Cutoff Frequency
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's
frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than passing through. Typically in electronic systems such as
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...
and
communication channel A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for informa ...
s, cutoff frequency applies to an edge in a
lowpass A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter ...
,
highpass A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
,
bandpass A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two-po ...
, or
band-stop In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter. A notch filter is a b ...
characteristic – a frequency characterizing a boundary between a
passband A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
and a stopband. It is sometimes taken to be the point in the filter response where a
transition band The transition band, also called the skirt, is a range of frequencies that allows a transition between a passband and a stopband of a signal processing filter. The transition band is defined by a passband and a stopband cutoff frequency or corner ...
and passband meet, for example, as defined by a
half-power point The half-power point is the point at which the output power has dropped to half of its peak value; that is, at a level of approximately -3  dB. In filters, optical filters, and electronic amplifiers, the half-power point is also known as hal ...
(a frequency for which the output of the circuit is −3  dB of the nominal passband value). Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB. In the case of a
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
or an
antenna Antenna ( antennas or antennae) may refer to: Science and engineering * Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves * Antennae Galaxies, the name of two collid ...
, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.


Electronics

In
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, cutoff frequency or corner frequency is the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
either above or below which the power output of a circuit, such as a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
, amplifier, or
electronic filter Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components ...
has fallen to a given proportion of the power in the
passband A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
. Most frequently this proportion is one half the passband power, also referred to as the 3  dB point since a fall of 3 dB corresponds approximately to half power. As a voltage ratio this is a fall to \sqrt \ \approx \ 0.707 of the passband voltage. Other ratios besides the 3 dB point may also be relevant, for example see below. Far from the cutoff frequency in the transition band, the rate of increase of attenuation (
roll-off Roll-off is the steepness of a Transfer function, transfer function with frequency, particularly in network analysis (electrical circuits), electrical network analysis, and most especially in connection with filter (signal processing), filter cir ...
) with logarithm of frequency is
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
to a constant. For a
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
network, the roll-off is −20 dB per
decade A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement that "du ...
(−6 dB per
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
.)


Single-pole transfer function example

The
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
for the simplest
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
, H(s) = \frac , has a single
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
at . The magnitude of this function in the plane is \left , H(j\omega) \right , = \left , \frac \right , =\sqrt. At cutoff \left , H(j\omega_ \mathrm c) \right , = \frac = \sqrt. Hence, the cutoff frequency is given by \omega_ \mathrm c = \frac . Where is the
s-plane 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 compl ...
variable, is
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
and is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.


Chebyshev filters

Sometimes other ratios are more convenient than the 3 dB point. For instance, in the case of the
Chebyshev filter Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error betw ...
it is usual to define the cutoff frequency as the point after the last peak in the frequency response at which the level has fallen to the design value of the passband ripple. The amount of ripple in this class of filter can be set by the designer to any desired value, hence the ratio used could be any value.


Radio communications

In
radio communication Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmit ...
,
skywave In radio communication, skywave or skip refers to the propagation of radio waves reflected or refracted back toward Earth from the ionosphere, an electrically charged layer of the upper atmosphere. Since it is not limited by the curvature of ...
communication is a technique in which
radio wave Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz (GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (short ...
s are transmitted at an angle into the sky and reflected back to Earth by layers of charged particles in the
ionosphere The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays an ...
. In this context, the term ''cutoff frequency'' refers to the
maximum usable frequency In radio transmission maximum usable frequency (MUF) is the highest radio frequency that can be used for transmission between two points via reflection from the ionosphere (skywave or "skip" propagation) at a specified time, independent of transm ...
, the frequency above which a radio wave fails to reflect off the ionosphere at the incidence angle required for transmission between two specified points by reflection from the layer.


Waveguides

The cutoff frequency of an electromagnetic waveguide is the lowest frequency for which a mode will propagate in it. In
fiber optics An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
, it is more common to consider the cutoff wavelength, the maximum
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
that will propagate in an
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
or
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
. The cutoff frequency is found with the characteristic equation of the
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...
for electromagnetic waves, which is derived from the
electromagnetic wave equation The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form ...
by setting the longitudinal
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, should be taken to be the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
of light in whatever material fills the waveguide. For a rectangular waveguide, the cutoff frequency is \omega_ = c \sqrt, where m,n \ge 0 are the mode numbers for the rectangle's sides of length a and b respectively. For TE modes, m,n \ge 0 (but m = n = 0 is not allowed), while for TM modes m,n \ge 1 . The cutoff frequency of the TM01 mode (next higher from dominant mode TE11) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by \omega_ = c \frac = c \frac, where r is the radius of the waveguide, and \chi_ is the first root of J_(r), the
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
of the first kind of order 1. The dominant mode TE11 cutoff frequency is given by \omega_ = c \frac = c \frac However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide. For a
single-mode optical fiber In fiber-optic communication, a single-mode optical fiber (SMF), also known as fundamental- or mono-mode, is an optical fiber designed to carry only a single mode of light - the transverse mode. Modes are the possible solutions of the Helmholtz ...
, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405.


Mathematical analysis

The starting point is the wave equation (which is derived from the
Maxwell equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
), \left(\nabla^2-\frac\frac\right)\psi(\mathbf,t)=0, which becomes a
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...
by considering only functions of the form \psi(x,y,z,t) = \psi(x,y,z)e^. Substituting and evaluating the time derivative gives \left(\nabla^2 + \frac\right) \psi(x,y,z) = 0. The function \psi 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 In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
can further be reduced by considering only functions of the form \psi(x,y,z,t) = \psi(x,y)e^, where k_z is the longitudinal
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
, resulting in \left(\nabla_^2 - k_^2 + \frac\right) \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_e^. Thus for the rectangular guide the Laplacian is evaluated, and we arrive at \frac = 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 cross-section with dimensions and : k_ = \frac, k_ = \frac, where and are the two integers representing a specific eigenmode. Performing the final substitution, we obtain \frac = \left(\frac\right)^2 + \left(\frac\right)^2 + k_^2, which is the
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
in the rectangular waveguide. The cutoff frequency \omega_ is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber k_ is zero. It is given by \omega_ = c \sqrt 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 and the wave is thus evanescent.


See also

*
Full width at half maximum In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...
*
High-pass filter A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
*
Miller effect In electronics, the Miller effect accounts for the increase in the equivalent input capacitance of an inverting voltage amplifier due to amplification of the effect of capacitance between the input and output terminals. The virtually increased input ...
*
Spatial cutoff frequency In optics, spatial cutoff frequency is a precise way to quantify the smallest object resolvable by an optical system. Due to diffraction at the image plane, all optical systems act as low pass filters with a finite ability to resolve detail. If i ...
(in optical systems) *
Time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...


References

* {{refend


External links


Calculation of the center frequency with geometric mean and comparison to the arithmetic mean solution
* ttp://mathworld.wolfram.com/BesselFunction.html Mathematical definition of and information about the Bessel functions Filter theory