Wheatstone bridge for strain gauges

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.

          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

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.