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

Input impedance for this resonator with length $l$ is ${Z}_{in}={Z}_{0}\mathrm{tan}h(\alpha +j\beta )l={Z}_{0}\frac{1\u2013j\mathrm{tan}h\alpha lcot\beta l}{\mathrm{tan}h\alpha l\u2013jcot\beta l}$.

If $l=\frac{\lambda}{4},\omega ={\omega}_{0},and\omega ={\omega}_{0}+\u2206\omega $.

Here we can conclude that $\beta l=\frac{\pi}{2}+\frac{\pi \u2206\omega}{2{\omega}_{0}}$, then $cot\beta l\approx \u2013\frac{\pi \u2206\omega}{2{\omega}_{0}}$. And $\mathrm{tan}h\alpha l\approx \alpha l$, then ${Z}_{in}\approx \frac{{Z}_{0}}{\alpha l+{\displaystyle \frac{\mathrm{j\pi}\u2206\mathrm{\omega}}{2{\omega}_{0}}}}$