Cutoff frequency

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

The term cutoff frequency is applicable to a wide range of systems. However, its meaning is consistent in a general sense across fields of study. The cutoff frequency of a system is a boundary in the input spectrum at which energy entering the system begins to be attenuated or reflected instead of transmitted. The following sections describe the specific meaning and usage of the term in various applications.

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[edit] Electronics

In electronics, cutoff frequency (fc) 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. Also called the knee frequency, due to a frequency response curve's physical appearance.

A bandpass circuit has two cutoff frequencies and their geometric mean is the center frequency.

[edit] 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.

[edit] Physics

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In physics, 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.

The wave equation (which is derived from the Maxwell equations)

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

becomes a Helmholtz equation by considering only functions of the form

ψ(x,y,z,t) = ψ(x,y,z)eiωt

After substituting and evaluating the time derivative, we arrive at a Helmholtz equation:

(\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.

Note that we will consider the cartesian z-coordinate to represent the axial direction of the waveguide, and the x- and y-coordinates will represent the transverse directions.

The "longitudinal" derivative in the Laplacian can further be reduced by considering only functions of the form

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

resulting in

(\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} y\right)}

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 whole numbers which represent a specific eigenmode. Performing the final substitution,

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

which incidentally is also the dispersion relation in the rectangular waveguide.

Finally, the cutoff frequency ωc is the

critical frequency between propagation and attenuation, which corresponds to the

frequency at which the longitudinal wavenumber kz is zero, yielding the equation

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

or

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

It is important to note that for a frequency

ω < ωc, the longitudinal wave number is imaginary. Thus, the previously oscillatory dependence

\psi \propto e^{i k_{z} z}

becomes an exponential decay relationship

\psi \propto e^{- Re(k_{z}) z}

The cutoff frequency for other regular waveguide geometries is also calculable.

For instance, the cutoff frequency of the

TM01 mode in a waveguide of circular

crossection (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

χ01 is the first root of

J0(r), the bessel function of the first

kind of order 1.

For single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405.

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

[edit] See also

[edit] External links

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