Cutoff frequency

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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.)
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 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.

Contents

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 1/2\, the power of the passband. Because power is proportional to the square of voltage, the voltage signal is \sqrt{1/2} of the passband voltage at the corner frequency. Hence, the corner frequency is also known as the −3 dB point because \sqrt{1/2} is close to −3 decibels. A bandpass circuit has two corner frequencies; their geometric mean is called the center frequency.

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 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

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

The starting point is 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,

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

ψ(x,y,z,t) = ψ(x,y,z)eiωt.

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

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

where kz is the longitudinal wavenumber, 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 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 ωc is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber kz is zero. It is given by

\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.

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