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# Short-circuited quarter wave transmission line

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

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 $\omega ={\omega }_{0}+∆\omega$.  Then $\beta =\frac{\pi }{2}+\frac{\pi ∆\omega }{2{\omega }_{0}}$.

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

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.

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