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[]
- ^ 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.
Categories:
- Circuit theorems
- Two-port networks