupled Waveguide
Junctions for Filter Applications
Jingliang Zheng and Ming Yu, Senior Member, IEEE
Abstract—An accurate and efficient method for analyzing cir-
cular to multiple off-center rectangular side-coupled waveguide
T-junctions is developed based on a rigorous mode-matching tech-
nique. The method is very general and not limited by symmetry or
other dimensional constraints to the junction. A new way to match
fields on the curved interface between subregions is described in
detail. The computer code developed based on this theory is highly
efficient. The numerical results match with the results obtained by
other methods including experimental ones. The method is used to
design a side-coupled circular waveguide dual-mode filter.
Index Terms—Circular waveguide T-junction, mode-matching
method.
I. INTRODUCTION
T
satellite splitting and combing networks because of its compact
size and excellent performance. Traditionally, this type of filter
is coupled at the ends of the circular cavities. In some cases,
this may not be possible since the ends may be inaccessible,
or may contain an adjustment mechanism for temperature
compensation or frequency tuning. In the new generation of
dual-mode waveguide filters introduced by Yu et al. [1], the
input/output coupling and/or coupling between the circular
cavities are realized at the sides of the circular cavities. In these
filters, there are rectangular irises and/or rectangular waveguide
sections on the sidewalls of the circular cavities (Fig. 1). To
design such filters more efficiently, an accurate and efficient
method to simulate circular to rectangular side-coupled wave-
guide T-junctions, as shown in Figs. 2 and 3, is desired.
The mode-matching technique [2]–[7] is a powerful electro-
magnetic (EM) modal analyzing method and is widely used for
simulating different waveguide circuits and solving other EM
problems.Forproblemswherethediscontinuityinterfaceispar-
allel to the coordinate planes at both sides of the discontinuity,
mode-matching formulas are straightforward to derive and the
method is extremely efficient and convenient. However, if the
discontinuity interface is not parallelto one side’s or both sides’
coordinate plane, the matching between the fields of both sides
becomes much more complicated. In order to obtain an efficient
HE DUAL-MODE circular waveguide filter using the
TE11n mode is one of the most important filter types for
Manuscript received May 16,2007; revised August 28, 2007.
The authors are with Com Dev Ltd., Cambridge, ON, Canada N1R 7H6
(e-mail: jingliang.zheng@comdev.ca; ming.yu@ieee.org).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMTT.2007.908662
Fig. 1. Side-coupled dual-mode filter.
Fig. 2. Side-coupled circular to single rectangular waveguide T-junction.
solution, the authors developed different ways to choose a con-
venient surface or region for matching the fields.
Melloni et al. [8] analyzed a waveguide bandpass filter using
the mode-matching technique. The main building blocks of the
filter are the cylindrical resonators coupled by rectangular irises
from the sidewalls. When matching the fields in the junction,
the curvature effect of the cylinder is neglected. Therefore, the
accuracy of this method will be questionable if the size of the
rectangular waveguide is large.
0018-9480/$25.00 © 2007 IEEE

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2366IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007
Fig. 3. Subregions and the geometry of side-coupled circular to multiple rect-
angular waveguide T-junction.
Krauss and Arndt [9] divided the whole T-junction structure
into six subregions, as shown in Fig. 2(a): two circular wave-
guide regions at two circular ports (outside of surfaces
), one rectangular waveguide region at the rectangular port
(outside of surface
), one circular cavity region between sur-
faces
and, and two additional regions, which are the “in-
homogeneous region” between surfaces
finitely thin artificial intermediate region on surface
“inhomogeneous region” contains the half-moon-shaped area.
The field in this region is calculated by the boundary-contour
mode-matching method [10]. The existence of the “inhomoge-
neous region” requires a minimum length of rectangular wave-
guide such that the rectangular port is not too close to the junc-
tion.
GentiliandMelloni[11]dividedthejunctionpartofthestruc-
ture into subregions using a similar approach to Krauss and
Arndt [9]. The generalized admittance matrix (GAM) has been
used and the infinitely thin artificial intermediate region on sur-
face
is no longer needed. Although minimal detailed deriva-
tion was given, a different method from [9] for the analysis of
thetransitionregionhasbeenemployed.Whiletherearenolim-
itations on the physical dimensions, the transition subregion be-
tween surfaces
andis still needed.
