Resonant stub tank circuits in
vacuum tube backpack UHF
transceiver, 1938. About 1/8 wavelength long:
(left) 200 MHz stub is 19 cm,
(right) 300 MHz stub is 12.5 cm
In microwave and radiofrequency engineering, a stub or resonant stub is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left opencircuit or (always in the case of waveguides) shortcircuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.
Stubs work by means of standing waves of radio waves along their length. Their reactive properties are determined by their physical length in relation to the wavelength of the radio waves. Therefore stubs are most commonly used in UHF or microwave circuits in which the wavelengths are short enough that the stub is conveniently small.^{[1]} They are often used to replace discrete capacitors and inductors, because at UHF and microwave frequencies lumped components perform poorly due to parasitic reactance.^{[1]} Stubs are commonly used in antenna impedance matching circuits, frequency selective filters, and resonant circuits for UHF electronic oscillators and RF amplifiers.
Stubs can be constructed with any type of transmission line: parallel conductor line (where they are called Lecher lines), coaxial cable, stripline, 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.
Contents

Short circuited stub 1

Open circuited stub 2

Resonant stub 3

Stub matching 4

References 5

See also 6
Short circuited stub
The input impedance of a lossless short circuited line is,


Z_\mathrm{SC} = j Z_0 \tan(\beta l)\,\!
where j is the imaginary unit, Z_0 is the characteristic impedance of the line, \beta = 2\pi/\lambda\, is the phase constant 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 of \omega is then given by:


l = \frac{1}{\beta} \left[(n+1)\pi  \arctan \left(\frac{1}{\omega C Z_0}\right) \right]
The length of a stub to act as an inductor L at the same frequency is given by:


l = \frac{1}{\beta} \left[ n \pi + \arctan\left(\frac{\omega L}{Z_0}\right) \right]
Open circuited stub
The input impedance of a lossless open circuit stub is given by


Z_\mathrm{OC} = j Z_0 \cot (\beta l) \,\!
It follows that 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{1}{\beta} \left[(n+1)\pi  \arccot\left(\frac{\omega L}{Z_0}\right) \right]
The length of an open circuit stub to act as a capacitor C at the same frequency is:


l = \frac{1}{\beta} \left[n \pi + \arccot\left(\frac{1}{\omega C Z_0}\right) \right]
Resonant stub
Stubs are often used as resonant circuits in oscillators and distributed element filters. 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 = {\omega \over v}
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 {\pi v}{2 l}
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 actually an antiresonance 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 totally failed. This response of the stub continues to repeat with increasing frequency alternating between resonance and antiresonance. 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 passbands are produced.^{[2]}
Similarly, a short circuit stub is an antiresonator at \scriptstyle \pi /2, that is, it behaves as a parallel resonant circuit, but again fails as \scriptstyle 3 \pi /2 is approached.^{[2]}
Stub matching
In a stripline circuit, a stub may be placed just before an output connector to compensate for small mismatches due to the device's output load or the connector itself.
Stubs can be used to 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. That is, the stub is made capacitive or inductive according to whether the main line is presenting 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 as well as the resistive part. Matching stubs can be made adjustable so that matching can be corrected on test.^{[3]}
A single stub will only achieve a perfect match at one specific frequency. For wideband matching several stubs may be used spaced along the main transmission line. The resulting structure is filterlike 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 like the Chebyshev filter, but the ripples never reach 0dB insertion loss at any point in the passband, as they would do for the standard filter.^{[4]}
References

^ ^{a} ^{b} Shuart, George W. (October 1934). "New high impedance lines replace coils" (PDF). Short Wave Craft (New York: Popular Book Corp.) 5 (6): 332–333. Retrieved March 24, 2015.

^ ^{a} ^{b} Ganesh Prasad Srivastava, Vijay Laxmi Gupta, Microwave Devices and Circuit Design, pp.2931, PHI Learning, 2006 ISBN 8120321952.

^ F.R. Connor, Wave Transmission, pp.3234, Edward Arnold Ltd., 1972 ISBN 0713132787.

^ Matthaei, G.; Young, L.; Jones, E. M. T., Microwave Filters, ImpedanceMatching Networks, and Coupling Structures, pp.681713, McGrawHill 1964.
See also
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