# Cutoff frequency

Luncat ka: pituduh, sungsi
Amplitudo réspon frékuénsi tina sahiji bandpass filter kalayan cutoff frequency handap, f1 atawa ${\displaystyle \omega }$c1/2${\displaystyle \pi }$, 3dB sarta cutoff frequency luhur, f2 atawa ${\displaystyle \omega }$c2/2${\displaystyle \pi }$, 3dB
Sahiji bode plot pikeun réspon frékuénsi Butterworth filter. (Kamiringan −20 dB per dékade ogé sarua jeung −6 dB per oktap.)

Dina élmu fisika jeung téknik listrik, istilah cutoff frequency luhur atawa handap, frékuénsi juru, jeung break frequency ngagambarkeun hiji wates dina réspon frékuénsi hiji sistem dimana énergi nu asup kana sistem kasebut mimiti diaténuasi atawa malah dipantulkeun lain ditransmisikeun. Conto umumna:

 Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantosanna diantos kanggo narjamahkeun.
• a cutoff frequency of an ideal lowpass, highpass, bandpass or Band-stop filter, i.e. discrete point in the magnitude transfer function, a frequency where a passband and a stopband meets.
• the passband frequency limits of an electronic circuit such as a filter or an amplifier, i.e. the point in the filter specification where the transition band and the passband meets; defined as the 3 dB cutoff frequencies, i.e. the lowest or the highest frequency for which the output of the circuit deviates less than typically 3 dB from the nominal passband value.
• the stopband corner frequencies or limit frequencies of a filter, i.e. the point where the transit band and the stopband meets; either the lowest or the highest frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.
• the upper or lower 3dB cutoff frequencies of a communication channel. In the case of a waveguide or an antenna, this corresponds to the lower and upper cutoff wavelengths respectively.
• the 3dB cutoff frequency of a signal spectrum, for example of a baseband signal or a passband signal.

The cutoff frequency can also refer to the plasma frequency, or to some concepts related to renormalization in quantum field theory.

## Electronics

In electronics, cutoff frequency or corner frequency is the frequency either above which or below which the power output of a circuit, such as a line, amplifier, or electronic filter is ${\displaystyle 1/2\,}$ the power of the passband.[1] Because power is proportional to the square of voltage, the voltage signal is ${\displaystyle {\sqrt {1/2}}}$ of the passband voltage at the corner frequency. Hence, the corner frequency is also known as the −3 dB point because ${\displaystyle {\sqrt {1/2}}}$ is close to −3 decibels. A bandpass circuit has two corner frequencies; their geometric mean is called the center frequency.

## 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.

## 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.

For a rectangular waveguide, the cutoff frequency is

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

where ${\displaystyle n,m\geq 0}$ 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-magnetic mode with no angular dependence and lowest radial dependence) is given by

${\displaystyle \omega _{c}=c{\frac {\chi _{01}}{r}}=c{\frac {2.4048}{r}},}$

where ${\displaystyle r}$ is the radius of the waveguide, and ${\displaystyle \chi _{01}}$ is the first root of ${\displaystyle J_{0}(r)}$, the bessel function of the first kind of order 1.

For a single-mode optical fiber, 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),

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

which becomes a Helmholtz equation by considering only functions of the form

${\displaystyle \psi (x,y,z,t)=\psi (x,y,z)e^{i\omega t}.}$

Substituting and evaluating the time derivative gives

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

The function ${\displaystyle \psi }$ 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 can further be reduced by considering only functions of the form

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

where ${\displaystyle k_{z}}$ is the longitudinal wavenumber, resulting in

${\displaystyle \left(\nabla _{T}^{2}-k_{z}^{2}+{\frac {\omega ^{2}}{c^{2}}}\right)\psi (x,y,z)=0,}$

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

${\displaystyle \psi (x,y,z,t)=\psi _{0}e^{i\left(\omega t-k_{z}z-k_{x}x-k_{y}y\right)}.}$

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

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

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

${\displaystyle k_{x}={\frac {n\pi }{a}},}$
${\displaystyle k_{y}={\frac {m\pi }{b}},}$

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

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

which is the dispersion relation in the rectangular waveguide. The cutoff frequency ${\displaystyle \omega _{c}}$ is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber ${\displaystyle k_{z}}$ is zero. It is given by

${\displaystyle \omega _{c}=c{\sqrt {\left({\frac {n\pi }{a}}\right)^{2}+\left({\frac {m\pi }{b}}\right)^{2}}}}$

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.

## Rujukan

1. Van Valkenburg, M. E. Network Analysis (3rd edition ed.). pp. pp. 383–384. ISBN 0-13-611095-9. Retrieved 2008-06-22.