3940 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 11, NOVEMBER 2006
Coupling-Matrix Design of Dual
and Triple Passband Filters
Marjan Mokhtaari, Jens Bornemann, Fellow, IEEE, K. Rambabu, and Smain Amari, Member, IEEE
Abstract—The concept of the conventional coupling matrix is
extended to include designs of dual- and triple-band filters. The
multiband response is created by either placing transmission zeros
within the bandwidth of a wideband filter or using higher order
resonances. Realizable topologies both in planar and waveguide
technologies can be imposed and associated coupling coefficients
enforced during optimization. The design process is verified by
measurements and comparison with results of commercially avail-
able field solvers.
Index Terms—Dual-band filters, filters, microstrip filters, wave-
guide filters.
I. INTRODUCTION
R
ECENT developments in microwave filters have focused
on many different topics, one of them being the design
of dual- and triple-band filters. They are in heavy demand due
to applications in modern wireless and satellite systems. Usu-
ally, theoretical design techniques are related to individual filter
technologies. For instance, lumped-element or stepped-imped-
ances approaches are used in low-temperature co-fired ceramic
(LTCC) applications, e.g., [1], [2]. So-called dual-behavior res-
onators [3], [4] create attenuation poles at specific frequencies
in order to separate individual passbands. The coupling between
two modified open-loop resonators is used to create a dual-band
filter in microstrip technology [5]. Polynomial approaches and
coupling matrices are applied to the design of dual-band band-
pass [6] and bandstop filters [7] in waveguide technology.
Common to all such design procedures is the fact that they
cannot immediately be used if the filter topology changes. More-
over, many approaches are limited with respect to the number of
transmission zeros and locations over the frequency band of in-
terest.
Therefore, this paper focuses on the design of dual- and triple-
band filters by employing the coupling matrix and the optimiza-
tion of its entries. One of the major advantages of this approach
is that topologies and certain coupling elements can be con-
trolled from the onset [8]. The basic approach of this method
was introduced in [9]. However, measurements failed to con-
firm the transmission zeros between individual passbands.
Manuscript received March 22, 2006; revised August 3, 2006.
M. Mokhtaari and J. Bornemann are with the Department of Electrical and
Computer Engineering, University of Victoria, Victoria, BC, Canada V8W 3P6.
K. Rambabu is with the Institute for Infocomm Research, Singapore 117674.
S. Amari is with the Department of Electrical and Computer Engineering,
Royal Military College of Canada, Kingston, ON, Canada K7K 7B4.
Color versions of Figs. 3, 4, 5(a), and 6(a) are available online at http://iee-
explore.ieee.org.
Digital Object Identifier 10.1109/TMTT.2006.884687
In this paper, we present a new design and measurements,
which validate the design approach in microstrip technology.
Moreover, the same theory is applied to dual-band waveguide
filters and shows that the inclusion of higher order mode reso-
nances aids in the design. It is thus demonstrated that the de-
sign process is generally applicable to multiband filter designs
in varying topologies and technologies.
We are using hairpin resonators [10] and open-loop res-
onators [11] as triple/dual-band examples in microstrip tech-
nology. Dual-band waveguide designs include folded cavity
filters, e.g., [12], and inline dual-mode configurations, e.g.,
[13], [14]. Folded waveguide filters make use of higher order
resonances in
addition to the dual-band design, whereas the
dual-band effect in inline configurations is created because of
higher order resonances.
II. D
ESIGN PROCEDURE
The basic approach to create a coupling matrix for given spec-
ifications of a single bandpass filter is presented in [8] and [15].
(The reader is referred to [8] and [15] for further details.) The
principal advantages of this technique are, first, that the topology
of any scheme of coupled resonators can be specified in advance
and, secondly, that the signs and limits of coupling coefficients
can be strictly enforced during optimization.
For the design of dual- and triple-band filters, we assume first
that a single wideband filter will be constructed whose band-
width covers all bandwidths of the dual- and/or triple-band fil-
ters. Since the maximum number
of realizable transmission
zeros is dictated by the topology, any number
of trans-
mission zeros can now be placed within the initial broad pass-
band in order to separate individual passbands. Optimization
[15] is then employed to adjust the entries of the coupling ma-
trix.
