Editing Cutoff frequency

{{Short description|Frequency response boundary}}
[[File:Bandwidth.svg|thumb|right|320px|Magnitude transfer function of a [[bandpass filter]] with lower 3&nbsp;dB cutoff frequency ''f''<sub>1</sub> and upper 3&nbsp;dB cutoff frequency ''f''<sub>2</sub>]]
[[File:Butterworth response.svg|right|thumb|320px|[[Bode plot]] (a logarithmic [[frequency response]] plot) of any first-order low-pass filter with a normalized cutoff frequency at <math>\omega</math>=1 and a [[unity gain]] (0 dB) passband.]]

In [[physics]] and [[electrical engineering]], a '''cutoff frequency''', '''corner frequency''', or '''break frequency''' is a boundary in a system's [[frequency response]] at which energy flowing through the system begins to be reduced ([[attenuation|attenuated]] or reflected) rather than passing through.

Typically in electronic systems such as [[Filter (signal processing)|filters]] and [[communication channel]]s, cutoff frequency applies to an edge in a [[lowpass]], [[highpass]], [[bandpass]], or [[band-stop]] characteristic – a frequency characterizing a boundary between a [[passband]] and a [[stopband]]. It is sometimes taken to be the point in the filter response where a [[transition band]] and passband meet, for example, as defined by a [[half-power point]] (a frequency for which the output of the circuit is approximately −3.01&nbsp;[[Decibel|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&nbsp;dB or 100&nbsp;dB.

In the case of a [[waveguide]] or an [[antenna (radio)|antenna]], the cutoff frequencies correspond to the lower and upper '''cutoff wavelengths'''.

==Electronics==
In [[electronics]], cutoff frequency or corner frequency is the [[frequency]] either above or below which the power output of a [[Electronic circuit|circuit]], such as a [[telephone line|line]], amplifier, or [[electronic filter]] has fallen to a given proportion of the power in the [[passband]].  Most frequently this proportion is one half the passband power, also referred to as the 3&nbsp;[[decibel|dB]] point since a fall of 3&nbsp;dB corresponds approximately to half power. As a voltage ratio this is a fall to <math display="inline"> \sqrt{1/2} \ \approx \ 0.707</math> of the passband voltage.<ref>
{{cite book
|title=Network Analysis
|edition=3rd
|last=Van Valkenburg
|first=M. E.
|year=1974
|pages=[https://archive.org/details/networkanalysis00vanv/page/383 383–384]
|isbn=0-13-611095-9
|url=https://archive.org/details/networkanalysis00vanv/page/383
|access-date=2008-06-22
|url-access=registration
}}</ref> Other ratios besides the 3&nbsp;dB point may also be relevant, for example see {{section link|#Chebyshev filters}} below. Far from the cutoff frequency in the transition band, the rate of increase of attenuation ([[roll-off]]) with logarithm of frequency is [[asymptotic]] to a constant. For a [[First-order linear differential equation|first-order]] network, the roll-off is −20&nbsp;dB per [[decade (log scale)|decade]] (approximately −6&nbsp;dB per [[octave]].)

===Single-pole transfer function example===
The [[transfer function]] for the simplest [[low-pass filter]],
<math display="block">H(s) = \frac {1}{1+\alpha s},</math>
has a single [[Pole (complex analysis)|pole]] at {{math|1=''s'' = −1/''α''}}. The magnitude of this function in the {{math|''jω''}} plane is
<math display="block">\left | H(j\omega) \right | = \left |  \frac {1}{1+\alpha j \omega} \right | =\sqrt{ \frac {1}{1 + \alpha^2\omega^2}}.</math>

At cutoff
<math display="block">\left | H(j\omega_ \mathrm c) \right | = \frac {1}{\sqrt{2}} = \sqrt{ \frac {1}{1 + \alpha^2\omega_\mathrm c ^2}}.</math>

Hence, the cutoff frequency is given by
<math display="block">\omega_ \mathrm c = \frac {1}{\alpha}.</math>

Where {{mvar|s}} is the [[s-plane]] variable, {{math|ω}} is [[angular frequency]] and {{math|''j''}} is the [[imaginary unit]].

===Chebyshev filters===
Sometimes other ratios are more convenient than the 3&nbsp;dB point. For instance, in the case of the [[Chebyshev filter]] 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.<ref>Mathaei, Young, Jones ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'', pp.85-86, McGraw-Hill 1964.</ref>

==Radio communications==
In [[radio communication]], [[skywave]] communication is a technique in which [[radio wave]]s are transmitted at an angle into the sky and reflected back to Earth by layers of charged particles in the [[ionosphere]].  In this context, the term ''cutoff frequency'' refers to the [[maximum usable frequency]], 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 [[waveguide (electromagnetism)|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 (optics)|waveguide]]. The cutoff frequency is found with the [[Characteristic equation (calculus)|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
<math display="block"> \omega_{c} = c \sqrt{\left(\frac{m \pi}{a}\right)^2 + \left(\frac{n \pi}{b}\right) ^2}, </math>
where <math>m,n \ge 0</math> are the mode numbers for the rectangle's sides of length <math>a</math> and <math>b</math> respectively. For TE modes, <math> m,n \ge 0</math> (but <math> m = n = 0</math> is not allowed), while for TM modes <math> m,n \ge 1 </math>.

