Short-circuited half wavelength line resonator

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 $Z_{in}=Z_0\frac{\tan h\alpha l+j\tan \beta l}{1+j\tan \beta l*\tan h\alpha l}$.

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 $\omega ={\omega }_0+∆\omega$, here $∆\omega$ is small.

Then $\beta l=\frac{{\omega }_{0}l}{v_p}+\frac{∆\omega l}{v_p}$, $v_p$ 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 $\omega ={\omega }_0$, $\beta l=\pi +\frac{∆\omega \pi }{{\omega }_0}$, and $Z_{in}\simeq Z_0(\alpha l+j\frac{∆\omega \pi }{{\omega }_0})$.  Input impedance can be presented as $Z_{in}=R+2jL∆\omega$. 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 $∆\omega =0$.

Unloaded quality factor for this transmission line is $Q_0=\frac{\beta }{2\alpha }$.

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