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The Pseudo-McMillan Degree of Implicit Transfer Functions of RLC Networks

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Abstract

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.

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Notes

  1. The node potential \(\eta _i\) expresses the voltage between the ith node in the network graph and the ground node.

  2. \(G(s)\in {\mathbb {R}}(s)^{n\times n}\) is proper, if \(\lim _{s\rightarrow \infty } G(s) \in {\mathbb {R}}^{n\times n}\) exists.

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Correspondence to Thomas Berger.

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This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft) via the grant BE 6263/1-1.

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Berger, T., Karcanias, N. & Livada, M. The Pseudo-McMillan Degree of Implicit Transfer Functions of RLC Networks. Circuits Syst Signal Process 38, 967–985 (2019). https://doi.org/10.1007/s00034-018-0921-6

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  • DOI: https://doi.org/10.1007/s00034-018-0921-6

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