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where the symbol *T denotes the transpose of a matrix. To put
it alternatively, the diagonal elements of [S] dictate the symmetry of the network, while the off-diagonal ones, the reciprocity
of it. For the sake of completeness, we note that the passivity of
the medium is determined by the negativeness of the matrix
� 2� CT
, where the operator *CT corresponds to the conjugate
2j
transpose of a matrix. By comparing the complex values of certain elements from matrix [S], we realized that the component
which is parallel to the incident field remains the same regardless of the side that is excited. More specifically, we obtain the
equalities:
xx
Sxx
21 5S12 ;
yy
Syy
21 5S12
(6)
for arbitrary combination of the parameters (rxx, rxy, ryx, ryy).
For this reason, to evaluate the nonreciprocity of the effective
medium, we focus on the other four pairs of the off-diagonal
elements of matrix [S], and we define the following difference
quantities normalized to unity:
xy
yx
jSxx
jSxy
11 2S11 j
12 2S21 j
;
yx
xy ; DS12 5 xy
jS11 j1jS11 j
jS12 j1jSyx
21 j
jSxy 2Syx j
jSyx 2Sxy j
DS21 5 xy21 12yx ; DS22 5 yx22 22xy :
jS21 j1jS12 j
jS22 j1jS22 j
DS11 5
(7)
What is most impressive is that the variations of the aforementioned four quantities (DS11, DS12, DS21, DS22) are numerically equal to each other, with respect to arbitrary selection of
the relative permittivities (rxx, rxy, ryx, ryy). As a result, the
degree of nonreciprocity of the medium is solely expressed by
one positive quantity DS varying between 0 and 1:
05NRI 5DS11 5DS12 5DS21 5DS22 < 1;
(8)
which we call ?nonreciprocity index? NRI. Obviously, for
NRI 5 0, we have a reciprocal medium and in case NRI ! 1,
giant nonreciprocity is occurred.
4.
NUMERICAL RESULTS
Through extensive numerical simulations, we concluded that the
function NRI possesses the following properties: (i) It is independent from the electrical (and the physical) thickness of the
slab which is natural, since the nonreciprocity is not related to
the size of the layer. (ii) It is numerically independent from the
diagonal permittivities (rxx, ryy) of the matrix [r], because they
are related to the symmetry and not to the reciprocity of the
device.
(iii)
The
contour levels of NRI with respect to
(< 2ryx ; = 2ryx ) are dependent basically on the magnitude of
the other off-diagonal component jrxy j. (iv) The dependence of
NRI on the argument h of rxy is trivial; in particular, the twodimensional distribution for an arbitrary rxy with magnitude jrxy j
and argument h is formulated as a rotation of the corresponding
one for real positive rxy 5 jrxy j (h 5 0) by the angle h.
For these reasons, we represent
(Fig.
3) the contour plots
of NRI with respect to (< 2ryx ; = 2ryx ) for various magnitudes
jrxy j with h 5 0. By inspection of the graphs, one can clearly
notice that NRI 5 0 when rxy 5 ryx and far from this point, the
depicted quantity gets increased (and restored at NRI 5 0:9)
with more rapid pace, the smaller is the magnitude jrxy j. Furthermore, the isocontour surfaces are symmetric with respect to
the line arg(ryx ) 5 h. The shortest path on the considered map
in order to reach the maximal nonreciprocity index NRI is to
follow the line arg(ryx ) 5 h toward the origin. In other words, a
DOI 10.1002/mop
specific asymmetry of [S] with the minimum asymmetry of [r ]
is achieved when ryx possesses magnitudes very close to zero.
REFERENCES
1. Y. Ayasli, Non-ferrite, non-reciprocal phase shifter and circulator,
US Patent, 4801901, 1989.
2. D. Sounas, C. Caloz, and A. Alu, Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,
Nat Commun 4 (2013), 2407.
3. K. Nishimura, Nonreciprocity of electromagnetic wave propagation
characteristics in a grounded ferrite slab waveguide with a metallic
strip grating, In: Proceedings of the 39th European Microwave Conference, Rome, Italy, 2009, pp. 727?730.
