In microwave and radio-frequency engineering, a stub or resonant stub is a transmission line or waveguide 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. They are often used to replace discrete capacitors and inductors, because at UHF and microwave frequencies lumped components perform poorly due to parasitic reactance. 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,
Z s c = j Z 0 tan ( β ℓ ) {\displaystyle Z_{\mathsf {sc}}~=~j\ Z_{0}\ \tan(\ \beta \ell \ )~}
where
j {\displaystyle \ j\ } is the imaginary unit ( j 2 ≡ − 1 {\displaystyle \ j^{2}\equiv -1\ } ),
Z 0 {\displaystyle \ Z_{0}\ } is the characteristic impedance of the line,
β = 2 π / λ {\displaystyle \ \beta =2\pi /\lambda \ } is the phase constant of the line, and
ℓ {\displaystyle \ \ell \ } is the physical length of the line. Thus, depending on whether tan ( β ℓ ) {\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 ω {\displaystyle \ \omega \ } is then given by:
ℓ = 1 β [ ( n + 1 ) π − arctan ( 1 ω C Z 0 ) ] ; {\displaystyle \ell ~=~{\frac {1}{\ \beta \ }}\left[\ (n+1)\ \pi \ -\ \arctan \left({\frac {1}{\ \omega CZ_{0}\ }}\right)\ \right]~;}
the length of a stub to act as an inductor L at the same frequency is given by:
ℓ = 1 β [ n π + arctan ( ω L Z 0 ) ] , {\displaystyle \ell ~=~{\frac {1}{\ \beta \ }}\left[\ n\ \pi \ +\ \arctan \left({\frac {\ \omega L\ }{\ Z_{0}\ }}\right)\ \right]~,}
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
Z o c = − j Z 0 cot ( β ℓ ) , {\displaystyle Z_{\mathsf {oc}}=-j\ Z_{0}\ \cot(\ \beta \ell \ )~,}
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