Driving-point Immittance Research Articles

Cascade Synthesis of a Class of Two-Element-Order Fractional Immittance Functions

The passive synthesis of the fractional-order driving-point immittance functions is of great significance for the modeling and design of fractional-order circuits. This paper studies the synthesis of a class of two-element-order fractional immittance functions based on variable substitution. First, the method of realizing the two-variable immittance functions corresponding to this kind of two-element-order fractional immittance functions is established; second, we perform the inverse substitution of variables to realize the two-element-order fractional immittance network; finally, the synthesis program of this kind of two-element-order fractional immittance functions is proposed, and specific examples are given for simulation verification. This synthesis method overcomes the disadvantages of using a multi-port transformer, and is more conducive to network analysis.

Passive Synthesis of a Class of Fractional Immittance Function Based on Multivariable Theory

Passive synthesis of fractional driving-point immittance function is one of the most important research topics in fractional circuits. The synthesis method for a class of fractional driving-point immittance function is proposed in this paper. Firstly, the immittance function of passive fractional circuit is transformed to multivariable positive-real function by proper variable substitution. Secondly, to synthesize the function, except one variable, other high-order variables are transformed to the combination of products of different one-order variables by introducing new variables. Thirdly, synthesize the new function by traditional method. At last, substitute elements inversely twice and the final circuit for fractional driving-point immittance function can be achieved. The method is verified by giving synthesis demonstration.

Properties of purely reactive Foster and non‐Foster passive networks

The mathematical concept of strongly real functions of positive and negative types is introduced to network theory for the first time. The driving-point reactance/susceptance of a pure Foster network, made up of only ideal positive inductance and capacitance elements, is a strongly real function of real frequency of positive type. As a corollary, for a pure non-Foster network made up of only ideal negative inductance and capacitance elements, the driving-point reactance/susceptance is a strongly real function of real frequency of negative type. It is shown that a condition for a purely reactive passive network to exhibit a positive or negative reactance/susceptance-frequency gradient is that the driving-point immittance should have alternating poles and zeroes lying on the real frequency axis. Finally, it is shown that either purely Foster or non-Foster networks can be constructed by combining ideal Foster and non-Foster reactive elements.

Synthesis of symmetric and asymmetric singly terminated elliptic ladder filters for multiplexing applications

An extension of the Cauer ladder development for synthesizing singly terminated filters with symmetric and asymmetric responses is presented. Basically, a driving-point immittance including reactive constant elements is carried out in such a way that provides the transmission zeros. The reactive constant elements are introduced into the synthesis for two reasons. The first is to consider the possibility of the asymmetric position of transmission zeros in the real frequency axis. The second one is to obtain canonical forms, i.e. networks with the minimum number of elements in the case of symmetrical responses. To validate the proposed method, a filter with asymmetrical response has been synthesized, comparing different topologies for its use in multiplexers. This fact is illustrated with a Ku-band elliptic response diplexer designed in H-plane rectangular waveguide. © 2009 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2009.

Explicit Solution of the Inverse Eigenvalue Problem of Real Symmetric Matrices and Its Application to Electrical Network Synthesis

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving‐point immittance functions of transformerless, common‐ground, two‐element‐kind RLC networks and in generation of their equivalent realizations.

Analysis of polarization dynamics by singularity decomposition method

The driving point immittance (impedance or admittance) function is commonly used in electrical characterization of polarized materials and interfaces. The immittance function typically attenuates following a power function dependence on frequency. This fact has been recognized as a macroscopic dynamical property manifested by strongly interacting dielectric, viscoelastic and magnetic materials and interfaces between different conducting substances. Linear interfacial polarization processes which occur at metal electrode-electrolyte interfaces have been represented by the Fractional Power Pole [FPP] function in single or multiple stages. The FPP function is referred to as the Davidson-Cole function in the dielectrics literature. A related function widely used in mathematical modeling of dielectric and viscoelastic polarization dynamics is the Cole-Cole function. The fractional power factor which parametrizes the FPP or the Davidson-Cole function has been shown earlier to equal the logarithmic ratio of the locations of the pole-zero singularities. In this paper we first review a modified form of the singularity decomposition of the FPP function accomplished within a prescribed error range. The distribution spectrum and the corresponding simulation by a cascade R-C network, as opposed to the synthesis by a ladder R-C network, are readily obtained as the next step in the simulation. The method is then applied to decompose the Cole-Cole function; the pole-zero placement of the singularity function is determined and the equivalent cascade R-C network is synthesized.

Optimisation of driving-point immittance with equiripple magnitude

A simple procedure for the design of a Chebyshev rational function is presented. Extremes of the equiripple magnitude function are defined by the accepted orthogonal polynomial, the ‘ minimum property’ of which guarantees a good group delay characteristic and small transient overshoot. A single parameter allows us to compromise between a broad bandwidth and a good group delay characteristic without changing the magnitude of the ripples. When compared with the published maximally flat approximations, the presented function proved superior in both the frequency and time domains.One of the possible applications of such rational functions, in the design of equalisers to cancel the effects of parasitic reactances, is discussed.

Driving-point immittance with equiripple magnitude

Reccurrence relations of the Ln polynomials of class L filters are derived. Using them, it is found that the ratio of the two consecutive odd order monotonic Ln polynomials form a Chebyshev rational function, whose maxima and minima coincide with the zeros of the tabulated orthogonal Legendre polynomials and their derivatives.One of the possible applications of such rational functions, in the design of equalisers for parasitic reactances, is discussed. The network has a larger gain-bandwidth product and a smaller step response overshoot than other known networks with maximally flat magnitudes.

