**This post tells about short-circuited half wavelength line resonator. Half wavelength line resonator is depicted below, it is short circuited at one end.**

This transmission line as characterised with characteristic impedance ${Z}_{0}$, propagation constant $\beta $, attenuation constant $\alpha $, length $l=\frac{\lambda}{2}$, and resonant frequency $\omega ={\omega}_{0}$.

Input impedance here ${Z}_{in}={Z}_{0}\frac{\mathrm{tan}h\alpha l+j\mathrm{tan}\beta l}{1+j\mathrm{tan}\beta l*\mathrm{tan}h\alpha l}$.

In practice it is usually used low-losses transmission lines where $\alpha l\ll 1$, then $tahn\alpha l\simeq \alpha l$. Let’s consider the transmission line in the conditions close to resonant $\omega ={\omega}_{0}+\u2206\omega $, here $\u2206\omega $ is small.

Then $\beta l=\frac{{\omega}_{0}l}{{v}_{p}}+\frac{\u2206\omega l}{{v}_{p}}$, ${v}_{p}$ is a velocity of the transmission line. As we have half wavelength line resonator $l=\frac{\lambda}{2}=\frac{\pi {v}_{p}}{{\omega}_{0}}$.

For the condition of resonance we have $\omega ={\omega}_{0}$, $\beta l=\pi +\frac{\u2206\omega \pi}{{\omega}_{0}}$, and ${Z}_{in}\simeq {Z}_{0}(\alpha l+j\frac{\u2206\omega \pi}{{\omega}_{0}})$. Input impedance can be presented as ${Z}_{in}=R+2jL\u2206\omega $. So this transmission line can be presented as series RLC circuit, where $R={Z}_{0}\alpha l$, $L=\frac{{Z}_{0}\pi}{2{\omega}_{0}}$, and $C=\frac{1}{{{\omega}_{0}}^{2}L}$.

Resonance in short-circuited transmission line also happens for $l=\frac{n\lambda}{2},n=1,2...$ when $\u2206\omega =0$.

Unloaded quality factor for this transmission line is ${Q}_{0}=\frac{\beta}{2\alpha}$.

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