---
title: "Wheatstone bridge for strain gauges"
subject: "Instrumentation and Measurement"
chapter: 6
key: "semester_4/instrumentation_and_measurement:wheatstone-bridge-strain-gauge-circuits"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/wheatstone-bridge-strain-gauge-circuits
site: BCT Engineering
---

# Wheatstone bridge for strain gauges

Instrumentation and Measurement · Chapter 6: Strain Gage

- Asked in: year not given, 8 marks
- Video: [Wheatstone  Bridge with strain gauge explained](https://www.youtube.com/watch?v=GjmM6KKhgp0) — learnchannel
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/wheatstone-bridge-strain-gauge-circuits
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-6

## Question

Derive the output of a Wheatstone bridge with all four arms active. Hence explain the quarter-bridge, half-bridge and full-bridge arrangements for strain measurement and how temperature compensation is obtained.

## Short answer

For four active arms, Vo = Vex/4(dR1/R - dR2/R + dR3/R - dR4/R), or Vex G/4(eps1 - eps2 + eps3 - eps4). Quarter bridge uses one active gauge, half bridge two, full bridge four; adjacent arms subtract and opposite arms add, giving temperature compensation.

## Answer

A Wheatstone bridge converts the small resistance changes of strain gauges into a measurable voltage.

#### Bridge output
Four arms $R_1$ (between A and B), $R_2$ (B-C), $R_3$ (C-D), $R_4$ (D-A). The excitation $V_{ex}$ is applied across A and C, and the output is measured between B and D.

```text
          A
        /   \
      R1     R4
      /       \
     B         D     output V_o between B and D
      \       /
      R2     R3
        \   /
          C   (V_ex across A and C)
```

$$
V_o = V_{ex}\left[\frac{R_1}{R_1 + R_2} - \frac{R_4}{R_3 + R_4}\right]
$$

The bridge is **balanced** ($V_o = 0$) when $R_1R_3 = R_2R_4$.

Let each arm be $R$ initially and change by $\Delta R_i$. For small changes:

$$
V_o = \frac{V_{ex}}{4}\left[\frac{\Delta R_1}{R} - \frac{\Delta R_2}{R} + \frac{\Delta R_3}{R} - \frac{\Delta R_4}{R}\right]
$$

With $\Delta R/R = G\varepsilon$:

$$
V_o = \frac{V_{ex}G}{4}\left(\varepsilon_1 - \varepsilon_2 + \varepsilon_3 - \varepsilon_4\right)
$$

Adjacent arms subtract and opposite arms add, which is the basis of compensation.

#### Arrangements

| Bridge | Active gauges | Output | Remarks |
|---|---|---|---|
| Quarter | 1 | $V_{ex}G\varepsilon/4$ | Dummy gauge in adjacent arm for temperature compensation |
| Half | 2 (one tension, one compression) | $V_{ex}G\varepsilon/2$ | Bending beam; compensates temperature and axial load |
| Full | 4 (two tension, two compression) | $V_{ex}G\varepsilon$ | Highest output, best compensation |

#### Temperature compensation
Temperature changes the resistance of all gauges equally (the same $\Delta R_T$). Because adjacent arms subtract, this common change cancels. In a quarter bridge a **dummy gauge** (same type, same temperature, mounted on unstrained material) is placed in the adjacent arm. In half and full bridges the active gauges act as each other's compensators. Strain from bending or axial load can similarly be separated by placing gauges in the correct arms (e.g. on top and bottom of a beam).

#### Comments
- Output of a quarter bridge is slightly nonlinear for large $\Delta R/R$; half and full bridges are more linear.
- A three-wire connection compensates the lead resistance.
- Balancing and calibration (shunt calibration) are done before use.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. When is the Wheatstone bridge balanced (Vo = 0) in the answer's notation?
   A. R1 + R3 = R2 + R4
   B. R1 R3 = R2 R4
   C. R1 = R2 + R3
   D. R1 R2 = R3 R4
   Answer: B. From Vo = Vex[R1/(R1+R2) - R4/(R3+R4)], Vo = 0 when R1 R3 = R2 R4. (Key point: Balance condition)

2. For the small-change output, how do strains in different arms combine?
   A. All arms add
   B. Adjacent arms subtract and opposite arms add
   C. Adjacent arms add and opposite arms subtract
   D. Only arm 1 contributes
   Answer: B. Vo = (Vex G/4)(e1 - e2 + e3 - e4), so adjacent arms subtract, opposite arms add. (Key point: Adjacent and opposite arms)

3. For equal strain magnitude, which ordering of outputs is correct?
   A. Quarter = Vex G e, Half = Vex G e / 2, Full = Vex G e / 4
   B. Quarter = Vex G e / 4, Half = Vex G e / 2, Full = Vex G e
   C. All three give the same output
   D. Quarter is largest because it has one gauge
   Answer: B. Output scales with the number of active gauges: one, two or four quarters of Vex G e. (Key point: Output of arrangements)

4. How does a dummy gauge compensate temperature in a quarter bridge?
   A. It is placed in the opposite arm so the changes add
   B. It is the same type at the same temperature on unstrained material in the adjacent arm, so the common change cancels
   C. It is heated to a fixed temperature
   D. It is placed under strain to double the output
   Answer: B. Temperature changes both gauges equally and adjacent arms subtract, so only the strain signal remains. (Key point: Temperature compensation)

## Related in Instrumentation and Measurement

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- [Gauge resistance change and quarter-bridge output](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Agauge-resistance-change-quarter-bridge) — Instrumentation and Measurement, chapter 6
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