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.A. Morelli was supported by the MathWorks studentship - a Cambridge University Trust fund

On the Synthesis of Passive Networks without Transformers

This thesis is concerned with the synthesis of passive networks, motivated by the recent invention of a new mechanical component, the inerter, which establishes a direct analogy between mechanical and electrical networks. We investigate the minimum numbers of inductors, capacitors and resistors required to synthesise a given impedance, with a particular focus on transformerless network synthesis. The conclusions of this thesis are relevant to the design of compact and cost-effective mechanical and electrical networks for a broad range of applications. In Part 1, we unify the Laplace-domain and phasor approach to the analysis of transformerless networks, using the framework of the behavioural approach. We show that the autonomous part of any driving-point trajectory of a transformerless network decays to zero as time passes. We then consider the trajectories of a transformerless network, which describe the permissible currents and voltages in the elements and at the driving-point terminals. We show that the autonomous part of any trajectory of a transformerless network is bounded into the future, but need not decay to zero. We then show that the value of the network's impedance at a particular point in the closed right half plane can be determined by finding a special type of network trajectory. In Part 2, we establish lower bounds on the numbers of inductors and capacitors required to realise a given impedance. These lower bounds are expressed in terms of the extended Cauchy index for the impedance, a property defined in that part. Explicit algebraic conditions are also stated in terms of a Sylvester and a Bezoutian matrix. The lower bounds are generalised to multi-port networks. Also, a connection is established with continued fraction expansions, with implications for network synthesis. In Part 3, we first present four procedures for the realisation of a general impedance with a transformerless network. These include two known procedures, the Bott-Duffin procedure and the Reza-Pantell-Fialkow-Gerst simplification, and two new procedures. We then show that the networks produced by the Bott-Duffin procedure, and one of our new alternatives, contain the least possible number of reactive elements (inductors and capacitors) and resistors, for the realisation of a certain type of impedance (called a biquadratic minimum function), among all series-parallel networks. Moreover, we show that these procedures produce the only series-parallel networks which contain exactly six reactive elements and two resistors and realise a biquadratic minimum function. We further show that the networks produced by the Reza-Pantell-Fialkow-Gerst simplification, and the second of our new alternatives, contain the least possible number of reactive elements and resistors for the realisation of almost all biquadratic minimum functions among the class of transformerless networks. We group the networks obtained by these two procedures into two quartets, and we show that these are the only quartets of transformerless networks which contain exactly five reactive elements and two resistors and realise all of the biquadratic minimum functions. Finally, we investigate the minimum number of reactive elements required to realise certain impedances, of greater complexity than the biquadratic minimum function, with series-parallel networks.Funded in part by the EPSRC Programme Grant on Control For Energy and Sustainabilit

On a concept of genericity for RLC networks

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordA 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 nMathWork

The pseudo McMillan degree of implicit transfer functions of RLC networks

We study the structure of a given RLC network without sources. Since the McMillan degree of the implicit transfer function is not a suitable measure of complexity of the network, we introduce the pseudo McMillan degree to overcome these shortcomings.Using modified nodal analysis models, which are linked directly to the natural network topology, we shaw that the pseudo-McMillan degree equals the sum of the number of capacitors and inductors minus the number of fundamental loops of capacitors and fundamental cutsets of inductors; this is the number of independent dynamic elements in the network. Exploiting this representation we derive a minimal realization of the given RLC network, that is one where the number of independent differential equations equals the pseudo McMillan degree

Behavioral realizations using companion matrices and the smith form

This is the author accepted manuscript. The final version is available from Society for Industrial and Applied Mathematics via the DOI in this record.Classical procedures for the realization of transfer functions are unable to represent uncontrollable behaviors. In this paper, we use companion matrices and the Smith form to derive explicit observable realizations for a general (not necessarily controllable) linear time-invariant be- havior. We then exploit the properties of companion matrices to efficiently compute trajectories, and the solutions to Lyapunov equations, for the realizations obtained. The results are motivated by the important role played by uncontrollable behaviors in the context of physical systems such as passive electrical and mechanical networks

