In microwave and radio-frequency 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 open-circuit, or short-circuited (as is always the case for waveguides). 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.

The behaviour of stubs is due to standing 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.

Short circuited stub

The input impedance of a lossless, short circuited line is,

${\displaystyle Z_{\mathsf {sc}}~=~j\ Z_{0}\ \tan(\ \beta \ell \ )~}$

where

${\displaystyle \ j\ }$ is the imaginary unit (${\displaystyle \ j^{2}\equiv -1\ }$),

${\displaystyle \ Z_{0}\ }$ is the characteristic impedance of the line,

${\displaystyle \ \beta =2\pi /\lambda \ }$ is the phase constant of the line, and

${\displaystyle \ \ell \ }$ is the physical length of the line.

Thus, depending on whether ${\displaystyle \ \tan(\beta \ell )\ }$ is positive or negative, the short circuited stub will be inductive or capacitive, respectively.

The length of a stub to act as a capacitor C at an angular frequency of ${\displaystyle \ \omega \ }$ is then given by:

${\displaystyle \ell ~=~{\frac {1}{\ \beta \ }}\left~;}$

the length of a stub to act as an inductor L at the same frequency is given by:

${\displaystyle \ell ~=~{\frac {1}{\ \beta \ }}\left~,}$

where in both equations, 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

${\displaystyle Z_{\mathsf {oc}}=-j\ Z_{0}\ \cot(\ \beta \ell \ )~,}$

where the symbols ${\displaystyle \ Z_{0},\beta ,\ell ,\omega ,\ }$ etc. used in this section have the same meaning as in the section above.

It follows that depending on whether ${\displaystyle \cot(\beta \ell )}$ 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 ${\displaystyle \ \omega \ }$ is:

${\displaystyle \ell ~=~{\frac {1}{\ \beta \ }}\left~=~{\frac {1}{\ \beta \ }}\left~;}$

the length of an open circuit stub to act as a capacitor C at the same frequency is:

${\displaystyle \ell ~=~{\frac {1}{\ \beta \ }}\left~=~{\frac {1}{\ \beta \ }}\left~,}$

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

Resonant stubedit

Stubs are often used as resonant circuits in oscillators and distributed element filters. An open circuit stub of length ${\displaystyle \scriptstyle l}$ will have a capacitive impedance at low frequency when ${\displaystyle \scriptstyle \beta l<\pi /2}$. Above this frequency the impedance is inductive. At precisely ${\displaystyle \scriptstyle \beta l=\pi /2}$ 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,

${\displaystyle \beta ={\omega \over v}}$

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,

${\displaystyle \omega _{0}={\frac {\pi v}{2l}}}$

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 ${\displaystyle \scriptstyle \omega _{0}\,}$, they resonate at multiples of this frequency: ${\displaystyle \scriptstyle n\omega _{0}\,}$. 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 ${\displaystyle \scriptstyle \beta l=\pi }$ 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 ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Stub_(electronics)
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