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,^[1]

\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

References

↑ 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.

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