The cutoff frequency of the TM<sub>01</sub> mode (next higher from dominant mode TE<sub>11</sub>) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by 
<math display="block"> \omega_{c} = c \frac{\chi_{01}}{r} = c \frac{2.4048}{r},</math>
where <math>r</math> is the radius of the waveguide, and  <math>\chi_{01}</math> is the first root of <math>J_{0}(r)</math>, the [[Bessel function]] of the first kind of order 1.

The dominant mode TE<sub>11</sub> cutoff frequency is given by<ref>{{Cite book |last=Hunter |first=I. C. |title=Theory and design of microwave filters |date=2001 |publisher=Institution of Electrical Engineers |others=Institution of Electrical Engineers |isbn=978-0-86341-253-0 |location=London |url=https://www.worldcat.org/oclc/505848355 |oclc=505848355 |pages=214}}</ref>
<math display="block"> \omega_{c} = c \frac{\chi_{11}}{r} = c \frac{1.8412}{r}</math>

However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide.<ref>{{Cite journal |last1=Modi |first1=Anuj Y. |last2=Balanis |first2=Constantine A. |date=2016-03-01 |title=PEC-PMC Baffle Inside Circular Cross Section Waveguide for Reduction of Cut-Off Frequency |url=https://ieeexplore.ieee.org/document/7422717 |journal=IEEE Microwave and Wireless Components Letters |volume=26 |issue=3 |pages=171–173 |doi=10.1109/LMWC.2016.2524529 |s2cid=9594124 |issn=1531-1309|url-access=subscription }}</ref> For a [[single-mode optical fiber]], the cutoff wavelength is the wavelength at which the [[normalized frequency (fiber optics)|normalized frequency]] is approximately equal to 2.405.

===Mathematical analysis===
The starting point is the wave equation (which is derived from the [[Maxwell equations]]),
<math display="block">
  \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial{t}^2}\right)\psi(\mathbf{r},t)=0,
</math>
which becomes a [[Helmholtz equation]] by considering only functions of the form 
<math display="block">
  \psi(x,y,z,t) = \psi(x,y,z)e^{i \omega t}.
</math>
Substituting and evaluating the time derivative gives
<math display="block">
  \left(\nabla^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0.
</math>
The function <math> \psi </math> 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 
<math display="block">
  \psi(x,y,z,t) = \psi(x,y)e^{i \left(\omega t - k_{z} z \right)},
</math>
where <math>k_z</math> is the longitudinal [[wavenumber]], resulting in
<math display="block">
  \left(\nabla_{T}^2 - k_{z}^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0,
</math>
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 
<math display="block">
  \psi(x,y,z,t) = \psi_{0}e^{i \left(\omega t - k_{z} z - k_{x} x - k_{y} y\right)}.
</math>
Thus for the rectangular guide the Laplacian is evaluated, and we arrive at
<math display="block">
  \frac{\omega^2}{c^2} = k_x^2 + k_y^2 + k_z^2
</math>
The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry cross-section with dimensions {{mvar|a}} and {{mvar|b}}:
<math display="block"> k_{x} = \frac{n \pi}{a},</math>
<math display="block"> k_{y} = \frac{m \pi}{b},</math>
where {{mvar|n}} and {{mvar|m}} are the two integers representing a specific eigenmode. Performing the final substitution, we obtain
<math display="block"> \frac{\omega^2}{c^2} = \left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2 + k_{z}^2,</math>
which is the [[dispersion relation]] in the rectangular waveguide. The cutoff frequency <math>\omega_{c}</math> is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber <math>k_{z}</math> is zero. It is given by
<math display="block"> \omega_{c} = c \sqrt{\left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2}</math>
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 wave|evanescent]].

== See also ==
*[[Full width at half maximum]]
*[[High-pass filter]]
*[[Miller effect]]
*[[Spatial cutoff frequency]] (in optical systems)
*[[Time constant]]

== References ==
{{reflist}}
{{refbegin}} 
*{{FS1037C MS188}}
{{refend}}

== External links ==
*[http://www.sengpielaudio.com/calculator-geommean.htm Calculation of the center frequency with geometric mean and comparison to the arithmetic mean solution]
*[http://www.sengpielaudio.com/calculator-timeconstant.htm Conversion of cutoff frequency f<sub>c</sub> and time constant τ]
*[http://mathworld.wolfram.com/BesselFunction.html Mathematical definition of and information about the Bessel functions]

[[Category:Filter theory]]