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The Versatile Strain Gauge Load Cell

All About Force Measurement, Knowledge Base Directory

Preface

In load measurement applications, strain gauge load cells are a popular choice. Strain gauge type load cells are cost-effective, accurate and highly configurable. This article explains the underlying technology behind these versatile devices.

What is a Strain Gauge?

A bonded strain gauge consists of a thin wire etched in a back-and-forth pattern onto a non-conductive substrate material with connectors at each end of the wire (Figure 1). The length of wire is the total length of all the loops; the end loops (labeled) are wider to make negligible any change in resistance from the same length of straight wire. Additionally, note the alignment marks indicating the direction of a normal strain (the top and bottom marks in the figure) and an orthogonal one.

From  a circuit standpoint, the wire acts as a resistor. The strain gauge is a function of the elastic properties of the wire. When the wire is stretched, its length increases, its cross section decreases and therefore its resistance goes up. When the wire is compressed, the opposite occurs. An analogy of this is flexible tubing. When stretched, it lengthens and narrows in cross section which restricts the fluid flow compared to the original tube.

The ratio of the change in length of the wire divided by the original length of the wire is known as its mechanical strain. It is expressed as the equation:

    \[ \epsilon=\frac{\Delta{L}}{L}, \]

where \epsilon is the conventional symbol for strain, \Delta{L} is the change in length of the wire and L is the original length of the wire.

Figure 1. Strain Gauge Diagram

The relationship between the change in resistance of the strain gauge, and the strain in the etched wire due to an applied load, is approximately linear within the elastic limit of the wire. This relationship is expressed as a ratio known as the gauge factor (G).

    \[ G=\frac{ \frac{\Delta{R}}{R}}{ \frac{\Delta{L}}{L}} \; or \; G=\frac{ \frac{\Delta{R}}{R}}{ \epsilon} \]

This linear relationship is the key to the strain gauge’s operation. The applied load creates a strain which in turn alters the resistance. Consequently, a voltage drop through the strain gauge will change depending on whether the gauge is under load conditions or not. This change in voltage due to a change in applied strain produces a corresponding change in output voltage of a strain gauge load cell.

Unbonded strain gauges also exist but the bonded type are most common and therefore this article focuses on the latter.

Strain Gauge Load Cell Circuitry

Strain gauges have load limits. The strain on the wire cannot exceed the point where the metal will no longer return to its original length but instead permanently deforms. For most metals, this change is very small. For example, for steel the elastic limit is a strain of 0.001 mm/mm.

The gauge factor tells us the resistance change in a metal is proportional to the strain. For most metals, the gauge factor is around the order of 2. This means for a strain gauge of 100 ohms unloaded, the maximum change in resistance possible within the elastic limit is about 0.2 ohms.

Obviously this resistance change is very small. In fact, if a simple ohm-meter were to measure it, the change in resistance in the strain gauge would be so small that it would fall within the percent error of the meter, and therefore would be imperceptible. To accurately measure this resistance change, a strain gauge load cell employs a simple circuit known as a Wheatstone Bridge.

The Wheatstone Bridge

A Wheatstone bridge is simply two voltage dividers wired in parallel arms of a circuit with a common voltage source. Figure 2 shows a representation of this circuit.

Figure 2. Simple Wheatstone Bridge Circuit

The input voltage, V_{ex}, is the excitation voltage. The variable resistor at R + \Delta{R} represents the strain gauge. R, R_1, R_2, and R_3 have equal resistances and \Delta{R} has a value of zero under no load. Therefore when the strain gauge bears no load, the voltages at the nodes V_{o-} and V_{o+} are equivalent and the output of the bridge circuit, which is the voltage difference across these two nodes, is zero volts.

