# Gauge Factor of Strain Gauge: Formula, Materials, Advantage

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Gauge Factor of Strain Gauge: The ratio of change in resistance per unit (△R/R) and change per unit length (△L/L) of the metal conductor is called the Gauge factor. The gauge factor is very important to determine the sensitivity of the strain gauge.

text { Gauge Factor, } G_{f}=frac{frac{△R}{R}}{frac{△L}{L}} …………… (i)

Strain can be explained in part by changes in resistance values based on dimensional variations in simple elastic materials. If a strip of elastic material is stretched or a positive strain is applied, its longitudinal dimension will increase and its lateral dimension will decrease.

Thus, when a gauge is subjected to positive strain, its length increases, and its cross-sectional area decreases. Since the resistance of a conductor is proportional to its length and inversely proportional to its area, the positive strain will increase the resistance of the gauge.

Consider, that a strain gauge consists of round wire and wire length = L, area = A, diameter = D, (before straining) if the resistivity of the material is ρ.

So the resistance of the unstrained gauge, R=frac{rho L}{A} …………….. (ii)

Suppose, a tensile stress S is applied to the wire. This will produce positive strain, increasing the length and decreasing the area. As a result, if a wire is strained, its length and width will change. If the change in length is △L, change in area = △A, change in diameter △D, and change in resistance = △R. However, by partial differentiation of equation (ii) with respect to S, we get,

frac{d R}{d S}=frac{rho}{A} frac{delta L}{delta S}-frac{rho L}{A^{2}} frac{delta A}{delta S}+frac{L}{A} frac{partial rho}{delta S}

Dividing both sides by R=frac{rho L}{A} we get,

frac{1}{R} frac{d R}{d S}=frac{1}{L} frac{delta L}{delta S}-frac{1}{A} frac{delta A}{delta S}+frac{1}{rho} frac{delta rho}{delta S} …………… (iii)

Here is the area, A=frac{pi D^{2}}{4} ……………. (iv)

Change in length per unit, = frac{Delta L}{L} and changes in area, = frac{Delta A}{A}

Differentiating equation (iv) with respect to S we get,

frac{delta A}{delta S}=2 frac{pi}{4} D cdot frac{delta D}{delta S} ……………. (v)

Now dividing equation (iii) by equation (ii) we get,

frac{1}{A} frac{d A}{d S}=frac{left(frac{2 pi}{4}right) D}{left(frac{pi}{4}right) D^{2}} cdot frac{delta D}{delta S}=frac{2}{D} cdot frac{delta D}{delta S}

Hence from equation (iii) we get,

frac{1}{R} frac{d R}{d S}=frac{1}{L} frac{delta L}{delta S}-frac{2}{D} cdot frac{delta D}{delta S}+frac{1}{rho} cdot frac{delta rho}{delta S} ……………….. (vi)

We know that,

Poisson’s ratio, frac{Lateral strain}{Longitudinal strain} =frac{shape distortion}{Length distortion}

=frac{frac{delta D}{D}}{frac{delta L}{L}}or, frac{delta D}{D}= -gammatimesfrac{delta L}{L}

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(vi) Substituting the value of equation frac{delta D}{D} , we get,

frac{1}{R}.frac{dR}{dS}=frac{1}{L} frac{delta L}{delta S}- gamma frac{2}{L}. frac{delta L}{delta S}+frac{1}{rho}. frac{deltarho}{delta S} ……………. (vii)

For small value changes this equation can be written,

frac{Delta R}{R}=frac{Delta L}{L}+ 2gamma frac{Delta L}{L}+frac{Deltarho}{rho} ……………… (viii)

We know that,

Gauge factor, G_f =frac{frac{Delta R}{R}}{frac{Delta L}{L}}

Or, frac{Delta R}{R}=G_f timesfrac{Delta L}{L} =G_ftimesin                               [Here in= ] Strain =frac{Delta L}{L}

Now, dividing equation (viii) by , we get,

G_f=frac{frac{Delta R}{R}}{frac{Delta L}{L}} =1+2gamma +frac{frac{Deltarho}{rho}}{frac{Delta L}{L}} =1+2gamma +frac{frac{Deltarho}{rho}}{in}                        [Since, frac{Delta L}{L} = in ]

Here 1 is Resistance change due to change of length,

2γ is Resistance change due to change of area

And frac{frac{Deltarho}{rho}}{in} Resistance change due to change of resistivity or piezoresistive effect.

If the resistivity change is assumed to be negligible, we can write equation (ix),

k = 1 + 2γ .............................. (x)

## Materials used in strain gauges

Materials Composition Gauge Factor Resistivity Ωm Resistance Temperature Co-Efficient/C Upper Temperature /C
Nichrome Ni : 80%
Cr : 20%
2.5 100times10^{-8} 0.1times10^{-3} 1200
Constantan Ni : 45%
Cu : 55%
2.1 48times10^{-8} pm0.02times10^{-3} 400
Isoclastic Ni : 36%
Cr : 8%
Mo : 0.5%
etc.
3.6 106times10^{-8} pm0.175times10^{-3} 1200
Nickel - -12 6.5times10^{-8} 6.8times10^{-3} -
Platinum - 4.8 10times10^{-8} 4.0times10^{-3} -

## Advantage of high quality gauge factor

Gauge Factor Of Strain Gauge: The sensitivity depends on the gauge factor of the strain gauge. That is, the higher the gauge factor, the greater the change in resistance. And this change in resistance can be easily and accurately determined. As a result, it will be possible to measure the measured quantity very precisely. So the gauge factor of the strain gauge should be as high as possible. Gauge factor values of different materials have different values. So strain gauges need to be made from materials with high gauge factor.

## Effect of temperature on strain gauge

When the temperature of the strain gauge changes, its resistance changes. As a result, it can cause error in strain measurement. Therefore, the temperature effect of all materials on resistance change is very small. Strain gauge needs to be made with all those materials.

That is, the Temperature co-efficient of resistance of the strain gauge material must be very low. Dummy strain gauge is used to eliminate error due to temperature variation. As a result, the effect of temperature change between the main gauge and the dummy gauge is equal, so the error is very less observed.