Belevitch's theorem explained

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

;

S(p)=\begin{bmatrix}s₁₁&s₁₂\ s₂₁&s₂₂\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,

\scriptstyleS(p)

represents a lossless rational network if and only if,^[1]

S(p)=

	1
	g(p)

\begin{bmatrix}h(p)&f(p)\ \pmf(-p)&\mph(-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

g(p)g(-p)=f(p)f(-p)+h(p)h(-p)

for all

\scriptstylep\inC

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 .

Notes and References

Rockmore et al., pp.35-36