4. C.L. Hogan, The microwave gyrator, Bell Syst Tech J 31 (1952), 1?
31.
5. A.G. Fox, S.E. Miller, and M.T. Weiss, Behavior and applications of
ferrites in the microwave region, Bell Syst Tech J 34 (1954), 1?31.
6. Philips semiconductors, circulators and isolators, unique passive
devices application notes, 98035.
7. C. Wenzel, Low frequency circulator/isolator uses no ferrite or magnet, RF Design, 1991.
8. D. G. Hoag, Electromagnetic isolator/actuator system, US Patent,
4849666, 1989.
9. H. Lira, Z. Yu, S. Fan, and M. Lipson, Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,
Phys Rev Lett 109 (2102), 033901.
10. D. Sounas and C. Caloz, Electromagnetic nonreciprocity and gyrotropy of graphene, Appl Phys Lett 98 (2011), 021911.
11. C. He, X.-C. Sun, Z. Zhang, C.-S. Yuan, M.-H. Lu, Y.-F. Chen, and
C. Sun, Nonreciprocal resonant transmission/reflection based on a
one-dimensional photonic crystal adjacent to the magneto-optical
metal film, Opt Express 21 (2013), 28934.
12. Y. Leviatan and G.S. Sheaffer, Analysis of inductive dielectric posts
in rectangular waveguide, IEEE Trans Microwave Theory Tech 35
(1987), 48?59.
C 2014 Wiley Periodicals, Inc.
V
IMPLEMENTATION OF CROSS
COUPLINGS IN MICROWAVE BANDPASS
FILTERS
B. A. Belyaev1,2,3, A. M. Serzhantov1,2, Y. F. Bal?va1,3,
V. V. Tyurnev1,2, A. A. Leksikov1,2,3, and R. G. Galeev4
1
Kirensky Institute of Physics, Siberian Branch, Russian Academy of
Sciences, Krasnoyarsk, Russia; Corresponding author:
belyaev@iph.krasn.ru
2
Institute of Engineering Physics and Radio Electronics, Siberian
Federal University, Krasnoyarsk, Russia
3
Reshetnev Siberian State Aerospace University, Krasnoyarsk,
Russia
4
OJSC Scientific Production Enterprise ?Radiosviaz?, Krasnoyarsk,
Russia
Received 16 January 2014
ABSTRACT: New quasi-lumped resonator bandpass filters with cross
couplings are presented. Coaxial and suspended stripline structures are
considered. These filters are notable for their extremely wide and deep
upper stopbands at small dimensions. They also have transmission zeros
near the passband that significantly improve frequency selectivity. Photographs and measured frequency responses for the four fabricated filC 2014 Wiley Periodicals, Inc. Microwave Opt
ters are presented. V
Technol Lett 56:2021?2025, 2014; View this article online at
wileyonlinelibrary.com. DOI 10.1002/mop.28507
Key words: microwave filters; elliptic-function bandpass filters; transmission zeros; cross couplings
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 9, September 2014
2021
Figure 1 Direct (a) and indirect (b) cross couplings in the resonator
quadruplet
Figure 2 Layout of the coaxial filter No. 1
1.
pling topologies are well-known. These topologies are good for
cascading to obtain the higher order frequency response with
more number of transmission zeros.
Every trisection stage in the cascaded filter generates only
one transmission zero. Its frequency is situated below or above
the passband depending on the type of the cross coupling (i.e.,
sign of k13) between the first and third resonators. At that, the
stronger the cross coupling, the closer to the passband the transmission zero situates.
Every quadruplet stage generates none or two transmission
zeros. They arise if the sign of the cross-coupling coefficient k14
between the first and fourth resonators is opposite to the sign of
the product of all nearest coupling coefficients k12k23k34. Two
transmission zeros situate symmetrically on both sides of the
passband if the frequency dispersion of all the coupling coefficients kij is not too strong near the passband. They also move
closer to the passband when the cross coupling (k14) is made
stronger. In case of a strong frequency dispersion of k14, the
transmission zeros may be both situated below or above the
passband if k14 inverts its sign near the passband.
