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 of degree ;

where,
p is the complex frequency variable and may be replaced by 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, represents a lossless rational network if and only if,[1]

where,
, and are real polynomials
is a strict Hurwitz polynomial of degree not exceeding
for all .

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.
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