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 $d$;

:$\mathbf S(p) = \begin s_ & s_ \\ s_ & s_ \end$
: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,Rockmore ''et al.'', pp.35-36

:$\mathbf S(p) = \frac \begin h(p) & f(p) \\ \pm f(-p) & \mp h(-p) \end$
: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 {{ISBN, 0-521-82706-X.

Circuit theorems
Two-port networks