Synthesis of electrical and mechanical networks of restricted complexity

This dissertation is concerned with the synthesis of linear passive electrical and mechanical networks. The main objective is to gain a better understanding of minimal realisations within the simplest non-trivial class of networks of restricted complexity--the networks of the so-called "Ladenheim catalogue"--and thence establish more general results in the field of passive network synthesis. Practical motivation for this work stems from the recent invention of the inerter mechanical device, which completes the analogy between electrical and mechanical networks. A full derivation of the Ladenheim catalogue is first presented, i.e. the set of all electrical networks with at most two energy storage elements (inductors or capacitors) and at most three resistors. Formal classification tools are introduced, which greatly simplify the task of analysing the networks in the catalogue and help make the procedure as systematic as possible. Realisability conditions are thus derived for all the networks in the catalogue, i.e. a rigorous characterisation of the behaviours which are physically realisable by such networks. This allows the structure within the catalogue to be revealed and a number of outstanding questions to be settled, e.g. regarding the network equivalences which exist within the catalogue. A new definition of "generic" network is introduced, that is a network which fully exploits the degrees of freedom offered by the number of elements in the network itself. It is then formally proven that all the networks in the Ladenheim catalogue are generic, and that they form the complete set of generic electrical networks with at most two energy storage elements. Finally, a necessary and sufficient condition is provided to efficiently test the genericity of any given network, and it is further shown that any positive-real function can be realised by a generic network.The MathWorks Studentship in Engineerin

The inerter: a retrospective

The paper provides an introduction and overview of the inerter concept and device. Careful attention is given to the distinction between the inerter as an ideal modelling element and devices that approximate the ideal behaviour. The background is given to the formal definition of the inerter as a mechanical one-port with terminal forces proportional to the relative acceleration between them. Four major methods of construction are described and modelled. The discussion focuses particularly on: the notion of terminals; the distinction between a device and an effect; sign reversals; back-driving in geared systems; the conceptual aspects of the modelling step for inerter embodiments; the problem of reverse engineering to discover a purpose. The paper includes an analysis and discussion of the rotational inerter. A brief review of the ideas of passive network synthesis that led to the inerter concept are provided. A discussion and analysis is given on several examples of integrated mechanical devices. The article concludes with an imaginary dialogue between the author and an interlocutor on the understanding and purpose of the inerter

Behavioral realizations using companion matrices and the smith form

Classical procedures for the realization of transfer functions are unable to represent uncontrollable behaviors. In this paper, we use companion matrices and the Smith form to derive explicit observable realizations for a general (not necessarily controllable) linear time-invariant behavior. We then exploit the properties of companion matrices to efficiently compute trajectories, and the solutions to Lyapunov equations, for the realizations obtained. The results are motivated by the important role played by uncontrollable behaviors in the context of physical systems such as passive electrical and mechanical networks.This is the author accepted manuscript. The final version is available from SIAM via https://doi.org/ 10.1137/14099191

Implicit network descriptions of RLC networks and the problem of re-engineering

The thesis deals with aspects of Systems Re-engineering specialised to the case of passive electrical networks. Re-engineering is a problem different from traditional control problems and this emerges when it is realised that the systems designed in the past cannot perform according to the new performance requirements and such performance cannot be improved by traditional control activities. Re-engineering implies that we intervene in early stages of system design involving sub-processes, values of physical elements, interconnection topology, selection of systems of inputs and outputs and of course retuning of control structures. This is a very challenging problem which has not been addressed before in a systematic way and needs fundamental new thinking, based on understanding of structure evolution during the stages of integrated design. A major challenge in the study of this problem is to have a system representation that allows study of evolution of system properties as well as structural invariants. For linear systems the traditional system representations, such as transfer functions, state space models and polynomial type models do not provide a suitable framework for study structure and property evolutions, since for every change we need to compute again these models and the transformations we have used do not appear in an explicit form in such models. It is for this reason, for a general system, such system representations are not suitable for study of system representations on re-engineering. It has been recognized that for the special family of systems defined by the passive electrical networks (RLC), there exists a representation introduced by the loop/ nodal analysis, expressed by the impedance/admittance integral-differential models, which have the property of re-engineering transformations of the following type: 1. Changing the values or possible nature of existing elements without changing the network topology, 2. Modifying the network topology without changing network cardinality, that is number of independent loops or nodes, 3. Augmenting or reducing the network by addition or deletion of sub-networks, 4. Combination of all the above transformations. These kinds of transformations may be represented as perturbations on the original impedance/admittance models. The above indicates that impedance/admittance integral-differential models, which from now on will be referred to as Implicit Network Descriptions is the natural vehicle for studying re-engineering on electrical networks. Although issues related to realisation of impedance/admittance transfer functions within RLC topologies, has been the topic of classical network synthesis, the system aspects of such descriptions have not been properly considered. Addressing problems of network re-engineering requires the development of the fundamental system aspects of such new descriptions in terms of McMillan degree, regularity and a number of other properties. Certain problems of evolution (of system properties) are linked to Frequency Assignment, as far as natural frequencies under re-engineering and this requires use of techniques developed within control theory for Frequency Assignment Problems