The cross coupling in most conventional designs of the
cross-coupled filters is direct, that is, it is fulfilled without the
use of a complementary coupling element between nonadjacent
resonators [Fig. 1(a)]. The direct cross coupling is frequently
made with inductive or capacitive iris in the metal wall of the
waveguide [2,3] or coaxial filters [4]. As for planar filters, the
cross-coupled resonators often have a common section of
coupled transmission lines [1]. The direct cross coupling implementation is complicated and such filters are difficult to tune.
When the distance between the nonadjacent resonators is too
long to ensure the required cross coupling, designers use complementary pins, or transmission line sections [5,6]. Such cross
coupling is indirect [Fig. 1(b)].
In this letter, we present some new constructions of the
highly selective cross-coupled bandpass filters where the cross
coupling is implemented with a complementary nonresonant
transmission line section.
INTRODUCTION
Elliptic-function bandpass filters are known for their high selectivity. They have transmission zeros in stopbands, which significantly improve frequency responses of the filters. One of the
ways to realize transmission zeros is the arrangement of additional signal paths in the filter by means of additional couplings
between nonadjacent resonators (i.e., cross couplings). A transmission zero arises when the signals passing the two parallel
paths have the same amplitudes and opposite phases. Various
constructions of the cross-coupled microstrip filters are
described in the monograph [1]. Trisection and quadruplet cou-
2. QUASI-LUMPED COAXIAL FILTER
A new quasi-lumped resonator was recently proposed [7]. It
consists of two coaxial tubular conductors each of them is
grounded by one of its ends to the metal case. It differs in high
ratio of the resonant frequencies for the first spurious mode and
the fundamental mode. It also differs in small size and
high quality factor. The frequency responses of the crosscoupling free bandpass filters of the forth order were studied in
Refs. 8,9.
Figure 3 Measured frequency response of filter No. 1. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
2022
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 9, September 2014
DOI 10.1002/mop
Figure 4 Frequency responses of filter No. 1 with and without cross
coupling. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
Figure 5 Layout of the suspended stripline filter No. 2. [Color figure
can be viewed in the online issue, which is available at wileyonlinelibrary.com]
An example of an indirect cross coupling implemented in the
quasi-lumped coaxial filter is shown in Figure 2. Filter No. 1
has a row of six EM-coupled coaxial resonators where inductive
(positive) couplings prevail over corresponding capacitive (negative) ones. Here the coaxial tubular conductors in every resonator are insulated by a polytetrafluoroethylene pipe (er52.1).
Note that the resonators in the filter are very short. Their
relative length lr/k 5 0.054 where lr is the resonator length
and k is wavelength in free space at the resonant frequency.
The first and the sixth resonators are tapped to the filter
ports. The second and the fifth resonators have a cross coupling that is implemented with the use of a piece of insulated wire and two capacitors mounted on the resonators
(Fig. 2). This means that the cross-coupling coefficient k25 is
negative. The filter case has the inner dimensions of 143 3
44 3 31 mm3.
Filter No. 1 and all other filters that are presented below
were manually designed using a 3D EM simulator and a special
optimization method based on universal physical rules [10,11].
We did not use equivalent circuits, coupling coefficients, and
coupling matrices during the filter design.
Figure 3 shows the photograph and the measured frequency
response of the fabricated coaxial filter. The passband has the
center frequency f05368 MHz, the 3-dB bandwidth Df 5 70
MHz (19%), the minimum insertion loss L 5 0.23 dB, and the
minimum return loss R 5 17 dB. Besides two transmission
zeros at the frequencies fz1 5 321 MHz and fz2 5 407 MHz, the
cross coupling generates a series of spurious transmission
peaks at all resonant frequencies of the wire ensuring the cross
coupling. Two first peaks of 242 and 241 dB are situated at
the frequencies fp15642 MHz and fp251810 MHz,
respectively.
