Question:
Published on: 23 June, 2022

Define gauge factor of a strain gauge & obtain its expression.

Gauge factor:

It is defined as the ratio of per unit change in resistance to per unit change in length.

Expression: Gauge factor $$\left(G_f\right)=\frac{\Delta R/R}{\Delta L/L}$$

Where, ΔR=Corresponding charge in resistance R

ΔL=Change in length per unit length L

The resistance of the wire of strain gauge R is given by

$$R=\frac{\rho L}{A}$$

Where $$\rho$$ =Resistivity of the material of wire

L=Length of wire

A=cross-sectional area

=KD2, K & D being a constant and diameter of wire respectively

When the wire is strained, its length increases and lateral dimension is reduced as a function of poisson’s ratio (m); consequently there is an increase in resistance.

$$R=\frac{\rho L}{KD^2}$$

Differentiating

$$dR=\frac{KD^2\left(\rho dL+Ld\rho\right)+\rho L\left(2KD.dD\right)}{\left(KD\right)^2}=\frac{1}{\left(KD\right)^2}[\left(\rho.dL+L.d\rho\right)+2\rho L\frac{dD}{D}$$

$$\frac{dR}{R}=\frac{\frac{1}{\left(KD\right)^2}\left[\rho.dL+L.d\rho-2\rho L\frac{dD}{D}\right]}{\frac{\rho L}{KD^2}}=\frac{dL}{L}+\frac{d\rho}{\rho}-2\frac{dD}{D}$$

Poisson’s ratio $$\left(\mu\right)=\frac{Lateral\ strain}{Longitudinal\ strain}=\frac{-\frac{dD}{D}}{\frac{dL}{L}}$$

$$\frac{dD}{D}=\mu\ast\frac{dL}{L}$$

For small variation, the above relationship can be written as

$$\frac{dR}{R}=\frac{dL}{L}+2\mu\frac{dL}{L}+\frac{d\rho}{\rho}$$

Gauge factor $$\left(G_f\right)=\frac{\Delta R/R}{\Delta L/L}$$

$$\frac{dR}{R}=G_f\frac{dL}{L}=G_f\ast e$$, Where e=strain $$\frac{dL}{L}$$

Gauge factor $$\left(G_f\right)=1 + 2 \mu + \frac{\Delta \rho/\rho}{\Delta L/L}$$

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