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 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

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

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 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