View source for Engineering:Stub (electronics)

{{Short description|Short electrical transmission line}}
[[File:Resonant stubs in UHF transceiver 1938.jpg|thumb|Resonant stub tank circuits in [[Physics:Vacuum tube|vacuum tube]] backpack UHF [[Transceiver|transceiver]], 1938. About 1/8 wavelength long: ''(left)'' 200 MHz stub is 19 cm, ''(right)'' 300 MHz stub is 12.5 cm]]  
[[Image:FM radio transmitter resonant lines 1947.jpg|thumb|10 kW [[Engineering:Transmitter|FM broadcast transmitter]] from 1947 showing quarter-wave resonant stub plate tank circuit]]

In [[Physics:Microwave|microwave]] and radio-frequency engineering, a '''stub''' or '''resonant stub''' is a length of [[Engineering:Transmission line|transmission line]] or [[Physics:Waveguide (electromagnetism)|waveguide]] 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 [[Physics:Input impedance|input impedance]] of the stub is purely [[Engineering:Reactance (electronics)|reactive]]; either capacitive or [[Physics:Inductance|inductive]], depending on the [[Electrical length|electrical length]] of the stub, and on whether it is open or short circuit. Stubs may thus function as [[Engineering:Capacitor|capacitor]]s, [[Engineering:Inductor|inductor]]s and resonant circuits at radio frequencies.

The behaviour of stubs is due to [[Physics:Standing wave|standing wave]]s along their length.   Their [[Engineering:Reactance (electronics)|reactive]] properties are determined by their physical length in relation to the [[Wavelength|wavelength]] of the radio waves.  Therefore, stubs are most commonly used in UHF or [[Physics:Microwave|microwave]] circuits in which the wavelengths are short enough that the stub is conveniently small.<ref name="Shuart">{{cite journal
  | last1  = Shuart
  | first1 = George W.
  | title = New high impedance lines replace coils
  | journal = Short Wave Craft
  | volume = 5
  | issue = 6
  | pages = 332–333
  | publisher = Popular Book Corp.
  | location = New York
  | date = October 1934
  | url = http://www.americanradiohistory.com/Archive-Short-Wave-Television/30s/SW-TV-1934-10.pdf
  | issn =
  | doi =
  | id =
  | accessdate = March 24, 2015}}</ref>  They are often used to replace discrete capacitors and inductors, because at UHF and microwave frequencies lumped components perform poorly due to parasitic reactance.<ref name="Shuart" />    Stubs are commonly used in [[Engineering:Antenna (radio)|antenna]] [[Impedance matching|impedance matching]] circuits, frequency selective [[Engineering:Distributed element filter|filters]], and resonant circuits for UHF [[Engineering:Electronic oscillator|electronic oscillator]]s and [[Engineering:Amplifier|RF amplifiers]].

Stubs can be constructed with any type of [[Engineering:Transmission line|transmission line]]: [[Engineering:Ladder line|parallel conductor line]] (where they are called Lecher lines),  [[Engineering:Coaxial cable|coaxial cable]], [[Engineering:Stripline|stripline]], [[Physics:Waveguide|waveguide]], and dielectric waveguide.  Stub circuits can be designed using a [[Smith chart]], a graphical tool which can determine what length line to use to obtain a desired reactance.

==Short circuited stub<span class="anchor" id="short_circuited_stub_anchor"></span>==
The [[Engineering:Reflections of signals on conducting lines#Input impedance|input impedance]] of a lossless, short circuited line is,
:<math>
 Z_\mathsf{sc} ~=~ j\ Z_0\ \tan(\ \beta \ell\ ) ~
</math>

where
:: <math>\ j\ </math> is the [[Imaginary unit|imaginary unit]] (<math>\ j^2 \equiv -1\ </math>),
:: <math>\ Z_0\ </math> is the [[Physics:Characteristic impedance|characteristic impedance]] of the line,
:: <math>\ \beta = 2\pi/\lambda\ </math> is the phase constant of the line, and
:: <math>\ \ell\ </math> is the physical length of the line.

Thus, depending on whether <math>\ \tan(\beta \ell)\ </math> is positive or negative, the short circuited stub will be inductive or capacitive, respectively.

The length of a stub to act as a capacitor {{mvar|C}} at an [[Physics:Angular frequency|angular frequency]] of <math>\ \omega\ </math> is then given by:
:<math>
\ell ~=~ \frac{1}{\ \beta\ } \left[\ (n+1)\ \pi\ -\ \arctan \left(\frac{1}{\ \omega C Z_0\ } \right)\ \right] ~;
</math>

the length of a stub to act as an inductor {{mvar|L}} at the same frequency is given by:
:<math>
\ell ~=~ \frac{1}{\ \beta\ } \left[\ n\ \pi\ +\ \arctan\left(\frac{\ \omega L\ }{\ Z_0\ }\right)\ \right] ~,
</math>

where in both equations, {{mvar|n}} is an integer number of half-wavelengths (possibly zero) that can be arbitrarily added to the line without changing the impedance.

