Advanced Coupling Matrix and Admittance Function Synthesis Techniques for Dissipative Microwave Filters

In this paper a new approach to the synthesis of coupling matrices for microwave filters is presented. The new approach represents an advance on existing direct and optimization methods for coupling matrix synthesis in that it will exhaustively discover all possible coupling matrix solutions for a network if more than one exists. This enables a selection to be made of the set of coupling values, resonator frequency offsets, parasitic coupling tolerance etc that will be best suited to the technology it is intended to realize the microwave filter with. To demonstrate the use of the method, the case of the recently – introduced ‘extended box’ (EB) coupling matrix configuration is taken. The EB represents a new class of filter configuration featuring a number of important advantages, one of which is the existence of multiple coupling matrix solutions for each prototype filtering function, eg 16 for 8th degree cases. This case is taken as an example to demonstrate the use of the synthesis method – yielding one solution suitable for dual-mode realization and one where some couplings are small enough to neglec

Direct synthesis technique for coupling matrix of mixed topology filters that contain cross-coupled structures and a series of extracted-pole sections is proposed and demonstrated. Instead of evaluating elements one by one in conventional synthesis, the canonical coupling matrix, including all the cross-coupled and extracted-pole couplings, can be solved altogether one time by using the proposed technique. The direct synthesis is based on a process for the admittance function construction of a filter, yielding a mixed topology prototype where one or more extracted-pole sections are generated at the source port naturally. A sixth-order filter is carried out as an example, which verifies the validity of the proposed technique.

The equations of motion for pressure and displacement fields in a waveguide have been used to derive an energy-conserving, one-way coupled mode propagation model. This model has three important properties: First, since it is based on the equations of motion, rather than the wave equation, instead of two coupling matrices, it only contains one coupling matrix. Second, the resulting coupling matrix is anti-symmetric, which implies that the energy among modes is conserved. Third, the coupling matrix can be computed using the local modes and their depth derivatives. The model has been applied to two range-dependent cases: Propagation in a wedge, where range dependence is due to variations in water depth and propagation through internal waves, where range dependence is due to variations in water sound speed. In both cases the solutions are compared with those obtained from the parabolic equation (PE) method.

ABSTRACTA novel coupling matrix synthesis method for dual‐band bandpass filter is presented in this article. The output admittance poles and zeros of the single‐band filter prototype are firstly obtained and then transformed into dual‐band prototype. Then the coupling matrix with predefined topology is optimized according to these poles and zeroes directly. This method can be achieved with no need to construct the characteristic function of dual‐band filter and reconfigure the coupling matrix. Symmetric and asymmetric dual‐band bandpass filter coupling matrix synthesis has been carried out to illustrate and validate this method. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 58:603–606, 2016

The group E6 for grand unification is combined with the generation symmetry group SO(3)g × Z(2). The coupling matrices in the Yukawa interaction are identified with the vacuum expectation values of scalar flavons. The symmetric part of this 3 × 3 coupling matrix can be identified with the known up‐quark hierarchy. The antisymmetric part is responsible for fermion mixings and CP violation. Numerical fits with only few parameters reproduce quantitatively all known fermion properties. The model predicts an inverted neutrino hierarchy, CP violation properties as well as the 0 ν 2 β decay parameter. It also predicts that the masses of the two lightest of six ‘right handed’ neutrinos lie in the low TeV region.

In this paper, we propose a novel analytical framework that is able to capture inter-cell interactions and relevant variations in WCDMA systems using the so-called coupling matrix. Coupling matrix properties related to its spectral radius are analyzed and used to introduce a pertinent indicator which is able to dynamically optimize spectrum efficiency in the context of cognitive networks. Herein, an advanced spectrum management methodology based on coupling matrices is proposed as an illustrative example. Simulation results have shown that the proposed methodology has the capability to adapt to relevant changes in the network and outperforms the frequency allocation in classical hierarchical cell structure planning.

This paper presents a new analytical approach for a problem of coupling matrix (CM) synthesis. The approach is based on reformulation of well-known equations representing filter scattering parameters S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> and S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</inf> for a given coupling matrix. With the use of relation connecting inverse matrix its determinant and its adjugate matrix the system of equations is formed. The equations are reformulated in a way allowing direct comparison of polynomials coefficients for numerator and denominator of S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</inf> and numerator of S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</inf> . This enables to form the system of polynomial equations, where unknowns are all entries of searched coupling matrix. In the proposed approach no conversion from S-parameters to Y-parameters is needed like in most of published techniques. SINGULAR software (GPL license) using Groebner basis for solution of such a system of polynomial equations was used. Two numerical experiments have been presented. First one CM synthesis for 4th order (one cross-coupling) filter. Second one is CM synthesis for 8th order (two cross-couplings) filter. Both experiments showed excellent results obtained with the proposed method. The method works for S-parameters of filters with arbitrary topologies.

Although the synthesis of symmetrical dual-band bandpass filters has been studied, little seems to be known about the general asymmetrical case. In this paper, a procedure for the synthesis of general asymmetrical dual-band bandpass filters implemented with inline dual-mode cavities is proposed. The inline architecture, as well as the asymmetrical nature of the response, lead naturally to the choice of an extended box topology or generalizations of the latter. It was recently shown that these topologies possess the property of multiple solutions, meaning that the related coupling matrix synthesis problem admits several solutions. On one hand, this multiplicity offers some flexibility to the designer, but on the other hand, working with multiple solutions may lead to ambiguities during the tuning process. Our procedure takes the best of both worlds: using the list of equivalent coupling matrices, simplifications of the original topology can be obtained by canceling some particular couplings. The locations of cancelled couplings are chosen so as to preserve the electrical response and to provide some hardware simplifications. It is also shown that the resulting simplified topology no longer has the multiple solution property, therefore solving ambiguity problems. The procedure is detailed and demonstrated on two examples.