The Normal Family of Linear Systems

The normal family of linear (time-variant) n-ports is introduced as the most natural extension of ordinary scalar driving-point immittance behaviour. The normality induces a certain symmetry and simplicity which merits due emphasis in future n-port research investigations. As an application of the concept of normality the overdamped-underdamped criterion is extended to n-ports. Also, an interesting problem of maximal power transfer is solved and the absolute energy invariant index calculated. This paper invites attention to the development of circuit synthesis procedures based on the algebraic structure of the normal family.

Determination of a Driving-Point Function from its Specified Real Part

The imaginary part of a minimum susceptance function has similar number of terms as the poles, and at all poles s i, the real and the imaginary parts are equal. Using these properties, an expression for determining the driving-point immittance from a given real part is derived.

Properties and synthesis of distributed‐parameter circuits composed of resistors and unit elements

AbstractThe circuit composed of alternatively cascaded resistors and unit elements is the most fundamental among distributed‐parameter circuits containing resistors. The necessary and sufficient conditions for realization of the driving point immittance in which element parameters satisfy given relations have been reported. As for R (resistor)‐terminated circuits composed of alternatively cascaded n resistors Rj's and characteristic impedances Zj's, there are three types of circuit configurations: 1) R1 > Z1, Rν+1 > Zν + Zν+1, R > Zn, 2) R1 = Z1, R̄ν+1 = Zν + Zν+1, R = Zn; 3) R1 < Z1, Rν+1 < Zν + Zν+1, R < Zn (ν = 1, 2, …, n‐1). For 1) the realization condition was reported. For 2) and 3), the circuit properties and synthesis methods have not yet been investigated. In this paper the driving point immittance for 2) is derived in case that the element parameters satisfy given relations. This immittance function is that for the RL or RC circuit which is degenerated with respect to Richard's variable p. A new synthesis method which differs from 1) is proposed.

Linearly variable networks— I. analysis

A linearly variable element is any passive two-terminal network element in which the immittance varies linearly with respect to an independent (of frequency) real variable, x. A definite set of fundamental passive two terminal network elements (F- elements) consisting of fixed passive elements and linearly variable elements is presented. It is shown that any network consisting of only F-elements has a driving point immittance, D(s,x), that is positive real for s complex, x real and positive real for x complex, s real. Conditions on the variable coefficients, degree and location of zeros and poles of D(s,x) are established. A method of testing whether D(s,x) is positive real for one complex and one real variable is developed. This testing is accomplished by extending the Hurwitz and Sturm test to one complex and one real variable.

Minimal realisation of 1st-order RC transfer immittances

Minimal realisations of 1st-order RC transfer immittances are derived. The proposed method of realisation works in the following two steps. First it realises an open-circuit RC voltage-transfer function obtained through the ratio of the specified transfer immittance and a suitably chosen RC driving-point immittance. Next it restores the location of the pole of the driving-point immittance chosen by scaling/adding a resistor in a suitable manner. It gives balanced realisation for nonminimum phase functions and unbalanced realisation for the remaining functions. In each case the realisation requires one, and only one, grounded capacitor except in one special case.

Stability measure for negative-resistance amplifiers

A measure of the stability of reflection-type negative-resistance amplifiers is defined as the distance between the actual driving-point immittance of the active one-port and its value that would cause instability. The measure has a simple physical interpretation and demonstrates the gain-stability trade-off.

Determination of Elements of Band-pass and Band-elimination Networks

New algorithms for determining the elements of band-pass and band-elimination lossless networks with and without resistive termination from the given driving-point immittance functions are suggested. These algorithms require less computation than the existing ones.

A multiport network synthesis using a matrix continued fraction

The matrix continued fractions of three Cauer matrix forms are applied to the synthesis of two-element kind multiport networks given their driving-point immittance matrices. The network realizations of these forms as presented in this paper are canonical.

On stability and realizability of derived FDNR networks

A proof is given for the empirical fact that resistance-capacitance-supercapacitance networks, as well as superinductanceinductance-resistance networks are unconditionally stable. This is in spite of the second degree poles at zero and at infinity. Realizability conditions are given for a function of frequency to be the driving point immittance of such networks.

Maximally flat magnitude driving-point immittance functions with equi-ripple real part

A class of maximally flat magnitude driving-point immittance functions whose real part on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j\omega</tex> axis has on equi-ripple variation is identified, and their application in the design of equalizers is indicated. These functions exhibit a better frequency response over a limited band than other known designs and lend themselves to convenient realization in the form of equal-element ladder networks.

Necessary and sufficient conditions for the realizability of biquadratic functions

The necessary and sufficient condition for the positiverealness of a general biquadratic function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> is presented. It is shown that for each given pair of conjugate-complex poles of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> with negative-real parts, the zeros of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> are graphically restricted in a realizability region, which is an open region bounded by two curves. One of these curves is the locus of the zeros for which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> is a minimum positive-real function. A theorem is given stating the realizability conditions for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> as a driving-point immittance with passive elements.

Normal Transmission-Line Networks and Their Lumped LC Presentation

A previous result is extended and a new presentation is described for Richards' transmission-line (TL) networks. The new presentation is in a lumped LC form; therefore, classical analysis and synthesis techniques are directly applicable. The class of networks for which this presentation is possible is studied in detail. This class of networks, called normal TL networks, is found to be a necessary and sufficient condition for driving-point immittance functions to be rational positive real functions of /spl lambda/ =tanh /spl tau//spl rho/. The effectiveness of this representation is further demonstrated by applying it to graph-transformation analysis method. It simplifies the procedure considerably and reveals additional physical insight into the TL network.