The Use of Flexural Resonator Thermal Noise for Simultaneous Measurements of Fluid Density, Viscosity and Dielectric Constant
Publication Date: 2005-Aug-15
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Collecting the noise signal from the flexural resonator and fitting the spectrum of this noise to the response of the resonator's equivalent circuit allows the measurement of fluid density, viscosity and dielectric constant. This method does not require any excitation source for the resonator.
Collecting the noise signal from the flexural resonator and fitting the spectrum of this noise to the response of the resonator’s equivalent circuit allows the measurement of fluid density, viscosity and dielectric constant. This method does not require any excitation source for the resonator
The experiment below is a realistic simulation of what can be observed using spectrum analyzer.
1. Thermodynamic limits to density, viscosity and dielectric constant detection
The fundamental limits of a sensor, such as a tuning fork, ability to accurately record the resonant curve and hence measure the liquid parameters are defined by the intrinsic system noise. It is possible to determine these limits experimentally by measuring the output noise with no test current passing through the tuning forks. A noise analysis of the equivalent circuit shows that the two primary noise sources are the Brownian noise associated with mechanical dissipation in the fork, as manifested by the R in the series arm of the equivalent circuit and the Brownian noise associated with the viscous dissipation in the surrounding liquid, as represented by the real part of the impedance Z (see Fig. 1).
Fig 1. Sensor equivalent circuit
Equivalent Johnson noise of the equivalent mechanical loss resistor, (4kBTR) (1/2) in units of V/Hz which does not depend on the frequency and the equivalent Johnson noise of the viscous impedance Z is (4kBTB), which is frequency dependent. Noise density of the both sources are calculated for the typical tuning fork values of R = 41740.2403, B= 42678.7, 0.863.5. The sensor loss produces noise of 3.1E-7 V/ÖHz and the viscous noise is shown in Fig. 2.
For this particular sensor in this particular liquid the viscous noise is about 20 to 30 times higher than the loss noise in the sensor itself. The viscous noise scales with the square root of the viscosity density product.
To determine the influence of these two sources of noise on the sensor impedance measurements it is necessary to take into account the fact that they are buried inside the equivalent circuit (see Fig.) and the transfer function G() of the tuning fork has to be determined, which is:
As it can be seen from the above formula, both the sensor transfer function and the noise voltage depend on the viscosity and the density of the liquid. For the same typical values mentioned above and also Cs = 3.51E-15, L = 6722.7542, Cp = 4.37E-12 and A = 1643.4 this transfer function is shown in Fig. 3:
Fig 2. Viscous noise density
Fig 3. Transfer function G()
Taking into account that the two noise sources are uncorrelated, the noise voltage produced at the output contacts of the senor can now be calculated as follows:
To estimate an error of the sensor impedance measurements, the RMS voltage produced by both noise sources...