Belevitch's theorem

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 $\scriptstyle p\,\in \,\mathbb {C}$.

Bibliography

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