An inexpensive pressure gauge can be made either to demonstrate physics and engineering principles in the classroom or as a substitute for more expensive gauges in locations where the accuracy of a commercial gauge is not required. You could convert a Bourdon gauge for use with a data acquisition system by attaching a strain gauge to it or -- as discussed in this blog -- create a pressure gauge by placing the strain gauge on a disk that has been welded to the end of a pipe. The gauge that is created here is used in the Multidisciplinary Engineering Lab at the Colorado School of Mines.
The theoretical stress on the center of a circular surface (disk) whose outer edge is supported and has a uniform load over the entire area is given by:
Where p is the pressure, r is the radius, and T is the thickness of the circular surface. The strain at the center would be the stress divided by the Young's Modulus (E) of the disk.
This would result in a change of resistance on the strain gauge of:
Where Gf is the gauge factor of the strain gauge. Now consider what happens when this strain gauge is placed in a quarter bridge circuit, as illustrated below (where Vs is the power supply voltage.
In this case, R2M is the strain gauge with a value of R+Δr. The remaining resistors are "fixed" or constant and are all assumed to be of value R. Based on this:
Now we can substitute the ΔR/R from the previous equation.
Solving the algebra for pressure (p), we find that:
For our device:
- E, Young's modulus for the brass plate = 15*106 PSI
- T, the thickness of the brass plate = 0.016 inches
- r, the radius of the brass plate = 0.75 inches
- Vs, the power supply to the quarter bridge = 5.00 Volts
- Gf, the gauge factor of the strain gauge = 2.09
This calculates to:
This gives a sensitivity of approximately 11 PSI per millivolt. The actual sensitivity will vary from this somewhat due to the fact that the entire strain gauge is not located in the center of the disk. (The strain drops by half once you get to the edge of the disk.) Also, the glue may add thickness to the plate, but it's a good starting point.
A potential thermal drift problem also exists when using a quarter bridge, since both the brass plate itself and the strain gauge have thermal coefficients of expansion. In order to compensate for those changes, another strain gauge mounted on another brass plate (which is at the same temperature as the gauge but not subject to any stress) would need to be added in a half-bridge configuration. You could add a resistor in series with the bridge circuit -- effectively lowering the power supply voltage (Vs). This resistance can be adjusted to give a desired sensitivity.
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