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Foundations of network theory

Albrecht Reibiger (Faculty of Electrical and Computer Engineering, Institute of Circuits and Systems, University of Technology Dresden, Dresden, Germany)



Different axiomatizations of network theory are considered. Kirchhoff networks are regarded with priority. Multipole and multiport networks are introduced as alternative variants. Additionally, Minty and Paynter networks, which are always dualizable are discussed briefly. The latter are special cases of Kirchhoff as well as of multiport networks. This paper seeks to discuss these issues.


The paper develops network theory inside of set theory, i.e. networks, multipoles, etc. are defined as objects of set theory. As such objects we use preferably ordered pairs, ordered triples, etc. The objects of network theory are then separated from the class of all these set theoretical objects by means of some defining conditions. These conditions are the axioms of our approaches to network theory.


It is shown that all presented variants of axiomatizations can be developed on the basis of a uniform representation for the time functions for voltages and currents. All these variants allow interdisciplinary applications of network theory and they can be generalized to multidimensional networks. An interesting byproduct is the relationship between multiport networks, networks in Belevitch normal form, Paynter networks, and bond graphs.


For applications it is essential that Kirchhoff and multipole networks are with respect to their modeling capability of equal value. But from the foundational point of view the multipole terminology has a number of crucial disadvantages compared with that based on Kirchhoff networks. This fact is important both for the conception of circuit simulation software packages and for the development of basic circuit theory curricula.



Reibiger, A. (2011), "Foundations of network theory", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 30 No. 4, pp. 1319-1332.



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