Resonators for accurate dielectric measurements in conducting liquids

The compact, rugged, re-entrant radio-frequency resonator [A. R. H. Goodwin, J. B. Mehl, and M. R. Moldover, Rev Sci. Instrum. 67, 4294 (1996)] was modified for accurate measurements of the zero-frequency dielectric constant (relative electric permittivity) εr of moderately conducting liquids such as impure water. The modified resonator has two modes with frequencies near 216/εr MHz and 566/εr MHz. The results for εr at both frequencies were consistent within 0.0002εr, verifying that the low-frequency limit had been attained with water samples with conductivities in the range 100–2500 μS/m. The results for water and for the insulating liquid cyclohexane were within 0.0005εr of literature values. The present analysis is based on a simplified equivalent circuit that accounts for the loading of the resonator by the external instrumentation. This circuit can easily be generalized for a resonator with three or more modes. The present resonator has a thick gold plating on its interior surfaces. With the plating, the quality factors Q of the resonances varied in a predictable way with frequency and temperature. Predictable Qs were essential for obtaining accurate values of εr.