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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 scattering parameters, 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 Electrical resistance, 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 of a polynomial, degree d; : \mathbf S(p) = \begin s_ & s_ \\ s_ & s_ \end :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 (indu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitold Belevitch
Vitold Belevitch (2 March 1921 – 26 December 1999) was a Belgian mathematician and electrical engineer of Russian origin 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. Belevitch is responsible for a number of circuit theorems and introduced the now well-known scattering parameters. Belevitch had an interest in languages and found a mathematical derivation of Zipf's law. He also published on machine languages. Another field of interest was transmission lines, where he published on line coupling. He worked on telephone conferencing and introduced the mathematical construct of the conference matrix. Early life Belevitch was born 2 March 1921 in Terijoki, Karelia, now incorporated into Russia, but at the time part of Finland. Belevitch's parents were Russian and his mother was an ethnic Pole. They were attempting to flee from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributed Elements
: ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.'' In electrical engineering, the distributed-element model or transmission-line model of electrical Electrical network, circuits assumes that the attributes of the circuit (Electrical resistance, resistance, capacitance, and inductance) are distributed continuously throughout the material of the circuit. This is in contrast to the more common lumped-element model, which assumes that these values are lumped into electrical components that are joined by perfectly conducting wires. In the distributed-element model, each circuit element is infinitesimally small, and the wires connecting elements are not assumed to be perfect Electrical conduction, conductors; that is, they have impedance. Unlike the lumped-element model, it assumes nonuniform current along each branch and nonuniform voltage along each wire. The distributed model is used w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz Polynomial
In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose root of a function, roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the complex number, real part of every root is zero or negative. Such a polynomial must have coefficients that are positive real numbers. The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the imaginary axis (i.e., a Hurwitz stable polynomial). A polynomial function ''P''(''s'') of a complex variable ''s'' is said to be Hurwitz if the following conditions are satisfied: :1. ''P''(''s'') is real when ''s'' is real. :2. The roots of ''P''(''s'') have real parts which are zero or negative. Hurwitz polynomials are important in control system, control systems theory, because they represent the Characteristic polynomial#Characteristic equation, characteristic equations of Stability theory, stable linear systems. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 mathematics, as well as in physics, engineering, signal processing and many other fields. Formulation Its most basic form as a function of time (''t'') is: y(t) = A\sin(2 \pi f t + \varphi) = A\sin(\omega t + \varphi) where: * ''A'', ''amplitude'', the peak deviation of the function from zero. * ''f'', '' ordinary frequency'', the ''number'' of oscillations (cycles) that occur each second of time. * ''ω'' = 2''f'', ''angular frequency'', the rate of change of the function argument in units of radians per second. * \varphi, '' phase'', specifies (in radians) where in its cycle the oscillation is at ''t'' = 0. When \varphi is non-zero, the entire waveform appears to be shifted in time by the amount ''φ''/''ω'' seconds. A negative value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 complex frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions ''f'', the Laplace transform is the integral \mathcal\(s) = \int_0^\infty f(t)e^ \, dt. History The Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions in ''Essai philosophique sur les probabilités'' (1814), and the integral form of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as the condenser, a term still encountered in a few compound names, such as the ''condenser microphone''. The physical form and construction of practical capacitors vary widely and many types of capacitor are in common use. Most capacitors contain at least two electrical conductors often in the form of metallic plates or surfaces separated by a dielectric medium. A conductor may be a foil, thin film, sintered bead of metal, or an electrolyte. The nonconducting dielectric acts to increase the capacitor's c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Parameters
Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals. The parameters are useful for several branches of electrical engineering, including electronics, communication systems design, and especially for microwave engineering. The S-parameters are members of a family of similar parameters, other examples being: Y-parameters, Z-parameters, H-parameters, T-parameters or ABCD-parameters. They differ from these, in the sense that ''S-parameters'' do not use open or short circuit conditions to characterize a linear electrical network; instead, matched loads are used. These terminations are much easier to use at high signal frequencies than open-circuit and short-circuit terminations. Contrary to popular belief, the quantities are not measured in terms of power (except in now-obsolete six-port network analyzers). Modern vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 coil. When the current flowing through the coil changes, the time-varying magnetic field induces an electromotive force (''emf'') (voltage) in the conductor, described by Faraday's law of induction. According to Lenz's law, the induced voltage has a polarity (direction) which opposes the change in current that created it. As a result, inductors oppose any changes in current through them. An inductor is characterized by its inductance, which is the ratio of the voltage to the rate of change of current. In the International System of Units (SI), the unit of inductance is the henry (H) named for 19th century American scientist Joseph Henry. In the measurement of magnetic circuits, it is equivalent to . Inductors have values that typically ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 rational numbers; they may be taken in any field ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the variables may be taken in any field ''L'' containing ''K''. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is ''L''. The set of rational functions over a field ''K'' is a field, the field of fractions of the ring of the polynomial functions over ''K''. Definitions A function f(x) is called a rational function if and only if it can be written in the form : f(x) = \frac where P\, and Q\, are polynomial functions of x\, and Q\, is not the zero function. The domain of f\, is the set of all valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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