Inductor Cutsets Research Articles

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 network transfer function is not a suitable measure for the 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 show 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 involved (independent) differential equations equals the pseudo-McMillan degree.

Generalizations of Blakesley's Source Shift Theorem

Abstract. It is shown that the well-known Voltage Source Shift Theorem due to Blakesley and its dual version, the Current Source Shift Theorem as well as the rules for the transformation of networks with loops of capacitors or cut sets of inductors into networks without such loops or cutsets, resp., and the relationships between capacitance coefficients and partial capacitors are special cases of general theorems on the terminal behavior of networks. The proof of these theorems is based on the theory of terminal behavior of networks. For these proofs we do not need the substitution theorem with its strong uniqueness assumptions. This fact is an essential advantage in comparison to the original proof given by Chua and Green.

Birkhoffian formulation of the dynamics of LC circuits

We present a formulation of general nonlinear LC circuits within the framework of Birkhoffian dynamical systems on manifolds. We develop a systematic procedure which allows, under rather mild non-degeneracy conditions, to write the governing equations for the mathematical description of the dynamics of an LC circuit as a Birkhoffian differential system. In order to illustrate the advantages of this approach compared to known Lagrangian or Hamiltonian approaches we discuss a number of specific examples. In particular, the Birkhoffian approach includes networks which contain closed loops formed by capacitors, as well as inductor cutsets. We also extend our approach to the case of networks which contain independent voltage sources as well as independent current sources. Also, we derive a general balance law for an associated "energy function".

Consistent initial conditions of linear switched networks

The analysis of switched networks encounters a major difficulty due to the discontinuity of network variables and the presence of impulsive voltages and currents at the switching instants. The problem of obtaining consistent initial conditions for linear networks with ideal switched is addressed. The method used is based on numerical Laplace transform theory and is applicable to any circuit configuration. Capacitor loops, inductor cutsets, ideal switches with zero impedance or admittance, and impulsive sources do not present problems. Although initial conditions are normally required for time-domain analysis, an even more important application is in the frequency-domain analysis of general periodically switched linear networks. The algorithms are easily implemented in a computer program. >

On equivalent dynamic networks: Elimination of capacitor loops and inductor cutsets

Equivalence of two (n + 1) -terminal dynamic networks is defined. An algorithm is presented which shows how networks with loops of capacitors and/or cutsets of inductors can be transformed into equivalent networks without such loops and cutsets. Implications of this result are discussed.

Graph-theoretic properties of dynamic nonlinear networks

Graph-theoretic concepts are used to deduce properties of nonlinear networks and properties of nonlinear resistive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports. The basic result is that if the ports of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port form no cutsets (resp., loops), then each port voltage (resp., port current) is a linear function of the internal element voltages (resp., currents) only; i.e., no other external port voltage (resp., port current) is involved. This result is very general in the sense that it is independent of the constitutive relations of the internal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port elements. It is also a rather subtle result because it forms the basis of a large number of new network and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port theorems. For example, in examining the closure properties of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ports, this result is used to show, among other things, that if the ports of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port form neither loops nor cutsets, and if the internal resistors are strictly passive, or strictly increasing, then the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port is also strictly passive, or strictly increasing. Moreover, many of these conclusions remain valid when the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port contains independent voltage and current sources. The main result is extended to a network containing capacitors, inductors, resistors, and sources. Here, graph-theoretic conditions are given such that the voltage and current waveforms of the capacitors and inductors are functions of the resistor and source voltage and current waveforms. Dynamic nonlinear networks containing capacitors, inductors, resistors, and sources such that there are loops of capacitors, or cutsets of inductors are shown to be equivalent to networks without such loops or cutsets. Explicit analytical expressions are given for specifying the constitutive relations of the elements of the equivalent circuit. This result allows the generalization of many previous results in nonlinear networks which exclude capacitor loops and inductor cutsets.