Rong and Zaki [12] solved a more complicated, but symmet-
rical problem, where two circular cylinders with circular dielec-
tric bodies at their centers are connected by a rectangular wave-
guide. The cross-sectional geometry of the structure is symmet-
ricalinboththe -and -directions[seeFig.2(b)].Atthecurved
connecting interface between circular and rectangular parts, no
additional region was used and the fields were matched directly
ontheinterface.Usingthesymmetrycondition,theauthorssim-
plified the problem by putting a perfect electric/magnetic con-
ducting wall on both symmetry planes.
Wuetal.[13]solveda1-Dsymmetricalproblem,asshownin
Fig. 2(c). No additional region was used at the curved interface,
and the rectangular region was extended into the cavity region.
Instead of using the interfaces
artificial interface
. In order to avoid the numerical integra-
tions where the integrand includes Bessel functions, the finite
plane-waveseries-expansion technique [14] wasused toexpand
the modes in the cavity region. Since the boundary conditions
on the half-moon-shaped area between the circular waveguide
region and rectangular waveguide region were ignored, the ac-
curacy becomes poor if the rectangular waveguide size “b” is
large.
and
and, and an in-
. The
, the fields are matched on the
Zheng and Yu [16] uses the mode-matching method to ana-
lyze a side-coupled circular waveguide to multiple rectangular
waveguide T-junctions without any symmetry, as shown in
Fig. 3. The rectangular waveguides may have different dimen-
sions and not necessarily be centered on the circular cavity.
Since the longitudinal field components and the longitudinal
factor of the transversal field are included in the equations
obtained from the boundary conditions, choosing an appro-
priate weighting function is an essential part of the solution
process. Theweighting functions chosenin thisstudy ledto two
important features. First, the fields are matched directly on the
curved interface
with neither artificial intermediate region,
nor transition region being used, which reduces the complexity
of the problem. Second, there are no Bessel functions in the
integrand of the numerical integrations so that the numerical
computation is much faster. The approach of selecting the
weighting function in this study is applicable to either flat or
curved interfaces.
Thispaperdiscussesindetailthemethodforanalyzingaside-
coupled circular waveguide to multiple rectangular waveguide
T-junctionsdescribedbrieflyin[16].Theresultantformulations
are listed in the Appendix. The convergence problem is also
discussed. The method is used to design a side-coupled circular
waveguide dual-mode filter, and is validated by numerical and
experimentalresults.Itisdemonstratedthatthemethodishighly
efficient, and has minimal limitations on the geometry of the
waveguide sections.
II. FORMULATIONS
A. Sub-Regions
A few rectangular waveguides with different cross sections
connected to a circular waveguide from its side are shown in
Fig. 3. Angle
andcan be any reasonable value where the
subscript indicates the th rectangular waveguide. Fig. 2(a) is
again used to represent the 3-D view of Fig. 3, although only
one rectangular waveguide branch is shown. The whole T-junc-
tion is divided into three types of subregions, i.e.: 1) circular
waveguideregions
;2)rectangularwaveguideregions
3) cavity region
. There are two circular waveguide regions,
i.e.,
and, from interfaces
respectively. The number of rectangular waveguide regions
which span from interface
to one rectangular port, is equal
to the number of rectangular waveguides connected to the junc-
tion. The cavity region
is in the middle of the structure, con-
necting with the circular waveguide subregions at the interface
andand the rectangular waveguide subregions at the in-
terface. All circular and rectangular waveguide subregions
do not connect with each other directly. The EM fields in dif-
ferent subregions are expressed by their own modal function se-
ries. The fields in neighboring subregions are matched on their
common interface
,, or
;and
andto two circular ports,
,
.
B. Field Modal Functions in Subregions
The modalfunctions for EMfields incircular and rectangular
waveguide can be found in standard textbooks, which are re-
peated here for completeness. In order to stay concise, only the
longitudinal magnetic field component
for TE modes and

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ZHENG AND YU: RIGOROUS MODE-MATCHING METHOD OF CIRCULAR TO OFF-CENTER RECTANGULAR SIDE-COUPLED WAVEGUIDE JUNCTIONS2367
the longitudinal electric field component
expressed in this paper. All transverse field components can be
derived from longitudinal field components by the formula
for TM modes are
(1)
where
junction,
mode in the waveguide, and
ator
isthewavenumberofthemediumfillingthewaveguide
is the propagation constant of the
is the transverse gradient oper-
(Cartesian coordinate)
(cylindrical coordinate).
