General Topological Formulas for Linear Network Functions

Coates, C. L.

doi:10.1109/tct.1958.1086422

Cited by 40 publications

(6 citation statements)

References 3 publications

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“…Coates' flow graphs, 3,4 closely related to Mason's flow graphs 5,6 and to transitor graphs (see Starzyk 2 pp. 16 and 27), as well as conjugated graph pair (two-graph) approaches like current-voltage 7 and nullator-norator graphs.…”

Section: And 17)mentioning

confidence: 99%

“…There are a few directed graph representations of linear circuits, which include unistor graphs, 1 dispersor graphs (see Starzyk 2 pp. 16 and 17) Coates' flow graphs, 3,4 closely related to Mason's flow graphs 5,6 and to transitor graphs (see Starzyk 2 pp. 16 and 27), as well as conjugated graph pair (two‐graph) approaches like current‐voltage 7 and nullator‐norator graphs 8,9 .…”

Section: Introductionmentioning

confidence: 99%

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A unified approach to autonomous and nonautonomous models of circuit elements

Nałęcz

2020

Circuit Theory & Apps

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This paper proposes a unified approach to both autonomous models of circuit elements and nonautonomous models which contain nullator-norator pairs. The unification is based on voltage-controlled current sources as the only kind of element utilized in autonomous models. It is shown that nonautonomous models are special (limit) cases of autonomous ones as some transadmittances of these sources tend from one to infinity. A systematic approach to the creation of models which consist solely of voltage-controlled current sources and are based on multiport equations is presented and illustrated by examples. By replacing the sources with their unistor models and then by application of the known transformations to the resulting unistor graph, autonomous and nonautonomous unistor models of several basic circuit elements are obtained. Some of these models seem to be new.

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Section: And 17)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A unified approach to autonomous and nonautonomous models of circuit elements

Nałęcz

2020

Circuit Theory & Apps

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“…Although symbolic determinants can be evaluated, the process is slow and complicated. Topological formulas [Si, [9], [lo], provide an effective solution to the problem.…”

Section: Network Topology Programsmentioning

confidence: 99%

“…. (9) where T is the increment of time, x(T) is the state vector which contains transient response, and eAT is the transition matrix. Since eAT is expanded in an infinite series, the truncation error of the matrix series and the roundoff error do propagate with increasing time.…”

Section: Time and Frequency Domain Analysismentioning

confidence: 99%

Network analysis by digital computer

Kuo¹

1966

Proc. IEEE

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This article gives a brief survey of various methods for network analysis by digital computer. Topics discussed include methods and programs for ladder networks, mesh and nodal analysis, network topology, electronic circuit analysis, statevariable analysis, n-port hybrid matrix analysis, and nonlinear circuit analysis. Also given is a brief discussion concerning algorithms for inverse Laplace transformation, and methods for obtaining magnitude, phase and delay responses in the frequency domain.KUO: NETIVORK ASALYSIS BY COMPUTER 829

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“…[30] We emphasise, therefore, that, in the present work, there is certainly no intention on our part to attempt to supplant previous thorough and comprehensive treatments -such as, for example, the formal accounts of Bryant, [31,32] Slepian, [33] Chen [34] and Bondy & Murty [35] nor the extensive coverage in the electrical engineering literature, such as Refs. [36][37][38][39][40][41][42]. Rather, our emphasis here will be on giving a simple, easily comprehensible, exposition of the essence of the matter -illustrated, for convenience, by extended consideration of just one specific example -and on connecting Kirchhoff's work with late 20 th -century extensions that enable the spanning trees of a planar graph to be counted by considering its inner dual, [21,[43][44][45][46] as well as with 21 st -century developments concerning spanning-tree counts in graphs that may, in general, be non-planar.…”

Section: Introductionmentioning

confidence: 99%

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

Kirby¹,

Mallion²,

Pollak³

et al. 2016

Croat. Chem. Acta

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What Kirchhoff actually did concerning spanning trees in the course of his classic paper in the 1847 Annalen der Physik und Chemie has, to some extent, long been shrouded in myth in the literature of Graph Theory and Mathematical Chemistry. In this review, Kirchhoff's manipulation of the equations that arise from application of his two celebrated Laws of electrical circuits -formulated in the middle of the 19 th century -is related to 20 th -and 21 st -century work on the enumeration of spanning trees. It is shown that matrices encountered in an analysis of what Kirchhoff really did include (a) the Kirchhoff (Laplacian, Admittance) matrix, K, that features in the well-known Matrix Tree Theorem, (b) the matrix G encountered in the theorem of Gutman, Mallion & Essam (1983), applicable only to planar graphs, and (c) the analogous matrix M that arises in the Cycle Theorem (Kirby et al. 2004), a theorem that applies to graphs of any genus. It is concluded that Kirchhoff himself was not interested in counting spanning trees, and, accordingly, he did not explicitly do so. Nevertheless, it is shown how the modulus of the determinant of a certain matrix (here denoted by the label C') -associated with the linear equations arising from application of Kirchhoff's two Laws -is numerically equal to the number of spanning trees in the graph representing the connectivity of the electrical network being studied. Kirchhoff did, however, invoke the concept of spanning trees, introducing them in a complementary fashion by referring to the chords that must be removed from the original graph in order to form such trees. It is further emphasised that, in choosing the cycles in the network being studied, around which to apply his circuit Law, Kirchhoff explicitly selected what would now be called a 'Fundamental System of Cycles'.

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General Topological Formulas for Linear Network Functions

Cited by 40 publications

References 3 publications

A unified approach to autonomous and nonautonomous models of circuit elements

A unified approach to autonomous and nonautonomous models of circuit elements

Network analysis by digital computer

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

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