**This post tells about short-circuited quarter wave transmission line. Quarter wave line resonator is depicted below, it is short-circuited at one end and opened at the other. The length of this transmission line is $\frac{\lambda}{4}$.**

This transmission line as characterised with characteristic impedance ${Z}_{0}$, propagation constant $\beta $, attenuation constant $\alpha $, length $l=\frac{\lambda}{4}$, and resonant frequency $\omega ={\omega}_{0}$.

The input impedance of the short-circuited quarter wave transmission line is ${Z}_{in}={Z}_{0}\frac{1\u2013j*\mathrm{tan}h\alpha l*cot\beta l}{\mathrm{tan}h\alpha l\u2013j*cot\beta l}$.

In practice it is usually used low-losses transmission lines where $\alpha l\ll 1$ , then $\mathrm{tan}h\alpha l\simeq \alpha l$. If we have a transmission line with length $l=\frac{\lambda}{4}$at resonance $\omega ={\omega}_{0}$. Let’s consider a close to resonance situation $\omega ={\omega}_{0}+\u2206\omega $. Then $\beta =\frac{\pi}{2}+\frac{\pi \u2206\omega}{2{\omega}_{0}}$.

After simplification ${Z}_{in}\simeq \frac{{Z}_{0}}{\alpha l+{\displaystyle \frac{j\pi \u2206\omega}{2{\omega}_{0}}}}$. This result can be interpreted like the impedance of parallel RLC circuit ${Z}_{in}=\frac{1}{{\displaystyle \frac{1}{R}}+2j\u2206\omega C}$. Resistance, capacitance and inductance here can be represented as: $R=\frac{{Z}_{0}}{\alpha l},C=\frac{\pi}{4{\omega}_{0}{Z}_{0}},L=\frac{1}{{{\omega}_{0}}^{2}C}$.

Unloaded quality factor for this resonator is ${Q}_{0}={\omega}_{0}RC=\frac{\pi}{4\alpha l}=\frac{\beta}{2\alpha}$.

The application of quarter wave transmission line is impedance matching and impedance inversion in size relevant electronic structures, another application in RF/DC coupling in transistor amplifiers.

More educational content can be found at our Reddit community** r/ElectronicsEasy.**