STRAIN BASICS
Strain gages are sensing devices used in a variety of physical test and measurement applications. They change resistance at their output terminals when stretched or compressed. Because of this characteristic, the gages are typically bonded to the surface of a solid material and measure its minute dimensional changes when put in compression or tension. Strain gages and strain gage principles are often used in devices for measuring acceleration, pressure, tension, and force. Strain is a dimensionless unit, defined as a change in length per unit length. For example, if a 1-m long bar stretches to 1.000002 m, the strain is defined as 2 microstrains. Strain gages have a characteristic gage factor, defined as the fractional change in resistance divided by the strain. For example, 2 microstrain applied to a gage with gage factor of 2 produces a fractional resistance change of (2x2)10^-6 = 4x10^-6, or 4 µW. Common gage resistance values typically range from 120 to 350 W, but some devices can be as low as 30 W or as high as 3 kW.

STRAIN MEASUREMENT CONFIGURATIONS
Wheatstone Bridge
To make an accurate strain measurement, extremely small resistance changes must be measured. The Wheatstone bridge circuit is widely used to convert the gage’s microstrain into a voltage change that can be fed to the input of the ADC (See Figure 7.01). When all four resistors in the bridge are absolutely equal, the bridge is perfectly balanced and V_out = 0. But when any one or more of the resistors change value by only a fractional amount, the bridge produces a significant, measurable voltage. When used with an instrument, a strain gage replaces one or more of the resistors in the bridge, and as the strain gage undergoes dimensional changes (because it is bonded to a test specimen), it unbalances the bridge and produces an output voltage proportional to the strain.

Full-Bridge Circuits
Although half-bridge and quarter bridge circuits are often used, the full bridge circuit is the optimal configuration for strain gages. It provides the highest sensitivity and the fewest error components, and because the full bridge produces the highest output, noise is a less significant factor in the measurement. For these reasons, the full bridge is recommended when possible.

A full bridge contains four strain gages mounted on a test member (See Figure 7.01). Two gages are mounted on the surface under tension and the other two are mounted on the opposite surface under compression. As the member deflects, the two gages in tension increase in resistance while the other two decrease, unbalancing the bridge and producing an output proportional to the displacement. The bridge output voltage is given by:

EQN: 7.01. Full-Bridge Output Voltage

V_o = (V_ex)(X)
Where:
V_o = bridge output voltage, V
V_ex = excitation voltage applied to the bridge, V
X = relative change in resistance, DR/R

The bridge nulls out potential error factors such as temperature changes because all four strain gages have the same temperature coefficient and are located in close proximity on the specimen. The resistance of the lead wire does not affect the accuracy of the measurement as long as the input amplifier has high input impedance. For example, an amplifier with a 100-MW input impedance produces negligible current flow through the measurement leads, minimizing voltage drops due to lead resistance.

Half-Bridge Circuits
When physical conditions do not allow mounting a full-bridge gage, a half bridge might fit. Typically, two strain gages are mounted on a test member, and two discrete resistors complete the bridge. The output voltage is:

EQN. 7.02. Half-Bridge Output Voltage

V_o = V_ex (X/2)
Where:
V_o = bridge output voltage, V
V_ex = excitation voltage applied to the bridge, V
X = relative change in resistance, DR/R

For a large DR, half bridge and quarter-bridge circuits can introduce an additional nonlinearity error (See Figure 7.02). Also, the readings are not accurate when the temperature coefficients between bridge completion resistors and strain gages are different and the resistances do not change proportionally with temperature. Furthermore, bridge completion resistors are not usually located near the strain gages, so temperature differences contribute additional errors. In systems with long lead wires, the bridge completion resistors should be attached close to the gages; however, this may not always be practical due to test fixture limitations or other physical conditions.

Quarter-Bridge Circuits
A quarter-bridge circuit uses one strain gage and three bridge completion resistors. The output voltage is:

EQN. 7.03. Quarter-Bridge Output Voltage

V_o = V_ex (X/4)
Where:
V_o = bridge output voltage, V
V_ex = excitation voltage applied to the bridge, V
X = relative change in resistance, DR/R

This arrangement has the smallest output, so noise is a potential problem. Furthermore, all the error sources and limitations in the half-bridge apply to the quarter bridge circuit (See Figure 7.03).