The basic design steps are shown here at the example of a
triple-band filter with six resonators and four transmission zeros,
each two of which are located between adjacent passbands. The
individual passbands are centered at 2.65, 3, and 3.35 GHz, and
more than 50-MHz bandwidth is to be retained in each of the
bands.
We first design a standard Chebyshev response over the en-
tire triple-band frequency range using 3 GHz as the center fre-
quency, 800-MHz bandwidth, and 24-dB return loss. This de-
sign is shown in Fig. 1 as dashed lines. To allow a coupling ma-
trix to be optimized, we then require, first, a prototype function
for the triple-band filter and, secondly, an initial coupling ma-
trix to start the optimization. One possibility to obtain the pro-
totype function is to optimize the coefficients of the numerator
and denominator of the filtering function. This is usually done
0018-9480/$20.00 © 2006 IEEE
MOKHTAARI et al.: COUPLING-MATRIX DESIGN OF DUAL AND TRIPLE PASSBAND FILTERS 3941
Fig. 1. Wideband Chebyshev and triple-band prototype responses of design
example.
to determine the prototype function and the positions of the re-
flection and transmission zeros in particular. Our approach is
different though. We obtain the prototype function from piecing
together the functions of three individual filters, each of which
is designed according to the single bandpass filter approach in
[15]. For the triple-band filter considered in this example, the
left-most response is that of a two-pole filter with a transmission
zero to the right, the center response corresponds to a two-pole
elliptic-function filter (one transmission zero on each side), and
the right-most response is that of a two-pole filter with one trans-
mission zero on the left. The so-obtained overall prototype func-
tion is shown in Fig. 1 as solid lines. The three different parts
are clearly distinguished by the discontinuous points between
the passbands at 2.78 and 3.23 GHz.
The choice of an initial coupling matrix involves the actual
circuit topology. Let us assume that the triple-band filter be de-
signed for microstrip technology on RT6006 substrate and that
six hairpin resonators be employed. The four transmission zeros
are to be created by cross couplings between hairpin resonators
1 and 6, as well as 2 and 5, thus specifying a symmetric folded
filter configuration. One of the simplest initial coupling matrices
is that of the standard wideband Chebyshev filter (cf. Fig. 1,
dashed line) and allowing for the additional cross couplings 1–6
and 2–5. As the sign of the coupling is determined by the ori-
entation of the hairpin resonators, we allow both magnetic and
electric couplings in the optimization. For the given example,
a quick calculation using the closed-form expressions in [16]
advises that the magnitude of normalized inline coupling coeffi-
cients be less that 0.9 and that of the cross couplings be less than
0.5. With this initial coupling matrix, the optimization produces
the following coupling matrix (including source and load), as
shown in (1) at the bottom of this page, whose performance is
shown as dashed lines in Fig. 2. Depending on the initial values,
different coupling matrices are obtained. For instance, using di-
rect and cross couplings to be 0.5 and 0.25 in magnitude, re-
spectively, and specifying two couplings
and as neg-
ative, a different final matrix is obtained, as shown in (2) at the
bottom of this page, and its performance is shown as solid lines
in Fig. 2. Note that both approaches result in a very small cou-
pling between resonators 3 and 4; it is actually much smaller
than 0.0001 in (2) and, therefore, appears as zero.
Whereas both matrices
and adhere to the restrictions
specified above, the filter governed by
is attractive due to
the fact that one of the coupling coefficients
vanishes. A
disadvantage compared to that given by
is the reduced re-
turn loss in the center band. This filter can now be designed by
translating the coupling coefficients into line dimensions on an
RT6006 substrate using commercial field solvers. The Ansoft
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(2)
3942 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 11, NOVEMBER 2006
Fig. 2. Responses of triple-band coupling matrices obtained by optimization
from different initial values.
Fig. 3. Performance of triple-band filter using hairpin resonators on RT 6006
substrate. Ansoft Designer (solid lines, data from [9]) and IE3D (dotted lines).
