Electronic – How to calculate permittivity of a fluid

analogcapacitorsensor

I am confused about calculating permittivity of a fluid. Permittivity differs from one fluid to another.

$$\epsilon=\epsilon_r\epsilon_0$$

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 the electric relative permittivity factor of a dielectric medium, which can be expressed as a ratio of capacity \$C_x\$ of a capacitor whose space between and around the electrodes is completely filled with the medium, to capacity \$C_0\$ of the same electrodes in vacuum.

$$\epsilon_r= C_x/C_0$$

I know and measured \$C_0\$ value. I am facing problems with calculating \$C_x\$ of the capacitor whose space between and around the electrodes is completely filled with the medium.

I am using method as described below.

I am applying an A.C signal (125 KHz) to the capacitive sensor which is filled with some fluid; in response I am getting an A.C. signal with some phase difference. I am able to measure the amplitudes of the sensor input and sensor output signals and their phase difference also. I am trying to make an equation that will give \$C_x\$ value from above known values (\$V_g\$ (input), \$V_r\$ (output), \$\phi\$ (phase difference)).

One approach is to compare it with a known capacitor C1 in a bridge. If you will excuse very crude ASCII circuits...

Either of the following combinations would be suitable

AC --- C1 --- V1 --- CX --- Gnd
|----- R1 --- V2 --- RX --- Gnd


or

AC --- RX --- V1 --- CX --- Gnd
|----- R1 --- V2 --- C1 --- Gnd


C1 should be about the centre of the expected permittivity range times C0.

In either case, adjust RX until V1=V2 (the output of a differential amplifier calculating V2-V1 is 0) at which point RX/R1 = CX/C1 giving you an easy reading of permittivity.

It will also tell you if the medium is lossy (e.g. if the fluid is conductive to some extent). In that case, there is no zero on V2-V1, but just a minimum, where V2-V1 is 90 degrees out of phase with Vin. The magnitude of the out-of-phase component gives some estimate of the loss.