Resonators for accurate dielectric measurements in conducting liquids

Abstract

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.

Characterisation of a microwave re-entrant cavity resonator for phase-equilibrium measurements and new dew-point data for a (0.25 argon + 0.75 carbon dioxide) mixture

“…where 1/Qs is the contribution to the losses due to the surface resistivity of the bounding conductor and 1/Qe is the contribution to the losses due to the external measurement circuit. The 1/Qs contribution associated with the electrical resistivity of the gold plating matched that expected by theory only when Hamelin et al [20] increased the plating thickness to over 30 m. In part, this was required by their use of non-magnetic stainless steel for the resonator's body (which was also the pressure vessel), and because they conducted measurements with water, which lowered the cavity's resonant frequencies and increased the maximum mi-crowave penetration depth to about 16 m.…”

“…In this work, we were unable to manipulate the magnetic field probes but instead determined the loading of the resonance by the external circuit by measuring each of the complex scattering parameters S11, S22, S12, as well as S21 both at the resonance frequency and far from it. The analysis method described by Luiten [21] was then used to determine each probe's coupling coefficient, which in turn allowed the value of 1/Qe  1.7  10 -4 to be calculated for the lowest frequency mode; this is similar to the value found by Hamelin et al [20]. Accounting for this contribution to the losses of Mode 1, the value of Qs  1680 is obtained from the experimental Q-factor measurement of that resonance, which is about 30 % closer to the FEA calculated value.…”

“…Hamelin et al [20], who found that when their respective contributions were considered, the agreement between the measured and calculated Q-factors was within the combined uncertainties. Hamelin et al [20] calculated the Q-factors of their modes using the formula…”

“…Hamelin et al [20] measured the 1/Qe contribution associated with the external measurement circuit (VNA) by varying the orientation of their magnetic field coupling loops and reported 1/Qe  1.5  10 -4 for the lowest mode. In this work, we were unable to manipulate the magnetic field probes but instead determined the loading of the resonance by the external circuit by measuring each of the complex scattering parameters S11, S22, S12, as well as S21 both at the resonance frequency and far from it.…”

Accurate High‐Pressure Measurements of Carbon Monoxide's Electrical Properties

“…For the determination of the complex dielectric permittivity e r of the fluid loaded into the resonator we used the implicit model developed by Hamelin et al [37] According to this model, the complex dielectric permittivity is related to the measured complex resonant frequency via the following equation [Eq. (6)]:…”