Limitations of the transmission line theory in the simulation of ultra thin wire conductivities with coaxial resonators

This work aims to illustrate how the transmission line theory may yield to strong errors in predicting the reflection coefficient at the input of a short - circuited coaxial transmission line when the inner conductor is either thin enough or lossy enough to have a measurable DC ohmic resistance. These situations are found in the measurement of wire conductivities at RF frequencies with coaxial resonators. The transmission line theory is based on the assumption that the current flowing through a conductor is located at its surface. However, this hypothesis is not accurate in a number of practical situations. In order to identify some of them, we have made a comparison between the simulations and measurements of the reflection coefficient at the input of a short circuited coaxial resonator where the centre conductor is a thin wire made of either a good conductor (high purity platinum) or a poor one (carbon fibre). The transmission line theory is not able to predict accurately the conductivity of any of these wires at frequencies below 300-500 MHz, where there is a strong frequency dependence of the resistance used in the modelling of the distributed conductor losses. In order to account for this effect, a model is proposed, which is based on modifying the distributed resistance, which account for conductor losses. This model is computationally effortless and was verified with measurements of different materials and with different thicknesses.