Belevitch's theorem
title: "Belevitch's theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["circuit-theorems", "two-port-networks"] topic_path: "general/circuit-theorems" source: "https://en.wikipedia.org/wiki/Belevitch's_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
Belevitch's theorem is a theorem in electrical network analysis due to the Russo-Belgian mathematician Vitold Belevitch (1921–1999). The theorem provides a test for a given S-matrix to determine whether or not it can be constructed as a lossless rational two-port network.
Lossless implies that the network contains only inductances and capacitances – no resistances. Rational (meaning the driving point impedance Z(p) is a rational function of p) implies that the network consists solely of discrete elements (inductors and capacitors only – no distributed elements).
The theorem
For a given S-matrix \mathbf S(p) of degree d;
: \mathbf S(p) = \begin{bmatrix} s_{11} & s_{12} \ s_{21} & s_{22} \end{bmatrix} :where, :p is the complex frequency variable and may be replaced by i \omega in the case of steady state sine wave signals, that is, where only a Fourier analysis is required :d will equate to the number of elements (inductors and capacitors) in the network, if such network exists.
Belevitch's theorem states that, \scriptstyle \mathbf S(p) represents a lossless rational network if and only if,
: \mathbf S(p) = \frac {1}{g(p)} \begin{bmatrix} h(p) & f(p) \ \pm f(-p) & \mp h(-p) \end{bmatrix} :where, :f(p), g(p) and h(p) are real polynomials :g(p) is a strict Hurwitz polynomial of degree not exceeding d :g(p)g(-p) = f(p)f(-p) + h(p)h(-p) for all \scriptstyle p , \in , \mathbb C .
References
Bibliography
- Belevitch, Vitold Classical Network Theory, San Francisco: Holden-Day, 1968 .
- Rockmore, Daniel Nahum; Healy, Dennis M. Modern Signal Processing, Cambridge: Cambridge University Press, 2004 .
References
- Rockmore ''et al.'', pp.35-36
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::