# Using a strain-gauge transducer in a Wheatstone-bridge configuration

A Wheatstone bridge circuit makes a very good general-purpose measurement circuit. Some of the physical quantities that it can be used to measure are stress and strain, pressure, fluid flow and, temperature. This article discusses using a strain gauge in conjunction with a Wheatstone bridge to measure load, weight or force. It briefly discusses the principles of the Wheatstone bridge circuit and the strain gauge and then explains how the two are used together in a measurement circuit. It then goes a step further and explains how to interface the Wheatstone bridge output with a suitable precision amplifier to generate an accurate, amplified-measurement reading.

Principle of the strain gauge
A strain gauge is used to measure the strain on an object. It is a very thin, flat coil of conductive wire. (Editor's note : strain “gauge” is often spelled as “gage”; why this is so, and which is better, is the subject of spirited and sometimes stressful arguments, but we won't go into those here.) It is 'glued' onto the object on which the strain is to be applied. In this particular circuit, the strain gauge is formed by a very thin printed circuit board (PCB) trace laid out on the board.

When strain is applied to the PCB, the resistance of the gauge changes in accordance with the strain applied to bend the board. The strain can be tensile or compressive as distinguished by a positive or negative change in the nominal resistance of the gauge. When used in a Wheatstone bridge configuration, this property of the strain gauge is exploited to convert the change in resistance of the strain gauge to a voltage which corresponds to the strain applied.

Principle of operation of the Wheatstone bridge

The Wheatstone bridge, Figure 1 , consists of four resistances (R1 , R2 , R3 , and R4 ), an excitation voltage, and an output voltage. Figure 1: Basic Wheatstone-bridge configuration.

Generally, one or more of the resistances are variable and change in accordance with some physical phenomenon, such as strain in this case. The Wheatstone bridge then converts this change in resistance to a change in voltage.

In the circuit, Vin is the excitation voltage and Vg is the output voltage. The voltage Vg measures the potential difference between the outputs of the voltage dividers ADC and ABC, connected across the excitation voltage source. The voltage at terminal D is equal to the drop across resistor R4 and the voltage at terminal B is equal to the drop across R1 .

Initially, the bridge is balanced, which implies that the output voltage Vg is zero. This is because the resistors are chosen in such a way that the ratio of R1 :R2 is equal to the ratio R4 :R3 . As a result, the voltage at terminals B and D is equal. If there is a change in any of the resistances, the output voltage Vg is no longer zero, due to an imbalance in the bridge.

First, a brief mathematical calculation of the functioning of the Wheatstone bridge is useful. The voltage drop in the arms AB and AD is given by: The equations are obtained by using the current-divider rule to determine the current flowing through arms ABC and ADC of the bridge, IABC and IADC , respectively. The output voltage Vg can then be obtained from the following equations: Simplifying gives This equation shows that if the ratio R1 :R2 is equal to the ratio R4 :R3 , then Vg will be zero. Any change in the values of any of the resistances will disturb this balance and the output voltage will no longer be zero.

Strain-gauge transducer in a Wheatstone-bridge configuration

The principle of the Wheatstone bridge is used to design a strain-gauge transducer, where the gauge is laid out in the form of a PCB trace on the circuit board. Two of the four resistors in the Wheatstone bridge are replaced by identical strain gauges, which are laid out on opposite sides of the PCB. The strain gauge laid out on the top of the board will be referred to as the 'actual' strain gauge, and the one laid out on the bottom of the board will be the 'dummy' strain gauge.

In the absence of any strain, the Wheatstone bridge output voltage is zero, since the bridge is balanced in the 'no strain' condition. To balance, the bridge the circuit needs to be calibrated for the 'no strain' condition. (The methods of calibration are discussed below in detail.)

