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,


 * $$ \mathbf S(p) = \frac {1}{g(p)} \begin{bmatrix} h(p) & f(p) \\ \pm f(-p) & \mp h(-p) \end{bmatrix} $$
 * 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 $$\scriptstyle p \, \in \, \mathbb C $$.