Excitation Source
Accurate measurements depend on a stable, regulated, and low-noise excitation source. A regulated source is necessary because the output voltage of a strain gage is also proportional to the excitation voltage. Therefore, fluctuations in the excitation voltage produce inaccurate output voltages.

An ideal data acquisition system provides an excitation source for each channel, independently adjustable from 0.5 to 10.5 V with a current limit of 100 mA. An excitation voltage, V, used with a strain gage of resistance, R, requires a current of I = V/R. The resistance of a Wheatstone bridge measured between any two symmetrical terminals equals the value of one of the resistance arms. For example, four 350-W arms make a 350-W bridge. The load current equals the excitation voltage divided by the bridge resistance; in this case, 10 V/350 = 0.029 A = 29 mA.

Heating
Resistive heating in strain gages also should be considered because the gages respond to temperature as well as stress. In most standard circuits, the heat that each gage dissipates is less than 100 mW, so it’s not usually a problem. This is especially true when the strain gage is bonded to a material that conducts heat quickly, such as metal. However, because materials such as wood, plastic, or glass do not conduct heat away as rapidly, use the lowest excitation voltage possible without introducing noise problems. Also, heat can become a problem when the strain gages are uncommonly small, or numerous gages occupy a limited space.

Consider a Kelvin connection for applying the excitation voltage. Because the excitation leads carry a small current, they drop a correspondingly small voltage; V = I/R_L, which reduces the voltage reaching the bridge terminals. As illustrated in Figure 7.04, Kelvin connections eliminate this drop with a pair of leads added at the excitation terminals to measure and regulate the bridge voltage. For example, when i_e = 50 mA, R_L = 5 W, and the combined voltage drop in the two leads is 500 mV, no voltage drops in the sense wires.

A commercial unit uses a Kelvin connection to measure and regulate the voltage at the bridge. It supplies the voltage to the strain gage with one pair of leads and measures it with another pair as shown in Figure 7.05. The six wires are used in pairs for Sense, Excite, and Measure. The Sense lead is a feedback loop to ensure that the Excite voltage is constantly held within specifications.

Strain Gage Signal Conditioning
Most strain gage-based transducers and load cells are assigned units of measure for weight, force, tension, pressure, torque, and deflection with a full-scale value measured in mV/V of excitation. For example, a load cell with a 10-V excitation supply and a 2-mV/V-gain factor generates an output of 20 mV at full load, whether the load cell was designed to handle 10, 100, or 1,000 lbs. The difference is in the resolution of the system. That is, the small 10-lb load cell produces 0.5 lbs/mV, and the large 1,000-lb load cell produces 50 lbs/mV.

Conductors carrying such low level signals are susceptible to noise interference and should be shielded. Low-pass filters, differential voltage measurements, and signal averaging are also effective techniques for suppressing noise interference. Furthermore, instrumentation amplifiers usually condition the extremely low strain gage signals before feeding them to ADCs. For example, a 10-V full-scale input provides 156 µV of resolution for a 16-bit ADC. The instrumentation amplifier gain should be adjusted to provide the full-scale output of the strain gage or load cell over the entire range of the ADC.

Force and pressure transducers typically generate an offset output signal when no external force is applied. Instrumentation amplifiers usually contain a control to adjust this offset to zero and let the load cell cover the full range of the ADC. Most instruments also provide adjustable excitation, and gain.

Common Mode Rejection Ratio
A high common mode rejection ratio (CMRR) is essential for strain gage amplifiers (See Chapter 5). A strain-gage signal in a Wheatstone bridge is superimposed on a common-mode voltage equal to half the excitation voltage. CMRR is a measure of how well the amplifier rejects common-mode voltages. For example, consider a 10-V excitation supply (V_max = 5 V) for a strain gage with 2 mV/V (V_s = 20 mV) at full scale and an amplifier with a CMRR of 90 dB (See Figure 7.06). The amplifier can introduce 0.158 mV of error, corresponding to about 0.80% full scale, which may not be acceptable:

EQN: 7.04. Common-Mode Rejection Ratio

dB = 20 log₁₀(V_s/V_e)
V_max/V_e = log₁₀^-1 (dB/20)
               = log₁₀^-1 (90/20)
               = 31,622
V_e = V_max/log₁₀^-1 (dB/20)
               = 5.00/31,622
               = 0.158 mV
% error = (V_e /V_s)100
               = (0.158/20)100
               = 0.79%
Where:
V_e = error voltage, 0.158 mV
V_s = signal voltage, 20 mV
Vmax = maximum voltage, 5V
CMRR = 90 dB

By comparison, a CMRR of 115 dB introduces only 9 µV of error, which corresponds to only 0.04% of full scale.

