Calculating the electric relative permittivity of fluid or medium?

Question

I'm unsure of how to calculate the permittivity of a fluid. Permittivity differs from one fluid to another:

$$\varepsilon=\varepsilon_r\varepsilon_0$$

Since it is an electrical property combined with an electrical capacity, it is possible to measure it indirectly in a capacitive sensor. I have used a capacitive sensor to measure electric relative permittivity factor of a dielectric medium can be expressed as a ration of capacity, $C_x$ of capacitor, which space between and around the electrodes is completely filled with the medium, to capacity $C_0$ of the same electrodes in vacuum.

$$\varepsilon_r=\frac{C_x}{C_0} $$

I know and measured $C_0$ value. I am facing problems with calculating $C_x$ of capacitor, which space between and around the electrodes is completely filled with the medium. I am using method as described below. I am applying an AC signal ($125\: \mathrm{kHz}$) to the capacitive sensor which is filled with some fluid, in response I am getting AC signal with some phase difference. I am able to measure the amplitudes of sensor input, sensor output signals and their phase difference also. I am trying to make equation that will give $C_x$ value from above known values ($V_g$ (input), $V_r$ (output), $\phi$ (phase difference)).

Answer 1

I would use a capacitance bridge. This is by far the most accurate way to measure capacitance. If you're at a university there's probably someone around who has a commercial capacitance bridge. If not, you can easily make one.