GATE 2020 IN Q.45: solution and answer

GATE 2020 · INQ.45 · 2 marks · NAT

A metallic strain gauge of resistance $R_{x}$ with a gauge factor of 2 is bonded to a structure made of a metal with modulus of elasticity of 200 GN/m $^{2}$ . The value of $R_{x}$ is 1 k $Ω$ when no stress is applied. $R_{x}$ is a part of a quarter bridge with three identical fixed resistors of 1 k $Ω$ each. The bridge is excited from a DC voltage of 4 V. The structure is subjected to a stress of 100 MN/m $^{2}$ . Magnitude of the output of the bridge (in mV, rounded off to two decimal places) is ____

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Answer and solution

ANSWER

The magnitude of the bridge output is about 1.00 mV (0.9995 mV exactly, quarter bridge, 4 V excitation).

Official answer: 0.95 to 1.05Source: Official answer key, GATE 2020 IN (IIT Delhi)

Strain from Hooke's law: $ε = \frac{σ}{E} = \frac{100 \times 1 0 ^{6} N/m ^{2}}{200 \times 1 0 ^{9} N/m ^{2}} = 5 \times 1 0^{- 4}$ .
Gauge factor $G = \frac{Δ R / R}{ε}$ , so $\frac{Δ R}{R} = 2 \times 5 \times 1 0^{- 4} = 1 \times 1 0^{- 3}$ . With $R_{x} = 1 k Ω$ this gives $Δ R = 1 Ω$ .
Quarter bridge: the strain gauge $R_{x}$ sits in one arm and the other three arms are fixed $R = 1 k Ω$ resistors. The output is the difference of the two divider voltages: $V_{o} = V [\frac{R + Δ R}{2 R + Δ R} - \frac{1}{2}] = V \frac{Δ R}{2 ( 2 R + Δ R )}$
Substitute $V = 4 V$ , $R = 1000 Ω$ , $Δ R = 1 Ω$ : $V_{o} = 4 \times \frac{1}{2 \times 2001} = 0.9995 \times 1 0^{- 3} V = 0.9995 mV$
Check with the small-change form for a quarter bridge: $V_{o} \approx \frac{V}{4} \frac{Δ R}{R} = \frac{4}{4} \times 1 0^{- 3} V = 1 mV$ , which agrees.
Rounded to two decimal places, the magnitude of the output is $1.00 mV$ .

KEY IDEA

Convert stress to strain with Hooke's law, strain to $Δ R / R$ with the gauge factor, then use the quarter-bridge output $V_{o} \approx \frac{V}{4} \frac{Δ R}{R}$ (exact form $V \frac{Δ R}{2 ( 2 R + Δ R )}$ ).

WATCH OUT FOR

Using the half-bridge or full-bridge factor (V/2 or V instead of V/4) for a quarter bridge, or mixing units so that GN/m^2 and MN/m^2 are off by a factor of 1000 in the strain.

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GATE 2020 IN Q.45: solution and answer

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