Formulas and measurements of the complex resonances envelope of a 2-conductor transmission line including skin effect, proximity effect, and dielectric losses

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In practical energy and communication applications a transmission line is not terminated with its characteristic impedance. Consequently, to avoid reflections it is utilised in frequencies where its physical length is significantly smaller or larger than the wavelength along it. Common examples are low-voltage energy transmission across the residential AC network, cables of household equipment and internal circuitry such as printed circuit boards. The resonance frequencies of the input impedance of any transmission line pose significant worst-case scenario cases where its magnitude becomes significantly larger or smaller than the magnitude of its characteristic impedance. Even though it is commonly accepted that the input impedance resonances attenuate and asymptotically reach the characteristic impedance with increasing frequency, analytical calculation of this attenuation at the intermediate bandwidth where resonances occur, remains a challenge. In this work it is shown that when the skin effect, proximity effect, and dielectric losses are considered, measurement and calculation can approximately coincide. Furthermore derivation of formulas that describe this complex attenuating envelope, is performed. This envelope indicates above which frequency the input impedance can be approximated with a constant complex characteristic impedance. Finally, a comparison with measurements is presented, exhibiting the significance of all frequency-dependent transmission line properties.