Positive-real Impedance Research Articles

One Port Impedance Quantization For a Class of Annihilation Operator Linear Quantum Systems

This paper provides a procedure for building a one port impedance quantization involving annihilation operators only for a class of linear quantum systems having a positive real impedance transfer function matrix. Based on the positive real properties of these quantum systems, it is shown that it is possible to use the Brune algorithm in order to find an electrical circuit that can physically implement these quantum systems. This theory, illustrated for one-port circuits may be useful for the implementation of superconducting microwave circuits used in quantum filters found in the field of quantum computing.

On a concept of genericity for RLC networks

A recent definition of genericity for resistor–inductor–capacitor (RLC) networks is that the realisability set of the network has dimension one more than the number of elements in the network. We prove that such networks are minimal in the sense that it is not possible to realise a set of dimension n with fewer than n−1 elements. We provide an easily testable necessary and sufficient condition for genericity in terms of the derivative of the mapping from element values to impedance parameters, which is illustrated by several examples. We show that the number of resistors in a generic RLC network cannot exceed k+1 where k is the order of the impedance. With an example, we show that an impedance function of lower order than the number of reactive elements in the network need not imply that the network is non-generic. We prove that a network with a non-generic subnetwork is itself non-generic. Finally we show that any positive-real impedance can be realised by a generic network. In particular we show that sub-networks that are used in the important Bott–Duffin synthesis method are in fact generic.

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u (Studia Math., 88 (1988)).

Use of Inerter-based Vibration Absorbers for Suppressing Multiple Cable Modes

A common approach to the mitigation of vibrations of structural cables, e.g. on cable-stayed bridges, is to install a viscous damper transverse to the cable axis quite close to one end. The damping coefficient can be optimized for suppressing vibrations in one cable mode but the damper is then sub-optimal for other modes. This paper proposes the use of a passive inerter-based vibration absorber for suppressing multiple unwanted cable vibration modes. The inerter has the property that the force between its two terminals is proportional to their relative acceleration. A previous study has shown that inerter-based vibration absorber configurations can provide greater modal damping ratios than a viscous damper alone for vibration modes around the first undamped natural frequency. In this study, a finite-element model of a taut cable, together with a generic absorber is built. The absorber is located close to one end of the cable and represented by a general positive-real impedance function. The absorber layouts for maximizing the modal damping ratios over lower-frequency modes while perverting deterioration those for higher-frequency modes are identified. It will be shown that, compared with traditional viscous dampers, the proposed inerter-based vibration absorbers can give enhanced damping performance over multiple modes.

X-parameters ® -based closed-form expressions for evaluating power-dependent fundamental negative and positive real impedance boundaries in oscillator design

In this study, an analytical design tool, applicable under large-signal operation, based on X-parameters closed-form expressions, to determine the large-signal boundary between the negative (energy delivered) and positive (energy absorbing) real input impedance regions of the non-linear oscillator block, is presented to be used prior to and/or during non-linear oscillator circuit design. The consistency of the approach is demonstrated since in small-signal operation; the proposed expressions for the gamma equal to unity loci converges to the classic expression, generally referred to as the input or output stability circles. At large-signal levels, while we are not claiming that this formulation provides by itself similar general stability guidance, it can provide relevant information for oscillator design. The validity and usefulness of the proposed expressions as a real-time design aid, thus either minimising or avoiding the necessity for complex and time-consuming harmonic balance simulations, is demonstrated in the study.

Synthesis of mechanical networks: the inerter

The paper is concerned with the problem of synthesis of (passive) mechanical one-port networks. One of the main contributions of the paper is the introduction of a device, which win be called the inerter, which is the true network dual of the spring. This contrasts with the mass element which, by definition, always has one terminal connected to ground. The inerter allows electrical circuits to be translated over to mechanical ones in a completely analogous way. The inerter need not have large mass. This allows any arbitrary positive-real impedance to be synthesized mechanically using physical components which may be assumed to have small mass compared to other structures to which they may be attached. The possible application of the inerter is considered to a vibration absorption problem, a suspension strut design, and as a simulated.

