Using Capacitors to Measure Fluid Level

Measuring the level of a fluid is useful for a variety of applications, and the technology for the techniques employed in the acquisition of this measurement has progressed far beyond the use of sight glasses and mechanical floats. In fact, a widely used method to measure the amount of fuel in a gas tank is with a device that floats on top of the fuel combined with a sensor, the fuel gauge sending unit, that translates the angle of the float to the amount of fluid in the tank. With this method, the gauge tends to change position with the angle of the car as well as the angle of the float relative to the fluid, so a lot of the time the gauge position can be misleading. A more modern technique of measuring fluid level involves capacitors, and this article will detail the concepts and mathematics behind the relationship with fluid height.

Conceptual Background

Capacitors

For more details, see capacitor.

A parallel plate capacitor with conductive plates of area [math]\displaystyle{ A }[/math] separated by a dielectric with a gap distance of [math]\displaystyle{ d }[/math].

A capacitor consists of two conductors, e.g. conducting plates, separated by some kind of insulator. The insulator--a dielectric--within the gap between the two conductors can be air, plastic, glass, etc. Additionally, the conductors have to be connected to some sort of power supply in order to acquire a buildup of charge on the surface of the conductors.

The magnitude of the electric field within a capacitor is [math]\displaystyle{ \left| \vec{E}_{capacitor} \right| = \frac{Q/A}{\epsilon_{0}} }[/math], where the gap is only occupied by free space. [math]\displaystyle{ Q }[/math] is the charge on a plate and [math]\displaystyle{ A }[/math] is the area of a plate. Additionally, [math]\displaystyle{ \epsilon_{0} }[/math] is the permittivity of free space as detailed here.

With a dielectric involved, we apply the equation [math]\displaystyle{ \vec{E}_{dielectric} = \frac{\vec{E}_{applied}}{K} }[/math], where [math]\displaystyle{ K }[/math] is the dielectric constant of the material, and get [math]\displaystyle{ \left| \vec{E}_{dielectric} \right| = \frac{Q/A}{K\epsilon_{0}} }[/math]. The capacitance, [math]\displaystyle{ C=\frac{Q}{V} }[/math] (where [math]\displaystyle{ V }[/math] is the voltage) or [math]\displaystyle{ C=\frac{\epsilon_0 A K}{d} }[/math], therefore changes with a varying dielectric constant.

Relating Capacitance to Fluid Level

When multiple materials are between the gap, for instance water and air, the overall capacitance would be [math]\displaystyle{ C=\epsilon_0 A (\frac{K_{water}}{d_{water}}+\frac{K_{air}}{d_{air}}) }[/math], where [math]\displaystyle{ d=d_{water}+d_{air} }[/math] and [math]\displaystyle{ K_{air} \approx 1 }[/math].

A Computational Model

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