Cutoff frequency

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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 (attenuated or reflected) rather than passing through.

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Electrical engineering is a technical discipline concerned with the study, design and application of equipment, devices and systems which use electricity, electronics, and electromagnetism. It emerged as an identified activity in the latter half of the 19th century after commercialization of the electric telegraph, the telephone, and electrical power generation, distribution and use.

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input. Also for a linear system, doubling the amplitude of the input will double the amplitude of the output. In addition, if the system is time-invariant, then the frequency response also will not vary with time. Thus for LTI systems, the frequency response can be seen as applying the system's transfer function to a purely imaginary number argument representing the frequency of the sinusoidal excitation.

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Typically in electronic systems such as filters and communication channels, 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 −3  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 dB or 100 dB.

In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist. Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication, in radio, television, audio recording, radar, control systems, music synthesis, image processing, and computer graphics.

A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used to convey an information signal, for example a digital bit stream, from one or several senders to one or several receivers. A channel has a certain capacity for transmitting information, often measured by its bandwidth in Hz or its data rate in bits per second.

The transition band, also called the skirt, is a range of frequencies that allows a transition between a passband and a stopband of a signal processing filter. The transition band is defined by a passband and a stopband cutoff frequency or corner frequency.

In the case of a waveguide or an antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

In radio engineering, an antenna is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an electric current to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves. In reception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to be amplified. Antennas are essential components of all radio equipment.

Electronics

In electronics, cutoff frequency or corner frequency is the frequency either above or below which the power output of a circuit, such as a 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  dB point since a fall of 3 dB corresponds approximately to half power. As a voltage ratio this is a fall to ${\displaystyle \scriptstyle {\sqrt {1/2}}\ \approx \ 0.707}$ of the passband voltage. [1] Other ratios besides the 3 dB point may also be relevant, for example see Chebyshev Filters below.

Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter.

An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. To be referred to as electronic, rather than electrical, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.

A telephone line or telephone circuit is a single-user circuit on a telephone communication system. This is the physical wire or other signaling medium connecting the user's telephone apparatus to the telecommunications network, and usually also implies a single telephone number for billing purposes reserved for that user. Telephone lines are used to deliver landline telephone service and Digital subscriber line (DSL) phone cable service to the premises. Telephone overhead lines are connected to the public switched telephone network.

Single-pole transfer function example

The transfer function for the simplest low-pass filter,

In engineering, a transfer function of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, this function is a two-dimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.

A low-pass filter (LPF) is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a high-cut filter, or treble-cut filter in audio applications. A low-pass filter is the complement of a high-pass filter.

${\displaystyle H(s)={\frac {1}{1+\alpha s}},}$

has a single pole at s = -1/α. The magnitude of this function in the jω plane is

${\displaystyle \left|H(j\omega )\right|=\left|{\frac {1}{1+\alpha j\omega }}\right|={\sqrt {\frac {1}{1+\alpha ^{2}\omega ^{2}}}}.}$

At cutoff

${\displaystyle \left|H(j\omega _{\mathrm {c} })\right|={\frac {1}{\sqrt {2}}}={\sqrt {\frac {1}{1+\alpha ^{2}\omega _{\mathrm {c} }^{2}}}}.}$

Hence, the cutoff frequency is given by

${\displaystyle \omega _{\mathrm {c} }={\frac {1}{\alpha }}.}$

Where s is the s-plane variable, ω is angular frequency and j is the imaginary unit.

Chebyshev filters

Sometimes other ratios are more convenient than the 3 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. [2]

In radio communication, "skip" or "skywave" communication is a technique in which radio waves 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 means 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.

Waveguides

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

${\displaystyle \omega _{c}=c{\sqrt {\left({\frac {n\pi }{a}}\right)^{2}+\left({\frac {m\pi }{b}}\right)^{2}}},}$

where the integers ${\displaystyle n,m\geq 0}$ are the mode numbers, and a and b the lengths of the sides of the rectangle. For TE modes, ${\displaystyle n,m\geq 0}$ (but ${\displaystyle n=m=0}$ is not allowed), while for TM modes ${\displaystyle n,m\geq 1}$.

The cutoff frequency of the TM01 mode (next higher from dominant mode TE11) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by

${\displaystyle \omega _{c}=c{\frac {\chi _{01}}{r}}=c{\frac {2.4048}{r}},}$

where ${\displaystyle r}$ is the radius of the waveguide, and ${\displaystyle \chi _{01}}$ is the first root of ${\displaystyle J_{0}(r)}$, the bessel function of the first kind of order 1.

The dominant mode TE11 cutoff frequency is given by

${\displaystyle \omega _{c}=c{\frac {\chi _{11}}{r}}=c{\frac {1.8412}{r}}}$ [3]

However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide. [4] For a single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405.

Mathematical analysis

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

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

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

${\displaystyle \psi (x,y,z,t)=\psi (x,y,z)e^{i\omega t}.}$

Substituting and evaluating the time derivative gives

${\displaystyle \left(\nabla ^{2}+{\frac {\omega ^{2}}{c^{2}}}\right)\psi (x,y,z)=0.}$

The function ${\displaystyle \psi }$ 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. 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

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

where ${\displaystyle k_{z}}$ is the longitudinal wavenumber, resulting in

${\displaystyle (\nabla _{T}^{2}-k_{z}^{2}+{\frac {\omega ^{2}}{c^{2}}})\psi (x,y,z)=0,}$

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

${\displaystyle \psi (x,y,z,t)=\psi _{0}e^{i\left(\omega t-k_{z}z-k_{x}x-k_{y}y\right)}.}$

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

${\displaystyle {\frac {\omega ^{2}}{c^{2}}}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$

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

${\displaystyle k_{x}={\frac {n\pi }{a}},}$
${\displaystyle k_{y}={\frac {m\pi }{b}},}$

where n and m are the two integers representing a specific eigenmode. Performing the final substitution, we obtain

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

which is the dispersion relation in the rectangular waveguide. The cutoff frequency ${\displaystyle \omega _{c}}$ is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber ${\displaystyle k_{z}}$ is zero. It is given by

${\displaystyle \omega _{c}=c{\sqrt {\left({\frac {n\pi }{a}}\right)^{2}+\left({\frac {m\pi }{b}}\right)^{2}}}}$

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 and the wave is thus evanescent.

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References

1. Van Valkenburg, M. E. (3rd ed.). pp. 383–384. ISBN   0-13-611095-9 . Retrieved 2008-06-22.
2. Mathaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp.85-86, McGraw-Hill 1964.
3. I. C. Hunter, Theory and Design of Microwave Filters, p.214 IET, 2001 ISBN   0-85296-777-2.
4. A. Y. Modi and C. A. Balanis, "PEC-PMC Baffle Inside Circular Cross Section Waveguide for Reduction of Cut-Off Frequency," in IEEE Microwave and Wireless Components Letters, vol. 26, no. 3, pp. 171-173, March 2016. doi : 10.1109/LMWC.2016.2524529