A new study of the problem of compatible impedances

It is a classical result, known as Darlington's theorem, that every rational positive-real function z(p) is realizable as the input impedance of a lumped reciprocal reactance two-port tuner Nt closed at the far end on 1 Ω. The theorem is evidently false if the 1 Ω termination is replaced by some prescribed non-constant positive-real impedance z1(p) . Any z(p) synthesizable in this more restrictive manner is said to be compatible with z1(p) and we write z∼z1 to indicate the correspondence. The determination of necessary and sufficient conditions for the validity of z∼z1 is the problem of compatible impedances. Of the four better-known network treatments, only that of Schoeffler IRE Trans. Circuit Theory, CT-8, 131–137 (1961) is completely correct, although severely restricted in scope. In particular, the remaining three contain a common error which appears to have propagated because a constraint on the ratio of even parts ze(p)/z1e(p) derived by Schoeffler is unnecessary if z(p) is not minimum-reactance. The main theorems of Wohlers (IEEE Trans. Circuit Theory, CT-12, 528–535 (1965)) and Satyanaryana and Chen (J. Franklin Inst., 309, 267–280 (1980)) are very similar in structure to our Theorem 3 but considerably more complex and do not provide a sufficiently explicit description of the associated tuner Nt. In fact (Theorem 1), with the exception of a physically irrelevant degeneracy (which is easily detected), Nt, when it exists, must possess an impedance matrix Z(p). Moreover, the latter can be effectively parametrized in terms of z(p), z1(p) and a regular-allpass b(p) found as the solution of a standard interpolation problem of the Nevalinna–Pick type. Three fully worked examples clarify the theory and also illustrate many of the numerical steps. © 1997 by John Wiley & Sons, Ltd.

On consistency of the complex normalized scattering matrices of multi-port networks

Given an n-port network, it is shown that its scattering matrix, normalized to n non-Foster positive real impedances, is closely related to the scattering matrix of the augmented network normalized to n unit resistances within all-pass functions.

Broad-band impedance matching of the RLC generator and load

This paper presents a simple theory for the design of a broad-band impedance matching network which, when operating between a given RLC generator z 1(s) and an RLC load z 2(s), yields a Chebyshev low-pass or band-pass response. Using realizability conditions of double frequency-dependent terminations, we present a new theorem which states that if there exists a second-order all-pass function such that the resulting reflection coefficients are bounded-real and satisfy Youla's conditions at each zero of transmission of z 1(s) and z 2(s), a ladder equalizer exists. A method to realize such a ladder is described. After choosing the order n of the transfer function not smaller than the total number of the energy storage elements of z 1(s) and z 2(s), the result can easily be extended to any all-pole low-pass response for any non-Foster positive-real impedances z 1(s) and z 2(s) except for the case where z 1(s) and z 2(s) have a common finite zero on the jω-axis.

A theorem on complex-normalized reflection coefficient and its application (multiport networks)

The authors examine the scattering matrix of a lossless reciprocal n-port network terminated in n non-Foster positive-real impedances and its augmented n-port network. A theorem relating the complex-normalized scattering matrix of the n-port network to that of its augmented n-port normalizing to the n 1- Omega resistors is given, from which the scattering matrix of a diplexer or a multiplexer is obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Realizability of compatible impedances using transformerless ladder two-port networks

This paper studies the compatibility of two non-Foster positive-real impedances using a lossless reciprocal transformerless ladder two-port network. Illustrative examples are presented.

On a complex normalized scattering matrix and its application to broadband matching of multiport networks

The necessary and sufficient conditions that an n x n matrix be the scattering matrix of a lumped, lossless n-port normalizing to n non-Foster positive—real impedances are presented in matrix form in terms of coefficients of the Laurent series of the parameters. Also, a method is introduced for constructing a generalized scattering matrix of a lossless reciprocal n-port equalizer from the preassigned transducer power-gain characteristics and a set of n passive terminations. The significance of this work is that it provides a means for testing the realizability of an n x n scattering matrix by dealing with the coefficients of the Laurent series expansions of the parameters. An example of a three-port network is given to show the procedure for constructing a scattering matrix possessing the para-unitary property, and achieving the preassigned transducer power-gain characteristics when it is terminated in the given impedances.

