Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 45 min ago
PreprintCC BY 4.0arXiv

Representation of arbitrary rational functions and polynomials by non-Foster resistor-capacitor-inductor networks

  • Andrew Mackay
Open at sourcePDFDOICite

Abstract

The Bott-Duffin theorem shows that any arbitrary positive real rational function of finite order, representing a passive network, can be represented by a finite number of positive value resistors, capacitors and inductors. It is shown here that if the positivity restriction is lifted, any rational function or polynomial of any order may be represented by a network of (possibly negative) inductors, capacitors and resistors. When there are no repeated roots this is a sum of simple sub-circuits each of which contains no more than four components. When there are repeated roots a more complex circuit is required, here featuring iterated Wheatstone bridges of special form.

Related

  1. Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions
    Hengfeng Liu, Sihem Mesnager, Chunming Tang +1
    arXivPreprintCC BY 4.0Full textGeneral mathematicsarXiv
  2. The average number of rational preperiodic points of polynomials over ℚ
    Jungin Lee
    arXivPreprintarXiv licenceFull textAlgebra & number theoryDynamical systemsarXiv
  3. Exact barycentric rational interpolation of analytic functions with an application to the Bessel functions
    Brett R. Green
    arXivPreprintarXiv licenceFull textAnalysis & PDEsNumerical analysis & scientific computingarXiv
  4. Rational and integral values of rational functions at rational points
    Pietro Corvaja, Umberto Zannier, appendix by D. Masser
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Real rational Herglotz-Nevanlinna functions in two variables and a class of extremals
    Ludvig Fagrell, Annemarie Luger
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{mackay2026representation,
  title = {{Representation of arbitrary rational functions and polynomials by non-Foster resistor-capacitor-inductor networks}},
  author = {Andrew Mackay},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06047},
  eprint = {2609.06047},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06047},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Representation of arbitrary rational functions and polynomials by non-Foster resistor-capacitor-inductor networks
AU  - Andrew Mackay
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06047
UR  - https://arxiv.org/abs/2609.06047
AB  - The Bott-Duffin theorem shows that any arbitrary positive real rational function of finite order, representing a passive network, can be represented by a finite number of positive value resistors, capacitors and inductors. It is shown here that if the positivity restriction is lifted, any rational function or polynomial of any order may be represented by a network of (possibly negative) inductors, capacitors and resistors. When there are no repeated roots this is a sum of simple sub-circuits each of which contains no more than four components. When there are repeated roots a more complex circuit is required, here featuring iterated Wheatstone bridges of special form.
ER  -