Designer response (solid lines in Fig. 3) is obtained by equating
the coupling coefficients of (2) with those in the actual structure
(inset of Fig. 3). Of course, the achievable precision is limited
in this step due to comparison of an equivalent circuit (coupling
matrix) with a full-wave electromagnetic (EM) model. There-
fore, slight differences between responses of the coupling ma-
trix and that of the actual circuit must be accepted. Some minor
fine-tuning in the EM-based code is also usually required. The
design is verified by a different commercial package, i.e., IE3D,
whose response is denoted as dashed lines in Fig. 3. (Note that
slight differences between Ansoft Designer and IE3D results
have been previously observed [17] and are attributed to the dif-
ferent approaches of the method-of-moments implementation
in both packages.) The passband insertion losses, as simulated
with Ansoft Designer, are approximately 2.0, 2.3 and 2.1 dB;
the minimum passband return loss is slightly below 10 dB.
Depending on the actual filter topology and technology,
tighter restrictions might have to be imposed on magnitudes
and signs of individual coupling coefficients. For planar cir-
cuits, closed-form expressions for the electric and magnetic
coupling coefficients of, for example, open-loop resonators
[16], are used. For waveguide-based filters, maximum aperture
dimensions are determined in advance and related coupling co-
efficients are calculated from simple mode-matching routines,
e.g., [18].
III. R
ESULTS
The above theory is now applied to dual-band filters in
microstrip and waveguide technologies. A few microstrip
examples are presented in [9] and will not be repeated here.
However, due to manufacturing tolerances, the transmission
zeros between passbands are not experimentally confirmed in
[9]. Therefore, a six-resonator dual-band microstrip filter with
open-loop resonators is designed on an RT5880 substrate with
a height of 508
m. The design parameters are GHz
and
MHz and the transmission zeros at 2.64, 2.94,
3.06, and 3.42 GHz. The initial coupling values in the coupling
scheme shown in the inset of Fig. 4(a) are all 0.5, except for
. The optimized coupling matrix is given in (3),
as shown at the bottom of this page. Fig. 4(a) shows the com-
parison between the performances of the coupling matrix and
that of the actual circuits using Ansoft Designer. A prototype
filter was built, and its response is also shown in Fig. 4(b)
together with a photograph and the Ansoft Designer data for
comparison.
The measurements confirm the basic shape of the computed
filter characteristic, especially the existence of the four transmis-
sion zeros. However, the entire measurement is slightly shifted
towards higher frequencies. After investigation, it was deter-
mined that this shift can be partly attributed to the fact that along
the tracks (transmissions lines), the manufacturing process pro-
duces slightly deeper cuts into the dielectric. Therefore, the ef-
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MOKHTAARI et al.: COUPLING-MATRIX DESIGN OF DUAL AND TRIPLE PASSBAND FILTERS 3943
Fig. 4. Performance of a dual-band filter using six open-loop resonators on
RT 5880 substrate. (a) Comparison of coupling-matrix theory (dashed lines)
with Ansoft Designer (solid lines). (b) Prototype measurements (solid lines) and
photograph (Ansoft Designer data (dashed lines) for comparison).
fective dielectric constant of the line will be lower than ex-
pected. Note that this effect is more pronounced for the thin
lines used to form the open-loop resonators whose linewidths
are much smaller than those of the 50-
input and output sec-
tions. The remaining discrepancies are attributed to manufac-
turing tolerances, which compare well with measurements pre-
sented in [9] for higher permittivity substrates. Measured pass-
band insertion losses are 2.1 and 2.4 dB.
Single passband waveguide filters are traditionally modeled
by coupling matrices, e.g., [19]. We are extending the dual-band
design of Section II to waveguide technology by making use of
two principles. The first example uses the same procedure as ap-
plied to the microstrip filters, but takes into account additional
transmission zeros produced by the actual filter. In the second
example, we will use higher order resonances in an inline con-
figuration to create the second passband. This principle was first
applied in [13].
