NationMaster - Encyclopedia: Cutoff frequency

A bode plot of the Butterworth filter's frequency response, with corner frequency labeled. (The slope −20 dB per decade also equals −6 dB per octave.)

In physics and electrical engineering, the term cutoff frequency or corner frequency represents a boundary in the system response at which energy entering the system begins to be attenuated or reflected instead of transmitted. Common examples are The frequency response of a Butterworth filter with logarithmic axes. ... The frequency response of a Butterworth filter with logarithmic axes. ... The Bode plot for a first-order Butterworth filter A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show... The Butterworth filter is one type of electronic filter design. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... Electrical Engineers design power systemsâ€¦ â€¦ and complex electronic circuits. ...

the cutoff frequency of an electronic circuit: either the lowest or the highest frequency for which the output of the circuit deviates less than 3 dB from the nominal value.
the lowest frequency or longest wavelength which can be transmitted through a waveguide.

The cutoff frequency can also refer to the plasma frequency, or to some concepts related to renormalization in quantum field theory. Look up waveguide in Wiktionary, the free dictionary. ... In physics, plasma oscillations, often referred to as Langmuir waves or plasma waves, are periodic oscillations of charge density in conducting media such as plasmas or metals. ... Figure 1. ... Quantum field theory (QFT) is the quantum theory of fields. ...

1 Electronics
2 Communications
3 Waveguides
- 3.1 Mathematical analysis
4 See also
5 External links

Electronics

In electronics, cutoff frequency or corner frequency (f_c) is the frequency either above which or below which the power output of a circuit, such as a line, amplifier, or electronic filter is $1 / 2$ the power of the passband, and since voltage V² is proportional to power P, V is $sqrt{1/2}$ of the V in the passband. This happens to be close to −3 decibels, and the cutoff frequency is frequently referred to as the −3 dB point. Electronics is the study of the flow of charge through various materials and devices such as, semiconductors, resistors, inductors, capacitors, nano-structures, and vacuum tubes. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... // Information processing In information processing, output is the process of transmitting information by an object (verb usage). ... An electronic circuit is an electrical circuit that also contains active electronic devices such as transistors or vacuum tubes. ... A telephone line (or just line) is a single-user circuit on a telephone communications system. ... Television signal splitter consisting of a hi-pass and a low-pass filter. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ... For delivered electrical power, see Electrical power industry. ... The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power) relative to a specified or implied reference level. ...

A bandpass circuit has two cutoff frequencies and their geometric mean is the center frequency. The frequency axis of this symbolic diagram would be logarithmically scaled. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... The frequency axis of this symbolic diagram would be logarithmically scaled. ...

Communications

In communications, the term cutoff frequency can mean the frequency below which a radio wave fails to penetrate a layer of the ionosphere at the incidence angle required for transmission between two specified points by reflection from the layer. The term communications is used in a number of disciplines: Communications, also known as communication studies is the academic discipline which studies communication, generally seen as a mixture between media studies and linguistics. ... Radio frequency, or RF, refers to that portion of the electromagnetic spectrum in which electromagnetic waves can be generated by alternating current fed to an antenna. ... Layer may refer to: Look up Layer in Wiktionary, the free dictionary. ... Relationship of the atmosphere and ionosphere The ionosphere is the uppermost part of the atmosphere, distinguished because it is ionized by solar radiation. ... In telecommunications, transmission is the act of transmitting electrical messages (and the associated phenomena of radiant energy that passes through media). ... The reflection of a bridge in Indianapolis, Indianas Central Canal. ...

Waveguides

The cutoff frequency of an electromagnetic waveguide is the lowest frequency for which a mode will propagate in it. In fiber optics, it is more common to consider the cutoff wavelength, the maximum wavelength that will propagate in an optical fiber or waveguide. The cutoff frequency is found with the characteristic equation of the Helmholtz equation for electromagnetic waves, which is derived from the electromagnetic wave equation by setting the longitudinal wave number equal to zero and solving for the frequency. Thus, any exciting frequency lower than the cutoff frequency will attenuate, rather than propagate. The following derivation assumes lossless walls. The value of c, the speed of light, should be taken to be the group velocity of light in whatever material fills the waveguide. It has been suggested that this article or section be merged with Waveguide (optics). ... Fiber Optic strands An optical fiber in American English or fibre in British English is a transparent thin fiber for transmitting light. ... The wavelength is the distance between repeating units of a wave pattern. ... Optical fibers An optical fiber (or fibre) is a glass or plastic fiber designed to guide light along its length by confining as much light as possible in a propagating form. ... In electromagnetics and communications engineering, a waveguide is defined as any physical structure that guides electromagnetic waves. ... In linear algebra, the characteristic equation of a square matrix A is the equation in one variable λ where I is the identity matrix. ... The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ... Lasers used for visual effects during a musical performance. ... Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic... The group velocity of a wave is the velocity with which the variations in the shape of the waves amplitude (known as the modulation or envelope of the wave) propagate through space. ...

