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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 filter, is reduced to ½ 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 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( abla^2- rac rac ight)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: ( abla^2 + rac) psi(x,y,z) = 0 The function 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^ resulting in ( abla_^2 - k_^2 + rac) 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^ Thus for the rectangular guide the Laplacian is evaluated, and we arrive at rac = 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_ = rac k_ = rac where n and m are the two whole numbers which represent a specific eigenmode. Performing the final substitution, rac = left( rac ight)^2 + left( rac ight)^2 + k_^2 which incidentally is also the dispersion relation in the rectangular waveguide. Finally, the cutoff frequency is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber is zero, yielding the equation rac = left( rac ight)^2 + left( rac ight)^2 or omega_ = c sqrt It is important to note that for a frequency , the longitudinal wave number is imaginary. Thus, the previously oscillatory dependence becomes an exponential decay relationship The cutoff frequency for other regular waveguide geometries is also calculable. For instance, the cutoff frequency of the mode in a waveguide of circular crossection (the transverse-electric mode with no angular dependence and lowest radial dependence) is given by omega_ = c rac = c rac where is the radius of the waveguide, and is the first root of , 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. See also | ||||||||
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