RF Electronic Devices and Systems

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

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 $\omega ={\omega }_{0}+∆\omega$, 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}$$\beta l=\pi +\frac{∆\omega \pi }{{\omega }_{0}}$, and ${Z}_{in}\simeq {Z}_{0}\left(\alpha l+j\frac{∆\omega \pi }{{\omega }_{0}}\right)$.  Input impedance can be presented as ${Z}_{in}=R+2jL∆\omega$. So this transmission line can be presented as series RLC circuit, where $R={Z}_{0}\alpha l$$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  when $∆\omega =0$.

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

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