Figure 4 shows the simulated frequency responses of filter
No. 1 in the presence and absence of the cross coupling. The filters in both cases were synthesized for the same passband as the
passband of the frequency response in Figure 3. The solid line
in Figure 4 refers to the filter with the cross coupling, and the
dash line refers to the filter without that. Similarly, simulated
frequency responses were obtained for all other filters that are
presented below.
3. STRIPLINE FILTERS ON SINGLE-LAYER SUSPENDED
SUBSTRATE
A new quasi-lumped suspended stripline resonator was proposed
in [12]. This resonator has the same advantages as the quasilumped coaxial resonator. It consists of two strip conductors
oppositely placed on both sides of a single-layer dielectric substrate that is suspended inside a metal case. Both strip conductors are grounded by one end in each of the conductors to the
opposite walls of the metal case. The frequency responses of the
cross-coupling free suspended stripline bandpass filters were
studied in [12]. The study of similar filters but filled with
Figure 6 Measured frequency response of filter No. 2. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
DOI 10.1002/mop
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 9, September 2014
2023
Figure 7 Layout of the suspended stripline filter No. 3. [Color figure
can be viewed in the online issue, which is available at wileyonlinelibrary.com]
Figure 9 Layout of the suspended stripline filter No. 4. [Color figure
can be viewed in the online issue, which is available at wileyonlinelibrary.com]
homogeneous dielectric and realized with the use of LTCC technology was presented in [13].
An implementation example of an indirect cross coupling in
the quasi-lumped suspended stripline filter is shown in Figure 5.
Filter No. 2 has a metal case with inner dimensions of 100 3
40 3 24 mm3. The single-layer substrate of filter No. 2 is made
of FAF-4D (foil-coated reinforced polytetrafluoroethylene sheet)
with the thickness hd 5 0.5 mm and permittivity er 2.5. The
filter has six EM-coupled suspended stripline resonators where
inductive couplings prevail over corresponding capacitive ones.
This means that all the coupling coefficients between the nearest
resonators are positive. All the strip conductors in the resonators
have the length ls 5 35 mm. All the resonators have the relative
length lr/k 5 0.049. The first and sixth resonators are tapped to
the filter ports. The second and the fifth resonators have a cross
coupling that is implemented with the use of a narrow strip conductor terminated by rectangular electrodes. Each of the electrodes together with the resonator strip conductor on opposite
surface of the substrate forms a capacitor. Thus, the indirect
cross coupling between the second and fifth resonators is
capacitive (i.e., k25<0).
Figure 6 shows the photograph and the measured frequency
response of the fabricated suspended stripline filter No. 2. Its
passband has the center frequency f0 5 369 MHz, the 3-dB
bandwidth Df 5 75 MHz (20%), the minimum insertion loss
L 5 0.35 dB, and the minimum return loss R 5 15.4 dB. Besides
two transmission zeros at the frequencies fz1 5 321 MHz and
fz2 5 407 MHz, the cross coupling generates a series of spurious
transmission peaks at all its resonant frequencies. Three first
peaks of 273, 245, and 238 dB are situated at the frequencies
fp1 5 714 MHz, fp2 5 1900 MHz, and fp3 5 2948 MHz,
respectively.
Another implementation example of the indirect cross coupling in the quasi-lumped suspended stripline filter is shown in
Figure 7. Filter No. 3 has a row of five EM-coupled resonators
where inductive (positive) couplings prevail over corresponding
capacitive (negative) ones. The filter substrate is made of
ceramics barium-niobium-strontium titanate with er 80. It has
the thickness hd 5 1 mm. The metal case of the filter has the
inner dimensions of 44 3 5.5 3 6 mm3. The first and fifth resonators are inductively coupled with the filter ports. Each of suspended stripline resonators has the relative length lr/k 5 0.026.
Filter No. 3 has two cross couplings and they are symmetrical. Every coupling is implemented with the use of a narrow Pshaped strip conductor. The first coupling is arranged between
the first and fourth resonators, and its conductor is placed on
one substrate surface. The second coupling is arranged between
the second and fifth resonators, and its conductor is placed on
the opposite substrate surface.
