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

In practice it is usually used low-losses transmission lines where $\alpha l\ll 1$ , then $\tan h\alpha l\simeq \alpha l$. 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 $Z_{in}\simeq \frac{Z_0}{\alpha l+{\displaystyle \frac{j\pi ∆\omega }{2{\omega }_0}}}$. This result can be interpreted like the impedance of parallel RLC circuit $Z_{in}=\frac{1}{{\displaystyle \frac{1}{R}}+2j∆\omega C}$. 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.

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