A Test for the Stability of Networks

Agnieszka Rowinska-Schwarzweller; Christoph Schwarzweller

Formalized Mathematics (2013)

  • Volume: 21, Issue: 1, page 47-53
  • ISSN: 1426-2630

Abstract

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A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].

How to cite

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Agnieszka Rowinska-Schwarzweller, and Christoph Schwarzweller. "A Test for the Stability of Networks." Formalized Mathematics 21.1 (2013): 47-53. <http://eudml.org/doc/267528>.

@article{AgnieszkaRowinska2013,
abstract = {A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].},
author = {Agnieszka Rowinska-Schwarzweller, Christoph Schwarzweller},
journal = {Formalized Mathematics},
language = {eng},
number = {1},
pages = {47-53},
title = {A Test for the Stability of Networks},
url = {http://eudml.org/doc/267528},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Agnieszka Rowinska-Schwarzweller
AU - Christoph Schwarzweller
TI - A Test for the Stability of Networks
JO - Formalized Mathematics
PY - 2013
VL - 21
IS - 1
SP - 47
EP - 53
AB - A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].
LA - eng
UR - http://eudml.org/doc/267528
ER -

References

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  1. [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  2. [2] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990. 
  3. [3] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990. 
  4. [4] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990. 
  5. [5] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  6. [6] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  7. [7] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. 
  8. [8] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990. 
  9. [9] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990. 
  10. [10] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001. 
  11. [11] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001. 
  12. [12] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001. 
  13. [13] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001. 
  14. [14] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990. 
  15. [15] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991. 
  16. [16] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997. 
  17. [17] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001. 
  18. [18] Christoph Schwarzweller. Introduction to rational functions. Formalized Mathematics, 20 (2):181-191, 2012. doi:10.2478/v10037-012-0021-1.[Crossref] Zbl1285.26027
  19. [19] Christoph Schwarzweller and Agnieszka Rowinska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135-142, 2006. doi:10.2478/v10037-006-0017-9.[Crossref] 
  20. [20] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. FormalizedMathematics, 1(3):445-449, 1990. 
  21. [21] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. 
  22. [22] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990. 
  23. [23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990. 
  24. [24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  25. [25] Rolf Unbehauen. Netzwerk- und Filtersynthese: Grundlagen und Anwendungen. Oldenbourg-Verlag, fourth edition, 1993. 
  26. [26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990. 
  27. [27] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992. 

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