Active strain gauges are connected in a half-bridge configuration to measure strain in a cantilever element. When powered by a voltage source of $10$ V, the voltage at the output of the bridge is $1$ mV. If the gauge factor is $2.5$, the strain in the cantilever element is ________ microstrain.

Question

Suresh said, "I did it yesterday."
Which one of the following options is the correct form of this sentence in indirect speech?

Answer 1

This method starts from how each gauge's resistance changes and builds the bridge output from there, instead of quoting the half-bridge sensitivity formula directly.

Resistance change per gauge: The gauge factor is defined by $GF = \dfrac{\Delta R/R}{\varepsilon}$, so each gauge's fractional resistance change is $\dfrac{\Delta R}{R} = GF \cdot \varepsilon$. On a bending cantilever, the gauge on the stretched face increases its resistance by $\Delta R$, and the gauge on the compressed face decreases its resistance by the same amount $\Delta R$.
Placing the gauges in the bridge: With the two gauges placed in adjacent arms of the Wheatstone bridge, one arm becomes $R+\Delta R$ and the neighboring arm becomes $R-\Delta R$, so the imbalance from the two gauges adds together instead of partially cancelling. This is what makes the circuit a half-bridge rather than a quarter-bridge.
Bridge output: For small $\Delta R/R$, the standard result for this two-active-arm arrangement is $$V_o = V_{ex} \cdot \frac{1}{2}\cdot\frac{\Delta R}{R} = V_{ex}\cdot\frac{GF\cdot\varepsilon}{2}$$
Substitute the numbers: $$1\times 10^{-3} = 10 \times \frac{2.5\,\varepsilon}{2}$$ $$1\times 10^{-3} = 12.5\,\varepsilon$$ $$\varepsilon = \frac{1\times 10^{-3}}{12.5} = 8\times 10^{-5}$$

Converting to microstrain, since $1$ microstrain equals $1\times 10^{-6}$ strain: $$\varepsilon = 8\times 10^{-5} = 80\times 10^{-6} = 80 \text{ microstrain}$$

Let's summarize:

A half-bridge with two gauges responding oppositely doubles the sensitivity of a single quarter-bridge gauge.
Mixing up the quarter-bridge, half-bridge, and full-bridge sensitivity factors ($4$, $2$, and $1$ in the denominator) is what produces the other answer options.

So the strain in the cantilever element is 80 microstrain, option (B).

Answer 2

This method starts from how each gauge's resistance changes and builds the bridge output from there, instead of quoting the half-bridge sensitivity formula directly.

Resistance change per gauge: The gauge factor is defined by $GF = \dfrac{\Delta R/R}{\varepsilon}$, so each gauge's fractional resistance change is $\dfrac{\Delta R}{R} = GF \cdot \varepsilon$. On a bending cantilever, the gauge on the stretched face increases its resistance by $\Delta R$, and the gauge on the compressed face decreases its resistance by the same amount $\Delta R$.
Placing the gauges in the bridge: With the two gauges placed in adjacent arms of the Wheatstone bridge, one arm becomes $R+\Delta R$ and the neighboring arm becomes $R-\Delta R$, so the imbalance from the two gauges adds together instead of partially cancelling. This is what makes the circuit a half-bridge rather than a quarter-bridge.
Bridge output: For small $\Delta R/R$, the standard result for this two-active-arm arrangement is $$V_o = V_{ex} \cdot \frac{1}{2}\cdot\frac{\Delta R}{R} = V_{ex}\cdot\frac{GF\cdot\varepsilon}{2}$$
Substitute the numbers: $$1\times 10^{-3} = 10 \times \frac{2.5\,\varepsilon}{2}$$ $$1\times 10^{-3} = 12.5\,\varepsilon$$ $$\varepsilon = \frac{1\times 10^{-3}}{12.5} = 8\times 10^{-5}$$

Converting to microstrain, since $1$ microstrain equals $1\times 10^{-6}$ strain: $$\varepsilon = 8\times 10^{-5} = 80\times 10^{-6} = 80 \text{ microstrain}$$

Let's summarize:

A half-bridge with two gauges responding oppositely doubles the sensitivity of a single quarter-bridge gauge.
Mixing up the quarter-bridge, half-bridge, and full-bridge sensitivity factors ($4$, $2$, and $1$ in the denominator) is what produces the other answer options.

So the strain in the cantilever element is 80 microstrain, option (B).

Show Hint

The Correct Option is B

Solution and Explanation

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