For a rectangular waveguide, the cutoff frequency is

$omega_{c} = c sqrt{left(frac{n pi}{a}right)^2 + left(frac{m pi}{b}right) ^2},$

where $n,m ge 0$ are the mode numbers and a and b the lengths of the sides of the rectangle.

The cutoff frequency of the TM₀₁ mode in a waveguide of circular cross-section (the transverse-electric mode with no angular dependence and lowest radial dependence) is given by

$omega_{c} = c frac{chi_{01}}{r} = c frac{2.4048}{r},$

where $r$ is the radius of the waveguide, and $χ 01$ is the first root of $J 0 (r)$ , the bessel function of the first kind of order 1. In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number Î±. The most common and important special case is where Î± is an integer n, then Î± is referred to...

For a single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405. In fiber optics, a single-mode optical fiber is an optical fiber in which only the lowest order bound mode can propagate at the wavelength of interest. ... In telecommunication, the term normalized frequency ( V ) has several meanings; it may refer to the ratio between an actual frequency and a reference value, or to the ratio between an actual frequency and its nominal value. ...

Mathematical analysis

The starting point is the wave equation (which is derived from the Maxwell equations), Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

$left(nabla^2-frac{1}{c^2}frac{partial^2}{partial{t}^2}right)psi(mathbf{r},t)=0,$

which becomes a Helmholtz equation by considering only functions of the form The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ...

ψ(x, y, z, t) = ψ(x, y, z) e i ω t .

Substituting and evaluating the time derivative gives

$(nabla^2 + frac{omega^2}{c^2}) psi(x,y,z) = 0.$

The function $ψ$ here refers to whichever field (the electric field or the magnetic field) has no vector component in the longitudinal direction - the "transverse" field. It is a property of all the eigenmodes of the electromagnetic waveguide that at least one of the two fields is transverse. The z axis is defined to be along the axis of the waveguide.

The "longitudinal" derivative in the Laplacian can further be reduced by considering only functions of the form In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

$psi(x,y,z,t) = psi(x,y)e^{i left(omega t - k_{z} z right)},$

where $k z$ is the longitudinal wavenumber, resulting in Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (mâˆ’1). ...

$(nabla_{T}^2 - k_{z}^2 + frac{omega^2}{c^2}) psi(x,y,z) = 0,$

where subscript T indicates a 2-dimensional transverse Laplacian. The final step depends on the geometry of the waveguide. The easiest geometry to solve is the rectangular waveguide. In that case the remainder of the Laplacian can be evaluated to its characteristic equation by considering solutions of the form

$psi(x,y,z,t) = psi_{0}e^{i left(omega t - k_{z} z - k_{x} x - k_{y} yright)}.$

Thus for the rectangular guide the Laplacian is evaluated, and we arrive at

$frac{omega^2}{c^2} = k_{x}^2 + k_{y}^2 + k_{z}^2$

The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry crossection with dimensions a and b:

$k_{x} = frac{n pi}{a},$

$k_{y} = frac{m pi}{b},$

where n and m are the two integers representing a specific eigenmode. Performing the final substitution, we obtain

$frac{omega^2}{c^2} = left(frac{n pi}{a}right)^2 + left(frac{m pi}{b}right)^2 + k_{z}^2,$

which is the dispersion relation in the rectangular waveguide. The cutoff frequency $ω c$ is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber $k z$ is zero. It is given by The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ...

$omega_{c} = c sqrt{left(frac{n pi}{a}right)^2 + left(frac{m pi}{b}right)^2}$

The wave equations are also valid below the cutoff frequency, where the longitudinal wave number is imaginary. In this case, the field decays exponentially along the waveguide axis.

External links

Calculation of the center frequency with geometric mean and comparison to the arithmetic mean solution
Conversion of cutoff frequency f_c and time constant τ
Mathematical definition of and information about the Bessel functions

Categories: Electronics terms | Filter theory

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Contents

Electronics var ACE_AR = {Site: '756743', Size: '300250'};

Communications

Waveguides

Mathematical analysis

See also

External links

Electronics