Solution - Measurement
he following is schematic
diagram of the oscillator circuit used to determine the capacitance Cx.
The formula for the oscillator circuit’s frequency is:
when Cx is not attached.
When Cx is switched into the circuit the formula becomes:
Solving for CX:
Which gives us a relative capacitance factor that we use to determine the water level.
Using this equation, the only factors in the circuit are the two capacitors. C1 was chosen to have a very small temperature coefficient. It is a polyester capacitor. This capacitor acts as a reference capacitor; it cannot change value (with temperature or time) or the measured results will be wrong.
The software uses this value to determine the level of the water in the system. The following is a mathematical model of the capacitance due to the probe array. The model closely resembles a coaxial cable structure. The closely spaced outer probes act as a solid outer ground to the middle conductor.
This above configuration can be approximated as a coaxial cable due to the relatively close outer conductor arrangement. The following diagram shows the electric and magnetic field in a coaxial cable.
The formula for capacitance in a coaxial conductor is:
Where K is the dielectric constant, L is the length, a is the radius of the inner conductor and b is the radius of the outer conductor. Our variable is the length. The system can be modeled as two capacitors placed in parallel. One of these capacitors has oil as its dielectric, and the other has water, since oil has a dielectric constant of approximately 3 and water has a K of approximately 80 we are able to determine the oil/water transition level.
When a theoretical plot of the capacitance vs. the water’s height is made we get the following result:
This plot shows the linear relationship of capacitance vs. height.
An experiment performed before the completion of the system (and therefore with different absolute values for capacitance yielded the following result.
The blue line represents the measured values, the red represents the linear best fit line. This experiment verified the linear nature of capacitance as a function of water height, and therefore the ability of the system to determine the water height based on measured capacitance.