What happens when a force or load is applied and \Delta{R} is non-zero? Let’s look at the general equation for the voltage at each output node. Recall ohm’s law states that the voltage between two nodes in a series circuit is equal to the current through it multiplied by the total resistance in that path, or more commonly seen as V=IR. Rearranging, I = \frac{V}{R}, the current on the left half of the bridge circuit in Figure 2 is then equal to \frac{V_{ex}}{(R_1 + R_2)}. The two resistors along this path divide the excitation voltage at V_{o-}. Substituting our expression for current in the left path for I in the ohm’s law equation, this voltage at the V_{o-} node is:

    \[ V_{o-} = V_{ex}\frac{R_2}{R_1 + R_2} \]

Similarly, the voltage at the V_{o+} node is:

    \[ V_{o+} = V_{ex}\frac{R + \Delta{R}}{R + \Delta{R} + R_3} \]

The total output voltage is simply the difference between these two:

    \begin{gather*} V_o = V_{o+} - V_{o-} \\ \\ V_o = V_{ex}\frac{(R + \Delta{R})}{(R + \Delta{R}) + R_3} - V_{ex}\frac {R_2}{R_1 + R_2} \\ \\ V_o = V_{ex}\frac{(R + \Delta{R})R_1 - R_2R_3}{(R + \Delta{R} + R_3)(R_1 + R_2)} \\ \end{gather*}

The reason this setup improves accuracy is that now the voltage drop across the strain gauge is being compared to the voltage drop across a similar resistance for the same excitation voltage, instead of being compared to the much larger excitation voltage itself. That means the change in voltage across the strain gauge will be the same order of magnitude as the comparison voltage and any error will be a fraction of this. Also, any changes in strain due to temperature or other environmental factors will affect all of the resistors in the circuit equally.

The Quarter, Half, and Full-Bridge Configurations

All Wheatstone bridges have four resistive elements. However in load cell design, the number of those elements that are strain gauges vs non-variable resistors is flexible. If only one of the resistive elements in the bridge is a strain gauge, it is a quarter bridge. If two are strain gauges, it is a half bridge. And in the case where all four resistive elements are strain gauges, the Wheatstone bridge is a full bridge. See Figures 3-5 below.

Figure 3. Quarter bridge configuration
Figure 3. Quarter Bridge Configuration
Figure 4. Half bridge configuration
Figure 4. Half Bridge Configuration
Figure 5. Full bridge configuration
Figure 5. Full Bridge Configuration

The pros and cons of each bridge configuration motivate the choice for a particular application. In general, fewer gauges mean cheaper construction and easier installation. However, additional gauges increase bridge output, allow for temperature compensation, and cancel unwanted strain components.

To describe an example, one must know the property of the Wheatstone Bridge that strains on adjacent positions in the bridge cancel, while strains on opposite bridge arms sum. Now let us assume that we want to cancel a bending strain in a loaded beam and just measure any tensile strain. This can be achieved by bonding a strain gauge to both the top and the bottom (vertically aligned) of the beam in a half bridge configuration with the gauges on opposite bridge arms. Because the strain measured at the top gauge will have the load strain plus tensile strain due to bending, and the bottom gauge will have a compressive bending component, similar in magnitude but opposite to the tensile component at the top, the output of the bridge will add the common strain components but will cancel the opposite ones.

Likewise, strain components due to temperature add equally to all gauges in a bridge configuration, so by placing a non-loaded gauge in a position adjacent to a load-bearing one, the strain components due to temperature will cancel. In addition, bonding gauges at angles to each other on a measuring device can account for additional (non-axial) strain components and determine the angle of maximum strain.

Conclusion

This article touches on the underlying principles of the bonded strain gauge, the main component used in the Tacuna Systems load cell product line. Strain gauges have many advantages in load cell construction, including low cost, high accuracy (especially due to the wide variety of bridge configurations and bonding geometries) and durability. Because they can be bonded to a wide array of supporting structures and can handle a broad range of loads, strain gauge type load cells are practical for many applications. They are truly versatile.

The complementary article, “Choosing the Right Load Cell For Your Job,” explains the construction of various load cell types in the Tacuna Systems product line, and their uses.