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}

.

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gollark: It is not.

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gollark: Hmm, didn't respond to itself.

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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[1] Rockmore et al., pp.35-36