Fig. 5(a) shows a dual-band filter in folded waveguide tech-
nology. The design parameters are
GHz and
Fig. 5. Dual-band filter in folded waveguide technology. (a) Three-dimen-
sional (3-D) view of filter. (b) Coupling scheme. (c) Performances obtained
from theory (dashed lines) and EM-based software (solid and dotted lines).
GHz and transmission zeros at 12.94, 14.85, 15.09, and 16.31
GHz. In order to convert the optimized coupling matrix ele-
ments to actual aperture dimensions, we follow an approach
given in [12]. Upon inspection of a first waveguide design, addi-
tional transmission zeros were observed resulting from the dis-
tance between the source/load aperture and the rest of the filter.
The two resonances were included in the coupling scheme [see
Fig. 5(b)] as detuned nodes
and similar to [20]. The op-
timized coupling matrix is shown in (4) at the bottom of the
following page, and its response is shown via dashed lines in
Fig. 5(c). Good agreement is obtained with the actual filter as
modeled by the mode-matching technique (MMT) (solid lines)
3944 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 11, NOVEMBER 2006
and the WaveWizard (dotted lines), especially with respect to
the location of the two passbands and the number of transmis-
sion zeros.
Fig. 6(a) shows an inline dual-mode filter using
and
modes in the two cavities. This type of filter was intro-
duced in [14]. Here we present a design for dual-band opera-
tion at 10.5 and 11.75 GHz. The second passband results from
the next higher order modes in the two cavities. In the coupling
scheme of Fig. 6(b), they are modeled as detuned resonators
and in an attempt to maintain symmetry of the coupling ma-
trix. Note that this design differs from all previous ones in this
paper due to the fact that the second (higher) passband cannot
be controlled independently. The higher order resonances de-
pend on the two cavities, whose dimensions are determined to
obtain the first (lower) passband. Therefore, the midband fre-
quency and bandwidth in this design are those of the lower
passband (
GHz, MHz). In the coupling
scheme [see Fig. 6(b)], the straight line from input to output
including 1–6 cross-coupling forms a standard quadruplet for
the lower frequency band. The addition and connection of the
higher order resonances in Fig. 6(b) are based on the following
considerations.
In the first section of the dual-mode resonators,
and
are well below cutoff even in the upper frequency band.
In the second (larger) section of the dual-mode resonators, both
and are above cutoff. However, this section is too
short for a
resonance to occur and, therefore, nodes
and represent resonances. Therefore, both fun-
damental resonances
and couple to the
higher order resonance. Above 11.4 GHz, the input/output
irises become propagating. That means that the input/output
can couple directly to the
resonances . Since
the center iris of the filter is centered with respect to the two
adjacent cavities, there is no
– coupling through
the center iris, i.e.,
).
The coupling matrix obtained from following this scheme
is given in (5). Its response is shown in Fig. 6(c) via dashed
lines. The actual design as computed with the High-Frequency
Structure Simulator (HFSS) (dotted lines) and the coupled in-
tegral-equation technique (CIET) (solid lines) agrees relatively
well with the coupling-matrix prediction, shown in (5) at the top
of the following page.
Except for the small deviations between coupling matrix and
full-wave codes addressed earlier, the only notable discrepancy
is the location of the transmission zero between 11–11.5 GHz.
Fig. 6. Inline dual-band waveguide filter based on fundamental and higher
order mode resonances. (a) 3-D-view of filter. (b) Coupling scheme. (c) Per-
formances obtained from theory (dashed lines) and EM-based software (solid
and dotted lines, which are almost indistinguishable).
This is attributed to the fact that the – coupling
through the center iris is highly frequency dependent in this fre-
quency range. Such dependence is, of course, not captured by
a coupling matrix approach, which assumes constant coupling
coefficients. This is a limitation not only of this coupling-ma-
trix design procedure, but of other coupling matrix designs as
well. Nevertheless, this example demonstrates the flexibility of
the coupling-matrix design routine presented in this paper.
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![Fig. 3. Performance of triple-band filter using hairpin resonators on RT 6006 substrate. Ansoft Designer (solid lines, data from [9]) and IE3D (dotted lines).](/figures/fig-3-performance-of-triple-band-filter-using-hairpin-27eeqlwk.webp)
