On the Effective Impedance of Finite and Infinite Networks

Abstract

In this paper we deal with the notion of the effective impedance of alternating current (AC) networks consisting of resistances, coils and capacitors. Mathematically such a network is a locally finite graph whose edges are endowed with complex-valued weights depending on a complex parameter λ (by the physical meaning, λ = iω, where ω is the frequency of the AC). For finite networks, we prove some estimates of the effective impedance. Using these estimates, we show that, for infinite networks, the sequence of impedances of finite graph approximations converges in certain domains in \(\mathbb {C}\) to a holomorphic function of λ, which allows us to define the effective impedance of the infinite network.

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Acknowledgements

Firstly, the author thanks her scientific advisor, Professor Alexander Grigor’yan, for helpful comments related to this work.

Secondly, the author thanks an anonymous referee for careful reading of the paper and for his/her helpful comments and remarks which highly contributed to the clarity of the paper.

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Correspondence to Anna Muranova.

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This research was supported by IRTG 2235 Bielefeld-Seoul “Searching for the regular in the irregular: Analysis of singular and random systems”.

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Muranova, A. On the Effective Impedance of Finite and Infinite Networks. Potential Anal (2021). https://doi.org/10.1007/s11118-021-09901-8

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Keywords

  • Weighted graphs
  • Laplace operator
  • Kirchhoff’s equations
  • Electrical network
  • Effective impedance
  • Ladder network

Mathematics Subject Classification (2010)

  • 34B45
  • 39A12
  • 05C22