When some strain is applied to the circuit board, the nominal resistance of the strain gauges changes in accordance with the pressure applied. As a result, the Wheatstone bridge is unbalanced and an output voltage proportional to the change in resistance of the strain gauges, and hence proportional to the applied strain, is generated at the output of the bridge. The change in resistance of each strain gauge is typically very small, of the order of micro-ohms. Therefore, the output voltage of the bridge is also very small. Thus this output voltage needs to be amplified using a low noise precision amplifier. The design shown here uses the LMP2011, which is part of National's LMPTM precision amplifier family.

Reason for the “dummy” strain gauge
Figure 2 shows the Wheatstone bridge used in our circuit. Figure 2: Wheatstone bridge with “actual” and “dummy” strain gauges.

Excitation voltage VB is 5 V. Resistors R1 , R2 , R3 , and R4 form the arms of the bridge, with R1 and R2 both replaced by strain gauges. R1 is the 'actual' strain gauge, while R2 is the 'dummy' strain gauge. R2 is laid out on the bottom of the circuit board while R1 is laid out on the top of the circuit board.

The problem with the strain gauge is that its resistance varies with temperature and it is difficult to track this temperature change using a discrete resistor of the same value. The dummy strain gauge tracks the change in temperature of the actual strain gauge accurately, as it is constructed from the identical material as the actual strain gauge and has the same value.

Since the dummy strain gauge and the actual strain gauge are on opposite sides of the board, the two of them will experience strain equal in magnitude but opposite in direction. That is, if the board is bent, then one of them will experience tension while the other one will experience compression. The gauge that experiences tension will undergo an increase in the nominal resistance and the one that experiences compression will undergo a decrease in the nominal resistance. The nominal resistance of this particular strain gauge is 7.6 O. With tension, let's assume that the resistance increases to 7.8 O and, with compression, assume that the resistance decreases to 7.4 O. Therefore, R1 and R2 have the same magnitude change of 0.2 O but the change is in opposite directions. There is a larger change in the ratio R1 :R2 than there would have been if the dummy strain gauge had not been used.

To emphasize this point, let's use the equation from the Wheatstone bridge discussion above. Initially, when no strain is applied, R1 = R2 and R4 = R3 and the bridge output Vg is 0 V. With strain applied, R1 and R2 both experience a change, while R3 and R4 still remain constant. Then the output voltage Vg is given by: with the dummy strain gauge and without the dummy strain gauge.

As a result, the output voltage of the pressure sensor circuit is essentially doubled. This helps to increase the sensitivity of the circuit since it gives a larger output voltage for the same gain setting.

Layout consideration for strain gauges

The layout of the strain gauge on the PCB can affect the accuracy of the measurement. The two strain gauges should be laid out in such a manner that the surface area which is perpendicular to the direction of strain is larger than the surface area parallel to the direction of the strain. Doing so increases the change in the resistance value of the strain gauge for any given strain. Both the actual and the dummy strain gauges are laid out this way.

Assume that the strain applied is perpendicular to the paper, i.e. it is going into the paper. In Figure 3a , the area in the direction perpendicular to the strain is much smaller than that in Figure 3b .  Figure 3: Orientation of gauge affects sensitivity and resultant change in resistance.

Thus a strain gauge laid out as shown in Figure 3b will produce a larger change in resistance, compared to that laid out as in Figure 3a. A larger change in the resistance yields a larger output voltage.

Circuit calibration

Balancing the bridge or calibrating the circuit for the 'no strain' condition is important. If the bridge is not balanced, there will be a difference voltage at the input of the difference amplifier even when no external pressure is applied. This voltage will be amplified and will erroneously indicate that input strain is applied.

The main cause for this imbalance in the Wheatstone bridge is resistor mismatch. To correct this resistor mismatch several precautions must be taken while selecting the resistors. As the 'actual' strain gauge and the 'dummy' strain gauge are made from PCB traces laid out on the board, and are made of the same material, they are quite closely matched. Their temperature coefficients are also presumably the same, so they closely track changes in temperature. However, the discrete resistors R4 and R3 will have significant mismatch relative to the magnitude of the output signal, if they are not selected properly.