==Open circuited stub==
The input impedance of a lossless open circuit stub is given by
:<math>
 Z_\mathsf{oc} = -j\ Z_0\ \cot(\ \beta \ell\ ) ~,
</math>
where the symbols <math>\ Z_0, \beta, \ell, \omega,\ </math> etc. used in this section have the same meaning as in the section above.

It follows that depending on whether <math>\cot(\beta \ell)</math> is positive or negative, the stub will be capacitive or inductive, respectively.

The length of an open circuit stub to act as an inductor {{mvar|L}} at an angular frequency of <math>\ \omega\ </math> is:
:<math>
\ell ~=~ \frac{1}{\ \beta\ } \left[\ (n+1)\ \pi\ -\ \arccot\left(\frac{\ \omega L\ }{Z_0}\right)\ \right] ~=~ \frac{1}{\ \beta\ } \left[\ (n+1)\ \pi\ -\ \arctan\left(\frac{Z_0}{\ \omega L\ }\right)\ \right] ~;
</math>

the length of an open circuit stub to act as a capacitor {{mvar|C}} at the same frequency is:
:<math>
\ell ~=~ \frac{1}{\ \beta\ } \left[\ n\ \pi\ +\ \arccot\left(\frac{1}{\ \omega C Z_0\ }\right)\ \right] ~=~ \frac{1}{\ \beta\ } \left[\ n\ \pi\ +\ \arctan\left( \ \omega C Z_0\ \right)\ \right]  ~,
</math>

where again, {{mvar|n}} is an arbitrary whole number of half-wavelengths that can be inserted into the segment (including zero).

==Resonant stub==
Stubs are often used as resonant circuits in [[Engineering:Electronic oscillator|oscillators]] and [[Engineering:Distributed element filter|distributed element filter]]s.  An open circuit stub of length <math>\scriptstyle l</math> will have a capacitive impedance at low frequency when <math>\scriptstyle \beta l < \pi /2</math>.  Above this frequency the impedance is inductive.  At precisely <math>\scriptstyle \beta l = \pi /2</math> 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,

::<math>\beta = {\omega \over v}</math>

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,

::<math>\omega_0 = \frac {\pi v}{2 l}</math>

While stubs function as resonant circuits, they differ from [[Physics:Lumped element model|lumped element]] resonant circuits in that they have multiple resonant frequencies; in addition to the fundamental resonant frequency <math>\scriptstyle \omega_0\,</math>, they resonate at multiples of this frequency: <math>\scriptstyle n\omega_0\,</math>.  The impedance will not continue to rise [[Monotonic function|monotonically]] with frequency after resonance as in a lumped tuned circuit.  It will rise until the point where <math>\scriptstyle \beta l = \pi </math> 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 <math>\scriptstyle \beta l = 3 \pi /2\,</math> 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|passband]]s are produced.<ref name=Gupta>Ganesh Prasad Srivastava, Vijay Laxmi Gupta, ''Microwave Devices and Circuit Design'', pp.29-31, PHI Learning, 2006 {{ISBN|81-203-2195-2}}.</ref>

Similarly, a short circuit stub is an anti-resonator at <math>\scriptstyle \pi /2</math>, that is, it behaves as a parallel resonant circuit, but again fails as <math>\scriptstyle 3 \pi /2</math> is approached.<ref name=Gupta/>

==Stub matching==
[[File:Stripline stub matching (v1).svg|thumb|300px|In a stripline circuit, a stub may be placed just before an output connector to compensate for minor mismatches due to the device's output load or the connector itself.]]<!-- to be uploaded when complete - don't delete -->
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.<ref>F.R. Connor, ''Wave Transmission'', pp.32-34, Edward Arnold Ltd., 1972 {{ISBN|0-7131-3278-7}}.</ref>

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]] but is optimised for impedance matching instead of passband transmission.  The resulting transmission function of the network has a passband [[Ripple (electrical)#Frequency-domain ripple|ripple]] like the Chebyshev filter, but the ripples never reach 0&nbsp;dB insertion loss at any point in the passband, as they would do for the standard filter.<ref>Matthaei, G.; Young, L.; Jones, E. M. T., ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'', pp.681-713, McGraw-Hill 1964.</ref>

==Radial stub==
[[File:Microstrip Low Pass Bowtie Stub Filter.jpg|thumb|A microstrip filter using butterfly stubs]]
Radial stubs are a planar component that consists of a sector of a circle rather than a constant-width line.  They are used with [[Engineering:Planar transmission line|planar transmission line]]s 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.<ref>Jia-Shen G. Hong, M. J. Lancaster, ''Microstrip Filters for RF/Microwave Applications'', pp. 188-190, Wiley, 2004 {{ISBN|0471464201}}.</ref>

{{Antennas|components}}

==References==
{{reflist}}

==See also==
* Quarter wave impedance transformer

[[Category:Antennas (radio)]]
[[Category:Distributed element circuits]]
[[Category:Telecommunications equipment]]
[[Category:Radio electronics]]

{{Sourceattribution|Stub (electronics)}}