# Fano Resonance in Transmission Line

## DOI:

https://doi.org/10.20535/RADAP.2021.84.88-94## Keywords:

resonance characteristic, Fano resonance , transmission line## Abstract

**Introduction. **Wave phenomena are fundamental in our world. One of the key wave phenomena is resonance, in which the amplitude of the forced oscillations increases sharply. The resonance characteristic of an oscillator — an oscillatory structure with one resonant frequency — is symmetrical with a good approximation. In 1961, Ugo Fano obtained a formula for the asymmetric resonance, later called the Fano resonance. The Fano resonance is due to two-wave interference and is observed in structures of various wave nature. The uniqueness of the Fano resonance is due to the combination of full transmission and full reflection in it with a sharp transition between them, which explains the significant interest in it. For electromagnetic waves of the optical range, Fano resonance is considered in many articles and only a few papers are devoted to the Fano resonance in the radio range. The purpose of this article is to theoretically substantiate the possibility of Fano resonance characteristic forming in a transmission line (TL), as well as to compare a Fano resonance TL with a classical half-wave resonator.

**1 Symmetric resonance and Fano characteristics**. In many cases, at resonance, the frequency dependence of the normalized oscillation amplitude is approximated by a symmetric universal resonance curve. Initially, the Fano resonance was discovered for quantum-mechanical waves. According to the Planck formula as a result of the transformations, the relationship between universal resonance curve and Fano characteristic arguments is shown.

**2 Features of the Fano characteristic.** Symmetric resonance curve is determined by single parameter — the quality factor, and Fano characteristic — by two: the quality and the asymmetry factors. Comparison of Fano asymmetric resonance curve and symmetric universal resonance curve has been performed. For convenience of comparison of symmetric resonance curve and Fano characteristic we superpose their maxima shifting Fano characteristic to the left by a value of asymmetry factor to the minus one power and present the resulting expression in the form of symmetric resonance curve expression. If the modulus of the asymmetry factor is much greater than one, symmetric resonance curve and Fano characteristic in the passband are close. Fano characteristic maximum and minimum correspond to the full transmission and full reflection of the wave. To achieve the full reflection special solutions are needed. In such wave structures as reflectors, filters based on TL sections, limited periodic structures relations of the full reflection can only be approached, and their exact fulfillment requires physically unattainable conditions, for example, unlimited periodic structure.

**3 Full reflection in the transmission line with a stub**. The impedance conditions of the full reflection required for the formation the zero of Fano characteristic are analyzed. One of the solutions is achievable in a TL with open-circuited or short-circuited stub. Physical features of full reflection in a TL with open-circuited or short-circuited stub are considered.

**4 Fano resonance in the transmission lines based on stub and section. **Transmissin lines based on open-circuited or short-circuited stub and TL section are considered. Based on TL model mathematical analysis of transmission coefficient frequency characteristics of these structures is performed. It is shown that the Fano characteristic is formed in such structures. Transmission line parameters and calculated frequency characteristics are given. Comparison of obtained characteristics with Fano resonance curve as well as with classical half-wave TL resonator characteristic is made. Compared to half-wave resonator, TLs with Fano’s characteristic permit sufficiently increase quality factor, suppress responses on near harmonics and to decrease structure size.

**5 Fano resonance in the transmission lines based on two stubs.** Transmissin lines based on two open-circuited or open-circuited and short-circuited stubs are considered. Transmission line parameters and calculated frequency characteristics are given. It is shown, as in the previous case, that the Fano characteristic is formed in such structures and their selectivity is higher compared to half-wave resonator. Possibility of ultra-narrowband frequency characteristic realization by means of the structure based on open-circuited and short-circuited stubs is demonstrated.

**Results discussion. **The theoretical component of the obtained results is the established possibility of forming the Fano characteristic in TL with stubs, and the practical one is the possibility of increasing the quality factor and reducing the size of resonant structures based on TL with Fano characteristic against half-wave resonator. Of considerable interest is the possibility of implementing ultra-narrowband frequency characteristics with quality factor .

**Conclusion**. The Fano asymmetric resonance characteristic can be described by an expression, which shape is the same as for the symmetric universal resonance curve. Fano characteristic can be formed by TLs based on open-circuited or short-circuited stub with low-impedance TL section as well as by TLs based on two open-circuited stubs or one open-circuited and one short-circuited stubs. Compared to half-wave resonator in the first and the last two TLs quality factor is greater in 5.5, 11.6 and 47.4 times, dimensions are less in 1.5, 1.3 and 1.6 times respectively; there are no responses at the nearest harmonics.

## References

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*Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia*, (84), pp. 88-94. doi: 10.20535/RADAP.2021.84.88-94.

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