Belevitch's theorem - Alchetron, The Free Social Encyclopedia

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 S ( p ) of degree d ;

S ( p ) = [ s 11 s 12 s 21 s 22 ] where,p is the complex frequency variable and may be replaced by i ω in the case of steady state sine wave signals, that is, where only a Fourier analysis is requiredd will equate to the number of elements (inductors and capacitors) in the network, if such network exists.

Belevitch's theorem states that, S ( p ) represents a lossless rational network if and only if,

S ( p ) = 1 g ( p ) [ h ( p ) f ( p ) ± f ( − p ) ∓ h ( − p ) ] 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 p ∈ C .

References

Belevitch's theorem Wikipedia

(Text) CC BY-SA