Cutoff frequency

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Electronics

In electronics, cutoff frequency (f_c) is the frequency either above which or below which the power output of a circuit, such as a line, amplifier, or filter, is reduced to 1/2 of the passband power; the half-power point. This is equivalent to a voltage (or amplitude) reduction to 70.7% of the passband, because voltage V² is proportional to power P. 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.

Communications

In communications, the term cutoff frequency can also 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.

Physics

In physics, the cutoff frequency of a 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 perfectly conductive 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\frac^2}\right)\psi(\mathbf,t)=0]

becomes a Helmholtz equation by considering only functions of the form

[\psi(x,y,z,t) = \psi(x,y,z)e^]

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

[(\nabla^2 + \frac) \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.

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^ z \right)}]

resulting in

[(\nabla_^2 - k_^2 + \frac) \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^ z - k_ x - k_ y\right)}]

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

[\frac = k_^2 + k_^2 + k_^2]

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

[k_ = \frac]

[k_ = \frac]

where n and m are the two whole numbers which represent a specific eigenmode. Performing the final substitution,

[\frac = \left(\frac\right)^2 + \left(\frac\right)^2 + k_^2]

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

Finally, 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, yielding the equation

[\frac = \left(\frac\right)^2 + \left(\frac\right)^2]

or

[\omega_ = c \sqrt\right)^2 + \left(\frac\right)^2}]

The cutoff frequency for other regular waveguide geometries is also calculable. For instance, the cutoff frequency of the [ TE_] mode in a waveguide of circular crossection (the transverse-electric 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 of the first kind of order 1.

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

See also

• Low-pass filter
• High-pass filter
• Time constant
• Angular frequency

External links

• [Calculation of the center frequency with geometric mean and comparison to the arithmetic mean solution]
• [Conversion of cutoff frequency f_c and time constant τ]

 

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