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microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
and
radio-frequency Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upp ...
engineering, a stub or resonant stub is a length of
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
or
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
that is connected at one end only. The free end of the stub is either left open-circuit, or short-circuited (as is always the case for waveguides). Neglecting transmission line losses, the
input impedance The input impedance of an electrical network is the measure of the opposition to current ( impedance), both static ( resistance) and dynamic ( reactance), into the load network that is ''external'' to the electrical source. The input admittance (the ...
of the stub is purely
reactive Reactive may refer to: *Generally, capable of having a reaction (disambiguation) *An adjective abbreviation denoting a bowling ball coverstock made of reactive resin *Reactivity (chemistry) *Reactive mind *Reactive programming See also *Reactanc ...
; either
capacitive A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a c ...
or inductive, depending on the
electrical length In electrical engineering, electrical length is a dimensionless parameter equal to the physical length of an electrical conductor such as a cable or wire, divided by the wavelength of alternating current at a given frequency traveling through t ...
of the stub, and on whether it is open or short circuit. Stubs may thus function as
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s,
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s and
resonant circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
s at radio frequencies. The behaviour of stubs is due to standing waves along their length. Their
reactive Reactive may refer to: *Generally, capable of having a reaction (disambiguation) *An adjective abbreviation denoting a bowling ball coverstock made of reactive resin *Reactivity (chemistry) *Reactive mind *Reactive programming See also *Reactanc ...
properties are determined by their physical length in relation to the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
of the radio waves. Therefore, stubs are most commonly used in UHF or
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
circuits in which the wavelengths are short enough that the stub is conveniently small. They are often used to replace discrete capacitors and inductors, because at UHF and microwave frequencies
lumped component The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete e ...
s perform poorly due to parasitic reactance. Stubs are commonly used in antenna
impedance matching In electronics, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize si ...
circuits, frequency selective filters, and
resonant circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
s for UHF electronic oscillators and RF amplifiers. Stubs can be constructed with any type of
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
: parallel conductor line (where they are called
Lecher lines In electronics, a Lecher line or Lecher wires is a pair of parallel wires or rods that were used to measure the wavelength of radio waves, mainly at VHF, UHF and microwave frequencies. They form a short length of balanced transmission line ...
), coaxial cable, stripline,
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
, and
dielectric waveguide An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Common types of optical waveguides include optical fiber waveguides, transparent dielectric waveguides made of plastic and glass, liquid light ...
. Stub circuits can be designed using a
Smith chart The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assis ...
, a graphical tool which can determine what length line to use to obtain a desired reactance.


Short circuited stub

The
input impedance The input impedance of an electrical network is the measure of the opposition to current ( impedance), both static ( resistance) and dynamic ( reactance), into the load network that is ''external'' to the electrical source. The input admittance (the ...
of a lossless short circuited line is, :: Z_\mathrm = j Z_0 \tan(\beta l)\,\! where j is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, Z_0 is the characteristic impedance of the line, \beta = 2\pi/\lambda\, is the
phase constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
of the line, and l is the physical length of the line. Thus, depending on whether \tan(\beta l) is positive or negative, the stub will be inductive or capacitive, respectively. The length of a stub to act as a capacitor ''C'' at an
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of \omega is then given by: :: l = \frac \left n+1)\pi - \arctan \left(\frac\right) \right The length of a stub to act as an inductor ''L'' at the same frequency is given by: :: l = \frac \left n \pi + \arctan\left(\frac\right) \right


Open circuited stub

The input impedance of a lossless open circuit stub is given by :: Z_\mathrm = -j Z_0 \cot (\beta l) \,\! It follows that depending on whether \cot(\beta l) is positive or negative, the stub will be capacitive or inductive, respectively. The length of an open circuit stub to act as an inductor ''L'' at an angular frequency of \omega is: :: l = \frac \left n+1)\pi - \arccot\left(\frac\right) \right The length of an open circuit stub to act as a capacitor ''C'' at the same frequency is: :: l = \frac \left \pi + \arccot\left(\frac\right) \right


