Belevitch's theorem is a theorem in electrical network analysis due to the Russo-Belgian mathematician Vitold Belevitch

Vitold Belevitch

Vitold Belevitch was a Belgian mathematician and electrical engineer of Russian extraction who produced some important work in the field of electrical network theory. Born to parents fleeing the Bolsheviks, he settled in Belgium where he worked on early computer construction projects...

 (1921–1999). The theorem provides a test for a given S-matrix

Scattering parameters

Scattering parameters or S-parameters describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals....

 to determine whether or not it can be constructed as a lossless rational two-port network

Two-port network

A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...

.

Lossless implies that the network contains only inductance

Inductance

In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

s and capacitance

Capacitance

In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

s - no resistance

Electrical resistance

The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

s. Rational (meaning the driving point impedance Z(p) is a rational function

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

 of p) implies that the network consists solely of discrete elements (inductor

Inductor

An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

s and capacitor

Capacitor

A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

s only - no distributed elements

Distributed element model

In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...

).

The theorem

For a given S-matrix of degree

Degree of a polynomial

The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

 ;

where,

p is the complex frequency variable and may be replaced by in the case of steady state sine wave

Sine wave

The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

 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,

where,, and are real polynomials is a strict Hurwitz polynomial

Hurwitz polynomial

In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative...

of degree not exceeding for all .