Two-variable Reactance Research Articles

Synthesis of passive fractional‐order LCn‐port with three element orders

Fractional-order circuits find a widespread use in different engineering applications. The problem of realising fractional-order circuits has been discussed by several authors, however, it is far from being solved. Realising fractional-order resistorless passive network with three element orders is been studied. At first, this study extends the two-variable reactance matrix synthesis method to three-variable case, and then proposes a synthesis method of fractional-order reactance matrix with three element orders by variable substitution. The process in above methods mainly involves variable substitution, decomposition of three-variable reactance matrix, extraction of unit inductors, Laurent series expansion, spectral factorisation of two-variable positive semidefinite Hermitian matrix and synthesis of univariable reactance matrix. Then the above-mentioned synthesis process is illustrated by two examples.

Realization of a class of two-dimensional zero-phase recursive digital filters with variable cutoff boundary

Digital realization of a class of two-dimensional, zero-phase filters with variable cutoff boundary is given. The derivation of the 2-D transfer function is based on the transformation of a 1-D analog reference filter using a two-variable reactance function. Digital realization of the resulting 2-D transfer function is derived by using the generalized delay unit concept and cascade connections of two-port networks represented by their scattering matrices.

Stable fan filter design using two-variable reactance functions

Two-variable reactance transformation, as proposed by King et al., for the design of stable two-dimensional recursive fan filters with narrow stopbands, is compared with a design method, proposed by Bruton et al., using numerical optimisation of a resistively-terminated (2 + n1 + n2)-port network.

Simple design technique for inherently stable fan filters

The use of two-variable reactance functions for the design of fan filters is discussed. It is shown that this design approach is much simpler than those which require an optimisation process. The resulting filter is inherently stable and near optimal. An example is given to illustrate the effectiveness of the proposed method.

Characterization and synthesis of homogeneous two-variable reactance functions

It is shown in this paper that Homogeneous Two-variable Reactance Polyno- mials (HTRP's) can be expressed as a product of linear factors in the two-variables. Further, a Homogeneous Two-variable Reactance Function (HTRF) can be characterized in terms of the alteration property involving the coefficients of the linear factors of the HTRP in the numerator and denominator. Synthesis procedures for HTRF's are discus- sed; these realizations follow from the single variable RL- or RC-functions associated with a HTRF. Additional properties of HTRF, which will enable us to generate Two- variable Positive Real Functions and Two-variable Reactance Functions are also discussed.

Residue matrices and related properties in two-variable reactance functions

In this paper, the characterization of a two-variable reactance polynomial φ(λ,μ) is given in terms of the residue matrices of a single variable reactance matrix, Y(λ). Specificially, if Y(λ) is expressed in terms of its partial fraction as Y(λ)=λH ∞+Σ H i+jβ i λ+jω i +G where the residue matrices in general are p.s.d. Hermitian, then the ranks of these residue matrices are fixed in relation to the construction of φ(λ,μ) as the determinant of the two-variable reactance matrix μ1+Y(λ). Three theorems concerning these ranks—one each corresponding to the finite poles, poles at ∞ and the behaviour at λ=0 of Y(λ) are stated and proved. Several properties following from these theorems are studied. Also, implications of these theorems from a network theoretic point of view, like the minimum number of gyrators required to synthesize Y(λ) to yield the specific type of φ(λ,μ) etc., are studied. In the sequel, the concept of “generalized compact pole conditions” is introduced. Finally, these results are applied for the generation of two-variable reactance functions and matrices.

On the ranks of certain matrices

A few basic results regarding the ranks of certain types of matrices are presented. Also, some bounds on the ranks of the real and imaginary parts of a positive semi-definite Hermitian matrix are worked out. Applications of particular cases of these results in the areas like passive network synthesis, generation of two-variable reactance matrices, etc., are pointed out.

Frequency transformations for two-dimensional recursive digital filters

This letter discusses frequency transformations for two-dimensional recursive digital filters. It is shown that a stable transformation g(z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> , z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ) can be obtained directly from a two-variable reactance function in a simple manner via the double bilinear z-transform (z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> - 1)/(z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> + 1), i = 1, 2.

Synthesis of a class of two-variable positive-real functions†

Abstract A synthesis procedure for realizing a class of two-variable positive-real functions bilinear in one variable will be given here. The synthesis technique can be considered as a generalization of Ansell's synthesis procedure of two-variable reactance functions. Two examples will be given to illustrate the applicability of the above result in the synthesis of lumped-distributed structures and variable parameter networks.

Minimal Synthesis of Two-Variable Reactance Matrices*

A simple algebraic method stemming from ideas in minimal state-variable realization theory is developed for the synthesis of two-variable reactance matrices. The method rests mainly on the factorization of a one variable polynomial matrix which is para-Hermitian and positive semidefinite on the imaginary axis, and always yields a realization minimal in both variables.

Synthesis of Finite Passive n-Ports with Prescribed Two-Variable Reactance Matrices

This paper solves the problem of synthesizing twovariable reactance functions and matrices, symmetric or nonsymmetric. This problem has not been solved so far even for twovariable reactance functions except for the case where the functions are prescribed as bilinear functions of one of the two variables. As the basis for solving the problem, several theorems are derived from the definitions of the two-variable reactance function and matrix. It is shown that an arbitrarily prescribed two-variable reactance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(\lamda,\mu )</tex> of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is realizable as the immittance matrix of a passive "two-variable n-port", which is bilateral or nonbilateral accordingly as W is symmetric or nonsymmetric. Synthesis of a two-variable reactance function is included as a special case where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = 1</tex> . Related problems are discussed in addition.

On Certain Two-Variable Generalizations of Circuit Theory, with Applications to Networks of Transmission Lines and Lumped Reactances

This paper presents two results concerning the two-variable generalizations of circuit theory concepts discussed by Ozaki and Kasami [8], and indicates applications to finite networks combining commensurate delay transmission lines with lumped reactances. The first of the two contributions to this two-variable circuit theory is a necessary and sufficient test of the two-variable reactance property of any given nonzero two-variable real rational expression. The test requires a finite number of arithmetic computations. The second is a set of necessary and sufficient conditions on the driving-point impedance function for two-variable lossless one-port realizability. The formulation is in terms of the existence of a two-variable polynomial decomposition with certain properties. (A synthesis based on such a formulation would require, at least, the development of techniques of finding the relevant polynomial decomposition, whose existence is assured by network realizability.) An application of each of these results to the transmission linelumped reactance networks, is noted. In connection with the first result, it is observed how the formal replacement of the exponential portions of the driving-point impedance expression, by expressions introducing an independent variable, defines a function of two complex variables with the two-variable reactance property. The first result thus provides a test of a property necessary for drivingpoint impedance realizability using commensurate-delay transmission lines and lumped reactances. The second result provides a realizability formulation (of the type described above) for these networks.