Question

A catheter based arterial blood pressure measurement device uses a flexible diaphragm mounted with four identical strain gauges in a Wheatstone bridge configuration as shown in the figure. Assume that the strain gauges have a nominal resistance value of \(R_G = 10\ \text{k}\Omega\), Gauge Factor \(G = 40\) and Young’s Modulus \(E = 10\ \text{MPa}\). Blood pressure variations result in small finite change in strain \(\varepsilon (\varepsilon>0)\). If \(V_o\) is the output voltage of the Wheatstone bridge and \(\sigma\) is the stress in MPa, the sensitivity \[ \frac{V_o}{\sigma} \] is ________ V.MPa\(^{-1}\).

Hint:

The sensitivity of a strain gauge in a Wheatstone bridge is directly proportional to the gauge factor and inversely proportional to the Young's Modulus.

Answer 1

Correct Answer: 8

The strain gauge resistance variation is related to the applied strain \(\varepsilon\) by the formula: \[ \Delta R = G R_G \varepsilon \] In the Wheatstone bridge configuration, the output voltage is given by: \[ V_o = \frac{2 V}{R_G} \times \Delta R = \frac{2 V}{R_G} \times G R_G \varepsilon \] Simplifying: \[ V_o = 2 V \times G \varepsilon \] The sensitivity is the output voltage per unit stress: \[ \frac{V_o}{\sigma} = 2 V \times G \frac{\varepsilon}{\sigma} \] Given Young's Modulus \(E = 10\ \text{MPa}\), the strain is: \[ \varepsilon = \frac{\sigma}{E} \] Thus, \[ \frac{V_o}{\sigma} = 2 V \times G \times \frac{1}{E} \] Substitute the given values: \[ \frac{V_o}{\sigma} = 2 \times 40 \times \frac{1}{10} = 8\ \text{V.MPa}^{-1} \] Thus, the sensitivity is: \[ \boxed{8} \] Final Answer: 8 V.MPa\(^{-1}\)