Strain Gauge: Working Principle & Diagram

In this lecture, we are going to learn about the Strain Gauge and its working principle. Then we will see the detailed derivation of the Gauge factor (Gf). So let’s start with the definition and basic details of Strain Gauge.

Strain Gauge

• The “Strain Gauge” is an example of a passive transducer that converts a mechanical displacement into a change in resistance. A strain gauge is a thin, wafer-like device that can be attached to a variety of materials to measure applied strain.

• In this transducer principle used is the “piezoresistive effect“.

• If a metal conductor is stretched or compressed, its resistance changes on account of the fact that both the length and diameter of the conductor change. Also, there is a change in the value of resistivity of the conductor when it is strained and this property is called the piezoresistive effect.

• Therefore, resistance strain gauges are also known as piezoresistive gauges. resistance of a metallic conductor changes if there are dimension changes due to the applied strain.

Derivation of Gauge Factor (G_f) |Strain Gauge Factor Derivation

• Consider a strain gauge made of circular wire having length ‘L’, area ‘A’, and diameter ‘D’ before being strained. The material of the wire has a resistivity \rho.

\therefore Resistance of unstrained gauge (R) =\frac{\rho L}{A} …..Eq.(1)

When a tensile force ‘F’ is applied the circular wire changes its dimensions as in the figure below.

• \Delta L \;\rightarrow Change in length
• \Delta A \; \rightarrow Change in area
• \Delta D \; \rightarrow Changes in diameter
• \Delta R \; \rightarrow Change in resistance

Longitudinal Strain =\frac{\partial L}{L} in the direction of stress applied.

Lateral Strain =-\frac{\partial D}{D}, in the perpendicular direction of stress.

Poisson’s ratio=\frac{Lateral\;strain}{Longitudinal\;strain}

v=-\frac{\rho D/D}{\partial L/L}

From Equation (1)

R=\frac{\rho L}{A} \; and \; A=\pi r^2,

also, r=\frac{D}{2},

A=\pi \frac{D^2}{4}

R=\frac{4}{\pi}. \frac{\rho L}{D^2} …… Eq. (2)

Due to change in dimension inter atomic distance changes and hence the time between successive collisions and hence mean free path i.e. between two successive collision changes. This effect is said to be a piezoresistive effect. But here we consider negligible piezoresistive effect for sake of simplicity.

Taking natural log in the equation (2)

\ln R=\ln \left ( \frac{4 \rho}{\pi} \right )+ \ln l - 2\ln D

\ln \left( \frac{4 \rho}{\pi} \right ) is constant.

Change is resistance, i.e. (differentiate equation 6.8)

\frac{\partial R}{R}=\frac{\partial L}{L}- 2 \frac{\partial D}{D}

\frac{\Delta R}{R}=\frac{\Delta L}{L}- 2 \frac{\Delta D}{D}

Divide the above equation by \frac{\Delta L}{L}

\frac{\Delta R/R}{\Delta L/L}=1-2 \frac{\Delta D/D}{\Delta L/L}

Now from the equation, we get,

\frac{\Delta R/ R}{\Delta L/L}=1 + 2v

The Gauge factor (Gf) is defined as the ratio of per unit change in resistance per unit change in length.

\boxed{G_f=1+2v}

Also, \frac{\Delta R}{R}=G_f.\frac{\Delta L}{L}

\boxed{\frac{\Delta R}{R}=G_f.\epsilon}

Where, \epsilon=strain=\frac{\Delta L}{L}

if we consider a change in resistivity due to strain or piezoresistive effect then Gauge Factor is

Frequently Asked Questions on Strain Gauge

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