Resonant stub

Stubs are often used as resonant circuits in oscillators and
distributed element filter A distributed-element filter is an electronic filter in which capacitance, inductance, and resistance (the elements of the circuit) are not localised in discrete capacitors, inductors, and resistors as they are in conventional filters. Its purpo ...
s. An open circuit stub of length \scriptstyle l will have a capacitive impedance at low frequency when \scriptstyle \beta l < \pi /2. Above this frequency the impedance is inductive. At precisely \scriptstyle \beta l = \pi /2 the stub presents a short circuit. This is qualitatively the same behaviour as a series resonant circuit. For a lossless line the phase change constant is proportional to frequency, ::\beta = where ''v'' is the velocity of propagation and is constant with frequency for a lossless line. For such a case the resonant frequency is given by, ::\omega_0 = \frac While stubs function as resonant circuits, they differ from lumped element resonant circuits in that they have multiple resonant frequencies; in addition to the fundamental resonant frequency \scriptstyle \omega_0\,, they resonate at multiples of this frequency: \scriptstyle n\omega_0\,. The impedance will not continue to rise monotonically with frequency after resonance as in a lumped tuned circuit. It will rise until the point where \scriptstyle \beta l = \pi at which point it will be open circuit. After this point (which is an anti-resonance point), the impedance will again become capacitive and start to fall. It will continue to fall until at \scriptstyle \beta l = 3 \pi /2\, it again presents a short circuit. At this point, the filtering action of the stub has failed. This response of the stub continues to repeat with increasing frequency alternating between resonance and anti-resonance. It is not only a characteristic of stubs but of all distributed element filters that there is some frequency beyond which the filter fails and multiple unwanted
passband A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
s are produced. Ganesh Prasad Srivastava, Vijay Laxmi Gupta, ''Microwave Devices and Circuit Design'', pp.29-31, PHI Learning, 2006 . Similarly, a short circuit stub is an anti-resonator at \scriptstyle \pi /2, that is, it behaves as a parallel resonant circuit, but again fails as \scriptstyle 3 \pi /2 is approached.


Stub matching

Stubs can match a load impedance to the transmission line characteristic impedance. The stub is positioned a distance from the load. This distance is chosen so that at that point, the resistive part of the load impedance is made equal to the resistive part of the characteristic impedance by impedance transformer action of the length of the main line. The length of the stub is chosen so that it exactly cancels the reactive part of the presented impedance. The stub is made capacitive or inductive according to whether the main line presents an inductive or capacitive impedance, respectively. This is not the same as the actual impedance of the load since the reactive part of the load impedance will be subject to impedance transformer action and the resistive part. Matching stubs can be made adjustable so that matching can be corrected on test. A single stub will only achieve a perfect match at one specific frequency. Several stubs may be used spaced along the main transmission line for wideband matching. The resulting structure is filter-like, and filter design techniques are applied. For instance, the matching network may be designed as a
Chebyshev filter Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error betwee ...
but is optimised for impedance matching instead of passband transmission. The resulting transmission function of the network has a passband
ripple Ripple may refer to: Science and technology * Capillary wave, commonly known as ripple, a wave traveling along the phase boundary of a fluid ** Ripple, more generally a disturbance, for example of spacetime in gravitational waves * Ripple (electri ...
like the Chebyshev filter, but the ripples never reach 0 dB insertion loss at any point in the passband, as they would do for the standard filter.


Radial stub

Radial stubs are a planar component that consists of a sector of a circle rather than a constant-width line. They are used with planar transmission lines when a low impedance stub is required. Low characteristic impedance lines require a wide line. With a wide line, the junction of the stub with the main line is not at a well-defined point. Radial stubs overcome this difficulty by narrowing to a point at the junction. Filter circuits using stubs often use them in pairs, one connected to each side of the main line. A pair of radial stubs so connected is called a butterfly stub or a bowtie stub.Jia-Shen G. Hong, M. J. Lancaster, ''Microstrip Filters for RF/Microwave Applications'', pp. 188-190, Wiley, 2004 .


References


See also

{{commons category, Stubs * Quarter wave impedance transformer Antennas (radio) Distributed element circuits Telecommunications equipment Radio electronics