Multiphase particle-in-cell method

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 $d$;

$\mathbf{S}(p)=\left[\begin{array}{cc}{s}_{11}& s_{12}\\ s_{21}& s_{22}\end{array}\right]$

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, $\mathbf{S}(p)$ represents a lossless rational network if and only if,

$\mathbf{S}(p)=\frac{1}{g(p)}\left[\begin{array}{cc}h(p)& f(p)\\ \pm f(-p)& \mp h(-p)\end{array}\right]$

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 $p\in \mathbb{ℂ}$.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Bibliography

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