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 In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...

to determine whether or not it can be constructed as a lossless rational

two-port network A two-port network (a kind of four-terminal network or quadripole) is an electrical network (circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satis ...

. Lossless implies that the network contains only

inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of t ...

s and

capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized a ...

s – no

resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''T ...

s. Rational (meaning the

driving point impedance Driving is the controlled operation and movement of a vehicle, including cars, motorcycles, trucks, buses, and bicycles. Permission to drive on public highways is granted based on a set of conditions being met and drivers are required to f ...

''Z''(''p'') is a

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

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

inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a ...

s and

capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...

s only – no distributed elements).

The theorem

For a given S-matrix

\mathbf S(p)

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

d

; :

\mathbf S(p) = \begin s_ & s_ \\ s_ & s_ \end

:where, :''p'' is the

complex frequency In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...

variable and may be replaced by

i \omega

in the case of steady state

sine wave A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...

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,Rockmore ''et al.'', pp.35-36 :

\mathbf  S(p) = \frac  \begin h(p) & f(p) \\ \pm f(-p) & \mp h(-p) \end

: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

References

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 {{ISBN, 0-521-82706-X. Circuit theorems Two-port networks