Strain gage signal-conditioning modules usually provide a regulated excitation source with optional Kelvin excitation. Onboard bridge-completion resistors may be connected for quarter and half-bridge strain gages. Instrumentation amplifiers provide input and scaling gain adjustments and an offset adjustment nulls large quiescent loads. This lets input signals use the full range of the data acquisition system and the measurements cover the full resolution of the ADC.

Some strain gage signal conditioners provide fixed gain, offset, and excitation settings, but fixed settings do not take advantage of the maximum dynamic range of the ADC. It decreases the actual available resolution of the measurement. For example, many generic strain gage signal conditioner modules can be set to a fixed 3-mV/V rating. At 10 V, the excitation, offset, and gain trimming are all fixed and no adjustments can be made.

An excitation adjustment lets users set the excitation voltage to the maximum allowed by the manufacturer, which maximizes the bridge’s output. Also, the offset adjustment lets users zero the output offset produced by either a small bridge imbalance or a quiescent deformation of the mechanical member. And the gain adjustment lets users set a gain that provides a full-scale output under maximum load, which optimizes the dynamic range of the ADC.

Calibration
The signal-conditioning module also typically provides a shunt calibration feature (See Figure 7.07). It lets users switch their own shunt resistors into either one of the two lower legs of the bridge under software control. For example, a shunt resistor can be calculated to simulate a full load. Applying a shunt resistor is a convenient way to simulate an imbalance without having to apply a physical load. For any balanced bridge, a specific resistor can be connected in parallel with one of the four bridge elements to obtain a predictable imbalance and output voltage. For example, a 350-W, 2-mV/V strain gage delivers full output when one leg drops by 0.8% to 347.2 W. A 43.75 kW resistor shunted across one or the other lower bridge elements swings the output to full positive or full negative.

An equation for calculating the shunt calibration resistor value is:

EQN. 7.05. Shunt Calibration Resistor for Transducers

R_s = R_ba[V_ex/4(V_o)]
R_s = 350[10/4(0.020)] = 43.75 k
Where:
R_s = shunt resistor, W
R_ba = bridge arm resistor, W
R_s >> R_ba
V_ex = excitation voltage, V
V_o = bridge output voltage, V

Many products include calibration software with a Windows-based program that provides several calibration methods, online instruction, and a diagnostic screen for testing the calibrated system.

Transducers and Load Cells
Strain gages are commercially available in prefabricated modules such as load cells that measure force, tension, compression, and torque. Load cells typically use a full-bridge configuration and contain four leads for bridge excitation and measurement. The manufacturers provide calibration and accuracy information.

Strain Diaphragm Pressure Gages
A strained-diaphragm pressure gage consists of two or four strain gages mounted on a thin diaphragm. The gages are wired in a Wheatstone bridge circuit, including bridge completion resistors when needed, so the pressure gage is electrically equivalent to a load cell. The output voltage is specified in mV/V of excitation for a full-scale pressure differential across the diaphragm.

When one side of the diaphragm (called the reference pressure side) is open to the ambient atmosphere, the gage compares the inlet pressure to the ambient pressure, which is about 14.7 psi at sea level. When the gage measures ambient pressure, the reference chamber must be sealed with either a vacuum reference (near zero psi) or the sea-level reference.

Temperature variations can affect the accuracy of the strain diaphragm pressure gages. A pressure gage with a sealed non-zero reference pressure exhibits temperature variations consistent with the ideal gas law. For example, a 5°C change in ambient temperature near normal room temp (25°C) produces an error of 1.7% in the pressure measurement. Temperature variations can also affect the performance of the strain gages themselves. Transducers must contain temperature compensation circuits to maintain accurate pressure measurements in environments with widely varying temperatures.

All strained-diaphragm pressure gages require a regulated excitation source. Some gages contain internal regulators, so users can connect an unregulated voltage from a power supply. Some strained-diaphragm pressure gages also employ internal signal conditioning, which amplifies the mV signal output of the Wheatstone bridge to a full-scale voltage from 5 to 10 V. Gages of this type have low-impedance outputs. In contrast, other pressure gages have no internal signal conditioning so their output impedance equals the Wheatstone bridge resistance (several kW for semiconductor types), and their full-scale output is in mV.

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