Broadband matching of multi-port networks terminated in frequency-dependent loads

This paper extends the problem of double-port matching of two frequencydependent non-Foster positive real impedances to that of multi-port matching. Givenn non-Foster positive real impedances and the preassigned transducer power-gain characteristics, necessary and sufficient conditions are given for the existence of a lossless, reciprocaln-port equalizer terminated in these impedances at the ports satisfying the prescribed transducer power-gain characteristics.

Nonuniform Semi-Infinite Grounded Grids

Semi-infinite resistive grounded grids are countably infinite electrical networks that arise from the discretization of the partial differential equation governing the minority-carrier density in a doped semiconductor. If the doping varies with depth from the surface of the semiconductor, the grid’s resistances also vary with distance from the inputs to the grid. This nonuniformity prevents the use of the characteristic-resistance method for determining currents and voltages. A computational method for making such a determination is presented herein. It is based upon the theory of infinite continued fractions whose entries are positive operators on a Hilbert space. It is also know that the solution given by the method is precisely that solution for which the power dissipated in the network is finite. Finally, the method is extended to RLC networks, and this allows the computation of transient responses in semi-infinite grounded grids of positive-real impedances.

Partial derivative properties of multivariable cascaded networks

Some partial derivative properties of the impedance function of a resistively-terminated network which is a cascade of m lossless two-ports of variables p 1 to p m dre established. Those cases where some of the lossless two-ports are lowpass or highpass ladder networks with all of their transmission zeros at the origin or at infinity are also examined. Necessary and sufficient conditions involving partial derivatives and under which an m-variable reactance or positive real function can be realized as the impedance function of a p i-variable lowpass or highpass ladder network, with all of its transmission zeros at p i = 0 or p i = ∞, and terminated in a reactance or positive real impedance function in the other variables are derived.

Minimal gyrator lossless synthesis

A synthesis procedure is presented for a lossless positive real impedance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> . The procedure, based on a choice of a suitable state-space representation for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(s)</tex> , uses the minimum possible number of reactive elements and, simultaneously, the minimum possible number of gyrators.

Impedance synthesis via state-space techniques

The paper deals with the synthesis of passive networks and relies on general systems theory and control concepts. The network-synthesis problem is first interpreted in state-variable terminology, solved as a control problem and the solution is then translated back into network-theory terms. After a review of state-space formulations, an algebraic theory of synthesis is developed, beginning with a minimal state-space realisation, perhaps obtained through control-theory procedures, from which a synthesis of rational positive-real impedance matrices is obtained through a transformation on the state. The method rests upon an appropriate basis change, in the state-space, obtained by factoring the Pmatrix of the control-theory positive real lemma. The minimum number of resistors and reactive elements is used. The paper also serves as a review of the ‘state-of-the-art’ for formal nport synthesis; the results lead to new methods of attacking open problems, as well as to methods of analysis and synthesis via digital computers.

On the construction of a positive real impedance from a given even part

On the construction of a positive real impedance from a given even part

A new approach to active RC network synthesis

This paper presents several lemmas which are of importance in the synthesis of one-port and n-port active RC networks. Syntheses of such networks are viewed from the equivalent active LC network point of view by making use of RC-LC transformation. The first part deals with one-ports. Replacement of the resistance in the Darlington realization of a positive real impedance by an appropriate negative LC impedance results in the odd part of the positive real function at the input. An inverse LC-RC transformation yields the decomposition, partitioning, and consequently the synthesis, of active RC impedances. Admittances are treated by a dual procedure. This approach is extended in the second part to the synthesis of active RC immittance matrices of order n. It is shown that in the latter case, the number of active elements needed is at most equal to n.

Polynomial Decomposition in Active Network Synthesis

This paper deals with the use of polynomial decomposition as a basis for the synthesis of active RC networks. The first part briefly reviews the formation and important properties of the Horowitz decomposition. Next, the relationships between the decompositions of two polynomials are presented in the form of three basic theorems. Significant in the development of these theorems is the use made of the theory of positive real functions. The active network function is treated as the even or the odd part of a positive real impedance, and hence the wealth of information available on positive real functions is brought to bear on the problem of active synthesis. The tie between active and passive synthesis is then illustrated by means of a method of active driving point synthesis, which closely parallels the synthesis of lossless terminated passive networks.