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 ${\displaystyle \mathbf {S} (p)}$ of degree ${\displaystyle d}$;

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

where,

p is the complex frequency variable and may be replaced by ${\displaystyle i\omega }$ 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, ${\displaystyle \scriptstyle \mathbf {S} (p)}$ represents a lossless rational network if and only if,[1]

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

where,

${\displaystyle f(p)}$, ${\displaystyle g(p)}$ and ${\displaystyle h(p)}$ are real polynomials

${\displaystyle g(p)}$ is a strict Hurwitz polynomial of degree not exceeding ${\displaystyle d}$

${\displaystyle g(p)g(-p)=f(p)f(-p)+h(p)h(-p)}$ for all ${\displaystyle \scriptstyle p\,\in \,\mathbb {C} }$.

References

1. Rockmore et al., pp.35-36

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.