## Strain Gauge Basics – Part 2 – Shunt Calibration

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The use of strain gauges with Prosig’s data acquisition systems is well understood and has been used in many real-world applications over the years [see What Is A Strain Gauge? by James Wren here on the blog and also available as Strain Gauges Explained in the Prosig Noise and Vibration Measurement Handbook].

The following post is the second part of a series that provides a recap and update to that article.

## Shunt Calibration of Strain Gauges

As mentioned in the previous post, in order to relate the measured voltage difference across the bridge circuit to a strain measurement, we need a way of calibrating the system. In most systems, including the Prosig acquisition systems, this is done by including a high-precision shunt resistor across one of the legs in the bridge. For example, the following diagram shows a quarter-bridge circuit with a shunt calibration resistor, RS, across the bridge completion resistor R2.

As the shunt resistor (RS) is a known value, its imbalance effect on the bridge can be calculated. By measuring the voltage difference across points A and B with and without the shunt resistor added, we can relate the change in voltage to the change in resistance and hence the strain based on the following equation from the previous post.

$\frac{\mathrm{\Delta}R}{R_{G}} = \ F_{G}\ \varepsilon$

The equivalent resistance at R2 when we switch in the shunt resistor in parallel across R2 is given by:

$\frac{1}{R_E} = \frac{1}{R_2} + \frac{1}{R_S}$ or by rearranging RE = R2RS / (R2 + RS)

The change in resistance is

${\Delta}R = R_2 - R_E$

or

${\Delta}R = R_2 - \frac{{R_2}.{R_S}}{(R_2 + R_S)}$

$\frac{{\Delta}R}{R_2} = 1 - \frac{R_S}{(R_2 + R_S)}$

and so

$\frac{{\Delta}R}{R_2} = \frac{(R_2 + R_S)}{(R_2 + R_S)} - \frac{R_S}{(R_2 + R_S)}$

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Therefore

$\frac{{\Delta}R}{R_2} = \frac{R_2}{(R_2 + R_S)}$

From equation (3) in the previous post – $\frac{\mathrm{\Delta}R}{R_{G}} = \ {F_{B}F}_{G}\ \varepsilon$ (3)

We get

$\varepsilon = \frac{{\Delta}R}{{R_2}{F_G}{F_B}}$ and so $\varepsilon = \frac{R_2}{(R_2 + R_S){F_G}{F_B}}$

Therefore, the equivalent change in strain, with or without the shunt resistor applied is given by:

${\Delta}\varepsilon = \frac{R_2}{(R_2 + R_S){F_G}{F_B}} - 0 = \frac{R_2}{(R_2 + R_S){F_G}{F_B}}$ (4)

By measuring the difference in voltage, ${\Delta}V$, when the shunt is applied and when not, we can calculate the sensitivity factor, $\varphi$ , from

${\Delta}V = {\varphi}{\Delta}\varepsilon$

$\varphi = \frac{{\Delta}V(R_2 + R_S){F_G}{F_B}}{R_2}$

The voltage difference will be very small and usually measured in mV, the resultant sensitivity will also be very small and so the sensitivity is usually represented in units of  $mV\/{\mu}{\varepsilon}$. The final equation becomes

$\varphi = \frac{{\Delta}V(R_2 + R_S){F_G}{F_B}}{{R_2}{10^6}}$ (5)

As an example, for a 120$\Omega$ strain gauge with a gauge factor, FG = 2, configured as a quarter bridge (FB = 1) with completion resistors of 120$\Omega$ and a shunt resistance of 126,000$\Omega$ the equivalent resistance at R2 when the shunt is applied becomes 119.8858$\Omega$.

With an excitation voltage of 10V and we can calculate the voltage difference with and without the shunt resistor applied using equation (1) from the previous post

$V_{0} = V_{A} - V_{B} = Vext\ \left( \frac{R_{2}}{R_{G}+R_{2}} - \frac{R_{3}}{R_{4} + \ R_{3}} \right)$

This gives us the expected ${\Delta}V$ as 2.3798 mV, and finally, from equation (5), the sensitivity factor becomes 0.005002$mV\/{\mu}{\varepsilon}$

We can check the operation using the above equations to calculate the effective micro-strain when reapplying the shunt resistor with the calculated sensitivity and reading the measured strain. For the above example, the expected strain when the shunt is applied will be 475.73${\mu}{\varepsilon}$

In future posts, we will consider lead-wire resistance compensation and look at how we make real-world measurements with strain gauges.

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#### Don Davies

Applications Group Manager at Prosig
Don Davies graduated from the Institute of Sound and Vibration Research (ISVR) at Southampton University in 1979. Don specialises in the capture and analysis of vibration data from rotating machines such as power station turbine generators. He created and developed the PROTOR system and is Applications Group Manager at Prosig. Don is a member of the British Computer Society.

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