Dimensional analysis of a permittivity measurement probe

Open-ended coaxial-line probes provide a convenient means of determining the dielectric properties of many materials over a relatively wide frequency range. Because of this, much attention has been given to understanding the interaction of the probe and the material which 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 permittivity measurement probe where the admittance of the probe is a function of permittivity, frequency, and any number of linear dimensions.