One way to reduce the mismatch between the resistors is to use 0.01% wirewound resistors. The advantage of using these resistors is that they have very low flicker noise, which is mainly present at dc and lower frequencies. Since this is a dc application, elimination of flicker noise from the resistor network is very beneficial.

The test circuit, Figure 4 , used 0.01% wire wound resistors for R3 and R4 . Figure 4: Complete test circuit used for evaluation
(Click to enlarge image)

If possible, the thermal coefficients of these resistors should match. Although the resistors may have the same magnitude of the thermal coefficient of resistance, the direction may not be the same and this will add significant error to the measurement.

A tedious way to match the thermal coefficients of the resistors is to check the direction of change in resistance of the different resistors by subjecting them to the same changes in temperature and then to select those resistors which have the closest change in resistance with temperature with respect to magnitude and direction.

In spite of taking the previous steps, there might still be a small mismatch between the resistors. To account for this mismatch, two 100 O potentiometers are connected in parallel with R3 and R4 . The purpose of these potentiometers is to null the output of the Wheatstone bridge in the 'no strain' condition. By calibrating the circuit appropriately, the 'no strain' output voltage of the circuit is set to zero volts.

After calibrating the circuit using these steps, it is safe to assume that errors in the strain measurement due to inherent circuit mismatches are removed. However, airflow can vary the resistance of the strain gauge, again leading to erroneous readings of strain. This can be corrected simply by concealing the strain gauge in a box with a lid.

The output voltage of the Wheatstone bridge, generated in response to some strain, is very small and needs to be amplified using a low noise precision amplifier such as the LMP2011 amplifier used here. The LMP2011 is configured as a difference amplifier, so it amplifies the difference in the potentials between its inverting and non-inverting inputs. The two outputs of the bridge can be connected to the appropriate terminals of the amplifiers using the jumpers provided. This is to ensure that voltage at the non-inverting terminal is higher than that at the inverting terminal of the LMP2011 as it operates on single supply.

It is important to ensure that the resistors R7 , R8 , R9 , and R10 , which are used to set up the gain in the difference amplifier, are well matched and track each other over temperature, to reduce error in the measurement. Resistor R8 is equal to R7 , and R9 is equal to R10 , to ensure input bias-current compensation.

Since the output of the Wheatstone bridge is very small, the amplifier used must be a very low-noise, precision device. Any additional noise added to the circuit will result in a very poor signal-to-noise ratio. The small signal output of the Wheatstone bridge also makes it necessary to apply a large gain to the signal. The gain is set to 100 in this circuit.

However, the larger gain causes an increase in the thermal noise of the gain-setting resistors. Capacitors C4 and C6 filter the thermal noise of the gain-setting resistors R7 and resistor R8, respectively. As this is essentially a dc application, there must be very low noise at dc to ensure that the small output signal can be detected. At dc, most op amps have what is known as flicker noise. The LMP2011, however, is an auto-zero amplifier and it eliminates the low-frequency flicker noise. The absence of flicker noise makes this amplifier a good match for this application. With the elimination of flicker noise, which is quite large at dc, the accuracy of the pressure-sensor application is improved.

To some extent, the accuracy of the measurement also depends on the accuracy of the excitation voltage VB . To ensure the accuracy and stability of the excitation voltage, the LM340 and the LM2734 are used here, with the LM340 powered by a 9-V battery.

After calibrating the circuit, and setting up the appropriate gain, one final step remains before the circuit can be used: to calibrate the output voltage for different values of strain. To determine a relationship between the applied strain and the resultant output voltage of the circuit, a few known strains can be applied to the board and the output voltage can then be calibrated for the value of the strain over a given range. Once this is done, the strain applied to the board can be measured in the desired units by multiplying by the proportionality constant.