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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