Measurements are performed with strain gauges in mechanical stress analysis to examine loading and fatigue. In addition to the desired measurement signal indicating mechanical strain, each strain gauge also produces a temperature-dependent measurement signal. This signal, called the apparent strain, is superimposed on the actual measure.

Various effects contribute to the apparent strain:

• Thermal expansion of the measurement object (i.e. strain due entirely to temperature with no mechanical loading as the cause)
• Temperature-dependent change in the strain gauge resistance
• Thermal contraction of the strain gauge measuring grid foil
• Temperature response of the connection wires

Temperature effects can be compensated for in strain gauge measurements, for example, by connecting several strain gauges together to form a half or full bridge. This approach makes use of the effect that in the Wheatstone bridge circuit that is typically used, strains on different strain gauges are reflected in the measurement signal with opposite signs (i.e. positive and negative). By skillfully arranging the strain gauges, the resulting bridge voltage will represent only the mechanical load, and the temperature-dependent effects will cancel each other out.

Temperature compensation with a half or full bridge will not be further discussed in this paper, as the present topic is temperature response matching of strain gauges. The case considered below, with a quarter bridge strain gauge circuit, involves all four of the effects named above. Temperature-dependent apparent strain can be reduced by temperature response matching.

## Temperature Response Matching

The apparent strain that comes into play as the temperature changes can be represented in a simplified manner as follows:

Where:
εs = apparent strain of the strain gauge
αr = temperature coefficient of the electrical resistance of the measuring grid foil
αb = thermal expansion coefficient of the measurement object
αm = thermal expansion coefficient of the measuring grid material
k = gauge factor (sometimes called k factor) of the strain gauge
Δϑ = temperature difference that triggers the apparent strain

Measures can be taken when manufacturing strain gauges to minimize apparent strain. The temperature coefficient of the electrical resistance of the measuring grid foil is adapted by technical production measures so that the terms of the equation cancel each other out; thus αr = (αm - αb) • k.

Accordingly, there are different types of strain gauges that are identical in terms of geometry and resistance values, but differ in temperature response matching for the material on which the strain gauge is installed. Temperature response matching to a wide range of thermal expansion coefficients is available (for example, to ferritic steel with a thermal expansion coefficient of 10.8 • 10-6/K or aluminum with 23 • 10-6/K). The strain gauge is referred to in this case as a "strain gauge with adapted temperature coefficient," or more succinctly, as a "self-compensated strain gauge."

The equation for apparent strain is a simplified representation, containing only linear components. Residual errors in the form of non-linear variables must also be taken into consideration. To keep the error as small as possible, the residual error is adjusted to the lowest level possible in the range around room temperature.

The apparent strain is printed on each package of HBM strain gauges as a diagram. A polynomial is also specified -- typically a third-degree polynomial. The polynomial can be used for computational compensation. The diagram below shows an example from a strain gauge data sheet.

Of course, this compensation only works if the thermal expansion coefficient of the material matches the adaptation of the strain gauge. If this condition is met and the temperature is measured parallel to the strains, the residual error can be removed by calculation with the appropriate software, either during the measurement (online) or after (post-process).

As the curve shows, the need for compensation to reduce temperature-related measurement errors increases as the temperature range grows larger. The converse also hold true.  This kind of computational compensation is not needed if the temperature changes only slightly during the measurement; for example, because the measurement is very short in duration or because the environment is climate-controlled.

Computational compensation of the residual error is illustrated below based on an example using catman measurement data acquisition software.

## Computational Compensation

The catman data acquisition software1 can be used to set parameters, adjust measurement parameters, and represent measured values, all with a few mouse clicks. Parameters can be set to indicate to the software that temperature compensation is required.

To implement temperature compensation, the following information must be provided to the software for each channel to be compensated:

• Reference to the corresponding temperature channel
• Polynomial for apparent strain as indicated on the strain gauge package

Channels with identical parameters for the corresponding temperature channel and polynomial can be handled together. Strain gauges from the same production batch always have identical polynomials.

When defining the temperature channels, note that the actual temperature of the material must be measured at the measuring point. Depending on the application, several temperature measurement points may be necessary.

In catman AP, the configuration dialog for a strain gauge can be accessed from the central "measurement channels" worksheet. To do this, mark the channels to be adapted and right-click to open the "Sensor adaptation" dialog.

All settings relevant to the strain gauge -- especially the gauge factor -- can be made in this strain gauge configuration dialog.

Other parameters related to temperature compensation must also be adjusted:

• The coefficient of the polynomial (as indicated on the strain gauge package)
• Thermal expansion coefficient of the measurement object (ideally identical to the adaptation of the strain gauge)
• Thermal expansion coefficient to which the strain gauge is matched (as indicated on the strain gauge package)
• Reference temperature (typically 20°C)
• Corresponding temperature channel

The catman AP software now returns compensated measured values directly.

## Conclusion

When strain gauges are used in mechanical stress analysis, they are frequently combined to form a quarter bridge circuit. Temperature-dependent effects are included in the measurements, thereby distorting the measurement results.

To compensate for these effects, strain gauges are available with a temperature response matching. This compensates for at least the linear components of the error.

The residual error, due to the non-linear components, can be described by an error curve, and eliminated by mathematical processes in the software.

The following conditions must be met to ensure successful compensation for temperature-dependent effects:

• The thermal expansion coefficient of the material must be known and a correspondingly adapted strain gauge must be used
• The temperature at the measuring point must be measured in parallel
• Software with a corresponding mathematical algorithm must be used

1 This description of online temperature compensation also applies to the smaller software package catman Easy. The only difference is that unlike catman AP, compensation in the analysis range (post process) is not possible with catman Easy.