Dimensional analysis of an open-ended coaxial-line probe

Open-ended coaxial-line probes provide a convenient means of determining the dielectric properties of many materials over a relativity wide frequency range. Because of this, much attention has been given to understanding the interaction of the probe and the material it is inserted into. In this paper a dimensional analysis was performed on a generalized open-ended coaxial-line probe. Applying the Buckingham Π-theorem revealed that the admittance of the probe/dielectric interface, scaled by the frequency, is a function of a single dimensionless variable. This fact greatly simplifies the modeling of the probe. The problem is reduced from fitting a model of two variables, frequency and permittivity, to one dimensionless variable. In addition the dimensional analysis also revealed that the same results hold for any dielectric probe where the admittance of the probe is a function of permittivity, frequency and any number of linear dimensions