To automate the pressure measurement in a mercury-filled U tube, a Wheatstone Bridge can be fabricated by connecting two external resistance to a high-resistance wire threading the interior of the U tube, as shown in the schematic below.
U Tube Pressure Sensor
The resistance of the U tube wire is proportional to its current-carrying length. The two parts of the wire external to the mercury will carry current and therefore will impart resistances to the circuit. However, the immersed portion of the wire carries no current, since the current will instead travel through the highly-conductive mercury. The U tube wire is effectively separated into two separate resistances, each resistance dependent upon the wire length above the mercury. As a result, the difference in the resistance of these two wire segments will be proportional to the pressure difference across the U tube,
where c and k = cRw are factors that can be obtained during calibration.
For an initially balanced Wheatstone Bridge, the voltage output is given by,
where r = Rw / RRef is the efficiency of the bridge circuit.
Thus, the unknown gas pressure (with respect to the reference pressure) is proportional to the output voltage,