Belevitch's theorem

The theorem of Belevitch ( English Belevitch's theorem is) one after Vitold Belevitch named and in the electric circuit theory used theorem to determine whether a given S-matrix of a two-port network is lossless and rationally feasible in principle.

In this context, loss-free means that the two-port inside can only be implemented with ideal capacitors and coils without lossy ohmic resistors . Rational means that only rational functions are used to describe the S matrix . It is implicitly stipulated that only concentrated components (discrete components) in the form of capacitors and coils occur in the two-port.

General

The given S-matrix of a two-port of degree is in the form: ${\displaystyle \mathbf {S} (s)}$ ${\displaystyle d}$

${\displaystyle \mathbf {S} (s)={\begin{bmatrix}s_{11}&s_{12}\\s_{21}&s_{22}\end{bmatrix}}}$

with the complex frequency , in the steady state is reduced to , with the angular frequency . The degree indicates, if it exists, the number of concentrated capacitors and coils contained in the two-port. ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle \mathrm {j} \omega }$ ${\displaystyle \omega }$ ${\displaystyle d}$

Belevitch's theorem now states that such a lossless and rational two-port exists if and only if the S-matrix can be represented in the following form: ${\displaystyle \mathbf {S} (s)}$

${\displaystyle \mathbf {S} (s)={\frac {1}{g(s)}}{\begin{bmatrix}h(s)&f(s)\\\pm f(-s)&\mp h(-s)\end{bmatrix}}}$

with the following conditions:

1. ${\displaystyle f(s)}$, and are real polynomials ${\displaystyle g(s)}$ ${\displaystyle h(s)}$
2. ${\displaystyle g(s)}$is a Hurwitz polynomial of degree less than or equal . ${\displaystyle d}$
3. ${\displaystyle g(s)g(-s)=f(s)f(-s)+h(s)h(-s)}$

for everyone . ${\displaystyle s\in \mathbb {C} }$

literature

• Vitold Belevitch: Classical Network Theory . Holden-Day, San Francisco 1968, OCLC 413916 .
• Daniel Nahum Rockmore and Dennis M. Healy: Modern Signal Processing . Cambridge University Press, 2004, ISBN 0-521-82706-X .