Hyperreal Currents and Voltages in Transfinite Networks

Armen H. Zemanian
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Armen H. Zemanian: State University of New York at Stony Brook, Department of Electrical Engineering

Chapter 6 in Graphs and Networks, 2004, pp 99-131 from Springer

Abstract: Abstract Kirchhoff’s laws can fail to hold in various, conventionally infinite or transfinite, resistive networks. This was noted in Example 5.15-2. See also [50, Sec. 1.6] or [51, Examples 5.1-6 and 5.1-7] for several other examples of this anomaly. A basic problem is that standard calculus does not always allow the order of applying limiting processes to be reversed. Moreover, some severe restrictions have to be placed on the infinite networks in order to get convergent expressions for the voltages and currents. Such, for instance, is the requirement of finite total power generated or dissipated, but other restrictions are also used for the same purpose. See, for example, [11], [12], [13], [15], [16], [18], [19], [23], [28], [39], [41], [46], [50], [51], [54].

Keywords: Shunt Resistor; Nonstandard Analysis; Node Voltage; Maximal Node; Infinite Network (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-0-8176-8178-4_6

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