(2)
Theupperandlowersigninsymbols
formulasinthispaperalwaysindicatethewavepropagatingfor-
wards and backwards, respectively.
Incircularwaveguideregion
andin(1)andother
,thefieldmodalfunctionsare
(3)
where
are the th zero of the
tive, respectively,
and
indicate the TE and TM mode, and
-and-
coordinate factor.
Incircularwaveguideregion
almost the same as (3). The only difference, which is needed for
added convenience during field matching, is that the coordinate
in (3) has to be replaced with
between interfaces
and .
In rectangular waveguide regions
is the Bessel function of order
th-order Bessel function and its deriva-
is the radius of the circular waveguide, and
are the normalization factors. Subscripts
,and
and
andrepresent the
mode. Subscript indicates the transverse
,thefieldmodalfunctionsare
, where is the distance
,
(4)
where
are local coordinates in the th rectangular waveguide with
as the longitudinal coordinate and
circular cylinder in the structure.
andare the normalization factors.,, and
parallel to the axis of the
and are the dimensions
of the th rectangular waveguide in the
spectively.
There are three different sets of modal functions in circular
cylindricalcavityregion
.Thefirstmodalfunctionsetcontains
the modes for the circular cylinder shorted at interface
and
- and-direction, re-
(5)
where the transverse coordinate factors
and
, their normalization factorsand , and
the propagation constants
as they were in (3) for the circular waveguide region since the
radius is the same for both regions.
The second modal function set in the cavity region includes
themodesinthecircularcylindershortedatinterface
and. Its expressions can be obtained from (5) directly by
substituting
with .
The third modal function set in the cavity region is the mode
set for the space between two parallel plates at the interface
and, respectively,
andare exactly the same
(6)
Each mode normalization factor
determined by forcing the vector product between
andof that mode over the entire cavity side surface
to be 1, while all other normalization factors in (3)–(5)
are determined by the vector product over the cross-sectional
surface of the corresponding waveguide.
The propagation constants in (3)–(6) are
orin (6) will be
(7)

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2368IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007
C. Field Matching Procedure
Arranging the mode sequence in each modal function set ac-
cording to their cutoff frequencies, the double mode indices
andcan be replaced by the mode sequence’s number in the
formulas. In this paper, , , , ,
and TM-mode indices for different mode sets instead of
.
The fields in each subregion are the superposition of the
modal functions in that subregion. Taking only the first finite
terms, the superposition can be expressed in the following
matrix expressions. Since the electrical and magnetic fields
have a similar form, only the
-field expressions will be shown
here. In subregion
and, the total field is
andwill be used as TE-
and
(8)
where superscript
can be and
...
...
...
...
(9)
and aresimilarto
, and the coefficient matrix
and,hasasimilar
form to
(10)
and are similar toand,has a similar
form to
In subregion
.
, the fields have similar expressions
(11)
where
...
...
...
...
(12)
(13)
and coefficient matrix
(12) and (13).
In cavity region
have expressions similar to
, the fields contain three groups of modes
(14)
where
pressions, as shown in (9) and (10).
The matrices
cient matrices, which will be determined by the boundary con-
ditions on all ports and all interfaces between subregions. There
are
unknown coefficient matrices altogether,
where
indicates the number of rectangular waveguides in
the structure and
is the total port number.
matrix equations are needed to obtain the relationship between
the incoming and outgoing wave modes on all ports, which may
bepresentedintheformofageneralscatteringmatrix.Fromthe
electrical boundary conditions on the interface between subre-
gions, three matrix equations can be obtained. Another
equations will be derived from the magnetic boundary condi-
tions.
