SQRMEM,wdh,thc,fsp,ymod,pois,sel
Piezoresistive pressure sensor with a square diaphragm

wdh       side length of the diaphragm in µm
thc         thickness of the diaphragm in µm
fsp         full scale pressure in psi
ymod    Young's modulus of the material in GPa
pois       Poisson's ratio
sel         number denoting the selected result.
              Use 1 for sensitivity, 2 for burst pressure and 3 for non-linearity

Notes

Diaphragms or membranes as it is sometimes called, form the basic structural element of MEMS pressure sensors. They are easy to fabricate, offer good dynamic response and can be used over a wide range of pressures. 

In this model, the outer edges of a square diaphragm are held firmly and a uniform pressure amounting to the full scale pressure is applied from top. The applied pressure will deflect the diaphragm till the elastic forces balance  the pressure. When the applied pressure is small, the deflection of the membrane is linear. But at higher pressures, the strain in the neutral axis will increase to a level which cannot be neglected. This will introduce a nonlinearity term in the calculation of deflection at higher pressures.

The piezoresistor should be located at maximum stress locations near the edge. The relative change in resistance of the piezoresistor is related to the stress, the piezoresistive coefficients and the angle of placement with respect to the edge of the membrane. Typically four piezoresistors, two placed longitudinally to the stress and two placed transversely are connected in a Wheatstone bridge.  The relative change in resistance of the piezoresistors are registered as an output voltage proportional to the supply voltage connected to the bridge. The supply voltage is assumed to be 5V. The piezoresistors are assumed to be of p-type with rectangular shape.

This design interface can be used to determine the sensitivity, deflection nonlinearity and burst pressure of a square diaphragm. Sensitivity is defined as the change in output voltage for a unit change in pressure at 5V supply. The output at any other supply voltage can be calculated from this. Deflection nonlinearity is defined as the difference in center deflection due to the nonlinearity term explained earlier, expressed as a percentage of the linear deflection. Burst pressure is a theoretical limit for the pressure that can be applied to the diaphragm without rupturing the diaphragm. The maximum allowable stress is assumed to be 2GPa for this calculation. This could vary depending on the features peculiar to the given membrane like etch deformities and defects.

The plot shows the span of the pressure sensor as a function of membrane thickness. Span is the output of the sensor at the full scale pressure and 5V supply to the Wheatstone bridge circuit. Using the crosshair tool, the span at any diaphragm thickness can be estimated.

Assumptions

-The default material is (100) Silicon with a Young's modulus of 180GPa and Poisson's ratio of 0.3.
-The diaphragm has uniform thickness.
-The diaphragm is assumed to be held rigidly at the outer edges. In reality it may be slightly flexible.
-The property of the material remain constant across the diaphragm.
-The diaphragm is not subjected to any pre-stress and the pressure is the only load acting on it.
-The piezoresistors are placed at the maximum stress locations oriented to give the maximum change in resistance.
-The piezoresistors are connected in a Wheatstone bridge with 5V supply.
-The influence of temperature variation is not considered
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