This paper presents a comparison between two techniques of S band interdigital diplexer design with i.e. with generalized multiplexer theory and coupling matrix synthesis. The design based on GMT, doubly terminated Chebyshev low pass filter response is used. The first two resonators of each channel are modified using formulas in GMT. The design is verified though MathCAD plots. In coupling matrix synthesis design, the general coupling matrix of each filter and coupling band width for two adjacent resonators are calculated. A novel technique is used for diplexer input design. The group delays are measured at the centre frequencies of channel filters. The physical diplexer design round rod type inter digital filter is selected. Both designs are simulated in HFSS with 2100 and 2250MHz with 100MHz channel band width.

Coupling matrix synthesis has been recognized as a widely employed and powerful tool for microwave filter modeling and design. Generally, to obtain a coupling matrix whose topology maps to the actual physical structure, the rotation matrix is used for the matrix reduction and elimination. However, it is difficult to obtain a precise simplified matrix through the rotation matrix method for the corresponding planar resonant circuit filter structure. To overcome the shortcomings of conventional rotation matrices and similar equivalent simplification methods, based on the idea of optimization, we propose a simplification of the coupling matrix using a hybrid optimization algorithm. By the proposed algorithm, a simplified matrix that accurately consistent with the physical structure can be obtained. Firstly, we use the matrix M R that after the conventional equivalent transformation of the coupling matrix as the initial value of the hybrid optimization. Secondly, in order to further obtain the mapped coupling matrix, the scattering parameters corresponding to the coupling matrix is used to calculate the optimization fitness index, and the filter topology is used as the optimization constraint. Then, a global optimization algorithm is used to obtain a matrix M G which is similar to the initial coupling matrix engineering and the matrix structure is similar to the topology of the microwave filter. Finally, use a local optimization algorithm to further iteratively optimize the M G until it converges to meet the engineering accuracy requirements. Then, the simplified matrix M D that is approximately equivalent to the initial coupling matrix and corresponds to the filter topology can be obtained.

This paper considers the problem of synchronization of differential dynamical systems with time-varying coupling. The temporal variation of the couplings we consider here is rather general and includes variations in both the network structure and the reaction dynamics. For example, driven by a metric dynamical system. Inspired by the author's previous work [1], the generalized Hajnal diameter is introduced to study the stability of synchronization manifold via the underlying variational equation, which can be proved to equal to e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup> , where λ is the largest one of all Lyapunov exponents of the underlying variational equation, corresponding to the space transverses to the synchronization manifold. As an application, these results are used to investigate the synchronization of linearly coupled ordinary differential systems (LCODEs) with identity inner coupling matrix. In this case, the Hajnal diameter of the linear system induced by the time-varying coupling matrices can be used to measure the synchronizability of the time-varying coupling process. The corresponding network can synchronize some chaotic attractors if and only if there exists some T > 0 such that the joint union of the topologies across any T-length time interval has spanning trees.

Exploring synchronizability of complex dynamical networks from edges perspective

Global synchronization of coupled map lattices are investigated in this paper. A sufficient condition for global synchronization is derived. Unlike other approaches, the coupled lattices is not linearized in this paper. Instead, the sufficient condition for the synchronization is obtained here by introducing a reference state along with Lyapunov function method, and this condition is simply given in terms of the maximum absolute value of eigenvalues of the coupling matrices those are not equal to one. Moreover, we give how to construct the coupling matrix to guarantee global synchronization of coupled map lattices based on two comparing matrices, which is very convenient to use. A typical example of coupled map lattices with chaotic nodes is finally used to verify the theoretical results and the effectiveness of the proposed synchronization scheme.

In this brief, a universal coupled topology to design cascaded multiple passbands bandpass filters (BPFs) with numerous in-band poles and out-band transmission zeros (TZs) is presented. Based on the proposed topology, the coupling matrices are constructed to design multiband bandpass filters (MBPFs) with cascaded trisection (CT) BPFs’ filtering responses. To validate the theory and topologies, a quad-band CT BPFs with TZs located in each lower sub-band is successfully synthesized firstly. Then, a dual-band and tri-band CT BPF with TZs generated in the upper side of each passband are theoretically investigated by employing the coupling matrix. Finally, the proposed dual- and tri-band examples are implemented in compact planar multi-coupled lines structures. In all individual passbands, measured insertion losses (ILs) and measured return losses (RLs) are better than 2.82 dB and 15.9 dB, respectively. Good matching level between the simulated and experimental results have been achieved.

This paper further investigates the Experimenta Wave Intensity Analysis (EWIA) method, which the authors proposed in a previous paper. EWIA combines experimental data (internal and coupling loss factors) with Wave Intensity Analysis (WIA). This paper analyses the characteristics of the coupling matrix of EWIA models and discusses how EWIA parameters affect the coupling matrix, and thus, the energy predictions. First, the construction of the coupling matrices of EWIA, WIA, SEA and ESEA are compared. Next, the analysis focuses on the non-direct coupling loss factors characteristic of WIA and EWIA formulations by presenting a physical interpretation and comparing them with the SEA and ESEA case. Finally, the matrix condition of the methods is compared and it is shown that the WIA and EWIA tend to have higher matrix condition numbers relative to the SEA and Experimental SEA. This indicates that the WIA and EWIA methods become more sensitive to matrix operations as the complexity and size of the system increases. This paper shows that the EWIA method preserves both the computational efficiency of WIA and the simplicity of its application respect to WIA, SEA and ESEA. As a result, EWIA works well without any additional effort compared to the SEA, WIA and Experimental SEA methods.