The vector product of functions
are defined by the following integration:
,, and,,have similar ex-
in (8), (11), and (14) are unknown coeffi-
and on the surface
(15)
The electrical boundary condition on the interface
is
on(16)
where
,
functions, and taking the vector product on both sides of (16)
on
, it is derived that
is the unit vector in the -direction. Choosing
,, andas the weighting
(17)
where
larly,from theelectricalboundaryconditionontheinterface
the following equation is obtained:
is the diagonal matrix shown in the Appendix. Simi-
,
(18)
On the interface
, the electrical boundary condition is
on(19)
where
cavity. Using the weighting functions
, and
both sides of (19) on
containsthecompletesidesurfaceofthecylindrical
,,
, and taking the vector product on
, (20) is obtained as follows:
(20)
where matrices
Appendix. Most integrations included in the elements of ma-
trices
, except for a few special cases, cannot be solved an-
alytically. However, since the integrands do not contain Bessel
functions, numerical integration can be done quickly. That is
one of the advantages of choosing a cylindrical surface as the
and diagonal matrixare shown in the

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ZHENG AND YU: RIGOROUS MODE-MATCHING METHOD OF CIRCULAR TO OFF-CENTER RECTANGULAR SIDE-COUPLED WAVEGUIDE JUNCTIONS 2369
interface between the cavity and rectangular waveguide subre-
gions.
The magnetic boundary condition on the interface
shows
on (21)
Using weighting functions
and
(21) on the interface
,,,
, and taking the vector product on both sides of
, it is derived
(22)
Similarly, from the magnetic boundary condition on the in-
terface
, using similar weighting functions, it is obtained
(23)
The magnetic boundary condition on the interface
shows
on(24)
where
rectangular waveguide region.
The choice of weighting functions for performing the vector
productson bothsides of (24)is important.The weightingfunc-
tions should have the characteristics of both the forward and
backward propagating waves. If the interface is a flat plane such
as
or, and is parallel to the tangential field components,
field expression factors having the longitudinal coordinate are
constant on the interface, and thus the forward and backward
propagating waves differ by a constant factor. In the case of a
curvednonparallelinterface, fieldexpressionfactorshavingthe
longitudinal coordinate are not constant over the interface so
the difference between the forward and backward propagating
waves is no longer described by a constant factor. A possible
way to achieve the goal is to use the sum of the forward and
backward propagating waves as the weighting functions
is the interface between the cavity region and the th
(25)
Performing thevectorproduct toboth sidesof(24) withthese
weighting functions on
yields
(26)
where superscript
tion(17),(18),(20),(22),(23),and(26)aretogether
matrix equations. Combining these equations, unknown coef-
ficient matrices
,, and
desired matrix equations are obtained. These con-
tain
unknown coefficient matrices
representing the magnitude and phase of the forward and
backward propagating wave modes at each port. The general
scattering matrix for the structure can be obtained from these
equations.
Selecting the weighting function and taking the vector
product in order to get equations from the boundary conditions
indicatesthetransposeof thematrix.Equa-
can be eliminated and
,, and
is one of the essential steps of the mode-matching technique.
Actually, (25) is a general way to select the weighting func-
tions. It is valid for cases where the interface is parallel or is not
parallel to the tangential field components in the corresponding
subregions. Following this approach, the weighting function
for boundary condition (21) should be
(27)
Since the interface
cular waveguide,
is parallel to the cross section of the cir-
on
on(28)
By combining (27) and (28), the weighting function used to de-
rive (22) can be obtained.
III. CONVERGENCE PROBLEM
For practical purposes, all the field modal functions have to
be truncated in numerical calculation. It is well known in mode-
matching techniques that the choice of the number of the modes
and the ratio of these numbers between different subregions can
strongly influence the accuracy of the simulation. In this study,
a critical frequency is used to determine the mode numbers. All
modes that have cutoff frequencies lower than the critical fre-
quency must be taken into account.
The convergence of the example shown in Fig. 4 is studied. It
is worth pointing out that the aperture at the junction is large in
the circumferential direction of the cylinder. The
of the junction are simulated at frequency
different critical frequencies. Table I shows the mode numbers
and the simulated data versus the critical frequency. When the
critical frequency is selected as six times the frequency of
the simulated
-parameters converged. Although S11sin (the
return loss of sin mode at the circular port) varies rapidly in the
frequency band near
(see Fig. 4), its simulated results at
showed little change, while the critical frequency changed from
to.
-parameters
GHz using
,
IV. NUMERICAL RESULTS
A computer program based on the presented theory is de-
veloped to simulate side-coupled circular waveguide to mul-
tiple rectangular waveguide T-junctions. A series of examples
of such structures with different dimensions and different sym-
metries is simulated by the program and compared with the
measurements and other simulation tools. Three examples are
shown in Figs. 4–6.