Both conductor ends in every cross coupling are connected
to the metal case walls making the cross coupling inductive.
However, the cross-coupling coefficients k14 and k25 are negative because the P-shaped conductor is rather long for its resonant frequencies f1 and f2 to satisfy the inequality f1 < f0 < f2,
that is, the currents in both conductor ends flow in opposite
directions near the center frequency f0. The P-shaped conductors in narrowband filters turn out to be electrically long when
their suspended substrate has very high permittivity er.
Figure 8 shows the photograph and the measured frequency
response of the fabricated filter No. 3. Its passband has the center frequency f0 5 1.415 GHz, the 3-dB bandwidth Df 5 28 MHz
(2%), the minimum insertion loss L 5 4.35 dB, and the minimum return loss R 5 14 dB. Besides two transmission zeros at
the frequencies fz1 5 1.368 GHz and fz2 5 1.458 GHz, the cross
couplings generate a series of spurious transmission peaks at all
Figure 8 Measured frequency response of filter No. 3. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
2024
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 9, September 2014
DOI 10.1002/mop
Figure 10 Measured frequency response of filter No. 4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
resonant frequencies of the P-shaped conductors. Two first
peaks of 271 and 286 dB are situated at the frequencies
fp1 5 1.178 GHz and fp2 5 4.046 MHz, respectively.
4. STRIPLINE FILTER ON DOUBLE-LAYER SUSPENDED
SUBSTRATE
One more quasi-lumped suspended stripline resonator was proposed in [14,15]. This resonator has the same advantages as the
three first quasi-lumped resonators. It allows designing bandpass
filters with a deeper stopband. The resonator consists of a
double-layer suspended substrate and three parallel strip conductors. Two of them are oppositely placed on both outer surfaces
of the substrate and connected by one of their ends to one of
the sidewalls in the metal shielding case. The third conductor is
placed inside the double-layered substrate between the outer
conductors and connected by one of its ends to the opposite
sidewall. The rest of the strip conductors? ends do not reach the
sidewalls and remain open circuited.
Figure 9 shows an example of implementation of indirect
cross coupling in the stripline bandpass filter on the suspended
double-layer substrate. The metal case of filter No. 4 has the
inner dimensions of 47 3 12 3 5.4 mm3. The filter substrate is
made of RO4003CTM (hydrocarbon ceramic laminate) with the
thickness hd 5 0.203 mm and permittivity er 3.38. The strip
conductors in the resonators have the length ls 5 9.5 mm. The
resonators have the relative length lr/k 5 0.040. The first and
sixth resonators are tapped to the filter ports. The second and
fifth resonators have a cross coupling that is implemented with
a P-shaped strip conductor, which is placed on one of outer
surfaces of the double-layer substrate.
Figure 10 shows the photograph and the measured frequency
response of the fabricated filter No. 4. Its passband has the center
frequency f0 5 1.01 GHz, the 3-dB bandwidth Df 5 56 MHz
(5.5%), the minimum insertion loss L 5 4 dB, and the minimum
return loss R 5 13 dB. Besides two transmission zeros at the frequencies fz1 5 925 MHz and fz2 5 1077 MHz, the cross coupling
generates a series of spurious transmission peaks at all its resonant
frequencies, but their level is lower than the noise level and they
are not visible. The first visible peak of 242 dB at fp1 5 9.653
GHz is due to the third oscillation mode of the resonators [14].
5.
CONCLUSION
Thus, we have considered four examples of how cross couplings
in quasi-lumped bandpass filters may be implemented to generate transmission zeros that make the filter skirts significantly
sharper. These examples have been illustrated with photographs
of the fabricated filters and their measured frequency responses.
The presented filters are small in size. They have wide upper
stopband with high rejection.
DOI 10.1002/mop
ACKNOWLEDGMENT
This work was supported by the Siberian Branch of the Russian
Academy of Sciences, Integration Project No. 109.
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C 2014 Wiley Periodicals, Inc.
V
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 56, No. 9, September 2014
2025
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